diff --git a/-tFJT4oBgHgl3EQfpywl/content/tmp_files/2301.11601v1.pdf.txt b/-tFJT4oBgHgl3EQfpywl/content/tmp_files/2301.11601v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec6ba6f77c71384aed41ce9a228be26a605ba63c --- /dev/null +++ b/-tFJT4oBgHgl3EQfpywl/content/tmp_files/2301.11601v1.pdf.txt @@ -0,0 +1,1262 @@ +Improved Differential-neural Cryptanalysis for +Round-reduced Simeck32/64 ∗ +Liu Zhang1,3[0000−0001−6106−3767], Jinyu Lu2(�)[0000−0002−7299−0934], +Zilong Wang1,3[0000−0002−1525−3356], and Chao Li2,3[0000−0001−7467−7573] +1 School of Cyber Engineering, Xidian University, Xi’an 710126, China +{liuzhang@stu., zlwang@}xidian.edu.cn +2 College of Sciences, National University of Defense Technology, Hunan, Changsha +410073, China, jinyu_smile@foxmail.com, lichao_nudt@sina.com +3 State Key Laboratory of Cryptology, P.O.Box 5159, Beijing 100878, China +Abstract. In CRYPTO 2019, Gohr presented differential-neural crypt- +analysis by building the differential distinguisher with a neural network, +achieving practical 11-, and 12-round key recovery attack for Speck32/64. +Inspired by this framework, we develop the Inception neural network that +is compatible with the round function of Simeck to improve the accuracy +of the neural distinguishers, thus improving the accuracy of (9-12)-round +neural distinguishers for Simeck32/64. To provide solid baselines for neu- +ral distinguishers, we compute the full distribution of differences induced +by one specific input difference up to 13-round Simeck32/64. Moreover, +the performance of the DDT-based distinguishers in multiple ciphertext +pairs is evaluated. Compared with the DDT-based distinguishers, the 9-, +and 10-round neural distinguishers achieve better accuracy. Also, an in- +depth analysis of the wrong key response profile revealed that the 12-th +and 13-th bits of the subkey have little effect on the score of the neu- +ral distinguisher, thereby accelerating key recovery attacks. Finally, an +enhanced 15-round and the first practical 16-, and 17-round attacks are +implemented for Simeck32/64, and the success rate of both the 15-, and +16-round attacks is almost 100%. +Keywords: Neural Distinguisher, Wrong Key Response Profile, Key +Recovery Attack, Simeck32/64 +1 +Introduction +Lightweight block ciphers present trade-offs between appropriate security and +small resource-constrained devices, which is an essential foundation for data con- +fidentiality in resource-constrained environments. Therefore, the design require- +ments and security analysis of lightweight block ciphers are of great importance. +Combining traditional analysis methods with “machine speed” to efficiently and +∗ Supported by organization x. +First Author and Second Author contribute equally to this work. +arXiv:2301.11601v1 [cs.CR] 27 Jan 2023 + +intelligently evaluate the security of cryptographic algorithm components, is one +of the critical points and trends of current research. The development of Artificial +Intelligence (AI) provides new opportunities for cryptanalysis. +In CRYPTO 2019 [8], Gohr creatively combines deep learning with differ- +ential cryptanalysis and applies it to the Speck32/64, gaining the neural dis- +tinguisher (ND) can surpass the DDT-based distinguisher (DD). Then, a hy- +brid distinguisher (HD) consisting of a ND and a classical differential (CD) +with highly selective key search strategies result in forceful practical 11-, and +12-round key recovery attacks. In EUROCRYPT 2021 [7], Benamira et al. pro- +posed a thorough analysis of Gohr’s neural network. They discovered that these +distinguishers are basing their decisions on the ciphertext pair difference and the +internal state difference in penultimate and antepenultimate rounds. +To attack more rounds, the component CD or ND must be extended. In +ASIACRYPT 2022 +[4], Bao et al. devised the first practical 13-round and an +improved 12-round ND-based key recovery attacks for Speck32/64 by enhanc- +ing the CDs, which they deeply explored more generalized neutral bits of dif- +ferentials, i.e., conditional (simultaneous) neutral bit/bit-sets. In addition, they +obtained NDs up to 11-round Simon32/64 by using DenseNet and SENet, thus +launching the practical 16-round key recovery attack. Zhang et al. [16] focused +on improving the accuracy of ND and added the Inception composed of the +multiple-parallel convolutional layers before the Residual network to capture +information on multiple dimensions. Under the combined effect of multiple im- +provements, they reduced the time complexity of key recovery attacks for 12-, +and 13-round Speck32/64 and 16-round Simon32/64. They also devised the +first practical 17-round key recovery for Simon32/64. +The Simeck algorithm [15], which combines the good design components +from both Simon and Speck [5] designed by National Security Agency (NSA), +has received a lot of attention for its security. In 2022, Lyu et al. [13] improved +Gohr’s framework and applied it to Simeck32/64. They obtained (8-10)-round +NDs for Simeck32/64 and successfully accomplished attacks for (13-15)-round +Simeck32/64 with low data complexity and time complexity. In the same year, +Lu et al. [12] adopted the multiple ciphertext pairs (8 ciphertext pairs) to train +the SE-ResNet neural network fed with a new data format for Simon and +Simeck. Finally, they obtained (9-12)-round NDs for Simeck32/64. This raises +the question of whether the key recovery attack for Simeck can be enhanced. +Our Contribution. The contributions of this work are summarized as follows. +• We improved the Inception neural network proposed by zhang et al. [16] ac- +cording to the number of cyclic rotation in the round function of Simeck32/64. +Meanwhile, to capture the connections between ciphertext pairs, we use mul- +tiple ciphertext pairs forming a sample as the input of the neural network. +Therefore, we improved the accuracy of (9-12)-round NDs using the ba- +sic training method and staged training method. The result can be seen in +Table 3. + +• To provide solid baselines for NDs, the full distribution of differences induced +by the input difference (0x0000, 0x0040) is computed up to 13 rounds for +Simeck32/64. Also, to make a fair comparison with NDs, the accuracy of +the DDs with multiple ciphertext pairs under independent assumptions is +investigated. The comparison shows that the 9-, and 10-round NDs achieve +higher accuracy than the DDs, i.e., the ND contains more information than +the DDs (see Table 3). +• Based on the wrong key random hypothesis, we computed the score of the +ND for ciphertexts decrypted with different wrong keys and derived the +wrong key response profile (see Figure 3). Through a thorough study of the +wrong key response profile, we found that the 12-th and 13-th bit subkeys +have little effect on the score of the ND, but the ND is extremely sensitive +to the 14-th, and 15-th bit subkeys. Thus optimizing the Bayesian key search +algorithm (see Algorithm 3) and accelerating the key recovery attack. +• We enhanced the 15-round and launched the first practical 16-, 17-round +key recovery attacks for Simeck32/64 based on the ND. Table 1 provides a +summary of these results. +Table 1. Summary of key recovery attacks on Simeck32/64 +Attacks +R +Configure +Data +Time +Success Rate +Ref. +ND +13 +1+2+9+1 +216 +227.95+5⋆ +88% +[13] +14 +1+3+9+1 +223 +232.99+5⋆ +88% +[13] +15 +1+3+10+1 +224 +233.90+5⋆ +88% +[13] +1+3+10+1 +222 +235.309 +99.17% +Sect. 5 +16 +1+3+11+1 +224 +238.189 +100% +Sect. 5 +17 +1+3+12+1 +226 +245.037 +30% +Sect. 5 +1. +⋆: Time complexity is calculated in terms of the number of full rounds of +Simeck32/64 encryption per second of 223.304 in [13]. For a fair comparison, we +convert the time complexity to be calculated in terms of the number of 1-round +decryption performed per second. These two benchmarks differ by about 25. +2. Time complexity is calculated based on that one-second equals to 226.693 1-round +decryption per second in this paper. Also, 221.762 full-rounds of Simeck32/64 en- +cryption per second can be performed on our device. +Organization. The rest of the paper is organized as follows. Section 2 introduces +the design of Simeck and gives the preliminary on the ND model. Section 3 +gives the data format, network structure, training method, and result of NDs +The +experiment +is +conducted +by +Python +3.7.15 +and +Tensorflow +2.5.0 +in +Ubuntu +20.04. +The +device +information +is +Intel +Xeon +E5-2680V4*2 +with +2.40GHz, +256GB +RAM, +and +NVIDIA +RTX3080Ti +12GB*6. +The +source +code +is +available +on +GitHub +https://github.com/CryptAnalystDesigner/ +Differential-Neural-Cryptanalysis-Simeck32.git. + +for Simeck32/64. Section 4 describes the neutral bits and wrong key response +profiles used for key recovery attacks. Section 5 exhibits details of the (15-17)- +round key recovery attacks. Section 6 concludes this paper. +2 +Preliminary +In this paper, we denote an n-bit binary vector by x = (xn−1, . . . , x0), where +xi is the bit in position i with x0 the least significant one. ⊕ and ⊙ denote the +eXclusive-OR operation and the bitwise AND operation, respectively. x ≪ γ +or Sγ(x) represent circular left shift of x by γ bits. x ≫ γ or S−γ(x) represent +circular right shift of x by γ bits. x ∥ y represents the concatenation of bit strings +x and y. +2.1 +A Brief Description of Simeck +The Simeck family of lightweight block cipher was designed by Yang et al. in +CHES 2015 [15]. To develop even more compact and efficient block ciphers, it +incorporates good design components from both Simon and Speck designed by +NSA. A standardized approach for lightweight cryptography was proposed by +the National Institute of Standards and Technology (NIST) in 2019. Some ideas +for this project use modified Simeck as a fundamental module, such as ACE [1], +SPOC [2], and SPIX [3], which suggests that Simeck has more practical promise. +Simeck adopt the feistel structure to perform encryptions or decryptions on +2n-bit message blocks using a 4n-bit key, while n is the word size. The round +function of Simeck is defined as f5,0,1(x) = +� +S5 (x) ⊙ x +� +⊕ S1(x). Designers +reuse the round function in the key schedule to subkeys like Speck does. The +encryption algorithm of Simeck32/64 is listed in Algorithms 1. +Algorithm 1: Encryption of Simeck32/64. +Input: P = (x0, y0): the paintext, (k0, k1, · · · , k31): the round keys. +Output: C = (x32, y32): the ciphertext. +1 for r = 0 to 31 do +2 +xr+1 ← (xr ≪ 5) & xr ⊕ (xr ≪ 1) +3 +yr+1 ← xr +4 end +2.2 +Overview of Neural Distinguisher Model +The ND is a supervised model which distinguishes whether ciphertexts are en- +crypted by plaintexts that satisfies a specific input difference or by random +numbers. Given m plaintext pairs {(Pi,0, Pi,1), i ∈ [0, m − 1]} and target cipher, +the resulting ciphertext pairs {(Ci,0, Ci,1), i ∈ [0, m−1]} is regarded as a sample. +Each sample will be attached with a label Y : +Y = +�1, if Pi,0 ⊕ Pi,1 = ∆, i ∈ [0, m − 1] +0, if Pi,0 ⊕ Pi,1 ̸= ∆, i ∈ [0, m − 1] + +A large number of samples are fed into the neural network for training. Then, +the ND model can be described as: +Pr(Y = 1 | X0, . . . , Xm−1) = F (f(X0), · · · , f(Xm−1), ϕ(f(X0), · · · , f(Xm−1))) , +Xi = (Ci,0, Ci,1), i ∈ [0, m − 1], +Pr(Y = 1 | X0, · · · , Xm−1) ∈ [0, 1], +where f(Xi) represents the basic features of a ciphertext pair Xi, ϕ(·) is the +derived features, and F(·) is the new posterior probability estimation function. +3 +Neural Distinguisher for Simeck32/64 +It is crucial that a well-performing ND be obtained before a key recovery +can be conducted. In this section, we provided the state-of-the-art NDs for +Simeck32/64. More importantly, the DDs resulting from the input difference +(0x0000, 0x0040) are computed up to 13 rounds for Simeck32/64. These DDs +provide a solid baseline for NDs. +3.1 +Construction of the Dataset +Data quality is fundamentally the most important factor affecting the good- +ness of a model. Constructing a good dataset for NDs requires answering the +following questions: +• How to select a good input difference? +• What data format is used for a sample? +• How many ciphertext pairs are contained in a sample? +Input Difference. Numerous experiments have shown that the input difference +has a significant impact on the accuracy of the NDs/DDs [4,6,7,8,9,10,13,14]. +Simultaneously, obtaining better results for the key recovery attack depends on +whether the input difference of the NDs leads to better accuracy, while leading +to the prepended CDs with high probability. Therefore, it is also necessary to +consider the number of rounds and the neutral bits of the prepended CDs. +The choice of input difference of NDs varies depending on the block cipher. +For Simeck32/64, Lyu et al. [13] present two methods to select the input differ- +ence of the NDs. In the first method, the input difference for the NDs is selected +from the input difference of the classical differential trail of existing literature. +As part of the second method, the MILP model was used to find input differ- +ences for classical differential transitions that had high probabilities, then NDs +based on these input differences were trained with short epochs, and then the +NDs whose input differences had higher accuracy were selected for training long +epochs. But they did not consider the effect of the Hamming weight of the input +difference on the neural network. Lu et al. [12] studied the effect of the input +difference of NDs of Hamming weight less than or equal to 3 on the performance +of HDs, and their experiments showed that the input difference (0, ei) is a good + +choice to obtain a HD for Simon-like ciphers. Eventually, they built NDs for +Simeck32/64 up to 12 rounds with input difference (0x0000, 0x0040). +In this paper, we further explore the neutral bit of the input difference +(0x0000, 0x0040) (see Sect. 4.1) and, in a comprehensive comparison, chose this +input difference. +Data Format. In the process of training a ND, the format of the sample needs +to be specified in advance. This format is referred to as the ND’s data format +for convenience. The most intuitive data format is the ciphertext pair (C, C′) = +(xr, yr, x′ +r, y′ +r), which is used in Gohr’s network for Speck32/64 in [8,9]. As the +research progressed, Benamira et al. [7] constructed a new data format (xr ⊕ +x′ +r, xr ⊕x′ +r ⊕yr ⊕y′ +r, xr ⊕yr, x′ +r ⊕y′ +r) through the output of the first convolution +layer of Gohr’s neural network for Speck32/64, where xr ⊕ x′ +r represents the +left branch difference of the ciphertext, xr ⊕ x′ +r ⊕ yr ⊕ y′ +r represents the right +branch difference after decrypting one round of ciphertexts without knowing the +(r − 1)-th subkey according to the round function of Speck, xr ⊕ yr/x′ +r ⊕ y′ +r +represents the right branch ciphertext C/C′ of the penultimate round. It shows +that the data format is closely related to the structure of the ciphers. +Bao et al. [4] accepted data of the form (xr−1, x′ +r−1, yr−1 ⊕ y′ +r−1) for Si- +mon32/64. Since when the output of the r-th round (C, C′) = (xr, yr, x′ +r, y′ +r) +is known, one can directly compute (xr−1, x′ +r−1, yr−1 ⊕ y′ +r−1) without knowing +the (r−1)-th subkey according to the round function of Simon-like ciphers. Lu et +al. [12] further proposed a new data format (∆xr, ∆yr, xr, yr, x′ +r, y′ +r, ∆yr−1, p∆yr−2) +and obtained better performance. The details are illustrated in Fig. 1, and this +data format is used in this paper due to its superiority. +Using Multiple Ciphertext Pairs. Gohr et al. [9] showed that for a single +ciphertext pair, only their differences may provide information for Simon. One +option to surpass DDs is to use multiple ciphertext pairs simultaneously, us- +ing dependencies between the pairs, especially if the key is fixed. Therefore, in +order to surpass DDs, we use multiple ciphertext pairs for training, and the re- +sults (Section 3) confirm that multiple ciphertext pairs indeed help to surpass +DDs, albeit only in some rounds. One current trend in deep learning-assisted +cryptanalysis is the employment of multiple ciphertext pairs per sample, and +our results offer solid evidence in favor of this trend. +The three questions above have been addressed, and the dataset can be gen- +erated. Specifically, training and test sets were generated by using the Linux +random number generator to obtain uniformly distributed keys Ki and mul- +tiple plaintext pairs {(Pi,j,0, Pi,j,1), j ∈ [0, m − 1]} with the input difference +(0x0000, 0x0040) as well as a vector of binary-valued labels Yi. During the pro- +duction of the training or test sets for r-round Simeck32/64, the multiple plain- +text pairs were then encrypted for r rounds if Yi = 1, while otherwise, the second +plaintext of the pairs were replaced with a freshly generated random plaintext +and then encrypted for r rounds. Then use the r-round ciphertext pairs to gen- +erate samples with data of form (∆xr, ∆yr, xr, yr, x′ +r, y′ +r, ∆yr−1, p∆yr−2). + +∆xr−1 = ∆yr +∆yr−1 = yr−1 ⊕ y +′ +r−1 +Sa +Sb +Sc +kr−1 +∆xr = xr ⊕ x +′ +r +∆yr = yr ⊕ y +′ +r +∆xr−2 = ∆yr−1 +p∆yr−2 +Sa +Sb +Sc +kr−2 +Fig. 1. Notation of the data format for Simon-like ciphers, where yr−1 = Sa(yr) ⊙ +Sb(yr)⊕Sc(yr)⊕xr ⊕kr−1 ≜ A⊕kr−1, y +′ +r−1 = Sa(y +′ +r)⊙Sb(y +′ +r)⊕Sc(y +′ +r)⊕x +′ +r ⊕kr−1 ≜ +A +′ ⊕ kr−1, and p∆yr−2 = Sa(A) ⊙ Sb(A) ⊕ Sc(A) ⊕ yr ⊕ Sa(A +′) ⊙ Sb(A +′) ⊕ Sc(A +′) ⊕ y +′ +r +3.2 +Network Architecture +In CRYPTO 2019, Gohr [8] used the Residual Network to capture the dif- +ferential information between the ciphertext pairs, thus getting the ND for +Speck32/64. To learn the XOR relation at the same position of the cipher- +text, a one-dimensional convolution of kernel size 1 is used in Gohr’s network +architecture. Since there may be some intrinsic connection between several adja- +cent bits, Zhang et al. [16] added multiple one-dimensional convolutional layers +with different kernel sizes in front of the residual block according to the circular +shift operation in the round function of Speck32/64 and Simon32/64. In this +paper, we improved Zhang et al.’s neural network to fit with the round function +of Simeck to improve the accuracy of the NDs, the framework shown in Fig. 2. +Initial Convolution (Module 1). The input layer is connected to the initial +convolutional layer, which comprises two convolutional layers with Nf channels +of kernel sizes 1 and 5. The two convolution layers are concatenated at the chan- +nel dimension. Batch normalization is applied to the output of the concatenate +layers. Finally, rectifier nonlinearity is applied to the output of batch normaliza- +tion, and the resulting [m, ω, 2Nf] matrix is passed to the convolutional blocks +layer where m = 8, ω = 16 and Nf = 32. +Convolutional Blocks (Module 2). Each convolutional block consists of two +layers of 2Nf filters. Each block applies first the convolution with kernel size + +Output +Module 3 +Module 2 +Module 2 +Module 1 +Input +F(·) +f(·) +Module 1 +Conv, 1, Nf +Conv, 5, Nf +Concatenate, 2Nf +BN +Relu +Module 2 +ks = ks + 2 +Conv, ks, 2Nf +BN +Relu +Conv, ks, 2Nf +BN +Relu +⊕ +Module 3 +FC, d1 +BN +Relu +FC, d2 +BN +Relu +Output +FC, 1 +Sigmod +Fig. 2. The network architecture for Simeck32/64 +ks, then a batch normalization, and finally a rectifier layer. At the end of the +convolutional block, a skip connection is added to the output of the final rec- +tifier layer of the block to the input of the convolutional block. It transfers the +result to the next block. After each convolutional block, the kernel size ks in- +creases by 2 where ks = 3. The number of convolutional blocks is 5 in our model. +Prediction Head (Module 3 and Output). The prediction head consists of +two hidden layers and one output unit. The three fully connected layers comprise +d1, d2 units, followed by the batch normalization and rectifier layers where d1 = +512 and d2 = 64. The final layer consists of a single output unit using the +Sigmoid activation function. +3.3 +The Training method of Differential-Neural Distinguisher +The accuracy is the most critical indicator reflecting the performance of the neu- +ral distinguisher. The following training method was carried out to verify the +performance of our NDs. +Basic Training Scheme. We run the training for 20 epochs on the dataset for +N = 2∗107 and M = 2∗106. We set the batch size to 30000 and used Mirrored- +Strategy of TensorFlow to distribute it equally among the 6 GPUs. Optimization +was performed against mean square error loss plus a small penalty based on L2 +weights regularization parameter c = 10−5 using the Adam algorithm [11]. A +cyclic learning rate schedule was applied, setting the learning rate li for epoch i + +to li = α+ (n−i) mod (n+1) +n +·(β −α) with α = 10−4, β = 2×10−3 and n = 9. The +networks obtained at the end of each epoch were stored, and the best network +by validation loss was evaluated against a test set. +Training using the Staged Train Method. We use several stages of pre- +training to train an r-round ND for Simeck. First, we use our (r−1)-round dis- +tinguisher to recognize (r − 3)-round Simeck with the input difference (0x0140, +0x0080) (the most likely difference to appear three rounds after the input differ- +ence (0x0000, 0x0040). The training was done on 2 ∗ 107 instances for 10 epochs +with a cyclic learning rate schedule (2×10−3, 10−4). Then we trained the distin- +guisher to recognize r-round Simeck with the input difference (0x0000, 0x0040) +by processing 2 ∗ 107 freshly generated instances for 10 epochs with a cyclic +learning rate schedule (10−4, 10−5). Finally, the learning rate was dropped to +10−5 after processing another 2 ∗ 107 new instances for 10 epochs. +3.4 +Compared Result +We presented the state-of-the-art NDs for Simeck32/ 64. Meanwhile, we calcu- +late the DDs for Simeck32/64 triggered by the input difference (0x0000, 0x0040) +up to 13 rounds to give baselines for NDs (see Table 2). This is accomplished +through the use of the frameworks of Gohr’s implementation for Speck32/64 +and Bao et al.’s implementation for Simon32/64. The calculation is feasible on +Simeck32/64 but quite expensive. In fact, the calculation took about 939 core- +days of computation time and yielded about 34 gigabytes of distribution data +for each round, which was saved on disk for further studies. +Table 2. Accuracy of the DDs for Simeck32/64 with input difference (0x0000, 0x0040). +Combined means that the corresponding single pair distinguisher was used by combin- +ing the scores under independence assumption. For this, 2×106 samples, each consisting +of the given number of pairs m, were used to evaluating the accuracy. +R +m +1 +2 +4 +8 +16 +32 +64 +128 +256 +7 +0.9040 +0.9765 +0.9936 +0.9996 +1.0 +1.0 +1.0 +1.0 +1.0 +8 +0.7105 +0.7921 +0.8786 +0.9518 +0.9907 +0.9995 +1.0 +1.0 +1.0 +9 +0.5738 +0.6097 +0.6590 +0.7221 +0.8011 +0.8848 +0.9554 +0.9919 +0.9998 +10 +0.5194 +0.5299 +0.5462 +0.5677 +0.5984 +0.6403 +0.6977 +0.7690 +0.8517 +11 +0.5044 +0.5068 +0.5109 +0.5176 +0.5247 +0.5364 +0.5530 +0.5761 +0.6085 +12 +0.5010 +0.5017 +0.5025 +0.5039 +0.5055 +0.5083 +0.5121 +0.5176 +0.5259 +13 +0.5002 +0.5001 +0.5007 +0.5009 +0.5012 +0.5016 +0.5032 +0.5039 +0.5086 + +It is important to note that when multiple ciphertext pairs are used as a +sample in the NDs, comparing the accuracy of the DDs computed with a single +ciphertext pair as a sample is not fair. Actually, the accuracy of the DDs with +multiple ciphertext pairs per sample can be calculated. This calculation is im- +plicitly used by Gohr in [8], and later Gohr et al. [9] explicitly proposed rules for +combining probabilities/distinguisher responses (see Corollary 2 in [9]). One can +use this rule to explicitly convert a distinguisher for one ciphertext pair into one +for an arbitrary number of ciphertext pairs. Algorithm 2 gives the pseudo-code +for computing this distinguisher, and the results are shown in Table 2. +Algorithm 2: Convert the DD for one ciphertext pair into one for an +m number of ciphertext pairs. +Input: DDT: the R round DDT table; N: the number of samples for single +ciphertext pairs; m: the combined number of ciphertext pairs for one +sample. +Output: the combined Acc, TPR, TNR with m ciphertext pairs. +1 Y ← {} +2 for i = 1 to N do +3 +Y[i ∗ m] ← random{0, 1} +4 +for j = 1 to m − 1 do +5 +Y[i ∗ m − j] ← Y[i ∗ m] +6 +end +7 end +8 Randomly generate N ∗ m samples [x1, x2, · · · , xN∗m] according to Y +9 Z ← {} +10 for i = 1 to N ∗ m do +11 +Z[i] ← DDT[xi] +12 end +13 Z ← Z / (Z+2−32) +14 Z ← mean(Z.reshape(N,m), axis=1) +15 predict_Y ← {} +16 for i = 1 to N ∗ m do +17 +if +Z[i] > 0.5 then +18 +predict_Y[i] ← 1 +19 +end +20 +else +21 +predict_Y[i] ← 0 +22 +end +23 end +24 calculate Acc, TPR, TNR based on (Y, predict_Y) +25 return Acc, TPR, TNR +/* In our experiments, N takes 220 when m no more than 210. +*/ +In addition, r-round ND should be compared with (r − 1)-round DD. Since +the data fed to r-round ND is the value of the ciphertext, one can directly com- + +pute the differences on (r − 1)-round outputs without knowing the subkey. The +results are represented in Table 3, which shows that we improved the accuracy of +the NDs for Simeck32/64. More importantly, it is able to surpass the accuracy +of DDs for 9- and 10-round. +Table 3. Comparison of NDs on Simeck32/64 with 8 ciphertext pairs as a sample. +The input difference of ND/DD is (0x0000, 0x0040). *: The staged training method is +used to train ND. +R +Attack +Network +Acc +TPR +TNR +Ref. +9 +DD +DDT +0.9518 +0.9604 +0.9433 +Sect. 3 +ND +SE-ResNet +0.9952 +0.9989 +0.9914 +[12] +ND +Inception +0.9954 +0.9986 +0.9920 +Sect. 3 +10 +DD +DDT +0.7221 +0.7126 +0.7316 +Sect. 3 +ND +SE-ResNet +0.7354 +0.7207 +0.7501 +[12] +ND +Inception +0.7371 +0.7165 +0.7525 +Sect. 3 +11 +DD +DDT +0.5677 +0.5416 +0.5940 +Sect. 3 +ND +SE-ResNet +0.5646 +0.5356 +0.5936 +[12] +ND +Inception +0.5657 +0.5363 +0.5954 +Sect. 3 +ND +Inception +0.5666⋆ +0.5441 +0.5895 +Sect. 3 +12 +DD +DDT +0.5176 +0.4737 +0.5615 +Sect. 3 +ND +SE-ResNet +0.5146⋆ +0.4770 +0.5522 +[12] +ND +Inception +0.5161⋆ +0.4807 +0.5504 +Sect. 3 +4 +Neutral bits and Wrong Key Response Profile +In Sect. 3, we provided the state-of-the-art NDs for Simeck32/64, which use to +perform better key recovery attacks in the following section. In [8], Gohr provides +a framework of (1+s+r +1)-round key recovery attack (refer to Appendix A.1) +consisting of three techniques to increase the success rate and speed up the at- +tacks, where s is the length of the CD, and r is the length of the ND. Here is a +description of these techniques. +Neutral Bits. In the key recovery attack, multiple samples (formed into a ci- +phertext structure) decrypted by the guessed subkey are predicted using the +distinguisher. Then, the multiple scores are combined according to formula vk = +�nb +i=1 Zk +i/1−Zk +i as the final score of that guessed subkey to reduce the misjudgment +rate of the ND. Since the CD suspended in front of the ND are probabilistic, re- +sulting in sample entering the distinguisher not satisfying the same distribution. + +Multiple samples generated by neutral bits will have the same distribution. Also, +the lower the accuracy of the distinguisher, the more neutral bits are needed. +Priority of Ciphertext Structure. Spending the same amount of compu- +tation on every ciphertext structure is inefficient. Gohr used a generic method +(automatic exploitation versus exploration tradeoff based on Upper Confidence +Bounds) to focus the key search on the most promising ciphertext structures. +The priority score of each ciphertext structure is si = ωi +max + √nc · +� +log2(j)/ni +where denote by ωi +max the highest distinguisher score, ni the number of previous +iterations in which the ith ciphertext structure, j the number of the current +iteration and √nc the number of ciphertext structures available. +Wrong Key Response Profile. The key search policy based on Bayesian Op- +timization drastically reduces the number of trial decryptions. The basic idea +of this policy is the wrong key randomization hypothesis. This hypothesis does +not hold when only one round of trial decryption is performed, especially in a +lightweight cipher. The expected response of the ND upon wrong-key decryp- +tion will depend on the bitwise difference between the trial and real keys. This +wrong-key response profile can be captured in a precomputation. Give some +trial decryptions, the optimization step then trials to come up with a new set +of candidate keys to try. These new candidate keys are chosen to maximize the +probability of the observed distinguisher responses. +4.1 +Exploring Neutral Bits +To be able to attack more rounds with the ND, the CD is generally prepended +in front of the ND. For the resulting HD used in the key recovery attack, it is +not straightforward to aggregate enough samples of the same distribution fed to +the ND due to the prepended CD. To overcome this problem, Gohr [8] used the +neutral bits of the CD. The more neutral bits there are for the prepended CD, the +more samples of the same distribution could be generated for the ND. However, +generally, the longer the CD, the fewer the neutral bits. Finding enough neutral +bits for prepending a long CD over a weak ND becomes a difficult problem for +devising a key recovery to cover more rounds. To solve this problem, Bao et al. +exploited various generalized NBs to make weak ND usable again. Particularly, +they employed conditional simultaneous neutral bit-sets (CSNBS) and switching +bits for adjoining differentials (SBfAD), which are essential for achieving efficient +12-round and practical 13-round attacks for Speck32/64. +Thus, the first part of the key recovery attack focuses on finding various +types of neutral bits. Given a differential, in order to find the neutral bits, it is +generally divided into two steps: firstly, collect enough conforming pairs (correct +pairs); secondly, flip the target bits of the conforming pair, or flip all the bits +contained in the target set of bits, and check the probability that the new plain- +text pair is still the conforming pair. + +Finding SNBSs for 3-round Differential. For the prepended 3-round CD +(0x0140, 0x0200) → (0x0000, 0x0040) on top of the NDs, one can experimen- +tally obtain 14 deterministic NBs and 2 SNBSs (simultaneously complementing +up to 4 bits) using an exhaustive search. Concretely, for the 3-round differential +(0x0140, 0x0200) → (0x0000, 0x0040), (simultaneous-) neutral bits and bit-sets +are [3], [4], [5], [7], [8], [9], [13], [14], [15], [18], [20], [22], [24], [30], [0, 31], [10, 25]. +Finding SNBSs for 4-round Differential. For the prepended 4-round CD +(0x0300, 0x0440) → (0x0000, 0x0040) on top of the NDs, there are 7 com- +plete NB/SNBS: [2], [4], [6], [8], [14], [9, 24], [9, 10, 25]. Still, the numbers of +NBs/SNBSs are not enough for appending a weak neural network distinguisher. +Thus, conditional ones were searched using Algorithm 3 in paper [4], and the +obtained CSNBSs and their conditions are summarized together in Table 4. +Table +4. +CSNBS +for +4-round +Classical +Differential +(0x0300, 0x0440) +→ +(0x0000, 0x0040) of Simeck32/64 +Bit-set +C. +Bit-set +C. +x[0, 10] +x[2, 12] +[21] +00 +[23] +00 +[21, 5] +10 +[23, 12] +10 +[21, 10] +01 +[23, 7] +01 +[21, 10, 5] +11 +[23, 12, 7] +11 +C.: Condition on x[i, j], e.g., x[i, j] = 10 means x[i] = 1 and x[j] = 0. +4.2 +Wrong Key Response Profile +To calculate the r-round wrong key response profile, we generated 3000 random +keys and multiple input pairs {(Pi,0, Pi,1), i ∈ [0, m − 1]} for each difference +δ ∈ (0, 216) and encrypted for r +1 rounds to obtain ciphertexts {(Ci,0, Ci,1), i ∈ +[0, m − 1]}, where Pi,0 ⊕ Pi,1 = ∆. Denoting the final real subkey of each +encryption operation by k, we then performed single-round decryption to get +E−1 +k⊕δ({Ci,0, i ∈ [0, m − 1]}), E−1 +k⊕δ({Ci,1, i ∈ [0, m − 1]}) and had the resulting +partially decrypted ciphertext pair rated by an r-round ND. µδ and σδ were +then calculated as empirical mean and standard deviation over these 3000 trials. +We call the r-round wrong key response profile WKRPr. From the wrong key +Response Profile, we can find some rules to speed up the key recovery attack. +• Analysis of WKRP9. In Figure 3a, when the difference between guessed +key and real key δ is greater than 16384, the score of the distinguisher is +close to 0. This phenomenon indicates that the score of the distinguisher +is very low when the 14-th and 15-th bit is guessed incorrectly. When δ ∈ +{2048, 4096, 8192, 10240, 12288, 14436}, the score of the distinguisher is greater + +than 0.6. This indicates that when the 11-th, 12-th, and 13-th bits are guessed +incorrectly, it has little effect on the score of the distinguisher. +• Analysis of WKRP10 and WKRP11. It is clear from Figure 3b that +when the δ is greater than 32768, the score of the distinguisher is less than +0.45, i.e., the 15-th bit has a greater impact on the distinguisher score. When +δ ∈ {4096, 8192, 12288}, the score of the distinguisher is close to 0.55. This +indicates that when the 12-th and 13-th bits are guessed incorrectly, it has +little effect on the score of the distinguisher. It can also be observed from +Figure 3c that the 12-th and 13-th bits have less influence on the score of +the distinguisher, and the 14-th and 15-th bits have more influence on the +score of the distinguisher. +• Analysis of WKRP12. Despite the small difference in scores in Figure 3d, +it was found that when only the 12-th and 13-th bits are wrongly guessed, +the score of the distinguisher is still higher than the other positions. +(a) WKRP9 +(b) WKRP10 +(c) WKRP11 +(d) WKRP12 +Fig. 3. Wrong Key Response Profile for Simeck32/64. + +1.0 +0.50 +0.8 +Meanresponse +0.6 +0.4 +0.2 +0.0 +0 +4096 +Differencetorealkey0.50 +0.65 +0.60 +0.55 +response +0.50 +Mean +0.45 +0.40 +0.35 +0 +4096 +81921228816384204802457628672327683686440960450564915253248573446144065536 +Differenceto realkey0.520 +0.50 +0.515 +0.510 +0.505 +0.500 +Mean +0.495 +0.490 +0.485 +0.480 +0 +Differenceto realkey0.50 +0.5015 +0.5010 +0.5005 +response +0.5000 +Mean +0.4995 +0.4990 +0.4985 +0.4980 +0.4975 +0 +4096 +81921228816384204802457628672327683686440960450564915253248573446144065536 +Differencetoreal keyFrom the four wrong key response profiles, we can conclude that when the +14-th and 15-th bit subkeys are guessed incorrectly, it has a greater impact on +the score of the distinguisher; when the 12-th and 13-th bit subkeys are guessed +incorrectly, it has a smaller impact on the score of the distinguisher. According +to these phenomena, we can speed up the key recovery attack. +• Guess the 14-th and 15-th bit subkeys. Since the difference between +the score of the distinguisher of bits 14 and 15 in the case of correct and +incorrect guesses is relatively large, we can first determine the values of these +two bits. Before performing a Bayesian key search, a random set of subkeys +is guessed, then the 14-th and 15-th bits of the subkeys are traversed, and +the ciphertext is decrypted using the subkeys. Thus, the values of the 14-th +and 15-th bits can be determined based on the score of the distinguisher. +The Bayesian key search algorithm can easily recover these two bits even if +the values of these two bits are not determined in advance. +• Ignore the 12-th and 13-th bit subkeys. Since the 12-th and 13-th bit +subkeys have less influence on the score of the distinguisher, we first set +these two bits to 0 when generating the first batch of candidate subkeys and +then randomize the values of the two bits after completing the Bayesian key +sorting and recommending the new candidate subkeys. Previous researchers +have also exploited this feature to accelerate key recovery attacks, and the +14-th and 15-th bit subkeys have little impact on the score of the distin- +guisher when guessed incorrectly for Speck32/64 and Simon32/64[4,8,16]. +The Bayesian key search algorithm considering insensitive key bits is shown +in Algorithm 3. +5 +Practical Key Recovery Attack +When a fast graphics card is used, the performance of the implementation is not +limited by the speed of neural network evaluation but by the total number of +iterations on the ciphertext structures. We count a key guess as successful if the +sum of the Hamming weights of the differences between the returned last two +subkeys and the real two subkeys are at most two. The experimental parameters +for key recovery attacks are denoted as follows. +1. ncts: the number of ciphertext structure. +2. nb: the number of ciphertext pairs in each ciphertext structures. +3. nit: the total number of iterations on the ciphertext structures. +4. c1 and c2: the cutoffs with respect to the scores of the recommended last +subkey and second to last subkey, respectively. +5. nbyit1, ncand1 and nbyit2, ncand2: the number of iterations and number of key +candidates within each iteration in the BayesianKeySearch Algorithm for +guessing each of the last and the second to last subkeys, respectively. + +Algorithm 3: BayesianKeySearch Algorithm For Simeck32/64. +Input: Ciphertext structure C := {C0, · · · , Cnb−1}, a neural distinguisher +ND, and its wrong key response profile µ and σ, the number of +candidates to be generated within each iteration ncand, the number of +iterations nbyit +Output: The list L of tuples of recommended keys and their scores +1 S := {k0, k1, · · · , kncand−1} ← choose ncand values at random without +replacement from the set of all subkey candidates +2 S = S & 0xCFFF +3 L ← {} +4 for t = 1 to nbyit do +5 +for ∀ki ∈ S do +6 +for j = 0 to nb − 1 do +7 +C +′ +j,ki = F −1 +ki (Cj) +8 +vj,ki = ND(C +′ +j,ki) +9 +sj,ki = log2(vj,ki/(1 − vj,ki)) +10 +end +11 +ski = �nb−1 +j=0 sj,ki; /* the combined score of ki using neutral +bits. +*/ +12 +L ← L∥(ki, ski); +13 +mki = �nb−1 +j=0 vj,ki/nb +14 +end +15 +for k ∈ {0, 1, · · · , 216 − 1} & 0xCFFF do +16 +λk = �ncand−1 +i=0 +(mki − µki⊕k)2/σ2 +ki⊕k; /* using wrong key response +profile. +*/ +17 +end +18 +S ← argsortk(λ)[0 : ncand − 1]; +19 +r := {r0, r1, · · · , rncand−1} ← choose ncand values at (0, 4) at random +20 +r = r << 12; /* Randomize the 12-th and 13-th bit subkeys. +*/ +21 +S = S ⊕ r +22 end +23 return L + +5.1 +Complexity Calculation +Theoretical Data Complexity. The theoretical data complexity of the exper- +iment is calculated by the formula nb × nct × m × 2. In the actual experiment, +when the accuracy of the ND is high, the key can be recovered quickly and suc- +cessfully. Not all the ciphertext structure is used, so the actual data complexity +is lower than the theoretical. +Experimental Time Complexity. The time complexity calculation formula +in our experiments is 226.693 × rt × log1−sr 0.01, which is borrowed from [16]. +Our device can perform 226.693 1-round decryption per second. rt is the average +running time of multiple experiments. The success rate sr is the number of suc- +cessfully recovered subkeys divided by the number of experiments. We calculate +how many experiments need to be performed to ensure at least one successful +experiment. When the overall success rate is 99%, we consider the experiment +to be successful, and the number of experiments ne is: 1−(1−sr)ne = 0.99, i.e., +log1−sr 0.01. +5.2 +Key Recovery Attack on 15-round Simeck32/64 +Experiment 1: The components of key recovery attack ASimeck15R of 15-round +Simeck32/64 are as follows. +1. 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040). +2. neutral bits of generating multiple ciphertext pairs: [3], [4], [5]. +3. neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14], +[15], [18], [20]. +4. 10-round neural distinguisher NDSimeck10R and wrong key response profiles +NDSimon10R · µ and NDSimeck10R · δ. +5. 9-round distinguisher NDSimeck9R and wrong key response profiles NDSimon9R· +µ and NDSimeck9R · δ. +Concrete parameters used in our 15-round key recovery attack ASimeck15R are +listed as follows. +m = 8 +nb = 28 +ncts = 210 +nit = 211 +c1 = 10 +c2 = 10 +nbyit1 = nbyit2 = 5 +ncand1 = ncand2 = 32 +The theoretical data complexity is m×nb ×ncts ×2 = 222 plaintexts. The ac- +tual data complexity is 219.621. In total, 120 trials are running and 119 successful +trials. Thus, the success rate sr is 99.17%. The average running time of the exper- +iment rt is 407.901s. The time complexity is 226.693 × rt × log1−sr 0.01 = 235.309. +5.3 +Key Recovery Attack on 16-round Simeck32/64 +Experiment 2: The components of key recovery attack ASimeck16R of 16-round +Simeck32/64 are shown as follows. + +1. 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040). +2. neutral bits of generating multiple ciphertext pairs: [3], [4], [5]. +3. neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14], +[15], [18], [20], [22], [24]. +4. 11-round neural distinguisher NDSimeck11R and wrong key response profiles +NDSimeck11R · µ and NDSimeck11R · δ. +5. 10-round neural distinguisher NDSimeck10R and wrong key response profiles +NDSimeck10R · µ and NDSimeck10R · δ. +Concrete parameters used in our 16-round key recovery attack ASimeck16R are +listed as follows. +m = 8 +nb = 210 +ncts = 210 +nit = 211 +c1 = 10 +c2 = 10 +nbyit1 = nbyit2 = 5 +ncand1 = ncand2 = 32 +The theoretical data complexity is m × nb × ncts × 2 = 224 plaintexts. The +actual data complexity is 222.788. We use 6 processes, each running 20 experi- +ments. Since the memory limit was exceeded during the experiment, one process +was killed, leaving 100 experiments, 100 of which successfully recovered the key. +Thus, the success rate sr is 100%. The average running time of the experiment +rt is 2889.648s. The time complexity is 226.693 × rt = 238.189. +5.4 +Key Recovery Attack on 17-round Simeck32/64 +Experiment 3: The components of key recovery attack ASimeck17R of 17-round +Simeck32/64 are shown as follows. +1. 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040). +2. neutral bits of generating multiple ciphertext pairs: [3], [4], [5]. +3. neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14], +[15], [18], [20], [22], [24], [30], [0, 31]. +4. 12-round neural distinguisher NDSimeck12R and wrong key response profiles +NDSimeck12R · µ and NDSimeck12R · δ. +5. 11-round neural distinguisher NDSimeck11R and wrong key response profiles +NDSimeck11R · µ and NDSimeck11R · δ. +Concrete parameters used in our 17-round key recovery attack ASimeck17R are +listed as follows. +m = 8 +nb = 212 +ncts = 210 +nit = 211 +c1 = 20 +c2 = −120 +nbyit1 = nbyit2 = 5 +ncand1 = ncand2 = 32 +The theoretical data complexity is m × nb × ncts × 2 = 226 plaintexts. The +actual data complexity is 225.935. In total, trials are 50 running, and there are +15 successful trials. Thus, the success rate sr is 30%. The average running +time of the experiment rt is 25774.822s. The time complexity is 226.693 × rt × +log1−sr 0.01 = 245.037. + +Remark 1. There are two reasons why we do not launch a 17-round key recovery +attack using a 4-round CD and an 11-round ND. One is that the probability +of the 4-round CD (0x0300, 0x0440) → (0x0000, 0x0040) is about 212 (the prob- +ability of the 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040) is about 2−8), +resulting in too much data required, and the second is that there are not enough +neutral bits in the 4-round CD. +6 +Conclusion +In this paper, we show practical key recovery attacks up to 17 rounds of Simeck +32/64, raising the technical level of practical attacks by two rounds. We design +neural network that fits with the round function of Simeck to improve the ac- +curacy of the neural distinguishers, and is able to outperform the DDT-based +distinguisher in some rounds. To launch more rounds of the key recovery attack, +we make a concerted effort on the classical differential and the neural distin- +guisher to make both modules good. In addition, we optimize the key recovery +attack process by deeply analyzing the wrong key response profile, thus reducing +the complexity of the key recovery attack. +References +1. Aagaard, M., AlTawy, R., Gong, G., Mandal, K., Rohit, R.: Ace: An authenticated +encryption and hash algorithm. Submission to NIST-LWC (announced as round 2 +candidate on August 30, 2019) (2019) +2. AlTawy, R., Gong, G., He, M., Jha, A., Mandal, K., Nandi, M., Rohit, R.: Spoc: +an authenticated cipher submission to the nist lwc competition (2019) +3. 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In: International Conference on Information Security. pp. +443–463. Springer (2022) +14. Yadav, T., Kumar, M.: Differential-ml distinguisher: Machine learning based +generic extension for differential cryptanalysis. In: International Conference on +Cryptology and Information Security in Latin America. pp. 191–212. Springer +(2021) +15. Yang, G., Zhu, B., Suder, V., Aagaard, M.D., Gong, G.: The simeck family of +lightweight block ciphers. In: International Workshop on Cryptographic Hardware +and Embedded Systems. pp. 307–329. Springer (2015) +16. Zhang, L., Wang, Z., Wang, B.: Improving differential-neural cryptanalysis with +inception blocks. Cryptology ePrint Archive (2022) + +A +Appendix +A.1 +Procedure of (1 + s + r + 1)-round key recovery attack +The attack procedure is as follows. +1. Initialize variables Gbestkey ← (None, None), Gbestscore ← −∞. +2. Generate ncts random plaintext pairs with difference ∆P. +3. Using ncts plaintext pairs and log2 m neutral bit with probability one to +generate ncts multiple plaintext pairs. Every multiple plaintext pairs have +m plaintext pairs. +4. From the ncts multiple plaintext pairs, generate ncts plaintext structures +using nb generalized neutral bit. +5. Decrypt one round using zero as the subkey for all multiple plaintext pairs +in the structures and obtain ncts plaintext structure. +6. Query for the ciphertexts under (1 + s + r + 1)-round Simeck32/64 of the +ncts × nb × 2 plaintext structures, thus obtain ncts ciphertext structures, +denoted by {C1, . . . , Cncts}. +7. Initialize an array ωmax and an array nvisit to record the highest distinguisher +score obtained so far and the number of visits have received in the last subkey +search for the ciphertext structures. +8. Initialize variables bestscore ← −∞, bestkey ← (None, None), bestpos ← +None to record the best score, the corresponding best recommended values +for the two subkeys obtained among all ciphertext structures and the index +of this ciphertext structures. +9. For j from 1 to nit: +(a) Compute the priority of each of the ciphertext structures as follows: +si = ωmaxi + α · +� +log2 j/nvisiti, for i ∈ {1, . . . , ncts}, and α = √ncts; +The formula of priority is designed according to a general method in +reinforcement learning for achieving automatic exploitation versus ex- +ploration trade-off based on Upper Confidence Bounds. It is motivated +to focus the key search on the most promising ciphertext structures [8]. +(b) Pick the ciphertext structure with the highest priority score for further +processing in this j-th iteration, denote it by C, and its index by idx, +nvisitidx ← nvisitidx + 1. +(c) Run BayesianKeySearch Algorithm [8] with C, the r-round neural +distinguisher NDr and its wrong key response profile NDr ·µ and NDr · +σ, ncand1, and nbyit1 as input parameters; obtain the output, that is a +list L1 of nbyit1 × ncand1 candidate values for the last subkey and their +scores, i.e., L1 = {(g1i, v1i) : i ∈ {1, . . . , nbyit1 × ncand1}}. +(d) Find the maximum v1max among v1i in L1, if v1max > ωmaxidx, ωmaxidx ← +v1max. +(e) For each of recommended last subkey g1i ∈ L1, if the score v1i > c1, +i. Decrypt the ciphertext in C using the g1i by one round and obtain +the ciphertext structures C′ of (1 + s + r)-round Simeck32/64. + +ii. Run BayesianKeySearch Algorithm [8] with C′ , the neural dis- +tinguisher NDr−1 and its wrong key response profile NDr−1 · µ and +NDr−1·σ, ncand2, and nbyit2 as input parameters; obtain the output, +that is a list L2 of nbyit2×ncand2 candidate values for the last subkey +and their scores, i.e., L2 = {(g2i, v2i) : i ∈ {1, . . . , nbyit2 × ncand2}}. +iii. Find the maximum v2i and the corresponding g2i in L2, and denote +them by v2max and g2max. +iv. If v2max > bestscore, update bestscore ← v2max, bestkey ← (g1i, +g2max), bestpos ← idx. +(f) If bestscore > c2, go to Step 10. +10. Make a final improvement using VerifierSearch [8] on the value of bestkey +by examining whether the scores of a set of keys obtained by changing at +most 2 bits on top of the incrementally updated bestkey could be improved +recursively until no improvement obtained, update bestscore to the best score +in the final improvement; If bestscore > Gbestscore, update Gbestscore ← +bestscore, Gbestkey ← bestkey. +11. Return Gbestkey, Gbestscore. + diff --git a/-tFJT4oBgHgl3EQfpywl/content/tmp_files/load_file.txt b/-tFJT4oBgHgl3EQfpywl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4915986b0681bbdcb749ab451353dd019d2f203d --- /dev/null +++ b/-tFJT4oBgHgl3EQfpywl/content/tmp_files/load_file.txt @@ -0,0 +1,810 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf,len=809 +page_content='Improved Differential-neural Cryptanalysis for Round-reduced Simeck32/64 ∗ Liu Zhang1,3[0000−0001−6106−3767], Jinyu Lu2(�)[0000−0002−7299−0934], Zilong Wang1,3[0000−0002−1525−3356], and Chao Li2,3[0000−0001−7467−7573] 1 School of Cyber Engineering, Xidian University, Xi’an 710126, China {liuzhang@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', zlwang@}xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='cn 2 College of Sciences, National University of Defense Technology, Hunan, Changsha 410073, China, jinyu_smile@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='com, lichao_nudt@sina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='com 3 State Key Laboratory of Cryptology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='Box 5159, Beijing 100878, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In CRYPTO 2019, Gohr presented differential-neural crypt- analysis by building the differential distinguisher with a neural network, achieving practical 11-, and 12-round key recovery attack for Speck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Inspired by this framework, we develop the Inception neural network that is compatible with the round function of Simeck to improve the accuracy of the neural distinguishers, thus improving the accuracy of (9-12)-round neural distinguishers for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To provide solid baselines for neu- ral distinguishers, we compute the full distribution of differences induced by one specific input difference up to 13-round Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Moreover, the performance of the DDT-based distinguishers in multiple ciphertext pairs is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Compared with the DDT-based distinguishers, the 9-, and 10-round neural distinguishers achieve better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Also, an in- depth analysis of the wrong key response profile revealed that the 12-th and 13-th bits of the subkey have little effect on the score of the neu- ral distinguisher, thereby accelerating key recovery attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finally, an enhanced 15-round and the first practical 16-, and 17-round attacks are implemented for Simeck32/64, and the success rate of both the 15-, and 16-round attacks is almost 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Keywords: Neural Distinguisher, Wrong Key Response Profile, Key Recovery Attack, Simeck32/64 1 Introduction Lightweight block ciphers present trade-offs between appropriate security and small resource-constrained devices, which is an essential foundation for data con- fidentiality in resource-constrained environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Therefore, the design require- ments and security analysis of lightweight block ciphers are of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Combining traditional analysis methods with “machine speed” to efficiently and ∗ Supported by organization x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' First Author and Second Author contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='11601v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='CR] 27 Jan 2023 intelligently evaluate the security of cryptographic algorithm components, is one of the critical points and trends of current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The development of Artificial Intelligence (AI) provides new opportunities for cryptanalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In CRYPTO 2019 [8], Gohr creatively combines deep learning with differ- ential cryptanalysis and applies it to the Speck32/64, gaining the neural dis- tinguisher (ND) can surpass the DDT-based distinguisher (DD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Then, a hy- brid distinguisher (HD) consisting of a ND and a classical differential (CD) with highly selective key search strategies result in forceful practical 11-, and 12-round key recovery attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In EUROCRYPT 2021 [7], Benamira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' pro- posed a thorough analysis of Gohr’s neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' They discovered that these distinguishers are basing their decisions on the ciphertext pair difference and the internal state difference in penultimate and antepenultimate rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To attack more rounds, the component CD or ND must be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In ASIACRYPT 2022 [4], Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' devised the first practical 13-round and an improved 12-round ND-based key recovery attacks for Speck32/64 by enhanc- ing the CDs, which they deeply explored more generalized neutral bits of dif- ferentials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', conditional (simultaneous) neutral bit/bit-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In addition, they obtained NDs up to 11-round Simon32/64 by using DenseNet and SENet, thus launching the practical 16-round key recovery attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [16] focused on improving the accuracy of ND and added the Inception composed of the multiple-parallel convolutional layers before the Residual network to capture information on multiple dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Under the combined effect of multiple im- provements, they reduced the time complexity of key recovery attacks for 12-, and 13-round Speck32/64 and 16-round Simon32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' They also devised the first practical 17-round key recovery for Simon32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The Simeck algorithm [15], which combines the good design components from both Simon and Speck [5] designed by National Security Agency (NSA), has received a lot of attention for its security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In 2022, Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [13] improved Gohr’s framework and applied it to Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' They obtained (8-10)-round NDs for Simeck32/64 and successfully accomplished attacks for (13-15)-round Simeck32/64 with low data complexity and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In the same year, Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [12] adopted the multiple ciphertext pairs (8 ciphertext pairs) to train the SE-ResNet neural network fed with a new data format for Simon and Simeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finally, they obtained (9-12)-round NDs for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This raises the question of whether the key recovery attack for Simeck can be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Our Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The contributions of this work are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We improved the Inception neural network proposed by zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [16] ac- cording to the number of cyclic rotation in the round function of Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Meanwhile, to capture the connections between ciphertext pairs, we use mul- tiple ciphertext pairs forming a sample as the input of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Therefore, we improved the accuracy of (9-12)-round NDs using the ba- sic training method and staged training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The result can be seen in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To provide solid baselines for NDs, the full distribution of differences induced by the input difference (0x0000, 0x0040) is computed up to 13 rounds for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Also, to make a fair comparison with NDs, the accuracy of the DDs with multiple ciphertext pairs under independent assumptions is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The comparison shows that the 9-, and 10-round NDs achieve higher accuracy than the DDs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', the ND contains more information than the DDs (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Based on the wrong key random hypothesis, we computed the score of the ND for ciphertexts decrypted with different wrong keys and derived the wrong key response profile (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Through a thorough study of the wrong key response profile, we found that the 12-th and 13-th bit subkeys have little effect on the score of the ND, but the ND is extremely sensitive to the 14-th, and 15-th bit subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus optimizing the Bayesian key search algorithm (see Algorithm 3) and accelerating the key recovery attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We enhanced the 15-round and launched the first practical 16-, 17-round key recovery attacks for Simeck32/64 based on the ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Table 1 provides a summary of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Summary of key recovery attacks on Simeck32/64 Attacks R Configure Data Time Success Rate Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' ND 13 1+2+9+1 216 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='95+5⋆ 88% [13] 14 1+3+9+1 223 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='99+5⋆ 88% [13] 15 1+3+10+1 224 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='90+5⋆ 88% [13] 1+3+10+1 222 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='309 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='17% Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5 16 1+3+11+1 224 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='189 100% Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5 17 1+3+12+1 226 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='037 30% Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' ⋆: Time complexity is calculated in terms of the number of full rounds of Simeck32/64 encryption per second of 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='304 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For a fair comparison, we convert the time complexity to be calculated in terms of the number of 1-round decryption performed per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' These two benchmarks differ by about 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Time complexity is calculated based on that one-second equals to 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='693 1-round decryption per second in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Also, 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='762 full-rounds of Simeck32/64 en- cryption per second can be performed on our device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Section 2 introduces the design of Simeck and gives the preliminary on the ND model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Section 3 gives the data format, network structure, training method, and result of NDs The experiment is conducted by Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='15 and Tensorflow 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='0 in Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The device information is Intel Xeon E5-2680V4*2 with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='40GHz, 256GB RAM, and NVIDIA RTX3080Ti 12GB*6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The source code is available on GitHub https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='com/CryptAnalystDesigner/ Differential-Neural-Cryptanalysis-Simeck32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Section 4 describes the neutral bits and wrong key response profiles used for key recovery attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Section 5 exhibits details of the (15-17)- round key recovery attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Section 6 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2 Preliminary In this paper, we denote an n-bit binary vector by x = (xn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' , x0), where xi is the bit in position i with x0 the least significant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' ⊕ and ⊙ denote the eXclusive-OR operation and the bitwise AND operation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' x ≪ γ or Sγ(x) represent circular left shift of x by γ bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' x ≫ γ or S−γ(x) represent circular right shift of x by γ bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' x ∥ y represents the concatenation of bit strings x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1 A Brief Description of Simeck The Simeck family of lightweight block cipher was designed by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' in CHES 2015 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To develop even more compact and efficient block ciphers, it incorporates good design components from both Simon and Speck designed by NSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' A standardized approach for lightweight cryptography was proposed by the National Institute of Standards and Technology (NIST) in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Some ideas for this project use modified Simeck as a fundamental module, such as ACE [1], SPOC [2], and SPIX [3], which suggests that Simeck has more practical promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Simeck adopt the feistel structure to perform encryptions or decryptions on 2n-bit message blocks using a 4n-bit key, while n is the word size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The round function of Simeck is defined as f5,0,1(x) = � S5 (x) ⊙ x � ⊕ S1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Designers reuse the round function in the key schedule to subkeys like Speck does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The encryption algorithm of Simeck32/64 is listed in Algorithms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Algorithm 1: Encryption of Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Input: P = (x0, y0): the paintext, (k0, k1, · · · , k31): the round keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Output: C = (x32, y32): the ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1 for r = 0 to 31 do 2 xr+1 ← (xr ≪ 5) & xr ⊕ (xr ≪ 1) 3 yr+1 ← xr 4 end 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='2 Overview of Neural Distinguisher Model The ND is a supervised model which distinguishes whether ciphertexts are en- crypted by plaintexts that satisfies a specific input difference or by random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Given m plaintext pairs {(Pi,0, Pi,1), i ∈ [0, m − 1]} and target cipher, the resulting ciphertext pairs {(Ci,0, Ci,1), i ∈ [0, m−1]} is regarded as a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Each sample will be attached with a label Y : Y = �1, if Pi,0 ⊕ Pi,1 = ∆, i ∈ [0, m − 1] 0, if Pi,0 ⊕ Pi,1 ̸= ∆, i ∈ [0, m − 1] A large number of samples are fed into the neural network for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Then, the ND model can be described as: Pr(Y = 1 | X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' , Xm−1) = F (f(X0), · · · , f(Xm−1), ϕ(f(X0), · · · , f(Xm−1))) , Xi = (Ci,0, Ci,1), i ∈ [0, m − 1], Pr(Y = 1 | X0, · · · , Xm−1) ∈ [0, 1], where f(Xi) represents the basic features of a ciphertext pair Xi, ϕ(·) is the derived features, and F(·) is the new posterior probability estimation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 Neural Distinguisher for Simeck32/64 It is crucial that a well-performing ND be obtained before a key recovery can be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In this section, we provided the state-of-the-art NDs for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' More importantly, the DDs resulting from the input difference (0x0000, 0x0040) are computed up to 13 rounds for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' These DDs provide a solid baseline for NDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1 Construction of the Dataset Data quality is fundamentally the most important factor affecting the good- ness of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Constructing a good dataset for NDs requires answering the following questions: How to select a good input difference?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' What data format is used for a sample?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' How many ciphertext pairs are contained in a sample?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Input Difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Numerous experiments have shown that the input difference has a significant impact on the accuracy of the NDs/DDs [4,6,7,8,9,10,13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Simultaneously, obtaining better results for the key recovery attack depends on whether the input difference of the NDs leads to better accuracy, while leading to the prepended CDs with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Therefore, it is also necessary to consider the number of rounds and the neutral bits of the prepended CDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The choice of input difference of NDs varies depending on the block cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For Simeck32/64, Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [13] present two methods to select the input differ- ence of the NDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In the first method, the input difference for the NDs is selected from the input difference of the classical differential trail of existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' As part of the second method, the MILP model was used to find input differ- ences for classical differential transitions that had high probabilities, then NDs based on these input differences were trained with short epochs, and then the NDs whose input differences had higher accuracy were selected for training long epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' But they did not consider the effect of the Hamming weight of the input difference on the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [12] studied the effect of the input difference of NDs of Hamming weight less than or equal to 3 on the performance of HDs, and their experiments showed that the input difference (0, ei) is a good choice to obtain a HD for Simon-like ciphers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Eventually, they built NDs for Simeck32/64 up to 12 rounds with input difference (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In this paper, we further explore the neutral bit of the input difference (0x0000, 0x0040) (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1) and, in a comprehensive comparison, chose this input difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Data Format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In the process of training a ND, the format of the sample needs to be specified in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This format is referred to as the ND’s data format for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The most intuitive data format is the ciphertext pair (C, C′) = (xr, yr, x′ r, y′ r), which is used in Gohr’s network for Speck32/64 in [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' As the research progressed, Benamira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [7] constructed a new data format (xr ⊕ x′ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' xr ⊕x′ r ⊕yr ⊕y′ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' xr ⊕yr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' x′ r ⊕y′ r) through the output of the first convolution layer of Gohr’s neural network for Speck32/64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' where xr ⊕ x′ r represents the left branch difference of the ciphertext,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' xr ⊕ x′ r ⊕ yr ⊕ y′ r represents the right branch difference after decrypting one round of ciphertexts without knowing the (r − 1)-th subkey according to the round function of Speck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' xr ⊕ yr/x′ r ⊕ y′ r represents the right branch ciphertext C/C′ of the penultimate round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' It shows that the data format is closely related to the structure of the ciphers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [4] accepted data of the form (xr−1, x′ r−1, yr−1 ⊕ y′ r−1) for Si- mon32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since when the output of the r-th round (C, C′) = (xr, yr, x′ r, y′ r) is known, one can directly compute (xr−1, x′ r−1, yr−1 ⊕ y′ r−1) without knowing the (r−1)-th subkey according to the round function of Simon-like ciphers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [12] further proposed a new data format (∆xr, ∆yr, xr, yr, x′ r, y′ r, ∆yr−1, p∆yr−2) and obtained better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The details are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1, and this data format is used in this paper due to its superiority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Using Multiple Ciphertext Pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Gohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [9] showed that for a single ciphertext pair, only their differences may provide information for Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' One option to surpass DDs is to use multiple ciphertext pairs simultaneously, us- ing dependencies between the pairs, especially if the key is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Therefore, in order to surpass DDs, we use multiple ciphertext pairs for training, and the re- sults (Section 3) confirm that multiple ciphertext pairs indeed help to surpass DDs, albeit only in some rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' One current trend in deep learning-assisted cryptanalysis is the employment of multiple ciphertext pairs per sample, and our results offer solid evidence in favor of this trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The three questions above have been addressed, and the dataset can be gen- erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Specifically, training and test sets were generated by using the Linux random number generator to obtain uniformly distributed keys Ki and mul- tiple plaintext pairs {(Pi,j,0, Pi,j,1), j ∈ [0, m − 1]} with the input difference (0x0000, 0x0040) as well as a vector of binary-valued labels Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' During the pro- duction of the training or test sets for r-round Simeck32/64, the multiple plain- text pairs were then encrypted for r rounds if Yi = 1, while otherwise, the second plaintext of the pairs were replaced with a freshly generated random plaintext and then encrypted for r rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Then use the r-round ciphertext pairs to gen- erate samples with data of form (∆xr, ∆yr, xr, yr, x′ r, y′ r, ∆yr−1, p∆yr−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' ∆xr−1 = ∆yr ∆yr−1 = yr−1 ⊕ y ′ r−1 Sa Sb Sc kr−1 ∆xr = xr ⊕ x ′ r ∆yr = yr ⊕ y ′ r ∆xr−2 = ∆yr−1 p∆yr−2 Sa Sb Sc kr−2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Notation of the data format for Simon-like ciphers, where yr−1 = Sa(yr) ⊙ Sb(yr)⊕Sc(yr)⊕xr ⊕kr−1 ≜ A⊕kr−1, y ′ r−1 = Sa(y ′ r)⊙Sb(y ′ r)⊕Sc(y ′ r)⊕x ′ r ⊕kr−1 ≜ A ′ ⊕ kr−1, and p∆yr−2 = Sa(A) ⊙ Sb(A) ⊕ Sc(A) ⊕ yr ⊕ Sa(A ′) ⊙ Sb(A ′) ⊕ Sc(A ′) ⊕ y ′ r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='2 Network Architecture In CRYPTO 2019, Gohr [8] used the Residual Network to capture the dif- ferential information between the ciphertext pairs, thus getting the ND for Speck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To learn the XOR relation at the same position of the cipher- text, a one-dimensional convolution of kernel size 1 is used in Gohr’s network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since there may be some intrinsic connection between several adja- cent bits, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [16] added multiple one-dimensional convolutional layers with different kernel sizes in front of the residual block according to the circular shift operation in the round function of Speck32/64 and Simon32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In this paper, we improved Zhang et al.’s neural network to fit with the round function of Simeck to improve the accuracy of the NDs, the framework shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Initial Convolution (Module 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The input layer is connected to the initial convolutional layer, which comprises two convolutional layers with Nf channels of kernel sizes 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The two convolution layers are concatenated at the chan- nel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Batch normalization is applied to the output of the concatenate layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finally, rectifier nonlinearity is applied to the output of batch normaliza- tion, and the resulting [m, ω, 2Nf] matrix is passed to the convolutional blocks layer where m = 8, ω = 16 and Nf = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Convolutional Blocks (Module 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Each convolutional block consists of two layers of 2Nf filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Each block applies first the convolution with kernel size Output Module 3 Module 2 Module 2 Module 1 Input F(·) f(·) Module 1 Conv, 1, Nf Conv, 5, Nf Concatenate, 2Nf BN Relu Module 2 ks = ks + 2 Conv, ks, 2Nf BN Relu Conv, ks, 2Nf BN Relu ⊕ Module 3 FC, d1 BN Relu FC, d2 BN Relu Output FC, 1 Sigmod Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The network architecture for Simeck32/64 ks, then a batch normalization, and finally a rectifier layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' At the end of the convolutional block, a skip connection is added to the output of the final rec- tifier layer of the block to the input of the convolutional block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' It transfers the result to the next block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' After each convolutional block, the kernel size ks in- creases by 2 where ks = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The number of convolutional blocks is 5 in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Prediction Head (Module 3 and Output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The prediction head consists of two hidden layers and one output unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The three fully connected layers comprise d1, d2 units, followed by the batch normalization and rectifier layers where d1 = 512 and d2 = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The final layer consists of a single output unit using the Sigmoid activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='3 The Training method of Differential-Neural Distinguisher The accuracy is the most critical indicator reflecting the performance of the neu- ral distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The following training method was carried out to verify the performance of our NDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Basic Training Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We run the training for 20 epochs on the dataset for N = 2∗107 and M = 2∗106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We set the batch size to 30000 and used Mirrored- Strategy of TensorFlow to distribute it equally among the 6 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Optimization was performed against mean square error loss plus a small penalty based on L2 weights regularization parameter c = 10−5 using the Adam algorithm [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' A cyclic learning rate schedule was applied, setting the learning rate li for epoch i to li = α+ (n−i) mod (n+1) n (β −α) with α = 10−4, β = 2×10−3 and n = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The networks obtained at the end of each epoch were stored, and the best network by validation loss was evaluated against a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Training using the Staged Train Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We use several stages of pre- training to train an r-round ND for Simeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' First, we use our (r−1)-round dis- tinguisher to recognize (r − 3)-round Simeck with the input difference (0x0140, 0x0080) (the most likely difference to appear three rounds after the input differ- ence (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The training was done on 2 ∗ 107 instances for 10 epochs with a cyclic learning rate schedule (2×10−3, 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Then we trained the distin- guisher to recognize r-round Simeck with the input difference (0x0000, 0x0040) by processing 2 ∗ 107 freshly generated instances for 10 epochs with a cyclic learning rate schedule (10−4, 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finally, the learning rate was dropped to 10−5 after processing another 2 ∗ 107 new instances for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4 Compared Result We presented the state-of-the-art NDs for Simeck32/ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Meanwhile, we calcu- late the DDs for Simeck32/64 triggered by the input difference (0x0000, 0x0040) up to 13 rounds to give baselines for NDs (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This is accomplished through the use of the frameworks of Gohr’s implementation for Speck32/64 and Bao et al.’s implementation for Simon32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The calculation is feasible on Simeck32/64 but quite expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In fact, the calculation took about 939 core- days of computation time and yielded about 34 gigabytes of distribution data for each round, which was saved on disk for further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Accuracy of the DDs for Simeck32/64 with input difference (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Combined means that the corresponding single pair distinguisher was used by combin- ing the scores under independence assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For this, 2×106 samples, each consisting of the given number of pairs m, were used to evaluating the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' R m 1 2 4 8 16 32 64 128 256 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9765 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5086 It is important to note that when multiple ciphertext pairs are used as a sample in the NDs, comparing the accuracy of the DDs computed with a single ciphertext pair as a sample is not fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Actually, the accuracy of the DDs with multiple ciphertext pairs per sample can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This calculation is im- plicitly used by Gohr in [8], and later Gohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' [9] explicitly proposed rules for combining probabilities/distinguisher responses (see Corollary 2 in [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' One can use this rule to explicitly convert a distinguisher for one ciphertext pair into one for an arbitrary number of ciphertext pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Algorithm 2 gives the pseudo-code for computing this distinguisher, and the results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Algorithm 2: Convert the DD for one ciphertext pair into one for an m number of ciphertext pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Input: DDT: the R round DDT table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' N: the number of samples for single ciphertext pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' m: the combined number of ciphertext pairs for one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Output: the combined Acc, TPR, TNR with m ciphertext pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1 Y ← {} 2 for i = 1 to N do 3 Y[i ∗ m] ← random{0, 1} 4 for j = 1 to m − 1 do 5 Y[i ∗ m − j] ← Y[i ∗ m] 6 end 7 end 8 Randomly generate N ∗ m samples [x1, x2, · · · , xN∗m] according to Y 9 Z ← {} 10 for i = 1 to N ∗ m do 11 Z[i] ← DDT[xi] 12 end 13 Z ← Z / (Z+2−32) 14 Z ← mean(Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='reshape(N,m), axis=1) 15 predict_Y ← {} 16 for i = 1 to N ∗ m do 17 if Z[i] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5 then 18 predict_Y[i] ← 1 19 end 20 else 21 predict_Y[i] ← 0 22 end 23 end 24 calculate Acc, TPR, TNR based on (Y, predict_Y) 25 return Acc, TPR, TNR /* In our experiments, N takes 220 when m no more than 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' / In addition, r-round ND should be compared with (r − 1)-round DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since the data fed to r-round ND is the value of the ciphertext, one can directly com- pute the differences on (r − 1)-round outputs without knowing the subkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The results are represented in Table 3, which shows that we improved the accuracy of the NDs for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' More importantly, it is able to surpass the accuracy of DDs for 9- and 10-round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Comparison of NDs on Simeck32/64 with 8 ciphertext pairs as a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The input difference of ND/DD is (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' *: The staged training method is used to train ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' R Attack Network Acc TPR TNR Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 9 DD DDT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9433 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 ND SE-ResNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9914 [12] ND Inception 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='9920 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 10 DD DDT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7316 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 ND SE-ResNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7501 [12] ND Inception 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='7525 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 11 DD DDT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5940 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 ND SE-ResNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5936 [12] ND Inception 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5954 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 ND Inception 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5666⋆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5895 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 12 DD DDT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5615 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 ND SE-ResNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5146⋆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5522 [12] ND Inception 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5161⋆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5504 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3 4 Neutral bits and Wrong Key Response Profile In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3, we provided the state-of-the-art NDs for Simeck32/64, which use to perform better key recovery attacks in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In [8], Gohr provides a framework of (1+s+r +1)-round key recovery attack (refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1) consisting of three techniques to increase the success rate and speed up the at- tacks, where s is the length of the CD, and r is the length of the ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Here is a description of these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Neutral Bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In the key recovery attack, multiple samples (formed into a ci- phertext structure) decrypted by the guessed subkey are predicted using the distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Then, the multiple scores are combined according to formula vk = �nb i=1 Zk i/1−Zk i as the final score of that guessed subkey to reduce the misjudgment rate of the ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since the CD suspended in front of the ND are probabilistic, re- sulting in sample entering the distinguisher not satisfying the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Multiple samples generated by neutral bits will have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Also, the lower the accuracy of the distinguisher, the more neutral bits are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Priority of Ciphertext Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Spending the same amount of compu- tation on every ciphertext structure is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Gohr used a generic method (automatic exploitation versus exploration tradeoff based on Upper Confidence Bounds) to focus the key search on the most promising ciphertext structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The priority score of each ciphertext structure is si = ωi max + √nc · � log2(j)/ni where denote by ωi max the highest distinguisher score, ni the number of previous iterations in which the ith ciphertext structure, j the number of the current iteration and √nc the number of ciphertext structures available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Wrong Key Response Profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The key search policy based on Bayesian Op- timization drastically reduces the number of trial decryptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The basic idea of this policy is the wrong key randomization hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This hypothesis does not hold when only one round of trial decryption is performed, especially in a lightweight cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The expected response of the ND upon wrong-key decryp- tion will depend on the bitwise difference between the trial and real keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This wrong-key response profile can be captured in a precomputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Give some trial decryptions, the optimization step then trials to come up with a new set of candidate keys to try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' These new candidate keys are chosen to maximize the probability of the observed distinguisher responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1 Exploring Neutral Bits To be able to attack more rounds with the ND, the CD is generally prepended in front of the ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For the resulting HD used in the key recovery attack, it is not straightforward to aggregate enough samples of the same distribution fed to the ND due to the prepended CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To overcome this problem, Gohr [8] used the neutral bits of the CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The more neutral bits there are for the prepended CD, the more samples of the same distribution could be generated for the ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' However, generally, the longer the CD, the fewer the neutral bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finding enough neutral bits for prepending a long CD over a weak ND becomes a difficult problem for devising a key recovery to cover more rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To solve this problem, Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' exploited various generalized NBs to make weak ND usable again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Particularly, they employed conditional simultaneous neutral bit-sets (CSNBS) and switching bits for adjoining differentials (SBfAD), which are essential for achieving efficient 12-round and practical 13-round attacks for Speck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus, the first part of the key recovery attack focuses on finding various types of neutral bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Given a differential, in order to find the neutral bits, it is generally divided into two steps: firstly, collect enough conforming pairs (correct pairs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' secondly, flip the target bits of the conforming pair, or flip all the bits contained in the target set of bits, and check the probability that the new plain- text pair is still the conforming pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finding SNBSs for 3-round Differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For the prepended 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040) on top of the NDs, one can experimen- tally obtain 14 deterministic NBs and 2 SNBSs (simultaneously complementing up to 4 bits) using an exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Concretely, for the 3-round differential (0x0140, 0x0200) → (0x0000, 0x0040), (simultaneous-) neutral bits and bit-sets are [3], [4], [5], [7], [8], [9], [13], [14], [15], [18], [20], [22], [24], [30], [0, 31], [10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Finding SNBSs for 4-round Differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For the prepended 4-round CD (0x0300, 0x0440) → (0x0000, 0x0040) on top of the NDs, there are 7 com- plete NB/SNBS: [2], [4], [6], [8], [14], [9, 24], [9, 10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Still, the numbers of NBs/SNBSs are not enough for appending a weak neural network distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus, conditional ones were searched using Algorithm 3 in paper [4], and the obtained CSNBSs and their conditions are summarized together in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' CSNBS for 4-round Classical Differential (0x0300, 0x0440) → (0x0000, 0x0040) of Simeck32/64 Bit-set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Bit-set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' x[0, 10] x[2, 12] [21] 00 [23] 00 [21, 5] 10 [23, 12] 10 [21, 10] 01 [23, 7] 01 [21, 10, 5] 11 [23, 12, 7] 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': Condition on x[i, j], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', x[i, j] = 10 means x[i] = 1 and x[j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='2 Wrong Key Response Profile To calculate the r-round wrong key response profile, we generated 3000 random keys and multiple input pairs {(Pi,0, Pi,1), i ∈ [0, m − 1]} for each difference δ ∈ (0, 216) and encrypted for r +1 rounds to obtain ciphertexts {(Ci,0, Ci,1), i ∈ [0, m − 1]}, where Pi,0 ⊕ Pi,1 = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Denoting the final real subkey of each encryption operation by k, we then performed single-round decryption to get E−1 k⊕δ({Ci,0, i ∈ [0, m − 1]}), E−1 k⊕δ({Ci,1, i ∈ [0, m − 1]}) and had the resulting partially decrypted ciphertext pair rated by an r-round ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' µδ and σδ were then calculated as empirical mean and standard deviation over these 3000 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We call the r-round wrong key response profile WKRPr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' From the wrong key Response Profile, we can find some rules to speed up the key recovery attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Analysis of WKRP9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In Figure 3a, when the difference between guessed key and real key δ is greater than 16384, the score of the distinguisher is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This phenomenon indicates that the score of the distinguisher is very low when the 14-th and 15-th bit is guessed incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' When δ ∈ {2048, 4096, 8192, 10240, 12288, 14436}, the score of the distinguisher is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This indicates that when the 11-th, 12-th, and 13-th bits are guessed incorrectly, it has little effect on the score of the distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Analysis of WKRP10 and WKRP11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' It is clear from Figure 3b that when the δ is greater than 32768, the score of the distinguisher is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='45, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', the 15-th bit has a greater impact on the distinguisher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' When δ ∈ {4096, 8192, 12288}, the score of the distinguisher is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' This indicates that when the 12-th and 13-th bits are guessed incorrectly, it has little effect on the score of the distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' It can also be observed from Figure 3c that the 12-th and 13-th bits have less influence on the score of the distinguisher, and the 14-th and 15-th bits have more influence on the score of the distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Analysis of WKRP12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Despite the small difference in scores in Figure 3d, it was found that when only the 12-th and 13-th bits are wrongly guessed, the score of the distinguisher is still higher than the other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' (a) WKRP9 (b) WKRP10 (c) WKRP11 (d) WKRP12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Wrong Key Response Profile for Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='8 Meanresponse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='0 0 4096 Differencetorealkey0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='55 response 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='50 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='35 0 4096 81921228816384204802457628672327683686440960450564915253248573446144065536 Differenceto realkey0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='500 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='480 0 Differenceto realkey0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5005 response 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='5000 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4975 0 4096 81921228816384204802457628672327683686440960450564915253248573446144065536 Differencetoreal keyFrom the four wrong key response profiles, we can conclude that when the 14-th and 15-th bit subkeys are guessed incorrectly, it has a greater impact on the score of the distinguisher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' when the 12-th and 13-th bit subkeys are guessed incorrectly, it has a smaller impact on the score of the distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' According to these phenomena, we can speed up the key recovery attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Guess the 14-th and 15-th bit subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since the difference between the score of the distinguisher of bits 14 and 15 in the case of correct and incorrect guesses is relatively large, we can first determine the values of these two bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Before performing a Bayesian key search, a random set of subkeys is guessed, then the 14-th and 15-th bits of the subkeys are traversed, and the ciphertext is decrypted using the subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus, the values of the 14-th and 15-th bits can be determined based on the score of the distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The Bayesian key search algorithm can easily recover these two bits even if the values of these two bits are not determined in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Ignore the 12-th and 13-th bit subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since the 12-th and 13-th bit subkeys have less influence on the score of the distinguisher, we first set these two bits to 0 when generating the first batch of candidate subkeys and then randomize the values of the two bits after completing the Bayesian key sorting and recommending the new candidate subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Previous researchers have also exploited this feature to accelerate key recovery attacks, and the 14-th and 15-th bit subkeys have little impact on the score of the distin- guisher when guessed incorrectly for Speck32/64 and Simon32/64[4,8,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The Bayesian key search algorithm considering insensitive key bits is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5 Practical Key Recovery Attack When a fast graphics card is used, the performance of the implementation is not limited by the speed of neural network evaluation but by the total number of iterations on the ciphertext structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We count a key guess as successful if the sum of the Hamming weights of the differences between the returned last two subkeys and the real two subkeys are at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The experimental parameters for key recovery attacks are denoted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' ncts: the number of ciphertext structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' nb: the number of ciphertext pairs in each ciphertext structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' nit: the total number of iterations on the ciphertext structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' c1 and c2: the cutoffs with respect to the scores of the recommended last subkey and second to last subkey, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' nbyit1, ncand1 and nbyit2, ncand2: the number of iterations and number of key candidates within each iteration in the BayesianKeySearch Algorithm for guessing each of the last and the second to last subkeys, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Algorithm 3: BayesianKeySearch Algorithm For Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Input: Ciphertext structure C := {C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Cnb−1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' a neural distinguisher ND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' and its wrong key response profile µ and σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' the number of candidates to be generated within each iteration ncand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' the number of iterations nbyit Output: The list L of tuples of recommended keys and their scores 1 S := {k0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' kncand−1} ← choose ncand values at random without replacement from the set of all subkey candidates 2 S = S & 0xCFFF 3 L ← {} 4 for t = 1 to nbyit do 5 for ∀ki ∈ S do 6 for j = 0 to nb − 1 do 7 C ′ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki = F −1 ki (Cj) 8 vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki = ND(C ′ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki) 9 sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki = log2(vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki/(1 − vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki)) 10 end 11 ski = �nb−1 j=0 sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' /* the combined score of ki using neutral bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' / 12 L ← L∥(ki, ski);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 13 mki = �nb−1 j=0 vj,ki/nb 14 end 15 for k ∈ {0, 1, · · · , 216 − 1} & 0xCFFF do 16 λk = �ncand−1 i=0 (mki − µki⊕k)2/σ2 ki⊕k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' /* using wrong key response profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' / 17 end 18 S ← argsortk(λ)[0 : ncand − 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 19 r := {r0, r1, · · · , rncand−1} ← choose ncand values at (0, 4) at random 20 r = r << 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' /* Randomize the 12-th and 13-th bit subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' / 21 S = S ⊕ r 22 end 23 return L 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1 Complexity Calculation Theoretical Data Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The theoretical data complexity of the exper- iment is calculated by the formula nb × nct × m × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In the actual experiment, when the accuracy of the ND is high, the key can be recovered quickly and suc- cessfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Not all the ciphertext structure is used, so the actual data complexity is lower than the theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Experimental Time Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The time complexity calculation formula in our experiments is 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='693 × rt × log1−sr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='01, which is borrowed from [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Our device can perform 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='693 1-round decryption per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' rt is the average running time of multiple experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The success rate sr is the number of suc- cessfully recovered subkeys divided by the number of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We calculate how many experiments need to be performed to ensure at least one successful experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' When the overall success rate is 99%, we consider the experiment to be successful, and the number of experiments ne is: 1−(1−sr)ne = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='99, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', log1−sr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='2 Key Recovery Attack on 15-round Simeck32/64 Experiment 1: The components of key recovery attack ASimeck15R of 15-round Simeck32/64 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' neutral bits of generating multiple ciphertext pairs: [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14], [15], [18], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 10-round neural distinguisher NDSimeck10R and wrong key response profiles NDSimon10R · µ and NDSimeck10R · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 9-round distinguisher NDSimeck9R and wrong key response profiles NDSimon9R· µ and NDSimeck9R · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Concrete parameters used in our 15-round key recovery attack ASimeck15R are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' m = 8 nb = 28 ncts = 210 nit = 211 c1 = 10 c2 = 10 nbyit1 = nbyit2 = 5 ncand1 = ncand2 = 32 The theoretical data complexity is m×nb ×ncts ×2 = 222 plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The ac- tual data complexity is 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In total, 120 trials are running and 119 successful trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus, the success rate sr is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The average running time of the exper- iment rt is 407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='901s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The time complexity is 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='693 × rt × log1−sr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='01 = 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='3 Key Recovery Attack on 16-round Simeck32/64 Experiment 2: The components of key recovery attack ASimeck16R of 16-round Simeck32/64 are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' neutral bits of generating multiple ciphertext pairs: [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14], [15], [18], [20], [22], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 11-round neural distinguisher NDSimeck11R and wrong key response profiles NDSimeck11R · µ and NDSimeck11R · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 10-round neural distinguisher NDSimeck10R and wrong key response profiles NDSimeck10R · µ and NDSimeck10R · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Concrete parameters used in our 16-round key recovery attack ASimeck16R are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' m = 8 nb = 210 ncts = 210 nit = 211 c1 = 10 c2 = 10 nbyit1 = nbyit2 = 5 ncand1 = ncand2 = 32 The theoretical data complexity is m × nb × ncts × 2 = 224 plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The actual data complexity is 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We use 6 processes, each running 20 experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Since the memory limit was exceeded during the experiment, one process was killed, leaving 100 experiments, 100 of which successfully recovered the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus, the success rate sr is 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The average running time of the experiment rt is 2889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='648s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The time complexity is 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='693 × rt = 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='4 Key Recovery Attack on 17-round Simeck32/64 Experiment 3: The components of key recovery attack ASimeck17R of 17-round Simeck32/64 are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' neutral bits of generating multiple ciphertext pairs: [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' neutral bits of combined response of neural distinguisher: [7], [8], [9], [13], [14], [15], [18], [20], [22], [24], [30], [0, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 12-round neural distinguisher NDSimeck12R and wrong key response profiles NDSimeck12R · µ and NDSimeck12R · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 11-round neural distinguisher NDSimeck11R and wrong key response profiles NDSimeck11R · µ and NDSimeck11R · δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Concrete parameters used in our 17-round key recovery attack ASimeck17R are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' m = 8 nb = 212 ncts = 210 nit = 211 c1 = 20 c2 = −120 nbyit1 = nbyit2 = 5 ncand1 = ncand2 = 32 The theoretical data complexity is m × nb × ncts × 2 = 226 plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The actual data complexity is 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In total, trials are 50 running, and there are 15 successful trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Thus, the success rate sr is 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The average running time of the experiment rt is 25774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='822s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The time complexity is 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='693 × rt × log1−sr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='01 = 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' There are two reasons why we do not launch a 17-round key recovery attack using a 4-round CD and an 11-round ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' One is that the probability of the 4-round CD (0x0300, 0x0440) → (0x0000, 0x0040) is about 212 (the prob- ability of the 3-round CD (0x0140, 0x0200) → (0x0000, 0x0040) is about 2−8), resulting in too much data required, and the second is that there are not enough neutral bits in the 4-round CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 6 Conclusion In this paper, we show practical key recovery attacks up to 17 rounds of Simeck 32/64, raising the technical level of practical attacks by two rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' We design neural network that fits with the round function of Simeck to improve the ac- curacy of the neural distinguishers, and is able to outperform the DDT-based distinguisher in some rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' To launch more rounds of the key recovery attack, we make a concerted effort on the classical differential and the neural distin- guisher to make both modules good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In addition, we optimize the key recovery attack process by deeply analyzing the wrong key response profile, thus reducing the complexity of the key recovery attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Aagaard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', AlTawy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Gong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Mandal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Rohit, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': Ace: An authenticated encryption and hash algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Submission to NIST-LWC (announced as round 2 candidate on August 30, 2019) (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' AlTawy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Gong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Jha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Mandal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Nandi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Rohit, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': Spoc: an authenticated cipher submission to the nist lwc competition (2019) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' AlTawy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Gong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Mandal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Rohit, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': Spix: An authenticated cipher submission to the nist lwc competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Submitted to NIST Lightweight Standardization Process (2019) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Bao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Tu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': Enhancing differential-neural cryptanal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In: International Conference on the Theory and Application of Cryptology and Information Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Springer (2022) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Beaulieu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Shors, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Treatman-Clark, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Weeks, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Wingers, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': The simon and speck lightweight block ciphers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In: Proceedings of the 52nd annual design automation conference.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Aagaard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Gong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': The simeck family of lightweight block ciphers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' In: International Workshop on Cryptographic Hardware and Embedded Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 307–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Springer (2015) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=': Improving differential-neural cryptanalysis with inception blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Cryptology ePrint Archive (2022) A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='1 Procedure of (1 + s + r + 1)-round key recovery attack The attack procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Initialize variables Gbestkey ← (None, None), Gbestscore ← −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Generate ncts random plaintext pairs with difference ∆P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Using ncts plaintext pairs and log2 m neutral bit with probability one to generate ncts multiple plaintext pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Every multiple plaintext pairs have m plaintext pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' From the ncts multiple plaintext pairs, generate ncts plaintext structures using nb generalized neutral bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Decrypt one round using zero as the subkey for all multiple plaintext pairs in the structures and obtain ncts plaintext structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Query for the ciphertexts under (1 + s + r + 1)-round Simeck32/64 of the ncts × nb × 2 plaintext structures, thus obtain ncts ciphertext structures, denoted by {C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' , Cncts}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Initialize an array ωmax and an array nvisit to record the highest distinguisher score obtained so far and the number of visits have received in the last subkey search for the ciphertext structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Initialize variables bestscore ← −∞, bestkey ← (None, None), bestpos ← None to record the best score, the corresponding best recommended values for the two subkeys obtained among all ciphertext structures and the index of this ciphertext structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' For j from 1 to nit: (a) Compute the priority of each of the ciphertext structures as follows: si = ωmaxi + α · � log2 j/nvisiti, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' , ncts}, and α = √ncts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' The formula of priority is designed according to a general method in reinforcement learning for achieving automatic exploitation versus ex- ploration trade-off based on Upper Confidence Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' It is motivated to focus the key search on the most promising ciphertext structures [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' (b) Pick the ciphertext structure with the highest priority score for further processing in this j-th iteration, denote it by C, and its index by idx, nvisitidx ← nvisitidx + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' (c) Run BayesianKeySearch Algorithm [8] with C, the r-round neural distinguisher NDr and its wrong key response profile NDr ·µ and NDr · σ, ncand1, and nbyit1 as input parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' obtain the output, that is a list L1 of nbyit1 × ncand1 candidate values for the last subkey and their scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', L1 = {(g1i, v1i) : i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' , nbyit1 × ncand1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' (d) Find the maximum v1max among v1i in L1, if v1max > ωmaxidx, ωmaxidx ← v1max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' (e) For each of recommended last subkey g1i ∈ L1, if the score v1i > c1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Decrypt the ciphertext in C using the g1i by one round and obtain the ciphertext structures C′ of (1 + s + r)-round Simeck32/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Run BayesianKeySearch Algorithm [8] with C′ , the neural dis- tinguisher NDr−1 and its wrong key response profile NDr−1 · µ and NDr−1·σ, ncand2, and nbyit2 as input parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' obtain the output, that is a list L2 of nbyit2×ncand2 candidate values for the last subkey and their scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=', L2 = {(g2i, v2i) : i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' , nbyit2 × ncand2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Find the maximum v2i and the corresponding g2i in L2, and denote them by v2max and g2max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' If v2max > bestscore, update bestscore ← v2max, bestkey ← (g1i, g2max), bestpos ← idx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' (f) If bestscore > c2, go to Step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Make a final improvement using VerifierSearch [8] on the value of bestkey by examining whether the scores of a set of keys obtained by changing at most 2 bits on top of the incrementally updated bestkey could be improved recursively until no improvement obtained, update bestscore to the best score in the final improvement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' If bestscore > Gbestscore, update Gbestscore ← bestscore, Gbestkey ← bestkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} +page_content=' Return Gbestkey, Gbestscore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFJT4oBgHgl3EQfpywl/content/2301.11601v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 94847b96802de9fe770d9ba68ddff5a468d5e464..decc63323ce59d8e8282be54b074a2ae8688859c 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1793,3 +1793,78 @@ atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf 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a Isothermally Heated Rotating +Circular Cylinder +Amarjit Hatya, Rajendra K. Ray*a +aSchool of Mathematical and Statistical Sciences, Indian Institute of Technology +Mandi, Mandi, 175005, Himachal Pradesh, India +Abstract +The main objective of this paper is to study the flow characteristics of a rotat- +ing, isothermally heated circular cylinder with a vertical arc-shaped control plate +placed downstream. Stream function-Vorticity (ψ − ω) formulation of two di- +mensional (2-D) Navier-Stokes (N-S) equations is considered as the governing +equation and the simulations are performed for different distances of the control +plate (0.5, 1, 2, 3), rotational rates (0.5, 1, 2.07, 3.25) at Prandtl number 0.7 and +Reynolds number 150. The governing equations are discretized using the Higher +Order Compact (HOC) scheme and the system of algebraic equations, arising +from HOC discretization, is solved using the Bi-Conjugate Gradient Stabilized +approach. Present computed results show that the vortex shedding plane is shifted +upward from the centerline of the flow domain by the cylinder’s rotational mo- +tion. The structure of the wake varies based on the plate’s position. The size of +vortices is greatly reduced when the control plate is set at d/R0 = 3 and the rota- +tional rate is very high. At greater rotational rates, the impact of varied positions +of the arc-shaped control plate is very significant. The rotation of the cylinder and +the location of the plate can be used to lower or enhance the values of drag and +lift coefficients as well as the heat transfer from the surface of the cylinder. The +maximum value of the drag coefficient, which is about 3, is achieved for d/R0 = 2 +and α = 3.25. +Keywords: Navier-Stokes equations, Circular cylinder, Arc-shaped control plate, +Heat transfer, HOC +*Corresponding author: rajendra@iitmandi.ac.in + +1. Introduction +Active control of flow past a rotating circular cylinder has always been an in- +teresting topic in fluid dynamics. The wake behaviour for flow past a rotating +cylinder is more complicated than for flow past a stationary cylinder because the +rotation of the cylinder separates the shear layer and modifies the boundary layer. +In 1928, Bickley [1] was among the first to attempt analytical study of the viscous +flow over a rotating cylinder. He considered the potential flow created by a vortex +in the vicinity of a cylinder. The wake structure in flow past a cylinder is compli- +cated due to interactions between a boundary layer, a separating free shear layer, +and a wake. It has huge significance in engineering as the alternating shedding +pattern of the vortices in the wake causes considerable fluctuating pressure forces +in a direction transverse to the fluid flow, which can produce structural vibrations, +acoustic noise, or resonance, and in certain situations, structural collapse. In 1966, +Gerrard [2] experimentally studied the flow past bluff bodies along with flow past +circular cylinder with splitter plates for high Reynolds numbers. He found that +the shear layer was drawn by the vortex formation from the opposite side of the +wake across the center line of the wake, cutting off the vorticity supply to the ex- +panding vortex. He found that the width of the gap between the cylinder and a +splitter plate parallel to the flow, is the only relevant parameter than the position +of the trailing edge of the plate. He studied the effect of a plate normal to the +flow and found that the length of the effective vortex formation area equalled the +distance of the plate from the domain boundary. He observed a substantial cross- +flow velocity created near the plate when a vortex grew close behind it, facilitating +the shedding process and increasing the frequency. Pralits et al. [3] numerically +studied the flow past rotary cylinder and found that the increased rotational speed +caused two distinct instability in the flow. Kang et al. [4] found that the vortex +shedding was stopped completely when the cylinder rotation rate was set at twice +the velocity of free stream fluid. Diaz et al. [5] have experimentally studied the +flow past rotary cylinder for Reynolds number 9000. They saw a decrease in pe- +riodic vortex activity and a rise in random modulation of the shedding process, +which he attributed to the relocation of the stagnation point and the thickening of +the spinning fluid layer near the cylinder surface. They discovered that when the +rotating speed equals the free-stream speed, a regular periodic vortex shedding +occurs, and that the periodic vortex shedding is suppressed at large velocity ra- +tios. For velocity ratios equal to or greater than 1.5, they concluded that rotation +considerably alters the traditional Karman vortex shedding. Similar findings were +produced by Massons et al. [6] for flow past rotating cylinder. Stojkovic et al. [7] +2 + +studied the flow at greater rotation rates and discovered a second shedding mode +in a limited interval [4.85, 5.15] of rotation rate where the shedding frequency was +substantially lower than that of the traditional Von-Karman vortex shedding. At a +high Reynolds number (Re = 105), Roshko [8] investigated the impact of a splitter +plate positioned downstream of a bluff body and parallel to the free stream. By +bringing the plate closer to the cylinder, he observed that the shedding frequency +and base suction were reduced. Bearman [9] found that the separating shear flow +on the top of the surface is pushed to rejoin if the circular cylinder with an end +plate downstream is spun at a constant pace. As a result, the effects and vibrations +caused by boundary-layer development are diminished, and the vortex formation +is suppressed. Apelt et al. [10] used a horizontal splitter plate with varied lengths +to diameter ratios less than 2 to investigate the flow past a circular cylinder for +104 < Re < 5 × 104. The splitter plate considerably reduces drag by stabilising +separation points, lowers the Strouhal number, and increases base pressure by +roughly 50%, according to their research. They also discovered that when using +a splitter plate instead of a cylinder without one, the wake pattern narrows. Kwon +and Choi [11] indicated that there is a critical length of splitter plate that causes +vortex shedding to totally disappear, and that this critical length is proportional +to the Reynolds number. They also discovered that the Strouhal number rises as +the plate’s length increases until it equals the cylinder’s diameter. Bao and Tao +[12] analyzed the flow past a circular cylinder with twin parallel plates attached +and discovered that optimal positioning can outperform the standard splitter plate. +More studies with control plate can be found in [13–16]. +Along with studying the wake structure and pressure forces, force convective +heat transfer from rotating cylinders has been widely investigated by many re- +searchers for its many real-life applications and scientific interests. Drying cylin- +drical items [17]; cylindrical cooling devices in the plastics and glass industries; +drying and coating of papers using a hot spinning cylinder; chemical and food pro- +cessing industries; textile and paper manufacturing, and so on are some examples +of real-world uses. In an experiment, Anderson and Saunders [18] explored heat +convection in a confined room filled with air using an isothermally heated rotating +circular cylinder. Temperatures were elevated to 140 degrees Fahrenheit above the +ambient temperature while air pressure was maintained at 4 atm. The experiment +used three distinct cylinders, each with varying diameters (1, 1.8, and 3.9 inches) +but the same length (2 feet). They determined that heat exchange is nearly steady +when rotational speed is between 0 to a crucial value of 0.9, and past that point, +heat exchange increases in proportion to the rotational speed’s 2/3 power. Badr +3 + +and Dennis [19] conducted a numerical research on force convective heat transfer +from an unconfined rotary cylinder, concluding that increasing rotational speed re- +duces overall rate of heat transfer because the cylinder is isolated from the stream +by the spinning fluid layer. Mohanty et al. [20] performed experimental study on +heat transfer from rotating cylinder for high Reynolds numbers. They discovered +that rotational motion increases average heat transmission by roughly 30% when +compared to a fixed cylinder with a fixed Reynolds number. They also discovered +that as compared to stationary cylinders, rotational motion caused a lower heat +transfer rate at the front stagnation point. An analytical study was attempted by +Kendoush [21] and a formula, Nu = 0.6366(RePr)1/2, was proposed to compute +the local Nusselt number (Nu) for low Prandtl numbers (Pr), where Re denotes +the Reynolds number. With the help of the finite volume technique, Paramane and +Sharma [22] studied the heat transfer and fluid flow across a rotating cylinder for +Prandtl number of 0.7, low Reynolds numbers ranging from 20 to 160, and rotary +speeds of 0 ≤ α ≤ 6. They discovered that when rotary speeds rise, the average +Nusselt number falls while the Reynolds number rises. It was concluded that the +rotation could be employed to reduce drag and suppress heat transmission from +the cylinder. Sufyan et al. [23] discovered that low and medium rotary speeds +immediately reduce heat transmission, but that at higher rotational rates, the in- +creased size of the enclosing vortex causes even more heat transfer reduction. A +few more studies on this subject can be found on [24–27]. +After an extensive literature survey, it is found that many researchers worked +on heat transfer and flow across a rotating circular cylinder. There are numer- +ous works on the flow across a circular cylinder with splitter plates and attached +fins. Effect of curved fins and plates are studied by few researchers for missiles +[28] and formula−1 cars [29] and these are being used in real life. There are +some researchers who tried to study the wake structure and base pressure after +applying the rotation to the cylinder with attached splitter plates or fins, but the +effect of both rotation and the presence of control plates on the process of heat +transfer is not tested. Considering the importance, the current investigation is cen- +tred on the impact of a control plate on forced convective heat transfer and flow +across a rotating circular cylinder. It can be useful in electronic equipment cool- +ing and processing industries. We have taken into account an arc-shaped plate +with a vertical orientation since we are considering a polar coordinate system +with non-uniform grids. For this investigation, the Reynolds number is fixed at +150 and the Prandtl number is fixed at 0.7. The plate distance to cylinder radius +ratio varies between 0.5 and 3, while the rotational rates range from 0.5 to 3.25. +4 + +The two-dimensional unsteady Navier-Stokes equations and energy equation are +first non-dimensionalized and then discretized by using a Higher Order Compact +(HOC) scheme [30, 31] based on non-uniform polar grids. Temporal accuracy of +2nd order and spatial accuracy of atleast 3rd order are obtained through the ap- +plication of the finite difference scheme. To obtain a solution from a discretized +system, the Bi-conjugate Gradient Stabilized method approach is employed. +The paper is arranged as follows: in Section 2, we discuss the governing equa- +tions and initial and boundary conditions related to the current problem; in Section +3, the numerical scheme is described as well as the independence tests and valid- +ity of the numerical scheme are produced; results are discussed in Section 4; and +finally, we conclude our remarks in Section 5. +2. The governing equations and the problem +The considered system is represented in Fig. 1 as a two-dimensional unsteady, +incompressible, laminar, and viscous flow of a Newtonian fluid over an isother- +mally heated circular cylinder of radius R0. At ˆt = 0, the cylinder acquires the +surface temperature Ts impulsively. The following formulas are used to transform +dimensional parameters to dimensionless form: t = ˆtU∞ +R0 , r = +ˆr +R0, u = +ˆu +U∞, v = +ˆv +U∞, +ψ = ˆψU∞ +R0 , ω = ˆωR0 +U∞ , φ = (T−T∞) +(Ts−T∞). The control plate has unit arc length and a con- +stant thickness roughly equal to 0.18 times the cylinder radius and is situated at +a distance d from the cylinder surface. On the surface of the control plate, im- +permeability and no-slip boundary conditions are considered. The control plate is +kept constant at the same temperature as the free stream fluid. +The nondimensional stream-function-vorticity formulation of the 2-D Navier- +Stokes equations and energy equation in polar coordinates (r,θ) are given as, +∂ 2ω +∂r2 + 1 +r +∂ω +∂r + 1 +r2 +∂ 2ω +∂θ2 = Re +2 +� +u∂ω +∂r + v +r +∂ω +∂θ + ∂ω +∂t +� +(1) +∂ 2ψ +∂r2 + 1 +r +∂ψ +∂r + 1 +r2 +∂ 2ψ +∂θ2 = −ω +(2) +∂ 2φ +∂r2 + 1 +r +∂φ +∂r + 1 +r2 +∂ 2φ +∂θ2 = RePr +2 +� +u∂φ +∂r + v +r +∂φ +∂θ + ∂φ +∂t +� +(3) +5 + +Nomenclature +Re +Reynolds number (= 2R0U∞/ν) +Pr +Prandtl number (= ν/β) +R0 +Radius of the circular cylinder +R∞ +Radius of the far field boundary +d +Dimensional distance of the control plate +from the cylinder surface +U∞ +The free-stream fluid’s velocity +T∞ +The free-stream fluid’s temperature +ˆt, t +Time in dimensional and nondimensional form +Ts +Surface temperature of the cylinder in dimensional form +ˆα, α +Rotational velocity in dimensional and +nondimensional form (α = ˆαR0/U∞) +d +Distance of control plate from the surface of the cylinder +Nu, Nu, Nut +Nusselt number (local, average, and time-averaged total) +h, havg +Coefficients of heat transfer (local and average) +ν +The fluid’s kinematic viscosity +K +The fluid’s thermal conductivity +β +The fluid’s thermal diffusivity +Q′′ +Radial heat flux on the surface (Local) +ˆψ, ψ +Stream function in dimensional and nondimensional form +ˆω, ω +Vorticity in dimensional and nondimensional form +T, φ +Temperature in dimensional and nondimensional form +ˆu, u +Radial velocity in dimensional and nondimensional form +ˆv, v +Tangential velocity in dimensional and nondimensional form +ˆr, r +Radius in dimensional and nondimensional form +6 + +Figure 1: The schematic illustration of the current problem. +The velocities, v and u can be expressed as +v = −∂ψ +∂r +and +u = 1 +r +∂ψ +∂θ +(4) +ω can be written as +ω = 1 +r +� ∂ +∂r(vr)− ∂u +∂θ +� +(5) +The boundary conditions on the cylinder’s surface include impermeability, no- +slip, and constant temperature, i.e. +ψ = 0, +∂ψ +∂r = −α +and +φ = 1.0 +when +r = 1 +(6) +The condition of surface vorticity is provided by +ω = −∂ 2ψ +∂r2 +when +r = 1 +(7) +7 + +Uoo +Too +()0 +u.(r,0) = U +cos6 +d +u(r,0,t) =0 +0=0 +v(r,0,t) = 0 +ControlPlate +Ts +u +Uo +V(r,0) +sing +r2 +Too +V +Uoo +Roo +Rco +XIn the distant field, R∞, the vorticity’s resulting decay and the free-stream con- +dition are taken to constitute the boundary conditions, i.e. +ψ → +� +r − 1 +r +� +sin θ, +∂ψ +∂r → +� +1+ 1 +r2 +� +sin θ +and φ → 0 as r → R∞ +R0 +(8) +ω → 0 as r → R∞ +R0 +(9) +The criteria Eqs. (6) to (9) must be followed by all the parameters with 0 ≤ +θ ≤ 2π for all θ. In addition, all of the parameters are functions of θ with a period +of 2π. The initial conditions for the stream function are given by Eqs. (8) and (9). +The vorticity in the distant field is initially assumed to be zero. Eqs. (4) and (8) +provide the initial requirements for the velocities as follows: +u = +� +1− 1 +r2 +� +cos θ +and v = − +� +1+ 1 +r2 +� +sin θ +(10) +3. Numerical Scheme +Using a temporally second order accurate and spatially atleast third order accu- +rate higher order compact (HOC) finite difference technique [32–35], the govern- +ing equations of motion and the energy equation are discretized on non-uniform +polar grids in the circular region ([R0,R∞] × [0,2π]) with grid points (ri,θj). +The non-uniform grid concentrated around the cylinder is generated using the +stretching function ri = exp +�λπi +imax +� +, 0 ≤ i ≤ imax. The function θj is given by, +θj = 2π j +jmax +, 0 ≤ j ≤ jmax. The discretized equations can be written as [30, 36, 37]: +[X1i jδ 2 +r +X2i jδ 2 +θ +X3i jδr +X4i jδrδθ +X5i jδrδ 2 +θ ++X6i jδ 2 +r δθ +X7i jδ 2 +r δ 2 +θ ]ψi j = Gi j +(11) +[Y11i jδ 2 +r +Y12i jδ 2 +θ +Y13i jδr +Y14i jδθ +Y15i jδrδθ ++Y16i jδrδ 2 +θ +Y17i jδ 2 +r δθ +Y18i jδ 2 +r δ 2 +θ ]ωn+1 +i j += [Y21i jδ 2 +r +Y22i jδ 2 +θ +Y23i jδr +Y24i jδθ +Y25i jδrδθ ++Y26i jδrδ 2 +θ +Y27i jδ 2 +r δθ +Y28i jδ 2 +r δ 2 +θ ]ωn +i j +(12) +8 + +and +[Z11i jδ 2 +r +Z12i jδ 2 +θ +Z13i jδr +Z14i jδθ +Z15i jδrδθ ++Z16i jδrδ 2 +θ +Z17i jδ 2 +r δθ +Z18i jδ 2 +r δ 2 +θ ]φn+1 +i j += [Z21i jδ 2 +r +Z22i jδ 2 +θ +Z23i jδr +Z24i jδθ +Z25i jδrδθ ++Z26i jδrδ 2 +θ +Z27i jδ 2 +r δθ +Z28i jδ 2 +r δ 2 +θ ]φn +i j +(13) +The coefficients X1i j, X2i j,..., X7i j; Gi j; Y11i j, Y12i j,..., Y18i j; Y21i j, Y22i j,..., +Y28i j; Z11i j, Z12i j,..., Z18i j and Z21i j, Z22i j,..., Z28i j are the functions of the +parameters r and θ. [30, 36, 37] provide the expressions for the non-uniform +central difference operators δθ, δ 2 +θ , δr and δ 2 +r , as well as the notations θf , θb, r f, rb +and the coefficients. The Bi-conjugate Gradient Stabilized approach is employed +in order to solve the discretized problem. +3.1. Drag and lift coefficients +The forces acting on a circular cylinder submerged in fluids for uniform flow +are generally caused by surface friction and surface pressure distribution. The +expressions for drag (CD) and lift (CL) coefficients are adopted from [30, 36]. The +expressions are as follows, +CD = 1 +Re +� 2π +0 +��∂ω +∂r +� +R0 +−ωR0 +� +cosθdθ +(14) +CL = 1 +Re +� 2π +0 +��∂ω +∂r +� +R0 +−ωR0 +� +sinθdθ +(15) +The integral values are calculated using Simpson’s 1/3 method. The time- +averaged drag, CD is expressed as, +CD = +1 +t1 −t2 +� t2 +t1 +CDdt +(16) +When the flow achieves a periodic mode and executes numerous cycles, the time +span between t1 and t2 is selected. +9 + +3.2. The heat transfer parameters +Initially, heat conduction happens from the cylinder surface to the adjacent +fluid, and subsequently it convects away with the flow. The heat conduction path +follows the radius of the cylinder surface. The dimensionless local heat flux in the +radial direction is the local Nusselt number, Nu, defined by, +Nu = 2hR0 +k += Q′′(2R0) +k(Ts −T∞) +(17) +where h represents the local heat transfer coefficient, k represents the thermal +conductivity of the fluid, and Q′′ represents the surface local radial heat flux. Q′′ +is expressed as, Q′′ = −k ∂T +∂r |r=R0. The average Nusselt number, denoted by Nu, +used to represent the dimensionless heat transfer from the cylinder’s surface, is +expressed as +Nu = 2havgR0 +k += 1 +2π +� 2π +0 +Nudθ +(18) +The average heat transfer coefficient (havg) is expressed as havg = +1 +2π +� 2π +0 hdθ. +Nut, the time-averaged total Nusselt number is given as, +Nut = +1 +t1 −t2 +� t2 +t1 +Nudt +(19) +When the flow achieves a periodic mode and executes numerous cycles, the time +span between t1 and t2 is selected. +3.3. Validation +The computational domain is discretized using non-uniform grids. The grid +independence test is performed in Fig. 2(a) with three different grid sizes (181 × +181), (191 ×202) and (351 ×341), with a set time step ∆t = 0.01, a fixed 25 : 1 +domain-to-cylinder-radius ratio, Pr = 0.7, Re = 150, α = 1 and d/R0 = 1. All the +grid sizes seem to produce almost same results. The grid size (191×202) is cho- +sen for future computations. For grid size (181 × 181) and time step ∆t = 0.01, +the domain independence test is performed in Fig. 2(b) for three distinct radii, +15, 25 and 35 of the outer boundary, other parameter values, on the other hand, +are treated the same as the grid independence test. This test demonstrates that +a domain radius of 25 is adequate to provide the best possible results. Finally, +with a set grid size (181 ×181) and the far field border defined at 25 : 1 domain- +to-cylinder-radius ratio, the time independence test is conducted in Fig. 2(c) for +10 + +(a) +(b) +(c) +Figure 2: Variation of local Nusselt number distribution, Nu (a) grid independence test with grid +sizes 181 × 181, 191 × 202, 351 × 341, (b) space independence test with outer boundary radius +15, 25, 35 and (c) time independence test with time steps 0.001, 0.005, 0.01 at instant t = 10 for +Re = 150, Pr = 0.7, α = 1 and d = 1. +time increments ∆t = 0.001, 0.005, 0.01, 0.02. For later computations, we used +R∞ +R0 = 25 and ∆t = 0.01, as suggested by these test findings. +There hasn’t been any research towards controlling heat and flow transfer from +a rotating cylinder using an arc-shaped vertical control plate placed across a free +stream of uniform flow. To prove the correctness of our code and model, we be- +gin by comparing our findings to those of previous studies of heat transfer from +11 + +15 +At=0.001 +At=0.005 +At=0.01 +12 +9 +nN +6 +0 +60 +120 +180 +240 +300 +360 +015 +15 +25 +35 +10 +nN +60 +120 +180 +240 +300 +360 +015 +181 X 181 +191 X 202 +8 +351 X 341 +10 +Nu +5 +11 +1 +0 +60 +120 +180 +240 +300 +360 +0Table 1: Comparison of the current computation with the equivalent time-averaged total Nusselt +number computed by Paramane & Sharma [22] for Re = 40, 100, Pr = 0.7, α = 1, and isother- +mally heated cylinder. +Re +40 +100 +Nut (Current) +3.276112 +4.936597 +Nut (Paramane & Sharma) +3.213 +4.991 +Di f ference(%) +1.964% +1.09% +Table 2: Comparison between time-averaged drag results from the current study to Kwon and +Choi’s work [11] for Re = 160. +Length of splitter plate +1 +2 +Time-averaged Drag (Present Study) +1.133021 +1.056131 +Time-averaged Drag (Kwon and Choi) +1.10162 +1.08812 +Di f ference(%) +2.85% +2.94% +rotating cylinders [22], then it is compared with the results of flow past circular +cylinder with a splitter plate attached [11]. When the flow becomes periodic, the +mean drag coefficients are used to determine the time-averaged drag coefficient +on the cylinder surface in Table 2. According to Table 1, the maximum difference +of time-averaged total Nusselt number is 1.964%, which is within a reasonable +range. Also, Table 2 shows the maximum difference of time-averaged Drag coef- +ficients from the results of the current study and the previously published works is +2.94% which is also within a considerable range. As a result, the current findings +are consistent with earlier studies. +4. Results and Discussions +Reynolds number (Re), Prandtl number (Pr), angular velocity of the cylinder +(α), and control plate distance (d/R0) are all well-known factors that influence +flow and heat fields. Fig. 3 exhibits the Drag (CD) and lift (CL) coefficients, as +well as the variation of local Nusselt number (Nu) for d/R0 = 0 and 0.5 with +fixed α = 0.5, Re = 150, Pr = 0.7. The parameter value d/R0 = 0 corresponds +to the case without the arc-shaped control plate. Fig. 3(a) clearly demonstrates +that, the peak value of CD is significantly reduced as well as the amplitude of +CL with the introduction of control plate at a distance d/R0 = 0.5 downstream. +d/R0 = 0, indicating flow across the cylinder in the absence of the control plate. +By comparing Fig. 3(b) and Fig. 3(c), it is found that the introduction of the con- +12 + +(a) +(b) +(c) +Figure 3: (a) Drag (CD) and lift (CL) coefficients, (b) variation of local Nusselt number (Nu) for +d/R0 = 0 and (c) variation of local Nusselt number (Nu) for d/R0 = 0.5 with fixed α = 0.5. +trol plate slightly reduced the peak value of Nu approximately from 11.35 to 11.27 +at θ ≈ 192◦, but the local maximum peak is significantly increased at θ ≈ 30◦. +It means, although the heat transfer near the front stagnation is slightly decreased +by the control plate, the heat transfer is significantly increased near the rear stag- +nation point, which eventually increases the overall heat transfer from the upper +half of the cylinder surface. The control plate alters the vortex shedding process, +which in turn affects the thermal boundary layer and causes this effect. Realizing +the importance of the arc-shaped control plate, the current studies are performed +for Re = 150, 0.5 ≤ α ≤ 3.25 and 0.5 ≤ d/R0 ≤ 3, while Pr is maintained at 0.7. +The values of α are typically chosen in accordance with [31]. +For α = 0.5 and d/R0 = 1, Fig. 4 exhibits the isotherm, streamline and vor- +ticity at periodic phases. Two vortices are shed periodically from the upper and +lower sides of the cylinder, according to the vorticity and streamline. The upper +vortex is slightly larger than the lower vortex. The continuous and dashed lines +indicate the positive and negative contours, respectively. The vortex shedding +plane is shifted by approximately θ = 20◦ from the centerline or the x-axis due +to the rotation of the cylinder. The shear layer around the plate changes the nega- +13 + +16.5 +11.27 +t = t+(O)T +15 +t = t +(1/4)T +13.5 +1.265 +t = t,+(1/2)T +t = t,+(3/4)T +12 +11.26 +190 +192 +19. +t = t,+(1)T +10.5 +7.5 +6 +4.5 +3 +1.5 +60 +120 +240 +300 +36016.5 +t = t+(O)T +15 +11.35 +t = t +(1/4)T +13.5 +t = t,+(1/2)T +t = t,+(3/4)T +12 +11.3 +190 +195 +t = t,+(1)T +10.5 +7.5 +6 +4.5 +3 +1.5 +60 +120 +180 +240 +300 +360 +03 +Cp, d/R, = 0 +Cp, d/R, = 0.5 +CL, d/R. +=0 +d/R. += 0.5 +0 +50 +100 +150 +tt = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 4: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 0.5 +and d/R0 = 1 at different phases. +tive equi-vorticity lines that come from the surface of the cylinder, but the positive +equi-vorticity lines from the cylinder merge with the shear layer due to the rotation +of the cylinder. No recirculation zone or vortex is observed between the cylinder +and the plate. Two large vortices as lumps of hot fluid shed periodically from the +upper and bottom sides of the cylinder according to the isotherm contours. The +14 + +** +! +.......:2 +:i.. +: +2t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 5: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 1 and +d/R0 = 1 at different phases. +isotherm density is high near the front stagnation point, which indicates the higher +heat transfer rate in this region. Isotherm, streakline and vorticity are displayed +in Fig. 5 for α = 1 and d/R0 = 1. Streakline and vorticity indicate that two vor- +tices are periodically shed from the upper side and lower side of the cylinder. The +increase in rotational rate, increases the movement of the fluid around the control +15 + +: +.....t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 6: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 2.07 +and d/R0 = 1 at different phases. +plate which leads to thickening of shear layer around the control plate. It affects +the vorticity contour coming from the cylinder. Positive equi-vorticity lines from +the cylinder and the plate get merged together to shed a sleek, elongated vortex. +The positive equi-vorticity lines from the cylinder completely cover the control +plate, also dragging the negative equi-vorticity lines towards the bottom of the +16 + +t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 7: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 3.25 +and d/R0 = 1 at different phases. +cylinder. This increases the density in thermal boundary layein the upper half of +the cylinder, increasing the heat transfer. The upper vortex is much wider as com- +pared to the sleek bottom vortex. Because of the increased α, the vortex shedding +plane is shifted by approximately θ = 23◦ from the centerline. The isotherm con- +tours suggest that two warm blobs convect away periodically from the upper and +17 + +lower sides of the cylinder. There is no vortex or recirculation zone found be- +tween the cylinder and the plate. Isotherm, streakline and vorticity for α = 2.07 +and d/R0 = 1 are shown in Fig. 6. The streakline and vorticity suggest that two +vortices are periodically shed in the flow domain. One vortex is shed from the top +of the cylinder and the other one is shed from the back of the plate. The lower +vortex pushes the upper vortex due to the high rotation rate of the cylinder. As a +result, the upper vortex is shed much earlier than at lower rotational rates. Also, +the upper vortex becomes sleek and the bottom vortex becomes wide. Negative +equi-vorticity lines coming from the cylinder completely cover the control plate +as well as the positive vortex. Due to the high movement of fluid around the con- +trol plate, the shear layers get thickened and drag the negative equi-vorticity lines +from the cylinder to the bottom of the plate. This affects the thermal boundary +layer of the cylinder by thinning around the rear stagnation point. As a result, the +heat transfer is increased in this region. However, the high rotation of the cylin- +der thickens the thermal boundary layer around the front stagnation point, leading +to a decrease in heat transfer rate. Here, The vortex shedding plane is shifted +by approximately θ = 37◦ from the centerline. The isotherm contours suggest +that two warm blobs convect away periodically by the vortices generated in the +flow domain. Fig. 7 shows the isotherm, streakline and vorticity for α = 3.25 +and d/R0 = 1. Due to very high rotational rate, the negative equi-vorticity lines +completely cover the cylinder as well as the positive equi-vorticity lines originated +from the control plate. Two vortices are shed periodically, one from the top of the +cylinder and another from the back of the control plate. Due to the high rotational +speed, the bottom vortex pushes the upper vortex. As a result, the upper vortex is +shed much earlier. Also, the bottom vortex is much larger than the upper vortex. +One small negative vortex is formed behind the control plate, but it gets dissolved +into the positive vortex. After the negative vortex is shed, the shear layer from +the cylinder splits on top and bottom of the shear layer from the control plate. It +gradually merges and creates an elongated negative vortex. Between the cylinder +and the plate, no vortex or recirculation zone forms. The moving fluid around the +cylinder drags the shear layer from the control plate towards the top of the cylin- +der, which leads to the increased density of the isotherm contour. As a result, heat +transfer is boosted in this region. The vortex shedding plane is displaced from the +centerline by approximately θ = 50◦ at this rotational rate. The isotherm contours +suggest that the density of the isotherm around the cylinder becomes less than at +the lower rotational rates, which means that the high rotation rate is suppressing +the heat transfer rate from the cylinder surface. Additionally, two warm blobs pe- +riodically convect away from the cylinder’s upper side and the plate’s rear. The +18 + +t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 8: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 0.5 +and d/R0 = 2 at different phases. +top blob is sleek and the bottom one is wide, similar to the vortices. Figs. 4 to 7 +show that increasing rotational rates increased the size of vortices as well as the +angle of vortex shedding plane from the centerline for a fixed d/R0 = 1. +Fig. 8 shows the isotherm, streakline and vorticity for α = 0.5 and d/R0 = 2. +19 + +:: +:t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 9: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 3.25 +and d/R0 = 2 at different phases. +Two vortices are shed periodically from the upper and lower sides of the cylinder, +according to the streakline and vorticity. One recirculation zone is formed by the +interaction of the shear layers between the cylinder and the plate, near the top of +the plate. Positive equi-vorticity lines originated from the cylinder partially covers +the control plate. The isotherm contours show that two warm blobs convect away +20 + +with the shedding vortices. The vortex shedding plane is slightly higher than the +centerline by approximately θ = 15◦. This angle of the vortex shedding plane +is slightly lower than that of Fig. 4 due to the increase in d/R0. This happens +due to the interaction of shear layers around the control plate. Fig. 9 exhibits the +isotherm, streakline and vorticity for α = 3.25 and d/R0 = 2. Here, two vortices +are periodically shed. One is shed from the top of the cylinder, and the other one +is shed from behind the plate. One temporary recirculation zone is formed be- +tween the cylinder and the plate, which gradually merges with the upper vortex. +Due to the high rotational rate, the bottom vortex is pulled upwards and pushes +the upper vortex. As a result, the upper vortex is shed much earlier. The negative +equi-vorticity lines cover the positive equi-vorticity lines that originated from the +control plate. After the negative vortex is shed, the shear layer is split into two +by the positive vorticity contour. The shear layers from the top and bottom of the +control plate are squeezed together by the negative vorticity contour to form the +positive vortex. The vortex shedding plane is shifted by approximately θ = 40◦ +from the centerline, and this angle is also slightly lower than that of Fig. 7. The +widths of vortices are much larger than those at Fig. 7. The interaction between +the shear layer and the boundary layer of the cylinder thickens the thermal bound- +ary layer near the front stagnation point and increases the density of the isotherm +contour near the rear stagnation point and at the bottom of the cylinder. It leads +to the reduction of heat transfer near the front stagnation point and an increase in +heat transfer rate near the rear stagnation point and bottom of the cylinder. Figs. 8 +and 9 show that as α increases from 0.5 to 3.25 for d/R0 = 2, the vortices increase +in size. +Isotherm, streakline and vorticity are displayed in Fig. 10 for α = 0.5 and +d/R0 = 3. Two vortices shed periodically from the upper and lower sides of the +cylinder. The bottom vortex is slightly sleeker than the upper one. The positive +equi-vorticity lines coming from the cylinder, partially cover the control plate, +and the interaction between the shear layers sheds the positive vortex. One re- +circulation zone is formed between the cylinder and the plate, which gradually +merges with the upper vortex. The density of the isotherm contour is higher near +the front stagnation point, which means the rate of heat transfer is much higher in +this region. Also, two warm blobs convect away periodically from the upper and +lower sides of the cylinder. Also, the vortex shedding plane is at an angle of ap- +proximately θ = 8.5◦ with the centerline, which is much lower than the previous +placements of the control plate. It happens as the bottom shear layers are resisted +by the control plate to freely move upwards. Fig. 11 exhibits the isotherm, streak- +21 + +t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 10: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 0.5 +and d/R0 = 3 at different phases. +line and vorticity are displayed for α = 3.25 and d/R0 = 3. Two vortices shed +periodically behind the control plate. The rotational motion of the fluid surround- +ing the cylinder causes the negative equi-vorticity lines to surround the cylinder +as well as the positive equi-vorticity lines that originate from the control plate. +The positive equi-vorticity lines also cover the control plate. The shear layers that +22 + +sitt:t = t0 +(0)T +t = t0 +(1/4)T +t = t0 +(1/2)T +t = t0 +(3/4)T +t = t0 +(1)T +(a) +(b) +(c) +Figure 11: (a) Isotherm, (b) streakline and (c) vorticity contour for Pr = 0.7, Re = 150, α = 3.25 +and d/R0 = 3 at different phases. +originate from the cylinder get split after interaction with the shear layer around +the control plate, and they merge together during the shedding of the negative vor- +tex. Most of the fluid particles that flow across the cylinder are sucked down and +flow below the control plate. This complex flow dynamics is the combined effect +of the high rotational rate and the placement of the control plate. It reduces the +23 + +angle of the vortex shedding plane with the centerline to approximately θ = 12◦, +which is much less than the previous placements of the control plate with this +high rotational rate. Also, the size of the negative vortex is drastically reduced +and becomes extremely sleek due to the interaction of shear layers. The lower +vortex grows from the bottom of the plate and moves upwards. The density of the +isotherm contour around the cylinder is very low due to the high rotational rate. As +a result, the boundary layer thickens around the cylinder and suppresses the rate +of force convective heat transfer. The isotherm contours indicate that two warm +blobs periodically convect away from the upper side of the cylinder and the lower +end of the control plate. Therefore, the placement of the control plate, together +with the rotational rate, considerably suppressed the vortex shedding process as +well as the heat convection. Figs. 10 and 11 illustrate that the size of the vortices +grow as α increases from 0.5 to 3.25 for d/R0 = 3. Also, Figs. 4, 8 and 10 show +that the wake length of vortices increases with increasing distance of the control +plate from the cylinder surface at α = 0.5. It is also observed that the increasing +distance of the control plate significantly decreases the angle of the vortex shed- +ding plane with the centerline for respecting rotational rates. +The drag (CD) and lift (CL) coefficients at different α with varying d/R0 are +shown in Fig. 12. The figures show that the drag and lift coefficients are periodic +in nature. For α = 0.5, the drag coefficient gradually decreases with increasing +d/R0 and the lift coefficient is minimum at d/R0 = 0.5. The differences in the +drag coefficients for this α = 0.5 are very small. There is not much difference +in lift coefficient for d/R0 = 1, 2, and 3. When α = 1, gradual decrease in the +drag coefficient is found with increasing d/R0. The lift coefficient is found to be +minimum for d/R0 = 0.5. The maximum value of the lift coefficient is observed +for d/R0 = 1 and 2. When α = 2.07, maximum value of the drag coefficient is +found for d/R0 = 3 and minimum value is found for d/R0 = 0.5. Here, the maxi- +mum value of the lift coefficient is found for d/R0 = 1 and the minimum value is +found for d/R0 = 0.5. When the rotation rate is at its maximum, i.e., α = 3.25, +the amplitudes of the drag and lift coefficients increase drastically for all d/R0. +Here, the minimum values of lift and drag coefficients are found for d/R0 = 0.5 +and the minimum values are found d/R0 = 2. For d/R0 = 3, the amplitudes of the +drag and lift coefficients are the smallest. So, the impact of various positionings +of the arc-shaped control plate is significant at higher rotational rates. In Fig. 13, +the drag (CD) and lift (CL) coefficients at different d/R0 with varying α are shown. +At, d/R0 = 0.5, the maximum value of the CD is found for α = 1 and the mini- +mum value is found for α = 3.25. With increasing α, the maximum value of CL +24 + +α = 0.5 +α = 1 +α = 2.07 +α = 3.25 +(a) +(b) +Figure 12: (a) Drag coefficient CD and (b) lift coefficient CL with varying d/R0. +gradually decreases while the amplitude of CL gradually increases. The highest +amplitude of the drag and lift coefficients is observed for α = 3.25. When the +25 + +4 +d/R. = 0.5 +d/R。= 1 +2 +d/R。= 2 +d/R. = 3 +0 +-6 +008 +220 +240 +260 +280 +300 +t5 +d/R. = 0.5 +d/R。= 1 +d/R。= 2 +d/R. = 3 +3 +2 +200 +220 +240 +260 +280 +300 +t4 +d/R. = 0.5 +d/R。= 1 +2 +d/R。= 2 +d/R. = 3 +0 +6 +003 +220 +240 +260 +280 +300 +t5 +d/R. = 0.5 +d/R。= 1 +d/R。= 2 +d/R.= 3 +3 +2 +220 +240 +260 +280 +300 +t4 +d/R. = 0.5 +d/R。= 1 +2 +d/R。= 2 +d/R.= 3 +0 +-1.2 +-1.4 +-1.6 +-6 +-1.8 +260 +280 +300 +003 +220 +240 +260 +280 +300 +t5 +d/R. = 0.5 +d/R。= 1 +d/R。= 2 +4 +d/R. = 3 +1.3 +3 +D +1.2 +2 +250 +260 +270 +280 +290 +220 +240 +260 +280 +300 +t4 +d/R. = 0.5 +d/R。= 1 +2 +d/R。= 2 +d/R. = 3 +0 +-0.7 +-4 +-0.8 +-6 +-0.9 +260 +280 +300 +800 +220 +240 +260 +280 +300 +t5 +d/R. = 0.5 +d/R。= 1 +d/R。= 2 +4 +d/R. = 3 +3 +1.2 +D +1.15 +C +2 +1.1 +250 +260 +270 +280 +290 +220 +240 +260 +280 +300 +td/R0 = 0.5 +d/R0 = 1 +d/R0 = 2 +d/R0 = 3 +(a) +(b) +Figure 13: (a) Drag coefficient CD and (b) lift coefficient CL with varying α. +26 + +4 +α = 0.5 +α=1 +2 +α = 2.07 +α = 3.25 +0 +4 +-6 +003 +220 +240 +260 +280 +300 +tin +α = 0.5 +α=1 +4 +α = 2.07 +α = 3.25 +3 +D +2 +200 +220 +240 +260 +280 +300 +t4 +α = 0.5 +α=1 +2 +α = 2.07 +α = 3.25 +0 +-4 +-6 +003 +220 +240 +260 +280 +300 +t5 +α = 0.5 +α=1 +4 +α = 2.07 +α = 3.25 +3 +D +2 +220 +240 +260 +280 +300 +t4 +α = 0.5 +α=1 +2 +α = 2.07 +α = 3.25 +0 +A +-6 +800 +220 +240 +260 +280 +300 +t5 +α = 0.5 +α=1 +A +α = 2.07 +α = 3.25 +3 +2 +220 +240 +260 +280 +300 +t4 +α = 0.5 +α=1 +2 +α = 2.07 +α = 3.25 +0 +008 +220 +240 +260 +280 +300 +t5 +α = 0.5 +α=1 +A +α = 2.07 +α = 3.25 +3 +2 +220 +240 +260 +280 +300 +tθ +θ +θ +(a) +(b) +(c) +θ +θ +θ +(d) +(e) +(f) +θ +θ +(g) +(h) +Figure 14: Local Nusselt number variation at periodic phases for (a) d/R0 = 1, α = 0.5; (b) +d/R0 = 1, α = 1; (c) d/R0 = 1, α = 2.07; (d) d/R0 = 1, α = 3.25; (e) d/R0 = 2, α = 0.5; (f) +d/R0 = 2, α = 3.25; (g) d/R0 = 3, α = 0.5; and (h) d/R0 = 3, α = 3.25. +plate distance is increased to 1 and 2, the maximum value of CD and the minimum +value of CL are found for α = 3.25. Also, the amplitudes are maximum for the +highest rotational rate. When d/R0 = 3, CD gradually increases while CL gradu- +ally deceases as α increases. The lift coefficients suggest that the lock-on vortices +are shed under all the considered rotational rates and distances of the control plate. +Fig. 14 shows the variation of local Nusselt numbers at periodic phases for var- +27 + +ious rotational rates of the cylinder and different positioning of the plate. Fig. 14(a) +shows the variation of Nu for d/R0 = 1 and α = 0.5. It can be seen that the +maximum value of Nu is slightly shifted downwards from the front stagnation +point (θ = 180◦) approximately to θ = 192◦. It indicates the difference in heat +transfer processes between the upper and lower half of the cylinder surface. The +differences in values of Nu between the periodic phases are very small. A local +maximum peak of Nu is found at θ ≈ 30◦ which indicates the higher rate of heat +convection in this area. This is supported by the concentrated isotherm contours +in this area close to the cylinder surface shown in Fig. 4. As the α increases to +2 for d/R0 = 1 in Fig. 14(b), the differences in values are increased at different +phases. The highest point of Nu is found around θ = 204◦. It shows the difference +in heat transfer mechanisms from the upper and lower surfaces. A local maximum +peak of Nu is found at θ ≈ 42◦ indicating higher rate of heat convection in this +area. It is also supported by the highly concentrated isotherm contours in Fig. 5. +When α = 2.07 for d/R0 = 1, the maximum value of Nu slightly decreases in +Fig. 14(c) than that the previous cases and the maximum point of heat transfer is +around θ = 240◦. Local maximum peak is found to be changing position between +θ ≈ 42◦ and θ ≈ 78◦ at different periodic phases due to the complex vortex shed- +ding phenomenon. These areas convect a large amount of heat into the fluid. The +asymmetric Nu-distribution around the front stagnation point shows that the heat +transfer process from the upper part of the cylinder surface is far different from +the heat transfer process from the lower part of the cylinder surface. Fig. 14(d) +shows the variation of Nu with maximum rotation rate, α = 3.25 for d/R0 = 1 and +the maximum value of Nu drastically decreases and occurs at θ ≈ 72◦ i.e. near +the rear stagnation point. As many researchers previously mentioned, here too, +large rotational rates significantly reduce the maximum heat transfer rate from the +cylinder [19, 22, 23]. A local maximum peak is found at θ ≈ 252◦. The reduc- +tion of the maximum peak value of Nu at front stagnation point with increasing α +hints to the fact that more heat is transferred under conduction in this area. This +happens due to the thickening of the boundary layer around the cylinder surface +at the high rotational rate. Fig. 14(e) shows the variation of Nu with α = 0.5 and +d/R0 = 2. It shows that the maximum point of heat transfer is around θ = 191◦. +Also, the peak value is slightly lower than that of d/R0 = 1 due to the vortex shed- +ding process. A local maximum peak is found at θ ≈ 24◦ i.e. the heat transfer is +higher in this area. It is also supported by the respective dense isotherm contours. +In Fig. 14(f), α is increased to 3.25 for d/R0 = 2 and it is found that the highest +value of Nu significantly reduced than the previous case. The highest value of Nu +is observed around θ = 264◦ i.e. maximum heat transfer under convection occurs +28 + +Nut +α +α +α +α +α +Nut +(a) +(b) +Figure 15: (a) Nut for varying α, and (b) Nut for varying d/R0. +in this area. This is supported by the respective dense isotherm contour in this +region close to the cylinder surface. A local maximum value of Nu is found at +θ ≈ 60◦ at periodic phases t = t0 + (0)T, t0 + (1)T, i.e. the heat transfer is en- +hanced in this area under convection by the complex vortex shedding process. The +Nu-distribution at 180◦ ≤ θ ≤ 0◦ is significantly different than the Nu-distribution +at 360◦ ≤ θ ≤ 180◦. This demonstrates that the lower half of the cylinder surface +convects more heat than the upper half. Fig. 14(g) shows the variation of Nu for +α = 0.5 and d/R0 = 3. The highest value of Nu is observed around θ = 192◦. +The maximum value is slightly lower than that of d/R0 = 1, 2. A local maximum +value of Nu distribution is found at θ ≈ 24◦. The maximum heat transfer under +convection occurs in these areas. The highest value of Nu-distribution curve is +significantly reduced in Fig. 14(h) where α = 3.25 and d/R0 = 3 as compared to +Fig. 14(d) for d/R0 = 1 and Fig. 14(f) for d/R0 = 2. This indicates that the in- +creasing distance of the control plate significantly reduced the heat transfer under +convection for the fixed α. The highest value of Nu is shifted to θ ≈ 276◦ due +to the complex vortex shedding. The lowest value of Nu-distribution curve at the +front stagnation point indicates that a large amount of heat is transferred by con- +duction at this place. Also, the distribution curve at 180◦ ≤ θ ≤ 0◦ is significantly +different than the curve at 360◦ ≤ θ ≤ 180◦, which shows that the lower half of +the cylinder surface convects more heat than the upper half. +Fig. 15 exhibits the variation of time-averaged total Nusselt number (Nut) with +Fig. 15(a) varying α and Fig. 15(b) varying d/R0. The values of Nut for α = 0.5 +29 + +are 6.672265, 6.251865, 6.154835 and 6.074185 with d/R0 = 0.5, 1, 2 and 3 +respectively. It means that the increasing distance of control plate reduces the +heat transfer rate at α = 0.5. The values of Nut for α = 1 are 6.790804, 6.28877, +6.07076 and 5.89388 with d/R0 = 0.5, 1, 2 and 3 respectively. It means that the +increasing distance of control plate also reduces the heat transfer rate at α = 1. +The values of Nut for α = 2.07 are 6.686615, 5.774501, 5.6436 and 5.643545 +with d/R0 = 0.5, 1, 2 and 3 respectively. Here also, the increasing distance of +control plate reduces the heat transfer rate. The values of Nut for α = 3.25 are +6.68507, 5.899766, 4.931795 and 4.9757 with d/R0 = 0.5, 1, 2 and 3 respectively. +Again the increasing distance of the control plate reduces the heat transfer rate +except for d/R0 = 3. This occurs due to the interaction of high rotation and the +large distance of the control plate. Fig. 15(a) shows that Nut gradually deceases +with increasing α at d/R0 = 2, 3 and the maximum value of Nut is found for +d/R0 = 0.5, α = 0.5. Fig. 15(b) shows that increasing d/R0 significantly reduces +Nut within the range of 0.5 ≤ d/R0 ≤ 2 for all rotational rates. However, if we +place the plate further at a distance d/R0 = 3, not much change occurs. It is +found from the comparison of maximum and minimum values of Nut that certain +positioning of the control plate and rotational rate can enhance the heat transfer +rate by 37.69%. +5. Conclusion +We numerically examined the control of a uniform, viscous fluid flow past +circular cylinder by an arc-shaped plate positioned in the normal direction behind +an isothermally heated circular cylinder rotating in the cross stream. The gov- +erning equations are discretized using a HOC finite difference technique, and the +system of algebraic equations obtained by the HOC discretization is solved using +the Bi-conjugate gradient stabilised iterative method. According to the research, +the distance between the control plate and the cylinder surface has a considerable +impact on fluid flow along with the rotation of the cylinder. The structure of the +wake changes depending on the position of the plate. When α is less than 1 with +d/R0 = 1, two vortices as lumps of hot fluid are shed periodically from either +side of the cylinder; when α is greater than 2.07 with d/R0 = 1, a large nega- +tive vortex of heated fluid is shed from the upper side of the cylinder and another +positive vortex of hot fluid is shed behind the control plate on a periodic basis. +The increasing rotational rates increase the size of vortices and decrease the wake +length for all positions of the control plate. The vortex shedding plane is shifted +from the centerline by the cylinder’s rotational motion. For all rotational rates, the +30 + +increased distance of the control plate decreases the angle of the vortex shedding +plane with the centerline, but the angle is increased with increasing rotational rates +for all positions of the control plate. At higher rotational rates, the positive vortex +is pulled upwards due to the interaction of fluid, and it pushes the negative vortex, +causing an early shedding of it. Placing the control plate at d/R0 = 3 along with +a high rotational rate is found to significantly reduce the size of vortices. It is also +found that the impact of various positionings of the arc-shaped control plate is +significant at higher rotational rates. An additional recirculation zone is found for +(d/R0 = 2, α = 0.5, 3.25) and (d/R0 = 3, α = 3.25). Drag and lift coefficients +for all 0.5 ≤ d/R0 ≤ 3 and 0.5 ≤ α ≤ 3.25 have a periodic nature . The values of +drag and lift coefficients can be reduced or increased by utilising the rotation of the +cylinder and the placement of the plate. The maximum value of drag coefficient +is achieved for d/R0 = 2 and α = 3.25 which is about 3. All vortices shed are +locked-on under the scope of considered parameters. It is found that the rotational +rates relocate the highest point of heat transfer further from the front stagnation +point, i.e., increasing the heat transfer by conduction in this region. The increasing +distance of the control plate significantly reduced the heat transfer under convec- +tion for the fixed α. The combined effect of rotation and the positioning of the +control plate causes a different heat transfer mechanism at the upper half of the +cylinder surface than at the lower half. For fixed d/R0 = 2 and α = 3.25, the +maximum point of heat transfer is shifted towards the rear stagnation point from +the front stagnation point due to the complex vortex shedding. +Author Declarations +The authors have no conflicts to disclose. +Data Availability Statement +The data that support the findings of this study are available from the corre- +sponding author upon reasonable request. +References +[1] W. Bickley, “The influence of vortices upon the resitance experienced by +solids moving through a liquid,” Proceedings of the Royal Society of London. +Series A, Containing Papers of a Mathematical and Physical Character, vol. +119, no. 781, pp. 146–156, 1928. +31 + +[2] J. Gerrard, “The mechanics of the formation region of vortices behind bluff +bodies,” Journal of Fluid Mechanics, vol. 25, no. 2, pp. 401–413, 1966. +[3] J. O. Pralits, L. Brandt, and F. Giannetti, “Instability and sensitivity of the +flow around a rotating circular cylinder,” Journal of Fluid Mechanics, vol. +650, pp. 513–536, 2010. +[4] S. Kang, H. Choi, and S. Lee, “Laminar flow past a rotating circular cylin- +der,” Physics of Fluids, vol. 11, no. 11, pp. 3312–3321, 1999. +[5] F. Diaz, J. Gavaldà, J. Kawall, J. Keffer, and F. Giralt, “Vortex shedding from +a spinning cylinder,” The Physics of Fluids, vol. 26, no. 12, pp. 3454–3460, +1983. +[6] J. Massons, X. Ruiz, and F. Diaz, “Image processing of the near wakes of +stationary and rotating cylinders,” Journal of Fluid Mechanics, vol. 204, pp. +167–184, 1989. +[7] D. Stojkovi´c, M. Breuer, and F. Durst, “Effect of high rotation rates on the +laminar flow around a circular cylinder,” Physics of Fluids, vol. 14, no. 9, +pp. 3160–3178, 2002. +[8] A. Roshko, “On the wake and drag of bluff bodies,” Journal of the Aeronau- +tical Sciences, vol. 22, no. 2, pp. 124–132, 1955. +[9] P. Bearman, “Investigation of the flow behind a two-dimensional model with +a blunt trailing edge and fitted with splitter plates,” Journal of Fluid Mechan- +ics, vol. 21, no. 2, pp. 241–255, 1965. +[10] C. J. Apelt, G. S. West, and A. A. Szewczyk, “The effects of wake split- +ter plates on the flow past a circular cylinder in the range 104 1 suggests GAS of the Endemic +Equilibrium (from here onwards, EE). In more complex models, the aforementioned condi- +tions on R0 might not be sufficient to prove the GAS of either equilibria, especially in cases +in which the EE is not unique. Lyapunov functions often explicitly involve R0 to guarantee +the extinction of the disease or its endemicity over time. +Unfortunately, given a generic system of ODEs, there is no universal way of deriving a +Lyapunov function, nor to rule out the existence of one. However, there exist a few Lyapunov +functions which have proven quite effective in a variety of different models. +In this survey, we collect some of the most relevant functions available in the literature, to +provide the reader with a series of options to apply to the model of their interest, depending +on its formulation. We include an extensive bibliography to complement the essential infor- +mation of each model we present. This will provide the reader with a convenient starting +point to investigate the availability of a known Lyapunov function to analytically prove the +asymptotic behaviour of their system of ODEs. For the sake of brevity, we do not repeat +the proofs to show that any of the functions we present are, indeed, Lyapunov function +for the respective system of ODEs. These proofs can be found in the papers we cite when +introducing each model. +Consider a model with compartments X1, X2, . . . , Xn. Then, the DFE has coordinates +Xi = 0 for all i ∈ I, where I is the set of the indexes of infectious compartments, and the +EE, which we indicate with (X∗ +1, X∗ +2, . . . , X∗ +n), has all positive entries. A vast majority of +Lyapunov functions in epidemic modelling fall into one of the categories listed below. +2 + +1. Linear combination of infectious compartments. The Lyapunov function for the +DFE when R0 < 1 is of the form +L = +� +i≥2 +ciXi, +for some constants ci ≥ 0 to be determined [6, 16, 18, 21, 32, 36, 39, 45, 49, 50, 59, 64, +70]. To prove convergence of the system to the DFE in this case it is often required the +use of additional tools, such as LaSalle’s invariance principle, which we briefly recall +at the end of Section 2.1. +2. Goh-Lotka-Volterra. The Lyapunov function for the EE when R0 > 1 is of the form +L = +� +i +ci(Xi − X∗ +i ln Xi), +for some constants ci ≥ 0 to be determined [2, 5, 6, 20, 27, 29, 32, 33, 45, 49, 52, 53, +59, 63, 65]. These functions are adapted from a first integral of the notorious Lotka- +Volterra prey-predator system, and were popularized by Bean-San Goh in a series of +paper [12, 13, 14]. +3. Quadratic. The Lyapunov function for the EE when R0 > 1 is of the common form +L = +� +i +ci(Xi − X∗ +i )2, +for some constants ci ≥ 0 to be determined, or the composite form +L = +�� +i +Xi − X∗ +i +�2 +. +Some examples can be found in [40, 41, 60, 65, 66]. +4. Integral Lyapunov. Lyapunov functions given as integrals over the dynamics of the +model. +The integration interval often start at some EE value X∗ +i and ends at the +same Xi; this construction is very convenient if uniqueness of the EE is guaranteed, +but the exact values of the EE are hard (or impossible) to determine analytically. +Integral Lyapunov functions are particularly useful when the model includes multiple +stages of infection, and consequently the infectious period changes from an exponential +distribution to a gamma distribution [8, 11, 18, 38, 58, 61]. Integral Lyapunov functions, +albeit in different forms, are widely used in models which incorporate explicit delay, +such as systems of Delay Differential Equations (from here onwards, DDEs), and age- +structured models. However, these fall beyond the scope of this paper, and we will +briefly comment on them in Section 3. +5. Hybrid. +A linear combination of the above, which often includes the Goh-Lotka- +Volterra in at least a few of the compartments of the system [15, 27, 37, 47, 50, 53, 51, +63]. +3 + +For some high-dimensional models, proving convergence to the EE might require addi- +tional tools, such as the geometric approach used in [53, 64]. +Lastly, we must notice that not all compartmental models only exhibit convergence to +equilibrium. Some systems of autonomous ODEs may present stable or unstable limit cycles +[9, 54, 68], homoclinic orbits [54] or even chaos [57]. In such cases, clearly, no global Lyapunov +function may exist. +In the remainder of this survey, we will present various models and the corresponding +Lyapunov functions, covering all the cases listed above. +2 +Epidemic models +In this section, we present various compartmental epidemic models with the corresponding +Lyapunov function(s). We present the models from the smallest to the largest, in terms +of number of compartments. We refer to [1, 28] for a basic introduction on compartmental +epidemic models, and to [55] for a detailed exemplification of Lyapunov theory in this setting. +We provide a schematic representation of the flows in most of the systems we present. +Flow diagrams can be useful to provide a visual, intuitive interpretation of the parameters +involved in each system. Arrows between compartments indicate a change in the current +state of individuals with respect to the ongoing epidemics, whereas arrows inward/outward +the union of the compartments represent birth rate and death rate in the population. Often, +these last two rates are considered to be equal, as this assumption allows the population to +either remain constant or converge to a constant value, reducing the dimensionality of the +system and (hopefully) its analytical complexity. However, some models include additional +disease-induced mortality, to increase realism when modelling severe infectious diseases. We +uniform the notation throughout the various models we present in this survey as much as +possible, and provide a brief description of each parameter the first time it is encountered. +We remark that each variable is assumed to be non-negative, since it represents a fraction of +the population, but the biologically relevant region varies depending on the specific model +we are describing. +Moreover, we illustrate the corresponding Lyapunov functions for 2D models, showcasing +a selection of their power levels. The same procedure can be easily adapted to 3D models, +but the corresponding visualizations can be hard to interpret in a static image. +2.1 +SIS +The SIS model is characterized by the total absence of immunity after infection, i.e. the +recovery from infection is followed by an instantaneous return to the susceptible class. The +ODEs system which describes this situation is +dS +dt = γI − βSI +N , +dI +dt = βSI +N − γI, +(1) +S +I +β SI +N +γI +where β is the transmission rate and γ is the recovery rate. +4 + +Notice that the population N = S + I is constant, thus we can normalize it to N = 1. +Moreover, since S + I = 1, we can reduce the system to one ODE which involves only +infectious individuals +dI +dt = (β(1 − I) − γ)I. +System (1) always admits the DFE, i.e. E0 = (1, 0), and the EE, i.e. E∗ = +�γ +β , β − γ +β +� +, +which exists if and only if β > γ (or equivalently if R0 = β/γ > 1). Notice that, if R0 < 1, +then I is always decreasing in the biologically relevant interval [0, 1]. +A variation of model (1) can be obtained by adding demography to the system. This is +the example of [65], in which the authors consider a birth/immigration rate different from +the natural death rate; moreover, they include an additional disease-induced death rate from +infectious class. Thus, the population is not constant and the system of ODEs which describe +the model is +dS +dt = Λ + γI − βSI +N − µS, +dI +dt = βSI +N − (δ + γ + µ)I, +(2) +S +I +β SI +N +γI +Λ +µS +(δ + µ)I +where Λ represents the birth/immigration rate, µ the natural death rate and δ the disease- +induced mortality rate. System (2) always admits the DFE, namely E0 = (S0, 0) := +�Λ +µ, 0 +� +, +and the EE, namely E∗ = (S∗, I∗), where I∗ > 0 if and only if R0 = +Λβ +µ(µ + δ + γ) > 1. In +[65], a Lyapunov function for the DFE is defined as +V (S, I) := 1 +2 (S − S0 + I)2 + 2µ + δ +β +I, +(3) +whereas the Lyapunov function for the EE is built using a combination of the quadratic and +logarithmic functions +V (S, I) := 1 +2 (S − S∗ + I − I∗)2 + 2µ + δ +β +� +I − I∗ − I∗ ln +� I +I∗ +�� +. +(4) +The authors also construct two more examples of Lyapunov functions for the EE, namely +V (S, I) := 1 +2(S − S∗)2 + µ + δ +β +� +I − I∗ − I∗ ln +� I +I∗ +�� +, +(5) +and +V (S, I) :=1 +2 (S − S∗ + I − I∗)2 + S∗(δ + 2µ) +2γ +� +S − S∗ − S∗ ln +� S +S∗ +�� ++ S∗(δ + 2µ) +γ +� +I − I∗ − I∗ ln +� I +I∗ +�� +. +(6) +5 + +Power levels of the functions (3), (4), (5) and (6) are visualized if Figure 1. By definition +of a Lyapunov functions, orbits of the corresponding system (2) evolve on decreasing power +levels, and they tend to the corresponding equilibrium as t → +∞. +(a) +(b) +(c) +(d) +Figure 1: Power levels of Lyapunov functions (3) (a), (4) (b), (5) (c), and (6) (d). Values of +the parameters are Λ = 0.8, µ = 1, δ = 1, γ = 1 in all the figures, β = 1 in (a), so that R0 = +4/15 < 1, and β = 4 in (b), (c) and (d), so that R0 = 16/15 > 1. We represent V (S, I) = k, +with k ∈ {0.1, 0.25, 0.5, 1, 1.5, 2, 2.5} in (a), k ∈ {0.001, 0.01, 0.025, 0.05, 0.1, 0.2} in (b) and +(c), and k ∈ {0.01, 0.025, 0.05, 0.1, 0.2, 0.5} in (d). Black dots represent the globally stable +equilibrium the system converges to, and correspond to V (S, I) = 0. +In [66] the author found a simpler Lyapunov function for the DFE when R0 < 1, i.e. +V (I) = 1 +2I2. +(7) +However, this last Lyapunov function (7) only ensures that I → 0 as t → +∞. To complete +6 + +the proof of the converge of the system to the DFE, one needs in addiction to invoke LaSalle’s +theorem [35] (see also [31, Thm. 3.4]), as is indeed done in [66]. +Considering the importance of this theorem, especially when combined with the use of +Lyapunov functions, we include its statement here. +Theorem 2.1. (LaSalle’s invariance principle) Let X′ = f(X) be a system of n ODEs +defined on a positively invariant set Ω ⊂ Rn. +Assume the existence of a function V ∈ +C1(Ω, R) such that V ′(X) ≤ 0 for all X ∈ Ω. Let MV be the set of stationary points for V , +i.e. V ′(X) = 0 for all X ∈ MV , and let N be the largest invariant set of MV . Then, every +solution which starts in Ω approaches N as t → +∞. +In particular, this theorem implies that, if we can prove the approach of the disease to +the manifold describing absence of infection and the uniqueness of the DFE, then the DFE +is GAS. +2.2 +SIR/SIRS +The SIR model is characterized by the total immunity after the infections, i.e. recovered +individuals can not become susceptible again. A classical example for this scenario is measles. +The ODEs system which describes this situation is +dS +dt = −βSI +N , +dI +dt = βSI +N − γI, +dR +dt = γI, +(8) +S +I +R +β SI +N +γI +where β is the transmission rate and γ is the recovery rate. +If we assume that recovered individuals eventually lose their immunity, we obtain the +SIRS model. Denoting by α the immunity loss rate, we obtain the following ODEs system +dS +dt = −βSI +N + αR, +dI +dt = βSI +N − γI, +dR +dt = γI − αR. +(9) +S +I +R +β SI +N +γI +αR +It is clear that, if α = 0, system (9) coincides with system (8). +These models admit only the DFE; in order to have an EE, we need to add the demog- +raphy to model (8) or (9). +In [65], the authors consider the following ODEs system +7 + +dS +dt = Λ − βSI +N − µS + αR, +dI +dt = βSI +N − (γ + δ + µ)I, +dR +dt = γI − (α + µ)R. +(10) +S +I +R +β SI +N +γI +αR +Λ +µS +µR +(δ + µ)I +System (10) admits the DFE, E0 = (S0, 0, 0), and the EE, E∗ = (S∗, I∗, R∗), which exists if +and only if R0 = +βΛ +µ(µ + γ + δ) > 1. In [65], the Lyapunov function for the DFE is defined +as follows +V (S, I, R) := 1 +2 (S − S0 + I + R)2 + 2µ + δ +β +I + 2µ + δ +2γ +R2, +whereas the Lyapunov function for the EE is the combination of the composite quadratic, +common quadratic and logarithmic functions as follows +V (S, I, R) :=1 +2 (S − S∗ + I − I∗ + R − R∗)2 ++ 2µ + δ +β +� +I − I∗ − I∗ ln +� I +I∗ +�� ++ 2µ + δ +2γ +(R − R∗)2. +The authors also present other Lyapunov functions for SIR/SIRS models; in particular, +they also cite [3, 46] in which some variations of system (10) are showed. Other Lyapunov +functions for SIR/SIRS epidemic models are in [55], in which the authors use a graph- +theoretic approach. +In [66], the author proved that the quadratic Lyapunov function (7) of the SIS model +applies to the SIR and the SIRS, as well. +2.3 +SEIR/SEIS/SEIRS +In [32], the authors study both SEIR and SEIS models. Many real world examples present +a phase of exposition to the disease, between susceptibility and infectiousness. The models +presented thus far, albeit simpler to study, are unable to replicate this mechanism. +The authors first analyze a SEIR model with demography and constant population, in +which the disease is transmitted both horizontally and vertically. Individuals infected verti- +cally pass first in the exposed compartment. The ODEs system which describe the model is +dS +dt =µ − βSI − pµI − qµE − µS, +dE +dt =βSI + pµI − θE − µE + qµE, +dI +dt =θE − (δ + µ)I, +(11) +S +E +I +βSI +θE +µ(1 − pI − qE) +µS +µ(pI + qE) +µE +(δ + µ)I +and R = 1 − S − E − I. The vertical transmission of the disease is represented by the +probabilities p and q of being born directly in the Exposed compartment, rather than in the +Susceptible one, and is represented by the inward arrow in compartment E. +8 + +The authors first provide an equivalent system, performing the substitution (S, E, I) −→ +(P, E, I), where P := S + pµ +β . They then proceed to prove the GAS of the EE, using the +following Lyapunov function +V (P, E, I) :=(P − P ∗ ln P) + +θ + µ +θ + µ − qµ(E − E∗ ln E) ++ +θ + µ +θ + µ − qµ(I − I∗ ln I). +Later, the authors analyze a situation in which the recovery does not provide immunity, +namely the SEIS model. They also assume that a fraction r of offspring of the infective +hosts is born directly into the infective compartment. In this case, the ODEs system changes +accordingly describe the model is +dS +dt =µ − βSI + (δ − pµ − rµ)I − qµE − µS, +dE +dt =βSI + pµI − (θ + µ − qµ)E, +dI +dt =θE − (δ + µ − µr)I, +(12) +and S + E + I = 1. Notice that, due to the population remaining constant in system (12), +one could in principle reduce its dimensionality and consider it as a planar system. +The authors prove the GAS of the EE using the following Lyapunov function +V (S, E, I) :=(S − S∗ ln S) + µ1 − S∗ +βI∗S∗ (E − E∗ ln E) ++ µ1 − S∗ +θE∗ +� +1 + pρ0 +µ +β +� +(I − I∗ ln I). +A natural extension to these models is the SEIRS [22, 64], in which one can combine the +existence of an immune compartment and the loss of immunity. +It is described by the +following system of ODEs +dS +dt = − βg(I)S + µ − µS + αR, +dE +dt =βg(I)S − (θ + µ)E, +dI +dt =θE − (γ + µ)I, +dR +dt =γI − (α + µ)R, +(13) +S +E +I +R +βg(I)S +γI +αR +θE +µE +µ +µS +µR +µI +where g ∈ C3(0, 1], g(0) = 0 (meaning, in absence of infectious individuals, the disease does +not spread) and g(I) > 0 for I > 0. The classical choice is g(I) = I, as in systems (11) and +(12). Assuming moreover +lim +I→0+ +g(I) +I += c ∈ [0, +∞), +9 + +the authors of [22] derive R0 = +cβθ +(θ + µ)(γ + µ). They then prove GAS of the DFE of system +(13) through the use of the following linear Lyapunov function +V (E, I) = E + θ + µ +θ +I, +whereas the GAS of the EE is proved with a more complex geometrical method in [64]. +2.4 +SAIR/SAIRS +One of the main challenges of the Covid-19 pandemic was the presence of asymptomatic +individuals spreading the disease. Such individuals must clearly be somehow distinguished +from symptomatic infectious individuals, as they are likely to behave like a susceptible +individual. Even though their viral load, and hence infectiousness, might be smaller, they +are more likely to get in close contact with susceptible individuals. +In [53], the authors consider a SAIRS model. The main difference between this kind +of models and the SEIR is that both asymptomatic and symptomatic hosts may infect +susceptible individuals. +The immunity is not permanent, i.e. +recovered individuals will +become susceptible again after a certain period of time. Moreover, vaccination are included. +The ODEs system which describe this model is +dS +dt = µ − +� +βAA + βII +� +S − (µ + ν)S + γR, +dA +dt = +� +βAA + βII +� +S − (α + δA + µ)A, +dI +dt = αA − (δI + µ)I, +dR +dt = δAA + δII + νS − (γ + µ)R, +S +A +R +I +µ +µS +(βAA + βII)S +δII +γR +δAA +µA +αA +νS +µI +µR +The global stability analysis of the EE has been performed for two variations of the original +model, described in the following. +The first model analyzed is the SAIR model, i.e. the case in which recovery from the +disease grants permanent immunity. In this case, the corresponding Lyapunov function is +the combination of the Lokta-Volterra Lyapunov functions for S, A and I +V (S, A, I) :=c1S∗ +� S +S∗ − 1 − ln +� S +S∗ +�� ++ c2A∗ +� A +A∗ − 1 − ln +� A +A∗ +�� ++ I∗ +� I +I∗ − 1 − ln +� I +I∗ +�� +, +where c1, c2 > 0. +The second model is the SAIRS model, with homogeneous disease transmission and +recovery among A and I, i.e. βA = βI and δA = δI. In this case, it is possible to sum +10 + +equations for A and I, defining M := A + I, reducing the dimensionality of the system. +Thus, the Lyapunov function can be written as the combination of the square function and +the Lokta-Volterra as follows +V (S, M) := 1 +2(S − S∗)2 + w +� +M − M∗ − M∗ ln +� M +M∗ +�� +, +where w > 0. +The global stability in the most general case is proved similarly to [64]. +2.5 +More exotic compartmental models +The aforementioned models are some of the most commonly used in literature. In order +to capture additional disease-specific nuances, these model can be modified or extended by +adding new compartments. +Some diseases, for example, present different stages of infection. In this case, an infected +individual can progress between two or more stages before recovering. In [18], the authors +perform the global stability analysis via an integral Lyapunov function of a general class +of multistage models. +In their model, infectious individual can move both forward and +backward on the chain of stages, in order to incorporate both a natural disease progression +and the amelioration due to the effects of treatments. +The system of ODEs which describes the model is +dS +dt = θ(S) − f(N) +n +� +j=1 +gj(S, Ij), +dI1 +dt = f(N) +n +� +j=1 +gj(S, Ij) + +n +� +j=1 +φ1,j(Ij) − +n+1 +� +j=1 +φj,1(I1) − ζ1(I1), +dIi +dt = +n +� +j=1 +φi,j(Ij) − +n+1 +� +j=1 +φj,i(Ii) − ζi(Ii), +i = 2, 3, . . ., n, +where θ(S) is the growth function, f(N) +�n +j=1 gj(S, Ij) is the incidence term, ζi(Ii), 1 ≤ i ≤ n, +denote the removal rates of the Ii compartment. +Moreover, for any i, j = 1, . . . , n, the +functions φi,j(Ij) represent the rate of the disease progression if i > j and the amelioration +if i < j. +The corresponding Lyapunov function for the DFE is linear in the disease compartments, +i.e. +V (I1, . . ., In) = +n +� +i=1 +ciIi, +where c1 = R0 and ci ≥ 0 for all i = 2, . . . , n. For the global stability of the EE the authors +made some assumptions on the aforementioned functions. In particular, they consider the +following integral Lyapunov function +V (S, I1, . . . , In) = τ +� S +S∗ +Φ(ξ) − Φ(S∗) +Φ(ξ) +dξ + +n +� +i=1 +τi +� Ii +I∗ +i +ψi(ξ) − ψi(I∗ +i ) +ψi(ξ) +dξ, +11 + +where τ, τi > 0, for all i = 1, . . ., n. For a more in-depth explanation on the functions Φ(·) +and ψi(·) we refer to [18, Sect. 5]. +Diseases which present multiple virus strains, due to the existence of different serotypes +of the virus or due to a mutation of the original disease, may need to be modelled differently. +Dengue, tuberculosis and various sexually transmitted diseases are caused by more than one +strain of a pathogen. Influenza type A viruses mutate constantly: an infection with one of +its strains gives permanent immunity against that specific strain. However, the so called +“antigenic drift” produces new virus strains, thus the hosts only acquire partial immunity, or +no immunity at all. Modelling these types of diseases requires the inclusion of cross-protective +effects, in which the immunity acquired towards one strain offers partial protection towards +another strain based on their antigenic similarity. In [6], the authors consider an n strain +model, both without immunity and with immunity for all the strains. Moreover, they analyze +an MSIR model, in which the M compartment represents the proportion of newborns who +possess temporary passive immunity due to protection from maternal antibodies. For all the +three model, the authors use a linear Lyapunov function to prove the global stability of the +DFE and a logarithmic Lyapunov function to prove the global stability of the EE. +Other compartmental models include e.g. control strategies. For new ongoing epidemics, +the most immediate strategy is including quarantine and isolation of infectious individuals. +For well-known epidemics for which a vaccination is available, it is useful to incorporate a +vaccinated individuals compartment V to keep track of the two possible immunities, disease +and vaccine induced, respectively. Usually, vaccination does not confer permanent immu- +nity, and after a certain disease-dependent period individuals become susceptible again. An +example is [50], in which the authors analyze a SIRV epidemic model with non-linear inci- +dence rate. The global stability of the DFE is proved using as linear Lyapunov function the +infectious compatment I and the global stability of the EE, instead, using a combination of +a quadratic function in S and a logarithmic function in the compartments I and V . +3 +Conclusion +In this survey, we presented the most widely used Lyapunov functions in the field of epi- +demic compartmental models. We focused on systems expressed as autonomous systems of +ODEs. These models allow for various interesting generalizations, of which we provide a +non-comprehensive list below. +One extension of the classic compartmental epidemic models is the so-called multi-group +approach, see e.g. [34, 58]. These models describe n communities, interacting with each other, +and whose internal evolution follows a standard compartmental model. A first example of +such a model is presented in [10], in which the authors consider a n groups SIS model. In +order to prove the GAS of the EE, they use a results on Metzler matrices. In [55], the authors +consider a heterogeneous SIS disease model, for which they provide Lyapunov functions both +for the DFE and for the EE. For the latter, they use a complex graph-teoretic method, for +the details of which we refer to the original paper. Global stability of EE via Lyapunov +function for multi-group generalization can be found also for the SIR [19], SIRS [48], SEIR +[17] and SAIR/SAIRS model [52]. Notice that, due to the complexity of the models, some +of them require additional technical assumptions to prove the global stability of the endemic +equilibrium. +12 + +Other classes of models include interactions between human and vector population, i.e. +animals which transmit the disease to humans, or with the pathogens, such as viruses or +bacteria. In both cases, authors often include a compartmental structure for the non-human +population. Some examples of vector-host models are shown in [59, 62, 70]. Another example +can be found in [40], in which a SIR-B compartmental model is considered. Here the “B” +denotes the concentration of the pathogen in the environment. +All the models discussed thus far are described by only autonomous systems of ODEs. +However, in order to increase realism, it is possible to use non-autonomous systems to de- +scribe the spread of an infectious disease. This is the case of systems in which some param- +eters change in time [42, 56], to describe seasonal changes, or in which the state variables +depend on the previous state, i.e. the model includes a time delay [4, 67]. In these cases, +it is still possible to find Lyapunov functions to prove the global stability of the equilibria +using other techniques, described for example in [35]. +Another popular option is to explicitly include delay in the system, such as in [4, 23, +25, 26, 43, 63, 69]. +In the latter the authors perform the global stability analysis of a +SEIQR model, in which Q denotes the quarantined individuals. They explicitly include a +latent period for the infection, transforming two of the ODEs in DDEs. The corresponding +Lyapunov function includes the integration over an interval whose size is precisely the latent +period. +Lastly, a widely adopted strategy is to explicitly include the “time since infection” [7, +24, 44, 71, 72] in age-structured models. This allows to explicitly take into account time +heterogeneity in the spread of an infectious disease in a population. +These last cases we mentioned are outside of the scope of this project, and we leave them +as inspiration for future works. +Acknowledgments. +The authors are grateful to the organizers of the conference 100 Years +Unione Matematica Italiana - 800 Years Università di Padova, which made their scientific +cooperation possible. Moreover, they acknowledge Politecnico di Milano, Polish Academy of +Sciences, Inria and University of Trento for supporting their research. +References +[1] R.M. Anderson. 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Eng., 12(4):859, +2015. +18 + diff --git a/29FKT4oBgHgl3EQf8C4w/content/tmp_files/load_file.txt b/29FKT4oBgHgl3EQf8C4w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7476ce13288004d126cbd3ec54818a72a54f597c --- /dev/null +++ b/29FKT4oBgHgl3EQf8C4w/content/tmp_files/load_file.txt @@ -0,0 +1,918 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf,len=917 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='11947v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='DS] 27 Jan 2023 A survey on Lyapunov functions for epidemic compartmental models N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Cangiotti∗, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Capolli†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Sensi‡, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Sottile§ Abstract In this survey, we propose an overview on Lyapunov functions for a variety of com- partmental models in epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We exhibit the most widely employed functions, together with a commentary on their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Mathematics Subject Classification: 34D20, 34D23, 37N25, 92D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Keywords: Epidemic models, Lyapunov functions, Compartmental models, Global stability, Ordinary Differential Equations, Disease Free and Endemic Equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 1 Introduction Stemming from the pioneering work of Kermack and McKendrick [30], the mathematical modelling of infectious diseases has developed, over the last century, in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' An abundance of approaches and mathematical techniques have been employed to capture the many facets and details which describe the spread of an infectious disease in a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In particular, compartmental models remain one of the most widely employed approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In these models, a population is partitioned into compartments, characterizing each individ- ual with respect to its current state in the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' One can then write a system of Ordinary Differential Equations (from here onwards, ODEs) to study the evolution in time of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' ∗Politecnico di Milano, Department of Mathematics, via Bonardi 9, Campus Leonardo, 20133, Milan (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' E-mail: nicolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='cangiotti@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='it †Institute of Mathematics, Polish Academy of Sciences, Jana i Jedrzeja Sniadeckich 8, 00-656, Warsaw, (Poland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' E-mail: mcapolli@impan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='pl ‡MathNeuro Team, Inria at Université Côte d’Azur, 2004 Rte des Lucioles, 06410, Biot, (France).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' E-mail: mattia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='sensi@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='fr §Department of Mathematics, University of Trento, Via Sommarive 14, 38123, Povo, (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' E-mail: sara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='sottile@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='it 1 These models usually take their names from the compartments they consider, the most renowned one being the Susceptible-Infected-Recovered (SIR) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The SIR models can be extended to SIRS models by considering the acquired immunity to be temporary rather than permanent, allowing Recovered individuals to become Susceptible again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Various com- partments can be added, depending on the characteristic of the specific disease under study: Asymptomatic, Exposed, Waning immunity and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A remarkably useful tool for the study of this kind of models are Lyapunov functions, which ensure global (or, in some cases, local) asymptotic convergence towards one of the equilibria of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Given a system of n ODEs X′ = f(X) and an equilibrium point X∗, we call a scalar function V ∈ C1(Rn, R) a Lyapunov function if the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' V attains its minimum at X = X∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' V ′ = ∇V · f < 0 for X ̸= X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The classical definition of Lyapunov function requires also the conditions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' X∗ = 0 and V (X∗) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' however, these amount to a change of coordinates in Rn and a vertical translation of V , so we will accept the more general definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The existence of such a function guarantess the global stability of the equilibrium X∗, as orbits of the systems naturally evolve towards the minimum power level of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The Basic Reproduction Number R0 is a well know threshold in epidemics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Usu- ally, R0 < 1 suggests Global Asymptotic Stability (from here onwards, GAS) of the Disease Free Equilibrium (from here onwards, DFE), whereas R0 > 1 suggests GAS of the Endemic Equilibrium (from here onwards, EE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In more complex models, the aforementioned condi- tions on R0 might not be sufficient to prove the GAS of either equilibria, especially in cases in which the EE is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Lyapunov functions often explicitly involve R0 to guarantee the extinction of the disease or its endemicity over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Unfortunately, given a generic system of ODEs, there is no universal way of deriving a Lyapunov function, nor to rule out the existence of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' However, there exist a few Lyapunov functions which have proven quite effective in a variety of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In this survey, we collect some of the most relevant functions available in the literature, to provide the reader with a series of options to apply to the model of their interest, depending on its formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We include an extensive bibliography to complement the essential infor- mation of each model we present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' This will provide the reader with a convenient starting point to investigate the availability of a known Lyapunov function to analytically prove the asymptotic behaviour of their system of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For the sake of brevity, we do not repeat the proofs to show that any of the functions we present are, indeed, Lyapunov function for the respective system of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' These proofs can be found in the papers we cite when introducing each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Consider a model with compartments X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Then, the DFE has coordinates Xi = 0 for all i ∈ I, where I is the set of the indexes of infectious compartments, and the EE, which we indicate with (X∗ 1, X∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' , X∗ n), has all positive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A vast majority of Lyapunov functions in epidemic modelling fall into one of the categories listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Linear combination of infectious compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The Lyapunov function for the DFE when R0 < 1 is of the form L = � i≥2 ciXi, for some constants ci ≥ 0 to be determined [6, 16, 18, 21, 32, 36, 39, 45, 49, 50, 59, 64, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' To prove convergence of the system to the DFE in this case it is often required the use of additional tools, such as LaSalle’s invariance principle, which we briefly recall at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Goh-Lotka-Volterra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The Lyapunov function for the EE when R0 > 1 is of the form L = � i ci(Xi − X∗ i ln Xi), for some constants ci ≥ 0 to be determined [2, 5, 6, 20, 27, 29, 32, 33, 45, 49, 52, 53, 59, 63, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' These functions are adapted from a first integral of the notorious Lotka- Volterra prey-predator system, and were popularized by Bean-San Goh in a series of paper [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The Lyapunov function for the EE when R0 > 1 is of the common form L = � i ci(Xi − X∗ i )2, for some constants ci ≥ 0 to be determined, or the composite form L = �� i Xi − X∗ i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Some examples can be found in [40, 41, 60, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Integral Lyapunov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Lyapunov functions given as integrals over the dynamics of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The integration interval often start at some EE value X∗ i and ends at the same Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' this construction is very convenient if uniqueness of the EE is guaranteed, but the exact values of the EE are hard (or impossible) to determine analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Integral Lyapunov functions are particularly useful when the model includes multiple stages of infection, and consequently the infectious period changes from an exponential distribution to a gamma distribution [8, 11, 18, 38, 58, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Integral Lyapunov functions, albeit in different forms, are widely used in models which incorporate explicit delay, such as systems of Delay Differential Equations (from here onwards, DDEs), and age- structured models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' However, these fall beyond the scope of this paper, and we will briefly comment on them in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A linear combination of the above, which often includes the Goh-Lotka- Volterra in at least a few of the compartments of the system [15, 27, 37, 47, 50, 53, 51, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 3 For some high-dimensional models, proving convergence to the EE might require addi- tional tools, such as the geometric approach used in [53, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Lastly, we must notice that not all compartmental models only exhibit convergence to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Some systems of autonomous ODEs may present stable or unstable limit cycles [9, 54, 68], homoclinic orbits [54] or even chaos [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In such cases, clearly, no global Lyapunov function may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In the remainder of this survey, we will present various models and the corresponding Lyapunov functions, covering all the cases listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2 Epidemic models In this section, we present various compartmental epidemic models with the corresponding Lyapunov function(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We present the models from the smallest to the largest, in terms of number of compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We refer to [1, 28] for a basic introduction on compartmental epidemic models, and to [55] for a detailed exemplification of Lyapunov theory in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We provide a schematic representation of the flows in most of the systems we present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Flow diagrams can be useful to provide a visual, intuitive interpretation of the parameters involved in each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Arrows between compartments indicate a change in the current state of individuals with respect to the ongoing epidemics, whereas arrows inward/outward the union of the compartments represent birth rate and death rate in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Often, these last two rates are considered to be equal, as this assumption allows the population to either remain constant or converge to a constant value, reducing the dimensionality of the system and (hopefully) its analytical complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' However, some models include additional disease-induced mortality, to increase realism when modelling severe infectious diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We uniform the notation throughout the various models we present in this survey as much as possible, and provide a brief description of each parameter the first time it is encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We remark that each variable is assumed to be non-negative, since it represents a fraction of the population, but the biologically relevant region varies depending on the specific model we are describing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Moreover, we illustrate the corresponding Lyapunov functions for 2D models, showcasing a selection of their power levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The same procedure can be easily adapted to 3D models, but the corresponding visualizations can be hard to interpret in a static image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='1 SIS The SIS model is characterized by the total absence of immunity after infection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' the recovery from infection is followed by an instantaneous return to the susceptible class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The ODEs system which describes this situation is dS dt = γI − βSI N , dI dt = βSI N − γI, (1) S I β SI N γI where β is the transmission rate and γ is the recovery rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 4 Notice that the population N = S + I is constant, thus we can normalize it to N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Moreover, since S + I = 1, we can reduce the system to one ODE which involves only infectious individuals dI dt = (β(1 − I) − γ)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' System (1) always admits the DFE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' E0 = (1, 0), and the EE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' E∗ = �γ β , β − γ β � , which exists if and only if β > γ (or equivalently if R0 = β/γ > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Notice that, if R0 < 1, then I is always decreasing in the biologically relevant interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A variation of model (1) can be obtained by adding demography to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' This is the example of [65], in which the authors consider a birth/immigration rate different from the natural death rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' moreover, they include an additional disease-induced death rate from infectious class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Thus, the population is not constant and the system of ODEs which describe the model is dS dt = Λ + γI − βSI N − µS, dI dt = βSI N − (δ + γ + µ)I, (2) S I β SI N γI Λ µS (δ + µ)I where Λ represents the birth/immigration rate, µ the natural death rate and δ the disease- induced mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' System (2) always admits the DFE, namely E0 = (S0, 0) := �Λ µ, 0 � , and the EE, namely E∗ = (S∗, I∗), where I∗ > 0 if and only if R0 = Λβ µ(µ + δ + γ) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [65], a Lyapunov function for the DFE is defined as V (S, I) := 1 2 (S − S0 + I)2 + 2µ + δ β I, (3) whereas the Lyapunov function for the EE is built using a combination of the quadratic and logarithmic functions V (S, I) := 1 2 (S − S∗ + I − I∗)2 + 2µ + δ β � I − I∗ − I∗ ln � I I∗ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (4) The authors also construct two more examples of Lyapunov functions for the EE, namely V (S, I) := 1 2(S − S∗)2 + µ + δ β � I − I∗ − I∗ ln � I I∗ �� , (5) and V (S, I) :=1 2 (S − S∗ + I − I∗)2 + S∗(δ + 2µ) 2γ � S − S∗ − S∗ ln � S S∗ �� + S∗(δ + 2µ) γ � I − I∗ − I∗ ln � I I∗ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (6) 5 Power levels of the functions (3), (4), (5) and (6) are visualized if Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' By definition of a Lyapunov functions, orbits of the corresponding system (2) evolve on decreasing power levels, and they tend to the corresponding equilibrium as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 1: Power levels of Lyapunov functions (3) (a), (4) (b), (5) (c), and (6) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Values of the parameters are Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='8, µ = 1, δ = 1, γ = 1 in all the figures, β = 1 in (a), so that R0 = 4/15 < 1, and β = 4 in (b), (c) and (d), so that R0 = 16/15 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We represent V (S, I) = k, with k ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='5, 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='5} in (a), k ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='2} in (b) and (c), and k ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='5} in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Black dots represent the globally stable equilibrium the system converges to, and correspond to V (S, I) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [66] the author found a simpler Lyapunov function for the DFE when R0 < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' V (I) = 1 2I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (7) However, this last Lyapunov function (7) only ensures that I → 0 as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' To complete 6 the proof of the converge of the system to the DFE, one needs in addiction to invoke LaSalle’s theorem [35] (see also [31, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='4]), as is indeed done in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Considering the importance of this theorem, especially when combined with the use of Lyapunov functions, we include its statement here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (LaSalle’s invariance principle) Let X′ = f(X) be a system of n ODEs defined on a positively invariant set Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Assume the existence of a function V ∈ C1(Ω, R) such that V ′(X) ≤ 0 for all X ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Let MV be the set of stationary points for V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' V ′(X) = 0 for all X ∈ MV , and let N be the largest invariant set of MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Then, every solution which starts in Ω approaches N as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In particular, this theorem implies that, if we can prove the approach of the disease to the manifold describing absence of infection and the uniqueness of the DFE, then the DFE is GAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='2 SIR/SIRS The SIR model is characterized by the total immunity after the infections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' recovered individuals can not become susceptible again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A classical example for this scenario is measles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The ODEs system which describes this situation is dS dt = −βSI N , dI dt = βSI N − γI, dR dt = γI, (8) S I R β SI N γI where β is the transmission rate and γ is the recovery rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' If we assume that recovered individuals eventually lose their immunity, we obtain the SIRS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Denoting by α the immunity loss rate, we obtain the following ODEs system dS dt = −βSI N + αR, dI dt = βSI N − γI, dR dt = γI − αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (9) S I R β SI N γI αR It is clear that, if α = 0, system (9) coincides with system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' These models admit only the DFE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' in order to have an EE, we need to add the demog- raphy to model (8) or (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [65], the authors consider the following ODEs system 7 dS dt = Λ − βSI N − µS + αR, dI dt = βSI N − (γ + δ + µ)I, dR dt = γI − (α + µ)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' (10) S I R β SI N γI αR Λ µS µR (δ + µ)I System (10) admits the DFE, E0 = (S0, 0, 0), and the EE, E∗ = (S∗, I∗, R∗), which exists if and only if R0 = βΛ µ(µ + γ + δ) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [65], the Lyapunov function for the DFE is defined as follows V (S, I, R) := 1 2 (S − S0 + I + R)2 + 2µ + δ β I + 2µ + δ 2γ R2, whereas the Lyapunov function for the EE is the combination of the composite quadratic, common quadratic and logarithmic functions as follows V (S, I, R) :=1 2 (S − S∗ + I − I∗ + R − R∗)2 + 2µ + δ β � I − I∗ − I∗ ln � I I∗ �� + 2µ + δ 2γ (R − R∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The authors also present other Lyapunov functions for SIR/SIRS models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' in particular, they also cite [3, 46] in which some variations of system (10) are showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Other Lyapunov functions for SIR/SIRS epidemic models are in [55], in which the authors use a graph- theoretic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [66], the author proved that the quadratic Lyapunov function (7) of the SIS model applies to the SIR and the SIRS, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='3 SEIR/SEIS/SEIRS In [32], the authors study both SEIR and SEIS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Many real world examples present a phase of exposition to the disease, between susceptibility and infectiousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The models presented thus far, albeit simpler to study, are unable to replicate this mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The authors first analyze a SEIR model with demography and constant population, in which the disease is transmitted both horizontally and vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Individuals infected verti- cally pass first in the exposed compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The ODEs system which describe the model is dS dt =µ − βSI − pµI − qµE − µS, dE dt =βSI + pµI − θE − µE + qµE, dI dt =θE − (δ + µ)I, (11) S E I βSI θE µ(1 − pI − qE) µS µ(pI + qE) µE (δ + µ)I and R = 1 − S − E − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The vertical transmission of the disease is represented by the probabilities p and q of being born directly in the Exposed compartment, rather than in the Susceptible one, and is represented by the inward arrow in compartment E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 8 The authors first provide an equivalent system, performing the substitution (S, E, I) −→ (P, E, I), where P := S + pµ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' They then proceed to prove the GAS of the EE, using the following Lyapunov function V (P, E, I) :=(P − P ∗ ln P) + θ + µ θ + µ − qµ(E − E∗ ln E) + θ + µ θ + µ − qµ(I − I∗ ln I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Later, the authors analyze a situation in which the recovery does not provide immunity, namely the SEIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' They also assume that a fraction r of offspring of the infective hosts is born directly into the infective compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In this case, the ODEs system changes accordingly describe the model is dS dt =µ − βSI + (δ − pµ − rµ)I − qµE − µS, dE dt =βSI + pµI − (θ + µ − qµ)E, dI dt =θE − (δ + µ − µr)I, (12) and S + E + I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Notice that, due to the population remaining constant in system (12), one could in principle reduce its dimensionality and consider it as a planar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The authors prove the GAS of the EE using the following Lyapunov function V (S, E, I) :=(S − S∗ ln S) + µ1 − S∗ βI∗S∗ (E − E∗ ln E) + µ1 − S∗ θE∗ � 1 + pρ0 µ β � (I − I∗ ln I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A natural extension to these models is the SEIRS [22, 64], in which one can combine the existence of an immune compartment and the loss of immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' It is described by the following system of ODEs dS dt = − βg(I)S + µ − µS + αR, dE dt =βg(I)S − (θ + µ)E, dI dt =θE − (γ + µ)I, dR dt =γI − (α + µ)R, (13) S E I R βg(I)S γI αR θE µE µ µS µR µI where g ∈ C3(0, 1], g(0) = 0 (meaning, in absence of infectious individuals, the disease does not spread) and g(I) > 0 for I > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The classical choice is g(I) = I, as in systems (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Assuming moreover lim I→0+ g(I) I = c ∈ [0, +∞), 9 the authors of [22] derive R0 = cβθ (θ + µ)(γ + µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' They then prove GAS of the DFE of system (13) through the use of the following linear Lyapunov function V (E, I) = E + θ + µ θ I, whereas the GAS of the EE is proved with a more complex geometrical method in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='4 SAIR/SAIRS One of the main challenges of the Covid-19 pandemic was the presence of asymptomatic individuals spreading the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Such individuals must clearly be somehow distinguished from symptomatic infectious individuals, as they are likely to behave like a susceptible individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Even though their viral load, and hence infectiousness, might be smaller, they are more likely to get in close contact with susceptible individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [53], the authors consider a SAIRS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The main difference between this kind of models and the SEIR is that both asymptomatic and symptomatic hosts may infect susceptible individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The immunity is not permanent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' recovered individuals will become susceptible again after a certain period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Moreover, vaccination are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The ODEs system which describe this model is dS dt = µ − � βAA + βII � S − (µ + ν)S + γR, dA dt = � βAA + βII � S − (α + δA + µ)A, dI dt = αA − (δI + µ)I, dR dt = δAA + δII + νS − (γ + µ)R, S A R I µ µS (βAA + βII)S δII γR δAA µA αA νS µI µR The global stability analysis of the EE has been performed for two variations of the original model, described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The first model analyzed is the SAIR model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' the case in which recovery from the disease grants permanent immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In this case, the corresponding Lyapunov function is the combination of the Lokta-Volterra Lyapunov functions for S, A and I V (S, A, I) :=c1S∗ � S S∗ − 1 − ln � S S∗ �� + c2A∗ � A A∗ − 1 − ln � A A∗ �� + I∗ � I I∗ − 1 − ln � I I∗ �� , where c1, c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The second model is the SAIRS model, with homogeneous disease transmission and recovery among A and I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' βA = βI and δA = δI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In this case, it is possible to sum 10 equations for A and I, defining M := A + I, reducing the dimensionality of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Thus, the Lyapunov function can be written as the combination of the square function and the Lokta-Volterra as follows V (S, M) := 1 2(S − S∗)2 + w � M − M∗ − M∗ ln � M M∗ �� , where w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The global stability in the most general case is proved similarly to [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='5 More exotic compartmental models The aforementioned models are some of the most commonly used in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In order to capture additional disease-specific nuances, these model can be modified or extended by adding new compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Some diseases, for example, present different stages of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In this case, an infected individual can progress between two or more stages before recovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [18], the authors perform the global stability analysis via an integral Lyapunov function of a general class of multistage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In their model, infectious individual can move both forward and backward on the chain of stages, in order to incorporate both a natural disease progression and the amelioration due to the effects of treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The system of ODEs which describes the model is dS dt = θ(S) − f(N) n � j=1 gj(S, Ij), dI1 dt = f(N) n � j=1 gj(S, Ij) + n � j=1 φ1,j(Ij) − n+1 � j=1 φj,1(I1) − ζ1(I1), dIi dt = n � j=1 φi,j(Ij) − n+1 � j=1 φj,i(Ii) − ζi(Ii), i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=', n, where θ(S) is the growth function, f(N) �n j=1 gj(S, Ij) is the incidence term, ζi(Ii), 1 ≤ i ≤ n, denote the removal rates of the Ii compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Moreover, for any i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' , n, the functions φi,j(Ij) represent the rate of the disease progression if i > j and the amelioration if i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The corresponding Lyapunov function for the DFE is linear in the disease compartments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' V (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=', In) = n � i=1 ciIi, where c1 = R0 and ci ≥ 0 for all i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For the global stability of the EE the authors made some assumptions on the aforementioned functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In particular, they consider the following integral Lyapunov function V (S, I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' , In) = τ � S S∗ Φ(ξ) − Φ(S∗) Φ(ξ) dξ + n � i=1 τi � Ii I∗ i ψi(ξ) − ψi(I∗ i ) ψi(ξ) dξ, 11 where τ, τi > 0, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For a more in-depth explanation on the functions Φ(·) and ψi(·) we refer to [18, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Diseases which present multiple virus strains, due to the existence of different serotypes of the virus or due to a mutation of the original disease, may need to be modelled differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Dengue, tuberculosis and various sexually transmitted diseases are caused by more than one strain of a pathogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Influenza type A viruses mutate constantly: an infection with one of its strains gives permanent immunity against that specific strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' However, the so called “antigenic drift” produces new virus strains, thus the hosts only acquire partial immunity, or no immunity at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Modelling these types of diseases requires the inclusion of cross-protective effects, in which the immunity acquired towards one strain offers partial protection towards another strain based on their antigenic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [6], the authors consider an n strain model, both without immunity and with immunity for all the strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Moreover, they analyze an MSIR model, in which the M compartment represents the proportion of newborns who possess temporary passive immunity due to protection from maternal antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For all the three model, the authors use a linear Lyapunov function to prove the global stability of the DFE and a logarithmic Lyapunov function to prove the global stability of the EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Other compartmental models include e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' control strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For new ongoing epidemics, the most immediate strategy is including quarantine and isolation of infectious individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For well-known epidemics for which a vaccination is available, it is useful to incorporate a vaccinated individuals compartment V to keep track of the two possible immunities, disease and vaccine induced, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Usually, vaccination does not confer permanent immu- nity, and after a certain disease-dependent period individuals become susceptible again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' An example is [50], in which the authors analyze a SIRV epidemic model with non-linear inci- dence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The global stability of the DFE is proved using as linear Lyapunov function the infectious compatment I and the global stability of the EE, instead, using a combination of a quadratic function in S and a logarithmic function in the compartments I and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 3 Conclusion In this survey, we presented the most widely used Lyapunov functions in the field of epi- demic compartmental models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' We focused on systems expressed as autonomous systems of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' These models allow for various interesting generalizations, of which we provide a non-comprehensive list below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' One extension of the classic compartmental epidemic models is the so-called multi-group approach, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' [34, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' These models describe n communities, interacting with each other, and whose internal evolution follows a standard compartmental model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' A first example of such a model is presented in [10], in which the authors consider a n groups SIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In order to prove the GAS of the EE, they use a results on Metzler matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In [55], the authors consider a heterogeneous SIS disease model, for which they provide Lyapunov functions both for the DFE and for the EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' For the latter, they use a complex graph-teoretic method, for the details of which we refer to the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Global stability of EE via Lyapunov function for multi-group generalization can be found also for the SIR [19], SIRS [48], SEIR [17] and SAIR/SAIRS model [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Notice that, due to the complexity of the models, some of them require additional technical assumptions to prove the global stability of the endemic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 12 Other classes of models include interactions between human and vector population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' animals which transmit the disease to humans, or with the pathogens, such as viruses or bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In both cases, authors often include a compartmental structure for the non-human population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Some examples of vector-host models are shown in [59, 62, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Another example can be found in [40], in which a SIR-B compartmental model is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Here the “B” denotes the concentration of the pathogen in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' All the models discussed thus far are described by only autonomous systems of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' However, in order to increase realism, it is possible to use non-autonomous systems to de- scribe the spread of an infectious disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' This is the case of systems in which some param- eters change in time [42, 56], to describe seasonal changes, or in which the state variables depend on the previous state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' the model includes a time delay [4, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In these cases, it is still possible to find Lyapunov functions to prove the global stability of the equilibria using other techniques, described for example in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Another popular option is to explicitly include delay in the system, such as in [4, 23, 25, 26, 43, 63, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' In the latter the authors perform the global stability analysis of a SEIQR model, in which Q denotes the quarantined individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' They explicitly include a latent period for the infection, transforming two of the ODEs in DDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The corresponding Lyapunov function includes the integration over an interval whose size is precisely the latent period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Lastly, a widely adopted strategy is to explicitly include the “time since infection” [7, 24, 44, 71, 72] in age-structured models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' This allows to explicitly take into account time heterogeneity in the spread of an infectious disease in a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' These last cases we mentioned are outside of the scope of this project, and we leave them as inspiration for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' The authors are grateful to the organizers of the conference 100 Years Unione Matematica Italiana - 800 Years Università di Padova, which made their scientific cooperation possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Moreover, they acknowledge Politecnico di Milano, Polish Academy of Sciences, Inria and University of Trento for supporting their research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Biosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=', 12(4):859, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQf8C4w/content/2301.11947v1.pdf'} diff --git a/2tFAT4oBgHgl3EQfDhzt/content/tmp_files/2301.08417v1.pdf.txt b/2tFAT4oBgHgl3EQfDhzt/content/tmp_files/2301.08417v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..817b09604ce6e3a3f23d537a4e0eafb81f8e59de --- /dev/null +++ b/2tFAT4oBgHgl3EQfDhzt/content/tmp_files/2301.08417v1.pdf.txt @@ -0,0 +1,727 @@ +Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback +Suppression of laser beam’s polarization and intensity fluctuation via a +Mach-Zehnder interferometer with proper feedback +Xiaokai Hou,1 Shuo Liu,1, a) Xin Wang,1 Feifei Lu,1 Jun He,1, 2 and Junmin Wang1, 2, b) +1)State Key Laboratory of Quantum Optics and Quantum Optics Devices, and Institute of Opto-Electronics, Shanxi University, +Tai Yuan 030006, Shanxi Province, China +2)Collaborative Innovation Center of Extreme Optics, Shanxi University, Tai Yuan 030006, Shanxi Province, +China +(Dated: 23 January 2023) +Long ground-Rydberg coherence lifetime is interesting for implementing high-fidelity quantum logic gates, many-body +physics, and other quantum information protocols. But, the potential well formed by a conventional far-off-resonance +red-detuned optical-dipole trap that is attractive for ground-state cold atoms is usually repulsive for Rydberg atoms, +which will result in the rapid loss of atoms and low repetition rate of the experimental sequence. Moreover, the coher- +ence time will be sharply shortened due to the residual thermal motion of cold atoms. These issues can be addressed +by an one-dimensional magic lattice trap and it can form a deeper potential trap than the traveling wave optical dipole +trap when the output power is limited. And these common techniques for atomic confinement generally have certain +requirement on the polarization and intensity stability of the laser. Here, we demonstrated a method to suppress both +the polarization drift and power fluctuation only based on the phase management of the Mach-Zehnder interferometer +for one-dimensional magic lattice trap. With the combination of three wave plates and the interferometer, we used the +instrument to collect data in the time domain, analyzed the fluctuation of laser intensity, and calculated the noise power +spectral density. We found that the total intensity fluctuation composed of laser power fluctuation and polarization drift +was significantly suppressed, and the noise power spectral density after closed-loop locking with typical bandwidth +1-3000 Hz was significantly lower than that under the free running of the laser system. Typically, at 1000 Hz, the +noise power spectral density after locking was about 10 dB lower than that when A Master Oscillator Power Amplifier +(MOPA) system free running. The intensity-polarization control technique provides potential applications for atomic +confinement protocols that demand for fixed polarization and intensity +I. +INTRODUCTION +For various atomic manipulation experiments, such as sin- +gle photon source1−5, quantum dynamics based on Rydberg +states 6−10 and electric field detection based on atoms 11−13, +strong confinement optical dipole trap (ODT) of atoms is usu- +ally employed. In these applications,high power laser with +fixed polarization and relatively stabled intensity normally is +used to confine atoms. Common experimental setup for the +laser power stabilization were based on the active feedback +loop which used acousto-optic modulator (AOM) 14−17 or +electro-optic modulator (EOM) 18 as the actuator. In 2020, +AOM and EOM were combined to broaden the bandwidth of +laser intensity noise stabilization to 1MHz by Ni et. al.19. At +present, the feedback loop based on AOM has some disadvan- +tages. For example, bragg diffraction of AOM will seriously +affect the spot quality of first-order diffraction light, and the +power utilization of the system will be limited by the diffrac- +tion efficiency of AOM. The common electro-optic intensity +modulator (EOIM) with input and output tailed fiber is effi- +cient, but not suitable for high-power applications. Moreover, +the schemes mentioned above can observably suppress the +power fluctuation of laser beam, but the reduction for the drift +a)Present Address: Key Laboratory of Laser & Infrared System of Ministry +of Education, Shandong University, Qing Dao 266000, Shandong Province, +China. +b)corresponding author. E-mail: wwjjmm@sxu.edu.cn ORCID : 0000-0001- +8055-000X +of laser’s polarization is still not be effectively achieved . Here +we demonstrate an experimental scheme based on the Mach- +Zehnder interferometer (MZI) for actively suppress both the +fluctuation of power and polarization of laser beam. By prop- +erly manipulated phase difference between two paths, the out- +put fraction of MZI account for the majority laser power while +its intensity fluctuation in the time domain has been reduced +dozens of times compared with the free-running case, and the +noise power spectral density (NPSD) has been decreased in +the range of 1-3000 Hz in the frequency domain. Such a sta- +ble system can certainly meet the needs of various applica- +tions, such as experiments where the lifetime of cold atoms is +highly desirable. +II. +THEORETICAL BACKGROUND +1. +Magic optical dipole trap for cesium 6S1/2 ground state +and 84P3/2 Rydberg state +Recently, a new experimental scheme which used interfer- +ometer as the actuator of the feedback loop has been proposed +20. Considering the light intensity requirement of the ODT, +the MZI can satisfy the power requirement of ODT without af- +fecting spot quality of output light, therefore the experimental +setups of constructing the blue-detuning optical trap reported +by Yelin et. al 21 and Isenhower et. al 22 both concentrate on +the MZI. The intensity of the output laser mainly depends on +the phase difference between two arms of the MZI, therefore it +can be used as a power stabilizer in some experiments 23. Due +arXiv:2301.08417v1 [physics.optics] 20 Jan 2023 + +Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback +2 +to the particularity of the output fraction of MZI, the com- +bination of interferometer and polarizer can realize the fixed +polarization, high proportion output and high intensity stabil- +ity. It is obviously useful for the experiment of optical trap. +The potential of ODT U can be expressed as: +U = − α +2ε0c +2P +πω2 +0 +(1) +Where α is the induced polarizability of the target state, ε0 +is the permittivity of vacuum, c is the speed of light, P is the +intensity of laser, ω0 is the radius of the spot at the focal point +after the laser is focused by a lens . As shown in the Eq. (1), +if the power of the 1879 nm laser is fluctuant, the resulting +trap depth will be changed. Thus, the lifetime of the trapped +atom will be severely affected by the presence of the heating +mechanism5,17,24. +FIG. 1. Diagram of the light shift induced by the ODT and MODT. +The intensity of laser which is intensly focused is still Gaussian, +and the closer to the center of beam, the stronger the intensity of +laser. The resulting trap depth or light shift is spatially dependent +(a) The ODT is attractive for ground states, but usually repulsive +for highly-excited Rydberg states because almost all strong dipole +transitions connected Rydberg state and the lower states have longer +wavelength than that of ODT laser. (b) The direct single-photon ex- +citation scheme from cesium |g⟩=|6S1/2⟩ to |r⟩=|84P3/2⟩ coupled by +a 319 nm ultraviolet laser. A 1879.43 nm laser is also tuned to the +blue side of the |r⟩ ⇐⇒ |a⟩=|7D5/2⟩ auxiliary transition to equalize +the trapping potential depth of the |g⟩ and |r⟩ state, which is so called +magic ODT (MODT). +In most of the experiments of cold atoms involving confine- +ment of ground-state atoms in an ODT and Rydberg excita- +tion, cold atomic sample is prepared in an ODT to hold them +in a fixed position in a significantly long time. The poten- +tial formed by a conventional far off-resonance red-detuned +ODT is attractive for the ground-state atoms, but usually re- +pulsive for highly-excited Rydberg atoms, leading that Ryd- +berg atoms normally cannot be confined in the conventional +ODT (Fig. 1(a)). Therefore, in the follow-up experiments, we +will face the following two problems: (1) if switching off the +ODT during Rydberg excitation and coherent manipulation, it +will result in atomic dephasing due to the thermal diffusion +of the atoms and the extremely low repetition rate of the ex- +perimental sequence; (2) if the ODT remains operation, it may +cause a low Rydberg excitation efficiency of atoms as the tran- +sition frequency is spatially position-dependent on the excita- +tion laser. The solution is to find an ODT such that the ground- +state atoms and the desired highly-excited Rydberg atoms can +experience the same potential, that is, the potential generated +by the ODT is a potential well for both the ground-state atoms +and the desired highly-excited Rydberg atoms, and is attrac- +tive to atoms in both states. So, the above-mentioned aspects +(1) and (2) can be solved. In Fig.1(b), the direct single-photon +excitation scheme from cesium |g⟩=|6S1/2⟩ to |r⟩=|84P3/2⟩ +coupled by a 319 nm ultraviolet laser. A 1879.43 nm laser +is also tuned to the blue side of the |r⟩ ⇐⇒ |a⟩=|7D5/2⟩ aux- +iliary transition to equalize the trapping potential depth of the +|g⟩ and |r⟩ state. The specific calculation process is not de- +scribed here. For details, please refer to the reference 25,26. +2. +Theoretical analysis of MZI +It is obvious that the MODT is not enough to meet the +need of extremely long coherence time in subsequent experi- +ments. The cold atoms trapped in the MODT still have resid- +ual thermal motion, which causes violent collisions that heat +the atoms and cause them to escape from the trap. We will fur- +ther construct one-dimensional magic lattice trap (1D-MLT), +and combine the advantages of lattice and magic conditions, +so as to prolong the coherence time of the ground-Rydberg +state of cold atoms. Of course, the 1D-MLT also needs to +suppress its power fluctuation. Because the power of the laser +used in the 1D-MLT fluctuates in the time domain, will di- +rectly shorten the coherence lifetime of the cold atom. There- +fore, we use the MZI to suppress the power fluctuation. +As shown in Fig. 2 (a), Iout1 and Iout2 are the intensity of +two output paths of the interferometer respectively; R1, T1, R2, +T2 are the reflectivity and transmittance of input and output +beam splitters plate respectively. The two output channels of +the interferometer can be expressed as Eq. (2) and (3): +Iout1 = R2 +1R2 +2 +T 2 +1 T 2 +2 +2R1R2T1T2cos(2∆L +λ ++π) +(2) +Iout2 = R2 +1R2 +2 +T 2 +1 T 2 +2 +2R1R2T1T2cos(2∆L +λ ) +(3) +Therefore, the laser intensity output of the interferometer can +be controlled by adjusting the driving voltage of PZT due to +the correlation between the output transmittance I and optical +path difference ∆L. In Fig. 2 (b), the interference fringes +generated by splitters with different splitter ratio is simulated +and analyzed by Mathematica. The splitter ratio shown by the +first line of Fig. 2(b) is 90/10, the second is 70/30, the third is +60/40, and the last is 50/50. +III. +EXPERIMENTAL SETUP +The laser intensity stabilization setup is shown in Fig. 3. A +MOPA system consists of a 1879-nm butterfly packaged laser +diode and a Thulium Doped Fiber Amplifier (TmDFA) which +has maximum output ∼ 3 W. With a free space polarization + +I(r +84P +3/2 +1879.43nm +319nm +ODT +1879.43nm +g +(a) +6S1/2 +(b)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback +3 +FIG. 2. Diagram of MZI and interference fringes of two channels of +the MZI are simulated and analyzed theoretically. (a) MZI consists +of two beam splitter plates (BS1 and BS2) and two high-reflectivity +mirrors (M1 and M2). Iin is the intensity of the incident light field, +Iout1 and Iout2 are the intensity of the outgoing light field at BS2. (b) +Normalized signal as a function of difference of optical path ∆L for +different splitter ratio.This ratio is both R1/T1 and R2/T2, because +the BS1 and BS2 that are used in the MZI are same. The solid red +and black lines represent the interference fringes of the two output +channels of MZI, respectively. +controller based on three waveplates ( λ/4, λ/2 and λ/4 ), +polarization fluctuation of 1879 nm beam is suppressed ini- +tially. The laser is injected into a MZI which is constructed by +a 50/50 beam splitter plate (BS1) that divides the incident light +into two beams with equal intensity and the different phase, a +high-reflectivity mirror (M1) that reflects one beam, a mirror +(M2) attached to a PZT that emits the other, and a beam split- +ter plate (BS2) that the two beams are finally combined. The +interferometer has two output channels and each channel can +be used for dynamic feedback to make the system more sta- +ble, and the output of this channel can then be used for sub- +sequent experiments. The photodetector (PD1) is mounted +behind a glass slice (GS1) of 1879 nm for sampling a little +fraction of light for in-loop feedback. The DC voltage signal +output by the PD1 is injected into Proportional Integral Dif- +ferential (PID) amplifier after passing through a low-pass filter +(LPF). The input signal of PID controller is subtracted from +the PID Set Point, which is an artificially set reference DC +voltage. The output signal of PID, that is the real-time differ- +ence between the detector signal and the reference DC volt- +age is added with the scanning signal (triangular wave) and +amplified by the high voltage (HV) amplifier as the driving +voltage of the PZT. The output power of interferometer can +therefore be controlled by manipulating the driving voltage of +PZT, and we expect that both power and polarization fluctua- +tion for 1879 nm laser are suppressed. And another photode- +tector (PD2) is mounted in order to independently monitor the +intensity stability of the output linear polarization laser. The +output signal of PD2 is then injected into the Data Acquisition +System (Keithley, DAQ-6510) in order to analyze and moni- +tor the intensity fluctuation of the laser in the time domain and +calculate the NPSD based on the measured optical power fluc- +tuation data. Undoubtedly, the little fraction of the far-infrared +laser is reflected by glass slice (GS2) and received by the PD2 +and the majority of laser is transmitted and focused in a ce- +sium magneto-optical trap (Cs-MOT) for the construction of +the ODT. +FIG. 3. Experimental setup for intensity stabilization system. The +dynamic stability of laser intensity of 1879 nm MOPA system is re- +alized by MZI, and the fluctuation of laser intensity is monitored +and analyzed in time domain and frequency domain.λ/2: half-wave +plate; λ/4: quarter-wave plate; PBS: polarization beam splitting +cube; BS: beam splitting plate; GS: glass slice; M1/M2: high- +reflectivity mirror; PD: photodetector; LPF: low-pass filter; PID: +Proportional Integral Differential amplifier; HVA: high voltage am- +plifier. +IV. +EXPERIMENTAL RESULTS AND DISCUSSION +Fig. 4 shows, the interference fringes obtained by scanning +triangular waves with 50/50 beam splitter ratio in the experi- +ment, in which the interference contrast is 95%. In theoretical +simulation, an interference fringe with an interference con- +trast of 99.9% can be obtained by using a 50/50 beam splitter +plate, but the best interference contrast is not achieved in ex- +periment, probably due to the following two reasons: first, the +spatial mode of the two lasers is not exactly same; second, the +polarization of the two lasers may be slightly different. +Considering the requirement of constructing dipole trap +with this laser source, the polarization of 1879 nm laser should +be fixed, so PBS is usually inserted in the light path to fixed +the polarization of light. Even though the scheme is effective, +an inevitable defect exists in this scheme is that the polariza- +tion fluctuation of light will couple with the intensity fluctu- +ation through this polarization element. As the measurement +of which the intensity for 1879 nm laser after a PBS, although +the power fluctuation of 1879 nm TmDFA itself is not obvi- +ous, the intensity fluctuation behind the PBS becomes obvious +and the results is shown in Fig. 5(a). We monitor the laser in- +tensity for about 30 minutes in the time domain, with a large +fluctuation of about ±14.2%. The huge intensity fluctuation + +90/10 +70/30 +60/40 +50/50LSuppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback +4 +FIG. 4. Interference fringe of the MZI. In the experiment, a 50/50 +beam splitter plate is used, and the PZT is driven by scanning tri- +angular wave, so that the phase difference between the two arms is +generated, then the interference fringes are generated. +will significantly affect the power utilization of the stable sys- +tem. To maximize the power utilization, three wave plates +are used to suppressed the power fluctuation initially. After +proper adjustment, measurement result of laser intensity fluc- +tuation after PBS is shown in Fig. 5 (b). Fluctuation of laser +polarization has been reduced significantly. Then the initial- +stabled laser has been injected in the combine system of MZI +and another PBS, here the transmittance of the interferometer +is locked up to 90% in order to improve the power utilization. +Then the intensity fluctuation probed by the out-of-loop detec- +tor PD2 is shown in Fig. 6. As shown below, the intensity fluc- +tuation of output linear polarized laser is reduced to ±0.3%, +that is much better than the fluctuation of direct TmDFA-PBS +output. At this stability, both fluctuation of laser power and +polarization will no longer have a significant influence on the +parameter of dipole trap. +As shown in Table 1, for the 1879 nm 1D-MLT, if the laser +is focused through a lens to ∼ 20 µm. and the incident laser +power at the cold atom is about 1.5 W, so the maximum depth +of the 1D-MLT is −1000 µK and the typical trap depth fluc- +tuation is ±140 µK. When the laser power decreases after the +initial suppression of the wave plate group or the closed-loop +locking of the MZI, the corresponding typical trap depth is +about −800 µK and −700 µK respectively. And the effec- +tive temperature of the cold atoms which are transferred from +MOT to 1D-MLT will be slightly higher, about 100 µK, but +the decrease of trap depth caused by the suppression of power +fluctuation will not affect the capture of the cold atoms. How- +ever, the residual fluctuation of laser power still exists, which +will lead to the typical trap depth fluctuation of ±45 µK and +±2 µK respectively. +The collected time-domain voltage signals are used to cal- +culate the NPSD. As shown in Fig. 7, the horizontal range +is determined by the sampling rate. In the experiment, we +selected sampling rate of 10000 Hz according to the actual +situation, so the horizontal axis in Fig. 7 ranges from 1 to +5000Hz. In addition, we believe that the feedback bandwidth +of the system should be at the level of kilohertz due to the +limitation of PZT in the MZI. Therefore, the sampling rate +can fully meet the requirement of representing the feedback +bandwidth of the system. +The NPSD after closed-loop locking from 1-3000Hz is sig- +nificantly lower than that under the free running of the MOPA +system. It can be proved that the MZI plays an obvious role in +the power stability of the system. In order to further broaden +the feedback bandwidth and improve the inhibitory effect, we +assume that the arm length of the MZI is L and the angular fre- +quency of the laser is ω0, then the distance of the laser going +through the MZI is L and the phase shift generated is27 +Φ0(t) = ω0t = ω0 +L +c +(4) +Φ0 is a constant, and the magnitude is proportional to L . +When the PZT is scanned, we introduce to characterize small +changes in phase. For simplicity, we assume that a sine wave +is used to scan the PZT, and the amplitude of the sine wave is +h0 and the angular frequency is ωs, so the sine wave can be +expressed as +h(t) = h0cos(ωst) +(5) +So, the phase shift of the entire system can be written as +Φ = Φ0(t)+δφ += ω0L +c ++ ω0 +2 +� t +t− L +c +h0cos(ωst)dt += ω0L +c ++ h0 +2 +ω0 +ωs +� +sin(ωs +L +c )−sin[ωs(t − L +c )] +� += ω0L +c ++h0 +ω0 +ωs +sin(ωs +L +2c)cos[ωs +2 (t − L +c )] +(6) +because L +2c ≪ 1 +h0 +ω0 +ωs +sin(ωs +L +2c) = h0ω0 +2 +L +c +(7) +δφ ∼ h0ω0 +2 +L +c +(8) +As shown in Eq. (8), if the arm length L of the MZI is +increased, δφ of the system can be increased. Thus, the de- +tection sensitivity of the system can be improved and the de- +tection effect of the MZI for phase can be better. Increasing +the arm length of the MZI will cause extra noise due to the +insufficient stability of the system. +However, such noise can be solved through the isolation +platform and system temperature control. We can add F-P +cavity on the two arms of the MZI. F-P cavity can fold up the +optical path, greatly increase the distance of light in the MZI, +and do not need to occupy a large area. + +0.35 +0.28 +Locked point +Voltage (V) +0.21 +0.14 +0.07 +0.00 +0.00 +0.02 +0.04 +0.06 +0.08 +Time (s)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback +5 +FIG. 5. (a) The power fluctuation of free running 1879 nm MOPA system . Through 30 mins of measurement, the power fluctuation is roughly +±14.2%. The inset is zoomed in on the vertical axis to 2.00∼3.00 W, and shows the intensity fluctuation in 30 mins. (b) The power fluctuation +after three wave plates. We thought that the polarization fluctuation is initially suppressed by these plates. Similarly, after 30 minutes of +measurement, the intensity fluctuation is approximately ±5.7%. And, the vertical axis range of the inset becomes 1.75∼2.25 W, the range of +horizontal axis is still 0∼30 mins. +Category +PODT (mW) +∆P(mW) +Gaussian radius after focused (µm) Udip(µK) ∆Udip(µK) +MOPA free running +1500 +±213.0 (±14.2%) +20 +−1000 +±140 +With wave plate group +1200 +±68.4 (±5.7%) +20 +−800 +±45 +After MZI is locked +1100 +±3.3 (±0.3%) +20 +−700 +±2 +TABLE I. The typical maximum trap depth and fluctuation of 1879.43nm 1D-MLT for cesium atoms under different power fluctuations are +calculated. +FIG. 6. The intensity fluctuation of 1879 nm laser on the bright fringe +of the MZI. By inter-of-loop locking, the phase difference between +the two arms is dynamically compensated , and the power fluctuation +is significantly suppressed , through 30 mins of measurement, the +power fluctuation is roughly ±0.3%. The vertical axis of the inset +has been enlarged with a range of 1.85∼1.88 W, and shows the 30- +min measurement. +FIG. 7. Intensity noise of 1879 nm laser as a function of analyze +frequency. (a): The solid black line represents the NPSD when the +1879nm laser system is running freely without passing through the +wave plate group. (b): The solid blue line represents the NPSD of +the 1879nm laser system after closed-loop locking by the MZI. + +3.00 +3.00 +2.50 +2.50 F +Power fluctuation with wave plates group : ±5.7% +Power fluctuaticn when MOPA free running : ±14.2% +2.00 +2.00 +(W) +M +3.00 +2.25 +Power ( +Power ( +1.50 +1.50 +Power (w) +() +Power ( +2.50 +2.00 +1.00 +1.00 +0.50 +0.50 +2.00 +(a) +1.75 , +(b) +15 +20 +25 +30 +20 +25 +30 +T ime (min) +Time (min) +0.00 +0.00 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +0 +Time (min) +Time (min)3.00 +2.50 +Power fluctuation after Mzl is locked : ±0.3% +Powewr (W) +2.00 +1.88 +1.50 +1.00 +1.86 +0.50 +1.85, +10 +15 +20 +25 +30 +Time (min) +0.00 +0 +5 +10 +15 +20 +25 +30 +Time (min)100 +3000Hz +10 +NPSD (W/NHz) +10 +(b) +10-8 +MOPA free runming +After MzI is locked +10-9 +100 +101 +102 +103 +Frequency (Hz)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback +6 +V. +CONCLUSIONS +In summary, we have demonstrated the reduction of the +power and polarization fluctuation for 1879 nm laser based +on the cooperation of three wave plates and a MZI. The in- +tensity fluctuation ∼ ±14.2% after the combination of MOPA +system and PBS is reduced to ∼ ±0.3% with locked MZI. +And after MZI is locked, the NPSD is lower than that under +free running in the range of 1-3000 Hz. Typically, at 1000 Hz, +the NPSD after MZI is locked is about 10 dB lower than that +when MOPA free running. The system can not only withstand +high power injecting laser, but also can stabilize both power +fluctuation and polarization fluctuation without affecting the +quality of light beam for the low-loss output light. The laser +power utilizing efficiency can be further improved by improv- +ing the transmittance of locked interferometer or improving +the interference visibility. +It is expected that Rydberg atoms can have long coherence +lifetime in subsequent experiments involving Rydberg dressed +ground state. On one hand, we can use the 1879-nm MOPA +system to implement a 1D-MLT, which can both eliminate the +position-dependent light shift to capture Rydberg-state atoms +in optical tweezer like the ground-state atoms and attenuate +collisions between cold atoms caused by residual thermal mo- +tion to prolong the coherence time of the Rydberg atoms. On +the other hand, we propose an upgraded interferometer, that +is, adding a F-P cavity to each arm of the interferometer, and +using the reflection of beam in the cavity, the arm length can +be extended at least dozens of times, to improve the phase +measurement sensitivity of the interferometer and improve the +power stability. +FUNDING +This research was financially funded by the National Key R +& D Program of China (2021YFA1402002), the National Nat- +ural Science Foundation of China (11974226, and 61875111). +REFERENCES +1 M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. +Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic, M. 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Progress in Astronomy, 32, +348, (2014).(In Chinese) + diff --git a/2tFAT4oBgHgl3EQfDhzt/content/tmp_files/load_file.txt b/2tFAT4oBgHgl3EQfDhzt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1bd79b36a36a97cf5ef836fcae1db8d691426dc --- /dev/null +++ b/2tFAT4oBgHgl3EQfDhzt/content/tmp_files/load_file.txt @@ -0,0 +1,617 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf,len=616 +page_content='Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback Xiaokai Hou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='1 Shuo Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' a) Xin Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='1 Feifei Lu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='1 Jun He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2 and Junmin Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' b) 1)State Key Laboratory of Quantum Optics and Quantum Optics Devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' and Institute of Opto-Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Shanxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Tai Yuan 030006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Shanxi Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' China 2)Collaborative Innovation Center of Extreme Optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Shanxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Tai Yuan 030006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Shanxi Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' China (Dated: 23 January 2023) Long ground-Rydberg coherence lifetime is interesting for implementing high-fidelity quantum logic gates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' many-body physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' and other quantum information protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' But, the potential well formed by a conventional far-off-resonance red-detuned optical-dipole trap that is attractive for ground-state cold atoms is usually repulsive for Rydberg atoms, which will result in the rapid loss of atoms and low repetition rate of the experimental sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Moreover, the coher- ence time will be sharply shortened due to the residual thermal motion of cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' These issues can be addressed by an one-dimensional magic lattice trap and it can form a deeper potential trap than the traveling wave optical dipole trap when the output power is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' And these common techniques for atomic confinement generally have certain requirement on the polarization and intensity stability of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Here, we demonstrated a method to suppress both the polarization drift and power fluctuation only based on the phase management of the Mach-Zehnder interferometer for one-dimensional magic lattice trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' With the combination of three wave plates and the interferometer, we used the instrument to collect data in the time domain, analyzed the fluctuation of laser intensity, and calculated the noise power spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' We found that the total intensity fluctuation composed of laser power fluctuation and polarization drift was significantly suppressed, and the noise power spectral density after closed-loop locking with typical bandwidth 1-3000 Hz was significantly lower than that under the free running of the laser system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Typically, at 1000 Hz, the noise power spectral density after locking was about 10 dB lower than that when A Master Oscillator Power Amplifier (MOPA) system free running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The intensity-polarization control technique provides potential applications for atomic confinement protocols that demand for fixed polarization and intensity I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' INTRODUCTION For various atomic manipulation experiments, such as sin- gle photon source1−5, quantum dynamics based on Rydberg states 6−10 and electric field detection based on atoms 11−13, strong confinement optical dipole trap (ODT) of atoms is usu- ally employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In these applications,high power laser with fixed polarization and relatively stabled intensity normally is used to confine atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Common experimental setup for the laser power stabilization were based on the active feedback loop which used acousto-optic modulator (AOM) 14−17 or electro-optic modulator (EOM) 18 as the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In 2020, AOM and EOM were combined to broaden the bandwidth of laser intensity noise stabilization to 1MHz by Ni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' At present, the feedback loop based on AOM has some disadvan- tages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' For example, bragg diffraction of AOM will seriously affect the spot quality of first-order diffraction light, and the power utilization of the system will be limited by the diffrac- tion efficiency of AOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The common electro-optic intensity modulator (EOIM) with input and output tailed fiber is effi- cient, but not suitable for high-power applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Moreover, the schemes mentioned above can observably suppress the power fluctuation of laser beam, but the reduction for the drift a)Present Address: Key Laboratory of Laser & Infrared System of Ministry of Education, Shandong University, Qing Dao 266000, Shandong Province, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' b)corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' E-mail: wwjjmm@sxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='cn ORCID : 0000-0001- 8055-000X of laser’s polarization is still not be effectively achieved .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Here we demonstrate an experimental scheme based on the Mach- Zehnder interferometer (MZI) for actively suppress both the fluctuation of power and polarization of laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' By prop- erly manipulated phase difference between two paths, the out- put fraction of MZI account for the majority laser power while its intensity fluctuation in the time domain has been reduced dozens of times compared with the free-running case, and the noise power spectral density (NPSD) has been decreased in the range of 1-3000 Hz in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Such a sta- ble system can certainly meet the needs of various applica- tions, such as experiments where the lifetime of cold atoms is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' THEORETICAL BACKGROUND 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Magic optical dipole trap for cesium 6S1/2 ground state and 84P3/2 Rydberg state Recently, a new experimental scheme which used interfer- ometer as the actuator of the feedback loop has been proposed 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Considering the light intensity requirement of the ODT, the MZI can satisfy the power requirement of ODT without af- fecting spot quality of output light, therefore the experimental setups of constructing the blue-detuning optical trap reported by Yelin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' al 21 and Isenhower et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' al 22 both concentrate on the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The intensity of the output laser mainly depends on the phase difference between two arms of the MZI, therefore it can be used as a power stabilizer in some experiments 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Due arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='08417v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='optics] 20 Jan 2023 Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback 2 to the particularity of the output fraction of MZI, the com- bination of interferometer and polarizer can realize the fixed polarization, high proportion output and high intensity stabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' It is obviously useful for the experiment of optical trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The potential of ODT U can be expressed as: U = − α 2ε0c 2P πω2 0 (1) Where α is the induced polarizability of the target state, ε0 is the permittivity of vacuum, c is the speed of light, P is the intensity of laser, ω0 is the radius of the spot at the focal point after the laser is focused by a lens .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' As shown in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (1), if the power of the 1879 nm laser is fluctuant, the resulting trap depth will be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Thus, the lifetime of the trapped atom will be severely affected by the presence of the heating mechanism5,17,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Diagram of the light shift induced by the ODT and MODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The intensity of laser which is intensly focused is still Gaussian, and the closer to the center of beam, the stronger the intensity of laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The resulting trap depth or light shift is spatially dependent (a) The ODT is attractive for ground states, but usually repulsive for highly-excited Rydberg states because almost all strong dipole transitions connected Rydberg state and the lower states have longer wavelength than that of ODT laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (b) The direct single-photon ex- citation scheme from cesium |g⟩=|6S1/2⟩ to |r⟩=|84P3/2⟩ coupled by a 319 nm ultraviolet laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' A 1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='43 nm laser is also tuned to the blue side of the |r⟩ ⇐⇒ |a⟩=|7D5/2⟩ auxiliary transition to equalize the trapping potential depth of the |g⟩ and |r⟩ state, which is so called magic ODT (MODT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In most of the experiments of cold atoms involving confine- ment of ground-state atoms in an ODT and Rydberg excita- tion, cold atomic sample is prepared in an ODT to hold them in a fixed position in a significantly long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The poten- tial formed by a conventional far off-resonance red-detuned ODT is attractive for the ground-state atoms, but usually re- pulsive for highly-excited Rydberg atoms, leading that Ryd- berg atoms normally cannot be confined in the conventional ODT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Therefore, in the follow-up experiments, we will face the following two problems: (1) if switching off the ODT during Rydberg excitation and coherent manipulation, it will result in atomic dephasing due to the thermal diffusion of the atoms and the extremely low repetition rate of the ex- perimental sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (2) if the ODT remains operation, it may cause a low Rydberg excitation efficiency of atoms as the tran- sition frequency is spatially position-dependent on the excita- tion laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The solution is to find an ODT such that the ground- state atoms and the desired highly-excited Rydberg atoms can experience the same potential, that is, the potential generated by the ODT is a potential well for both the ground-state atoms and the desired highly-excited Rydberg atoms, and is attrac- tive to atoms in both states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' So, the above-mentioned aspects (1) and (2) can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='1(b), the direct single-photon excitation scheme from cesium |g⟩=|6S1/2⟩ to |r⟩=|84P3/2⟩ coupled by a 319 nm ultraviolet laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' A 1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='43 nm laser is also tuned to the blue side of the |r⟩ ⇐⇒ |a⟩=|7D5/2⟩ aux- iliary transition to equalize the trapping potential depth of the |g⟩ and |r⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The specific calculation process is not de- scribed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' For details, please refer to the reference 25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Theoretical analysis of MZI It is obvious that the MODT is not enough to meet the need of extremely long coherence time in subsequent experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The cold atoms trapped in the MODT still have resid- ual thermal motion, which causes violent collisions that heat the atoms and cause them to escape from the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' We will fur- ther construct one-dimensional magic lattice trap (1D-MLT), and combine the advantages of lattice and magic conditions, so as to prolong the coherence time of the ground-Rydberg state of cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Of course, the 1D-MLT also needs to suppress its power fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Because the power of the laser used in the 1D-MLT fluctuates in the time domain, will di- rectly shorten the coherence lifetime of the cold atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' There- fore, we use the MZI to suppress the power fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2 (a), Iout1 and Iout2 are the intensity of two output paths of the interferometer respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' R1, T1, R2, T2 are the reflectivity and transmittance of input and output beam splitters plate respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The two output channels of the interferometer can be expressed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (2) and (3): Iout1 = R2 1R2 2 +T 2 1 T 2 2 +2R1R2T1T2cos(2∆L λ +π) (2) Iout2 = R2 1R2 2 +T 2 1 T 2 2 +2R1R2T1T2cos(2∆L λ ) (3) Therefore, the laser intensity output of the interferometer can be controlled by adjusting the driving voltage of PZT due to the correlation between the output transmittance I and optical path difference ∆L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2 (b), the interference fringes generated by splitters with different splitter ratio is simulated and analyzed by Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The splitter ratio shown by the first line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2(b) is 90/10, the second is 70/30, the third is 60/40, and the last is 50/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' EXPERIMENTAL SETUP The laser intensity stabilization setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' A MOPA system consists of a 1879-nm butterfly packaged laser diode and a Thulium Doped Fiber Amplifier (TmDFA) which has maximum output ∼ 3 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' With a free space polarization I(r 84P 3/2 1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='43nm 319nm ODT 1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='43nm g (a) 6S1/2 (b)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Diagram of MZI and interference fringes of two channels of the MZI are simulated and analyzed theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (a) MZI consists of two beam splitter plates (BS1 and BS2) and two high-reflectivity mirrors (M1 and M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Iin is the intensity of the incident light field, Iout1 and Iout2 are the intensity of the outgoing light field at BS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (b) Normalized signal as a function of difference of optical path ∆L for different splitter ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='This ratio is both R1/T1 and R2/T2, because the BS1 and BS2 that are used in the MZI are same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The solid red and black lines represent the interference fringes of the two output channels of MZI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' controller based on three waveplates ( λ/4, λ/2 and λ/4 ), polarization fluctuation of 1879 nm beam is suppressed ini- tially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The laser is injected into a MZI which is constructed by a 50/50 beam splitter plate (BS1) that divides the incident light into two beams with equal intensity and the different phase, a high-reflectivity mirror (M1) that reflects one beam, a mirror (M2) attached to a PZT that emits the other, and a beam split- ter plate (BS2) that the two beams are finally combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The interferometer has two output channels and each channel can be used for dynamic feedback to make the system more sta- ble, and the output of this channel can then be used for sub- sequent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The photodetector (PD1) is mounted behind a glass slice (GS1) of 1879 nm for sampling a little fraction of light for in-loop feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The DC voltage signal output by the PD1 is injected into Proportional Integral Dif- ferential (PID) amplifier after passing through a low-pass filter (LPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The input signal of PID controller is subtracted from the PID Set Point, which is an artificially set reference DC voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The output signal of PID, that is the real-time differ- ence between the detector signal and the reference DC volt- age is added with the scanning signal (triangular wave) and amplified by the high voltage (HV) amplifier as the driving voltage of the PZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The output power of interferometer can therefore be controlled by manipulating the driving voltage of PZT, and we expect that both power and polarization fluctua- tion for 1879 nm laser are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' And another photode- tector (PD2) is mounted in order to independently monitor the intensity stability of the output linear polarization laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The output signal of PD2 is then injected into the Data Acquisition System (Keithley, DAQ-6510) in order to analyze and moni- tor the intensity fluctuation of the laser in the time domain and calculate the NPSD based on the measured optical power fluc- tuation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Undoubtedly, the little fraction of the far-infrared laser is reflected by glass slice (GS2) and received by the PD2 and the majority of laser is transmitted and focused in a ce- sium magneto-optical trap (Cs-MOT) for the construction of the ODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Experimental setup for intensity stabilization system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The dynamic stability of laser intensity of 1879 nm MOPA system is re- alized by MZI, and the fluctuation of laser intensity is monitored and analyzed in time domain and frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='λ/2: half-wave plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' λ/4: quarter-wave plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' PBS: polarization beam splitting cube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' BS: beam splitting plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' GS: glass slice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' M1/M2: high- reflectivity mirror;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' PD: photodetector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' LPF: low-pass filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' PID: Proportional Integral Differential amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' HVA: high voltage am- plifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' EXPERIMENTAL RESULTS AND DISCUSSION Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 4 shows, the interference fringes obtained by scanning triangular waves with 50/50 beam splitter ratio in the experi- ment, in which the interference contrast is 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In theoretical simulation, an interference fringe with an interference con- trast of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='9% can be obtained by using a 50/50 beam splitter plate, but the best interference contrast is not achieved in ex- periment, probably due to the following two reasons: first, the spatial mode of the two lasers is not exactly same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' second, the polarization of the two lasers may be slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Considering the requirement of constructing dipole trap with this laser source, the polarization of 1879 nm laser should be fixed, so PBS is usually inserted in the light path to fixed the polarization of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Even though the scheme is effective, an inevitable defect exists in this scheme is that the polariza- tion fluctuation of light will couple with the intensity fluctu- ation through this polarization element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' As the measurement of which the intensity for 1879 nm laser after a PBS, although the power fluctuation of 1879 nm TmDFA itself is not obvi- ous, the intensity fluctuation behind the PBS becomes obvious and the results is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' We monitor the laser in- tensity for about 30 minutes in the time domain, with a large fluctuation of about ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The huge intensity fluctuation 90/10 70/30 60/40 50/50LSuppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Interference fringe of the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In the experiment, a 50/50 beam splitter plate is used, and the PZT is driven by scanning tri- angular wave, so that the phase difference between the two arms is generated, then the interference fringes are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' will significantly affect the power utilization of the stable sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' To maximize the power utilization, three wave plates are used to suppressed the power fluctuation initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' After proper adjustment, measurement result of laser intensity fluc- tuation after PBS is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Fluctuation of laser polarization has been reduced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Then the initial- stabled laser has been injected in the combine system of MZI and another PBS, here the transmittance of the interferometer is locked up to 90% in order to improve the power utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Then the intensity fluctuation probed by the out-of-loop detec- tor PD2 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' As shown below, the intensity fluc- tuation of output linear polarized laser is reduced to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='3%, that is much better than the fluctuation of direct TmDFA-PBS output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' At this stability, both fluctuation of laser power and polarization will no longer have a significant influence on the parameter of dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' As shown in Table 1, for the 1879 nm 1D-MLT, if the laser is focused through a lens to ∼ 20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' and the incident laser power at the cold atom is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='5 W, so the maximum depth of the 1D-MLT is −1000 µK and the typical trap depth fluc- tuation is ±140 µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' When the laser power decreases after the initial suppression of the wave plate group or the closed-loop locking of the MZI, the corresponding typical trap depth is about −800 µK and −700 µK respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' And the effec- tive temperature of the cold atoms which are transferred from MOT to 1D-MLT will be slightly higher, about 100 µK, but the decrease of trap depth caused by the suppression of power fluctuation will not affect the capture of the cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' How- ever, the residual fluctuation of laser power still exists, which will lead to the typical trap depth fluctuation of ±45 µK and ±2 µK respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The collected time-domain voltage signals are used to cal- culate the NPSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 7, the horizontal range is determined by the sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In the experiment, we selected sampling rate of 10000 Hz according to the actual situation, so the horizontal axis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 7 ranges from 1 to 5000Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In addition, we believe that the feedback bandwidth of the system should be at the level of kilohertz due to the limitation of PZT in the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Therefore, the sampling rate can fully meet the requirement of representing the feedback bandwidth of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The NPSD after closed-loop locking from 1-3000Hz is sig- nificantly lower than that under the free running of the MOPA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' It can be proved that the MZI plays an obvious role in the power stability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' In order to further broaden the feedback bandwidth and improve the inhibitory effect, we assume that the arm length of the MZI is L and the angular fre- quency of the laser is ω0, then the distance of the laser going through the MZI is L and the phase shift generated is27 Φ0(t) = ω0t = ω0 L c (4) Φ0 is a constant, and the magnitude is proportional to L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' When the PZT is scanned, we introduce to characterize small changes in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' For simplicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' we assume that a sine wave is used to scan the PZT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' and the amplitude of the sine wave is h0 and the angular frequency is ωs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' so the sine wave can be expressed as h(t) = h0cos(ωst) (5) So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' the phase shift of the entire system can be written as Φ = Φ0(t)+δφ = ω0L c + ω0 2 � t t− L c h0cos(ωst)dt = ω0L c + h0 2 ω0 ωs � sin(ωs L c )−sin[ωs(t − L c )] � = ω0L c +h0 ω0 ωs sin(ωs L 2c)cos[ωs 2 (t − L c )] (6) because L 2c ≪ 1 h0 ω0 ωs sin(ωs L 2c) = h0ω0 2 L c (7) δφ ∼ h0ω0 2 L c (8) As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (8), if the arm length L of the MZI is increased, δφ of the system can be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Thus, the de- tection sensitivity of the system can be improved and the de- tection effect of the MZI for phase can be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Increasing the arm length of the MZI will cause extra noise due to the insufficient stability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' However, such noise can be solved through the isolation platform and system temperature control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' We can add F-P cavity on the two arms of the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' F-P cavity can fold up the optical path, greatly increase the distance of light in the MZI, and do not need to occupy a large area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='28 Locked point Voltage (V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='08 Time (s)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (a) The power fluctuation of free running 1879 nm MOPA system .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Through 30 mins of measurement, the power fluctuation is roughly ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The inset is zoomed in on the vertical axis to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 W, and shows the intensity fluctuation in 30 mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (b) The power fluctuation after three wave plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' We thought that the polarization fluctuation is initially suppressed by these plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Similarly, after 30 minutes of measurement, the intensity fluctuation is approximately ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' And, the vertical axis range of the inset becomes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='75∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='25 W, the range of horizontal axis is still 0∼30 mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Category PODT (mW) ∆P(mW) Gaussian radius after focused (µm) Udip(µK) ∆Udip(µK) MOPA free running 1500 ±213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='0 (±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='2%) 20 −1000 ±140 With wave plate group 1200 ±68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='4 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='7%) 20 −800 ±45 After MZI is locked 1100 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='3 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='3%) 20 −700 ±2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The typical maximum trap depth and fluctuation of 1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='43nm 1D-MLT for cesium atoms under different power fluctuations are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The intensity fluctuation of 1879 nm laser on the bright fringe of the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' By inter-of-loop locking, the phase difference between the two arms is dynamically compensated , and the power fluctuation is significantly suppressed , through 30 mins of measurement, the power fluctuation is roughly ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The vertical axis of the inset has been enlarged with a range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='85∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='88 W, and shows the 30- min measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Intensity noise of 1879 nm laser as a function of analyze frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (a): The solid black line represents the NPSD when the 1879nm laser system is running freely without passing through the wave plate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' (b): The solid blue line represents the NPSD of the 1879nm laser system after closed-loop locking by the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 F Power fluctuation with wave plates group : ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='7% Power fluctuaticn when MOPA free running : ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='2% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 (W) M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='25 Power ( Power ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 Power (w) () Power ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='75 , (b) 15 20 25 30 20 25 30 T ime (min) Time (min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 5 10 15 20 25 30 0 5 10 15 20 25 30 0 Time (min) Time (min)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 Power fluctuation after Mzl is locked : ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='3% Powewr (W) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='85, 10 15 20 25 30 Time (min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='00 0 5 10 15 20 25 30 Time (min)100 3000Hz 10 NPSD (W/NHz) 10 (b) 10-8 MOPA free runming After MzI is locked 10-9 100 101 102 103 Frequency (Hz)Suppression of laser beam’s polarization and intensity fluctuation via a Mach-Zehnder interferometer with proper feedback 6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' CONCLUSIONS In summary, we have demonstrated the reduction of the power and polarization fluctuation for 1879 nm laser based on the cooperation of three wave plates and a MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The in- tensity fluctuation ∼ ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='2% after the combination of MOPA system and PBS is reduced to ∼ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content='3% with locked MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' And after MZI is locked, the NPSD is lower than that under free running in the range of 1-3000 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' Typically, at 1000 Hz, the NPSD after MZI is locked is about 10 dB lower than that when MOPA free running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The system can not only withstand high power injecting laser, but also can stabilize both power fluctuation and polarization fluctuation without affecting the quality of light beam for the low-loss output light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' The laser power utilizing efficiency can be further improved by improv- ing the transmittance of locked interferometer or improving the interference visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' It is expected that Rydberg atoms can have long coherence lifetime in subsequent experiments involving Rydberg dressed ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' On one hand, we can use the 1879-nm MOPA system to implement a 1D-MLT, which can both eliminate the position-dependent light shift to capture Rydberg-state atoms in optical tweezer like the ground-state atoms and attenuate collisions between cold atoms caused by residual thermal mo- tion to prolong the coherence time of the Rydberg atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' On the other hand, we propose an upgraded interferometer, that is, adding a F-P cavity to each arm of the interferometer, and using the reflection of beam in the cavity, the arm length can be extended at least dozens of times, to improve the phase measurement sensitivity of the interferometer and improve the power stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' FUNDING This research was financially funded by the National Key R & D Program of China (2021YFA1402002), the National Nat- ural Science Foundation of China (11974226, and 61875111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFAT4oBgHgl3EQfDhzt/content/2301.08417v1.pdf'} +page_content=' REFERENCES 1 M.' metadata={'source': 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+aDepartment of Mathematics, King’s College London, +The Strand, London WC2R 2LS, U.K. +bUniversit`a del Salento, Dipartimento di Matematica e Fisica Ennio De Giorgi, and I.N.F.N. - sezione +di Lecce, Via Arnesano, I-73100 Lecce, Italy +E-mail: pietro.benetti genolini, alejandro.cabo bizet, sameer.murthy +@kcl.ac.uk +Abstract: We exhibit an infinite family of supersymmetric phases in the three-dimensional +ABJM superconformal field theory and the dual asymptotically AdS4 gravity. +They are +interpreted as partially deconfined phases which generalize the confined/pure AdS phase and +deconfined/supersymmetric black hole phase. Our analysis involves finding a family of saddle- +points of the superconformal index labelled by rational points (equivalently, roots of unity), +separately in the bulk and boundary theories. In the ABJM theory we calculate the free +energy of each saddle by the large-N asymptotic expansion of the superconformal index to all +orders in perturbation theory near the saddle-point. We find that this expansion terminates +at finite order. +In the gravitational theory we show that there is a corresponding family +of solutions, constructed by orbifolding the eleven-dimensional uplift of the supersymmetric +black hole. The on-shell gravitational action of each orbifold agrees with the free energy of the +corresponding saddle in the SCFT. We find that there are two saddles in the ABJM theory +with the same entropy as the supersymmetric black hole, corresponding to the two primitive +fourth-roots of unity, which causes macroscopic oscillations in the microcanonical index. +arXiv:2301.00763v1 [hep-th] 2 Jan 2023 + +Contents +1 +Introduction and summary of results +1 +2 +The superconformal index of ABJM theory +8 +3 +ABJM index near rational points +14 +3.1 +Generalized Cardy limits to roots of unity +17 +3.2 +The large N saddle-point analysis +18 +3.3 +Subleading effects +25 +4 +Black holes and supersymmetric solutions +27 +4.1 +Kerr–Newman-AdS black hole +27 +4.2 +Supersymmetry +31 +5 +A family of saddles in AdS +37 +5.1 +Uplift to eleven dimensions +37 +5.2 +AdS/CFT comparison and orbifold solutions +39 +6 +Black holes in non-minimal gauged supergravity +43 +6.1 +Supersymmetric black holes in the X0X1 model +43 +6.2 +Uplift to eleven dimensions and AdS/CFT +47 +A Special functions and asymptotic limit formulas +52 +B Factorization of the integrand in the matrix integral +53 +C Saddle-point analysis of the large-N index +56 +D Killing spinor equations +61 +E Four-dimensional index +63 +1 +Introduction and summary of results +A fruitful way to learn about the collective behavior of a quantum statistical system is to study +the thermodynamic behavior of the theory as a function of external macroscopic sources (tem- +perature, angular velocities, chemical potentials) which couple to conserved charges (energy, +spin, global charges). Of particular interest are phase transitions that occur upon chang- +ing these external parameters. The profound insight provided by AdS/CFT is that phase +– 1 – + +transitions in the bulk and boundary theories—which, a priori, have completely different +mechanisms—map to one another. This insight has led to dramatic discoveries such as the +relation between the deconfinment transition in large-N gauge theory and the Hawking–Page +transition [1] in asymptotically AdS gravity [2]. +Recent progress on the superconformal index, initiated in [3–5], has allowed us to re- +visit this line of thought in the supersymmetric setting, which provides better quantitative +control. Recall that the superconformal index is defined as the Witten index of a certain +(complex) supercharge of an SCFT, refined by chemical potentials which commute with the +given supercharge. By the usual argument of pairing of bosonic and fermionic states, the in- +dex is invariant under small changes of the coupling [6]. Therefore, assuming no wall-crossing, +the values at weak and strong coupling are exactly equal, and we can compare the weakly +coupled field theory index with the strongly coupled gravitational answer. These investiga- +tions have led to the discovery of a rich phase structure in the supersymmetric AdS5/CFT4 +context—from the point of view of gauge theory as well as gravity. +On the gauge theory side, one finds complex saddle points of the large-N index when a +background chemical potential T ∈ H coupling to a certain combination of charges approaches +rational points [7–15]. On the gravitational side, one has a set of solutions with horizons in +asymptotically AdS5 space, whose free energy agrees with that of the corresponding micro- +scopic saddles [16]. Together, these results are interpreted as supersymmetric phases of the +AdS/CFT, generalizing the AdS black hole/deconfined phase and pure AdS/confined phase +to partially confined phases [7, 8, 14]. The perturbative series for the free energy near each +saddle terminates after a finite number of terms. Therefore, as N → ∞, the phase bound- +aries become sharp and the various transitions generalize the Hawking–Page/deconfinement +transition. +In this paper, we develop a similar picture of supersymmetric phases in the setting of +AdS4/CFT3. +The main observable that we focus on is the superconformal index of a 3d +SCFT on S2, with independent analyses of the bulk and boundary theories. In the boundary +theory we study the ABJM theory [17] with U(N)1 × U(N)−1 gauge group and N = 8 +superconformal symmetry, and in the bulk we study 4d gauged supergravity. We find an +infinite set of saddles labelled by rational points in large-N ABJM theory, and a corresponding +infinite set of asymptotically AdS4 gravitational orbifold solutions in the dual supergravity. +The expansion of the statistical free energy of the field theory around the saddles agrees +precisely with the corresponding gravitational free energy of the AdS4 orbifolds. +In the rest of the introduction we summarize the main points of the paper, drawing +parallels and contrasts with the AdS5/CFT4 situation wherever possible. +CFT analysis of the superconformal index +The most general superconformal index in ABJM theory receives contributions from +states that preserve two real supersymmetries of the theory, and can be expressed as a Witten +index over the Hilbert space HBPS(N) refined by four chemical potentials. The simplest such +1 +16-BPS index has one chemical potential T coupling to a combination 4J + 2r of angular +– 2 – + +momentum J and a certain R-symmetry r rotating the preserved supercharge. Since both +angular momentum and R-symmetry charges are quantized in this theory, the index is a +single-valued function of Q := e2πiT , and we have the Fourier expansion +IN(T) = TrHBPS(N) (−1)2J Q 4J+2r = +� +ℓ +dN(ℓ) Q ℓ . +(1.1) +We expect that the indexed degeneracy of states dN(ℓ) has an exponential growth reflecting +the existence of supersymmetric black hole solutions in the holographic dual theory. Indeed, +this expectation is borne out by numerical studies of four-dimensional N = 4 SYM [18, 19]. +At an analytic level, the problem is best solved in the grand canonical ensemble. It is clear +from inverting (1.1) that an exponential growth of IN(T) as T → 0 implies an exponential +growth of dN(ℓ) as ℓ → ∞. One therefore looks for saddle points of IN(T) near T → 0 +whose free energy is a negative power of T. Looking more carefully at the inversion of (1.1), +one realizes that an exponential growth of dN(ℓ) as ℓ → ∞ only implies that Im(T) → 0. +Moreover, in order to have a coherent addition, the Fourier series should split into a finite +number of congruence classes with the same phase, which implies that Re(T) ∈ Q [7]. We +thus conclude that the limits of interest are the +generalized Cardy limits : +T → D +C ∈ Q . +(1.2) +For four-dimensional N = 4 SYM, the saddle points of IN(T) in the generalized limits +have been investigated using various approaches, including asymptotic analysis [11, 20–27], +contour integrals and Bethe ansatz [28–30], and relations to special elliptic functions coming +from number theory [7, 9, 10, 12, 13, 15]. The results of these analyses are the following. +The leading contribution to the microcanonical growth of states comes from the saddles T → +± 1 +3, i.e. when Q approaches a primitive third root of unity. +(The quantization 1 +3 comes +from the quantization of R-charge in N = 4 SYM.) The magnitude of this contribution +grows as exp(SBH), where SBH is the entropy of the BPS black hole in five-dimensional +minimal gauged supergravity [31]. The contributions of these two leading saddles are complex +conjugates of each other, leading to macroscopic oscillations in the microcanonical growth of +states. More generally, the saddles (C, D) with gcd(C, D) = 1 contribute to an exponential +growth of states if and only if C is a multiple of 3. +For such saddles, the free energy is +FC,D(T) ∼ ±iπN2/9C (C T − D)2 if D = ±1 mod 3. +In this paper, we consider the generalized Cardy limits of the microscopic index of ABJM +theory. In the large-N limit, we find that the leading contribution to the microcanonical +growth of states comes from the saddles T → ±1/4, i.e. when Q approaches a primitive +fourth root of unity. (The quantization 1 +4 comes from the quantization of R-charge in ABJM +theory.) The magnitude of the growth of this contribution grows as the exponential of the +entropy of the supersymmetric Kerr–Newman-AdS black hole in four-dimensional minimal +gauged supergravity. The contributions of these two leading saddles are complex conjugates +of each other, leading to macroscopic oscillations in the microcanonical growth of states. +– 3 – + +More generally, the saddles (C, D) with gcd(C, D) = 1 contribute to an exponential growth +of states if and only if C is a multiple of 4 (the leading saddles being (C, D) = (4, ±1)). For +such saddles, the free energy is +FC,D(T) = ± 4 +C FBH(C T − D) , +FBH(ε) ∼ +π +3 +√ +2 +N +3 +2 +ε , +(1.3) +if D = ±1 mod 4. +In order to analyze the phase transitions, we summarize the above facts by (as N → ∞), +IN(T) ∼ +� +(C,D) +exp +� +−FC,D(T) +� +, +(1.4) +where the free energy of the saddle (C, D) is given by (1.3). We thus reach a picture of the T +plane divided into regions bounded by co-dimension-one walls. These regions are identified +with the phases and the phase boundaries occur at the walls where the neighboring entropies +become equal. In particular, since F4,1 is precisely the BH free energy, it is clear that the +transition between the BH phase and the pure AdS phase is precisely the Hawking–Page +phase transition. +In fact, we consider an all-order perturbation expansion in the small parameter ε = CT − +D and show that the asymptotic expansion of the free energy terminates at O(ε). This implies +that as N → ∞, the phase boundaries become sharp, similar to the phenomenon observed +in 4d SYM theory [11]. It is worth noting, though, that the saddle point equations in 4d +SYM theory are typically algebraic and can be solved easily at finite N, while in SCFT3 they +involve transcendental functions which can be solved in the N → ∞ limit using techniques +introduced in [32, 33]. +Gravitational analysis of the superconformal index +In the gravitational theory we do not have a good Hilbert space interpretation and, +instead, we try to interpret the index as a sum over saddle points. +The parameter N is +interpreted according to the standard AdS/CFT dictionary as proportional to the inverse +gravitational coupling and, similarly, other chemical potentials in the most general index +are interpreted as geometrical parameters in AdS space. +The saddles are defined in the +limit N → ∞ as solutions to the equations of motion of the semiclassical Euclidean gravita- +tional theory. The natural idea is to interpret the microscopic sum over saddles (1.4) as a +sum over gravitational solutions of the Euclidean bulk theory like the supersymmetric black +hole. We now summarize how this understanding is reached purely from bulk considerations. +Firstly, there is the question: “(How) does a supersymmetric black hole contribute to +the supersymmetric index?” The saddle points of the gravitational theory are solutions of +the effective low-energy supergravity with boundary conditions fixed to be asymptotically +AdS with conformal boundary S2 × S1. A first puzzle is that the (−1)2J in the trace (1.1) +naively implies that the fermions should have periodic boundary conditions around the S1, +– 4 – + +but this is in apparent tension with the fact that the S1 is contractible in the Euclidean black +hole geometry, which requires the spinor to be anti-periodic around the circle. +A second +issue is that supersymmetric black holes are extremal, so they have an infinite throat at the +horizon, leading to an infinite on-shell action. +These tensions were resolved in [3] in the +context of the supersymmetric AdS5 black holes. The same idea has been applied to black +holes asymptotically AdS4 [34, 35] (which are relevant for the current paper), as well as to +five-dimensional supersymmetric black holes in asymptotically locally flat space [36]. +The idea at the heart of the resolution is to allow complex gravitational solutions. The +first point is that the supersymmetric periodicity condition can be absorbed by an imaginary +shift of a chemical potential. In the concrete example of (1.1), we have +TrHBPS(N) (−1)2Je2πiT (4J+2r) = TrHBPS(N) e2πi(1+4T) J+4πiT r . +(1.5) +(One can equivalently shift the potential for the R-charge.) The shifted chemical potential +on the right-hand side is naturally interpreted in the gravitational black hole background +as complex boundary values of bosonic fields. Interpreting the index trace as a functional +integral, it is clear that only those configurations that extend the boundary Killing spinor to +the bulk contribute to the index, as otherwise there would be extra extra fermion zero modes +coming from broken supersymmetry which would kill the corresponding contribution of such +solutions to the path integral.1 +The second point is the definition of the contribution of the supersymmetric black hole +to the functional integral via a regularization procedure. One starts with a one-parameter +family of deformations of the Euclidean supersymmetric black hole with the above complex +boundary conditions, consisting of geometries that are not extremal but are supersymmetric. +Importantly, these configurations are inherently complex and they do not admit regular real +Lorentzian continuations. They have a real Euclidean section with the topology of the product +of a disc and a sphere, and are labelled by a continuous parameter β > 0 corresponding to +the boundary size of the Euclidean circle. In the limit β → ∞ we recover the Euclidean +supersymmetric extremal black hole with an infinite throat, which is the Wick-rotation of +the Lorentzian supersymmetric black hole solution. These complex supersymmetric solutions +also support a non-trivial gauge field, which vanishes at the origin of the disc, thus preserving +smoothness, and has a non-trivial holonomy around the boundary of the disc. +Since the +Killing spinor is charged under the gauge field, parallel transport around a circle in the +disc in the chosen gauge gives an anti-periodic spinor, consistently with topology, whereas +parallel transport with the fully covariant derivative does indeed lead to a periodic spinor, as +consistent with the naive expectation from supersymmetry. +One then calculates the holographically renormalized on-shell action of the complex su- +persymmetric solutions. Notably, this action, as a function of appropriately reduced chemical +potentials, is (a) finite, (b) independent of β, and (c) exactly equal to the logarithm of the +functional form of the microscopic grand-canonical index near the (4, ±1) saddle-point. In +1This has been demonstrated by an explicit calcuation in the asymptotically flat space in [36]. +– 5 – + +order to obtain the entropy of the supersymmetric extremal black hole, one does a Legen- +dre transform with respect to the reduced chemical potentials, and finds agreement with the +Bekenstein–Hawking area law for the black hole entropy [37]. This extends the prescriptions +of Euclidean quantum gravity [38] to supersymmetric black holes [3]. +The above procedure allows one to trace the relation between the Wick-rotated BPS +black hole and the (4, ±1) saddle of the field theory index. One then expects that the (C, D) +saddle of the index with gcd(C, 4) = 4 corresponds to supersymmetric solutions (regular- +ized as above) with the same conformal boundary conditions as the black hole, and on-shell +action equal to FBH/(C/4). In the context of the AdS/CFT correspondence applied to four- +dimensional N = 4 SYM, these solutions were described in [16]: they are supersymmetry- +preserving quotients of the ten-dimensional uplift on S5 to IIB of the Wick-rotated black hole +solution. These quotients crucially involve the Euclidean time circle and thus their Lorentzian +interpretation is not transparent. +In this paper, we construct the analogous solutions dual to the (C, D) saddles in the +large-N limit of the index of ABJM. We start with solutions to gauged supergravity with the +same conformal boundary conditions as the supersymmetric electrically charged black hole, +uplift them on S7 to eleven-dimensional supergravity, and then perform a ZC/4 quotient while +preserving supersymmetry. The solutions in this infinite family generalize the two Hawking– +Page-like phases, namely the supersymmetric Kerr–Newman-AdS black hole and pure AdS4, +and have on-shell action equal to the microscopic free energy (1.3) at that rational point. +Open questions and further directions +The structure of the sum over saddles (1.4), which we derive independently from field the- +ory and from gravity, is very interesting from the point of view of the underlying symmetries. +Such a sum over (C, D) saddles has previously appeared in the context of supersymmetric +AdS3/CFT2 [39, 40], and in the context of AdS2/CFT1 [41–44]. In both cases, there is an +underlying SL(2, Z) modular symmetry which is closely tied to the existence of such an ex- +act convergent formula.2 In contrast, the formula (1.4) for SCFT3 and the analogue in the +SCFT4 problem is not an exact formula: firstly, we do not know if this is an exhaustive set +of saddles (although there are indications that this may be so at large N); secondly, although +we have control over the all-order perturbation theory around each saddle, there is no claim +of convergence of the sum. Nevertheless, the observation that the index is arranged as a +sum over such saddles with a number-theoretic constraint on (C, D) is remarkable since the +superconformal index in 4d and in 3d are not SL(2, Z) modular forms. Perhaps it hints to an +approximate modular symmetry in these systems. +The phase structure of the 3d as well as the 4d theory and, in particular, the dominant +phase as one moves along the real axis in T has an erratic behavior. +It would be very +interesting if this fundamentally related to the appearance of randomness in gravitational +systems [45]. +2The sum over (C, D) here is really a sum over the equivalence classes Γ∞\ Γ /Γ∞ where Γ = SL(2, Z) and +Γ∞ is its subgroup stabilizing the point τ = i∞. +– 6 – + +The analysis of ABJM theory at level k > 1 is an interesting problem: in the gravity dual, +the ZC/4 quotient described above would be woven with the Zk quotient of the internal S7. +More generally, one could generalize the uplifts discussed here to uplifts on different seven- +dimensional Sasaki–Einstein spaces, where the quotient would identify points along the circle +orbit of the Reeb vector. These constructions would be dual to three-dimensional N = 2 +SCFTs obtained from arrangements of M2-branes. +In another direction, it would be interesting to study the N = 2 SCFTs obtained by +wrapping M5-branes on hyperbolic three-manifolds. In this case, the R-symmetry charge of +the states on S2 is quantized since it is a compact subgroup of the so(5) R-symmetry group +of the original six-dimensional theory, so the superconformal index has a similar structure +as that described around (1.1). The canonical Cardy limit describing the leading growth +of states has been calculated, and the large-N limit agrees with the gravity dual [35, 46]. +The gravity duals to the generalized Cardy limits would be constructed by quotienting the +eleven-dimensional geometry uplifting the supersymmetric Kerr–Newman-AdS black hole to +eleven dimensions on a fibration of S4 over the hyperbolic manifold [47]. +Another issue is the overabundance of eleven-dimensional geometries with boundary con- +ditions appropriate to describe the dual to the large-N saddles of the generalized Cardy limits +of ABJM. As pointed out in [16], their presence would make the gravitational grand-canonical +partition function diverge, but one can argue for their absence from the sum studying the +action of wrapped branes in the geometry. In this paper we include some comments on this +subject, and we plan to report on this in more detail in the near future. +Plan of the paper +In Section 2 we review the construction of the superconformal index for ABJM, highlight- +ing the interpretation as a functional integral over a complex background and the conditions +imposed on the chemical potentials in order to preserve supersymmetry. We also comment +on the subtleties arising from the fact that the index as usually defined is a multi-valued +function, and identify two choices of refinements that will be matched in gravity. In Section +3 we start from the expression of the index as a matrix model integral, and obtain the free +energy of the saddle points in the large-N limit, including subleading effects in the generalized +Cardy limit. +We then move to the dual gravity side. In Section 4 we begin by reviewing aspects of the +Euclidean Kerr–Newman-AdS black hole and its holographic renormalization, highlighting +the importance of the gauge choice for the Abelian gauge field. We then construct a family +of complex supersymmetric solutions deforming the Wick-rotation of the supersymmetric +black hole away from extremality and compute their on-shell action. In Section 5 we uplift +these solutions of four-dimensional minimal gauged supergravity on S7 to eleven dimensions, +describing the conformal boundary conditions and the matching with the supersymmetric +background of the boundary field theory. This leads us to the construction of the quotient +solutions, whose free energy matches the large-N limit of the (C, D) saddle of the unrefined +index of ABJM. Finally, in Section 6 we consider black holes electrically charged under two +– 7 – + +gauge fields, which are dual to saddles of the index of ABJM with R-symmetry fugacities being +pairwise equal. Tracing the same path as in the previous sections, we review the construction +of complex solutions, regularize the action of the BPS black hole, uplift the non-minimal +gauged supergravity to eleven dimensions, studying the conformal boundary conditions, and +finally take the ZC/4 quotients. +In multiple appendices we explain some details of the computations in the main text, and +review the analogy between the large-N behavior of the superconformal index of ABJM and +the superconformal index of 4d N = 4 SYM. +2 +The superconformal index of ABJM theory +In this section we introduce the superconformal index of ABJM theory in the Hamiltonian as +well as functional integral formalism, and discuss its behavior under shifts of various chemical +potentials. +ABJM theory [17] is a U(N)k × U(N)−k Chern–Simons-matter theory with N = 6 +superconformal symmetry for generic k. The superconformal algebra is osp(6|4) whose bosonic +subalgebra is so(3, 2)×so(6)R. In addition the theory has a u(1)b symmetry. The field content +in N = 2 language is as follows. There are two vector supermultiplets V, �V corresponding, +respectively, to the two gauge groups, and two chiral supermultiplets A1,2 in the (N, N) of +U(N) × U(N) and two chiral supermultiplets B1,2 in the anti-bifundamental (N, N). The +matter multiplets interact via a superpotential +W = 2π +k ϵabϵcd tr +� +AaBcAbBd +� +. +(2.1) +The N = 2 formalism makes the bosonic subalgebra su(2)A × su(2)B × u(1)b × u(1)R of +the global symmetry manifest. Here su(2)A,B rotates, respectively, the two pairs of chiral +multiplets A and B, u(1)R is the R-symmetry of the N = 2 supercharges (under which the +chiral multiplets have charge 1 +2) and u(1)b is the “baryonic symmetry” under which Aa have +charge 1 +2 and Ba have charge − 1 +2.3 It is convenient to group the components of the chiral +fields A = {A, ζ}, B = {B, ω} as +Y A = {Aa, B†a} , +Y † +A = {A† +a, Ba} , +ψA = {ϵabζbe−iπ/4, −ϵabω†beiπ/4} , +ψ†A = {−ϵabζ† +beiπ/4, ϵabωbe−iπ/4} . +(2.2) +The potential of the theory when written in terms of these combinations is invariant under +the full so(6)R×u(1)b symmetry provided Y A, ψ†A transform in the complex representation 4, +and Y † +A, ψA transform in the 4 [17, 49]. We focus on the k = 1 case, when the supersymmetry +3This symmetry is gauged, but it is related by the Chern–Simons term to the topological symmetry with +current J ∝ ∗ tr(F + �F). One can also construct the index refined by the topological symmetry, and then +perform a change of variables in the resulting matrix integral to obtain the same formula we have, see for +instance [33, 48] for some related comments. +– 8 – + +is enhanced to N = 8 by non-perturbative effects [50, 51], and the symmetry algebra so(6)R× +u(1)b combines into so(8)R. +In this theory we consider the superconformal index based on the N = 6 superconformal +algebra refined by the symmetry u(1)b. In order to define the index, we quantize the theory +on S2 obtaining the Hilbert space HS2. States in HS2 are labelled by the eigenvalues of the +Hamiltonian H and the angular momentum J (quantized as a half-integer), the weights of +the so(6)R R-symmetry, and the half-integer eigenvalues of the generator B of u(1)b. For +the u(1)3 Cartan subalgebra of so(6)R, we pick the “orthogonal” basis, consisting of the +generators of the rotations in the three orthogonal planes of R6, which we label H1, H2, H3, +having half-integer eigenvalues. Following [52, 53], we pick a supercharge Q with eigenvalues +� 1 +2, − 1 +2, 1, 0, 0, 0 +� +under (H, J, H1, H2, H3, B), with the anticommutation relation +{Q, Q†} = H − J − H1 . +(2.3) +On states annihilated by Q, which define the subspace HBPS, the right-hand side of the above +equation clearly vanishes, i.e. the energy H is determined by the values of H1 and J. In the +remaining five-dimensional subspace of bosonic charges, the subalgebra commuting with Q +and Q† is generated by J + 1 +2H1, H2, H3, B. Therefore, the refined index of ABJM theory of +interest here is +I(τ, ξ2, ξ3, ξB) = TrHS2(−1)2Je−β{Q,Q†}+2πiτ(J+ 1 +2 H1)+2πi(ξ2H2+ξ3H3+ξBB) += TrHBPS (−1)2Je2πiτ(J+ 1 +2 H1)+2πi(ξ2H2+ξ3H3+ξBB) . +(2.4) +The second equality follows, as usual, from the fact that the index only receives contributions +from states in HBPS [6]. Because of the half-integer quantization of the generators, the index +is invariant under the identifications τ ∼ τ + 4, and ξ2,3,B ∼ ξ2,3,B + 2. +It is useful to introduce a more symmetric basis of generators (and the corresponding +chemical potentials) +H1 = R1 + R2 + R3 + R4 +2 +, +H2 = −R1 + R2 − R3 + R4 +2 +, +H3 = −R1 + R2 + R3 − R4 +2 +, +B = −R1 − R2 + R3 + R4 +2 +; +ξ2 = −λ1 − λ3 , +ξ3 = λ2 + λ3 , +ξB = −λ1 − λ2 . +(2.5) +In terms of these, the refined index (2.4) takes the form +I = TrHS2(−1)2Je−β{Q,Q†}+2πiτ(J+ 1 +4 +�4 +a=1 Ra)+2πi �3 +i=1 λi(Ri−R4) += TrHS2 e−βH+βΩJ+�4 +a=1 βΦaRa +(2.6) +with +Ω = 1 + 2πi +β (τ + n0) , +(2.7) +– 9 – + +Fields +J +H +(H1, H2, H3, B) +(R1, R2, R3, R4) +Y 1 +0 +1 +2 +� 1 +2, 1 +2, − 1 +2, 1 +2 +� +(0, 0, 0, 1) +Y 2 +0 +1 +2 +� 1 +2, − 1 +2, 1 +2, 1 +2 +� +(0, 0, 1, 0) +Y 3 +0 +1 +2 +� +− 1 +2, 1 +2, 1 +2, 1 +2 +� +(−1, 0, 0, 0) +Y 4 +0 +1 +2 +� +− 1 +2, − 1 +2, − 1 +2, 1 +2 +� +(0, −1, 0, 0) +ψ1± +± 1 +2 +1 +� +− 1 +2, − 1 +2, 1 +2, 1 +2 +� +� +− 1 +2, − 1 +2, 1 +2, − 1 +2 +� +ψ2± +± 1 +2 +1 +� +− 1 +2, 1 +2, − 1 +2, 1 +2 +� +� +− 1 +2, − 1 +2, − 1 +2, 1 +2 +� +ψ3± +± 1 +2 +1 +� 1 +2, − 1 +2, − 1 +2, 1 +2 +� +� 1 +2, − 1 +2, 1 +2, 1 +2 +� +ψ4± +± 1 +2 +1 +� 1 +2, 1 +2, 1 +2, 1 +2 +� +� +− 1 +2, 1 +2, 1 +2, 1 +2 +� +Y † +1 +0 +1 +2 +� +− 1 +2, − 1 +2, 1 +2, − 1 +2 +� +(0, 0, 0, −1) +Y † +2 +0 +1 +2 +� +− 1 +2, 1 +2, − 1 +2, − 1 +2 +� +(0, 0, −1, 0) +Y † +3 +0 +1 +2 +� 1 +2, − 1 +2, − 1 +2, − 1 +2 +� +(1, 0, 0, 0) +Y † +4 +0 +1 +2 +� 1 +2, 1 +2, 1 +2, − 1 +2 +� +(0, 1, 0, 0) +ψ†1± +± 1 +2 +1 +� 1 +2, 1 +2, − 1 +2, − 1 +2 +� +� 1 +2, 1 +2, − 1 +2, 1 +2 +� +ψ†2± +± 1 +2 +1 +� 1 +2, − 1 +2, 1 +2, − 1 +2 +� +� 1 +2, 1 +2, 1 +2, − 1 +2 +� +ψ†3± +± 1 +2 +1 +� +− 1 +2, 1 +2, 1 +2, − 1 +2 +� +� +− 1 +2, 1 +2, − 1 +2, − 1 +2 +� +ψ†4± +± 1 +2 +1 +� +− 1 +2, − 1 +2, − 1 +2, − 1 +2 +� +� 1 +2, − 1 +2, − 1 +2, − 1 +2 +� +Q +− 1 +2 +1 +2 +(1, 0, 0, 0) +� 1 +2, 1 +2, 1 +2, 1 +2 +� +Table 1: Weights of the fields after the change of basis in the Cartan. +and +Φi = 1 +2 + 2πi +β +�τ +4 + λi +� +, +i = 1, 2, 3 , +Φ4 = 1 +2 + 2πi +β +� +τ +4 − +3 +� +i=1 +λi +� +. +(2.8) +Here, we have used the expression (2.3) for {Q, Q†} and written (−1)2J = e2πin0J for an +arbitrary odd integer n0. The charges of the fields (and the supercharge) under the spacetime +and global symmetry used to define the refined index are summarised in Table 1. It is clear +from this table that all states satisfy J = Ra mod 1 for all a = 1, . . . , 4, so that λi ∼ λi + 1. +By the usual relation between the Hamiltonian and the path integral quantization, we +can write the index (2.6) as a functional integral on the background S1 +β × S2 with metric +ds2 = dt2 +E + dθ2 + sin2 θ dφ2 , +(2.9) +where θ has the canonical periodicity π, and +(tE, φ) ∼ (tE, φ + 2π) ∼ (tE + β, φ − iΩβ) . +(2.10) +The fields f satisfy the following twisted boundary conditions around S1 +β, +f ( tE + β, x) = (−1)F eβΩJ+�4 +i=a βΦaRaf ( tE, x) +(2.11) +– 10 – + +Equivalently, we can write the index as the same functional integral, but over the fibered +background metric +ds2 = dtE2 + dθ2 + sin2 θ +� +d�φ − iΩ dtE +�2 +(2.12) +and with background gauge fields coupling to the currents for the symmetries generated by Ra +Aa = iΦa dtE . +(2.13) +In this case, the coordinates have the periodicities +� +tE, �φ +� +∼ +� +tE + β, �φ +� +∼ +� +tE, �φ + 2π +� +, +(2.14) +and the fields satisfy standard thermal boundary conditions. The metric (2.12) is real if Ω is +pure imaginary, but is otherwise complex-valued. +The background gauge field holonomies and the fibration parameter satisfy the following +constraint +β +� +1 − +� +a +Φa + Ω +� += 2πin0 . +(2.15) +Due to the relation J = Ra mod 1, the index is invariant under the following shifts, +Φa → Φa + 2πi +β na , +Ω → Ω + 2πi +β 2nΩ + 2πi +β +� +a +na , +na, nΩ ∈ Z . +(2.16) +Of course, these transformations preserve the parity of n0 in the right-hand side of (2.15), +since their effect is n0 → n0 − 2nΩ. +It will simplify part of the following analysis to include an additional chemical potential +λ4 defined by the constraint +4 +� +a=1 +λa = n1 ∈ Z . +(2.17) +This leads to the following expression for the index, obtained from (2.6) +I(τ; λ) = TrHBPS(−1)2Je2πiτ(J+ 1 +4 +� +a Qa)+2πi � +a λa(J+Ra) += TrHBPS(−1)2Je2πi � +a( τ +4 +λa)(J+Ra) +(2.18) +In these variables we can identify the fibration parameter Ω and the holonomies Φa (which +now have a symmetric form) for the background gauge fields +Ω = 1 + 2πi +β (τ + n0 + n1) , +Φa = 1 +2 + 2πi +β +�τ +4 + λa +� +, +a = 1, . . . , 4, , +(2.19) +which is related to (2.7) and (2.8) by a shift of the type (2.16). +The advantage of this +basis is that the chemical potentials couple to orthogonal generators of the Cartan of the +so(3) × so(8)R subalgebra of the N = 8 superalgebra. +Thus the fact that J = Ra mod +1 is seen as a consequence of the N = 8 superalgebra. +Indeed, both in the free N = 8 +superconformal theory and in the interacting BLG theory, we can find a triality frame for the +R-symmetry such that the supercharge sits in the 8∗ +s, the scalars in the 8v, and the spinors +in the 8s [54].4 +4This is different from the frame used in [53], where the supercharge is taken in the vector of so(8)R. +– 11 – + +The rewriting (2.18) shows that the index can be effectively rewritten as a function of +four variables τ/4 + λa, each defined modulo 1, since J = Ra mod 1. However, it will be +useful in later sections to consider shifts of τ alone, which lead to: +I(τ; λ) = TrHBPS(−1)2Je2πi � +a( τ +4 +λa)(J+Ra) , +I(τ + 1; λ) = TrHBPS(−1)H1e2πi � +a( τ +4 +λa)(J+Ra) ≡ IR(τ; λ) , +I(τ + 2; λ) = TrHBPS(−1)2(J+H1)e2πi � +a( τ +4 +λa)(J+Ra) , +I(τ + 3; λ) = IR(τ; λ) +(2.20) +In particular, the first and the third indices are graded by (−1)2J and (−1)2(J+H1), respec- +tively, both of which take values ±1 on the states of the ABJM theory, while the second and +the fourth ones are indices graded by (−1)H1 (H1 = 1 +2 +� +a Ra is the R-symmetry generator) +which takes values in the fourth roots of unity.5 These phases are reflected in the contribution +of the bosonic and fermionic states to the grand-canonical index. In particular, as we see in +section 3.2, the R-charge index has saddle points with exponential growth near τ = 0, while +the other two do not.6 7 +We can understand the constraint (2.15) in another equivalent manner. Recall that the +index can be computed as a functional integral over field configurations satisfying standard +thermal boundary conditions around the tE circle, in the background (2.12), (2.19). In order +to formulate a supersymmetric field theory on such a background, one couples it to off-shell +supergravity and then considers its rigid limit [56]. In this case, coupling the N = 8 field +theory to three-dimensional off-shell supergravity requires the existence of spinors solving +the (generalised) Killing equation obtained from the vanishing of the gravitino variation (see +e.g. [57]). Noting that the spinor is charged under the potentials Φa (for the u(1)4 ⊂ so(8) +gauge group of the supergravity), and that the spin connection is proportional to Ω, we see +that the tE dependence of the Killing spinor is +ei +� +1−� +a Φa+Ω +� +tE . +(2.21) +Now, from the fact that we impose anti-periodic boundary conditions for the fermions around +the tE circle, we obtain precisely the constraint (2.15) with n0 being odd. +When comparing with the gravity solutions, we shall consider less refined versions of the +index, obtained by setting certain combinations of the four chemical potentials associated to +the Cartan generators of so(8)R to be equal. We may first set the four chemical potentials +to be pairwise equal, that is λ1,2 ≡ σ1, λ3,4 ≡ σ2. +Since the chemical potentials λa are +5In fact, this quantization arises whenever u(1)R is part of a larger non-Abelian algebra. +6Of course, the the Fourier coefficients of all four indices in (2.20) have an exponential growth in magnitude, +with shifted phases. +7This phenomenon is analogous to the structure of the index of four-dimensional N = 4 SYM, where the +R-charges of the states are quantized in units of 1/3, so that there are three different indices, defined by shifts +of τ. Two of them lead to large growth as τ → 0, while the third does not [11, 55] (see Appendix E). +– 12 – + +constrained to satisfy � +a λa = n1, we have 2(σ1 + σ2) = n1. In this case, we find that the +index has the form +I(τ; λ1,2 = σ1, λ3,4 = σ2) = TrHBPS(−1)2Je2πiτ(J+ 1 +4 +� +a Ra)+2πi(σ1(R1+R2)+σ2(R3+R4)) . +(2.22) +We can identify two u(1) generators 2Q1 ≡ R1 + R2 and 2Q2 ≡ R3 + R4. In terms of these +generators, the interpretation as a functional integral over a fibered background requires the +following parameters +Ω = 1 + 2πi +β (τ + n0 + n1) , +Φ(QA) = 1 + 2πi +β +�τ +2 + 2σA +� +, +A = 1, 2 , +(2.23) +which are constrained by +β +� +1 − Φ(Q1) − Φ(Q2) + Ω +� += 2πin0 . +(2.24) +The shifts of the fibration parameters that leave the partition function invariant are +Φ(QA) → Φ(QA) + 2πi +β 2nQA , +Ω → Ω + 2πi +β 2nΩ , +n(RA), nΩ ∈ Z . +(2.25) +Unlike (2.16), these two shifts are now independent because the constraint between the charges +of the states in the theory, namely 2QA + 2J = 0 mod 1, is now trivially satisfied. Notice +that these two shifts preserve the parity of n0 in the right-hand side of (2.24). +Finally, we can further simplify this by setting all the generators equal: λa ≡ λ for all +1 ≤ a ≤ 4. Because of the constraint between the chemical potentials, we have 4λ = n1. In +this case, we find that the index has the form +I(τ; λ1,2,3,4 = λ) = TrHBPS(−1)2Je2πi(τ+n1)(J+ 1 +2 H1) . +(2.26) +In this form it is clear that, as observed in (2.20), n1 = � +a λa appears as a shift of τ, leading +to indices graded by different operators and thus with different asymptotic behaviors. +When we view the unrefined index as a thermal partition function over a fibered back- +ground, we find the parameters +Ω = 1 + 2πi +β (τ + n0 + n1) , +Φ(H1) = 1 + 2πi +β +τ + n1 +2 +, +(2.27) +constrained by +β +� +1 − 2Φ(H1) + Ω +� += 2πin0 . +(2.28) +As in the previous cases, we can shift the holonomy and fibration parameter while leaving +the partition function invariant +Φ(H1) → Φ(H1) + 2πi +β 2nH1 , +Ω → Ω + 2πi +β 2nΩ , +n(H1), nΩ ∈ Z . +(2.29) +– 13 – + +The shifts (2.29) and (2.25) will play an important role in the gravitational interpretation in +the later sections. +As we shall see in Section 4, the background with fibration parameters (2.27) is what +is found at the boundary of the supersymmetric electrically charged black hole in gauged +minimal supergravity, and so it is the correct background in order to match the gravity com- +putation in the bulk. However, it has a generically complex metric, which is not standard from +the field theory viewpoint. On the other hand, one can also define the index using a super- +symmetric background consisting of a real metric and background gauge field on S1 ×S2 (see +for instance [58, Sec. 7.2]). Since both backgrounds are used to compute the same supersym- +metric observable, which should only depend on the moduli of the transversely holomorphic +foliation, it is natural to conjecture that they are related by a Q-exact deformation.8 This is +similar to the discussion about four-dimensional backgrounds on S1 × S3 [11, 14, 55]. +3 +ABJM index near rational points +In this section we investigate the behavior of the superconformal index near rational points. +We consider the refined index (2.18), which is a periodic function of τ with period 4. In terms +of the variable q = exp(2πiτ), it is defined on a 4-sheeted cover of the complex plane. The +variable exp(2πiτ/4) lives on C and we consider the behavior of the index as this approaches +a primitive root of unity, i.e. 1 +4τ → d +c, c, d ∈ Z, c > 0 and gcd(c, d) = 1, further taking the +large-N limit of the result. +Recall from (2.18) that τ couples to J + 1 +4 +� +a Ra, and λ = (λ1, . . . , λ4), with � +a λ ∈ Z, +couple to the orthogonal generators of the Cartan of so(3) × so(8) in the definition of the +index. The index I is the N = 6 superconformal index further refined by the u(1)b. The index +without this refinement was computed in [52] using the free field theory in the background of +zero magnetic fluxes for the gauge groups, and in [59] using localization allowing for fluxes. +The refinement by the baryonic symmetry can be introduced referring to Table 1 (see also +footnote 3). An analogous expression for the refined index has been considered in [48, 60]. +As a matrix integral, the index (2.18) for ABJM at level k = 1 takes the form +I(τ; λ) = +�� +m [Du] +�� +�m [D�u] q +1 +2 +� +i,j |mi−�mj|− 1 +4 +� +i,j |mi−mj|− 1 +4 +� +i,j |�mi−�mj| +× Zclass(u, m, �u, �m) Zvec(u, m, �u, �m; τ) +4 +� +a=1 +Za +chi(u, m, �u, �m; τ, λ) . +(3.1) +Here the notation +�� +m [Du] ≡ +1 +N! +� +m ∈ ZN +N +� +i=1 +� +R/Z +dui +(3.2) +8One should be able to formulate the analogous discussion in extended supergravity for the background to +the index refined according to the N = 8 superalgebra. +– 14 – + +denotes the sum over the magnetic fluxes m = (m1, . . . mN) and the integral over the gauge +holonomies u = (u1, . . . , uN) for each gauge group. We introduce the fugacities q = e2πiτ, +ζa = e2πiλa, a = 1, . . . , 4, and xi = e2πiui, �xi = e2πi�ui. In all the products, i, j run from 1 +to N. +The various pieces in the integrand are the following. The contribution of the classical +action is +Zclass(u, m, �u, �m) = +� +i +exp +� +2πi +� +mi ui − �mi �ui +�� +, +(3.3) +the contribution of the two N = 2 vector multiplets for the U(N) × U(N) gauge group is +Zvec(u, m, �u, �m; τ) = +� +i̸=j +� +1 − xi x−1 +j q +1 +2 |mi−mj|� � +i̸=j +� +1 − �xi �x−1 +j q +1 +2 |�mi−�mj|� +(3.4) +and the contributions of the four N = 2 chiral multiplets are +Za +chi(u, m, �u, �m; τ, λ) = +� +i,j +� +x−1 +i +�xj ζ−1 +a +q +3 +4 + 1 +2 |mi−�mj| ; q +� +∞ +� +xi �x−1 +j +ζa q +1 +4 + 1 +2 |mi−�mj| ; q +� +∞ +, +a = 1, 2 , +Za +chi(u, m, �u, �m; τ, λ) = +� +i,j +� +xi �x−1 +j +ζ−1 +a +q +3 +4 + 1 +2 |mi−�mj| ; q +� +∞ +� +x−1 +i +�xj ζa q +1 +4 + 1 +2 |mi−�mj| ; q +� +∞ +, +a = 3, 4 . +(3.5) +It is manifest from the above expressions that I(τ; λ) is invariant under the separate shifts λa → +λa + 1, a = 1, . . . , 4, and τ → τ + 4. +We are interested in the behavior of I(τ; λ) for +τ = 4d +c + i ε +2πc ⇒ q = e2πi 4d +c e− ε +c , +gcd(c, d) = 1 , +ε ↘ 0 , +(3.6) +Although we begin our analysis for ε ∈ R+, all the statements that we make below hold +for ε tending to zero from any direction in the upper half-plane, and we continue to use the +symbol ε ↘ 0 to denote this more general limit. +In this limit we expect that the integral over u and the sum over m are both dominated by +saddle-points. In order to implement the saddle-point analysis, it is useful to change variables +so that the integrand factorizes into what are essentially holomorphic and anti-holomorphic +pieces. This is similar to the τ → 0 treatment in [60] but as we see below the τ → Q limit +has additional subtleties. It is useful to define the following variables +si = ui + imi +ε +4πc , +si = −ui + imi +ε +4πc , +�si = �ui + i�mi +ε +4πc , +�si = −�ui + i�mi +ε +4πc , +(3.7) +and the corresponding exponentiated variables zi = e2πisi, zi = e2πisi, and �zi = e2πi�si, �zi = +e2πi�si, with zi = z∗ +i , �zi = �z∗ +i . We set mij = mi − mj, �mij = �mi − �mj, and introduce +ξij ≡ exp +� +−2πi4d +c +mij +2 +� +, +�ξij ≡ exp +� +−2πi4d +c +�mij +2 +� +, +ξ′ +ij ≡ exp +� +−2πi4d +c +mi − �mj +2 +� +, +(3.8) +– 15 – + +which are roots of unity depending on mij. +As we show in Appendix B, in terms of these variables, the integrand of (3.1) essentially +factorizes into two parts9 +Zhol(z, �z; τ, λ) Zantihol(z, �z; τ, λ) , +(3.9) +with +Zhol(z, �z; τ, λ) = +� +i +� +exp +� +2πi4πc +4iε +� +s2 +i − �s2 +i +�� +z−1/2 +i +�z1/2 +i +ξ +′ 1 +2 +ii +× +� +a=1,2 +� +z−1 +i +�zi ξ′ +ii ζ−1 +a +q +3 +4 ; q +� +∞ +� +a=3,4 +� +z−1 +i +�zi ξ′ +ii ζa q +1 +4 ; q +� +∞ +� +× +� +i>j +� � +z−1 +i +zj ξij ; q +� +∞ +� +z−1 +i +zj ξij q ; q +� +∞ +� +�z−1 +i +�zj �ξij ; q +� +∞ +� +�z−1 +i +�zj �ξij q ; q +� +∞ +× +� +a=1,2 +� +z−1 +i +�zj ξ′ +ij ζ−1 +a +q +3 +4 ; q +� +∞ +� +zj �z−1 +i +ξ′−1 +ji +ζa q +1 +4 ; q +� +∞ +× +� +a=3,4 +� +zj �z−1 +i +ξ′−1 +ji +ζ−1 +a +q +3 +4 ; q +� +∞ +� +z−1 +i +�zj ξ′ +ij ζa q +1 +4 ; q +� +∞ +� +(3.10) +and +Zantihol(z, �z; τ, λ) = Zhol(z, �z; τ, λ) +�� +k→−k, zi→zi, �zi→�zi, ζ1↔ζ3, ζ2↔ζ4 . +(3.11) +Given this factorization of the integrand, we now use the idea of [62] to make a similar +split in the integration measure using a change of contour. Recall that zi = e−miε/2ce2πiui, +so that we identify e−miε/2c as the modulus of zi, and 2πui as its argument. The idea is to +exchange the domain of the sum over mi and integration over ui with the full complex plane, +i.e., +1 +ε +� +mi∈Z +ε∆mi +� 1 +0 +dui +ε→0 +−−−→ 1 +ε +� +∞ +−∞ +d(εmi) +� 1 +0 +dui += 2c +ε +� ∞ +0 +d|zi| +|zi| +� 2π +0 +dArg(zi) +2π += 2c +ε +� +C +d2zi +2π|zi|2 , +(3.12) +where d2zi = |zi| d|zi|dArg(zi) is the flat measure on the plane. We thus obtain +I(τ; λ) = +1 +(N!)2 +(2c)2N +ε2N +� N +� +i=1 +� +C2 +d2zi +2πzizi +d2�zi +2π�zi�zi +� +Zhol(z, �z; τ, ζa) Zantihol(z, �z; τ, ζa) . (3.13) +9More precisely, in the case c > 1, the presence of the terms ξij, �ξij and ξ′ +ij forbids us from declaring +that Zhol is a holomorphic function of z (and Zanti−hol of z). However, as we shall see in the next section, +at the leading order in the Cardy-like limit ε ↘ 0, the dependence on ξij, �ξij, ξ′ +ij drops and the integrand +indeed factorizes, even for c > 1 in the cases we are interested in. This factorization should be thought of +as a simplifying manipulation, and should not be crucial to the derivation. Indeed, in a related problem, the +authors of [61] directly compute the large-N limit of the twisted index, without using this manipulation. +– 16 – + +One then factorizes the full integral into “holomorphic” and “anti-holomorphic” pieces as +I(τ; λ) = (4πc)N +εN +� +[Ds] [D�s] Zhol(z, �z; τ, λ) × (4πc)N +εN +� +[Ds] [D�s] Zantihol(z, �z; τ, λ) +(3.14) +where the holomorphic variables s, �s are integrated +� +[Ds] ≡ +1 +N! +N +� +i=1 +� +dsi , +� +[D�s] ≡ +1 +N! +N +� +i=1 +� +d�si +(3.15) +over some contour. The corresponding anti-holomorphic variables are taken to be independent +and run over a possibly different contour. The contour integrals are then evaluated by a +saddle-point approximation, which we discuss now. +3.1 +Generalized Cardy limits to roots of unity +Our goal is to calculate the asymptotic behavior of the integral (3.14) in the generalized +Cardy limit q = e2πi 4d +c e− ε +c as ε ↘ 0. In the saddle-point approximation, we can analyze +the holomorphic and anti-holomorphic parts separately, which are built out of Pochhammer +symbols. +To study the asymptotics of these building blocks, we use a result of [63], whose details +we summarize in Appendix A. This requires q = ξme− ε +m , where ξm is a primitive root of +unity of order m. Therefore, we write (c, 4d) = gcd(c, 4d)(ℓc, ℓd) with gcd(ℓc, ℓd) = 1, so that +q = ξℓce− ε/ gcd(c,4d) +ℓc +, and we can now apply the result from [63] directly. The leading order +result for the holomorphic part of the integrand (3.10) as ε ↘ 0 is +log Zhol(z, �z, q, λ) ∼ +− gcd(c, 4d)2 +cε +�� +i +1 +2(2πiℓc)2(s2 +i − �s2 +i ) ++ +� +i>j +� � +a=1,2 +Li2 +� +z−ℓc +i +�zℓc +j ζ−ℓc +a +(ξ′ +ijξ +− 1 +4 +ℓc )ℓc� +− Li2 +� +zℓc +j �z−ℓc +i +ζℓc +a (ξ′−1 +ji ξ +1 +4 +ℓc)ℓc� ++ +� +a=3,4 +Li2 +� +zℓc +j �z−ℓc +i +ζ−ℓc +a +(ξ′−1 +ji ξ +− 1 +4 +ℓc )ℓc� +− Li2 +� +z−ℓc +i +�zℓc +j ζℓc +a (ξ′ +ijξ +1 +4 +ℓc)ℓc�� ++ +� +i +� � +a=1,2 +Li2 +� +z−ℓc +i +�zℓc +i ζ−ℓc +a +(ξ′ +iiξ +− 1 +4 +ℓc )ℓc� +− +� +a=3,4 +Li2 +� +z−ℓc +i +�zℓc +i ζℓc +a (ξ′ +iiξ +1 +4 +ℓc)ℓc��� +. +(3.16) +Note that the vector multiplet does not contribute in the Cardy-like limit. Upon rescaling +the integration variables +� +si, �si, si, �si +� +�→ 1 +ℓc +� +si, �si, si, �si +� +we obtain +log Zhol(z, �z, q, λ) ∼ −gcd(c, 4d)2 +cε +W(z, �z, λ) + O(1) , +(3.17) +– 17 – + +where +W(z, �z, λ) = +� +i +1 +2(2πi)2(s2 +i − �s2 +i ) ++ +� +i>j +� � +a=1,2 +� +Li2 +� +z−1 +i +�zjζ−ℓc +a +(ξ′ +ijξ +− 1 +4 +ℓc )ℓc� +− Li2 +� +zj�z−1 +i +ζℓc +a (ξ′−1 +ji ξ +1 +4 +ℓc)ℓc�� ++ +� +a=3,4 +� +Li2 +� +zj�z−1 +i +ζ−ℓc +a +(ξ′−1 +ji ξ +− 1 +4 +ℓc )ℓc� +− Li2 +� +z−1 +i +�zjζℓc +a (ξ′ +ijξ +1 +4 +ℓc)ℓc��� ++ +� +i +� � +a=1,2 +Li2 +� +z−1 +i +�ziζ−ℓc +a +(ξ′ +iiξ +− 1 +4 +ℓc )ℓc� +− +� +a=3,4 +Li2 +� +z−1 +i +�ziζℓc +a (ξ′ +iiξ +1 +4 +ℓc)ℓc�� +. +(3.18) +Note that the arguments of the dilogarithms above contain factors of the type (ξ′ +ijξ +− 1 +4 +ℓc )ℓc = +exp +� +−2πiℓd +� mi−�mj +2 ++ 1 +4 +�� +, which deserve a comment. Since we have gcd(c, d) = 1, there are +three possible cases, i.e., gcd(c, 4d) = gcd(c, 4) = 1, 2, or 4. When gcd(c, 4) = 1, all these +factors in (3.18) equals 1. When gcd(c, 4) = 2, they are equal to e−2πi d +2 . In this case it is clear +from (3.18) that this phase can be absorbed into a redefinition of ζa. When gcd(c, 4) = 4, +there is no such simplification, and we will restrict to the first two cases from now on. +The case c = 1 and d = 0, which is relevant for the q → 1 limit of the ABJM index, has +been studied in [60, 64] by the saddle-point approximation in the small parameter ε. Even +in this approximation, the presence of the dilogarithm functions in W makes it difficult to +perform an exact analysis. However, the potential W simplifies in the large-N limit. We +review the main points of this analysis in Section 3.2, and use the large-N method to analyze +the perturbation theory in ε to all orders in Section 3.3. +3.2 +The large N saddle-point analysis +Having expressed the ABJM superconformal index in the generalized Cardy limit in terms +of an effective potential W (3.17), in order to find its value in the large-N limit we should +extremize W. As already noticed in [60], the effective potential W obtained in the generalized +Cardy limit is a straightforward generalization, at a mathematical level, of the Bethe potential +introduced in [33] to describe the topologically twisted index which is, a priori, a different +problem, and one can therefore follow the analysis developed in [33]. +Recall that the large-N limit of a unitary matrix model can be implemented by re- +placing the discrete distribution of the N eigenvalues by a continuum of eigenvalues on the +interval [0, 1]. Sums over the discrete label of the eigenvalues i turn into integrals over the +interval, which one replaces by integrals over the space of eigenvalues by introducing a density +of eigenvalues. +In our case, we have two distributions s and �s for the integration variables that are both +complex. Justified by the study of the numerics in [33], one introduces the following single-cut +ansatz +s(x) = v(x) − iN +1 +2 x , +�s(x) = �v(x) − iN +1 +2 x , +(3.19) +– 18 – + +where x ∈ [x1, x2] ∈ R. The assumptions in the ansatz are that Im(�s) = Im(s) and that x2 − +x1, v(x), �v(x) are all O(1) quantities as N → ∞. The sums become integrals according to +N +� +i=1 +. . . �→ N +� x2 +x1 +dx ρ(x) . . . , +(3.20) +where the eigenvalue density ρ obeys +� x2 +x1 +dx ρ(x) = 1 . +(3.21) +We split the effective action W in (3.18) as a sum of three pieces +W = W1 + W2 + W3 , +(3.22) +with +W1 = N +� x2 +x1 +dx ρ(x) 1 +2(2πi)2� +s(x)2 − �s(x)2� +, +W2 = N2 +� x2 +x1 +dx ρ(x) +� x2 +x +dy ρ(y) +� +� � +a=1,2 +� +Li2 +��z(x) z(y)−1 +ζℓc +a +� +− Li2 +�z(x) �z(y)−1 +ζ−ℓc +a +�� ++ +� +a=3,4 +� +Li2 +�z(x) �z(y)−1 +ζℓc +a +� +− Li2 +��z(x) z(y)−1 +ζ−ℓc +a +��� +� , +W3 = N +� x2 +x1 +dx ρ(x) +� +� � +a=1,2 +Li2 +��z(x) z(x)−1 +ζℓc +a +� +− +� +a=3,4 +Li2 +��z(x) z(x)−1 +ζ−ℓc +a +� +� +� . +(3.23) +Here we have introduced the notation z(x) = e2πis(x) and �z(x) = e2πi�s(x). Our goal is to +extremize W subject to the constraint (3.21), or extremizing the quantity +Wµ := W + N +3 +2 µ i +�� x2 +x1 +dx ρ(x) − 1 +� +. +(3.24) +The resulting equations, namely +δρWµ = 0 , +δvWµ = 0 , +δ�vWµ = 0 , +∂µWµ = 0 , +(3.25) +should be solved for the distributions ρ, s(x) and �s(x). +We now evaluate the three terms W1,2,3 in (3.22) on the ansatz (3.19). As we will shortly +see, each piece has a simple term scaling as N +3 +2 , and subleading terms scaling as N, as N → ∞. +The first piece is +W1 ∼ −N +3 +2 +� x2 +x1 +dx ρ(x) (2π)2ix δv(x) +� +1 + O(1/N +1 +2 ) +� +, +(3.26) +– 19 – + +where δv(x) ≡ �v(x) − v(x). +The second piece contains terms of the following form +� x2 +x1 +dx ρ(x) +� x2 +x +dy ρ(y) Li2 +��z(x) z(y)−1 +ζℓc +a +� += +� x2 +x1 +dx ρ(x) +� x2 +x +dy ρ(y) Li2 +� +e2πN +1 +2 (x−y)+2πi +� +�v(x)−v(y)−λ′ +a +�� +, +(3.27) +where λ′ +a ≡ ℓcλa. +In matrix model language, these terms indicate non-local interactions +between the eigenvalues at x and y. However, at leading order in the large-N expansion, the +integrand simplifies further, and we are left only with local interactions at each x. To see +this, it is convenient to change integration variables from y to δy := N +1 +2 (y − x), after which +the right-hand side of (3.27) becomes +N− 1 +2 +� x2 +x1 +dx ρ(x) +� N +1 +2 (x2−x) +0 +dδy ρ(x + δy/N +1 +2 ) Li2 +� +e−2πδy+2πi +� +�v(x)−v(x+δy/N +1 +2 )−λ′ +a +�� +. (3.28) +At leading order in the large-N expansion (at fixed δy) this takes the asymptotic form +N− 1 +2 +� x2 +x1 +dx ρ(x)2 +� +∞ +0 +dδy Li2 +� +e−2πδy+2πi(δv(x)−λ′ +a)� +. +(3.29) +Then, using d +dxLi3(ex) = Li2(ex), one concludes that at leading order +� x2 +x1 +dx ρ(x) +� x2 +x +dy ρ(y) Li2 +��z(x) z(y)−1 +ζℓc +a +� +∼ N− 1 +2 +2π +� x2 +x1 +dx ρ(x)2 Li3 +� +e2πi(δv(x)−λ′ +a)� +. +(3.30) +Upon applying the above analysis to the four terms in W2, one obtains +W2 ∼ N +3 +2 +2π +� x2 +x1 +dx ρ(x)2 +� +� � +a=1,2 +� +Li3 +� +e2πi(δv(x)−λ′ +a)� +− Li3 +� +e−2πi(δv(x)−λ′ +a)�� +− +� +a=3,4 +� +Li3 +� +e2πi(δv(x)+λ′ +a)� +− Li3 +� +e−2πi(δv(x)+λ′ +a)�� +� +� . +(3.31) +We can simplify this expression further using the identity (A.9) +Li3 +� +e2πix� +− Li3 +� +e−2πix� += 4π3i +3 +B3(x) +(3.32) +where B3 is the third periodic Bernoulli polynomial that is defined in (A.7), obtaining +W2 ∼ 2π2i +3 +N +3 +2 +� x2 +x1 +dx ρ(x)2 +� +� � +a=1,2 +B3 +� +δv(x) − λ′ +a +� +− +� +a=3,4 +B3 +� +δv(x) + λ′ +a +� +� +� . +(3.33) +– 20 – + +Finally, the third piece W3 can be written as +W3 ∼ N +� x2 +x1 +dx ρ(x) +� +� � +a=1,2 +Li2 +� +e2πi(δv(x)−λ′ +a)� +− +� +a=3,4 +Li2 +� +e2πi(δv(x)+λ′ +a)� +� +� . +(3.34) +Note that the factor in front of the integral in (3.34) is N, instead of N +3 +2 as in W1, W2. +This means that W3 does not contribute to the on-shell value of the effective action W in +the large-N limit. However, it cannot be naively discarded, since it is relevant for the saddle +point equations, as the dilogarithm Li2(z) is not analytic at the branch point z = 1. Indeed, +the first derivative of the integrand in (3.34) with respect to δv(x) can grow as O(N +1 +2 ), if the +function approaches a branch point. In this case, W3 contributes at the same order as W1 +and W2 to the equation in (3.25) obtained by taking the derivative with respect to δv(x). +At this stage, the degree of difficulty of the original extremization problem is substantially +diminished, as the large-N form of the potential W given by the sum of (3.26), (3.33), (3.34), +has no non-local terms. Moreover, if one focuses on the contribution at order N +3 +2 to the +potential, one notices that in terms of the field variables ρ(x) and δv(x) the problem is +piecewise quadratic. Therefore, solutions to the variational problem (3.25) can be found by +using a linear ansatz ρ(x) = ρ0 + xρ1 for the density of eigenvalues. The solutions when all +the λas are equal are particularly simple, and we present them below. The details for generic +values of λas are discussed in Appendix C. +Unrefined index +We set all the chemical potentials to be equal (λa ≡ λ). +The con- +straint (2.17) implies that λ = n1/4. As for λ′ +a, recall from the discussion below (3.18), that +if gcd(c, 4) = 2, a phase is absorbed in ζa, so we write λ′ = (ℓcn1 + ℓd)/4, with the under- +standing that if gcd(c, 4) = 1, then ℓd = 4d and thus ℓd/4 can be removed from λ′ (which +is defined modulo 1), and if gcd(c, 4) = 2, then ℓd = 2d and ℓd/4 is non-trivial. We further +assume that δv(x) does not cross a branch point of the dilogarithm, so that we can effectively +ignore W3. This picture is also warranted by the numerics [33]. To leading order in N, the +function to be extremized Wµ defined in (3.24) takes the simple form +Wµ = N +3 +2 i +� x2 +x1 +dx ρ(x) +� +−4π2x δv(x) + 4π2 +3 ρ(x) (B3 (X−(x)) − B3 (X+(x))) +� ++ N +3 +2 µ i +�� x2 +x1 +dx ρ(x) − 1 +� +, +(3.35) +where +X±(x) ≡ δv(x) ± ℓcn1 + ℓd +4 ++ w± +(3.36) +for some integers w± such that +0 < X±(x) < 1 +(3.37) +In the following, we shall re-express these integers using Σ ≡ w++w− and ∆ ≡ w+−w−. The +single-cut solution is defined on a single sheet of the multi-valued polylogarithms provided +w± do not depend on x. +– 21 – + +Combining the first three extremization equations (3.25) gives (ignoring boundary terms) +0 = 4x + ρ(x)(ℓcn1 + ℓd + 2∆) (2(Σ − 1) + 3 δv(x)) , +0 = 48π2x δv(x) + π2ρ(x)(ℓcn1 + ℓd + 2∆) +� +8 + (ℓcn1 + ℓd + 2∆)2 ++ 12 Σ(Σ − 2) + 48 δv(x)(Σ − 1) + 48 δv(x)2� +− 12µ , +(3.38) +These equations are easily solved for ρ(x) and δv(x). Indeed, for fixed x, the first equation is +linear in ρ and δv, and upon substituting in the second equation the expression for ρ obtained +from the first (and assuming that ρ is finite), we find that the resulting equation is linear +in δv. The solutions are: +ρ(x) = +12µ + 24π2(Σ − 1)x +π2 (ℓcn1 + ℓd + 2∆ − 2) (ℓcn1 + ℓd + 2∆) (ℓcn1 + ℓd + 2∆ + 2) , +δv(x) = −π2x(ℓcn1 + ℓd + 2(∆ − Σ))(ℓcn1 + ℓd + 2(∆ + Σ)) − (Σ − 1) +� +6 + 8π2x(2Σ − 1) +� +12 (µ + 2π2x(Σ − 1)) +. +(3.39) +We focus on the solution with constant eigenvalue density, so we set Σ = 1, which means that +∆ must be odd.10 The conditions (3.37) imply +6|µ| +π2(ℓcn1 + ℓd + 2∆ + 2)(ℓcn1 + ℓd + 2∆ − 2) < x < +−6|µ| +π2(ℓcn1 + ℓd + 2∆ + 2)(ℓcn1 + ℓd + 2∆ − 2) , +−2 < ℓcn1 + ℓd + 2∆ < 2 . +(3.40) +In particular, the second inequality in (3.40) imposes that ℓcn1 + ℓd + 2∆ ∈ {−1, 0, 1}, but +the choice 0 (from the first equation in (3.39)) leads to a divergent density ρ unless µ = 0 +(which would reduce the support to a single point), and therefore should be discarded. Thus +ρ = − sgn(ℓcn1 + ℓd + 2∆) 4µ +π2 , +δv = π2 +4µx . +(3.41) +Clearly, in order to have a positive density of eigenvalues, we need µ ∈ R and sgn(µ) = +− sgn(ℓcn1 + ℓd + 2∆). +The first inequality in (3.40) is only a necessary condition for the range of x such that +(3.37) holds. In fact, upon substitution of ℓcn1 + ℓd + 2∆ = ±1, (3.37) reduces to +− |µ| +π2 < x < |µ| +π2 . +(3.42) +This is the smallest interval for x such that the single-cut solution does not cross a branch +cut of the polylogarithm, thus we set x1 = − |µ| +π2 and x2 = +|µ| +π2 . +Finally, we impose the +10The solution with constant eigenvalue density can also be obtained as a limit of the solutions to the saddle +point equations of the refined index in Appendix C. +– 22 – + +normalization of ρ, which is equivalent to extremizing Wµ with respect to µ, finding +ρ = +√ +2 , +δv = − sgn(ℓcn1 + ℓd + 2∆) x +√ +2 +[x1, x2] = +� +− 1 +2 +√ +2, +1 +2 +√ +2 +� +, +µ = − sgn(ℓcn1 + ℓd + 2∆) π2 +2 +√ +2 . +(3.43) +Overall, ∆ remains a free odd integer, and +W = ∓ iπ2 +3 +√ +2N +3 +2 +if ℓcn1 + ℓd = ±1 +mod 4. +(3.44) +We then substitute in (3.17) and perform the same analysis for the anti-holomorphic part +(3.11), reaching the following conclusion. The extremization problem for the unrefined index +has O(N +3 +2 ) scaling only if +ℓcn1 + ℓd = ±1 +mod 4 , +(3.45) +and +log I(τ; λ) ∼ ∓ π +3 +√ +2N +3 +2 +1 +ℓc (ℓcτ − ℓd) +as τ → 4d +c and ℓcn1 + ℓd = ±1 +mod 4 . +(3.46) +We recall that this holds if gcd(c, 4d) = 1, 2. +To summarize, we have the following picture: we start with the unrefined index, which +is a single-valued function of T +I(T) = TrHBPS(−1)2Je2πiT(4J+� +a Qa) = +� +n +dnQn , +(T ∼ T + 1 , +Q ≡ e2πiT ) +(3.47) +and to find dn we look at the limit where Q goes to a primitive root of unity, as explained in +Section 1, or T → D/C with gcd(C, D) = 1. In order to study the asymptotic behavior, we +further introduce 4T = ℓD/ℓc, for coprime ℓc, ℓD. In terms of these, we write the constraint +(3.45) as 4ℓcT = ℓD = ±1 mod 4. Since ℓD = 4D/ gcd(C, 4), this constraint can only be +solved if gcd(C, 4) = 4, in which case we have +log I(T) ∼ ∓ π +3 +√ +2N +3 +2 +1 +C +4 +� +CT − D +� +as T → D +C with D = ±1 mod 4. +(3.48) +This leads us to the conclusion that the leading growth for the index (3.47) is controlled by the +singularity near T = ± 1 +4, that is, when Q approaches the non-trivial primitive fourth roots of +unity. In terms of τ and n1, which is how the index has been presented in the literature, our +methods allow us to reach these singularities as τ → 0 and n1 = 1, 3, τ → 2 and n1 = 1, 3. We +are currently unable to study the case τ → 1, 3 as n1 = 0, but it is natural to conjecture that +it gives the same result. Since the index is a four-sheeted function of τ, the singularities in +the large-N limit appear on the first and third sheet, or taking the Cardy limit τ → 0 of the +R-charge index defined in (2.20). There is a clear analogy with the case of four-dimensional +N = 4, which is developed in Appendix E. +– 23 – + +Refined index +We now move to the most refined ABJM index, where the chemical poten- +tials λa are taken to be generic. In this case, the solutions to the extremization problem (3.25) +are more involved than (3.39), and have been addressed in [33] using the techniques of [32]. +Details of their construction are reviewed in Appendix C. Here, we notice that, assuming +λa ∈ R, a solution to the extremization problem (3.25) with scaling N +3 +2 in the large-N limit +exists in terms of a single-cut eigenvalue distributions only provided the chemical potentials +and ℓc satisfy +4 +� +a=1 +{ℓcλa + ℓd/4} = 1 or 3 . +(3.49) +Here, as in (3.36), we have introduced ℓd/4, which is either an integer or a half-integer, +depending on whether gcd(c, 4) = 1, 2, respectively. Notice that this constraint consistently +reduces to (3.45) upon setting all λa to be equal. +Combining the contributions from the extremization of the holomorphic and the anti- +holomorphic parts in (3.14), one finds that the leading result for the ABJM index, as ε = +2πi (cτ − 4d) ↘ 0, is +log I(τ; λ) ∼ ∓ 8π +√ +2 N +3 +2 +3 +� +{ℓcλ1 + ℓd/4}±{ℓcλ2 + ℓd/4}±{ℓcλ3 + ℓd/4}±{ℓcλ4 + ℓd/4}± +c(cτ − 4d) +. +(3.50) +where {x}+ := {x} and {x}− := {x} − 1 . The upper and lower signs correspond to the first +or second case in (3.49). +A few comments are in order. Firstly, studying the regions associated to different asymp- +totic behaviors of the fully refined index I(τ; λ) is quite involved. It is clear that (3.49) in +the case ℓc = 1 and ℓd = 0 reduces to n1 = 1, 3, and thus we obtain the same picture to the +one just described for the unrefined index: in the Cardy limit τ → 0, the non-trivial large-N +limit is found on the first and third sheet of the multivalued function I(τ), which corresponds +to the R-charge index defined in (2.20). More generally, though, the relation of (3.49) to the +shifts of τ is more involved.11 +Secondly, notice that (3.50) is invariant under a shift of any λa by +1 +ℓc , in addition to +the shift λa → λa + 1 due to the quantization of the charges Qa (see (2.18)). The emergent +periodicity suggests that in the expansion near roots of unity the superconformal index only +counts operators with charges dual to λa being an integer multiple of ℓc. That is, in such +an expansion the contribution to the trace (2.18) of operators with charges dual to λa not +being an integer multiple of ℓc is suppressed. Interestingly, the same phenomenon happens in +four-dimensions [11–13]. +Finally, recall that the partition function in the microcanonical ensemble is obtained by +taking the Laplace transform of (3.48). The saddles near T → ± 1 +4 (equivalently, c = 1 and +d = 0 and 2 for fixed n1 = 1) contribute equally in magnitude to the Laplace transform, +and with opposite phases. The logarithm of the real part agrees with the entropy of the +11Of course, the constraint (3.49) can be expressed without reference to n1, as it should from its definition. +– 24 – + +dual supersymmetric black hole, and the interference of the phases produces macroscopic +oscillations (of order N +3 +2 ) in the microcanonical index. +The same phenomenon has been +observed in related contexts in [10, 18, 19, 60, 65, 66]. +Index with pairwise equal fugacities +Finally, we discuss a further case, which is relevant +for the comparison with holographic duals in Section 6, namely the index (2.22) with pairwise +equal fugacities. We set λ1,2 ≡ σ1 and λ3,4 ≡ σ2. The result can be straightforwardly obtained +from the fully refined case just discussed (see appendix C). Assuming that σ1,2 ∈ R, the single- +cut eigenvalue distribution provides a solution to the extremization problem that scales like +N +3 +2 in the large-N limit provided +2 ({ℓcσ1 + ℓd/4} + {ℓcσ2 + ℓd/4}) = 1 or 3 . +(3.51) +Substituting the constraint 2(σ1 + σ2) = n1, it is straightforward to show that the left-hand +side of (3.51) has the same parity as ℓcn1 + ℓd, which in the cases we are interested in +corresponds to that of ℓcn1 (since ℓd is even). Therefore, ℓcn1 cannot be even, which singles +out the R-charge index among the sheets in (2.20). More precisely, if c is odd, then the +condition (3.51) is satisfied if cn1 is odd, and imposes {cσ1} ≤ 1 +2 for the right-hand side to +be 1, and {cσ1} > 1 +2 for the right-hand side to be 3. If instead gcd(c, 4) = 2, then one needs +cn1/2 odd, but now {cσ1} ≤ 1 +2 is consistent with the right-hand side being 3, and {cσ1} > 1 +2 +is consistent with the right-hand side being 1. +Upon substitution, in the case of c odd, we require n1 to be odd, and the resulting value of +the large-N partition function is +log I(τ; σ) ∼ −8π +√ +2 +3 +N +3 +2 {cσ1}{cσ2} +ℓc (cτ − 4d) +if τ → 4d +c and {cσ1} ≤ 1 +2 , +log I(τ; σ) ∼ +8π +√ +2 +3 +N +3 +2 (1 − {cσ1})(1 − {cσ2}) +c (cτ − 4d) +if τ → 4d +c and {cσ1} > 1 +2 . +(3.52) +3.3 +Subleading effects +Now we consider sub-leading effects in the asymptotic expansion in ε around any rational +point d/c. Our goal is to show that the expansion terminates at order ε i.e. all the terms of +order εk, k ≥ 2 vanish in the large-N limit. The same result was found in [67] for the case +of the black hole saddle i.e. (c, d) = (1, 0). We use the leading order analysis of the previous +subsection as a reference and discuss the new points that arise in the all-order analysis. +We begin again with the holomorphic part of the potential (3.10), expand it to all orders +in ε as given in (A.2), and run the steps of the large-N expansion as in Sections 3.1, and 3.2. +The potential W splits into three pieces W1,2,3 as in (3.22), (3.23), exactly as at leading +order. The term W1 is classical and does not depend on ε and therefore remains the same as +in (3.23). The other two pieces W2, W3 have, a priori, an infinite expansion governed by the +identity (A.2), with the leading-order term given by (3.23). +– 25 – + +Consider a general term of order εk, k ≥ 2, in the expansion of W2. This term has the +same structure as in the leading O(1/ε) term with Li1−k replacing Li2, with coefficients being +Bernoulli polynomials given by (A.4). +We first consider the case τ → 0, so that (c, d) = (1, 0). We use (A.2) with the value of w +and ν determined by the Pochhammer symbols in (3.10). We end up with a double integral +as in (3.23) with linear combinations of terms +Bk+1(−ν) Li1−k +��z(x) z(y)−1 +ζ +� +− Bk+1(1 + ν) Li1−k +�z(x) �z(y)−1 +ζ−1 +� +, +(3.53) +with ζ = ζℓc +a , a = 1, . . . , 4, and ν = 0 for vector multiplets and ν = − 1 +4 for hypermultiplets. +The large-N analysis of the term W2 proceeds as before for every k. The non-local interactions +drop out—effectively identifying x and y in (3.53)—and the arguments of the polylogarithms +in the combination (3.53), writing ζ = e2πiλ, become Z and 1/Z with Z = e2πi(δv(x) − λ). +Continuing onwards, we have, in the analog of the step (3.30), that Li1−k integrates +to Li2−k. This gives an integral as in (3.31) with the integrand consisting of linear combina- +tions of the differences +Bk+1(−ν) Li2−k(Z) − Bk+1(1 + ν) Li2−k(1/Z) , +ν = 0 , −1 +4 , +(3.54) +with Z = e2πi(δv(x) − λ) as above. Now, using +Bk(1 + ν) = (−1)kBk(−ν) , +k > 2 , +0 ≤ −ν ≤ 1 , +(3.55) +we see that the combination (3.54) equals +Bk+1(−ν) +� +Li2−k(Z) − (−1)k+1Li2−k(1/Z) +� +, +(3.56) +which vanishes due to the following classical identity satisfied by polylogarithms +Li−r(Z) + (−1)r Li−r(1/Z) = 0 , r ≥ 1 . +(3.57) +The case when τ → Q is not much different compared to when τ → 0. Consider τ → +d/c, c, d ∈ Z with gcd(c, d) = 1. The asymptotic expansion (A.4) now contains c terms, and +in lieu of (3.54) we have, with ν = 0, − 1 +4, +c +� +t=1 +Bk+1 +� +1 − t + ν +c +� +Li2−k +� +Z e2πi 2d +c (t+ν)� +− +c +� +t′=1 +Bk+1 +� +1 + ν + 1 − t′ +c +� +Li2−k +� +Z−1e−2πi 2d +c (ν+1−t′)� +. +(3.58) +Now, for each value of t in the first term, there corresponds exactly one value of t′ in the +second given by +t′ = c + 1 − t . +(3.59) +– 26 – + +Pairing up the two sums in this manner, we obtain that the coefficient of the term εk is +proportional to +c +� +t=1 +� +Bk+1 +� +1 − t + ν +c +� +Li2−k +� +Z e2πi 2d +c (t+ν)� +− Bk+1 +�t + ν +c +� +Li2−k +� +Z−1e−2πi 2d +c (ν+t)�� += +c +� +t=1 +Bk+1 +� +1 − t + ν +c +� � +Li2−k +� +Z e2πi 2d +c (t+ν)� +− (−1)k+1 Li2−k +� +Z−1e−2πi 2d +c (ν+t)�� += 0 . +(3.60) +Here, the first equality follows from (3.55) (and that 0 ≤ 1 − (t + ν)/c ≤ 1 for the above +values), and the second equality follows from the fact that each term in the sum vanishes due +to (3.57). +Finally, we turn to the analysis of W3. Now there is a significant difference compared to +the leading term. Recall from the discussion below (3.34) that the 1/N +1 +2 suppression of W3 +is compensated by the N +1 +2 -growth of the derivative of Li2. This phenomenon occured due to +the fact that the derivative Li′ +2(z) is large while Li2(z) itself is small is due to the logarithmic +non-analyticity of the function at z = 0. For the terms O(εk), the Li2 is replaced by Li1−k, +which, for k ≥ 2, are meromorphic functions and therefore do not have large derivative, as +can be explicitly checked. We conclude that the W3 term is suppressed at large N for k ≥ 2. +Collecting the above facts together, we reach the conclusion that the perturbation ex- +pansion of the large-N index around any rational point only contains terms multiplying ε−1, +ε0, and ε, and the O(εk) terms vanish for all k ≥ 2. +4 +Black holes and supersymmetric solutions +In this section we move to the bulk gravity computation. We briefly review the Kerr–Newman- +AdS black hole, then consider a related family of complex solutions to minimal gauged su- +pergravity, and finally connect to the BPS black hole. Similar studies of these solutions have +been made in [34, 35]. +4.1 +Kerr–Newman-AdS black hole +The Lorentzian action for Einstein–Maxwell theory with a cosmological constant is +S = +1 +16πG4 +� +Y4 +� +R + 6 − F2� +volG . +(4.1) +Here R is the Ricci scalar of the metric G, F is the curvature of the U(1) gauge field A, and +−3 is the cosmological constant. A black hole solution with rotation and electric charge has +been known for a long time [68]. In a frame that is non-rotating at infinity, which is more +– 27 – + +immediate for uses in AdS/CFT [69, 70], the solution is +ds2 = −∆r∆Θ +BΞ2 dt2 + sin2 Θ B +� +dφ + a∆Θ +∆r − (1 + r2)(r2 + a2) +BWΞ2 +dt +�2 ++ W +�dr2 +∆r ++ dΘ2 +∆Θ +� +, +A = mr sinh δ +WΞ +� +∆Θ dt − a sin2 Θ dφ +� ++ γ dt . +(4.2) +Here γ is a constant, Θ ∈ [0, π), φ ∈ [0, 2π), and +∆r = (r2 + a2)(1 + r2) − 2mr cosh δ + m2 sinh2 δ , +∆Θ = 1 − a2 cos2 Θ , +W = r2 + a2 cos2 Θ , +Ξ = 1 − a2 , +B ≡ ∆Θ(r2 + a2)2 − a2 sin2 Θ ∆r +WΞ2 +. +(4.3) +The black hole has an outer horizon at r = r+, the largest positive root of ∆r. On a slice +of constant t and r outside the horizon, the solution is topologically a sphere, and it is easy to +see that it closes off smoothly at Θ = 0, π with φ ∼ φ+2π. We then perform a Wick rotation +t = −itE and look near the horizon in geodesic coordinates. The space caps off smoothly at +the horizon only if we identify +(tE, φ) ∼ (tE, φ + 2π) ∼ (tE + β, φ − iΩβ) , +(4.4) +where the temperature and angular velocity of the horizon are +β = 4πa2 + r2 ++ +∆′r(r+) , +Ω = a 1 + r2 ++ +a2 + r2 ++ +. +(4.5) +The metric (4.2), once Wick rotated, is real if we take a to be pure imaginary and δ, m to +be real, whereas the gauge field A is pure imaginary provided γ is real. This also makes Ω +above pure imaginary, and the topology is that of the product of a disc and a 2-sphere [38]. +The metric (4.2) has two obvious commuting Killing vectors ∂t, ∂φ, and the surface {r = r+} +is a Killing horizon of the linear combination +V = ∂ +∂t + Ω ∂ +∂φ , +(4.6) +The electrostatic potential is12 +Φe := ιV A|r=r+ − ιV A|r→∞ += m sinh δ +r+ +a2 + r2 ++ +. +(4.7) +12More generally, we should only pick the Θ-independent part of ιV A|r→∞. +– 28 – + +In Euclidean signature, the horizon is a bolt for V , and regularity of the gauge field on the +disc requires that ιV A vanishes at the origin r = r+. This fixes the gauge choice in (4.2) to +be γ = −Φe. The Bekenstein–Hawking entropy of the horizon is +S = +π +G4 +r2 ++ + a2 +1 − a2 . +(4.8) +That the black hole is asymptotically anti-de Sitter can be shown by considering the +region r → ∞ and applying the following coordinate change in the asymptotic region [71] +cos θ +z += r cos Θ , +z−2 = r2∆Θ + a2 sin2 Θ +Ξ +, +(4.9) +obtaining, as z → 0, +ds2 ∼ dz2 +z2 + 1 +z2 +� +−dt2 + +� +dθ2 + sin2 θ dφ2�� +, +A ∼ −Φe dt . +(4.10) +The boundary metric in Euclidean signature is not just the round S1 × S2, due to the iden- +tifications (4.4). To see this explicitly, define tE = tE, �φ = φ + iΩtE, so that the boundary +metric has the fibered form +ds2 +bdry = dt2 +E + +� +dθ2 + sin2 θ (d�φ − iΩ dtE)2� +, +(4.11) +but now with the identifications +(tE, �φ) ∼ (tE, �φ + 2π) ∼ (tE + β, �φ) , +(4.12) +This metric is real for pure imaginary a, as consistent with the comments below (4.5). It is +clear that the metric (4.11) and the boundary gauge field A ∼ iΦe dtE match the metric of +the Euclidean background over which the index is computed as a functional integral (2.12), +and the background gauge field coupled to the u(1)R symmetry generated by H1, A(H1) = +iΦ(H1) dtE (2.27). +In order to compute the conserved charges, we can use the standard methods of holo- +graphic renormalization (see [72] for a review). We introduce a cutoff at z = δ ≥ 0 and +consider the geometry of the hypersurface ∂Yδ = {z = δ} ∩ Y4 ∼= M3, with induced metric h. +In order to make the problem well-defined and remove the divergences, we need to add to +the action the Gibbons–Hawking–York term and the counterterms, so that the renormalized +on-shell action is +I = lim +δ→0 +� +S + +1 +8πG4 +� +∂Yδ +� +K − 2 − 1 +2R +� +volh +� +. +(4.13) +We then define the holographic stress-energy tensor and electric current +⟨Tij⟩ = − +2 +√−g +δI +δgij , +⟨ji⟩ = +1 +√−g +δI +δAi +, +(4.14) +– 29 – + +where i, j are labels for the boundary coordinates, and g, A are, respectively, the boundary +metric and gauge field given by (4.11) and A = −Φe dt. These quantities satisfy the following +equations +∇i⟨Tij⟩ = Fji⟨ji⟩ , +∇i⟨ji⟩ = 0 , +⟨T i +i⟩ = 0 , +(4.15) +and for any boundary vector Kj generating a symmetry (that is LKg = 0 and LKA = 0) we +can construct a conserved current +∇i +� +(⟨T i +j⟩ + ⟨ji⟩Aj)Kj� += 0 , +(4.16) +where ∇ is the Levi-Civita connection of the boundary metric g. +Let C be a surface of +constant t, and ui be the future-directed unit normal to C. +Then, the conserved charge +associated to K is computed by +Q[K] = +� +C∩M3 +ui(⟨T i +j⟩ + ⟨ji⟩Aj)Kj volC∩M3 . +(4.17) +Notice that this definition is not invariant under gauge transformations of A (as stressed in +[73, 74]), but it still gives a conserved charge provided the gauge transformed A′ still satisfies +LKA′ = 0. The angular momentum is associated to K = −∂φ, so in our gauge +J = − +� +C∩M3 +ui⟨T i +φ⟩ volC∩M3 = am cosh δ +G4Ξ2 +. +(4.18) +We define the electric charge as +Qe = +� +C∩M3 +ui⟨ji⟩ volC∩M3 = − +1 +4πG4 +� +C∩M3 +∗4F = m sinh δ +G4Ξ +. +(4.19) +Finally, the energy is associated to K = ∂t, and in our gauge choice +E′ = +� +C∩M3 +ui⟨T i +t⟩ volC∩M3 − Φe +� +C∩M3 +ui⟨ji⟩ volC∩M3 = m cosh δ +G4Ξ2 +− ΦeQe += E − ΦeQe . +(4.20) +Here E = m cosh δ/G4(1 − a2) is the value of the energy in the gauge A = 0. +Moving to Euclidean signature, we can compute the holographically renormalized on-shell +action, obtaining +I = +β +2G4(1 − a2) +�r2 ++ + a2 +r+ +− m cosh δ + a2 m2 sinh2 δ +r+(r2 ++ + a2) +� +. +(4.21) +This action obeys the quantum statistical relation [72] +I = −S + β(Q[V ] − Φh Qe) += −S + β(Q[∂t] − Ω Q[−∂φ] − Φh Qe) +(4.22) +– 30 – + +where Φh = ιV A|r=r+ is by definition a constant. Even though each term in the relation above +is separately not gauge invariant, the overall relation is invariant under gauge transformations +of A. Concretely, it reduces to the canonical form in the A = 0 gauge +I = −S + β(E − Ω J − Φe Qe) . +(4.23) +Moreover, varying δ, a, m, we find that the first law of thermodynamics holds. Written +as above in terms of holographic conserved charges it reads [72] +dQ[∂t] = β−1 dS + Ω dQ[∂φ] + Φh dQe +(4.24) +or concretely +dE = β−1dS + Ω dJ + Φe dQe . +(4.25) +It follows from combining (4.23) and (4.25) that β, Ω, Φe are the chemical potentials conjugate +to the conserved charges, since +E = ∂I +∂β +���� +βΩ,βΦe +, +J = − 1 +β +∂I +∂Ω +���� +β,Ωe +, +Qe = − 1 +β +∂I +∂Φe +���� +β,Ωe +. +(4.26) +This shows that the on-shell action I = I(β, Ω, Φe) is minus the logarithm of the grand +canonical partition function, the Gibbs free energy. +4.2 +Supersymmetry +The Einstein–Maxwell action (4.1) describes the bosonic sector of four-dimensional minimal +gauged supergravity [75]. A solution to the equations of motion coming from (4.1) is super- +symmetric provided there is a non-zero Dirac spinor ϵ satisfying the equation +� +∇µ − iAµ + 1 +2Γµ + i +4FνρΓνρΓµ +� +ϵ = 0 . +(4.27) +The condition of supersymmetry on the solution (4.2) with parameters (a, δ, m) implies [76, 77] +E = J + Qe +⇔ +a = coth δ − 1 . +(4.28) +Thus, the family of supersymmetric solutions can be parametrized by the two parameters δ, m. +Supersymmetry and global regularity requires that in presence of non-zero electric charge the +angular momentum must be non-zero [77].13 +Imposing (4.28) reduces ∆r in (4.3) to a sum of squares +∆r|SUSY = +� +r2 − coth δ + 1 +�2 + coth2 δ +� +r − msinh2 δ +cosh δ +�2 +. +(4.29) +13The conclusion is different in presence of a magnetic charge [78]. +– 31 – + +Assuming reality of all parameters, both squares should vanish at the horizon, which fixes +the value of the horizon radius r∗ and m: +r2 +∗ = coth δ − 1 , +m2 = +cosh2 δ +eδ sinh5 δ , +(4.30) +leaving only one free parameter δ. +It is now easy to check that ∆′ +r|SUSY(r∗) = 0, that +is, supersymmetry and regularity of the Lorentzian metric imply extremality. +In order to define the gravitational partition function and reproduce the large-N behavior +of the supersymmetric index (2.18), we need to consider supersymmetric solutions connected +to the Euclidean solutions. As we observed around (4.5), the metric is real after Wick rotation +provided we choose a to be pure imaginary. Clearly, this can only be compatible with the +supersymmetry condition (4.28) if δ is complex, but this is itself incompatible with a real +Euclidean metric. Therefore, we conclude that the Wick rotation of a real supersymmetric +Lorentzian metric of the form (4.2) is generically complex.14 +We shall therefore focus on the family of complex metrics obtained by imposing the +supersymmetry condition (4.28) without requiring reality of the metric and gauge field. These +solutions arise from extending to complex parameters the Euclidean “black hole” solution with +topology R2 × S2. This approach was first suggested in five dimensions in [3] and elaborated +in other dimensions (including four) in [34]. Here, we have a two-parameter family, and it +is convenient to trade the parameter δ for r∗ using (4.30), and m for the largest root r+ +of (4.29), i.e., +m = +1 +sinh2 δ +� +r+ cosh δ ± i +� +sinh δ(1 + r2 ++) − cosh δ +�� +. +(4.31) +The thermodynamic quantities of the Euclidean supersymmetric solutions are generically +complex. The chemical potentials +β = ±2πi +r2 ++ + r4 +∗ +(r2 ++ − r2∗)(1 ± 2ir+ + r2∗) , +Ω = r2 +∗ +1 + r2 ++ +r2 ++ + r4∗ +, +Φe = r+ +r+r2 +∗ + r+ ± i(r+ − r∗)(r+ + r∗) +r2 ++ + r4∗ +, +(4.32) +and the conjugate charges +E = r+r2 +∗ + r+ ± i(r+ − r∗)(r+ + r∗) +G4(1 − r4∗)(1 − r2∗) +, +J = r2 +∗ +r+r2 +∗ + r+ ± i(r+ − r∗)(r+ + r∗) +G4(1 − r4∗)(1 − r2∗) +, +Qe = r+r2 +∗ + r+ ± i(r+ − r∗)(r+ + r∗) +G4(1 − r4∗) +(4.33) +14The bulk Killing spinor equation (4.27) is analytic in the supergravity fields, so it is still satisfied by the +Wick-rotated solution. +– 32 – + +can be computed directly or read off by imposing (4.28) in the corresponding expressions +for Wick-rotated non-supersymmetric solutions. Here the different sign choices refer to the +two branches of solutions to (4.31). The conserved charges satisfy the supersymmetry con- +dition (4.28) by construction, and we observe that the chemical potentials also satisfy the +constraint +β (1 − 2Φe + Ω) = ∓2πi . +(4.34) +If in addition to supersymmetry we impose extremality, we restrict to the Wick rotation +of the supersymmetric Lorentzian extremal solution discussed around (4.29), which we call +the BPS locus and indicate by an underscript ∗, as in [3]. The charges of the extremal solution +E∗ = +r∗ +G4(1 − r2∗)2 , +J∗ = +r3 +∗ +G4(1 − r2∗)2 , +Qe∗ = +r∗ +G4(1 − r2∗) +(4.35) +are real, and the extremal chemical potentials +Ω∗ = 1 , +Φe∗ = 1 +(4.36) +are real and fixed to constant values independent of the charges. Clearly the constraint (4.34) +stops being meaningful. It will be useful instead to define the “reduced chemical potentials” +for the supersymmetric solutions +τg ≡ β Ω − Ω∗ +2πi +, +ϕg ≡ β Φe − Φe∗ +2πi +, +(4.37) +which by construction satisfy the constraint +τg − 2ϕg = ∓1 . +(4.38) +This allows us to write the quantum statistical relation (4.23) as +I = β(E − J − Qe) − S − 2πiτg J − 2πiϕg Qe +⇒ +I|SUSY = −S − 2πiτg J − 2πiϕg Qe , +(4.39) +the second equation following from (4.28). In the following, we shall approach the BPS locus +via a limiting process taking β → ∞, Ω → Ω∗, Φe → Φe∗ keeping τg, ϕg fixed to a complex +value. +As shown above, the constraint (4.34) follows from imposing supersymmetry on the +parameters of the solution. It should be the case that it follows directly from the requirement +that the bulk Killing spinor satisfying (4.27) is anti-periodic when transported around the +circle generated by V [34]. The antiperiodicity of smooth Killing spinors is consistent with +the topological statement that in Euclidean solutions the circle generated by V shrinks in the +bulk. It was shown in [3] that this is indeed the case in the asymptotically AdS5 context. +However, to the best of our knowledge, an explicit expression for the bulk Killing spinor of the +four-dimensional black hole is not known (see [79] for a discussion of the supersymmetry of the +– 33 – + +metric). Nonetheless, we can draw some conclusions by looking near the conformal boundary +(see also [35]). As we show in Appendix D, the anti-periodicity of the boundary Killing spinor +around the circle that is contractible in the bulk leads to the following condition, +β (1 − 2Φe + Ω) = 2πin0 , +(4.40) +with odd n0, or equivalently +τg − 2ϕg = n0 . +(4.41) +It should be remarked, though, that at this level the topological argument is a formal state- +ment, since the non-extremal supersymmetric solutions are complex. In fact, the bulk solution +imposes a stronger condition, as the chemical potentials satisfy (4.40) with n0 = ∓1 depending +on the solution of (4.31) chosen (see (4.34)). +One of the main interests in the family of complex solutions is that they allow us to +define a regularization of the on-shell action of the extremal supersymmetric black hole. The +Wick-rotated extremal supersymmetric black hole has an infinite throat due to an H2 factor +in the metric, whereas the family described above is originated from an extension to complex +parameters of the Wick-rotation of the non-extremal supersymmetric black hole solution, +which has the topology of the product of a disc and a 2-sphere. Thus, the on-shell action +of the BPS solution can be defined as the limit of the on-shell action of the solutions in +the complex family, and this limit is well-defined [34]. One can compute the on-shell action +of the solutions by direct application of the holographic renormalization, as done above for +the non-supersymmetric case. In fact, it is possible to derive the supersymmetric result in +a shorter and elegant manner extending the methods in [80], as done in [74] for accelerating +black holes. +Notice that the boundary Killing vector ξi = �χEγiχE, computed from the boundary +Killing spinor (D.6), is +ξ = 2u�u +� ∂ +∂tE ++ i (Ω − 1) ∂ +∂ �φ +� +(4.42) +and can thus be extended to a bulk Killing vector, which is constructed as a bilinear of the +bulk Killing spinor. The on-shell action of any smooth supersymmetric solution of minimal +gauged supergravity with real metric and gauge field can be expressed in terms of topological +data of the circle action generated by the bulk “supersymmetric” Killing vector field on the +spacetime [80]. +In this case, we are considering a family of solutions outside the reality +assumptions, and yet the formula found in [80] remarkably still holds, as we now show. +Recall that the topology of the Wick-rotated black hole is R2 × S2, parametrized by +(r, tE) − (Θ, �φ). In order to find the generators of the rotations with unitary weight on the +two factors, we rescale the coordinates +ϕ1 = 2π +β tE , +ϕ2 = �φ , +ϕ1,2 ∼ ϕ1,2 + 2π . +(4.43) +– 34 – + +In these coordinates, the Killing vector has the form ξ = b1∂ϕ1 + b2∂ϕ2 with weights +b1 = 2u�u2π +β , +b2 = −2u�u2πτg +β +. +(4.44) +The Killing vector has isolated fixed points at the North and South pole of the S2 factor, so +the on-shell action of the solution is given by the sum of the contributions +I|SUSY = +π +2G4 +� +nuts∓ +±(b1 ± b2)2 +4b1b2 +, +(4.45) +where the label ± for the nut corresponds to the chirality of the bulk spinor there. In our +case, a solution in the ± branch of solutions to (4.31) has nuts with ± chirality. Substituting +in the formula the values of the weights and using the constraint (4.38), we find +I|SUSY = ± π +G4 +ϕ2 +g +τg +, +(4.46) +where again the ± refers to the branch of the solution. +As mentioned above, this result +is consistent with what is obtained by direct substitution of (4.28) in (4.13).15 +The final +result (4.46) for the action in terms of the variables τg, ϕg is independent of β and therefore +the limit β → ∞ is smooth. This is identified as the regulated on-shell action of the BPS +solution. +The above result is also consistent with the results of [80]: the on-shell action should +only depend on topological data of the circle action of ξ, which is well-defined for the complex +solutions and it is independent of the deformation parameter β. Even more concretely, observe +that the Killing vector (4.42), after a Wick rotation back to Lorentzian, coincides with the +null generator of the horizon of the BPS black hole (since it has the form ∂t + Ω∗ ∂φ, to be +compared with (4.6)). The derivation above and the analogous result for black holes with +orbifold horizons [74] suggest that it should be possible to extend the proof of [80] to complex +solutions, finding a general way of regularizing the on-shell action for extremal black holes in +minimal gauged supergravity.16 +Extending the principles of Euclidean quantum gravity to the family of supersymmetric +complex solutions, we can obtain the entropy of the BPS black hole in the microcanonical +ensemble by taking the Legendre transform of the on-shell action I|SUSY(τg, ϕg) (see also +[34, 35, 82]). In the following we briefly review it highlighting the assumptions made at each +stage. +15In four-dimensional minimal supergravity there is no need to add finite counterterms in order for holo- +graphic renormalization to be consistent with supersymmetry [81]. +16Notice that for static magnetically charged black holes with horizon homeomorphic to a Riemann surface +of genus higher than 1, an analogous derivation within a family of smooth real supersymmetric solutions has +been proposed in [80]. +– 35 – + +First, we observe that I|SUSY in (4.46) is a homogeneous function of degree one, so its +Legendre transform vanishes unless we impose the non-homogeneous constraint (4.38). In +order to find the constrained Legendre transform of I|SUSY(ϕg, ωg), we should extremize +f(τg, ϕg, Λ) = −I|SUSY(τg, ϕg) − 2πiτg J − 2πiϕg Qe + Λ (τg − 2ϕg − n0) , +(4.47) +where n0 = ∓1 represents the branch of gravitational solutions arising from (4.31). We notice +that at the critical point (τg∗, ϕg∗, Λ∗) the following holds +f(τg∗, ϕg∗, Λ∗) = −I|SUSY(τg∗, ϕg∗) + ∂I|SUSY +∂τg +(τg∗, ϕg∗)τg∗ + ∂I|SUSY +∂ϕg +(τg∗, ϕg∗)ϕg∗ +− n0Λ∗ , +(4.48) +and together with Euler’s theorem for I|SUSY(τg, ϕg), this leads to the (implicit) Legendre +transform +�f(J∗, Qe∗) = −n0Λ∗(J∗, Qe∗) . +(4.49) +Concretely, we can combine the critical point equations into the quadratic equation for +Λ∗(J∗, Qe∗) +0 = Λ2 +∗ + +� +2πiQe∗ − n0 +π +G4 +� +Λ∗ + +� +−π2Q2 +e∗ + n0 +2π2i +G4 +J∗ +� +. +(4.50) +We then impose the constraints +J∗, Qe∗ ∈ R , +Λ∗(J∗, Qe∗) ∈ R . +(4.51) +The first one says that the charges are real. +The second one leads to the entropy being +real, as we see shortly. With these constraints, we can write the real and imaginary parts of +equation (4.50) separately, finding +Λ∗ = −n0 +πJ∗ +G4Qe∗ +, +J∗ = Qe∗ +2 +� +−n0σ1 +� +1 + 4G4Q2∗ − 1 +� +, +(4.52) +where σ1 = ±1 represents the additional sign choice for the two solutions of the quadratic +equation (4.50). Substituting in �f gives +�f(J∗, Qe∗) = +πJ∗ +G4Qe∗ += +π +2G4 +� +−n0σ1 +� +1 + 4G2 +4Q2e∗ − 1 +� +. +(4.53) +The requirement that the entropy �f(J∗, Qe∗) should be positive implies that σ1 = −n0, so +that +�f(J∗, Qe∗) = +πJ∗ +G4Qe∗ += +π +2G4 +�� +1 + 4G2 +4Q2e∗ − 1 +� +. +(4.54) +This corresponds to the Bekenstein–Hawking entropy of the BPS black hole +S∗ = +πJ∗ +G4Qe∗ +, +(4.55) +and to the non-linear constraint between the charges imposed by supersymmetry +J∗ = Qe∗ +2 +�� +1 + 4G2 +4Q2e∗ − 1 +� +. +(4.56) +– 36 – + +5 +A family of saddles in AdS +In this section we introduce the gravitational dual to the generalised Cardy limits of Section 3, +where τ approaches a rational point. This gravitational construction is modelled after the +analogous one for five-dimensional black holes dual to N = 4 SYM introduced in [16]. +5.1 +Uplift to eleven dimensions +In order to appeal to the AdS/CFT dictionary and compare the gravitational results to +the field theory computation, we need to embed the four-dimensional gravitational solu- +tion (Y4, G(Y4), A) in eleven-dimensional supergravity. In particular, in order to match the +field theory limit taken in Section 3, we shall uplift the four-dimensional minimal gauged +supergravity on S7 to a solution of eleven-dimensional supergravity (Y11, G(Y11), C). +The +eleven-dimensional metric and gauge field can be locally written as a fibration [83] +G(Y11) = G(Y4) + 4 +�� +d �ψ + σ + 1 +2A +�2 ++ G(CP3) +� +, +dC = 3 vol(Y4) − 4 ∗4 F ∧ J . +(5.1) +Here we have used the local form of the metric on a seven-dimensional Sasaki–Einstein space +(such as S7): ∂ �ψ is the Reeb vector, J is the K¨ahler form on the K¨ahler–Einstein base CP3, +and is such that dσ = 2J. Moreover, G(CP3) is the Fubini–Study metric normalized with +Ric(G(CP3)) = 6 G(CP3), and the volume of S7 is Vol(S7) = π4/3. The adapted coordinate �ψ +is periodic with period 2π. We see that dC has an flux through the internal space quantized +as +N = − +1 +(2πℓP )6 +� +S7 ∗11dC = +128π4 +(2πℓP )6 . +(5.2) +Combining the above equation with the reduction of the Ricci scalar on the internal space, +we find the canonical identification for dual to ABJM theory: +1 +G4 += 2 +√ +2 +3 N +3 +2 . +(5.3) +Now we focus on the Wick rotation of the supersymmetric Kerr–Newman-AdS black hole +with chemical potentials (β, Ω, Φe). First, we observe that the non-vanishing holonomy of +the gauge field at the boundary (4.10), together with the presence of the gauge field in the +fibration term in (5.1), imply that Y11 does not have the same asymptotics as the direct +product AdS4 × S7: +G(Y11) ∼ dz2 +z2 + 1 +z2 +� +dt2 +E + dθ2 + sin2 θ (d�φ − iΩ dtE)2� ++ 4 +�� +d �ψ + σ + i +2Φe dtE +�2 ++ G(CP3) +� +. +(5.4) +– 37 – + +Regularity of the solution requires +(tE, �φ, �ψ) ∼ (tE + β, �φ, �ψ) ∼ (tE, �φ + 2π, �ψ) ∼ (tE, �φ, �ψ + 2π) , +(5.5) +but at the cost of having explicit fibration terms in the metric. Alternatively, we can shift +the realization of the chemical potentials to the twisting of the coordinates by defining ψ = +�ψ + iΦe tE/2, so that the metric is explicitly asymptotically locally EAdS4 × S7, but the +regularity of the solution then requires +(tE, φ, ψ) ∼ (tE + β, φ − iΩβ, ψ + iΦeβ/2) ∼ (tE, φ + 2π, ψ) ∼ (tE, φ, ψ + 2π) . +(5.6) +It is clear that the supersymmetric structure at the conformal boundary of the black +hole solution is the same as the one where the field theory is formulated on in Section 2, +with the same fibration parameter Ω and with Φe = Φ(H1) (see (2.27)). Furthermore, the +identifications in (5.6) can be combined to show that the same relations are satisfied by a +solution with the potentials +β′ = β , +Ω′ = Ω + 2πi +β n′ +Ω , +Φ′ +e = Φe + 2πi +β 2ne . +(5.7) +More precisely, though, if n′ +Ω is odd, the periodicity of the spinors around S1 changes, whereas +this doesn’t happen if n′ +Ω = 2nΩ (see the discussion of the Killing spinor in the previous section +and notice that n′ +Ω odd would change the parity of n0 in (4.40)). It is clear that these shifts +correspond to the shifts of the chemical potentials in the partition function of the dual field +theory (2.29). Observe that all the solutions (β′, Ω′, Φ′ +e) have the same boundary conditions as +the starting solution (β, Ω, Φe), and so they must be summed over in the functional integral, +with the reduced chemical potentials +τ ′ +g = τg + 2nΩ , +ϕ′ +g = ϕg + 2ne . +(5.8) +We expand on this below. From the eleven-dimensional perspective, all the shifts (5.7) are +obtained by combining conditions from regularity of the solution. From the effective viewpoint +on Y4, instead, the regularity of the solution (4.4) only leads to the shift of Ω in (5.7). The +shift of Φe follows from imposing the boundary condition for the bulk Abelian gauge field by +fixing its holonomy around the Euclidean circle, namely +exp +� +i +2 +� +S1 +β +A +� += exp +� +−β +2 Φe +� +. +(5.9) +The factor of 1 +2 is due to the fact that operators generically have half-integer R-charges.17 +17Recall that from the boundary viewpoint, A couples to the current corresponding to the symmetry gener- +ated by H1, which is integrated over the S2. +– 38 – + +5.2 +AdS/CFT comparison and orbifold solutions +In order to perform the match warranted by the AdS/CFT correspondence, we begin by +matching the chemical potentials on both sides of the correspondence using the boundary +conditions, given in (2.27) and (4.37), respectively. In these equations, we take n0 = ∓1 on +the positive/negative branch of solutions, which implies +τg ↔ τ + n1 ∓ 1 , +2ϕg ↔ τ + n1 . +(5.10) +The gravitational action (4.46) is singular as τg → 0 (and ϕg is finite there because of (4.38)). +The same singularity is reproduced by the field theory saddle with (c, d) = (1, 0) if we choose +n1 = ±1 (for the positive or negative branch of solutions, respectively). With this choice, the +on-shell action of a complex supersymmetric solution on the positive (resp. negative) branch +matches the large-N behavior of the unrefined index I(τ, λ) as τ → 0 and λ = 1/4 (resp. +λ = −1/4). Indeed, the gravitational action (4.46) expressed in terms of the field theory +parameters reads +I|SUSY(τg = τ) = ± π +3 +√ +2N +3 +2 (τ ± 1)2 +τ +, +(5.11) +whose singular part clearly matches (the negative of) (3.46) when ℓc = 1 and ℓd = 0. +Starting from the eleven-dimensional geometry uplifting a solution with chemical poten- +tials (�β, �Ω, �Φe), we can also define the following identification of the coordinates +(tE, φ, ψ) ∼ +� +tE + +�β +cg +, φ − i�Ω +�β +cg +− 2πr +cg +, ψ + i +2 +�Φe +�β +cg +− 2πs +cg +� +∼ (tE, φ + 2π, ψ) +∼ (tE, φ, ψ + 2π) , +(5.12) +or equivalently +� +tE, �φ, �ψ +� +∼ +� +tE + +�β +cg +, �φ − 2πr +cg +, �ψ − 2πs +cg +� +∼ +� +tE, �φ + 2π, �ψ +� +∼ +� +tE, �φ, �ψ + 2π +� +(5.13) +where r, s, cg are integers. It is clear from the construction that r and s are only defined +modulo cg, so that the metric is a Zcg quotient of the original solution.18 It is also clear from +the above identifications that this construction is different from the standard Zk quotient of +the internal sphere, which acts only on the Hopf fiber as �ψ ∼ �ψ + 2π/k, and changes the +dual field theory [17]. Since the construction of the solutions (5.12) crucially involves the +Euclidean circle, their Lorentzian interpretation is not transparent. Instead they represent +the gravitational duals to the Euclidean saddle-points of the field theory [16]. +18In fact, as we shall see, we need to allow r = 0, . . . , 2cg − 1, and s = 0, . . . , cg − 1 in order to preserve the +periodicity conditions for the fermions along the circle. A transformation by integer shifts (r, s) has order cg +in Zcg if and only if gcd(r, cg) = gcd(s, cg) = 1. +– 39 – + +In order to interpret these solutions, we first notice that the identifications (5.12) can be +neatly expressed in terms of the shifted potentials in (5.7) +(tE, φ, ψ) ∼ +� +tE + +�β′ +cg +, φ − i�Ω′ �β′ +cg +, ψ + i +2 +�Φ′ +e +�β′ +cg +� +∼ (tE, φ + 2π, ψ) +∼ (tE, φ, ψ + 2π) . +(5.14) +Comparing with (5.6), we see that these are the identifications required of a solution with +potentials (�β′/cg, �Ω′, �Φ′ +e). +In the saddle-point approximation to the gravity path integral, +we should sum over solutions with shifted chemical potentials, provided they have the same +boundary conditions. Although here �β ̸= �β′/cg, the supersymmetric index is independent of +the size of the thermal circle, and the boundary values of the holonomies of the gauge field +and the angular velocity (see discussion around (5.9)), which control the supersymmetric +index, are indeed the same. Thus, the Zcg quotient of a supersymmetric solution labelled +by (�β, �Ω, �Φe), or more appropriately for the supersymmetric locus (�τg, �ϕg), contributes to the +gravitational path integral with +β = +�β +cg +, +τg = �τg +cg +− r +cg +, +ϕg = �ϕg +cg ++ 2s +cg +. +(5.15) +In order to ensure that the orbifold (5.12) preserves supersymmetry, we need to check +that the Killing spinor is globally defined, i.e., that it is anti-periodic around S1 +β. This requires +that the chemical potentials of a gravitational solution satisfy the constraint (4.41), which +applied to (5.15), and combined with the assumption that �τg − 2�ϕg = ∓1 leads to +4s = −r ∓ 1 − cgn0 +(5.16) +for an appropriate odd n0. For every cg, there are cg combinations of (r, s, n0) solving the +equation, provided r = 0, . . . , 2cg −1, and s = 0, . . . , cg −1, and r and cg have opposite parity. +This is an extension of the earlier conclusions (5.8), which impose that r should be even if +cg = 1. +By construction, the on-shell action of the solution obtained via a Zcg quotient of the +supersymmetric solution with (�τg, �ϕg) is 1/cg the on-shell action of the original solution, that +is +I|SUSY (τg, ϕg) = +1 +cg +I|SUSY(�τg, �ϕg) = ± π +G4 +(cgϕg − 2s)2 +cg(cgτg + r) += ± π +4G4 +(cgτg + r ± 1)2 +cg(cgτg + r) +(5.17) +This represents the contribution of the orbifold to the gravitational sum dual to a grand +canonical partition function with parameters (τg, ϕg). In order to match with field theory, we +explain in detail the simplest case with c odd. The function I|SUSY is singular as τg → −r/cg, +and now the identification of the chemical potentials relates τg with τ +n0+n1 (where n0 is the +– 40 – + +odd number such that (5.16) holds). Therefore, using (5.16), the corresponding singularity +in field theory would appear as τ → (4s ± 1 − cgn1)/cg (where we remark again that the sign +refers to the branch of supersymmetric black holes that the orbifold is a quotient of). Indeed, +recall that in order for the index to have a large-N behavior O(N +3 +2 ), we need cn1 = ±1 mod +4, and we already established around (5.11) that for c = 1 the two signs are related to the +choice of branches for the dual gravity solution. Consistently, we establish the dictionary +cg ↔ c and s ↔ d, and we find that the singular behavior of the on-shell action of the Zc +quotient (5.12) of a solution in the positive (resp. negative) branch matches the large-N limit +of unrefined index I(τ; λ) as τ → 4d +c and cn1 = ±1 mod 4 +I|SUSY(cτg = cτ + cn0 ± 1) = ± π +3 +√ +2N +3 +2 (cτ − 4d ± 1)2 +c(cτ − 4d) +. +(5.18) +For comparison, consider (3.46), recalling that ℓc = c and ℓd = 4d if c is odd. +The only fixed points of the identifications (5.13) are at the horizon, that is r > r+, where +the circle generated by V in (4.6) shrinks. The remaining space in the eleven-dimensional +solution is the product of the S2 transverse in the black hole, and the internal S7 of which +�ψ is the Hopf coordinate. It is not possible for both r and s to be both vanishing while +preserving supersymmetry: the condition (5.16) would not be satisfied for cg > 1, and indeed +the transverse space to the fixed point would be C/Zcg, which does not supersymmetry. If +r = 0, then the quotient acts only on the Hopf fiber of S7 and the disc transverse to S2. +Since the Hopf fiber never shrinks, there are no fixed points. Finally, if s = 0, then we have +a quotient of the black hole only, and we have fixed point sets isomorphic to the round S7 at +the two poles of the S2 (where θ = 0, π and the circle generated by ∂�φ shrinks to zero size). +The transverse space is C2/Zcg, which preserves supersymmetry. +We conclude the discussion of the minimal gauged supergravity solutions with two com- +ments on further refinements of the gravitational path integral. First, it is straightforward +to compute corrections to the on-shell action subleading in N using the four-derivative cor- +rections to the minimal gauged supergravity action proposed in [84, 85]. These corrections +do not modify the Killing spinor equations, and the equations of motion derived from the +corrected action are a consequence of the two-derivative equations of motion.19 This implies +that the supersymmetric solutions to the higher-derivative theory are just those of the two- +derivative theory, so the on-shell action with the higher-derivative corrections follows again +from a localization principle [87]. For the supersymmetric family of complex solutions, we +have +IHD = ± +� +π +G4 +ϕ2 +g +τg ++ 64π2 +τg +� +−α1(1 ∓ ϕg)2 + α2ϕ2 +g +� +� += ± +�� π +2G4 ++ 32π2α2 − 32π2α1 +� (τg ± 1)2 +2τg +± 64π2α1 +� +. +(5.19) +19This circumstance can be explained appealing to field redefinitions [86]. +– 41 – + +Here α1,2 are constants introduced to parametrize the higher derivative corrections, respec- +tively a supersymmetrization of the Weyl squared action and the Gauss–Bonnet term, which +would be determined by the uplifting of the higher derivative action in eleven dimensions, or +comparing with the localized partition function, obtaining [85] +π +2G4 ++ 32π2α2 = 2 +3πN +3 +2 − +3 +8 +√ +2πN +1 +2 , +32π2α1 = − π +√ +2N +1 +2 , +(5.20) +and finally [48, 85] +IHD = ± +√ +2 +3 π +�� +N +3 +2 + 15 +16N +1 +2 +� (τg ± 1)2 +2τg +∓ 1 +3N +1 +2 +� +. +(5.21) +According to the prescriptions described around (5.17), the higher derivative contribution of +a Zcg quotient of a supersymmetric solution (�τg, �ϕg) would be +1 +cg IHD(�τg, �ϕg). +The second comment concerns the saddle point approximation to the gravitational path +integral. The AdS/CFT correspondence instructs us to compare the grand-canonical field +theory partition function in the limit of large N and fixed k with a sum over gravitational +saddles +ZGCE(Ω, Φ) ∼ +� +nΩ,na∈Z +e−I|SUSY +� +β, Ω+ 2πi +β 2nΩ+ 2πi +β +� +a na, Φa+ 2πi +β na +� +, +(5.22) +where the quantity in the exponent in the right-hand side is the on-shell action of the relevant +supergravity solution with the appropriate boundary conditions fixed by the radius of the +circle β, the boundary metric (2.12), and the holonomy of the gauge fields at the boundary +fixed by the form (2.13) to be +exp +� +i +2 +� +S1 +β +Aa +� += exp +� +−β +2 Φa +� +, +a = 1, 2, 3, 4 . +(5.23) +As we have seen, supersymmetry imposes the relation (2.15) between the chemical potentials, +and in fact it is appropriate to use the five reduced chemical potentials defined in analogy +with (4.37) +τg ≡ β Ω − 1 +2πi +, +ϕa ≡ β Φa − 1 +2 +2πi +, +(5.24) +which are constrained by +τg − +4 +� +a=1 +ϕa = n0 . +(5.25) +In terms of these the field theory partition function reads +Z = TrHS2 e−β{Q,Q†}+2πiτgJ+2πi � +a ϕaRa = TrHS2(−1)2Je−β{Q,Q†}+2πi � +a ϕa(J+Ra) . (5.26) +and so +ZGCE(ϕ) ∼ +� +na∈Z +e−I|SUSY(ϕa+na) , +(5.27) +– 42 – + +highlighting the fact that both functions only depend on four rather than five fugacities, and +that neither the field theory not the gravity depends on β. An important point stressed in +[16, 88] is that in the grand canonical ensemble it is not allowed to restrict to the case of +equal Φa = Φ(H1)/2: the sum over gravitational solutions on the right-hand side of (5.27) will +take us away from this locus. Therefore, one can either go to a mixed ensemble, as in [88], or +consider solutions in bulk gauged supergravity with multiple U(1) gauge vectors, as in [16]. +We now move to do this. +6 +Black holes in non-minimal gauged supergravity +The black hole solutions considered in the previous section are charged under a unique Abelian +gauge field, that is, they are solutions of minimal gauged supergravity in four dimensions. +Thus, they can only reproduce the behavior of the ABJM index when the fugacities for the +U(1)4 Cartan of the R-symmetry are set equal. However, there are also known rotating black +holes with multiple electric charges. +6.1 +Supersymmetric black holes in the X0X1 model +There is a rotating black hole solution with two different electric charges. We shall be quite +brief in reviewing its properties, and the reader can find similar discussions in [34]. +The bosonic part of the relevant Lorentzian action is +S = +1 +16πG4 +� +Y4 +� +R volG + 1 +2 dX2 +1 ∧ ∗dX2 +2 − 1 +2d(ϕX2 +1) ∧ ∗d(ϕX2 +1) ++ (4 + X2 +1 + X2 +2) volG − X−2 +1 (F1 ∧ ∗F1 + ϕX2 +1F1 ∧ F1) +− X−2 +2 +� +F2 ∧ ∗F2 − ϕX2 +1F2 ∧ F2 +� � +. +(6.1) +Here G is the metric, F1 and F2 are the curvature of two Abelian gauge fields, X1 and ϕ are +a scalar and a pseudo-scalar, respectively, and we have defined +X2 +2 = X−2 +1 ++ ϕ2X2 +1 . +(6.2) +There is a Z2 automorphism exchanging the two Abelian gauge factors +F1 ↔ F2 +X1 ↔ X2 +ϕX2 +1 ↔ −ϕX2 +1 . +(6.3) +The action reduces to minimal gauged supergravity (4.1) upon setting X1 = X2 = 1 and +F1 = F2 = F (which explains the non-canonical normalization of the kinetic terms for the +gauge fields). +The solution we are interested in, which again we present in the frame non-rotating at +infinity, is [89, 90] +ds2 = −∆Θ∆r +BΞ2 dt2 + sin2 Θ B +� +dφ + a∆Θ +∆r − (1 + r1r2)(a2 + r1r2) +BWΞ2 +dt +�2 ++ W +�dr2 +∆r ++ dΘ2 +∆Θ +� +. +(6.4) +– 43 – + +Here +ri = r + 2 m s2 +i , +∆r = r2 + a2 − 2 m r + r1 r2 +� +r1 r2 + a2� +, +∆Θ = 1 − a2 cos2 Θ , +W = r1 r2 + a2 cos2 Θ , +Ξ = 1 − a2 , +B = ∆Θ(r1r2 + a2)2 − a2 sin2 Θ ∆r +WΞ2 +, +(6.5) +and si = sinh δi, ci = cosh δi, i = 1, 2. The scalar fields are +X2 +1 = 1 + r1 (r1 − r2) +W +, +ϕ = a (r2 − r1) cos Θ +r2 +1 + a2 cos2 Θ , +X2 +2 = 1 + r2 (r2 − r1) +W +(6.6) +and the gauge fields +A1 = 2ms2c2r1 +WΞ +� +∆Θ dt − a sin2 Θ dφ +� ++ γ1 dt , +A2 = 2ms1c1r2 +WΞ +� +∆Θ dt − a sin2 Θ dφ +� ++ γ2 dt , +(6.7) +where γ1,2 are constant. +There is an outer horizon at the largest positive root of ∆r, and the Wick-rotated solution +(t = −itE) is smooth if we identify +(tE, φ) ∼ (tE, φ + 2π) ∼ (tE + β, φ − iΩβ) , +(6.8) +where the temperature and angular velocity of the horizon are +β = 4πa2 + r1+r2+ +∆′r(r+) +, +Ω = a 1 + r1+r2+ +a2 + r1+r2+ +, +(6.9) +and we defined ri+ ≡ r+ + 2ms2 +i . The horizon is a Killing horizon for V = ∂t + Ω∂φ. The +entropy is computed from the area of the horizon +S = +π +G4 +r1+r2+ + a2 +Ξ +(6.10) +The electrostatic potentials are +Φe,1 = 2ms2c2 +r1+ +a2 + r1+r2+ +, +Φe,2 = 2ms1c1 +r2+ +a2 + r1+r2+ +, +(6.11) +and we choose the gauge γ1 = −Φe,1 and γ2 = −Φe,2 in order to have smooth gauge fields +at the horizon. Notice that the Wick-rotated metric is real provided a is pure imaginary and +δ1,2, m are real. If these conditions are satisfied, the Euclidean metric has the topology of the +product of a disc and a 2-sphere. +The black hole is asymptotically AdS, and the boundary metric has the same form as +(4.11) and gauge fields A1 ∼ −Φe,1 dt, A2 ∼ −Φe,2 dt. The standard procedure of holographic +– 44 – + +renormalization then gives us the holographic conserved charges, as described in the previous +section. The difference with Einstein–Maxwell theory is the form of the counterterms, which in +presence of scalars are subtle. In light of the fact that we shall be intersted in supersymmetric +solutions, we fix the counterterms to be +I = lim +δ→0 +� +S + +1 +8πG4 +� +∂Yδ +� +K − 1 +2R − +� +2 + X2 +1 + X2 +2 +� +volh +� +. +(6.12) +The holographic stress-tensor and the electric current defined as +⟨Tij⟩ = − +2 +√−g +δI +δgij , +⟨ji +1⟩ = +1 +√−g +δI +δA1i +, +⟨ji +2⟩ = +1 +√−g +δI +δA2i +, +(6.13) +satisfy the conservation equation +∇i⟨Tij⟩ = F1ji⟨ji +1⟩ + F2ji⟨ji +2⟩ , +(6.14) +and we can define a conserved charge associated to any boundary vector K generating a +symmetry +Q[K] = +� +C∩M3 +ui +� +⟨T i +j⟩ + ⟨ji +1⟩A1j + ⟨ji +2⟩A2j +� +Kj volC∩M3 . +(6.15) +In particular, we have expressions for the angular momentum associated to −∂φ +J = am1 + s2 +1 + s2 +2 +G4Ξ2 +, +(6.16) +the electric charges +Qe,1 = +� +C∩M3 +ui⟨ji +1⟩ volC∩M3 = − +1 +8πG4 +� +C∩M3 +∗F1 = ms2c2 +G4Ξ , +Qe,2 = +� +C∩M3 +ui⟨ji +2⟩ volC∩M3 = − +1 +8πG4 +� +C∩M3 +∗F2 = ms1c1 +G4Ξ , +(6.17) +and finally the energy associated to ∂t (as in (4.20), we denote by E the energy in the gauge +where the boundary gauge fields vanish) +E′ = m1 + s2 +1 + s2 +2 +G4Ξ2 +− Φe,1 Qe,1 − Φe,2 Qe,2 ≡ E − Φe,1 Qe,1 − Φe,2 Qe,2 . +(6.18) +These quantities, together with the Euclidean on-shell action, satisty the quantum statistical +relation +I = −S + β(Q[V ] − Φe,1 Qe,1 − Φe,2 Qe,2) += −S + β(E − ΩJ − Φe,1 Qe,1 − Φe,2 Qe,2) , +(6.19) +which reduces to (4.23) upon setting equal the two gauge fields (notice that the charges are +defined to be half of the electric charge in minimal gauged supergravity). The same quantities +also satisfy a first law of thermodynamics +dE = β−1dS + Ω dJ + Φe,1 dQe,1 + Φe,2 dQe,2 . +(6.20) +– 45 – + +The Lagrangian (6.1) is the bosonic part of an N = 2 gauged supergravity model with +prepotential F = −iX0X1. The condition for having a supersymmetric solution is [90, 91] +E = J + Qe,1 + Qe,2 +⇔ +a = coth(δ1 + δ2) − 1 . +(6.21) +It is easy to check, studying ∆r|SUSY, that in Lorentzian signature, supersymmetry and +regularity imply extremality. However, as in Section 4.2, in order to reproduce the behavior +of the index we shall need to impose supersymmetry of the Wick-rotated solutions. This +necessarily leads us to consider a family of complex supersymmetric metrics obtained by +deforming the Euclidean metrics. We observe that imposing ∆r|SUSY = 0 leads to a quadratic +equation for m in terms of r+, δ1, δ2, and thus to two branches of solutions, labelled by the +choice of ± in the solution of the equation. +The BPS sublocus is obtained by imposing +extremality in addition to supersymmetry [34]. +The chemical potentials of the complex supersymmetric solutions satisfy +β(1 − Φe,1 − Φe,2 + Ω) = ∓2πi , +(6.22) +and if we define the “reduced chemical potential” by taking the difference with the value of +the BPS solutions, we find +τg ≡ β Ω − 1 +2πi +, +ϕg1 ≡ β Φe,1 − 1 +2πi +, +ϕg2 ≡ β Φe,2 − 1 +2πi +(6.23) +satisfying +τg − ϕg1 − ϕg2 = ∓1 . +(6.24) +The reduced chemical potentials for supersymmetric solutions are complex, but they remain +finite as we approach the BPS locus, whereas the chemical potentials and the conserved +charges of the BPS solutions become real. +The on-shell action of the supersymmetric solutions, computed using holographic renor- +malization, can be expressed using the reduced chemical potentials as20 +I|SUSY = ± π +G4 +ϕg1ϕg2 +τg +. +(6.25) +Since this expression is independent of β, it remains finite on the BPS locus, which allows +us to define the on-shell action of the supersymmetric extremal black holes. The relevance +of this object is two-fold. Firstly, interpreted as a Gibbs free energy in the grand canonical +ensemble, it allows us to obtain the entropy in the microcanonical ensemble via a Legendre +transform. Secondly, it can be related to the dual field theory partition function, and its +Cardy-like limits. +20The need to perform holographic renormalization while preserving supersymmetry explains the form of the +finite counterterms for the scalars in (6.12). For more details in the STU model or the Spin(4) supergravity, +of which (6.1) is a truncation, see [92–95]. +– 46 – + +To the first purpose, we point out again that for the complex supersymmetric solutions +the quantum statistical relation (6.19) can be written as +I|SUSY = −S − 2πiτg J − 2πiϕg1 Qe,1 − 2πiϕg2 Qe,2 , +(6.26) +and from this expression follows a Euclidean quantum gravity derivation of the extremization +procedure proposed in [82] that is analogous to that described in Section 4.2. From (6.26) +follows that the function to be extremized is +f(τg, ϕg1, ϕg2) = −I|SUSY(τg, ϕg1, ϕg2) − 2πiτg J − 2πiϕg1 Qe,1 − 2πiϕg2 Qe,2 ++ Λ(τg − ϕg1 − ϕg2 ± 1) . +(6.27) +Combining the equation satisfied by f at the critical point and Euler’s theorem for the +homogeneous function I|SUSY(τg, ϕg1, ϕg2), we obtain the value of the Legendre transform +�f(J∗, Qe,1∗, Qe,2∗) = ±Λ∗(J∗, Qe,1∗, Qe,2∗) . +(6.28) +Concretely, the equation to be solved to find Λ∗(J∗, Qe,1∗, Qe,2∗) is +0 = Λ2 +∗ + Λ∗ +� +2πi(Qe,1∗ + Qe,2∗) ± π +G4 +� ++ +� +−4π2Qe,1∗Qe,2∗ ∓ 2π2i +G4 +J∗ +� +. +(6.29) +We then impose the constraints that +J∗, Qe,1∗, Qe,2∗ ∈ R , +Λ∗(J∗, Qe,1∗, Qe,2∗) ∈ R . +(6.30) +Splitting real and imaginary part of the equation above leads us to the value of the Legendre +transform +�f(J, Qe,1, Qe,2) = +πJ +G4(Qe,1 + Qe,2) = +π +2G4 +�� +1 + 16G2 +4Q2 +e,1Q2 +e,2 − 1 +� +. +(6.31) +These values correspond, respectively, to the entropy of the extremal black hole in the U(1)2 +theory and to the non-linear constraint between its charges. +6.2 +Uplift to eleven dimensions and AdS/CFT +The theory (6.1) can be obtained from a consistent truncation of eleven-dimensional super- +gravity on S7 as described in [96] (for the bosonic sector). Geometrically, we write the metric +on S7 as a S3 × S3 fibered over the internal, and introduce a gauge field only along the Hopf +fiber inside each S3. Concretely, we write the eleven-dimensional solution as +G(Y11) = (Z1Z2) +1 +3 G(Y4) ++ 4(Z1Z2) +1 +3 +� +dΞ2 + cos2 Ξ +4Z1 +� +dΘ2 +1 + sin2 Θ1 dΦ1 + (d�Ψ1 + cos Θ1 dΦ1 + A1)2� ++ sin2 Ξ +4Z2 +� +dΘ22 + sin2 Θ2 dΦ2 + (d�Ψ2 + cos Θ2 dΦ2 + A2)2� � +, +dC = +� +2 + cos2 Ξ X2 +1 + sin2 Ξ X2 +2 +� +vol(Y4) ++ 2 cos Ξ sin Ξ +� 2 +X1 +∗4 dX1 − ϕX4 +1 ∗4 dϕ +� +∧ dΞ + d �A3 + �F ′ +4 +(6.32) +– 47 – + +where +Z1 = X2 +1 cos2 Ξ + sin2 Ξ , +Z2 = cos2 Ξ + X2 +2 sin2 Ξ , +�A3 = −8ϕX2 +1 +�cos4 Ξ +Z1 +Ω1(A1) − sin4 Ξ +Z2 +Ω2(A2) +� +, +�F ′ +4 = 2 cos Ξ +X2 +1 +� +sin Ξ dΞ ∧ +� +d�Ψ1 + cos Θ1 dΦ1 + A1 +� ++ cos Ξ +2 +sin Θ1 dΘ1 ∧ dΦ1 +� +∧ (∗4F1 + ϕX2 +1F1) +− 2 sin Ξ +X22 +� +cos Ξ dΞ ∧ +� +d�Ψ2 + cos Θ2 dΦ2 + A2 +� +− sin Ξ +2 +sin Θ2 dΘ2 ∧ dΦ2 +� +∧ (∗4F2 − ϕX2 +1F2) , +Ω1(A1) = 1 +8 sin Θ1(d�Ψ1 + cos Θ1 dΦ1 + A1) ∧ dΘ1 ∧ dΦ1 , +Ω2(A2) = 1 +8 sin Θ2(d�Ψ2 + cos Θ2 dΦ2 + A2) ∧ dΘ2 ∧ dΦ2 . +(6.33) +It is straightforward to see that this uplift reduces to (5.1) upon setting X1 = X2 = 1 and +A1 = A2 = A, and changing coordinates to +�ψ = 1 +4(�Ψ1 + �Ψ2) , +Λ = 1 +2(�Ψ1 − �Ψ2) . +(6.34) +The explicit expression for the CP3 quantities are +σ = 1 +2 +� +cos 2Ξ dΛ + cos2 Ξ cos Θ1 dΦ1 + sin2 Ξ cos Θ2 dΦ2 +� +, +G(CP3) = dΞ2 + cos2 Ξ +4 +� +dΘ2 +1 + sin2 Θ1 dΦ2 +1 +� ++ sin2 Ξ +4 +� +dΘ2 +2 + sin2 Θ2 dΦ2 +2 +� ++ sin2 Ξ cos2 Ξ +� +dΛ + cos Θ1 +2 +dΦ1 − cos Θ2 +2 +dΦ2 +� +(6.35) +As in Section 5, we can study the regularity of the uplift in a neighbourhood of the +conformal boundary, where the metric has the form +G(Y11) ∼ (Z1Z2) +1 +3 +�dz2 +z2 + 1 +z2 +� +dt2 +E + dΘ2 + sin2 Θ +� +d�φ − iΩ dtE +�2�� ++ 4(Z1Z2) +1 +3 +� +dΞ2 + cos2 Ξ +4Z1 +� +dΘ2 +1 + sin2 Θ1 dΦ1 + (d�Ψ1 + cos Θ1 dΦ1 + iΦe,1 dtE)2� ++ sin2 Ξ +4Z2 +� +dΘ22 + sin2 Θ2 dΦ2 + (d�Ψ2 + cos Θ2 dΦ2 + iΦe,2 dtE)2� � +. +(6.36) +– 48 – + +Regularity now requires +(tE, �φ, �Ψ1, �Ψ2) ∼ (tE + β, �φ, �Ψ1, �Ψ2) +∼ (tE, �φ + 2π, �Ψ1, �Ψ2) +∼ (tE, �φ, �Ψ1 + 4π, �Ψ2) ∼ (tE, �φ, �Ψ1, �Ψ2 + 4π) +(6.37) +while having explicit fibration terms in the metric, both in the S1 +β×S2, and also in the internal +space, due to the non-zero holonomies of the Abelian gauge fields. On the other hand, we can +twist the coordinates defining Ψ1 = �Ψ1 + iΦe,1tE (and analogously for Ψ2), thus absorbing +the chemical potentials and finding the regularity conditions +(tE, φ, Ψ1, Ψ2) ∼ (tE + β, φ − iΩβ, Ψ1 + iΦe,1β, Ψ2 + iΦe,2β) +∼ (tE, φ + 2π, Ψ1, Ψ2) +∼ (tE, φ, Ψ1 + 4π, Ψ2) ∼ (tE, φ, Ψ1, Ψ2 + 4π) . +(6.38) +Notice that we can combine the identifications above, showing that the following choice of +chemical potentials satisfy the same conditions +β′ = β , +Ω′ = Ω + 2πi +β n′ +Ω , +Φ′ +e,1/2 = Φe,1/2 + 2πi +β 2ne,1,2 . +(6.39) +As in the minimal gauged case, we should take n′ +Ω = 2nΩ in order to not change the periodicity +of the spinors. The identification for the electric potential follows from the boundary condition +for the Abelian gauge fields, which is imposed by fixing the holonomy +exp +� +i +2 +� +S1 +β +A1/2 +� += exp +� +−β +2 Φe,1/2 +� +. +(6.40) +Again, the factor of 1 +2 is due to the fact that the operators have half-integer charges under +the relevant symmetry: compare with (2.25). Moreover, the constraint (6.22) corresponds to +(2.24). +To compare with field theory, we should first match (2.23) with the definitions (6.23) +τg ↔ τ + n1 ∓ 1 , +ϕgA ↔ τ +2 + 2σA , +(6.41) +which of course satisfy (6.24). Consistently with the truncation to minimal gauged super- +gravity, we should set n1 = ±1 to discuss the on-shell action of the positive/negative branch +of solutions, in which case the gravity action (6.25) in field theory variables reads +I|SUSY(τg = τ) = ± π +3 +√ +2N +3 +2 (τ + 4σ1)(τ + 4σ2) +τ +. +(6.42) +The singular behavior as τ → 0 matches (the negative of) the index with pairwise equal +chemical potentials log I(τ; σ) in (3.52) on the saddle (c, d) = (1, 0) (since the constraint +(3.51) becomes n1 = ±1 mod 4 if c = 1). +– 49 – + +As in the case of the uplifted black holes with one electric charge, we can define a Zcg +quotient of the eleven-dimensional supergravity solution analogous to (5.12) and (5.13). We +define it by starting with an eleven-dimensional solution (�β, �Ω, �Φe,1, �Φe,2), and imposing the +identification +(tE, φ, Ψ1, Ψ2) ∼ +� +tE + +�β +cg +, φ − i�Ω +�β +cg +− 2πr +cg +, Ψ1 + i�Φe,1 +�β +cg +− 4πs +cg +, Ψ2 + i�Φe,2 +�β +cg +− 4πt +cg +� +∼ (tE, φ + 2π, Ψ1, Ψ2) +∼ (tE, φ, Ψ1 + 4π, Ψ2) ∼ (tE, φ, Ψ1, Ψ2 + 4π) , +(6.43) +where r, s, t are integers defined modulo cg.21 Starting from a solution with chemical potentials +(i.e. boundary conditions) (�β, �Ω, �Φe,1, �Φe,2), we can rewrite its orbifold solution above in terms +of the primed potentials (6.39) using +(tE, φ, Ψ1, Ψ2) ∼ +� +tE + +�β′ +cg +, φ − i�Ω′ �β′ +cg +, Ψ1 + i�Φ′ +e,1 +�β′ +cg +, Ψ2 + i�Φ′ +e,2 +�β′ +cg +� +∼ (tE, φ + 2π, Ψ1, Ψ2) +∼ (tE, φ, Ψ1 + 4π, Ψ2) ∼ (tE, φ, Ψ1, Ψ2 + 4π) . +(6.44) +Therefore, we conclude that the Zcg quotient of the solution (�β, �τ, �ϕ1, �ϕ2) contributes with +β = +�β +cg +, +τg = �τg +cg +− r +cg +ϕg1 = �ϕg1 +cg ++ 2s +cg +, +ϕg2 = �ϕg2 +cg ++ 2t +cg +. +(6.45) +Thus +I|SUSY(τg, ϕg1, ϕg2) = +1 +cg +I|SUSY(�τg, �ϕg1, �ϕg2) = ± π +G4 +(cgϕg1 − 2s)(cgϕg2 − 2t) +cg(cgτg + r) +. +(6.46) +Requiring that the supersymmetry of the original solution is preserved by the Zcg orbifold +imposes that +τg − ϕg1 − ϕg2 = n0 +⇔ +2s + 2t = −r ∓ 1 − cgn0 . +(6.47) +As in the case of minimal supergravity (5.16), this constraint can only be solved provided +r and cg have opposite parity. It is easy to convince ourselves that there is a fixed subset +preserving supersymmetry only if we choose s = t = 0, in which case the quotient only acts +on the black hole, and the fixed point sets develop on the horizon at the two poles of the +transverse S2. +We discuss in some detail the match with the field theory result (3.52) for c odd, the other +cases follow in a slightly more involved way. The boundary conditions identify ϕgA again with +21As mentioned in footnote 18, r = 0, . . . , 2cg − 1, whereas s, t = 0, . . . , cg − 1. +– 50 – + +τ/2+2σA, as in (6.41), but now τg is τ +n0 +n1 where n0 is the odd number such that (6.47) +holds. The on-shell action (6.46) is singular as τg → −r/cg, or τ → (2s + 2t ± 1 − cgn1)/c. +Consistently with the case of c = 1 and the minimal supergravity, we relate c ↔ cg, and the +quotient of a solution on the positive (resp. negative) branch with the field theory condition +cn1 = 1 mod 4 (resp. cn1 = −1 mod 4). With this choices, the singular behavior of the +on-shell action (6.46), expressed in field theory variables, is (in the minimal case) +I|SUSY = ±8 +√ +2π +3 +N +3 +2 +1 +c(cτ − 2(s + t)) +� +cσ1 + 2t − 2s +4 +� � +cσ2 + 2s − 2t +4 +� +. +(6.48) +Recall that s and t are defined modulo c, so once we account for the necessity of shifting +σA ∈ [0, 1) by integers, this matches (6.46). A more detailed match requires considerations +on the shifted chemical potentials as well. +Indeed, as pointed out at the end of Section 5, summing over gravity solutions dual to the +grand canonical field theory partition function necessarily involves summing over solutions +with the gauge-invariant same boundary conditions but shifted chemical potentials. However, +this leads to seemingly paradoxical results, as stressed in [16] for the four-dimensional case. +For instance, consider a complex supersymmetric solution and shift the reduced chemical +potentials while insisting that we remain on the same branch +τ ′ +g = τg + 2nΩ , +ϕ′ +g1 = ϕg1 + 2ne,1 , +ϕ′ +g2 = ϕg2 + 2nΩ − 2ne,1 +(6.49) +The action of the shifted solution is +I|SUSY = ±2 +√ +2π +3 +N +3 +2 (ϕg1 + 2ne,1)(ϕg2 + 2nΩ − 2ne,1) +τg + 2nΩ +. +(6.50) +Setting nΩ = 0 gives a value that diverges in the shift +Re (I|SUSY) = ∓n2 +e,1 +8 +√ +2π +3 +N +3 +2 Re 1 +τg ++ O(ne1) . +(6.51) +Therefore, the contribution e−I|SUSY of the first/second branch would diverge for Re(τg) posi- +tive/negative, respectively. As suggested in [16], though, it is possible to limit the summands +on the right-hand side of the AdS/CFT equation (5.27) by considering the contribution of +branes wrapping cycles in the eleven-dimensional geometry. We will report on this in the +future.22 +Finally, we mention that there are known BPS rotating black hole solutions to the U(1)4 +STU model with four different electric charges [97], which are dual to the fully refined field +theory index. Differently from the cases considered until now, only the extremal version of +these solutions is known, since by construction it is imposed that the near horizon geometry +has an infinite throat. Therefore, it is not possible to perform the previous analysis and define +the family of complex supersymmetric solutions away from extremality. +22An analogous problem has been considered in a different setting in [36], and solved by finding the presence +of zero-modes. +– 51 – + +Acknowledgments +We are grateful to Arash Arabi Ardehali, Davide Cassani, Zohar Komargodski, Dario Martelli, +Luigi Tizzano, Chiara Toldo, and Alberto Zaffaroni for helpful discussions. We would also +like to thank the organizers and participants of the SCGP workshop “Supersymmetric Black +Holes, Holography and Microstate Counting” for many interesting comments and discus- +sions. This work is supported by the ERC Consolidator Grant N. 681908, “Quantum black +holes: A macroscopic window into the microstructure of gravity”, and by the STFC grant +ST/P000258/1. ACB acknowledges financial support from the INFN grant GSS (Gauge The- +ories, Strings and Supergravity). +PBG gratefully acknowledges support from the Simons +Center for Geometry and Physics, Stony Brook University, at which some of the research for +this paper was performed. +A +Special functions and asymptotic limit formulas +The Pochhammer symbol is an entire function of z ∈ C, for q ∈ C, |q| < 1, defined as +(z; q)∞ := +∞ +� +n=0 +(1 − zqn) . +(A.1) +Its asymptotic expansion for when q approaches a root of unity is given as follows [63]. Let +w ∈ C with |w| < 1, q = ξme−ε/m where ξm is a primitive m-th root of unity and ε > 0, and +ν is a complex number such that νε = o(1) as ε → 0. Then +log +� +q w e−νε/m; q +� +∞ = − 1 +mεLi2(wm) − +� ν +m − 1 +2 +� +log(1 − wm) − εν2 +2m +wm +1 − wm +− 1 +m log Dξm(wm) − log(1 − w) + ψw,ξm(ν, ε) , +(A.2) +where +Dξm(x) ≡ +m−1 +� +t=1 +� +1 − ξt +mx +�t , +(A.3) +and ψw,ξm(ν, ε) has an asymptotic expansion as ε → 0 +ψw,ξm(ν, ε) ∼ − +� +r≥2 +m +� +t=1 +� +Br +� +1 − t + ν +m +� +− δr,2 +ν2 +m2 +� +Li2−r(ξt +mw) εr−1 +r! +. +(A.4) +The Bernoulli polynomials Br are defined by the generating function +t etx +et − 1 = +∞ +� +r=0 +Br(x) tr +r! , +(A.5) +– 52 – + +with the first few polynomials given by +B0 = 1 , +B1 = 1 +2 − x , +B2 = 1 +6 − x + x2 , +B3 = 1 +2x − 3 +2x2 + x3 . +(A.6) +They satisfy the relation B′ +r(x) = rBr−1(x). +The periodic Bernoulli polynomials Br(z) are defined, for z ∈ C, through their Fourier +series expansion, +− (2πi)j +j! +Br(z) = +� +k∈Z +′ e2πikz +kr +(z ∈ C , j ≥ 1) . +(A.7) +The prime in the above formula means that k = 0 has to be omitted, and that in the +j = 1 case—where the series is not absolutely convergent—the sum is in the sense of Cauchy +principal value. For x ∈ R we have that Br(x) = Br ({x}). +The polylogarithm functions Lin, n = 1, 2, . . . are defined as +Lin(z) = +∞ +� +k=1 +zk +kn , +|z| < 1 , +(A.8) +and is extended to C \ [1, ∞) by analytic continuation. They satisfy the relation +Lin(e2πix) + (−1)n Lin(e−2πix) = −(2πi)n +n! +Bn(x) . +(A.9) +B +Factorization of the integrand in the matrix integral +In this appendix, we review the factorization (3.9) of the integrand of the ABJM index (3.1). +We follow the treatment of [60], highlighting the differences due to the limit τ → 4d/c. Recall +that we have +si = ui + imi +ε +4πc , +si = −ui + imi +ε +4πc , +�si = �ui + i�mi +ε +4πc , +�si = −�ui + i�mi +ε +4πc , +(B.1) +and the corresponding exponentiated variables zi ≡ e2πisi and analogous. +The classical action (3.3) in terms of the variables (B.1) is +Zclass = +� +i +exp +� +2πi4πc +4iε +� +s2 +i − �s2 +i +� +− 2πi4πc +4iε +� +s2 +i − �s +2 +i +�� +. +(B.2) +Next we consider the contribution of the vector multiplets (3.4), first focusing on one of +the U(N) gauge groups. We shall use the following relation +� +i̸=j +(xix−1 +j qa(ij); qb)∞ +(x−1 +i xjqb+a(ij); qb)∞ += +� +i̸=j +�∞ +n=0(1 − xix−1 +j qa(ij)qbn) +�∞ +m=0(1 − x−1 +i xjqa(ij)qbm)(1 − x−1 +i xjqa(ij)) += +� +i̸=j +(1 − xix−1 +j qa(ij)) , +(B.3) +– 53 – + +valid for any b and for any coefficients a(ij) symmetric in i, j. Using this, we write the first +line on the right-hand side of (3.4) as +� +i̸=j +(1 − xix−1 +j q +1 +2 |mij|) = +� +i̸=j +(xix−1 +j q +1 +2 |mij|; q)∞ +(x−1 +i xjq1+ 1 +2 |mij|; q)∞ +. +(B.4) +It simplifies the following analysis to split the zero-point energy in (3.1) among the vector +and chiral contributions. Therefore, we write +vm ≡ +� +i̸=j +q− 1 +4 |mij| (xix−1 +j q +1 +2 |mij|; q)∞ +(x−1 +i xjq1+ 1 +2 |mij|; q)∞ +. +(B.5) +From the identity +(Xq +m+1 +2 ; q)∞ +(X−1q +m+1 +2 ; q)∞ += (−X)−m +(Xq +−m+1 +2 +; q)∞ +(X−1q +−m+1 +2 +; q)∞ +m ∈ Z , +(B.6) +valid for any X ∈ C (the case X = 1 should be treated with care), taking X = x−1y−1q +1−R +2 +it follows that [60, 98] +� +x−1y−1q +1−R +2 +� 1 +2 |m| (x−1y−1q +2−R+|m| +2 +; q)∞ +(xyq +R+|m| +2 +; q)∞ += (± sgn(m))m � +x−1y−1q +1−R +2 +�± 1 +2 m (x−1y−1q +2−R±m +2 +; q)∞ +(xyq +R±m +2 ; q)∞ +. +(B.7) +Using the identity (B.7) with y = 1, R = 0, the − sign for i > j and + sign for i < j, we +can rewrite (B.5) as +vm = +� +i>j +ξ +− 1 +2 +ij (zjz−1 +i +)− 1 +2 +� +zjz−1 +i +ξij; q +� +∞ +� +zjz−1 +i +ξijq; q +� +∞ +× ξ +− 1 +2 +ij (zjz−1 +i )− 1 +2 +� +zjz−1 +i ξij; q +� +∞ +� +zjz−1 +i ξijq; q +� +∞ +, +(B.8) +where +ξij ≡ e−2πi +mij +2 +4d +c . +(B.9) +A similar formula applies to the contribution of the other U(N) gauge group (the second line +on the right-hand side of (3.4)) with zi, zi replaced by �zi,�zi, respectively. +Finally, we consider the contributions of the N = 2 chiral multiplets, i.e. the product of +the expressions for a = 1, . . . , 4 given in (3.5). We first focus on the contribution of the first +line in (3.5), since the second one can be obtained simply by substituting xi → x−1 +i , �xj → �x−1 +j . +As in the contribution of the vector multiplet, we also introduce part of the zero-point energy, +defining +χm1 = +� +i,j +q +1 +8 |mi−� +mj| +� +x−1 +i +�xj ζ−1 +1 +q +3 +4 + 1 +2 |mi−�mj| ; q +� +∞ +� +xi �x−1 +j +ζ1 q +1 +4 + 1 +2 |mi−�mj| ; q +� +∞ +. +(B.10) +– 54 – + +We split the product into i = j and i ̸= j. For the first piece with i = j we use (B.7) with −, +and the result equals +4 +� +a=1 +χma +�� +i=j = +� +i +ξ +′ 1 +2 +ii (�ziz−1 +i +) +1 +2 +� +�ziz−1 +i +ξ′ +iiζ−1 +1 q +3 +4 ; q +� +∞ +� +�ziz−1 +i +ξ′ +iiζ3q +1 +4 ; q +� +∞ +� +�ziz−1 +i +ξ′ +iiζ−1 +2 q +3 +4 ; q +� +∞ +� +�ziz−1 +i +ξ′ +iiζ4q +1 +4 ; q +� +∞ +× +� +i +ξ +′ 1 +2 +ii (�ziz−1 +i ) +1 +2 +� +�ziz−1 +i ξ′ +iiζ−1 +3 q +3 +4 ; q +� +∞ +� +�ziz−1 +i ξ′ +iiζ1q +1 +4 ; q +� +∞ +� +�ziz−1 +i ξ′ +iiζ−1 +4 q +3 +4 ; q +� +∞ +� +�ziz−1 +i ξ′ +iiζ2q +1 +4 ; q +� +∞ +, +(B.11) +where ξ′ +ij ≡ exp +� +−2πi mi−�mj +2 +4d +c +� +. We split the product i ̸= j into i > j and i < j, and again +use the identity (B.7) with − for i > j and + for i < j. The result is +4 +� +a=1 +χma +�� +i̸=j = +� +i>j +(ξij �ξij) +1 +2 +�zj +zi +�zj +�zi +� 1 +2 +× +� +a=1,2 +� +�zjz−1 +i +ξ′ +ijζ−1 +a q +3 +4 ; q +� +∞ +� +�z−1 +i +zjξ′−1 +ji ζaq +1 +4 ; q +� +∞ +× +� +a=3,4 +� +�z−1 +i +zjξ′−1 +ji ζ−1 +a q +3 +4 ; q +� +∞ +� +�zjz−1 +i +ξ′ +ijζaq +1 +4 ; q +� +∞ +× (ξij �ξij) +1 +2 +� +zj +zi +�zj +�zi +� 1 +2 +× +� +a=1,2 +� +�z +−1 +i zjξ′−1 +ji ζ−1 +a q +3 +4 ; q +� +∞ +� +�zjz−1 +i ξ′ +ijζaq +1 +4 ; q +� +∞ +× +� +a=3,4 +� +�zjz−1 +i ξ′ +ijζ−1 +a q +3 +4 ; q +� +∞ +� +�z +−1 +i zjξ′−1 +ji ζaq +1 +4 ; q +� +∞ +. +(B.12) +Finally, upon putting together the various pieces (B.2), (B.8), (B.11), and (B.12), we +obtain that the integrand in (3.1) can be written as the product of +Zhol(z,�z; τ, λ) = +N +� +i=1 +exp +� +2πi4πc +4iε +� +s2 +i − �s2 +i +�� +(�ziz−1 +i +) +1 +2 ξ +′ 1 +2 +ii +× +� +�ziz−1 +i +ξ′ +iiζ−1 +1 q +3 +4 ; q +� +∞ +� +�ziz−1 +i +ξ′ +iiζ3q +1 +4 ; q +� +∞ +� +�ziz−1 +i +ξ′ +iiζ−1 +2 q +3 +4 ; q +� +∞ +� +�ziz−1 +i +ξ′ +iiζ4q +1 +4 ; q +� +∞ +× +N +� +i,j=1 +i>j +� +zjz−1 +i +ξij; q +� +∞ +� +zjz−1 +i +ξijq; q +� +∞ +� +�zj�z−1 +i +�ξij; q +� +∞ +� +�zj�z−1 +i +�ξijq; q +� +∞ +× +� +a=1,2 +� +�zjz−1 +i +ξ′ +ijζ−1 +a q +3 +4 ; q +� +∞ +� +�z−1 +i +zjξ′−1 +ji ζaq +1 +4 ; q +� +∞ +× +� +a=3,4 +� +�z−1 +i +zjξ′−1 +ji ζ−1 +a q +3 +4 ; q +� +∞ +� +�zjz−1 +i +ξ′ +ijζaq +1 +4 ; q +� +∞ +, +(B.13) +– 55 – + +and +Zantihol(z,�z; τ, λ) = +N +� +i=1 +exp +� +−2πi4πc +4iε +� +s2 +i − �s +2 +i +�� +(�ziz−1 +i ) +1 +2 ξ +′ 1 +2 +ii +× +� +�ziz−1 +i ξ′ +iiζ−1 +3 q +3 +4 ; q +� +∞ +� +�ziz−1 +i ξ′ +iiζ1q +1 +4 ; q +� +∞ +� +�ziz−1 +i ξ′ +iiζ−1 +4 q +3 +4 ; q +� +∞ +� +�ziz−1 +i ξ′ +iiζ2q +1 +4 ; q +� +∞ +× +N +� +i,j=1 +i>j +� +zjz−1 +i ξij; q +� +∞ +� +zjz−1 +i ξijq; q +� +∞ +� +�zj�z +−1 +i +�ξij; q +� +∞ +� +�zj�z +−1 +i +�ξijq; q +� +∞ +× +� +a=1,2 +� +�z +−1 +i zjξ′−1 +ji ζ−1 +a q +3 +4 ; q +� +∞ +� +�zjz−1 +i ξ′ +ijζaq +1 +4 ; q +� +∞ +× +� +a=3,4 +� +�zjz−1 +i ξ′ +ijζ−1 +a q +3 +4 ; q +� +∞ +� +�z +−1 +i zjξ′−1 +ji ζaq +1 +4 ; q +� +∞ +. +(B.14) +Notice that +Zantihol(z,�z; τ, λ) = Zhol(z,�z; τ, λ) +�� +k→−k,zi→zi,�zi→�zi,ζ1↔ζ3,ζ2↔ζ4 . +(B.15) +From these computations follow (3.9) and (3.10). +C +Saddle-point analysis of the large-N index +In this appendix we review the saddle-point solution to equations (3.25) in the general case +of generic λa [33]. +Due to the fact that the potential Wµ defined in (3.24) +Wµ := W +� +ρ(x), v(x), �v(x) +� ++ N +3 +2 µ i +�� x2 +x1 +dx ρ(x) − 1 +� +, +(C.1) +is piecewise polynomial, solutions the first three out of the four equations (3.25) +δρWµ = δvWµ = δ�vWµ = 0 , +(C.2) +can be found using a rather simple linear ansatz for the density of eigenvalues +ρ(x) = ρ0 + xρ1 . +(C.3) +We will assume k > 0, and [33] +− 1 < δv − λ′ +1,2 < 0 , +0 < δv + λ′ +3,4 < 1 . +(C.4) +where λ′ +a := ℓcλa + ℓd +4 . The conditions (C.4) comes from demanding δv not to cross branch +points of the polylogarithms in (3.31). +From the periodicity λ′ +a ∼ λ′ +a + 1 of the effective potential W we can assume 0 ≤ λ′ +a < 1 +without loss of generality. This assumption and the constraint λ1 + λ2 + λ3 + λ4 ∈ Z forces +λ′ +1 + λ′ +2 + λ′ +3 + λ′ +4 ∈ {0, 1, 2, 3}. +(C.5) +More general cases can be obtained using the periodicity λ′ +a ∼ λ′ +a + 1 . +– 56 – + +A first type of solutions to (C.2) +These are solutions that do not cross branch points +of polylogarithms (see (3.31)). Thus, they are only valid in domains of x +xleft < x < xright . +(C.6) +The xleft and xright are fixed by the inequalities obtained after plugging the explicit depen- +dence δv = δv(x) in the conditions (C.4). These solutions are the natural generalizations of +the saddle point solution found in section 3.2 for the unrefined index. +First, one solves δδvWµ = 0 for δv(x) in terms of ρ (we have reinstated k, which is set to +1 to obtain the results in the main text) +δv(x) = − +� +−λ′2 +1 + λ′ +1 − λ′2 +2 + λ′2 +3 + λ′2 +4 + λ′ +2 − λ′ +3 − λ′ +4 +� +ρ + kx +2 (λ′ +1 + λ′ +2 + λ′ +3 + λ′ +4 − 2) ρ +(C.7) +Plugging this solution and the ansatz ρ = ρ0 + xρ1 in the equation δρWµ = 0, and solving +for ρ0 and ρ1 , one obtains, under the assumption +λ′ +1 + λ′ +2 + λ′ +3 + λ′ +4 = 1 , +(C.8) +that +ρ0 = +µ/(4π2) +(λ′ +1 + λ′ +3) (λ′ +1 + λ′ +3 − 1) (λ′ +2 + λ′ +3) (λ′ +2 + λ′ +3 − 1) , +ρ1 = +kλ′ +3 − k (λ′ +1 + λ′ +3) (λ′ +2 + λ′ +3) +(λ′ +1 + λ′ +3) (λ′ +1 + λ′ +3 − 1) (λ′ +2 + λ′ +3) (λ′ +2 + λ′ +3 − 1) +(C.9) +which matches the constant unrefined solution in the family (3.39) when λ′ +a are set equal. +A second type of solutions to (C.2) +There is a second type of solutions to (C.2). These +are solutions that localize, at large N, around the non-analyticities of the polylogarithms +in (3.31). For these type of solutions the term of order N1/2 in the effective potential W +turns out to contribute non-trivially at order N3/2 to the saddle point equation δδvW = 0. 23 +Without loss of generality, these solutions take the form +δv(x) = ϵa +� +λ′ +a − 1 +2πe−N +1 +2 Ya(x)� +(C.10) +for a = 1 or 2, or 3, or 4 with ϵa = (1, 1, −1, −1) , and the real unknown function of x, +Ya(x) > 0. +(C.11) +These solutions exist in certain connected domains of x, +xleft < x < xright . +(C.12) +where xleft and xright are fixed by the positivity conditions ρ(x) > 0 and (C.11). +Plugging the ansatz (C.11) in δδv(x)Wµ one can then solve for Ya(x) as a function of ρ(x). +Then, plugging the answer in δρWµ = 0 one can proceed to solve for ρ0 and ρ1. Under the +assumption (C.8), this previous procedure leads to the solutions (C.16) and (C.18) below. +23This follows from the fact that ∂δv(x)Li2(ei(δv(x)∓λ′)) = i ei(δv(x)∓λ′)Li1(ei(δv(x)∓λ′)) grows as O(N +1 +2 ) +if δv(x) = ±(λ′ − e−N +1 +2 Y (x)) (in a large-N limit for which the function Y (x) > 0 remains finite). +– 57 – + +Constructing the dominant solution +Patching together solutions of (C.2) of the first +and second type, one can construct the dominant solution to +δρWµ = δvWµ = δ�vWµ = δµWµ = 0 . +(C.13) +Let us assume the following ordering +0 < λ′ +1 < λ′ +2 < λ′ +3 < λ′ +4 < 1 . +(C.14) +As it will be shown below, the last condition in (C.13) together with the choice (C.14), +implies µ ∈ R . For the moment let us further assume +µ > 0 . +(C.15) +The case µ < 0 can be worked out analogously. +Given the constraint (C.8) and the ordering (C.14), out of the four possible solutions of +the second type to (C.2), only two are consistent with the required positivity conditions ρ(x) > +0 , Ya > 0 . One of these two is +δvleft(x) = −λ′ +3 + +1 +2πe−N +1 +2 Y3(x) , +Y3(x) = µ + 4π2kλ′ +4x +2πλ′− +3,4 +ρleft(x) = − µ + 4π2kλ′ +3 x +(2π)3λ+ +1,3λ+ +2,3λ′− +3,4 +, +λ′± +i,j := λ′ +i ± λ′ +j . +(C.16) +This patch is consistent with the positivity constraints ρ(x) , Y3(x) > 0 in the domain +− +µ +4π2kλ′ +3 +< x < − +µ +4π2kλ′ +4 +. +(C.17) +The other consistent solution of the second type is +δvright(x) = λ′ +1 − +1 +2πe−N +1 +2 Y1(x) , +Y1(x) = µ − 4π2kλ′ +2x +2πλ′− +1,2 +ρright(x) = +−µ + 4π2kλ′ +1x +(2π)3λ′+ +1,3λ′+ +1,4λ′− +1,2 +, +(C.18) +which is consistent with the positivity constraints ρ(x) , Y1(x) > 0 in the domain +µ +4π2kλ′ +2 +< x < +µ +4π2kλ′ +1 +. +(C.19) +The left and right boundary of the domains (C.17) and (C.19), respectively, correspond to +the points x at which ρ(x) = 0: at every point in (C.17) and (C.19), ρ(x) > 0. The right and +left boundaries of the domains (C.17) and (C.19), respectively, correspond to the points x +at which Y3(x) and Y1(x) are equal to zero: at every point in (C.17) and (C.19), Y3(x) and +Y1(x) are larger than zero. +– 58 – + +At last, the solution of first type is such that δv matches the values (C.10) at the left and +right extrema of the interval +− +µ +4π2kλ′ +4 +< x < +µ +4π2kλ′ +2 +. +(C.20) +Such a solution is +δvcenter(x) = +− (λ′ +3λ′ +4 − λ′ +1λ′ +2) µ/4π2 + k +� +λ′ +3λ′ +4λ′ +1,2 + λ′ +1λ′ +2λ′ +3,4 +� +x +k (λ′ +3λ′ +4 − λ′ +1λ′ +2) x + λ′ +1,2,3,4 µ/4π2 +, +ρcenter(x) = λ′ +1,2,3,4 µ/4π2 + k (λ′ +3λ′ +4 − λ′ +1λ′ +2) x +2π λ′+ +1,3λ′+ +1,4λ′+ +2,3λ′+ +2,4 +, +(C.21) +again, with +λ′ +1,2,3,4 := λ′ +1 + λ′ +2 + λ′ +3 + λ′ +4 = 1 . +(C.22) +Patching together these three solutions to (C.2) one obtains a continuous solution, in the +lateral sense, which is defined in +� +− +µ +4π2kλ′ +3 +, − +µ +4π2kλ′ +4 +� +∪ +� +− +µ +4π2kλ′ +4 +, +µ +4π2kλ′ +2 +� +∪ +� +µ +4π2kλ′ +2 +, +µ +4π2kλ′ +1 +� +, +(C.23) +this is, in the strict large-N approximation, the left and right limits of the solution at the +two interior boundary points match each other. Outside (C.23), the solution vanishes which +means that the support of the eigenvalue distribution is given by +x1 = − +µ +4π2kλ′ +3 +, +x2 = +µ +4π2kλ′ +1 +. +(C.24) +The normalization condition for ρ – which follows from the fourth equation in (3.25) –, and +the assumption (C.15), fix the value of the Lagrange multiplier µ as follows +� x2 +x1 +dx ρ(x) = +µ2 +2 k (2π)4λ′ +1λ′ +2λ′ +3λ′ +4 += 1 =⇒ µ = +4π2� +2 k λ′ +1λ′ +2λ′ +3λ′ +4 +(C.25) +A computation shows that the value of W(ρ(x), v(x), �v) at the above-found solution is +W −→ − i2 +√ +2k +1 +2 N +3 +2 +3 +� +λ′ +1λ′ +2λ′ +3λ′ +4 +(C.26) +This result can be extended to λ′ +a ∈ R out of the previously-assumed domain (C.14). In +terms of the original variables λa = λ′ +a−ℓd/4 +ℓc +the answer can be written as +W −→ −i2 +√ +2k +1 +2 N +3 +2 +3 +(2π)2� +{ℓcλ1 + ℓd/4}{ℓcλ2 + ℓd/4}{ℓcλ3 + ℓd/4}{ℓcλ4 + ℓd/4} (C.27) +again, with +{ℓcλ1 + ℓd/4} + {ℓcλ2 + ℓd/4} + {ℓcλ3 + ℓd/4} + {ℓcλ4 + ℓd/4} = + 1 . +(C.28) +– 59 – + +The saddle-point solution in a different domain of λa’s +The saddle-point solution +takes a different form if one assumes the λa’s to belong to the following domain +− 1 < λ′ +4 < λ′ +3 < λ′ +2 < λ′ +1 < 0 . +(C.29) +Then, assuming +µ < 0 +(C.30) +together with (C.4), the expressions of δv and ρ in terms of µ and λ′ +i’s are the same as in the +previous case. Again, in these expressions there is one algebraic condition analogous to (C.22) +that in this case takes the form: +λ′ +1,2,3,4 = − 1. +(C.31) +The domains of the three different linear pieces remain as in (C.23), but this time with +µ = −4π2 � +2 k λ′ +1λ′ +2λ′ +3λ′ +4 , +(C.32) +and +W −→ + i2 +√ +2k +1 +2 N +3 +2 +3 +� +λ′ +1λ′ +2λ′ +3λ′ +4 . +(C.33) +This result can be extended to λ′ +a ∈ R out of the previously-assumed domain (C.14). In +terms of the original variables λa = λ′ +a−ℓd/4 +ℓc +the answer can be written as +W −→ + i2 +√ +2k +1 +2 N +3 +2 +3 +(2π)2� +{ℓcλ1 + ℓd/4}−{ℓcλ2 + ℓd/4}−{ℓcλ3 + ℓd/4}−{ℓcλ4 + ℓd/4}− +(C.34) +where {x}− := {x} − 1 again, and +{ℓcλ1 + ℓd/4} + {ℓcλ2 + ℓd/4} + {ℓcλ3 + ℓd/4} + {ℓcλ4 + ℓd/4} = + 3 . +(C.35) +Particular limit cases +Let us focus on branch one above and study the limits for which +some of the λa’s coincide. The simplest case is when only a single couple of λa’s coincide +λ′− +1,2 , λ′− +3,4 ̸= 0 , +λ′− +2,3 → 0 . +(C.36) +For this case the previous discussion applies trivially. The next case is when only two couples +of λa’s collide +λ′− +1,2 ̸= 0 , +λ′− +2,3 = λ′− +3,4 → 0 . +(C.37) +In this case the domain of the left patch, the first segment in (C.22), shrinks to zero. Naively, +one would say then that the saddle solution in this limit can be obtained by dropping the left +patch of the generic solution. Indeed, such an expectation is reinforced by the observation +that in the expansion λ3 → λ4 +� − +µ +4π2kλ′ +4 +x1 +dx ρleft(x) = O((λ3 − λ4)1) +(C.38) +– 60 – + +even though ρleft(x) blows up as λ3 → λ4 (see (C.16)). This means that the contribution +coming from the left patch solution to the normalization condition +� x2 +x1 dx ρ(x) = 1 vanishes +in the limit in which the left patch shrinks to zero. Consequently, in the limit λ3 → λ4 one can +drop the left patch of the generic solution and construct a solution by gluing the center and +right patches. This new solution is properly normalized as +� x2 +x1 dx ρ(x) = 1 as it is required. +Last, as in the previous case, in the limit for which all λa’s collide +λ′− +1,2 → λ′− +2,3 → λ′− +3,4 → 0 . +(C.39) +one has to drop not just the left but also the right patch, and as before, the new solution is +given by the center patch with +λ1 = λ2 = λ3 = λ4 = n1 +4 , +(C.40) +will be properly normalized. +D +Killing spinor equations +In Section 4, we remarked that the constraint (4.40) could be derived by studying the bulk +Killing spinor equation near the boundary and imposing the existence of a contractible circle. +Here we expand on those comments. +We shall first make some general considerations in Lorentzian signature. The bulk su- +persymmetry equation (4.27) induces a three-dimensional charged conformal Killing spinor χ +at the boundary. This spinor satisfies the equation of three-dimensional off-shell conformal +supergravity [99, 100] +(∇i − iAi) χ − 1 +3γiγj (∇j − iAj) χ = 0 , +(D.1) +where we are using the connection of the boundary metric g, γi generate the Clifford algebra +Cliff(1, 2), and A is interpreted as a background Abelian gauge field coupling to a u(1)R R- +symmetry. Here we have A = −Φe dt, for the Lorentzian boundary line element obtained +from (4.11) we choose the frame +e0 = dt , +e1 = dθ , +e2 = sin θ (d�φ + Ω dt) , +(D.2) +and γ0 = iσ1, γ1 = σ2, γ2 = σ3 (σi being the Pauli matrices). We find that the most general +– 61 – + +solution is χ = χ− + χ+ where +χ− = u1 exp +� +− i +2 +� +�φ + t (1 + 2Φe + Ω) +�� +e +i +2 θσ3 +� +1 +−1 +� ++ v1 exp +� i +2 +� +�φ − t (1 + 2Φe − Ω) +�� +e +i +2 θσ3 +� +1 +1 +� +, +χ+ = u2 exp +� +− i +2 +� +�φ − t (1 − 2Φe − Ω) +�� +e− i +2 θσ3 +� +1 +−1 +� ++ v2 exp +� i +2 +� +�φ + t (1 − 2Φe + Ω) +�� +e− i +2 θσ3 +� +1 +1 +� +, +(D.3) +and u1,2, v1,2 are arbitrary complex numbers. The two spinors satisfy the stronger charged +Killing spinor equation +(∇i − iAi)χ∓ = ∓ i +2γiγ0χ∓ . +(D.4) +The other key spinor used in the construction of the boundary rigid supersymmetric back- +ground is the spinor �χ that satisfies the conformal Killing spinor equation (D.1) with opposite +charge. If the gauge field is real, as it is in Lorentzian signature, we can take �χ = χc, where +the charge conjugate spinor is χc ≡ γ0C−1χ∗. From these two spinors, we can construct a +geometric background preserving two supercharges of opposite R-charge, and in particular a +bilinear vector ξi = �χγiχ, which is generically a conformal Killing vector.24 +Issues arise when we Wick rotate to Euclidean signature. As already pointed out, both +the bulk Killing spinor equation (4.27) and the conformal Killing spinor equation (D.1) are +analytic in the supergravity fields, so the Wick-rotated spinors are still solutions. However, +in Euclidean signature one is a priori not allowed to impose reality conditions on bosonic or +fermionic fields. This implies that the spinor �χ is independent of χ, and indeed it is well-known +that we can have Riemannian backgrounds supporting two independent supercharges with +opposite R-charge, such as the fibered S1×S2 considered in Section 2 [99, 101]. To derive this +condition holographically, one should then in principle consider Riemannian bulk solutions +with a supersymmetry obtained by doubling the Killing spinor equations, as stressed, for +instance, in [48, 92]. We shall take a simpler approach, often taken in the literature, and +Wick rotate the Lorentzian spinors to χ → χE and χc → �χE. The resulting spinors are +independent (i.e. �χE ̸= χc +E) and solve, respectively, the conformal Killing spinor equation +(D.1) and that with opposite charge. +24Here χ ≡ χT C, where the charge conjugation matrix C is the intertwiner satisfying +CT = −C , +C∗ = C , +C2 = −1 , +γT +i = −CγiC−1 . +With our choice of basis, one can choose C = iσ2. +– 62 – + +In fact, it follows from (D.4) that these spinors are also solutions to the new minimal +supergravity Killing spinor equations +0 = ∇iχE − iA(nm) +i +χE + iViχE + i +2V jγjiχE + 1 +2HγiχE , +0 = ∇i�χE + iA(nm) +i +�χE − iVi�χE − i +2V jγji�χE + 1 +2Hγi�χE , +(D.5) +with H = 0, an appropriate choice of Vi (depending on the choice χ±), and A(nm) = Ai + +3 +2Vi. Unsurprisingly, these are the Killing spinor equations of three-dimensional off-shell new +minimal supergravity that would give the backgrounds considered at the end of Section 2 +[99]. For our purposes it is consistent to choose +χE = u exp +�1 +2 +� +i�φ + tE (1 − 2Φe + Ω) +�� +e− i +2 θσ3 +� +1 +1 +� +, +�χE = �u exp +� +−1 +2 +� +i�φ + tE (1 − 2Φe + Ω) +�� +e +i +2 θσ3 +� +1 +−1 +� +, +(D.6) +where again u, �u are arbitrary complex numbers. These spinors solve the conformal Killing +spinor equation (D.1) with positive and negative R-charge, respectively. Moreover, they also +solve (D.5) with V = −i dtE and H = 0. In order to have well-defined global spinors on the +fibered S1 × S2 background (4.11), we should impose a constraint on the chemical potentials. +First, notice that under �φ → �φ + 2π, the spinors are antiperiodic as should be. However, in +order for the spinors to be antiperiodic as tE → tE + β we should require +β (1 − 2Φe + Ω) = 2πin0 , +(D.7) +with odd n0. This provides us with an additional way to argue for the constraint between +the chemical potentials. In Section 2, we found that the constraint (2.28) followed directly +from identifying the superconformal index as a thermal partition function. We also argued +around (2.21) that it could be derived by looking at the rigid supersymmetric background +and requiring thermal boundary conditions for the Killing spinor (which is the argument just +reproduced in detail to get to (4.40)). Whilst until now the constraint has been discussed +from the field theory viewpoint, we now find an argument from the regularity of the Euclidean +gravity solution: χE is the leading order spinor in the radial expansion of the bulk spinor +ϵ near the boundary, and the anti-periodicity of χE (and thus of ϵ) around S1 +β is needed in +order to be able to have a smooth disc filling S1 +β. +E +Four-dimensional index +In this appendix, we review the construction and limits of the four-dimensional supercon- +formal index for N = 4 SYM with SU(N) gauge group, emphasising the parallels with the +construction in three dimensions explained in Section 2. +– 63 – + +The four-dimensional superconformal index for N = 4 SYM counts the states on S3 +annihilated by a supercharge Q with +{Q, Q†} = H − J1 − J2 − +3 +� +i=1 +Ri , +(E.1) +where J1,2 are the generators of the Cartan of the su(2) × su(2) isometries on S3, and R1,2,3 +are generators of the Cartan of so(6)R in the orthogonal basis (so they have half-integer +eigenvalues). The remaining bosonic subalgebra commuting with Q, Q† is generated by +J1 + 1 +3 +� +i +Ri , J2 + 1 +3 +� +i +Ri , R1 − R3 , R2 − R3 . +(E.2) +The u(1)R commuting with the supercharge is generated by r ≡ 2 +3 +� +i Ri, with eigenvalues in +1 +3Z. We define the index as +I(τ1, τ2, λ1, λ2) = TrHS3 +� +(−1)2J1e−β{Q,Q†}+2πiτ1(J1+ 1 +3 +� +i Ri)+2πiτ2(J2+ 1 +3 +� +i Ri) +× e2πiλ1(R1−R3)+2πiλ2(R2−R3) +� +. +(E.3) +States in the theory satisfy J1,2 = R1,2,3 mod 1, so we find that the index is invariant under +the following shifts +τ1,2 → τ1,2 + 3 , +λ1,2 → λ1,2 + 1 . +(E.4) +Instead, observe that shifts of (say) τ1 by integers lead to indices graded by r, the R-charge +indices +I(τ1 + 1, τ2, λ1, λ2) = TrHS3 +� +(−1)re−β{Q,Q†}+2πiτ1(J1+ 1 +3 +� +i Ri)+2πiτ2(J2+ 1 +3 +� +i Ri) +× e2πiλ1(R1−R3)+2πiλ2(R2−R3) +� +≡ IR(τ1, τ2, λ1, λ2) , +I(τ1 + 2, τ2, λ1, λ2) = IR(τ1, τ2, λ1, λ2) . +(E.5) +It is convenient to introduce λ3 via the constraint +3 +� +i=1 +λi = n1 , +n1 ∈ Z . +(E.6) +This leads to +I(τ1, τ2; λ) = TrHS3(−1)2J1e−β{Q,Q†}+2πi[τ1(J1+ 1 +3 +� +i Ri)+τ2(J2+ 1 +3 +� +i Ri)+� +i λiRi−n1R3] += TrHS3(−1)2J1e−β{Q,Q†}+2πi[ +� +i( +τ1 +3 +λi)(J1+Ri)+τ2(J2+ 1 +3 +� +i Ri)] , +(E.7) +– 64 – + +so that the index is really a function of the four variables λi + τ1/3, τ2. +The functional integral formalism also works exactly in parallel to the discussion in Sec- +tion 2. The index can be seen as a functional integral over a fibered S1 ×S3 with background +gauge fields for the Cartan of the R-symmetry, and thermal boundary conditions for the +fields. In particular, looking at (E.7), we have fibration parameters +Ω1 = 1 + 2πi +β (τ1 + n0 + n1) , +Ω2 = 1 + 2πi +β τ2 , +(E.8) +and background gauge fields Ai = iΦi dtE with +Φi = 1 + 2πi +β +�τ1 + τ2 +3 ++ λi +� +. +(E.9) +Here n0 is an odd number taking care of the grading by (−1)2J1, and because of supersym- +metry, these quantities are not all independent +β +� +1 − +3 +� +i=1 +Φi + Ω1 + Ω2 +� += 2πin0 . +(E.10) +Moreover, thanks to the relation between the charges of the states in the theory, the partition +function is invariant under the following shifts +ΩA → ΩA + 2πi +β mA , +ΦA → ΦA + 2πi +β nA , +Φ3 → Φ3 + 2πi +β +� +2k − +2 +� +A=1 +(mA + nA) +� +(E.11) +with mA, nA, k ∈ Z. +To simplify the discussion, we reduce to the universal case by setting λ1 = λ2 = λ3 ≡ λ, +with the constraint 3λ = n1. In this case, it is +I(τ1, τ2; n1) = TrHS3(−1)2J1e−β{Q,Q†}+2πi[(τ1+n1)(J1+ 1 +2 r)+τ2(J2+ 1 +2 r)] . +(E.12) +Therefore, we immediately see that if n1 = ±1 mod 3 we have the R-charge index, graded by +the R-symmetry generator r satisfying 2J1 = 2J2 = 3r mod 2 on the states of N = 4 SYM. +We further simplify the discussion by setting τ1 = τ2 ≡ τ, obtaining the simplest index +receiving contributions only from states preserving two supercharges +�I(τ; n1) = TrHS3(−1)2J1e2πin1(J1+ r +2)+2πiτ(J1+J2+r) += TrHS3(−1)2J1e2πi(τ−n1)(J1+J2+r) , +(E.13) +where in the last equation we used the relation 2J2 = 3r mod 2. +For the same reason, +as mentioned, both τ and n1 are only defined modulo 3, so the unrefined index is really a +function of the C-valued variable exp(2πiT), with T ≡ (τ − n1)/3, and T ∼ T + 1. +We are interested in the generalized Cardy limit in which T → D/C with gcd(C, D) = +1. In order to study the asymptotic behavior of the index in this limit, we further write +– 65 – + +3T = ℓD/ℓC with gcd(ℓC, ℓD) = 1. If C is not a multiple of 3, then ℓD = 3D and ℓC = C, +whereas if C is a multiple of 3, then ℓD = D and ℓC = C/3. In terms of these variables, in +the generalized Cardy limit, log �I(τ; n1) has a leading O((3ℓCT − ℓD))−2) term provided that +C is a multiple of 3, in which case [7, 11] +log �I(T) ∼ ± iπ +27N2 +1 +C +3 (CT − D)2 +if D = ±1 mod 3. +(E.14) +Clearly, the smallest values for which the index has a leading singular behavior are T → ±1/3, +corresponding to exp(2πiT) approaching the primitive third roots of unity. +For historical reasons, the asymptotic behavior of the index has been studied splitting +T = (τ − n1)/3 in τ and n1. 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Seiberg, Exploring Curved Superspace, JHEP 08 +(2012) 141, [1205.1115]. +– 71 – + diff --git a/8tAyT4oBgHgl3EQf3PkC/content/tmp_files/load_file.txt b/8tAyT4oBgHgl3EQf3PkC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6450ed2deb029ed50eb1076c1b8ae0eac8a78c4 --- /dev/null +++ b/8tAyT4oBgHgl3EQf3PkC/content/tmp_files/load_file.txt @@ -0,0 +1,2467 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf,len=2466 +page_content='Prepared for submission to JHEP Supersymmetric phases of AdS4/CFT3 Pietro Benetti Genolini,a Alejandro Cabo-Bizet,a,b Sameer Murthya aDepartment of Mathematics, King’s College London, The Strand, London WC2R 2LS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' bUniversit`a del Salento, Dipartimento di Matematica e Fisica Ennio De Giorgi, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' - sezione di Lecce, Via Arnesano, I-73100 Lecce, Italy E-mail: pietro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='benetti genolini, alejandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='cabo bizet, sameer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='murthy @kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='uk Abstract: We exhibit an infinite family of supersymmetric phases in the three-dimensional ABJM superconformal field theory and the dual asymptotically AdS4 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' They are interpreted as partially deconfined phases which generalize the confined/pure AdS phase and deconfined/supersymmetric black hole phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Our analysis involves finding a family of saddle- points of the superconformal index labelled by rational points (equivalently, roots of unity), separately in the bulk and boundary theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the ABJM theory we calculate the free energy of each saddle by the large-N asymptotic expansion of the superconformal index to all orders in perturbation theory near the saddle-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We find that this expansion terminates at finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the gravitational theory we show that there is a corresponding family of solutions, constructed by orbifolding the eleven-dimensional uplift of the supersymmetric black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The on-shell gravitational action of each orbifold agrees with the free energy of the corresponding saddle in the SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We find that there are two saddles in the ABJM theory with the same entropy as the supersymmetric black hole, corresponding to the two primitive fourth-roots of unity, which causes macroscopic oscillations in the microcanonical index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='00763v1 [hep-th] 2 Jan 2023 Contents 1 Introduction and summary of results 1 2 The superconformal index of ABJM theory 8 3 ABJM index near rational points 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Generalized Cardy limits to roots of unity 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 The large N saddle-point analysis 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3 Subleading effects 25 4 Black holes and supersymmetric solutions 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Kerr–Newman-AdS black hole 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 Supersymmetry 31 5 A family of saddles in AdS 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Uplift to eleven dimensions 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 AdS/CFT comparison and orbifold solutions 39 6 Black holes in non-minimal gauged supergravity 43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Supersymmetric black holes in the X0X1 model 43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='Uplift to eleven dimensions and AdS/CFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='A Special functions and asymptotic limit formulas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='B Factorization of the integrand in the matrix integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='C Saddle-point analysis of the large-N index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='D Killing spinor equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='E Four-dimensional index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='Introduction and summary of results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='A fruitful way to learn about the collective behavior of a quantum statistical system is to study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='the thermodynamic behavior of the theory as a function of external macroscopic sources (tem- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='perature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' angular velocities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' chemical potentials) which couple to conserved charges (energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' spin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' global charges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Of particular interest are phase transitions that occur upon chang- ing these external parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The profound insight provided by AdS/CFT is that phase – 1 – transitions in the bulk and boundary theories—which, a priori, have completely different mechanisms—map to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This insight has led to dramatic discoveries such as the relation between the deconfinment transition in large-N gauge theory and the Hawking–Page transition [1] in asymptotically AdS gravity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recent progress on the superconformal index, initiated in [3–5], has allowed us to re- visit this line of thought in the supersymmetric setting, which provides better quantitative control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall that the superconformal index is defined as the Witten index of a certain (complex) supercharge of an SCFT, refined by chemical potentials which commute with the given supercharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' By the usual argument of pairing of bosonic and fermionic states, the in- dex is invariant under small changes of the coupling [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, assuming no wall-crossing, the values at weak and strong coupling are exactly equal, and we can compare the weakly coupled field theory index with the strongly coupled gravitational answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These investiga- tions have led to the discovery of a rich phase structure in the supersymmetric AdS5/CFT4 context—from the point of view of gauge theory as well as gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' On the gauge theory side, one finds complex saddle points of the large-N index when a background chemical potential T ∈ H coupling to a certain combination of charges approaches rational points [7–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' On the gravitational side, one has a set of solutions with horizons in asymptotically AdS5 space, whose free energy agrees with that of the corresponding micro- scopic saddles [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Together, these results are interpreted as supersymmetric phases of the AdS/CFT, generalizing the AdS black hole/deconfined phase and pure AdS/confined phase to partially confined phases [7, 8, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The perturbative series for the free energy near each saddle terminates after a finite number of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, as N → ∞, the phase bound- aries become sharp and the various transitions generalize the Hawking–Page/deconfinement transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this paper, we develop a similar picture of supersymmetric phases in the setting of AdS4/CFT3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The main observable that we focus on is the superconformal index of a 3d SCFT on S2, with independent analyses of the bulk and boundary theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the boundary theory we study the ABJM theory [17] with U(N)1 × U(N)−1 gauge group and N = 8 superconformal symmetry, and in the bulk we study 4d gauged supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We find an infinite set of saddles labelled by rational points in large-N ABJM theory, and a corresponding infinite set of asymptotically AdS4 gravitational orbifold solutions in the dual supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The expansion of the statistical free energy of the field theory around the saddles agrees precisely with the corresponding gravitational free energy of the AdS4 orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the rest of the introduction we summarize the main points of the paper, drawing parallels and contrasts with the AdS5/CFT4 situation wherever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' CFT analysis of the superconformal index The most general superconformal index in ABJM theory receives contributions from states that preserve two real supersymmetries of the theory, and can be expressed as a Witten index over the Hilbert space HBPS(N) refined by four chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The simplest such 1 16-BPS index has one chemical potential T coupling to a combination 4J + 2r of angular – 2 – momentum J and a certain R-symmetry r rotating the preserved supercharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since both angular momentum and R-symmetry charges are quantized in this theory, the index is a single-valued function of Q := e2πiT , and we have the Fourier expansion IN(T) = TrHBPS(N) (−1)2J Q 4J+2r = � ℓ dN(ℓ) Q ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) We expect that the indexed degeneracy of states dN(ℓ) has an exponential growth reflecting the existence of supersymmetric black hole solutions in the holographic dual theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, this expectation is borne out by numerical studies of four-dimensional N = 4 SYM [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' At an analytic level, the problem is best solved in the grand canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is clear from inverting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) that an exponential growth of IN(T) as T → 0 implies an exponential growth of dN(ℓ) as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One therefore looks for saddle points of IN(T) near T → 0 whose free energy is a negative power of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Looking more carefully at the inversion of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1), one realizes that an exponential growth of dN(ℓ) as ℓ → ∞ only implies that Im(T) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Moreover, in order to have a coherent addition, the Fourier series should split into a finite number of congruence classes with the same phase, which implies that Re(T) ∈ Q [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We thus conclude that the limits of interest are the generalized Cardy limits : T → D C ∈ Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) For four-dimensional N = 4 SYM, the saddle points of IN(T) in the generalized limits have been investigated using various approaches, including asymptotic analysis [11, 20–27], contour integrals and Bethe ansatz [28–30], and relations to special elliptic functions coming from number theory [7, 9, 10, 12, 13, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The results of these analyses are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The leading contribution to the microcanonical growth of states comes from the saddles T → ± 1 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' when Q approaches a primitive third root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (The quantization 1 3 comes from the quantization of R-charge in N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=') The magnitude of this contribution grows as exp(SBH), where SBH is the entropy of the BPS black hole in five-dimensional minimal gauged supergravity [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The contributions of these two leading saddles are complex conjugates of each other, leading to macroscopic oscillations in the microcanonical growth of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' More generally, the saddles (C, D) with gcd(C, D) = 1 contribute to an exponential growth of states if and only if C is a multiple of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For such saddles, the free energy is FC,D(T) ∼ ±iπN2/9C (C T − D)2 if D = ±1 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this paper, we consider the generalized Cardy limits of the microscopic index of ABJM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the large-N limit, we find that the leading contribution to the microcanonical growth of states comes from the saddles T → ±1/4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' when Q approaches a primitive fourth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (The quantization 1 4 comes from the quantization of R-charge in ABJM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=') The magnitude of the growth of this contribution grows as the exponential of the entropy of the supersymmetric Kerr–Newman-AdS black hole in four-dimensional minimal gauged supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The contributions of these two leading saddles are complex conjugates of each other, leading to macroscopic oscillations in the microcanonical growth of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 3 – More generally, the saddles (C, D) with gcd(C, D) = 1 contribute to an exponential growth of states if and only if C is a multiple of 4 (the leading saddles being (C, D) = (4, ±1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For such saddles, the free energy is FC,D(T) = ± 4 C FBH(C T − D) , FBH(ε) ∼ π 3 √ 2 N 3 2 ε , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) if D = ±1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to analyze the phase transitions, we summarize the above facts by (as N → ∞), IN(T) ∼ � (C,D) exp � −FC,D(T) � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) where the free energy of the saddle (C, D) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We thus reach a picture of the T plane divided into regions bounded by co-dimension-one walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These regions are identified with the phases and the phase boundaries occur at the walls where the neighboring entropies become equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In particular, since F4,1 is precisely the BH free energy, it is clear that the transition between the BH phase and the pure AdS phase is precisely the Hawking–Page phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In fact, we consider an all-order perturbation expansion in the small parameter ε = CT − D and show that the asymptotic expansion of the free energy terminates at O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This implies that as N → ∞, the phase boundaries become sharp, similar to the phenomenon observed in 4d SYM theory [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is worth noting, though, that the saddle point equations in 4d SYM theory are typically algebraic and can be solved easily at finite N, while in SCFT3 they involve transcendental functions which can be solved in the N → ∞ limit using techniques introduced in [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Gravitational analysis of the superconformal index In the gravitational theory we do not have a good Hilbert space interpretation and, instead, we try to interpret the index as a sum over saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The parameter N is interpreted according to the standard AdS/CFT dictionary as proportional to the inverse gravitational coupling and, similarly, other chemical potentials in the most general index are interpreted as geometrical parameters in AdS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The saddles are defined in the limit N → ∞ as solutions to the equations of motion of the semiclassical Euclidean gravita- tional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The natural idea is to interpret the microscopic sum over saddles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) as a sum over gravitational solutions of the Euclidean bulk theory like the supersymmetric black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We now summarize how this understanding is reached purely from bulk considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Firstly, there is the question: “(How) does a supersymmetric black hole contribute to the supersymmetric index?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The saddle points of the gravitational theory are solutions of the effective low-energy supergravity with boundary conditions fixed to be asymptotically AdS with conformal boundary S2 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A first puzzle is that the (−1)2J in the trace (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) naively implies that the fermions should have periodic boundary conditions around the S1, – 4 – but this is in apparent tension with the fact that the S1 is contractible in the Euclidean black hole geometry, which requires the spinor to be anti-periodic around the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A second issue is that supersymmetric black holes are extremal, so they have an infinite throat at the horizon, leading to an infinite on-shell action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These tensions were resolved in [3] in the context of the supersymmetric AdS5 black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The same idea has been applied to black holes asymptotically AdS4 [34, 35] (which are relevant for the current paper), as well as to five-dimensional supersymmetric black holes in asymptotically locally flat space [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The idea at the heart of the resolution is to allow complex gravitational solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The first point is that the supersymmetric periodicity condition can be absorbed by an imaginary shift of a chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the concrete example of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1), we have TrHBPS(N) (−1)2Je2πiT (4J+2r) = TrHBPS(N) e2πi(1+4T) J+4πiT r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) (One can equivalently shift the potential for the R-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=') The shifted chemical potential on the right-hand side is naturally interpreted in the gravitational black hole background as complex boundary values of bosonic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Interpreting the index trace as a functional integral, it is clear that only those configurations that extend the boundary Killing spinor to the bulk contribute to the index, as otherwise there would be extra extra fermion zero modes coming from broken supersymmetry which would kill the corresponding contribution of such solutions to the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 The second point is the definition of the contribution of the supersymmetric black hole to the functional integral via a regularization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One starts with a one-parameter family of deformations of the Euclidean supersymmetric black hole with the above complex boundary conditions, consisting of geometries that are not extremal but are supersymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Importantly, these configurations are inherently complex and they do not admit regular real Lorentzian continuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' They have a real Euclidean section with the topology of the product of a disc and a sphere, and are labelled by a continuous parameter β > 0 corresponding to the boundary size of the Euclidean circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the limit β → ∞ we recover the Euclidean supersymmetric extremal black hole with an infinite throat, which is the Wick-rotation of the Lorentzian supersymmetric black hole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These complex supersymmetric solutions also support a non-trivial gauge field, which vanishes at the origin of the disc, thus preserving smoothness, and has a non-trivial holonomy around the boundary of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since the Killing spinor is charged under the gauge field, parallel transport around a circle in the disc in the chosen gauge gives an anti-periodic spinor, consistently with topology, whereas parallel transport with the fully covariant derivative does indeed lead to a periodic spinor, as consistent with the naive expectation from supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One then calculates the holographically renormalized on-shell action of the complex su- persymmetric solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Notably, this action, as a function of appropriately reduced chemical potentials, is (a) finite, (b) independent of β, and (c) exactly equal to the logarithm of the functional form of the microscopic grand-canonical index near the (4, ±1) saddle-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In 1This has been demonstrated by an explicit calcuation in the asymptotically flat space in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 5 – order to obtain the entropy of the supersymmetric extremal black hole, one does a Legen- dre transform with respect to the reduced chemical potentials, and finds agreement with the Bekenstein–Hawking area law for the black hole entropy [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This extends the prescriptions of Euclidean quantum gravity [38] to supersymmetric black holes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The above procedure allows one to trace the relation between the Wick-rotated BPS black hole and the (4, ±1) saddle of the field theory index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One then expects that the (C, D) saddle of the index with gcd(C, 4) = 4 corresponds to supersymmetric solutions (regular- ized as above) with the same conformal boundary conditions as the black hole, and on-shell action equal to FBH/(C/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the context of the AdS/CFT correspondence applied to four- dimensional N = 4 SYM, these solutions were described in [16]: they are supersymmetry- preserving quotients of the ten-dimensional uplift on S5 to IIB of the Wick-rotated black hole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These quotients crucially involve the Euclidean time circle and thus their Lorentzian interpretation is not transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this paper, we construct the analogous solutions dual to the (C, D) saddles in the large-N limit of the index of ABJM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We start with solutions to gauged supergravity with the same conformal boundary conditions as the supersymmetric electrically charged black hole, uplift them on S7 to eleven-dimensional supergravity, and then perform a ZC/4 quotient while preserving supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The solutions in this infinite family generalize the two Hawking– Page-like phases, namely the supersymmetric Kerr–Newman-AdS black hole and pure AdS4, and have on-shell action equal to the microscopic free energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) at that rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Open questions and further directions The structure of the sum over saddles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4), which we derive independently from field the- ory and from gravity, is very interesting from the point of view of the underlying symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Such a sum over (C, D) saddles has previously appeared in the context of supersymmetric AdS3/CFT2 [39, 40], and in the context of AdS2/CFT1 [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In both cases, there is an underlying SL(2, Z) modular symmetry which is closely tied to the existence of such an ex- act convergent formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 In contrast, the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) for SCFT3 and the analogue in the SCFT4 problem is not an exact formula: firstly, we do not know if this is an exhaustive set of saddles (although there are indications that this may be so at large N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' secondly, although we have control over the all-order perturbation theory around each saddle, there is no claim of convergence of the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Nevertheless, the observation that the index is arranged as a sum over such saddles with a number-theoretic constraint on (C, D) is remarkable since the superconformal index in 4d and in 3d are not SL(2, Z) modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Perhaps it hints to an approximate modular symmetry in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The phase structure of the 3d as well as the 4d theory and, in particular, the dominant phase as one moves along the real axis in T has an erratic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It would be very interesting if this fundamentally related to the appearance of randomness in gravitational systems [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 2The sum over (C, D) here is really a sum over the equivalence classes Γ∞\\ Γ /Γ∞ where Γ = SL(2, Z) and Γ∞ is its subgroup stabilizing the point τ = i∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 6 – The analysis of ABJM theory at level k > 1 is an interesting problem: in the gravity dual, the ZC/4 quotient described above would be woven with the Zk quotient of the internal S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' More generally, one could generalize the uplifts discussed here to uplifts on different seven- dimensional Sasaki–Einstein spaces, where the quotient would identify points along the circle orbit of the Reeb vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These constructions would be dual to three-dimensional N = 2 SCFTs obtained from arrangements of M2-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In another direction, it would be interesting to study the N = 2 SCFTs obtained by wrapping M5-branes on hyperbolic three-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, the R-symmetry charge of the states on S2 is quantized since it is a compact subgroup of the so(5) R-symmetry group of the original six-dimensional theory, so the superconformal index has a similar structure as that described around (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The canonical Cardy limit describing the leading growth of states has been calculated, and the large-N limit agrees with the gravity dual [35, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The gravity duals to the generalized Cardy limits would be constructed by quotienting the eleven-dimensional geometry uplifting the supersymmetric Kerr–Newman-AdS black hole to eleven dimensions on a fibration of S4 over the hyperbolic manifold [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Another issue is the overabundance of eleven-dimensional geometries with boundary con- ditions appropriate to describe the dual to the large-N saddles of the generalized Cardy limits of ABJM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As pointed out in [16], their presence would make the gravitational grand-canonical partition function diverge, but one can argue for their absence from the sum studying the action of wrapped branes in the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this paper we include some comments on this subject, and we plan to report on this in more detail in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Plan of the paper In Section 2 we review the construction of the superconformal index for ABJM, highlight- ing the interpretation as a functional integral over a complex background and the conditions imposed on the chemical potentials in order to preserve supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We also comment on the subtleties arising from the fact that the index as usually defined is a multi-valued function, and identify two choices of refinements that will be matched in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In Section 3 we start from the expression of the index as a matrix model integral, and obtain the free energy of the saddle points in the large-N limit, including subleading effects in the generalized Cardy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We then move to the dual gravity side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In Section 4 we begin by reviewing aspects of the Euclidean Kerr–Newman-AdS black hole and its holographic renormalization, highlighting the importance of the gauge choice for the Abelian gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We then construct a family of complex supersymmetric solutions deforming the Wick-rotation of the supersymmetric black hole away from extremality and compute their on-shell action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In Section 5 we uplift these solutions of four-dimensional minimal gauged supergravity on S7 to eleven dimensions, describing the conformal boundary conditions and the matching with the supersymmetric background of the boundary field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This leads us to the construction of the quotient solutions, whose free energy matches the large-N limit of the (C, D) saddle of the unrefined index of ABJM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, in Section 6 we consider black holes electrically charged under two – 7 – gauge fields, which are dual to saddles of the index of ABJM with R-symmetry fugacities being pairwise equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Tracing the same path as in the previous sections, we review the construction of complex solutions, regularize the action of the BPS black hole, uplift the non-minimal gauged supergravity to eleven dimensions, studying the conformal boundary conditions, and finally take the ZC/4 quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In multiple appendices we explain some details of the computations in the main text, and review the analogy between the large-N behavior of the superconformal index of ABJM and the superconformal index of 4d N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 2 The superconformal index of ABJM theory In this section we introduce the superconformal index of ABJM theory in the Hamiltonian as well as functional integral formalism, and discuss its behavior under shifts of various chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' ABJM theory [17] is a U(N)k × U(N)−k Chern–Simons-matter theory with N = 6 superconformal symmetry for generic k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The superconformal algebra is osp(6|4) whose bosonic subalgebra is so(3, 2)×so(6)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In addition the theory has a u(1)b symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The field content in N = 2 language is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' There are two vector supermultiplets V, �V corresponding, respectively, to the two gauge groups, and two chiral supermultiplets A1,2 in the (N, N) of U(N) × U(N) and two chiral supermultiplets B1,2 in the anti-bifundamental (N, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The matter multiplets interact via a superpotential W = 2π k ϵabϵcd tr � AaBcAbBd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) The N = 2 formalism makes the bosonic subalgebra su(2)A × su(2)B × u(1)b × u(1)R of the global symmetry manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Here su(2)A,B rotates, respectively, the two pairs of chiral multiplets A and B, u(1)R is the R-symmetry of the N = 2 supercharges (under which the chiral multiplets have charge 1 2) and u(1)b is the “baryonic symmetry” under which Aa have charge 1 2 and Ba have charge − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3 It is convenient to group the components of the chiral fields A = {A, ζ}, B = {B, ω} as Y A = {Aa, B†a} , Y † A = {A† a, Ba} , ψA = {ϵabζbe−iπ/4, −ϵabω†beiπ/4} , ψ†A = {−ϵabζ† beiπ/4, ϵabωbe−iπ/4} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) The potential of the theory when written in terms of these combinations is invariant under the full so(6)R×u(1)b symmetry provided Y A, ψ†A transform in the complex representation 4, and Y † A, ψA transform in the 4 [17, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We focus on the k = 1 case, when the supersymmetry 3This symmetry is gauged, but it is related by the Chern–Simons term to the topological symmetry with current J ∝ ∗ tr(F + �F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One can also construct the index refined by the topological symmetry, and then perform a change of variables in the resulting matrix integral to obtain the same formula we have, see for instance [33, 48] for some related comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 8 – is enhanced to N = 8 by non-perturbative effects [50, 51], and the symmetry algebra so(6)R× u(1)b combines into so(8)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this theory we consider the superconformal index based on the N = 6 superconformal algebra refined by the symmetry u(1)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to define the index, we quantize the theory on S2 obtaining the Hilbert space HS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' States in HS2 are labelled by the eigenvalues of the Hamiltonian H and the angular momentum J (quantized as a half-integer), the weights of the so(6)R R-symmetry, and the half-integer eigenvalues of the generator B of u(1)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For the u(1)3 Cartan subalgebra of so(6)R, we pick the “orthogonal” basis, consisting of the generators of the rotations in the three orthogonal planes of R6, which we label H1, H2, H3, having half-integer eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Following [52, 53], we pick a supercharge Q with eigenvalues � 1 2, − 1 2, 1, 0, 0, 0 � under (H, J, H1, H2, H3, B), with the anticommutation relation {Q, Q†} = H − J − H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) On states annihilated by Q, which define the subspace HBPS, the right-hand side of the above equation clearly vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' the energy H is determined by the values of H1 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the remaining five-dimensional subspace of bosonic charges, the subalgebra commuting with Q and Q† is generated by J + 1 2H1, H2, H3, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, the refined index of ABJM theory of interest here is I(τ, ξ2, ξ3, ξB) = TrHS2(−1)2Je−β{Q,Q†}+2πiτ(J+ 1 2 H1)+2πi(ξ2H2+ξ3H3+ξBB) = TrHBPS (−1)2Je2πiτ(J+ 1 2 H1)+2πi(ξ2H2+ξ3H3+ξBB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) The second equality follows, as usual, from the fact that the index only receives contributions from states in HBPS [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Because of the half-integer quantization of the generators, the index is invariant under the identifications τ ∼ τ + 4, and ξ2,3,B ∼ ξ2,3,B + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is useful to introduce a more symmetric basis of generators (and the corresponding chemical potentials) H1 = R1 + R2 + R3 + R4 2 , H2 = −R1 + R2 − R3 + R4 2 , H3 = −R1 + R2 + R3 − R4 2 , B = −R1 − R2 + R3 + R4 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' ξ2 = −λ1 − λ3 , ξ3 = λ2 + λ3 , ξB = −λ1 − λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) In terms of these, the refined index (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) takes the form I = TrHS2(−1)2Je−β{Q,Q†}+2πiτ(J+ 1 4 �4 a=1 Ra)+2πi �3 i=1 λi(Ri−R4) = TrHS2 e−βH+βΩJ+�4 a=1 βΦaRa (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) with Ω = 1 + 2πi β (τ + n0) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) – 9 – Fields J H (H1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' H3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' B) (R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' R4) Y 1 0 1 2 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1) Y 2 0 1 2 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) Y 3 0 1 2 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) Y 4 0 1 2 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) ψ1± ± 1 2 1 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � ψ2± ± 1 2 1 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � ψ3± ± 1 2 1 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � ψ4± ± 1 2 1 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � Y † 1 0 1 2 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' −1) Y † 2 0 1 2 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) Y † 3 0 1 2 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) Y † 4 0 1 2 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) ψ†1± ± 1 2 1 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � ψ†2± ± 1 2 1 � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � ψ†3± ± 1 2 1 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � ψ†4± ± 1 2 1 � − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' − 1 2 � Q − 1 2 1 2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 0) � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 2 � Table 1: Weights of the fields after the change of basis in the Cartan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' and Φi = 1 2 + 2πi β �τ 4 + λi � , i = 1, 2, 3 , Φ4 = 1 2 + 2πi β � τ 4 − 3 � i=1 λi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) Here, we have used the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) for {Q, Q†} and written (−1)2J = e2πin0J for an arbitrary odd integer n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The charges of the fields (and the supercharge) under the spacetime and global symmetry used to define the refined index are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is clear from this table that all states satisfy J = Ra mod 1 for all a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 4, so that λi ∼ λi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' By the usual relation between the Hamiltonian and the path integral quantization, we can write the index (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) as a functional integral on the background S1 β × S2 with metric ds2 = dt2 E + dθ2 + sin2 θ dφ2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) where θ has the canonical periodicity π, and (tE, φ) ∼ (tE, φ + 2π) ∼ (tE + β, φ − iΩβ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) The fields f satisfy the following twisted boundary conditions around S1 β, f ( tE + β, x) = (−1)F eβΩJ+�4 i=a βΦaRaf ( tE, x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) – 10 – Equivalently, we can write the index as the same functional integral, but over the fibered background metric ds2 = dtE2 + dθ2 + sin2 θ � d�φ − iΩ dtE �2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) and with background gauge fields coupling to the currents for the symmetries generated by Ra Aa = iΦa dtE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) In this case, the coordinates have the periodicities � tE, �φ � ∼ � tE + β, �φ � ∼ � tE, �φ + 2π � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) and the fields satisfy standard thermal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) is real if Ω is pure imaginary, but is otherwise complex-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The background gauge field holonomies and the fibration parameter satisfy the following constraint β � 1 − � a Φa + Ω � = 2πin0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) Due to the relation J = Ra mod 1, the index is invariant under the following shifts, Φa → Φa + 2πi β na , Ω → Ω + 2πi β 2nΩ + 2πi β � a na , na, nΩ ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) Of course, these transformations preserve the parity of n0 in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15), since their effect is n0 → n0 − 2nΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It will simplify part of the following analysis to include an additional chemical potential λ4 defined by the constraint 4 � a=1 λa = n1 ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) This leads to the following expression for the index, obtained from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = TrHBPS(−1)2Je2πiτ(J+ 1 4 � a Qa)+2πi � a λa(J+Ra) = TrHBPS(−1)2Je2πi � a( τ 4 +λa)(J+Ra) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) In these variables we can identify the fibration parameter Ω and the holonomies Φa (which now have a symmetric form) for the background gauge fields Ω = 1 + 2πi β (τ + n0 + n1) , Φa = 1 2 + 2πi β �τ 4 + λa � , a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 4, , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) which is related to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) by a shift of the type (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The advantage of this basis is that the chemical potentials couple to orthogonal generators of the Cartan of the so(3) × so(8)R subalgebra of the N = 8 superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Thus the fact that J = Ra mod 1 is seen as a consequence of the N = 8 superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, both in the free N = 8 superconformal theory and in the interacting BLG theory, we can find a triality frame for the R-symmetry such that the supercharge sits in the 8∗ s, the scalars in the 8v, and the spinors in the 8s [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4 4This is different from the frame used in [53], where the supercharge is taken in the vector of so(8)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 11 – The rewriting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) shows that the index can be effectively rewritten as a function of four variables τ/4 + λa, each defined modulo 1, since J = Ra mod 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, it will be useful in later sections to consider shifts of τ alone, which lead to: I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = TrHBPS(−1)2Je2πi � a( τ 4 +λa)(J+Ra) , I(τ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = TrHBPS(−1)H1e2πi � a( τ 4 +λa)(J+Ra) ≡ IR(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) , I(τ + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = TrHBPS(−1)2(J+H1)e2πi � a( τ 4 +λa)(J+Ra) , I(τ + 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = IR(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) In particular, the first and the third indices are graded by (−1)2J and (−1)2(J+H1), respec- tively, both of which take values ±1 on the states of the ABJM theory, while the second and the fourth ones are indices graded by (−1)H1 (H1 = 1 2 � a Ra is the R-symmetry generator) which takes values in the fourth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5 These phases are reflected in the contribution of the bosonic and fermionic states to the grand-canonical index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In particular, as we see in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2, the R-charge index has saddle points with exponential growth near τ = 0, while the other two do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6 7 We can understand the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) in another equivalent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall that the index can be computed as a functional integral over field configurations satisfying standard thermal boundary conditions around the tE circle, in the background (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to formulate a supersymmetric field theory on such a background, one couples it to off-shell supergravity and then considers its rigid limit [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, coupling the N = 8 field theory to three-dimensional off-shell supergravity requires the existence of spinors solving the (generalised) Killing equation obtained from the vanishing of the gravitino variation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' [57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Noting that the spinor is charged under the potentials Φa (for the u(1)4 ⊂ so(8) gauge group of the supergravity), and that the spin connection is proportional to Ω, we see that the tE dependence of the Killing spinor is ei � 1−� a Φa+Ω � tE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) Now, from the fact that we impose anti-periodic boundary conditions for the fermions around the tE circle, we obtain precisely the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) with n0 being odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' When comparing with the gravity solutions, we shall consider less refined versions of the index, obtained by setting certain combinations of the four chemical potentials associated to the Cartan generators of so(8)R to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We may first set the four chemical potentials to be pairwise equal, that is λ1,2 ≡ σ1, λ3,4 ≡ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since the chemical potentials λa are 5In fact, this quantization arises whenever u(1)R is part of a larger non-Abelian algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 6Of course, the the Fourier coefficients of all four indices in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) have an exponential growth in magnitude, with shifted phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 7This phenomenon is analogous to the structure of the index of four-dimensional N = 4 SYM, where the R-charges of the states are quantized in units of 1/3, so that there are three different indices, defined by shifts of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Two of them lead to large growth as τ → 0, while the third does not [11, 55] (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 12 – constrained to satisfy � a λa = n1, we have 2(σ1 + σ2) = n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, we find that the index has the form I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ1,2 = σ1, λ3,4 = σ2) = TrHBPS(−1)2Je2πiτ(J+ 1 4 � a Ra)+2πi(σ1(R1+R2)+σ2(R3+R4)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) We can identify two u(1) generators 2Q1 ≡ R1 + R2 and 2Q2 ≡ R3 + R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of these generators, the interpretation as a functional integral over a fibered background requires the following parameters Ω = 1 + 2πi β (τ + n0 + n1) , Φ(QA) = 1 + 2πi β �τ 2 + 2σA � , A = 1, 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) which are constrained by β � 1 − Φ(Q1) − Φ(Q2) + Ω � = 2πin0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) The shifts of the fibration parameters that leave the partition function invariant are Φ(QA) → Φ(QA) + 2πi β 2nQA , Ω → Ω + 2πi β 2nΩ , n(RA), nΩ ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) Unlike (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16), these two shifts are now independent because the constraint between the charges of the states in the theory, namely 2QA + 2J = 0 mod 1, is now trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Notice that these two shifts preserve the parity of n0 in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, we can further simplify this by setting all the generators equal: λa ≡ λ for all 1 ≤ a ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Because of the constraint between the chemical potentials, we have 4λ = n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, we find that the index has the form I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ1,2,3,4 = λ) = TrHBPS(−1)2Je2πi(τ+n1)(J+ 1 2 H1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) In this form it is clear that, as observed in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20), n1 = � a λa appears as a shift of τ, leading to indices graded by different operators and thus with different asymptotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' When we view the unrefined index as a thermal partition function over a fibered back- ground, we find the parameters Ω = 1 + 2πi β (τ + n0 + n1) , Φ(H1) = 1 + 2πi β τ + n1 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) constrained by β � 1 − 2Φ(H1) + Ω � = 2πin0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) As in the previous cases, we can shift the holonomy and fibration parameter while leaving the partition function invariant Φ(H1) → Φ(H1) + 2πi β 2nH1 , Ω → Ω + 2πi β 2nΩ , n(H1), nΩ ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29) – 13 – The shifts (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) will play an important role in the gravitational interpretation in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As we shall see in Section 4, the background with fibration parameters (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) is what is found at the boundary of the supersymmetric electrically charged black hole in gauged minimal supergravity, and so it is the correct background in order to match the gravity com- putation in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, it has a generically complex metric, which is not standard from the field theory viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' On the other hand, one can also define the index using a super- symmetric background consisting of a real metric and background gauge field on S1 ×S2 (see for instance [58, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since both backgrounds are used to compute the same supersym- metric observable, which should only depend on the moduli of the transversely holomorphic foliation, it is natural to conjecture that they are related by a Q-exact deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8 This is similar to the discussion about four-dimensional backgrounds on S1 × S3 [11, 14, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 3 ABJM index near rational points In this section we investigate the behavior of the superconformal index near rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We consider the refined index (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18), which is a periodic function of τ with period 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of the variable q = exp(2πiτ), it is defined on a 4-sheeted cover of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The variable exp(2πiτ/4) lives on C and we consider the behavior of the index as this approaches a primitive root of unity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 1 4τ → d c, c, d ∈ Z, c > 0 and gcd(c, d) = 1, further taking the large-N limit of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) that τ couples to J + 1 4 � a Ra, and λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , λ4), with � a λ ∈ Z, couple to the orthogonal generators of the Cartan of so(3) × so(8) in the definition of the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The index I is the N = 6 superconformal index further refined by the u(1)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The index without this refinement was computed in [52] using the free field theory in the background of zero magnetic fluxes for the gauge groups, and in [59] using localization allowing for fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The refinement by the baryonic symmetry can be introduced referring to Table 1 (see also footnote 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' An analogous expression for the refined index has been considered in [48, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As a matrix integral, the index (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) for ABJM at level k = 1 takes the form I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = �� m [Du] �� �m [D�u] q 1 2 � i,j |mi−�mj|− 1 4 � i,j |mi−mj|− 1 4 � i,j |�mi−�mj| × Zclass(u, m, �u, �m) Zvec(u, m, �u, �m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ) 4 � a=1 Za chi(u, m, �u, �m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) Here the notation �� m [Du] ≡ 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' � m ∈ ZN N � i=1 � R/Z dui (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) 8One should be able to formulate the analogous discussion in extended supergravity for the background to the index refined according to the N = 8 superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 14 – denotes the sum over the magnetic fluxes m = (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' mN) and the integral over the gauge holonomies u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , uN) for each gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We introduce the fugacities q = e2πiτ, ζa = e2πiλa, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 4, and xi = e2πiui, �xi = e2πi�ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In all the products, i, j run from 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The various pieces in the integrand are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The contribution of the classical action is Zclass(u, m, �u, �m) = � i exp � 2πi � mi ui − �mi �ui �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) the contribution of the two N = 2 vector multiplets for the U(N) × U(N) gauge group is Zvec(u, m, �u, �m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ) = � i̸=j � 1 − xi x−1 j q 1 2 |mi−mj|� � i̸=j � 1 − �xi �x−1 j q 1 2 |�mi−�mj|� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) and the contributions of the four N = 2 chiral multiplets are Za chi(u, m, �u, �m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = � i,j � x−1 i �xj ζ−1 a q 3 4 + 1 2 |mi−�mj| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � xi �x−1 j ζa q 1 4 + 1 2 |mi−�mj| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ , a = 1, 2 , Za chi(u, m, �u, �m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = � i,j � xi �x−1 j ζ−1 a q 3 4 + 1 2 |mi−�mj| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � x−1 i �xj ζa q 1 4 + 1 2 |mi−�mj| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ , a = 3, 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) It is manifest from the above expressions that I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) is invariant under the separate shifts λa → λa + 1, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 4, and τ → τ + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We are interested in the behavior of I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) for τ = 4d c + i ε 2πc ⇒ q = e2πi 4d c e− ε c , gcd(c, d) = 1 , ε ↘ 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) Although we begin our analysis for ε ∈ R+, all the statements that we make below hold for ε tending to zero from any direction in the upper half-plane, and we continue to use the symbol ε ↘ 0 to denote this more general limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this limit we expect that the integral over u and the sum over m are both dominated by saddle-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to implement the saddle-point analysis, it is useful to change variables so that the integrand factorizes into what are essentially holomorphic and anti-holomorphic pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This is similar to the τ → 0 treatment in [60] but as we see below the τ → Q limit has additional subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is useful to define the following variables si = ui + imi ε 4πc , si = −ui + imi ε 4πc , �si = �ui + i�mi ε 4πc , �si = −�ui + i�mi ε 4πc , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) and the corresponding exponentiated variables zi = e2πisi, zi = e2πisi, and �zi = e2πi�si, �zi = e2πi�si, with zi = z∗ i , �zi = �z∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We set mij = mi − mj, �mij = �mi − �mj, and introduce ξij ≡ exp � −2πi4d c mij 2 � , �ξij ≡ exp � −2πi4d c �mij 2 � , ξ′ ij ≡ exp � −2πi4d c mi − �mj 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) – 15 – which are roots of unity depending on mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As we show in Appendix B, in terms of these variables, the integrand of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) essentially factorizes into two parts9 Zhol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) Zantihol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) with Zhol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = � i � exp � 2πi4πc 4iε � s2 i − �s2 i �� z−1/2 i �z1/2 i ξ ′ 1 2 ii × � a=1,2 � z−1 i �zi ξ′ ii ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � a=3,4 � z−1 i �zi ξ′ ii ζa q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � × � i>j � � z−1 i zj ξij ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � z−1 i zj ξij q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �z−1 i �zj �ξij ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �z−1 i �zj �ξij q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=1,2 � z−1 i �zj ξ′ ij ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � zj �z−1 i ξ′−1 ji ζa q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=3,4 � zj �z−1 i ξ′−1 ji ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � z−1 i �zj ξ′ ij ζa q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) and Zantihol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = Zhol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) �� k→−k, zi→zi, �zi→�zi, ζ1↔ζ3, ζ2↔ζ4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) Given this factorization of the integrand, we now use the idea of [62] to make a similar split in the integration measure using a change of contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall that zi = e−miε/2ce2πiui, so that we identify e−miε/2c as the modulus of zi, and 2πui as its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The idea is to exchange the domain of the sum over mi and integration over ui with the full complex plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=', 1 ε � mi∈Z ε∆mi � 1 0 dui ε→0 −−−→ 1 ε � +∞ −∞ d(εmi) � 1 0 dui = 2c ε � ∞ 0 d|zi| |zi| � 2π 0 dArg(zi) 2π = 2c ε � C d2zi 2π|zi|2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) where d2zi = |zi| d|zi|dArg(zi) is the flat measure on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We thus obtain I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = 1 (N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' )2 (2c)2N ε2N � N � i=1 � C2 d2zi 2πzizi d2�zi 2π�zi�zi � Zhol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, ζa) Zantihol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, ζa) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) 9More precisely, in the case c > 1, the presence of the terms ξij, �ξij and ξ′ ij forbids us from declaring that Zhol is a holomorphic function of z (and Zanti−hol of z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, as we shall see in the next section, at the leading order in the Cardy-like limit ε ↘ 0, the dependence on ξij, �ξij, ξ′ ij drops and the integrand indeed factorizes, even for c > 1 in the cases we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This factorization should be thought of as a simplifying manipulation, and should not be crucial to the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, in a related problem, the authors of [61] directly compute the large-N limit of the twisted index, without using this manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 16 – One then factorizes the full integral into “holomorphic” and “anti-holomorphic” pieces as I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = (4πc)N εN � [Ds] [D�s] Zhol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) × (4πc)N εN � [Ds] [D�s] Zantihol(z, �z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) where the holomorphic variables s, �s are integrated � [Ds] ≡ 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' N � i=1 � dsi , � [D�s] ≡ 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' N � i=1 � d�si (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) over some contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The corresponding anti-holomorphic variables are taken to be independent and run over a possibly different contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The contour integrals are then evaluated by a saddle-point approximation, which we discuss now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Generalized Cardy limits to roots of unity Our goal is to calculate the asymptotic behavior of the integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) in the generalized Cardy limit q = e2πi 4d c e− ε c as ε ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the saddle-point approximation, we can analyze the holomorphic and anti-holomorphic parts separately, which are built out of Pochhammer symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To study the asymptotics of these building blocks, we use a result of [63], whose details we summarize in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This requires q = ξme− ε m , where ξm is a primitive root of unity of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, we write (c, 4d) = gcd(c, 4d)(ℓc, ℓd) with gcd(ℓc, ℓd) = 1, so that q = ξℓce− ε/ gcd(c,4d) ℓc , and we can now apply the result from [63] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The leading order result for the holomorphic part of the integrand (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) as ε ↘ 0 is log Zhol(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' �z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) ∼ − gcd(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 4d)2 cε �� i 1 2(2πiℓc)2(s2 i − �s2 i ) + � i>j � � a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 Li2 � z−ℓc i �zℓc j ζ−ℓc a (ξ′ ijξ − 1 4 ℓc )ℓc� − Li2 � zℓc j �z−ℓc i ζℓc a (ξ′−1 ji ξ 1 4 ℓc)ℓc� + � a=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4 Li2 � zℓc j �z−ℓc i ζ−ℓc a (ξ′−1 ji ξ − 1 4 ℓc )ℓc� − Li2 � z−ℓc i �zℓc j ζℓc a (ξ′ ijξ 1 4 ℓc)ℓc�� + � i � � a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 Li2 � z−ℓc i �zℓc i ζ−ℓc a (ξ′ iiξ − 1 4 ℓc )ℓc� − � a=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4 Li2 � z−ℓc i �zℓc i ζℓc a (ξ′ iiξ 1 4 ℓc)ℓc��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) Note that the vector multiplet does not contribute in the Cardy-like limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Upon rescaling the integration variables � si, �si, si, �si � �→ 1 ℓc � si, �si, si, �si � we obtain log Zhol(z, �z, q, λ) ∼ −gcd(c, 4d)2 cε W(z, �z, λ) + O(1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) – 17 – where W(z, �z, λ) = � i 1 2(2πi)2(s2 i − �s2 i ) + � i>j � � a=1,2 � Li2 � z−1 i �zjζ−ℓc a (ξ′ ijξ − 1 4 ℓc )ℓc� − Li2 � zj�z−1 i ζℓc a (ξ′−1 ji ξ 1 4 ℓc)ℓc�� + � a=3,4 � Li2 � zj�z−1 i ζ−ℓc a (ξ′−1 ji ξ − 1 4 ℓc )ℓc� − Li2 � z−1 i �zjζℓc a (ξ′ ijξ 1 4 ℓc)ℓc��� + � i � � a=1,2 Li2 � z−1 i �ziζ−ℓc a (ξ′ iiξ − 1 4 ℓc )ℓc� − � a=3,4 Li2 � z−1 i �ziζℓc a (ξ′ iiξ 1 4 ℓc)ℓc�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) Note that the arguments of the dilogarithms above contain factors of the type (ξ′ ijξ − 1 4 ℓc )ℓc = exp � −2πiℓd � mi−�mj 2 + 1 4 �� , which deserve a comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since we have gcd(c, d) = 1, there are three possible cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=', gcd(c, 4d) = gcd(c, 4) = 1, 2, or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' When gcd(c, 4) = 1, all these factors in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' When gcd(c, 4) = 2, they are equal to e−2πi d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case it is clear from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) that this phase can be absorbed into a redefinition of ζa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' When gcd(c, 4) = 4, there is no such simplification, and we will restrict to the first two cases from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The case c = 1 and d = 0, which is relevant for the q → 1 limit of the ABJM index, has been studied in [60, 64] by the saddle-point approximation in the small parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Even in this approximation, the presence of the dilogarithm functions in W makes it difficult to perform an exact analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, the potential W simplifies in the large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We review the main points of this analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2, and use the large-N method to analyze the perturbation theory in ε to all orders in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 The large N saddle-point analysis Having expressed the ABJM superconformal index in the generalized Cardy limit in terms of an effective potential W (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17), in order to find its value in the large-N limit we should extremize W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As already noticed in [60], the effective potential W obtained in the generalized Cardy limit is a straightforward generalization, at a mathematical level, of the Bethe potential introduced in [33] to describe the topologically twisted index which is, a priori, a different problem, and one can therefore follow the analysis developed in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall that the large-N limit of a unitary matrix model can be implemented by re- placing the discrete distribution of the N eigenvalues by a continuum of eigenvalues on the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Sums over the discrete label of the eigenvalues i turn into integrals over the interval, which one replaces by integrals over the space of eigenvalues by introducing a density of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In our case, we have two distributions s and �s for the integration variables that are both complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Justified by the study of the numerics in [33], one introduces the following single-cut ansatz s(x) = v(x) − iN 1 2 x , �s(x) = �v(x) − iN 1 2 x , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) – 18 – where x ∈ [x1, x2] ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The assumptions in the ansatz are that Im(�s) = Im(s) and that x2 − x1, v(x), �v(x) are all O(1) quantities as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The sums become integrals according to N � i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' �→ N � x2 x1 dx ρ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) where the eigenvalue density ρ obeys � x2 x1 dx ρ(x) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) We split the effective action W in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) as a sum of three pieces W = W1 + W2 + W3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) with W1 = N � x2 x1 dx ρ(x) 1 2(2πi)2� s(x)2 − �s(x)2� , W2 = N2 � x2 x1 dx ρ(x) � x2 x dy ρ(y) � � � a=1,2 � Li2 ��z(x) z(y)−1 ζℓc a � − Li2 �z(x) �z(y)−1 ζ−ℓc a �� + � a=3,4 � Li2 �z(x) �z(y)−1 ζℓc a � − Li2 ��z(x) z(y)−1 ζ−ℓc a ��� � , W3 = N � x2 x1 dx ρ(x) � � � a=1,2 Li2 ��z(x) z(x)−1 ζℓc a � − � a=3,4 Li2 ��z(x) z(x)−1 ζ−ℓc a � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) Here we have introduced the notation z(x) = e2πis(x) and �z(x) = e2πi�s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Our goal is to extremize W subject to the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21), or extremizing the quantity Wµ := W + N 3 2 µ i �� x2 x1 dx ρ(x) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) The resulting equations, namely δρWµ = 0 , δvWµ = 0 , δ�vWµ = 0 , ∂µWµ = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) should be solved for the distributions ρ, s(x) and �s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We now evaluate the three terms W1,2,3 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) on the ansatz (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As we will shortly see, each piece has a simple term scaling as N 3 2 , and subleading terms scaling as N, as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The first piece is W1 ∼ −N 3 2 � x2 x1 dx ρ(x) (2π)2ix δv(x) � 1 + O(1/N 1 2 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) – 19 – where δv(x) ≡ �v(x) − v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The second piece contains terms of the following form � x2 x1 dx ρ(x) � x2 x dy ρ(y) Li2 ��z(x) z(y)−1 ζℓc a � = � x2 x1 dx ρ(x) � x2 x dy ρ(y) Li2 � e2πN 1 2 (x−y)+2πi � �v(x)−v(y)−λ′ a �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) where λ′ a ≡ ℓcλa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In matrix model language, these terms indicate non-local interactions between the eigenvalues at x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, at leading order in the large-N expansion, the integrand simplifies further, and we are left only with local interactions at each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To see this, it is convenient to change integration variables from y to δy := N 1 2 (y − x), after which the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) becomes N− 1 2 � x2 x1 dx ρ(x) � N 1 2 (x2−x) 0 dδy ρ(x + δy/N 1 2 ) Li2 � e−2πδy+2πi � �v(x)−v(x+δy/N 1 2 )−λ′ a �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) At leading order in the large-N expansion (at fixed δy) this takes the asymptotic form N− 1 2 � x2 x1 dx ρ(x)2 � +∞ 0 dδy Li2 � e−2πδy+2πi(δv(x)−λ′ a)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29) Then, using d dxLi3(ex) = Li2(ex), one concludes that at leading order � x2 x1 dx ρ(x) � x2 x dy ρ(y) Li2 ��z(x) z(y)−1 ζℓc a � ∼ N− 1 2 2π � x2 x1 dx ρ(x)2 Li3 � e2πi(δv(x)−λ′ a)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='30) Upon applying the above analysis to the four terms in W2, one obtains W2 ∼ N 3 2 2π � x2 x1 dx ρ(x)2 � � � a=1,2 � Li3 � e2πi(δv(x)−λ′ a)� − Li3 � e−2πi(δv(x)−λ′ a)�� − � a=3,4 � Li3 � e2πi(δv(x)+λ′ a)� − Li3 � e−2πi(δv(x)+λ′ a)�� � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) We can simplify this expression further using the identity (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) Li3 � e2πix� − Li3 � e−2πix� = 4π3i 3 B3(x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='32) where B3 is the third periodic Bernoulli polynomial that is defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7), obtaining W2 ∼ 2π2i 3 N 3 2 � x2 x1 dx ρ(x)2 � � � a=1,2 B3 � δv(x) − λ′ a � − � a=3,4 B3 � δv(x) + λ′ a � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='33) – 20 – Finally, the third piece W3 can be written as W3 ∼ N � x2 x1 dx ρ(x) � � � a=1,2 Li2 � e2πi(δv(x)−λ′ a)� − � a=3,4 Li2 � e2πi(δv(x)+λ′ a)� � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) Note that the factor in front of the integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) is N, instead of N 3 2 as in W1, W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This means that W3 does not contribute to the on-shell value of the effective action W in the large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, it cannot be naively discarded, since it is relevant for the saddle point equations, as the dilogarithm Li2(z) is not analytic at the branch point z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, the first derivative of the integrand in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) with respect to δv(x) can grow as O(N 1 2 ), if the function approaches a branch point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, W3 contributes at the same order as W1 and W2 to the equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) obtained by taking the derivative with respect to δv(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' At this stage, the degree of difficulty of the original extremization problem is substantially diminished, as the large-N form of the potential W given by the sum of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='33), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34), has no non-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Moreover, if one focuses on the contribution at order N 3 2 to the potential, one notices that in terms of the field variables ρ(x) and δv(x) the problem is piecewise quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, solutions to the variational problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) can be found by using a linear ansatz ρ(x) = ρ0 + xρ1 for the density of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The solutions when all the λas are equal are particularly simple, and we present them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The details for generic values of λas are discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Unrefined index We set all the chemical potentials to be equal (λa ≡ λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The con- straint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) implies that λ = n1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As for λ′ a, recall from the discussion below (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18), that if gcd(c, 4) = 2, a phase is absorbed in ζa, so we write λ′ = (ℓcn1 + ℓd)/4, with the under- standing that if gcd(c, 4) = 1, then ℓd = 4d and thus ℓd/4 can be removed from λ′ (which is defined modulo 1), and if gcd(c, 4) = 2, then ℓd = 2d and ℓd/4 is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We further assume that δv(x) does not cross a branch point of the dilogarithm, so that we can effectively ignore W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This picture is also warranted by the numerics [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To leading order in N, the function to be extremized Wµ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) takes the simple form Wµ = N 3 2 i � x2 x1 dx ρ(x) � −4π2x δv(x) + 4π2 3 ρ(x) (B3 (X−(x)) − B3 (X+(x))) � + N 3 2 µ i �� x2 x1 dx ρ(x) − 1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='35) where X±(x) ≡ δv(x) ± ℓcn1 + ℓd 4 + w± (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='36) for some integers w± such that 0 < X±(x) < 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) In the following, we shall re-express these integers using Σ ≡ w++w− and ∆ ≡ w+−w−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The single-cut solution is defined on a single sheet of the multi-valued polylogarithms provided w± do not depend on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 21 – Combining the first three extremization equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) gives (ignoring boundary terms) 0 = 4x + ρ(x)(ℓcn1 + ℓd + 2∆) (2(Σ − 1) + 3 δv(x)) , 0 = 48π2x δv(x) + π2ρ(x)(ℓcn1 + ℓd + 2∆) � 8 + (ℓcn1 + ℓd + 2∆)2 + 12 Σ(Σ − 2) + 48 δv(x)(Σ − 1) + 48 δv(x)2� − 12µ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38) These equations are easily solved for ρ(x) and δv(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, for fixed x, the first equation is linear in ρ and δv, and upon substituting in the second equation the expression for ρ obtained from the first (and assuming that ρ is finite), we find that the resulting equation is linear in δv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The solutions are: ρ(x) = 12µ + 24π2(Σ − 1)x π2 (ℓcn1 + ℓd + 2∆ − 2) (ℓcn1 + ℓd + 2∆) (ℓcn1 + ℓd + 2∆ + 2) , δv(x) = −π2x(ℓcn1 + ℓd + 2(∆ − Σ))(ℓcn1 + ℓd + 2(∆ + Σ)) − (Σ − 1) � 6 + 8π2x(2Σ − 1) � 12 (µ + 2π2x(Σ − 1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39) We focus on the solution with constant eigenvalue density, so we set Σ = 1, which means that ∆ must be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10 The conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) imply 6|µ| π2(ℓcn1 + ℓd + 2∆ + 2)(ℓcn1 + ℓd + 2∆ − 2) < x < −6|µ| π2(ℓcn1 + ℓd + 2∆ + 2)(ℓcn1 + ℓd + 2∆ − 2) , −2 < ℓcn1 + ℓd + 2∆ < 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) In particular, the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) imposes that ℓcn1 + ℓd + 2∆ ∈ {−1, 0, 1}, but the choice 0 (from the first equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39)) leads to a divergent density ρ unless µ = 0 (which would reduce the support to a single point), and therefore should be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Thus ρ = − sgn(ℓcn1 + ℓd + 2∆) 4µ π2 , δv = π2 4µx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='41) Clearly, in order to have a positive density of eigenvalues, we need µ ∈ R and sgn(µ) = − sgn(ℓcn1 + ℓd + 2∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) is only a necessary condition for the range of x such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In fact, upon substitution of ℓcn1 + ℓd + 2∆ = ±1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) reduces to − |µ| π2 < x < |µ| π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='42) This is the smallest interval for x such that the single-cut solution does not cross a branch cut of the polylogarithm, thus we set x1 = − |µ| π2 and x2 = |µ| π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, we impose the 10The solution with constant eigenvalue density can also be obtained as a limit of the solutions to the saddle point equations of the refined index in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 22 – normalization of ρ, which is equivalent to extremizing Wµ with respect to µ, finding ρ = √ 2 , δv = − sgn(ℓcn1 + ℓd + 2∆) x √ 2 [x1, x2] = � − 1 2 √ 2, 1 2 √ 2 � , µ = − sgn(ℓcn1 + ℓd + 2∆) π2 2 √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='43) Overall, ∆ remains a free odd integer, and W = ∓ iπ2 3 √ 2N 3 2 if ℓcn1 + ℓd = ±1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='44) We then substitute in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) and perform the same analysis for the anti-holomorphic part (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11), reaching the following conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The extremization problem for the unrefined index has O(N 3 2 ) scaling only if ℓcn1 + ℓd = ±1 mod 4 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='45) and log I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) ∼ ∓ π 3 √ 2N 3 2 1 ℓc (ℓcτ − ℓd) as τ → 4d c and ℓcn1 + ℓd = ±1 mod 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) We recall that this holds if gcd(c, 4d) = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To summarize, we have the following picture: we start with the unrefined index, which is a single-valued function of T I(T) = TrHBPS(−1)2Je2πiT(4J+� a Qa) = � n dnQn , (T ∼ T + 1 , Q ≡ e2πiT ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='47) and to find dn we look at the limit where Q goes to a primitive root of unity, as explained in Section 1, or T → D/C with gcd(C, D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to study the asymptotic behavior, we further introduce 4T = ℓD/ℓc, for coprime ℓc, ℓD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of these, we write the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='45) as 4ℓcT = ℓD = ±1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since ℓD = 4D/ gcd(C, 4), this constraint can only be solved if gcd(C, 4) = 4, in which case we have log I(T) ∼ ∓ π 3 √ 2N 3 2 1 C 4 � CT − D � as T → D C with D = ±1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='48) This leads us to the conclusion that the leading growth for the index (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='47) is controlled by the singularity near T = ± 1 4, that is, when Q approaches the non-trivial primitive fourth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of τ and n1, which is how the index has been presented in the literature, our methods allow us to reach these singularities as τ → 0 and n1 = 1, 3, τ → 2 and n1 = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We are currently unable to study the case τ → 1, 3 as n1 = 0, but it is natural to conjecture that it gives the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since the index is a four-sheeted function of τ, the singularities in the large-N limit appear on the first and third sheet, or taking the Cardy limit τ → 0 of the R-charge index defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' There is a clear analogy with the case of four-dimensional N = 4, which is developed in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 23 – Refined index We now move to the most refined ABJM index, where the chemical poten- tials λa are taken to be generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, the solutions to the extremization problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) are more involved than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39), and have been addressed in [33] using the techniques of [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Details of their construction are reviewed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Here, we notice that, assuming λa ∈ R, a solution to the extremization problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) with scaling N 3 2 in the large-N limit exists in terms of a single-cut eigenvalue distributions only provided the chemical potentials and ℓc satisfy 4 � a=1 {ℓcλa + ℓd/4} = 1 or 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49) Here, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='36), we have introduced ℓd/4, which is either an integer or a half-integer, depending on whether gcd(c, 4) = 1, 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Notice that this constraint consistently reduces to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='45) upon setting all λa to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Combining the contributions from the extremization of the holomorphic and the anti- holomorphic parts in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14), one finds that the leading result for the ABJM index, as ε = 2πi (cτ − 4d) ↘ 0, is log I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) ∼ ∓ 8π √ 2 N 3 2 3 � {ℓcλ1 + ℓd/4}±{ℓcλ2 + ℓd/4}±{ℓcλ3 + ℓd/4}±{ℓcλ4 + ℓd/4}± c(cτ − 4d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='50) where {x}+ := {x} and {x}− := {x} − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The upper and lower signs correspond to the first or second case in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A few comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Firstly, studying the regions associated to different asymp- totic behaviors of the fully refined index I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) is quite involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is clear that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49) in the case ℓc = 1 and ℓd = 0 reduces to n1 = 1, 3, and thus we obtain the same picture to the one just described for the unrefined index: in the Cardy limit τ → 0, the non-trivial large-N limit is found on the first and third sheet of the multivalued function I(τ), which corresponds to the R-charge index defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' More generally, though, the relation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49) to the shifts of τ is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11 Secondly, notice that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='50) is invariant under a shift of any λa by 1 ℓc , in addition to the shift λa → λa + 1 due to the quantization of the charges Qa (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The emergent periodicity suggests that in the expansion near roots of unity the superconformal index only counts operators with charges dual to λa being an integer multiple of ℓc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' That is, in such an expansion the contribution to the trace (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) of operators with charges dual to λa not being an integer multiple of ℓc is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Interestingly, the same phenomenon happens in four-dimensions [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, recall that the partition function in the microcanonical ensemble is obtained by taking the Laplace transform of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The saddles near T → ± 1 4 (equivalently, c = 1 and d = 0 and 2 for fixed n1 = 1) contribute equally in magnitude to the Laplace transform, and with opposite phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The logarithm of the real part agrees with the entropy of the 11Of course, the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49) can be expressed without reference to n1, as it should from its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 24 – dual supersymmetric black hole, and the interference of the phases produces macroscopic oscillations (of order N 3 2 ) in the microcanonical index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The same phenomenon has been observed in related contexts in [10, 18, 19, 60, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Index with pairwise equal fugacities Finally, we discuss a further case, which is relevant for the comparison with holographic duals in Section 6, namely the index (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) with pairwise equal fugacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We set λ1,2 ≡ σ1 and λ3,4 ≡ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The result can be straightforwardly obtained from the fully refined case just discussed (see appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Assuming that σ1,2 ∈ R, the single- cut eigenvalue distribution provides a solution to the extremization problem that scales like N 3 2 in the large-N limit provided 2 ({ℓcσ1 + ℓd/4} + {ℓcσ2 + ℓd/4}) = 1 or 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='51) Substituting the constraint 2(σ1 + σ2) = n1, it is straightforward to show that the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='51) has the same parity as ℓcn1 + ℓd, which in the cases we are interested in corresponds to that of ℓcn1 (since ℓd is even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, ℓcn1 cannot be even, which singles out the R-charge index among the sheets in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' More precisely, if c is odd, then the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='51) is satisfied if cn1 is odd, and imposes {cσ1} ≤ 1 2 for the right-hand side to be 1, and {cσ1} > 1 2 for the right-hand side to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' If instead gcd(c, 4) = 2, then one needs cn1/2 odd, but now {cσ1} ≤ 1 2 is consistent with the right-hand side being 3, and {cσ1} > 1 2 is consistent with the right-hand side being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Upon substitution, in the case of c odd, we require n1 to be odd, and the resulting value of the large-N partition function is log I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' σ) ∼ −8π √ 2 3 N 3 2 {cσ1}{cσ2} ℓc (cτ − 4d) if τ → 4d c and {cσ1} ≤ 1 2 , log I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' σ) ∼ +8π √ 2 3 N 3 2 (1 − {cσ1})(1 − {cσ2}) c (cτ − 4d) if τ → 4d c and {cσ1} > 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='52) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3 Subleading effects Now we consider sub-leading effects in the asymptotic expansion in ε around any rational point d/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Our goal is to show that the expansion terminates at order ε i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' all the terms of order εk, k ≥ 2 vanish in the large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The same result was found in [67] for the case of the black hole saddle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (c, d) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We use the leading order analysis of the previous subsection as a reference and discuss the new points that arise in the all-order analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We begin again with the holomorphic part of the potential (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10), expand it to all orders in ε as given in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2), and run the steps of the large-N expansion as in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The potential W splits into three pieces W1,2,3 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23), exactly as at leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The term W1 is classical and does not depend on ε and therefore remains the same as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The other two pieces W2, W3 have, a priori, an infinite expansion governed by the identity (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2), with the leading-order term given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 25 – Consider a general term of order εk, k ≥ 2, in the expansion of W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This term has the same structure as in the leading O(1/ε) term with Li1−k replacing Li2, with coefficients being Bernoulli polynomials given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We first consider the case τ → 0, so that (c, d) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) with the value of w and ν determined by the Pochhammer symbols in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We end up with a double integral as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) with linear combinations of terms Bk+1(−ν) Li1−k ��z(x) z(y)−1 ζ � − Bk+1(1 + ν) Li1−k �z(x) �z(y)−1 ζ−1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='53) with ζ = ζℓc a , a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 4, and ν = 0 for vector multiplets and ν = − 1 4 for hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The large-N analysis of the term W2 proceeds as before for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The non-local interactions drop out—effectively identifying x and y in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='53)—and the arguments of the polylogarithms in the combination (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='53), writing ζ = e2πiλ, become Z and 1/Z with Z = e2πi(δv(x) − λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Continuing onwards, we have, in the analog of the step (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='30), that Li1−k integrates to Li2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This gives an integral as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) with the integrand consisting of linear combina- tions of the differences Bk+1(−ν) Li2−k(Z) − Bk+1(1 + ν) Li2−k(1/Z) , ν = 0 , −1 4 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='54) with Z = e2πi(δv(x) − λ) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Now, using Bk(1 + ν) = (−1)kBk(−ν) , k > 2 , 0 ≤ −ν ≤ 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='55) we see that the combination (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='54) equals Bk+1(−ν) � Li2−k(Z) − (−1)k+1Li2−k(1/Z) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='56) which vanishes due to the following classical identity satisfied by polylogarithms Li−r(Z) + (−1)r Li−r(1/Z) = 0 , r ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='57) The case when τ → Q is not much different compared to when τ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Consider τ → d/c, c, d ∈ Z with gcd(c, d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The asymptotic expansion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) now contains c terms, and in lieu of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='54) we have, with ν = 0, − 1 4, c � t=1 Bk+1 � 1 − t + ν c � Li2−k � Z e2πi 2d c (t+ν)� − c � t′=1 Bk+1 � 1 + ν + 1 − t′ c � Li2−k � Z−1e−2πi 2d c (ν+1−t′)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='58) Now, for each value of t in the first term, there corresponds exactly one value of t′ in the second given by t′ = c + 1 − t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='59) – 26 – Pairing up the two sums in this manner, we obtain that the coefficient of the term εk is proportional to c � t=1 � Bk+1 � 1 − t + ν c � Li2−k � Z e2πi 2d c (t+ν)� − Bk+1 �t + ν c � Li2−k � Z−1e−2πi 2d c (ν+t)�� = c � t=1 Bk+1 � 1 − t + ν c � � Li2−k � Z e2πi 2d c (t+ν)� − (−1)k+1 Li2−k � Z−1e−2πi 2d c (ν+t)�� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='60) Here, the first equality follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='55) (and that 0 ≤ 1 − (t + ν)/c ≤ 1 for the above values), and the second equality follows from the fact that each term in the sum vanishes due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, we turn to the analysis of W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Now there is a significant difference compared to the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall from the discussion below (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) that the 1/N 1 2 suppression of W3 is compensated by the N 1 2 -growth of the derivative of Li2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This phenomenon occured due to the fact that the derivative Li′ 2(z) is large while Li2(z) itself is small is due to the logarithmic non-analyticity of the function at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For the terms O(εk), the Li2 is replaced by Li1−k, which, for k ≥ 2, are meromorphic functions and therefore do not have large derivative, as can be explicitly checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We conclude that the W3 term is suppressed at large N for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Collecting the above facts together, we reach the conclusion that the perturbation ex- pansion of the large-N index around any rational point only contains terms multiplying ε−1, ε0, and ε, and the O(εk) terms vanish for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 4 Black holes and supersymmetric solutions In this section we move to the bulk gravity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We briefly review the Kerr–Newman- AdS black hole, then consider a related family of complex solutions to minimal gauged su- pergravity, and finally connect to the BPS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Similar studies of these solutions have been made in [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Kerr–Newman-AdS black hole The Lorentzian action for Einstein–Maxwell theory with a cosmological constant is S = 1 16πG4 � Y4 � R + 6 − F2� volG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) Here R is the Ricci scalar of the metric G, F is the curvature of the U(1) gauge field A, and −3 is the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A black hole solution with rotation and electric charge has been known for a long time [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In a frame that is non-rotating at infinity, which is more – 27 – immediate for uses in AdS/CFT [69, 70], the solution is ds2 = −∆r∆Θ BΞ2 dt2 + sin2 Θ B � dφ + a∆Θ ∆r − (1 + r2)(r2 + a2) BWΞ2 dt �2 + W �dr2 ∆r + dΘ2 ∆Θ � , A = mr sinh δ WΞ � ∆Θ dt − a sin2 Θ dφ � + γ dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) Here γ is a constant, Θ ∈ [0, π), φ ∈ [0, 2π), and ∆r = (r2 + a2)(1 + r2) − 2mr cosh δ + m2 sinh2 δ , ∆Θ = 1 − a2 cos2 Θ , W = r2 + a2 cos2 Θ , Ξ = 1 − a2 , B ≡ ∆Θ(r2 + a2)2 − a2 sin2 Θ ∆r WΞ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) The black hole has an outer horizon at r = r+, the largest positive root of ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' On a slice of constant t and r outside the horizon, the solution is topologically a sphere, and it is easy to see that it closes off smoothly at Θ = 0, π with φ ∼ φ+2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We then perform a Wick rotation t = −itE and look near the horizon in geodesic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The space caps off smoothly at the horizon only if we identify (tE, φ) ∼ (tE, φ + 2π) ∼ (tE + β, φ − iΩβ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) where the temperature and angular velocity of the horizon are β = 4πa2 + r2 + ∆′r(r+) , Ω = a 1 + r2 + a2 + r2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) The metric (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2), once Wick rotated, is real if we take a to be pure imaginary and δ, m to be real, whereas the gauge field A is pure imaginary provided γ is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This also makes Ω above pure imaginary, and the topology is that of the product of a disc and a 2-sphere [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The metric (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) has two obvious commuting Killing vectors ∂t, ∂φ, and the surface {r = r+} is a Killing horizon of the linear combination V = ∂ ∂t + Ω ∂ ∂φ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) The electrostatic potential is12 Φe := ιV A|r=r+ − ιV A|r→∞ = m sinh δ r+ a2 + r2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) 12More generally, we should only pick the Θ-independent part of ιV A|r→∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 28 – In Euclidean signature, the horizon is a bolt for V , and regularity of the gauge field on the disc requires that ιV A vanishes at the origin r = r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This fixes the gauge choice in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) to be γ = −Φe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The Bekenstein–Hawking entropy of the horizon is S = π G4 r2 + + a2 1 − a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) That the black hole is asymptotically anti-de Sitter can be shown by considering the region r → ∞ and applying the following coordinate change in the asymptotic region [71] cos θ z = r cos Θ , z−2 = r2∆Θ + a2 sin2 Θ Ξ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) obtaining, as z → 0, ds2 ∼ dz2 z2 + 1 z2 � −dt2 + � dθ2 + sin2 θ dφ2�� , A ∼ −Φe dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) The boundary metric in Euclidean signature is not just the round S1 × S2, due to the iden- tifications (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To see this explicitly, define tE = tE, �φ = φ + iΩtE, so that the boundary metric has the fibered form ds2 bdry = dt2 E + � dθ2 + sin2 θ (d�φ − iΩ dtE)2� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) but now with the identifications (tE, �φ) ∼ (tE, �φ + 2π) ∼ (tE + β, �φ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) This metric is real for pure imaginary a, as consistent with the comments below (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is clear that the metric (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) and the boundary gauge field A ∼ iΦe dtE match the metric of the Euclidean background over which the index is computed as a functional integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12), and the background gauge field coupled to the u(1)R symmetry generated by H1, A(H1) = iΦ(H1) dtE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to compute the conserved charges, we can use the standard methods of holo- graphic renormalization (see [72] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We introduce a cutoff at z = δ ≥ 0 and consider the geometry of the hypersurface ∂Yδ = {z = δ} ∩ Y4 ∼= M3, with induced metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to make the problem well-defined and remove the divergences, we need to add to the action the Gibbons–Hawking–York term and the counterterms, so that the renormalized on-shell action is I = lim δ→0 � S + 1 8πG4 � ∂Yδ � K − 2 − 1 2R � volh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) We then define the holographic stress-energy tensor and electric current ⟨Tij⟩ = − 2 √−g δI δgij , ⟨ji⟩ = 1 √−g δI δAi , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) – 29 – where i, j are labels for the boundary coordinates, and g, A are, respectively, the boundary metric and gauge field given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) and A = −Φe dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These quantities satisfy the following equations ∇i⟨Tij⟩ = Fji⟨ji⟩ , ∇i⟨ji⟩ = 0 , ⟨T i i⟩ = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) and for any boundary vector Kj generating a symmetry (that is LKg = 0 and LKA = 0) we can construct a conserved current ∇i � (⟨T i j⟩ + ⟨ji⟩Aj)Kj� = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) where ∇ is the Levi-Civita connection of the boundary metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Let C be a surface of constant t, and ui be the future-directed unit normal to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Then, the conserved charge associated to K is computed by Q[K] = � C∩M3 ui(⟨T i j⟩ + ⟨ji⟩Aj)Kj volC∩M3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) Notice that this definition is not invariant under gauge transformations of A (as stressed in [73, 74]), but it still gives a conserved charge provided the gauge transformed A′ still satisfies LKA′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The angular momentum is associated to K = −∂φ, so in our gauge J = − � C∩M3 ui⟨T i φ⟩ volC∩M3 = am cosh δ G4Ξ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) We define the electric charge as Qe = � C∩M3 ui⟨ji⟩ volC∩M3 = − 1 4πG4 � C∩M3 ∗4F = m sinh δ G4Ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) Finally, the energy is associated to K = ∂t, and in our gauge choice E′ = � C∩M3 ui⟨T i t⟩ volC∩M3 − Φe � C∩M3 ui⟨ji⟩ volC∩M3 = m cosh δ G4Ξ2 − ΦeQe = E − ΦeQe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) Here E = m cosh δ/G4(1 − a2) is the value of the energy in the gauge A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Moving to Euclidean signature, we can compute the holographically renormalized on-shell action, obtaining I = β 2G4(1 − a2) �r2 + + a2 r+ − m cosh δ + a2 m2 sinh2 δ r+(r2 + + a2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) This action obeys the quantum statistical relation [72] I = −S + β(Q[V ] − Φh Qe) = −S + β(Q[∂t] − Ω Q[−∂φ] − Φh Qe) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) – 30 – where Φh = ιV A|r=r+ is by definition a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Even though each term in the relation above is separately not gauge invariant, the overall relation is invariant under gauge transformations of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Concretely, it reduces to the canonical form in the A = 0 gauge I = −S + β(E − Ω J − Φe Qe) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) Moreover, varying δ, a, m, we find that the first law of thermodynamics holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Written as above in terms of holographic conserved charges it reads [72] dQ[∂t] = β−1 dS + Ω dQ[∂φ] + Φh dQe (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) or concretely dE = β−1dS + Ω dJ + Φe dQe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) It follows from combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) that β, Ω, Φe are the chemical potentials conjugate to the conserved charges, since E = ∂I ∂β ���� βΩ,βΦe , J = − 1 β ∂I ∂Ω ���� β,Ωe , Qe = − 1 β ∂I ∂Φe ���� β,Ωe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) This shows that the on-shell action I = I(β, Ω, Φe) is minus the logarithm of the grand canonical partition function, the Gibbs free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 Supersymmetry The Einstein–Maxwell action (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) describes the bosonic sector of four-dimensional minimal gauged supergravity [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A solution to the equations of motion coming from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) is super- symmetric provided there is a non-zero Dirac spinor ϵ satisfying the equation � ∇µ − iAµ + 1 2Γµ + i 4FνρΓνρΓµ � ϵ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) The condition of supersymmetry on the solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) with parameters (a, δ, m) implies [76, 77] E = J + Qe ⇔ a = coth δ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) Thus, the family of supersymmetric solutions can be parametrized by the two parameters δ, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Supersymmetry and global regularity requires that in presence of non-zero electric charge the angular momentum must be non-zero [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13 Imposing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) reduces ∆r in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) to a sum of squares ∆r|SUSY = � r2 − coth δ + 1 �2 + coth2 δ � r − msinh2 δ cosh δ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29) 13The conclusion is different in presence of a magnetic charge [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 31 – Assuming reality of all parameters, both squares should vanish at the horizon, which fixes the value of the horizon radius r∗ and m: r2 ∗ = coth δ − 1 , m2 = cosh2 δ eδ sinh5 δ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='30) leaving only one free parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is now easy to check that ∆′ r|SUSY(r∗) = 0, that is, supersymmetry and regularity of the Lorentzian metric imply extremality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to define the gravitational partition function and reproduce the large-N behavior of the supersymmetric index (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18), we need to consider supersymmetric solutions connected to the Euclidean solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As we observed around (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5), the metric is real after Wick rotation provided we choose a to be pure imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Clearly, this can only be compatible with the supersymmetry condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) if δ is complex, but this is itself incompatible with a real Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, we conclude that the Wick rotation of a real supersymmetric Lorentzian metric of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) is generically complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14 We shall therefore focus on the family of complex metrics obtained by imposing the supersymmetry condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) without requiring reality of the metric and gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These solutions arise from extending to complex parameters the Euclidean “black hole” solution with topology R2 × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This approach was first suggested in five dimensions in [3] and elaborated in other dimensions (including four) in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Here, we have a two-parameter family, and it is convenient to trade the parameter δ for r∗ using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='30), and m for the largest root r+ of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=', m = 1 sinh2 δ � r+ cosh δ ± i � sinh δ(1 + r2 +) − cosh δ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) The thermodynamic quantities of the Euclidean supersymmetric solutions are generically complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The chemical potentials β = ±2πi r2 + + r4 ∗ (r2 + − r2∗)(1 ± 2ir+ + r2∗) , Ω = r2 ∗ 1 + r2 + r2 + + r4∗ , Φe = r+ r+r2 ∗ + r+ ± i(r+ − r∗)(r+ + r∗) r2 + + r4∗ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='32) and the conjugate charges E = r+r2 ∗ + r+ ± i(r+ − r∗)(r+ + r∗) G4(1 − r4∗)(1 − r2∗) , J = r2 ∗ r+r2 ∗ + r+ ± i(r+ − r∗)(r+ + r∗) G4(1 − r4∗)(1 − r2∗) , Qe = r+r2 ∗ + r+ ± i(r+ − r∗)(r+ + r∗) G4(1 − r4∗) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='33) 14The bulk Killing spinor equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) is analytic in the supergravity fields, so it is still satisfied by the Wick-rotated solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 32 – can be computed directly or read off by imposing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) in the corresponding expressions for Wick-rotated non-supersymmetric solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Here the different sign choices refer to the two branches of solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The conserved charges satisfy the supersymmetry con- dition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) by construction, and we observe that the chemical potentials also satisfy the constraint β (1 − 2Φe + Ω) = ∓2πi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) If in addition to supersymmetry we impose extremality, we restrict to the Wick rotation of the supersymmetric Lorentzian extremal solution discussed around (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29), which we call the BPS locus and indicate by an underscript ∗, as in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The charges of the extremal solution E∗ = r∗ G4(1 − r2∗)2 , J∗ = r3 ∗ G4(1 − r2∗)2 , Qe∗ = r∗ G4(1 − r2∗) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='35) are real, and the extremal chemical potentials Ω∗ = 1 , Φe∗ = 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='36) are real and fixed to constant values independent of the charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Clearly the constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) stops being meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It will be useful instead to define the “reduced chemical potentials” for the supersymmetric solutions τg ≡ β Ω − Ω∗ 2πi , ϕg ≡ β Φe − Φe∗ 2πi , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) which by construction satisfy the constraint τg − 2ϕg = ∓1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38) This allows us to write the quantum statistical relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) as I = β(E − J − Qe) − S − 2πiτg J − 2πiϕg Qe ⇒ I|SUSY = −S − 2πiτg J − 2πiϕg Qe , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39) the second equation following from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the following, we shall approach the BPS locus via a limiting process taking β → ∞, Ω → Ω∗, Φe → Φe∗ keeping τg, ϕg fixed to a complex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As shown above, the constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) follows from imposing supersymmetry on the parameters of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It should be the case that it follows directly from the requirement that the bulk Killing spinor satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) is anti-periodic when transported around the circle generated by V [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The antiperiodicity of smooth Killing spinors is consistent with the topological statement that in Euclidean solutions the circle generated by V shrinks in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It was shown in [3] that this is indeed the case in the asymptotically AdS5 context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, to the best of our knowledge, an explicit expression for the bulk Killing spinor of the four-dimensional black hole is not known (see [79] for a discussion of the supersymmetry of the – 33 – metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Nonetheless, we can draw some conclusions by looking near the conformal boundary (see also [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As we show in Appendix D, the anti-periodicity of the boundary Killing spinor around the circle that is contractible in the bulk leads to the following condition, β (1 − 2Φe + Ω) = 2πin0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) with odd n0, or equivalently τg − 2ϕg = n0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='41) It should be remarked, though, that at this level the topological argument is a formal state- ment, since the non-extremal supersymmetric solutions are complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In fact, the bulk solution imposes a stronger condition, as the chemical potentials satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) with n0 = ∓1 depending on the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) chosen (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One of the main interests in the family of complex solutions is that they allow us to define a regularization of the on-shell action of the extremal supersymmetric black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The Wick-rotated extremal supersymmetric black hole has an infinite throat due to an H2 factor in the metric, whereas the family described above is originated from an extension to complex parameters of the Wick-rotation of the non-extremal supersymmetric black hole solution, which has the topology of the product of a disc and a 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Thus, the on-shell action of the BPS solution can be defined as the limit of the on-shell action of the solutions in the complex family, and this limit is well-defined [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One can compute the on-shell action of the solutions by direct application of the holographic renormalization, as done above for the non-supersymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In fact, it is possible to derive the supersymmetric result in a shorter and elegant manner extending the methods in [80], as done in [74] for accelerating black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Notice that the boundary Killing vector ξi = �χEγiχE, computed from the boundary Killing spinor (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6), is ξ = 2u�u � ∂ ∂tE + i (Ω − 1) ∂ ∂ �φ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='42) and can thus be extended to a bulk Killing vector, which is constructed as a bilinear of the bulk Killing spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The on-shell action of any smooth supersymmetric solution of minimal gauged supergravity with real metric and gauge field can be expressed in terms of topological data of the circle action generated by the bulk “supersymmetric” Killing vector field on the spacetime [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, we are considering a family of solutions outside the reality assumptions, and yet the formula found in [80] remarkably still holds, as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall that the topology of the Wick-rotated black hole is R2 × S2, parametrized by (r, tE) − (Θ, �φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to find the generators of the rotations with unitary weight on the two factors, we rescale the coordinates ϕ1 = 2π β tE , ϕ2 = �φ , ϕ1,2 ∼ ϕ1,2 + 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='43) – 34 – In these coordinates, the Killing vector has the form ξ = b1∂ϕ1 + b2∂ϕ2 with weights b1 = 2u�u2π β , b2 = −2u�u2πτg β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='44) The Killing vector has isolated fixed points at the North and South pole of the S2 factor, so the on-shell action of the solution is given by the sum of the contributions I|SUSY = π 2G4 � nuts∓ ±(b1 ± b2)2 4b1b2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='45) where the label ± for the nut corresponds to the chirality of the bulk spinor there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In our case, a solution in the ± branch of solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) has nuts with ± chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Substituting in the formula the values of the weights and using the constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38), we find I|SUSY = ± π G4 ϕ2 g τg , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) where again the ± refers to the branch of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As mentioned above, this result is consistent with what is obtained by direct substitution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15 The final result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) for the action in terms of the variables τg, ϕg is independent of β and therefore the limit β → ∞ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This is identified as the regulated on-shell action of the BPS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The above result is also consistent with the results of [80]: the on-shell action should only depend on topological data of the circle action of ξ, which is well-defined for the complex solutions and it is independent of the deformation parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Even more concretely, observe that the Killing vector (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='42), after a Wick rotation back to Lorentzian, coincides with the null generator of the horizon of the BPS black hole (since it has the form ∂t + Ω∗ ∂φ, to be compared with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The derivation above and the analogous result for black holes with orbifold horizons [74] suggest that it should be possible to extend the proof of [80] to complex solutions, finding a general way of regularizing the on-shell action for extremal black holes in minimal gauged supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16 Extending the principles of Euclidean quantum gravity to the family of supersymmetric complex solutions, we can obtain the entropy of the BPS black hole in the microcanonical ensemble by taking the Legendre transform of the on-shell action I|SUSY(τg, ϕg) (see also [34, 35, 82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the following we briefly review it highlighting the assumptions made at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 15In four-dimensional minimal supergravity there is no need to add finite counterterms in order for holo- graphic renormalization to be consistent with supersymmetry [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 16Notice that for static magnetically charged black holes with horizon homeomorphic to a Riemann surface of genus higher than 1, an analogous derivation within a family of smooth real supersymmetric solutions has been proposed in [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 35 – First, we observe that I|SUSY in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) is a homogeneous function of degree one, so its Legendre transform vanishes unless we impose the non-homogeneous constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to find the constrained Legendre transform of I|SUSY(ϕg, ωg), we should extremize f(τg, ϕg, Λ) = −I|SUSY(τg, ϕg) − 2πiτg J − 2πiϕg Qe + Λ (τg − 2ϕg − n0) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='47) where n0 = ∓1 represents the branch of gravitational solutions arising from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We notice that at the critical point (τg∗, ϕg∗, Λ∗) the following holds f(τg∗, ϕg∗, Λ∗) = −I|SUSY(τg∗, ϕg∗) + ∂I|SUSY ∂τg (τg∗, ϕg∗)τg∗ + ∂I|SUSY ∂ϕg (τg∗, ϕg∗)ϕg∗ − n0Λ∗ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='48) and together with Euler’s theorem for I|SUSY(τg, ϕg), this leads to the (implicit) Legendre transform �f(J∗, Qe∗) = −n0Λ∗(J∗, Qe∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49) Concretely, we can combine the critical point equations into the quadratic equation for Λ∗(J∗, Qe∗) 0 = Λ2 ∗ + � 2πiQe∗ − n0 π G4 � Λ∗ + � −π2Q2 e∗ + n0 2π2i G4 J∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='50) We then impose the constraints J∗, Qe∗ ∈ R , Λ∗(J∗, Qe∗) ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='51) The first one says that the charges are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The second one leads to the entropy being real, as we see shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' With these constraints, we can write the real and imaginary parts of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='50) separately, finding Λ∗ = −n0 πJ∗ G4Qe∗ , J∗ = Qe∗ 2 � −n0σ1 � 1 + 4G4Q2∗ − 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='52) where σ1 = ±1 represents the additional sign choice for the two solutions of the quadratic equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Substituting in �f gives �f(J∗, Qe∗) = πJ∗ G4Qe∗ = π 2G4 � −n0σ1 � 1 + 4G2 4Q2e∗ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='53) The requirement that the entropy �f(J∗, Qe∗) should be positive implies that σ1 = −n0, so that �f(J∗, Qe∗) = πJ∗ G4Qe∗ = π 2G4 �� 1 + 4G2 4Q2e∗ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='54) This corresponds to the Bekenstein–Hawking entropy of the BPS black hole S∗ = πJ∗ G4Qe∗ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='55) and to the non-linear constraint between the charges imposed by supersymmetry J∗ = Qe∗ 2 �� 1 + 4G2 4Q2e∗ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='56) – 36 – 5 A family of saddles in AdS In this section we introduce the gravitational dual to the generalised Cardy limits of Section 3, where τ approaches a rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This gravitational construction is modelled after the analogous one for five-dimensional black holes dual to N = 4 SYM introduced in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Uplift to eleven dimensions In order to appeal to the AdS/CFT dictionary and compare the gravitational results to the field theory computation, we need to embed the four-dimensional gravitational solu- tion (Y4, G(Y4), A) in eleven-dimensional supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In particular, in order to match the field theory limit taken in Section 3, we shall uplift the four-dimensional minimal gauged supergravity on S7 to a solution of eleven-dimensional supergravity (Y11, G(Y11), C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The eleven-dimensional metric and gauge field can be locally written as a fibration [83] G(Y11) = G(Y4) + 4 �� d �ψ + σ + 1 2A �2 + G(CP3) � , dC = 3 vol(Y4) − 4 ∗4 F ∧ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) Here we have used the local form of the metric on a seven-dimensional Sasaki–Einstein space (such as S7): ∂ �ψ is the Reeb vector, J is the K¨ahler form on the K¨ahler–Einstein base CP3, and is such that dσ = 2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Moreover, G(CP3) is the Fubini–Study metric normalized with Ric(G(CP3)) = 6 G(CP3), and the volume of S7 is Vol(S7) = π4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The adapted coordinate �ψ is periodic with period 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We see that dC has an flux through the internal space quantized as N = − 1 (2πℓP )6 � S7 ∗11dC = 128π4 (2πℓP )6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) Combining the above equation with the reduction of the Ricci scalar on the internal space, we find the canonical identification for dual to ABJM theory: 1 G4 = 2 √ 2 3 N 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) Now we focus on the Wick rotation of the supersymmetric Kerr–Newman-AdS black hole with chemical potentials (β, Ω, Φe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' First, we observe that the non-vanishing holonomy of the gauge field at the boundary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10), together with the presence of the gauge field in the fibration term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1), imply that Y11 does not have the same asymptotics as the direct product AdS4 × S7: G(Y11) ∼ dz2 z2 + 1 z2 � dt2 E + dθ2 + sin2 θ (d�φ − iΩ dtE)2� + 4 �� d �ψ + σ + i 2Φe dtE �2 + G(CP3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) – 37 – Regularity of the solution requires (tE, �φ, �ψ) ∼ (tE + β, �φ, �ψ) ∼ (tE, �φ + 2π, �ψ) ∼ (tE, �φ, �ψ + 2π) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) but at the cost of having explicit fibration terms in the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Alternatively, we can shift the realization of the chemical potentials to the twisting of the coordinates by defining ψ = �ψ + iΦe tE/2, so that the metric is explicitly asymptotically locally EAdS4 × S7, but the regularity of the solution then requires (tE, φ, ψ) ∼ (tE + β, φ − iΩβ, ψ + iΦeβ/2) ∼ (tE, φ + 2π, ψ) ∼ (tE, φ, ψ + 2π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) It is clear that the supersymmetric structure at the conformal boundary of the black hole solution is the same as the one where the field theory is formulated on in Section 2, with the same fibration parameter Ω and with Φe = Φ(H1) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Furthermore, the identifications in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) can be combined to show that the same relations are satisfied by a solution with the potentials β′ = β , Ω′ = Ω + 2πi β n′ Ω , Φ′ e = Φe + 2πi β 2ne .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) More precisely, though, if n′ Ω is odd, the periodicity of the spinors around S1 changes, whereas this doesn’t happen if n′ Ω = 2nΩ (see the discussion of the Killing spinor in the previous section and notice that n′ Ω odd would change the parity of n0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is clear that these shifts correspond to the shifts of the chemical potentials in the partition function of the dual field theory (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Observe that all the solutions (β′, Ω′, Φ′ e) have the same boundary conditions as the starting solution (β, Ω, Φe), and so they must be summed over in the functional integral, with the reduced chemical potentials τ ′ g = τg + 2nΩ , ϕ′ g = ϕg + 2ne .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) We expand on this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' From the eleven-dimensional perspective, all the shifts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) are obtained by combining conditions from regularity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' From the effective viewpoint on Y4, instead, the regularity of the solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) only leads to the shift of Ω in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The shift of Φe follows from imposing the boundary condition for the bulk Abelian gauge field by fixing its holonomy around the Euclidean circle, namely exp � i 2 � S1 β A � = exp � −β 2 Φe � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) The factor of 1 2 is due to the fact that operators generically have half-integer R-charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17 17Recall that from the boundary viewpoint, A couples to the current corresponding to the symmetry gener- ated by H1, which is integrated over the S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 38 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 AdS/CFT comparison and orbifold solutions In order to perform the match warranted by the AdS/CFT correspondence, we begin by matching the chemical potentials on both sides of the correspondence using the boundary conditions, given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In these equations, we take n0 = ∓1 on the positive/negative branch of solutions, which implies τg ↔ τ + n1 ∓ 1 , 2ϕg ↔ τ + n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) The gravitational action (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) is singular as τg → 0 (and ϕg is finite there because of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The same singularity is reproduced by the field theory saddle with (c, d) = (1, 0) if we choose n1 = ±1 (for the positive or negative branch of solutions, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' With this choice, the on-shell action of a complex supersymmetric solution on the positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' negative) branch matches the large-N behavior of the unrefined index I(τ, λ) as τ → 0 and λ = 1/4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ = −1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, the gravitational action (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) expressed in terms of the field theory parameters reads I|SUSY(τg = τ) = ± π 3 √ 2N 3 2 (τ ± 1)2 τ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) whose singular part clearly matches (the negative of) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) when ℓc = 1 and ℓd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Starting from the eleven-dimensional geometry uplifting a solution with chemical poten- tials (�β, �Ω, �Φe), we can also define the following identification of the coordinates (tE, φ, ψ) ∼ � tE + �β cg , φ − i�Ω �β cg − 2πr cg , ψ + i 2 �Φe �β cg − 2πs cg � ∼ (tE, φ + 2π, ψ) ∼ (tE, φ, ψ + 2π) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) or equivalently � tE, �φ, �ψ � ∼ � tE + �β cg , �φ − 2πr cg , �ψ − 2πs cg � ∼ � tE, �φ + 2π, �ψ � ∼ � tE, �φ, �ψ + 2π � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) where r, s, cg are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is clear from the construction that r and s are only defined modulo cg, so that the metric is a Zcg quotient of the original solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18 It is also clear from the above identifications that this construction is different from the standard Zk quotient of the internal sphere, which acts only on the Hopf fiber as �ψ ∼ �ψ + 2π/k, and changes the dual field theory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since the construction of the solutions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) crucially involves the Euclidean circle, their Lorentzian interpretation is not transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Instead they represent the gravitational duals to the Euclidean saddle-points of the field theory [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 18In fact, as we shall see, we need to allow r = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 2cg − 1, and s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , cg − 1 in order to preserve the periodicity conditions for the fermions along the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A transformation by integer shifts (r, s) has order cg in Zcg if and only if gcd(r, cg) = gcd(s, cg) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 39 – In order to interpret these solutions, we first notice that the identifications (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) can be neatly expressed in terms of the shifted potentials in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) (tE, φ, ψ) ∼ � tE + �β′ cg , φ − i�Ω′ �β′ cg , ψ + i 2 �Φ′ e �β′ cg � ∼ (tE, φ + 2π, ψ) ∼ (tE, φ, ψ + 2π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) Comparing with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6), we see that these are the identifications required of a solution with potentials (�β′/cg, �Ω′, �Φ′ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In the saddle-point approximation to the gravity path integral, we should sum over solutions with shifted chemical potentials, provided they have the same boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Although here �β ̸= �β′/cg, the supersymmetric index is independent of the size of the thermal circle, and the boundary values of the holonomies of the gauge field and the angular velocity (see discussion around (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9)), which control the supersymmetric index, are indeed the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Thus, the Zcg quotient of a supersymmetric solution labelled by (�β, �Ω, �Φe), or more appropriately for the supersymmetric locus (�τg, �ϕg), contributes to the gravitational path integral with β = �β cg , τg = �τg cg − r cg , ϕg = �ϕg cg + 2s cg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) In order to ensure that the orbifold (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) preserves supersymmetry, we need to check that the Killing spinor is globally defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=', that it is anti-periodic around S1 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This requires that the chemical potentials of a gravitational solution satisfy the constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='41), which applied to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15), and combined with the assumption that �τg − 2�ϕg = ∓1 leads to 4s = −r ∓ 1 − cgn0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) for an appropriate odd n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For every cg, there are cg combinations of (r, s, n0) solving the equation, provided r = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 2cg −1, and s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , cg −1, and r and cg have opposite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This is an extension of the earlier conclusions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8), which impose that r should be even if cg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' By construction, the on-shell action of the solution obtained via a Zcg quotient of the supersymmetric solution with (�τg, �ϕg) is 1/cg the on-shell action of the original solution, that is I|SUSY (τg, ϕg) = 1 cg I|SUSY(�τg, �ϕg) = ± π G4 (cgϕg − 2s)2 cg(cgτg + r) = ± π 4G4 (cgτg + r ± 1)2 cg(cgτg + r) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) This represents the contribution of the orbifold to the gravitational sum dual to a grand canonical partition function with parameters (τg, ϕg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to match with field theory, we explain in detail the simplest case with c odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The function I|SUSY is singular as τg → −r/cg, and now the identification of the chemical potentials relates τg with τ +n0+n1 (where n0 is the – 40 – odd number such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16), the corresponding singularity in field theory would appear as τ → (4s ± 1 − cgn1)/cg (where we remark again that the sign refers to the branch of supersymmetric black holes that the orbifold is a quotient of).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, recall that in order for the index to have a large-N behavior O(N 3 2 ), we need cn1 = ±1 mod 4, and we already established around (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) that for c = 1 the two signs are related to the choice of branches for the dual gravity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Consistently, we establish the dictionary cg ↔ c and s ↔ d, and we find that the singular behavior of the on-shell action of the Zc quotient (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) of a solution in the positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' negative) branch matches the large-N limit of unrefined index I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) as τ → 4d c and cn1 = ±1 mod 4 I|SUSY(cτg = cτ + cn0 ± 1) = ± π 3 √ 2N 3 2 (cτ − 4d ± 1)2 c(cτ − 4d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) For comparison, consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46), recalling that ℓc = c and ℓd = 4d if c is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The only fixed points of the identifications (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) are at the horizon, that is r > r+, where the circle generated by V in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The remaining space in the eleven-dimensional solution is the product of the S2 transverse in the black hole, and the internal S7 of which �ψ is the Hopf coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is not possible for both r and s to be both vanishing while preserving supersymmetry: the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) would not be satisfied for cg > 1, and indeed the transverse space to the fixed point would be C/Zcg, which does not supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' If r = 0, then the quotient acts only on the Hopf fiber of S7 and the disc transverse to S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Since the Hopf fiber never shrinks, there are no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, if s = 0, then we have a quotient of the black hole only, and we have fixed point sets isomorphic to the round S7 at the two poles of the S2 (where θ = 0, π and the circle generated by ∂�φ shrinks to zero size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The transverse space is C2/Zcg, which preserves supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We conclude the discussion of the minimal gauged supergravity solutions with two com- ments on further refinements of the gravitational path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' First, it is straightforward to compute corrections to the on-shell action subleading in N using the four-derivative cor- rections to the minimal gauged supergravity action proposed in [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These corrections do not modify the Killing spinor equations, and the equations of motion derived from the corrected action are a consequence of the two-derivative equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19 This implies that the supersymmetric solutions to the higher-derivative theory are just those of the two- derivative theory, so the on-shell action with the higher-derivative corrections follows again from a localization principle [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For the supersymmetric family of complex solutions, we have IHD = ± � π G4 ϕ2 g τg + 64π2 τg � −α1(1 ∓ ϕg)2 + α2ϕ2 g � � = ± �� π 2G4 + 32π2α2 − 32π2α1 � (τg ± 1)2 2τg ± 64π2α1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) 19This circumstance can be explained appealing to field redefinitions [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 41 – Here α1,2 are constants introduced to parametrize the higher derivative corrections, respec- tively a supersymmetrization of the Weyl squared action and the Gauss–Bonnet term, which would be determined by the uplifting of the higher derivative action in eleven dimensions, or comparing with the localized partition function, obtaining [85] π 2G4 + 32π2α2 = 2 3πN 3 2 − 3 8 √ 2πN 1 2 , 32π2α1 = − π √ 2N 1 2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) and finally [48, 85] IHD = ± √ 2 3 π �� N 3 2 + 15 16N 1 2 � (τg ± 1)2 2τg ∓ 1 3N 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) According to the prescriptions described around (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17), the higher derivative contribution of a Zcg quotient of a supersymmetric solution (�τg, �ϕg) would be 1 cg IHD(�τg, �ϕg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The second comment concerns the saddle point approximation to the gravitational path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The AdS/CFT correspondence instructs us to compare the grand-canonical field theory partition function in the limit of large N and fixed k with a sum over gravitational saddles ZGCE(Ω, Φ) ∼ � nΩ,na∈Z e−I|SUSY � β, Ω+ 2πi β 2nΩ+ 2πi β � a na, Φa+ 2πi β na � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) where the quantity in the exponent in the right-hand side is the on-shell action of the relevant supergravity solution with the appropriate boundary conditions fixed by the radius of the circle β, the boundary metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12), and the holonomy of the gauge fields at the boundary fixed by the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) to be exp � i 2 � S1 β Aa � = exp � −β 2 Φa � , a = 1, 2, 3, 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) As we have seen, supersymmetry imposes the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) between the chemical potentials, and in fact it is appropriate to use the five reduced chemical potentials defined in analogy with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) τg ≡ β Ω − 1 2πi , ϕa ≡ β Φa − 1 2 2πi , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) which are constrained by τg − 4 � a=1 ϕa = n0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) In terms of these the field theory partition function reads Z = TrHS2 e−β{Q,Q†}+2πiτgJ+2πi � a ϕaRa = TrHS2(−1)2Je−β{Q,Q†}+2πi � a ϕa(J+Ra) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) and so ZGCE(ϕ) ∼ � na∈Z e−I|SUSY(ϕa+na) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) – 42 – highlighting the fact that both functions only depend on four rather than five fugacities, and that neither the field theory not the gravity depends on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' An important point stressed in [16, 88] is that in the grand canonical ensemble it is not allowed to restrict to the case of equal Φa = Φ(H1)/2: the sum over gravitational solutions on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) will take us away from this locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, one can either go to a mixed ensemble, as in [88], or consider solutions in bulk gauged supergravity with multiple U(1) gauge vectors, as in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We now move to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 6 Black holes in non-minimal gauged supergravity The black hole solutions considered in the previous section are charged under a unique Abelian gauge field, that is, they are solutions of minimal gauged supergravity in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Thus, they can only reproduce the behavior of the ABJM index when the fugacities for the U(1)4 Cartan of the R-symmetry are set equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, there are also known rotating black holes with multiple electric charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1 Supersymmetric black holes in the X0X1 model There is a rotating black hole solution with two different electric charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We shall be quite brief in reviewing its properties, and the reader can find similar discussions in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The bosonic part of the relevant Lorentzian action is S = 1 16πG4 � Y4 � R volG + 1 2 dX2 1 ∧ ∗dX2 2 − 1 2d(ϕX2 1) ∧ ∗d(ϕX2 1) + (4 + X2 1 + X2 2) volG − X−2 1 (F1 ∧ ∗F1 + ϕX2 1F1 ∧ F1) − X−2 2 � F2 ∧ ∗F2 − ϕX2 1F2 ∧ F2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) Here G is the metric, F1 and F2 are the curvature of two Abelian gauge fields, X1 and ϕ are a scalar and a pseudo-scalar, respectively, and we have defined X2 2 = X−2 1 + ϕ2X2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) There is a Z2 automorphism exchanging the two Abelian gauge factors F1 ↔ F2 X1 ↔ X2 ϕX2 1 ↔ −ϕX2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) The action reduces to minimal gauged supergravity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) upon setting X1 = X2 = 1 and F1 = F2 = F (which explains the non-canonical normalization of the kinetic terms for the gauge fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The solution we are interested in, which again we present in the frame non-rotating at infinity, is [89, 90] ds2 = −∆Θ∆r BΞ2 dt2 + sin2 Θ B � dφ + a∆Θ ∆r − (1 + r1r2)(a2 + r1r2) BWΞ2 dt �2 + W �dr2 ∆r + dΘ2 ∆Θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) – 43 – Here ri = r + 2 m s2 i , ∆r = r2 + a2 − 2 m r + r1 r2 � r1 r2 + a2� , ∆Θ = 1 − a2 cos2 Θ , W = r1 r2 + a2 cos2 Θ , Ξ = 1 − a2 , B = ∆Θ(r1r2 + a2)2 − a2 sin2 Θ ∆r WΞ2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) and si = sinh δi, ci = cosh δi, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The scalar fields are X2 1 = 1 + r1 (r1 − r2) W , ϕ = a (r2 − r1) cos Θ r2 1 + a2 cos2 Θ , X2 2 = 1 + r2 (r2 − r1) W (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) and the gauge fields A1 = 2ms2c2r1 WΞ � ∆Θ dt − a sin2 Θ dφ � + γ1 dt , A2 = 2ms1c1r2 WΞ � ∆Θ dt − a sin2 Θ dφ � + γ2 dt , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) where γ1,2 are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' There is an outer horizon at the largest positive root of ∆r, and the Wick-rotated solution (t = −itE) is smooth if we identify (tE, φ) ∼ (tE, φ + 2π) ∼ (tE + β, φ − iΩβ) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) where the temperature and angular velocity of the horizon are β = 4πa2 + r1+r2+ ∆′r(r+) , Ω = a 1 + r1+r2+ a2 + r1+r2+ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) and we defined ri+ ≡ r+ + 2ms2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The horizon is a Killing horizon for V = ∂t + Ω∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The entropy is computed from the area of the horizon S = π G4 r1+r2+ + a2 Ξ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) The electrostatic potentials are Φe,1 = 2ms2c2 r1+ a2 + r1+r2+ , Φe,2 = 2ms1c1 r2+ a2 + r1+r2+ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) and we choose the gauge γ1 = −Φe,1 and γ2 = −Φe,2 in order to have smooth gauge fields at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Notice that the Wick-rotated metric is real provided a is pure imaginary and δ1,2, m are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' If these conditions are satisfied, the Euclidean metric has the topology of the product of a disc and a 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The black hole is asymptotically AdS, and the boundary metric has the same form as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) and gauge fields A1 ∼ −Φe,1 dt, A2 ∼ −Φe,2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The standard procedure of holographic – 44 – renormalization then gives us the holographic conserved charges, as described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The difference with Einstein–Maxwell theory is the form of the counterterms, which in presence of scalars are subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In light of the fact that we shall be intersted in supersymmetric solutions, we fix the counterterms to be I = lim δ→0 � S + 1 8πG4 � ∂Yδ � K − 1 2R − � 2 + X2 1 + X2 2 � volh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) The holographic stress-tensor and the electric current defined as ⟨Tij⟩ = − 2 √−g δI δgij , ⟨ji 1⟩ = 1 √−g δI δA1i , ⟨ji 2⟩ = 1 √−g δI δA2i , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) satisfy the conservation equation ∇i⟨Tij⟩ = F1ji⟨ji 1⟩ + F2ji⟨ji 2⟩ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) and we can define a conserved charge associated to any boundary vector K generating a symmetry Q[K] = � C∩M3 ui � ⟨T i j⟩ + ⟨ji 1⟩A1j + ⟨ji 2⟩A2j � Kj volC∩M3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) In particular, we have expressions for the angular momentum associated to −∂φ J = am1 + s2 1 + s2 2 G4Ξ2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) the electric charges Qe,1 = � C∩M3 ui⟨ji 1⟩ volC∩M3 = − 1 8πG4 � C∩M3 ∗F1 = ms2c2 G4Ξ , Qe,2 = � C∩M3 ui⟨ji 2⟩ volC∩M3 = − 1 8πG4 � C∩M3 ∗F2 = ms1c1 G4Ξ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) and finally the energy associated to ∂t (as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20), we denote by E the energy in the gauge where the boundary gauge fields vanish) E′ = m1 + s2 1 + s2 2 G4Ξ2 − Φe,1 Qe,1 − Φe,2 Qe,2 ≡ E − Φe,1 Qe,1 − Φe,2 Qe,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) These quantities, together with the Euclidean on-shell action, satisty the quantum statistical relation I = −S + β(Q[V ] − Φe,1 Qe,1 − Φe,2 Qe,2) = −S + β(E − ΩJ − Φe,1 Qe,1 − Φe,2 Qe,2) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) which reduces to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) upon setting equal the two gauge fields (notice that the charges are defined to be half of the electric charge in minimal gauged supergravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The same quantities also satisfy a first law of thermodynamics dE = β−1dS + Ω dJ + Φe,1 dQe,1 + Φe,2 dQe,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) – 45 – The Lagrangian (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) is the bosonic part of an N = 2 gauged supergravity model with prepotential F = −iX0X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The condition for having a supersymmetric solution is [90, 91] E = J + Qe,1 + Qe,2 ⇔ a = coth(δ1 + δ2) − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) It is easy to check, studying ∆r|SUSY, that in Lorentzian signature, supersymmetry and regularity imply extremality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2, in order to reproduce the behavior of the index we shall need to impose supersymmetry of the Wick-rotated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This necessarily leads us to consider a family of complex supersymmetric metrics obtained by deforming the Euclidean metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We observe that imposing ∆r|SUSY = 0 leads to a quadratic equation for m in terms of r+, δ1, δ2, and thus to two branches of solutions, labelled by the choice of ± in the solution of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The BPS sublocus is obtained by imposing extremality in addition to supersymmetry [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The chemical potentials of the complex supersymmetric solutions satisfy β(1 − Φe,1 − Φe,2 + Ω) = ∓2πi , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) and if we define the “reduced chemical potential” by taking the difference with the value of the BPS solutions, we find τg ≡ β Ω − 1 2πi , ϕg1 ≡ β Φe,1 − 1 2πi , ϕg2 ≡ β Φe,2 − 1 2πi (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) satisfying τg − ϕg1 − ϕg2 = ∓1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) The reduced chemical potentials for supersymmetric solutions are complex, but they remain finite as we approach the BPS locus, whereas the chemical potentials and the conserved charges of the BPS solutions become real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The on-shell action of the supersymmetric solutions, computed using holographic renor- malization, can be expressed using the reduced chemical potentials as20 I|SUSY = ± π G4 ϕg1ϕg2 τg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) Since this expression is independent of β, it remains finite on the BPS locus, which allows us to define the on-shell action of the supersymmetric extremal black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The relevance of this object is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Firstly, interpreted as a Gibbs free energy in the grand canonical ensemble, it allows us to obtain the entropy in the microcanonical ensemble via a Legendre transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Secondly, it can be related to the dual field theory partition function, and its Cardy-like limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 20The need to perform holographic renormalization while preserving supersymmetry explains the form of the finite counterterms for the scalars in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For more details in the STU model or the Spin(4) supergravity, of which (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) is a truncation, see [92–95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 46 – To the first purpose, we point out again that for the complex supersymmetric solutions the quantum statistical relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) can be written as I|SUSY = −S − 2πiτg J − 2πiϕg1 Qe,1 − 2πiϕg2 Qe,2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) and from this expression follows a Euclidean quantum gravity derivation of the extremization procedure proposed in [82] that is analogous to that described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) follows that the function to be extremized is f(τg, ϕg1, ϕg2) = −I|SUSY(τg, ϕg1, ϕg2) − 2πiτg J − 2πiϕg1 Qe,1 − 2πiϕg2 Qe,2 + Λ(τg − ϕg1 − ϕg2 ± 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) Combining the equation satisfied by f at the critical point and Euler’s theorem for the homogeneous function I|SUSY(τg, ϕg1, ϕg2), we obtain the value of the Legendre transform �f(J∗, Qe,1∗, Qe,2∗) = ±Λ∗(J∗, Qe,1∗, Qe,2∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) Concretely, the equation to be solved to find Λ∗(J∗, Qe,1∗, Qe,2∗) is 0 = Λ2 ∗ + Λ∗ � 2πi(Qe,1∗ + Qe,2∗) ± π G4 � + � −4π2Qe,1∗Qe,2∗ ∓ 2π2i G4 J∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29) We then impose the constraints that J∗, Qe,1∗, Qe,2∗ ∈ R , Λ∗(J∗, Qe,1∗, Qe,2∗) ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='30) Splitting real and imaginary part of the equation above leads us to the value of the Legendre transform �f(J, Qe,1, Qe,2) = πJ G4(Qe,1 + Qe,2) = π 2G4 �� 1 + 16G2 4Q2 e,1Q2 e,2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) These values correspond, respectively, to the entropy of the extremal black hole in the U(1)2 theory and to the non-linear constraint between its charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 Uplift to eleven dimensions and AdS/CFT The theory (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) can be obtained from a consistent truncation of eleven-dimensional super- gravity on S7 as described in [96] (for the bosonic sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Geometrically, we write the metric on S7 as a S3 × S3 fibered over the internal, and introduce a gauge field only along the Hopf fiber inside each S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Concretely, we write the eleven-dimensional solution as G(Y11) = (Z1Z2) 1 3 G(Y4) + 4(Z1Z2) 1 3 � dΞ2 + cos2 Ξ 4Z1 � dΘ2 1 + sin2 Θ1 dΦ1 + (d�Ψ1 + cos Θ1 dΦ1 + A1)2� + sin2 Ξ 4Z2 � dΘ22 + sin2 Θ2 dΦ2 + (d�Ψ2 + cos Θ2 dΦ2 + A2)2� � , dC = � 2 + cos2 Ξ X2 1 + sin2 Ξ X2 2 � vol(Y4) + 2 cos Ξ sin Ξ � 2 X1 ∗4 dX1 − ϕX4 1 ∗4 dϕ � ∧ dΞ + d �A3 + �F ′ 4 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='32) – 47 – where Z1 = X2 1 cos2 Ξ + sin2 Ξ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Z2 = cos2 Ξ + X2 2 sin2 Ξ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' �A3 = −8ϕX2 1 �cos4 Ξ Z1 Ω1(A1) − sin4 Ξ Z2 Ω2(A2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' �F ′ 4 = 2 cos Ξ X2 1 � sin Ξ dΞ ∧ � d�Ψ1 + cos Θ1 dΦ1 + A1 � + cos Ξ 2 sin Θ1 dΘ1 ∧ dΦ1 � ∧ (∗4F1 + ϕX2 1F1) − 2 sin Ξ X22 � cos Ξ dΞ ∧ � d�Ψ2 + cos Θ2 dΦ2 + A2 � − sin Ξ 2 sin Θ2 dΘ2 ∧ dΦ2 � ∧ (∗4F2 − ϕX2 1F2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Ω1(A1) = 1 8 sin Θ1(d�Ψ1 + cos Θ1 dΦ1 + A1) ∧ dΘ1 ∧ dΦ1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Ω2(A2) = 1 8 sin Θ2(d�Ψ2 + cos Θ2 dΦ2 + A2) ∧ dΘ2 ∧ dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='33) It is straightforward to see that this uplift reduces to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) upon setting X1 = X2 = 1 and A1 = A2 = A, and changing coordinates to �ψ = 1 4(�Ψ1 + �Ψ2) , Λ = 1 2(�Ψ1 − �Ψ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) The explicit expression for the CP3 quantities are σ = 1 2 � cos 2Ξ dΛ + cos2 Ξ cos Θ1 dΦ1 + sin2 Ξ cos Θ2 dΦ2 � , G(CP3) = dΞ2 + cos2 Ξ 4 � dΘ2 1 + sin2 Θ1 dΦ2 1 � + sin2 Ξ 4 � dΘ2 2 + sin2 Θ2 dΦ2 2 � + sin2 Ξ cos2 Ξ � dΛ + cos Θ1 2 dΦ1 − cos Θ2 2 dΦ2 � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='35) As in Section 5, we can study the regularity of the uplift in a neighbourhood of the conformal boundary, where the metric has the form G(Y11) ∼ (Z1Z2) 1 3 �dz2 z2 + 1 z2 � dt2 E + dΘ2 + sin2 Θ � d�φ − iΩ dtE �2�� + 4(Z1Z2) 1 3 � dΞ2 + cos2 Ξ 4Z1 � dΘ2 1 + sin2 Θ1 dΦ1 + (d�Ψ1 + cos Θ1 dΦ1 + iΦe,1 dtE)2� + sin2 Ξ 4Z2 � dΘ22 + sin2 Θ2 dΦ2 + (d�Ψ2 + cos Θ2 dΦ2 + iΦe,2 dtE)2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='36) – 48 – Regularity now requires (tE, �φ, �Ψ1, �Ψ2) ∼ (tE + β, �φ, �Ψ1, �Ψ2) ∼ (tE, �φ + 2π, �Ψ1, �Ψ2) ∼ (tE, �φ, �Ψ1 + 4π, �Ψ2) ∼ (tE, �φ, �Ψ1, �Ψ2 + 4π) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) while having explicit fibration terms in the metric, both in the S1 β×S2, and also in the internal space, due to the non-zero holonomies of the Abelian gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' On the other hand, we can twist the coordinates defining Ψ1 = �Ψ1 + iΦe,1tE (and analogously for Ψ2), thus absorbing the chemical potentials and finding the regularity conditions (tE, φ, Ψ1, Ψ2) ∼ (tE + β, φ − iΩβ, Ψ1 + iΦe,1β, Ψ2 + iΦe,2β) ∼ (tE, φ + 2π, Ψ1, Ψ2) ∼ (tE, φ, Ψ1 + 4π, Ψ2) ∼ (tE, φ, Ψ1, Ψ2 + 4π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38) Notice that we can combine the identifications above, showing that the following choice of chemical potentials satisfy the same conditions β′ = β , Ω′ = Ω + 2πi β n′ Ω , Φ′ e,1/2 = Φe,1/2 + 2πi β 2ne,1,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39) As in the minimal gauged case, we should take n′ Ω = 2nΩ in order to not change the periodicity of the spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The identification for the electric potential follows from the boundary condition for the Abelian gauge fields, which is imposed by fixing the holonomy exp � i 2 � S1 β A1/2 � = exp � −β 2 Φe,1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) Again, the factor of 1 2 is due to the fact that the operators have half-integer charges under the relevant symmetry: compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Moreover, the constraint (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) corresponds to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To compare with field theory, we should first match (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) with the definitions (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) τg ↔ τ + n1 ∓ 1 , ϕgA ↔ τ 2 + 2σA , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='41) which of course satisfy (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Consistently with the truncation to minimal gauged super- gravity, we should set n1 = ±1 to discuss the on-shell action of the positive/negative branch of solutions, in which case the gravity action (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) in field theory variables reads I|SUSY(τg = τ) = ± π 3 √ 2N 3 2 (τ + 4σ1)(τ + 4σ2) τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='42) The singular behavior as τ → 0 matches (the negative of) the index with pairwise equal chemical potentials log I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' σ) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='52) on the saddle (c, d) = (1, 0) (since the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='51) becomes n1 = ±1 mod 4 if c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 49 – As in the case of the uplifted black holes with one electric charge, we can define a Zcg quotient of the eleven-dimensional supergravity solution analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We define it by starting with an eleven-dimensional solution (�β, �Ω, �Φe,1, �Φe,2), and imposing the identification (tE, φ, Ψ1, Ψ2) ∼ � tE + �β cg , φ − i�Ω �β cg − 2πr cg , Ψ1 + i�Φe,1 �β cg − 4πs cg , Ψ2 + i�Φe,2 �β cg − 4πt cg � ∼ (tE, φ + 2π, Ψ1, Ψ2) ∼ (tE, φ, Ψ1 + 4π, Ψ2) ∼ (tE, φ, Ψ1, Ψ2 + 4π) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='43) where r, s, t are integers defined modulo cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21 Starting from a solution with chemical potentials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' boundary conditions) (�β, �Ω, �Φe,1, �Φe,2), we can rewrite its orbifold solution above in terms of the primed potentials (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39) using (tE, φ, Ψ1, Ψ2) ∼ � tE + �β′ cg , φ − i�Ω′ �β′ cg , Ψ1 + i�Φ′ e,1 �β′ cg , Ψ2 + i�Φ′ e,2 �β′ cg � ∼ (tE, φ + 2π, Ψ1, Ψ2) ∼ (tE, φ, Ψ1 + 4π, Ψ2) ∼ (tE, φ, Ψ1, Ψ2 + 4π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='44) Therefore, we conclude that the Zcg quotient of the solution (�β, �τ, �ϕ1, �ϕ2) contributes with β = �β cg , τg = �τg cg − r cg ϕg1 = �ϕg1 cg + 2s cg , ϕg2 = �ϕg2 cg + 2t cg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='45) Thus I|SUSY(τg, ϕg1, ϕg2) = 1 cg I|SUSY(�τg, �ϕg1, �ϕg2) = ± π G4 (cgϕg1 − 2s)(cgϕg2 − 2t) cg(cgτg + r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) Requiring that the supersymmetry of the original solution is preserved by the Zcg orbifold imposes that τg − ϕg1 − ϕg2 = n0 ⇔ 2s + 2t = −r ∓ 1 − cgn0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='47) As in the case of minimal supergravity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16), this constraint can only be solved provided r and cg have opposite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' It is easy to convince ourselves that there is a fixed subset preserving supersymmetry only if we choose s = t = 0, in which case the quotient only acts on the black hole, and the fixed point sets develop on the horizon at the two poles of the transverse S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We discuss in some detail the match with the field theory result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='52) for c odd, the other cases follow in a slightly more involved way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The boundary conditions identify ϕgA again with 21As mentioned in footnote 18, r = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 2cg − 1, whereas s, t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , cg − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 50 – τ/2+2σA, as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='41), but now τg is τ +n0 +n1 where n0 is the odd number such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='47) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The on-shell action (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46) is singular as τg → −r/cg, or τ → (2s + 2t ± 1 − cgn1)/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Consistently with the case of c = 1 and the minimal supergravity, we relate c ↔ cg, and the quotient of a solution on the positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' negative) branch with the field theory condition cn1 = 1 mod 4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' cn1 = −1 mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' With this choices, the singular behavior of the on-shell action (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46), expressed in field theory variables, is (in the minimal case) I|SUSY = ±8 √ 2π 3 N 3 2 1 c(cτ − 2(s + t)) � cσ1 + 2t − 2s 4 � � cσ2 + 2s − 2t 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='48) Recall that s and t are defined modulo c, so once we account for the necessity of shifting σA ∈ [0, 1) by integers, this matches (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A more detailed match requires considerations on the shifted chemical potentials as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, as pointed out at the end of Section 5, summing over gravity solutions dual to the grand canonical field theory partition function necessarily involves summing over solutions with the gauge-invariant same boundary conditions but shifted chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, this leads to seemingly paradoxical results, as stressed in [16] for the four-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For instance, consider a complex supersymmetric solution and shift the reduced chemical potentials while insisting that we remain on the same branch τ ′ g = τg + 2nΩ , ϕ′ g1 = ϕg1 + 2ne,1 , ϕ′ g2 = ϕg2 + 2nΩ − 2ne,1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='49) The action of the shifted solution is I|SUSY = ±2 √ 2π 3 N 3 2 (ϕg1 + 2ne,1)(ϕg2 + 2nΩ − 2ne,1) τg + 2nΩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='50) Setting nΩ = 0 gives a value that diverges in the shift Re (I|SUSY) = ∓n2 e,1 8 √ 2π 3 N 3 2 Re 1 τg + O(ne1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='51) Therefore, the contribution e−I|SUSY of the first/second branch would diverge for Re(τg) posi- tive/negative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As suggested in [16], though, it is possible to limit the summands on the right-hand side of the AdS/CFT equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) by considering the contribution of branes wrapping cycles in the eleven-dimensional geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We will report on this in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22 Finally, we mention that there are known BPS rotating black hole solutions to the U(1)4 STU model with four different electric charges [97], which are dual to the fully refined field theory index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Differently from the cases considered until now, only the extremal version of these solutions is known, since by construction it is imposed that the near horizon geometry has an infinite throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, it is not possible to perform the previous analysis and define the family of complex supersymmetric solutions away from extremality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 22An analogous problem has been considered in a different setting in [36], and solved by finding the presence of zero-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 51 – Acknowledgments We are grateful to Arash Arabi Ardehali, Davide Cassani, Zohar Komargodski, Dario Martelli, Luigi Tizzano, Chiara Toldo, and Alberto Zaffaroni for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We would also like to thank the organizers and participants of the SCGP workshop “Supersymmetric Black Holes, Holography and Microstate Counting” for many interesting comments and discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This work is supported by the ERC Consolidator Grant N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 681908, “Quantum black holes: A macroscopic window into the microstructure of gravity”, and by the STFC grant ST/P000258/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' ACB acknowledges financial support from the INFN grant GSS (Gauge The- ories, Strings and Supergravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' PBG gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, at which some of the research for this paper was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A Special functions and asymptotic limit formulas The Pochhammer symbol is an entire function of z ∈ C, for q ∈ C, |q| < 1, defined as (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ := ∞ � n=0 (1 − zqn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) Its asymptotic expansion for when q approaches a root of unity is given as follows [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Let w ∈ C with |w| < 1, q = ξme−ε/m where ξm is a primitive m-th root of unity and ε > 0, and ν is a complex number such that νε = o(1) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Then log � q w e−νε/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ = − 1 mεLi2(wm) − � ν m − 1 2 � log(1 − wm) − εν2 2m wm 1 − wm − 1 m log Dξm(wm) − log(1 − w) + ψw,ξm(ν, ε) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) where Dξm(x) ≡ m−1 � t=1 � 1 − ξt mx �t , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) and ψw,ξm(ν, ε) has an asymptotic expansion as ε → 0 ψw,ξm(ν, ε) ∼ − � r≥2 m � t=1 � Br � 1 − t + ν m � − δr,2 ν2 m2 � Li2−r(ξt mw) εr−1 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) The Bernoulli polynomials Br are defined by the generating function t etx et − 1 = ∞ � r=0 Br(x) tr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) – 52 – with the first few polynomials given by B0 = 1 , B1 = 1 2 − x , B2 = 1 6 − x + x2 , B3 = 1 2x − 3 2x2 + x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) They satisfy the relation B′ r(x) = rBr−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The periodic Bernoulli polynomials Br(z) are defined, for z ∈ C, through their Fourier series expansion, − (2πi)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Br(z) = � k∈Z ′ e2πikz kr (z ∈ C , j ≥ 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) The prime in the above formula means that k = 0 has to be omitted, and that in the j = 1 case—where the series is not absolutely convergent—the sum is in the sense of Cauchy principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For x ∈ R we have that Br(x) = Br ({x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The polylogarithm functions Lin, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' are defined as Lin(z) = ∞ � k=1 zk kn , |z| < 1 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) and is extended to C \\ [1, ∞) by analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' They satisfy the relation Lin(e2πix) + (−1)n Lin(e−2πix) = −(2πi)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Bn(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) B Factorization of the integrand in the matrix integral In this appendix, we review the factorization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) of the integrand of the ABJM index (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We follow the treatment of [60], highlighting the differences due to the limit τ → 4d/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Recall that we have si = ui + imi ε 4πc , si = −ui + imi ε 4πc , �si = �ui + i�mi ε 4πc , �si = −�ui + i�mi ε 4πc , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) and the corresponding exponentiated variables zi ≡ e2πisi and analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The classical action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) in terms of the variables (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) is Zclass = � i exp � 2πi4πc 4iε � s2 i − �s2 i � − 2πi4πc 4iε � s2 i − �s 2 i �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) Next we consider the contribution of the vector multiplets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4), first focusing on one of the U(N) gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We shall use the following relation � i̸=j (xix−1 j qa(ij);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' qb)∞ (x−1 i xjqb+a(ij);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' qb)∞ = � i̸=j �∞ n=0(1 − xix−1 j qa(ij)qbn) �∞ m=0(1 − x−1 i xjqa(ij)qbm)(1 − x−1 i xjqa(ij)) = � i̸=j (1 − xix−1 j qa(ij)) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) – 53 – valid for any b and for any coefficients a(ij) symmetric in i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Using this, we write the first line on the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) as � i̸=j (1 − xix−1 j q 1 2 |mij|) = � i̸=j (xix−1 j q 1 2 |mij|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ (x−1 i xjq1+ 1 2 |mij|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) It simplifies the following analysis to split the zero-point energy in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) among the vector and chiral contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Therefore, we write vm ≡ � i̸=j q− 1 4 |mij| (xix−1 j q 1 2 |mij|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ (x−1 i xjq1+ 1 2 |mij|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) From the identity (Xq m+1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ (X−1q m+1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ = (−X)−m (Xq −m+1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ (X−1q −m+1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ m ∈ Z , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) valid for any X ∈ C (the case X = 1 should be treated with care), taking X = x−1y−1q 1−R 2 it follows that [60, 98] � x−1y−1q 1−R 2 � 1 2 |m| (x−1y−1q 2−R+|m| 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ (xyq R+|m| 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ = (± sgn(m))m � x−1y−1q 1−R 2 �± 1 2 m (x−1y−1q 2−R±m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ (xyq R±m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q)∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) Using the identity (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) with y = 1, R = 0, the − sign for i > j and + sign for i < j, we can rewrite (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) as vm = � i>j ξ − 1 2 ij (zjz−1 i )− 1 2 � zjz−1 i ξij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � zjz−1 i ξijq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × ξ − 1 2 ij (zjz−1 i )− 1 2 � zjz−1 i ξij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � zjz−1 i ξijq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) where ξij ≡ e−2πi mij 2 4d c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) A similar formula applies to the contribution of the other U(N) gauge group (the second line on the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4)) with zi, zi replaced by �zi,�zi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Finally, we consider the contributions of the N = 2 chiral multiplets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' the product of the expressions for a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' , 4 given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We first focus on the contribution of the first line in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5), since the second one can be obtained simply by substituting xi → x−1 i , �xj → �x−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As in the contribution of the vector multiplet, we also introduce part of the zero-point energy, defining χm1 = � i,j q 1 8 |mi−� mj| � x−1 i �xj ζ−1 1 q 3 4 + 1 2 |mi−�mj| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � xi �x−1 j ζ1 q 1 4 + 1 2 |mi−�mj| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) – 54 – We split the product into i = j and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For the first piece with i = j we use (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) with −, and the result equals 4 � a=1 χma �� i=j = � i ξ ′ 1 2 ii (�ziz−1 i ) 1 2 � �ziz−1 i ξ′ iiζ−1 1 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ3q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ−1 2 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ4q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � i ξ ′ 1 2 ii (�ziz−1 i ) 1 2 � �ziz−1 i ξ′ iiζ−1 3 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ1q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ−1 4 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ2q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) where ξ′ ij ≡ exp � −2πi mi−�mj 2 4d c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We split the product i ̸= j into i > j and i < j, and again use the identity (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) with − for i > j and + for i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The result is 4 � a=1 χma �� i̸=j = � i>j (ξij �ξij) 1 2 �zj zi �zj �zi � 1 2 × � a=1,2 � �zjz−1 i ξ′ ijζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �z−1 i zjξ′−1 ji ζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=3,4 � �z−1 i zjξ′−1 ji ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zjz−1 i ξ′ ijζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × (ξij �ξij) 1 2 � zj zi �zj �zi � 1 2 × � a=1,2 � �z −1 i zjξ′−1 ji ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zjz−1 i ξ′ ijζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=3,4 � �zjz−1 i ξ′ ijζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �z −1 i zjξ′−1 ji ζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) Finally, upon putting together the various pieces (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12), we obtain that the integrand in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) can be written as the product of Zhol(z,�z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = N � i=1 exp � 2πi4πc 4iε � s2 i − �s2 i �� (�ziz−1 i ) 1 2 ξ ′ 1 2 ii × � �ziz−1 i ξ′ iiζ−1 1 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ3q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ−1 2 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ4q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × N � i,j=1 i>j � zjz−1 i ξij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � zjz−1 i ξijq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zj�z−1 i �ξij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zj�z−1 i �ξijq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=1,2 � �zjz−1 i ξ′ ijζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �z−1 i zjξ′−1 ji ζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=3,4 � �z−1 i zjξ′−1 ji ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zjz−1 i ξ′ ijζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) – 55 – and Zantihol(z,�z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = N � i=1 exp � −2πi4πc 4iε � s2 i − �s 2 i �� (�ziz−1 i ) 1 2 ξ ′ 1 2 ii × � �ziz−1 i ξ′ iiζ−1 3 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ1q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ−1 4 q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �ziz−1 i ξ′ iiζ2q 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × N � i,j=1 i>j � zjz−1 i ξij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � zjz−1 i ξijq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zj�z −1 i �ξij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zj�z −1 i �ξijq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=1,2 � �z −1 i zjξ′−1 ji ζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �zjz−1 i ξ′ ijζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ × � a=3,4 � �zjz−1 i ξ′ ijζ−1 a q 3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ � �z −1 i zjξ′−1 ji ζaq 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' q � ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) Notice that Zantihol(z,�z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) = Zhol(z,�z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ, λ) �� k→−k,zi→zi,�zi→�zi,ζ1↔ζ3,ζ2↔ζ4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) From these computations follow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' C Saddle-point analysis of the large-N index In this appendix we review the saddle-point solution to equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) in the general case of generic λa [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Due to the fact that the potential Wµ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) Wµ := W � ρ(x), v(x), �v(x) � + N 3 2 µ i �� x2 x1 dx ρ(x) − 1 � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) is piecewise polynomial, solutions the first three out of the four equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) δρWµ = δvWµ = δ�vWµ = 0 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) can be found using a rather simple linear ansatz for the density of eigenvalues ρ(x) = ρ0 + xρ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) We will assume k > 0, and [33] − 1 < δv − λ′ 1,2 < 0 , 0 < δv + λ′ 3,4 < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) where λ′ a := ℓcλa + ℓd 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The conditions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) comes from demanding δv not to cross branch points of the polylogarithms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' From the periodicity λ′ a ∼ λ′ a + 1 of the effective potential W we can assume 0 ≤ λ′ a < 1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This assumption and the constraint λ1 + λ2 + λ3 + λ4 ∈ Z forces λ′ 1 + λ′ 2 + λ′ 3 + λ′ 4 ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) More general cases can be obtained using the periodicity λ′ a ∼ λ′ a + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 56 – A first type of solutions to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) These are solutions that do not cross branch points of polylogarithms (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Thus, they are only valid in domains of x xleft < x < xright .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) The xleft and xright are fixed by the inequalities obtained after plugging the explicit depen- dence δv = δv(x) in the conditions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These solutions are the natural generalizations of the saddle point solution found in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2 for the unrefined index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' First, one solves δδvWµ = 0 for δv(x) in terms of ρ (we have reinstated k, which is set to 1 to obtain the results in the main text) δv(x) = − � −λ′2 1 + λ′ 1 − λ′2 2 + λ′2 3 + λ′2 4 + λ′ 2 − λ′ 3 − λ′ 4 � ρ + kx 2 (λ′ 1 + λ′ 2 + λ′ 3 + λ′ 4 − 2) ρ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) Plugging this solution and the ansatz ρ = ρ0 + xρ1 in the equation δρWµ = 0, and solving for ρ0 and ρ1 , one obtains, under the assumption λ′ 1 + λ′ 2 + λ′ 3 + λ′ 4 = 1 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) that ρ0 = µ/(4π2) (λ′ 1 + λ′ 3) (λ′ 1 + λ′ 3 − 1) (λ′ 2 + λ′ 3) (λ′ 2 + λ′ 3 − 1) , ρ1 = kλ′ 3 − k (λ′ 1 + λ′ 3) (λ′ 2 + λ′ 3) (λ′ 1 + λ′ 3) (λ′ 1 + λ′ 3 − 1) (λ′ 2 + λ′ 3) (λ′ 2 + λ′ 3 − 1) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) which matches the constant unrefined solution in the family (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39) when λ′ a are set equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' A second type of solutions to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) There is a second type of solutions to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These are solutions that localize, at large N, around the non-analyticities of the polylogarithms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For these type of solutions the term of order N1/2 in the effective potential W turns out to contribute non-trivially at order N3/2 to the saddle point equation δδvW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 23 Without loss of generality, these solutions take the form δv(x) = ϵa � λ′ a − 1 2πe−N 1 2 Ya(x)� (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) for a = 1 or 2, or 3, or 4 with ϵa = (1, 1, −1, −1) , and the real unknown function of x, Ya(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) These solutions exist in certain connected domains of x, xleft < x < xright .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) where xleft and xright are fixed by the positivity conditions ρ(x) > 0 and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Plugging the ansatz (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) in δδv(x)Wµ one can then solve for Ya(x) as a function of ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Then, plugging the answer in δρWµ = 0 one can proceed to solve for ρ0 and ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Under the assumption (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8), this previous procedure leads to the solutions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 23This follows from the fact that ∂δv(x)Li2(ei(δv(x)∓λ′)) = i ei(δv(x)∓λ′)Li1(ei(δv(x)∓λ′)) grows as O(N 1 2 ) if δv(x) = ±(λ′ − e−N 1 2 Y (x)) (in a large-N limit for which the function Y (x) > 0 remains finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 57 – Constructing the dominant solution Patching together solutions of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) of the first and second type, one can construct the dominant solution to δρWµ = δvWµ = δ�vWµ = δµWµ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) Let us assume the following ordering 0 < λ′ 1 < λ′ 2 < λ′ 3 < λ′ 4 < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) As it will be shown below, the last condition in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) together with the choice (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14), implies µ ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For the moment let us further assume µ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15) The case µ < 0 can be worked out analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Given the constraint (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) and the ordering (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14), out of the four possible solutions of the second type to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2), only two are consistent with the required positivity conditions ρ(x) > 0 , Ya > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' One of these two is δvleft(x) = −λ′ 3 + 1 2πe−N 1 2 Y3(x) , Y3(x) = µ + 4π2kλ′ 4x 2πλ′− 3,4 ρleft(x) = − µ + 4π2kλ′ 3 x (2π)3λ+ 1,3λ+ 2,3λ′− 3,4 , λ′± i,j := λ′ i ± λ′ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16) This patch is consistent with the positivity constraints ρ(x) , Y3(x) > 0 in the domain − µ 4π2kλ′ 3 < x < − µ 4π2kλ′ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) The other consistent solution of the second type is δvright(x) = λ′ 1 − 1 2πe−N 1 2 Y1(x) , Y1(x) = µ − 4π2kλ′ 2x 2πλ′− 1,2 ρright(x) = −µ + 4π2kλ′ 1x (2π)3λ′+ 1,3λ′+ 1,4λ′− 1,2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='18) which is consistent with the positivity constraints ρ(x) , Y1(x) > 0 in the domain µ 4π2kλ′ 2 < x < µ 4π2kλ′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19) The left and right boundary of the domains (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19), respectively, correspond to the points x at which ρ(x) = 0: at every point in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19), ρ(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The right and left boundaries of the domains (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19), respectively, correspond to the points x at which Y3(x) and Y1(x) are equal to zero: at every point in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='17) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='19), Y3(x) and Y1(x) are larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 58 – At last, the solution of first type is such that δv matches the values (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) at the left and right extrema of the interval − µ 4π2kλ′ 4 < x < µ 4π2kλ′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='20) Such a solution is δvcenter(x) = − (λ′ 3λ′ 4 − λ′ 1λ′ 2) µ/4π2 + k � λ′ 3λ′ 4λ′ 1,2 + λ′ 1λ′ 2λ′ 3,4 � x k (λ′ 3λ′ 4 − λ′ 1λ′ 2) x + λ′ 1,2,3,4 µ/4π2 , ρcenter(x) = λ′ 1,2,3,4 µ/4π2 + k (λ′ 3λ′ 4 − λ′ 1λ′ 2) x 2π λ′+ 1,3λ′+ 1,4λ′+ 2,3λ′+ 2,4 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) again, with λ′ 1,2,3,4 := λ′ 1 + λ′ 2 + λ′ 3 + λ′ 4 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) Patching together these three solutions to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) one obtains a continuous solution, in the lateral sense, which is defined in � − µ 4π2kλ′ 3 , − µ 4π2kλ′ 4 � ∪ � − µ 4π2kλ′ 4 , µ 4π2kλ′ 2 � ∪ � µ 4π2kλ′ 2 , µ 4π2kλ′ 1 � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23) this is, in the strict large-N approximation, the left and right limits of the solution at the two interior boundary points match each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Outside (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23), the solution vanishes which means that the support of the eigenvalue distribution is given by x1 = − µ 4π2kλ′ 3 , x2 = µ 4π2kλ′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24) The normalization condition for ρ – which follows from the fourth equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) –, and the assumption (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='15), fix the value of the Lagrange multiplier µ as follows � x2 x1 dx ρ(x) = µ2 2 k (2π)4λ′ 1λ′ 2λ′ 3λ′ 4 = 1 =⇒ µ = +4π2� 2 k λ′ 1λ′ 2λ′ 3λ′ 4 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='25) A computation shows that the value of W(ρ(x), v(x), �v) at the above-found solution is W −→ − i2 √ 2k 1 2 N 3 2 3 � λ′ 1λ′ 2λ′ 3λ′ 4 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='26) This result can be extended to λ′ a ∈ R out of the previously-assumed domain (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of the original variables λa = λ′ a−ℓd/4 ℓc the answer can be written as W −→ −i2 √ 2k 1 2 N 3 2 3 (2π)2� {ℓcλ1 + ℓd/4}{ℓcλ2 + ℓd/4}{ℓcλ3 + ℓd/4}{ℓcλ4 + ℓd/4} (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) again, with {ℓcλ1 + ℓd/4} + {ℓcλ2 + ℓd/4} + {ℓcλ3 + ℓd/4} + {ℓcλ4 + ℓd/4} = + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) – 59 – The saddle-point solution in a different domain of λa’s The saddle-point solution takes a different form if one assumes the λa’s to belong to the following domain − 1 < λ′ 4 < λ′ 3 < λ′ 2 < λ′ 1 < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='29) Then, assuming µ < 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='30) together with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4), the expressions of δv and ρ in terms of µ and λ′ i’s are the same as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Again, in these expressions there is one algebraic condition analogous to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22) that in this case takes the form: λ′ 1,2,3,4 = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='31) The domains of the three different linear pieces remain as in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='23), but this time with µ = −4π2 � 2 k λ′ 1λ′ 2λ′ 3λ′ 4 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='32) and W −→ + i2 √ 2k 1 2 N 3 2 3 � λ′ 1λ′ 2λ′ 3λ′ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='33) This result can be extended to λ′ a ∈ R out of the previously-assumed domain (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of the original variables λa = λ′ a−ℓd/4 ℓc the answer can be written as W −→ + i2 √ 2k 1 2 N 3 2 3 (2π)2� {ℓcλ1 + ℓd/4}−{ℓcλ2 + ℓd/4}−{ℓcλ3 + ℓd/4}−{ℓcλ4 + ℓd/4}− (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='34) where {x}− := {x} − 1 again, and {ℓcλ1 + ℓd/4} + {ℓcλ2 + ℓd/4} + {ℓcλ3 + ℓd/4} + {ℓcλ4 + ℓd/4} = + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='35) Particular limit cases Let us focus on branch one above and study the limits for which some of the λa’s coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The simplest case is when only a single couple of λa’s coincide λ′− 1,2 , λ′− 3,4 ̸= 0 , λ′− 2,3 → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='36) For this case the previous discussion applies trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The next case is when only two couples of λa’s collide λ′− 1,2 ̸= 0 , λ′− 2,3 = λ′− 3,4 → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='37) In this case the domain of the left patch, the first segment in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='22), shrinks to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Naively, one would say then that the saddle solution in this limit can be obtained by dropping the left patch of the generic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Indeed, such an expectation is reinforced by the observation that in the expansion λ3 → λ4 � − µ 4π2kλ′ 4 x1 dx ρleft(x) = O((λ3 − λ4)1) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='38) – 60 – even though ρleft(x) blows up as λ3 → λ4 (see (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This means that the contribution coming from the left patch solution to the normalization condition � x2 x1 dx ρ(x) = 1 vanishes in the limit in which the left patch shrinks to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Consequently, in the limit λ3 → λ4 one can drop the left patch of the generic solution and construct a solution by gluing the center and right patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This new solution is properly normalized as � x2 x1 dx ρ(x) = 1 as it is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Last, as in the previous case, in the limit for which all λa’s collide λ′− 1,2 → λ′− 2,3 → λ′− 3,4 → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='39) one has to drop not just the left but also the right patch, and as before, the new solution is given by the center patch with λ1 = λ2 = λ3 = λ4 = n1 4 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) will be properly normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' D Killing spinor equations In Section 4, we remarked that the constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40) could be derived by studying the bulk Killing spinor equation near the boundary and imposing the existence of a contractible circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Here we expand on those comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We shall first make some general considerations in Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The bulk su- persymmetry equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) induces a three-dimensional charged conformal Killing spinor χ at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This spinor satisfies the equation of three-dimensional off-shell conformal supergravity [99, 100] (∇i − iAi) χ − 1 3γiγj (∇j − iAj) χ = 0 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) where we are using the connection of the boundary metric g, γi generate the Clifford algebra Cliff(1, 2), and A is interpreted as a background Abelian gauge field coupling to a u(1)R R- symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Here we have A = −Φe dt, for the Lorentzian boundary line element obtained from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) we choose the frame e0 = dt , e1 = dθ , e2 = sin θ (d�φ + Ω dt) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) and γ0 = iσ1, γ1 = σ2, γ2 = σ3 (σi being the Pauli matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We find that the most general – 61 – solution is χ = χ− + χ+ where χ− = u1 exp � − i 2 � �φ + t (1 + 2Φe + Ω) �� e i 2 θσ3 � 1 −1 � + v1 exp � i 2 � �φ − t (1 + 2Φe − Ω) �� e i 2 θσ3 � 1 1 � , χ+ = u2 exp � − i 2 � �φ − t (1 − 2Φe − Ω) �� e− i 2 θσ3 � 1 −1 � + v2 exp � i 2 � �φ + t (1 − 2Φe + Ω) �� e− i 2 θσ3 � 1 1 � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) and u1,2, v1,2 are arbitrary complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The two spinors satisfy the stronger charged Killing spinor equation (∇i − iAi)χ∓ = ∓ i 2γiγ0χ∓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) The other key spinor used in the construction of the boundary rigid supersymmetric back- ground is the spinor �χ that satisfies the conformal Killing spinor equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) with opposite charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' If the gauge field is real, as it is in Lorentzian signature, we can take �χ = χc, where the charge conjugate spinor is χc ≡ γ0C−1χ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' From these two spinors, we can construct a geometric background preserving two supercharges of opposite R-charge, and in particular a bilinear vector ξi = �χγiχ, which is generically a conformal Killing vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='24 Issues arise when we Wick rotate to Euclidean signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' As already pointed out, both the bulk Killing spinor equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='27) and the conformal Killing spinor equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) are analytic in the supergravity fields, so the Wick-rotated spinors are still solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, in Euclidean signature one is a priori not allowed to impose reality conditions on bosonic or fermionic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This implies that the spinor �χ is independent of χ, and indeed it is well-known that we can have Riemannian backgrounds supporting two independent supercharges with opposite R-charge, such as the fibered S1×S2 considered in Section 2 [99, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To derive this condition holographically, one should then in principle consider Riemannian bulk solutions with a supersymmetry obtained by doubling the Killing spinor equations, as stressed, for instance, in [48, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We shall take a simpler approach, often taken in the literature, and Wick rotate the Lorentzian spinors to χ → χE and χc → �χE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The resulting spinors are independent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' �χE ̸= χc E) and solve, respectively, the conformal Killing spinor equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) and that with opposite charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' 24Here χ ≡ χT C, where the charge conjugation matrix C is the intertwiner satisfying CT = −C , C∗ = C , C2 = −1 , γT i = −CγiC−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' With our choice of basis, one can choose C = iσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 62 – In fact, it follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) that these spinors are also solutions to the new minimal supergravity Killing spinor equations 0 = ∇iχE − iA(nm) i χE + iViχE + i 2V jγjiχE + 1 2HγiχE , 0 = ∇i�χE + iA(nm) i �χE − iVi�χE − i 2V jγji�χE + 1 2Hγi�χE , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) with H = 0, an appropriate choice of Vi (depending on the choice χ±), and A(nm) = Ai + 3 2Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Unsurprisingly, these are the Killing spinor equations of three-dimensional off-shell new minimal supergravity that would give the backgrounds considered at the end of Section 2 [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For our purposes it is consistent to choose χE = u exp �1 2 � i�φ + tE (1 − 2Φe + Ω) �� e− i 2 θσ3 � 1 1 � , �χE = �u exp � −1 2 � i�φ + tE (1 − 2Φe + Ω) �� e i 2 θσ3 � 1 −1 � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) where again u, �u are arbitrary complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' These spinors solve the conformal Killing spinor equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) with positive and negative R-charge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Moreover, they also solve (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) with V = −i dtE and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to have well-defined global spinors on the fibered S1 × S2 background (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11), we should impose a constraint on the chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' First, notice that under �φ → �φ + 2π, the spinors are antiperiodic as should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, in order for the spinors to be antiperiodic as tE → tE + β we should require β (1 − 2Φe + Ω) = 2πin0 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) with odd n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' This provides us with an additional way to argue for the constraint between the chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In Section 2, we found that the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='28) followed directly from identifying the superconformal index as a thermal partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We also argued around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='21) that it could be derived by looking at the rigid supersymmetric background and requiring thermal boundary conditions for the Killing spinor (which is the argument just reproduced in detail to get to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='40)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' Whilst until now the constraint has been discussed from the field theory viewpoint, we now find an argument from the regularity of the Euclidean gravity solution: χE is the leading order spinor in the radial expansion of the bulk spinor ϵ near the boundary, and the anti-periodicity of χE (and thus of ϵ) around S1 β is needed in order to be able to have a smooth disc filling S1 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' E Four-dimensional index In this appendix, we review the construction and limits of the four-dimensional supercon- formal index for N = 4 SYM with SU(N) gauge group, emphasising the parallels with the construction in three dimensions explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' – 63 – The four-dimensional superconformal index for N = 4 SYM counts the states on S3 annihilated by a supercharge Q with {Q, Q†} = H − J1 − J2 − 3 � i=1 Ri , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='1) where J1,2 are the generators of the Cartan of the su(2) × su(2) isometries on S3, and R1,2,3 are generators of the Cartan of so(6)R in the orthogonal basis (so they have half-integer eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The remaining bosonic subalgebra commuting with Q, Q† is generated by J1 + 1 3 � i Ri , J2 + 1 3 � i Ri , R1 − R3 , R2 − R3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='2) The u(1)R commuting with the supercharge is generated by r ≡ 2 3 � i Ri, with eigenvalues in 1 3Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We define the index as I(τ1, τ2, λ1, λ2) = TrHS3 � (−1)2J1e−β{Q,Q†}+2πiτ1(J1+ 1 3 � i Ri)+2πiτ2(J2+ 1 3 � i Ri) × e2πiλ1(R1−R3)+2πiλ2(R2−R3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='3) States in the theory satisfy J1,2 = R1,2,3 mod 1, so we find that the index is invariant under the following shifts τ1,2 → τ1,2 + 3 , λ1,2 → λ1,2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='4) Instead, observe that shifts of (say) τ1 by integers lead to indices graded by r, the R-charge indices I(τ1 + 1, τ2, λ1, λ2) = TrHS3 � (−1)re−β{Q,Q†}+2πiτ1(J1+ 1 3 � i Ri)+2πiτ2(J2+ 1 3 � i Ri) × e2πiλ1(R1−R3)+2πiλ2(R2−R3) � ≡ IR(τ1, τ2, λ1, λ2) , I(τ1 + 2, τ2, λ1, λ2) = IR(τ1, τ2, λ1, λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='5) It is convenient to introduce λ3 via the constraint 3 � i=1 λi = n1 , n1 ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='6) This leads to I(τ1, τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' λ) = TrHS3(−1)2J1e−β{Q,Q†}+2πi[τ1(J1+ 1 3 � i Ri)+τ2(J2+ 1 3 � i Ri)+� i λiRi−n1R3] = TrHS3(−1)2J1e−β{Q,Q†}+2πi[ � i( τ1 3 +λi)(J1+Ri)+τ2(J2+ 1 3 � i Ri)] , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7) – 64 – so that the index is really a function of the four variables λi + τ1/3, τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The functional integral formalism also works exactly in parallel to the discussion in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' The index can be seen as a functional integral over a fibered S1 ×S3 with background gauge fields for the Cartan of the R-symmetry, and thermal boundary conditions for the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In particular, looking at (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='7), we have fibration parameters Ω1 = 1 + 2πi β (τ1 + n0 + n1) , Ω2 = 1 + 2πi β τ2 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='8) and background gauge fields Ai = iΦi dtE with Φi = 1 + 2πi β �τ1 + τ2 3 + λi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='9) Here n0 is an odd number taking care of the grading by (−1)2J1, and because of supersym- metry, these quantities are not all independent β � 1 − 3 � i=1 Φi + Ω1 + Ω2 � = 2πin0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='10) Moreover, thanks to the relation between the charges of the states in the theory, the partition function is invariant under the following shifts ΩA → ΩA + 2πi β mA , ΦA → ΦA + 2πi β nA , Φ3 → Φ3 + 2πi β � 2k − 2 � A=1 (mA + nA) � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='11) with mA, nA, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' To simplify the discussion, we reduce to the universal case by setting λ1 = λ2 = λ3 ≡ λ, with the constraint 3λ = n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In this case, it is I(τ1, τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' n1) = TrHS3(−1)2J1e−β{Q,Q†}+2πi[(τ1+n1)(J1+ 1 2 r)+τ2(J2+ 1 2 r)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='12) Therefore, we immediately see that if n1 = ±1 mod 3 we have the R-charge index, graded by the R-symmetry generator r satisfying 2J1 = 2J2 = 3r mod 2 on the states of N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We further simplify the discussion by setting τ1 = τ2 ≡ τ, obtaining the simplest index receiving contributions only from states preserving two supercharges �I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' n1) = TrHS3(−1)2J1e2πin1(J1+ r 2)+2πiτ(J1+J2+r) = TrHS3(−1)2J1e2πi(τ−n1)(J1+J2+r) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='13) where in the last equation we used the relation 2J2 = 3r mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For the same reason, as mentioned, both τ and n1 are only defined modulo 3, so the unrefined index is really a function of the C-valued variable exp(2πiT), with T ≡ (τ − n1)/3, and T ∼ T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We are interested in the generalized Cardy limit in which T → D/C with gcd(C, D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In order to study the asymptotic behavior of the index in this limit, we further write – 65 – 3T = ℓD/ℓC with gcd(ℓC, ℓD) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' If C is not a multiple of 3, then ℓD = 3D and ℓC = C, whereas if C is a multiple of 3, then ℓD = D and ℓC = C/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of these variables, in the generalized Cardy limit, log �I(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' n1) has a leading O((3ℓCT − ℓD))−2) term provided that C is a multiple of 3, in which case [7, 11] log �I(T) ∼ ± iπ 27N2 1 C 3 (CT − D)2 if D = ±1 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='14) Clearly, the smallest values for which the index has a leading singular behavior are T → ±1/3, corresponding to exp(2πiT) approaching the primitive third roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' For historical reasons, the asymptotic behavior of the index has been studied splitting T = (τ − n1)/3 in τ and n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' In terms of the variable τ, the index of N = 4 SYM is clearly a three-sheeted function, and the leading singularities can be reached in multiple equivalent ways: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' τ → 1, 2 and n1 = 0, or, if we insist on the Cardy limit τ → 0, then n1 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' That is, we either need to take the limits on the first/second sheet, or take the Cardy limit τ → 0 of the R-charge index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' We derived this result using generalised Cardy limits τ → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQf3PkC/content/2301.00763v1.pdf'} +page_content=' However, the same behavior for the index is discovered when approaching its study using the Bethe ansatz method: in the large-N limit, one finds that the leading behavior is again on the first and second sheet, whereas on the zeroth sheet the large-N limit is undefined due to the competition of terms with the same 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As +surrogates of physical-model-based OPF solvers, neural network +(NN) solvers can accelerate the solving process. However, they +may be unreliable for “unseen” inputs when the training dataset +is unrepresentative. Enhancing the representativeness of the +training dataset for NN solvers is indispensable but is not well +studied in the literature. To tackle this challenge, we propose an +OPF solver based on a physical-model-integrated NN with worth- +learning data generation. The designed NN is a combination +of a conventional multi-layer perceptron (MLP) and an OPF- +model module, which outputs not only the optimal decision +variables of the OPF problem but also the constraints violation +degree. Based on this NN, the worth-learning data generation +method can identify feasible samples that are not well generalized +by the NN. By iteratively applying this method and including +the newly identified worth-learning samples in the training set, +the representativeness of the training set can be significantly +enhanced. Therefore, the solution reliability of the NN solver +can be remarkably improved. Experimental results show that the +proposed method leads to an over 50% reduction of constraint +violations and optimality loss compared to conventional NN +solvers. +Index Terms—Optimal power flow, physical-model-integrated +neural network, worth-learning data generation +I. INTRODUCTION +O +PTIMAL power flow (OPF) is a fundamental but chal- +lenging problem for power systems [1]. A typical OPF +problem usually involves determining the optimal power dis- +patch with an objective, e.g., minimizing total generation costs +or power loss, while satisfying nonlinear power flow equations +and other physical or engineering constraints [2]. Due to the +nonlinear interrelation of nodal power injections and voltages, +OPF is non-convex, NP-hard, and cannot be solved efficiently +[3]. With the increasing integration of renewable generation +and flexible demands, uncertainty and volatility have been +rising on both the demand and supply sides of modern power +systems [4], which requires OPF to be solved more frequently. +Thus, fast and reliable OPF solvers have become indispensable +to ensure effective operations of modern power systems and +have attracted surging interest in academia. +There is a dilemma between the solving efficiency and +solution reliability of OPF. Conventionally, OPF is solved +by iterative algorithms, such as interior point algorithms, +based on explicit physical models [5]. However, these methods +may converge to locally optimal solutions. Recently, some +researchers have made great progress in designing conic +relaxation models for OPF, which are convex and can be +efficiently solved [6]–[8]. Nevertheless, the exactness of these +relaxations may not hold in practical scenarios, and they may +obtain infeasible solutions [9]. In addition, the scalability of +the conic relaxation of alternating current optimal power flow +(AC-OPF) may still be a challenge, particularly in online, +combinatorial, and stochastic settings [10]. +To overcome the limitation of the aforementioned physical- +model-based solvers, some researchers propose surrogate OPF +solvers based on neural networks (NNs) [11]–[13]. These +solvers use NNs to approximate the functional mapping from +the operational parameters (e.g., profiles of renewable gen- +eration and power demands) to the decision variables (e.g., +power dispatch) of OPF. Compared to iterative algorithms, +they can introduce significant speedup because an NN is only +composed of simple fundamental functions in sequence [12], +[13]. However, one of the critical problems of NN solvers is +that they may be unreliable if not properly trained, especially +for “unseen” inputs in feasible regions due to NNs’ mystery +generalization mechanism [14]. +The generalization of NNs is mainly influenced by their +structures, loss functions, and training data. Most published +papers propose to enhance the generalization of NN OPF +solvers by adjusting the structures and loss functions. Various +advanced NN structures rather than conventional fully con- +nected networks are employed to imitate AC-OPF. For exam- +ple, Owerko et al. [15] use graph NNs to approximate a given +optimal solution. Su et al. [16] employ a deep belief network to +fit the generator’s power in OPF. Zhang et al. [17] construct a +convex NN solving DC-OPF to guarantee the generalization of +NNs. Jeyaraj et al. [18] employ a Bayesian regularized deep +NN to solve the OPF in DC microgrids. Some researchers +design elaborate loss functions that penalize the constraints +violation, combine Karush-Kuhn-Tucker conditions, or include +derivatives of decision variables to operational parameters. For +example, Pan et al. [11] introduce a penalty term related to the +inequality constraints into the loss function. This approach can +speed up the computation by up to two orders of magnitude +compared to the Gurobi solver, but 18.3% of its solutions are +infeasible. Ferdinando et al. [12] include a Lagrange item in +the loss function of NNs. Their method’s prediction errors +are as low as 0.2%, and its solving speed is faster than DC- +OPF by at least two orders of magnitude. Manish et al. [10] +include sensitivity information in the training of NN so that +only using about 10% to 25% of training data can attain the +same approximation accuracy as methods without sensitivity +information. Nellikkath et al. [19] apply physics-informed +arXiv:2301.03766v1 [cs.LG] 10 Jan 2023 + +2 +NNs to OPF problems, and their results have higher accuracy +than conventional NNs. +The above-mentioned studies have made significant progress +in designing elaborate network structures and loss functions. +However, little attention has been paid to the training set +generation problem. Specifically, they all adopt conventional +probability sampling methods to produce datasets for training +and testing, such as simple random sampling [10]–[13], [15], +[17], [20], Monte Carlo simulation [18], or Latin hypercube +sampling [16], [19]. These probability sampling methods can- +not provide a theoretical guarantee that a generated training +set can represent the input space of the OPF problem prop- +erly. As a result, probability sampling methods may generate +insufficient and unrepresentative training sets, so the trained +NN solvers may provide unreliable solutions. +It is important to create a sufficiently representative dataset +for training an NN OPF solver. A training set’s representative- +ness depends on its size and distribution in its feasible region +[21]. Taking a medium-scale OPF problem as an example, +millions of data samples may still be sparse given the high +dimension of the NN’s inputs (e.g., operational parameters of +the OPF problem: renewable generation and power demands +at all buses); in addition, because the OPF problem is non- +convex, the feasible region of the NN’s inputs is a complicated +irregular space. Thus, generating a representative training +set to cover all the feasible regions of the inputs with an +acceptable size is quite challenging. Without a representative +training set, it is difficult to guarantee that the NN OPF solver’s +outputs are reliable, especially given “unseen” inputs in the +inference process, as discussed in [22], [23]. +To address the above challenge, this study proposes +a physical-model-integrated deep NN method with worth- +learning data generation to solve AC-OPF problems. To the +best of our knowledge, this is the first study that has addressed +the representativeness problem of the training dataset for NN +OPF solvers. The major contributions of this study are twofold: +1) A novel physical-model-integrated NN is designed for +solving the AC-OPF problem. This NN is constructed by +a conventional MLP integrating an OPF-model module, +which outputs not only the optimal decision variables +of the OPF problem but also the violation degree of +constraints. By penalizing the latter in the loss function +during training, the NN can generate more reliable +decision variables. +2) Based on the designed NN, a novel generation method +for worth-learning training data is proposed, which can +identify samples in the input feasible region that are +not well generalized by the previous NN. By iteratively +applying this method during the training process, the +trained NN gradually generalizes to the whole feasible +region. As a result, the generalization and reliability of +the proposed NN solver can be significantly enhanced. +Furthermore, comprehensive numerical experiments are con- +ducted, which prove that the proposed method is effective in +terms of both reliability and optimality for solving AC-OPF +problems with high computational efficiency. +The remainder of this article is organized as follows. Section +II provides preliminary models and the motivations behind this +Fig. 1. The 3-bus system. +study. Section III introduces the proposed method. Section IV +details the experiments. Section V concludes this paper. +II. ANALYSIS OF APPROXIMATING OPF PROBLEMS BY NN +A. AC-OPF problem +The AC-OPF problem aims to determine the optimal power +dispatch (usually for generators) given specific operating con- +ditions of a power system, e.g., power loads and renewable +generation. A typical AC-OPF model can be formulated as +min +V , SG C +� +SG� +(1a) +s.t.: +[V ] Y∗ +busV ∗ = SG − SL, +(1b) +SG ≤ SG ≤ S +G, +(1c) +V ≤ V ≤ V, +(1d) +|YbV | ≤ ¯I, +(1e) +where Eq. (1a) is the objective, e.g., minimizing total genera- +tion costs, and Eqs. (1b) to (1e) denote constraints. Symbols +SG and SL are n × 1 vectors representing complex bus +injections from generators and loads, respectively, where n +is the number of buses. Symbol V is an n×1 vector denoting +node voltages. Symbol [.] denotes an operator that transforms +a vector into a diagonal matrix with the vector elements on +the diagonal. Symbol Ybus is a complex n × n bus admittance +matrix written as Y at other sections for convenience. Symbol +Yb is a complex nb × n branch admittance matrix, and nb is +the number of branches. The upper and lower bounds of any +variable x are represented by ¯x and x, respectively. Vector ¯I +denotes the current flow limit of branches. +B. AC-OPF mapping from loads to optimal dispatch +An NN model describes an input-output mapping. Specifi- +cally, for an NN model solving the AC-OPF problem shown in +Eq. (1), the input is the power demand SL, and the output is the +optimal generation SG *. Hence, an NN OPF solver describes +the mapping SG * = f OPF(SL). A well-trained NN should be +able to accurately approximate this mapping. +We provide a basic example of a 3-bus system, as shown in +Fig. 1, to illustrate how NN works for OPF problems and +explain the corresponding challenge for generalization. For +simplicity, we assume there is no reactive power in the system +and set r31 = r12 = 0.01 ; P i = 0 , P i = 4, for i ∈ {1, 3}, +P2 ∈ [−7, 0]; V i = 0.95 and V i = 1.05, for i ∈ {1, 2, 3}. + +Load +P2E[-7,0]3 +Fig. 2. Examples of an NN fitting the OPF of the 3-bus system based on (a) +simple random sampling, and (b) worth-learning data generation. +Then, the OPF model Eq. (1) is reduced to the following +quadratic programming: +min +V, PG P1 + 1.5 × P3 +(2a) +s.t.: P1 = V1 (V1 − V2) /0.01 + V1 (V1 − V3) /0.01, +(2b) +P2 = V2 (V2 − V1) /0.01, +(2c) +P3 = V3 (V3 − V1) /0.01, +(2d) +0.95 ≤ V3 ≤ 1.05, 0.95 ≤ V2 ≤ 1.05, +(2e) +0 ≤ P1 ≤ 4, 0 ≤ P3 ≤ 4, V1 = 1, +(2f) +where V is [V1 V2 V3]⊤, and P G is [P1 P3]⊤. +Given that P2 ranges from -7 to 0, the 3-bus OPF model can +be solved analytically. The closed-form solution of [P ∗ +1 P ∗ +3 ] = +f OPF +3-bus(P2), is formulated as follows: +P ∗ +1 = +� +50 − 50√0.04P2 + 1, +c1, +4, +c2, +(3) +P ∗ +3 = +� +� +� +0, +c1, +213 +� +1 − 0.34√0.04P2 + 1 +�2 ++50√0.04P2 + 1 − 146 +, +c2, +(4) +where c1 denotes condition 1: +−3.84 ≤ P2 ≤ 0, and c2 +denotes condition 2: −7 ≤ P2 < −3.84. +To further analyze the mapping f OPF +3-bus, we draw the +[P ∗ +1 P ∗ +3 ]–P2 curve according to Eqs. (3) and (4), shown in +Fig. 2. Both the P ∗ +1 –P2 and P ∗ +3 –P2 curves are piecewise +nonlinear functions, in which two oblique lines are nonlinear +because of the quadratic equality constraints. The reason +why the two curves above are piecewise is that the active +inequalities change the [P ∗ +1 P ∗ +3 ]–P2 relationship. From an +optimization perspective, each active inequality will add a +unique equality constraint to the relationship, so the pieces +in f OPF +3-bus are determined by the sets of active inequalities. In +this example, the two pieces in each curve correspond to two +sets of active inequalities: P1 ≤ 4 and 0 ≤ P3. Moreover, +the two intersection points are the critical points where these +inequalities are just satisfied as equalities. +For a general AC-OPF problem, its input is usually high- +dimensional (commonly determined by the number of buses), +and its feasible space is partitioned into some distinct regions +by different sets of active inequality constraints. From an +optimization perspective, a set of active constraints uniquely +characterizes the relationship SG * += f OPF(SL), and the +number of pieces theoretically increases with the number of +inequality constraints by exponential order [24]–[26]. There- +fore, there are massive regions, and each region corresponds +to a unique mapping relation, i.e., a piece of mapping function +f OPF. +C. Challenges of fitting OPF mapping by NN +As shown in Fig. 2(a), to fit the two-dimensional piecewise +nonlinear curve of f OPF +3-bus, we first adopt four data samples by +simple random sampling and then use an NN to learn the +curve. Obviously, there are significant fitting errors between +the fitting and the original lines. Because the training set lacks +the samples near the intersections in the curve (where p2 = +−0.384 in this case), the NN cannot accurately approximate +the mapping in the neighboring region of the intersections. +A training set representing the whole input space is a prereq- +uisite for an NN approximating the curve properly. However, +it is nontrivial to generate a representative training set by +probability sampling. As shown in Fig. 2(a), the intersections +of f OPF are key points for the representativeness, and the +number of intersections increases exponentially with that of +the inequality constraints, as analyzed in II-B. When each +sample is selected with a small possibility ρ, the generation +of a dataset containing all the intersection points are in a low +possibility event whose probability is equal to ρm, where m is +the number of intersections. In practice, the only way to collect +sufficient data representing the input space by probability +sampling is to expand the dataset as much as possible [27]. +This is impractical for large power networks. Therefore, the +conventional probability sampling in the literature can hardly +produce a representative dataset with a moderate size. +As shown in Fig. 2(b), if we are able to identify the two +intersections, i.e., (P2 = −0.384, P1 = 4) and (P2 = −0.384, +P3 = 0), and include them as new samples in the training +dataset, the corresponding large fitting errors of the NN +can be eliminated. These samples are termed as the worth- +learning data samples. The focus of this study is to propose a +worth-learning data generation method that can help identify +worth-learning data samples and overcome the aforementioned +disadvantage of conventional probability sampling (detailed in +the following section). +III. A PHYSICAL-MODEL-INTEGRATED NN WITH +WORTH-LEARNING DATA GENERATION +This section proposes a physical-model-integrated NN with +a worth-learning data generation method to solve AC-OPF +problems. The proposed NN is a combination of a fully- +connected network and a transformed OPF model. It outputs +not only the optimal decision variables of the OPF problem but +also the violation degree of constraints, which provides guid- +ance for identifying worth-learning data. The worth-learning +data generation method creates representative training sets to +enhance the generalization of the NN solver. + +P* +Data from simple random sampling +Fitting line +Fitting error Worth-learning data4 +START +Initialize a training set by randomly sampling +Train the NN on the current training set +Identify worth-learning data +for the current NN +Worth-learning data +are identified? +Output the current NN +END +Y +N +Add identified +data to the +training set +Fig. 3. Framework of the proposed training process. +A. Framework of the proposed method +The proposed data generation method has an iterative pro- +cess, as shown in Fig. 3. First, a training set is initialized by +random sampling; second, the physical-model-integrated NN +is trained on the training set, where an elaborate loss function +is utilized; third, worth-learning data for the current NN are +identified; fourth, if worth-learning data are identified, these +data are added to the training set and returns to the second +step; otherwise, the current NN is output. +The above training process converges until no worth- +learning data are identified. This means that the training set +is sufficiently representative of the input space of the OPF +problem. As a result, the NN trained based on this dataset +can generalize to the input feasible set well. The following +subsections introduce the proposed method in detail. +B. Physical-model-integrated NN +In the second step of the proposed method (Fig. 3), the NN +is trained to fit the mapping SG * = f OPF(SL). To obtain better +results, we design a physical-model-integrated NN structure +consisting of a conventional NN module and a physical-model +module, as shown in Fig. 4. The former is a conventional MLP, +while the latter is a computational graph transformed from the +OPF model. +1) Conventional NN module: This module first adopts a +conventional MLP with learnable parameters to fit the mapping +from the SL to the optimal decision variable V NN [28]. The +V NN has its box constraint defined in Eq. (1). To ensure that +the output V NN satisfies this constraint, we design a function +dRe() to adjust any infeasible output V NN into its feasible +region, which is formulated as follows: +x ← dRe(x, x, x) = ReLU(x − x) − ReLU(x − x) + x, +(5) +where ReLU(x) = max(x, 0); x is the input of the function, +and its lower and upper bounds are x and x, respectively. The +diagram of this function is illustrated in Fig. 5. +Input +… +… +Physical model module +Conventional NN module +Output +Fig. 4. The physical-model-integrated NN. +Applying dRe() as the activation function of the last layer +of the conventional MLP, the mathematical model of this +conventional NN module is formulated as follows: +V NN = MLP(SL), +(6) +V NN ← dRe(V NN, V , V ), +(7) +where Eq. (6) describes the conventional model of MLP and +Eq. (7) adjusts the output of the MLP. +2) Physical model module: This module receives V NN +from the previous module, and then it outputs the optimal +power generation SG +phm and the corresponding constraints +violation V iophm, where the subscript “phm” denotes the +physical model module. The first output SG +phm is the optimal +decision variable of the AC-OPF problem. It can be calculated +by V NN and SL, as follows: +SG +phm = [V NN]Y∗V ∗ +NN + SL. +(8) +The second output V iophm (termed as violation degree) +measures the quality of SG +phm and is the key metric to guide +the proposed worth-learning data generation (see details in +the following subsection III-C). Given V NN and SG +phm, the +violations of inequality constraints of the AC-OPF problem +V iophm are calculated as follows: +V ioS +phm = ReLU(SG +phm − S +G) + ReLU(SG − SG +phm), +(9a) +V ioI +phm = ReLU(|YfV NN| − ¯I), +(9b) +V iophm = (V ioS +phm +V ioI +phm)⊤, +(9c) +where V ioS +phm denotes the violation of the upper or lower +limit of SG +phm, and V ioI +phm represents the violation of branch +currents. +Remark 1. The physical-model-integrated NN is formed +by combining the conventional NN module and the physical +model module. It inputs SL and outputs SG +phm and V iophm, +as shown in Fig. 4. Its function is the same as conventional +OPF numerical solvers. In addition, it is convenient for users +Fig. 5. The dRe() function. + +5 +Feasible region +Label value +Region with tiny +predicted error +Predicted value +Effect of the proposed +loss function +Point 1 +Point 2 +Fig. 6. Illustration of the effectiveness of the three terms in the loss function. +to directly determine whether the result of the NN OPF solver +is acceptable or not based on the violation degree V iophm. In +contrast, most NN OPF solvers in the literature are incapable +of outputting the violation degree directly [10]–[12]. +3) Loss function: To enhance the training accuracy of the +physical-model-integrated NN, we design an elaborate loss +function, which consists of V NN from the conventional NN +module, and SG +phm and V iophm from the physical model +module. The formula is as follows: +loss = || ˆV − V NN||1 + || ˆ +SG − SG +phm||1 + V iophm, +(10) +where ˆV and ˆ +SG are label values from the training set, which +is a ground truth dataset from numerical solvers. +Combining the three terms in the loss function can help en- +hance fitting precision. As shown in Fig. 6, if the loss function +only has the first two items || ˆV − V NN||1+ || ˆ +SG − SG +phm||1 to +penalize conventional fitting errors, the predicted value will be +in a tiny square space (the red square in Fig. 6) around the +label value. From the optimization perspective, the optimal +label value is usually on the edge of its feasible region (the +blue polyhedron in Fig. 6). This edge through the label value +splits the square into two parts: the feasible (blue) part and +the infeasible (white) part. Intuitively, we would prefer the +predicted values to be in the feasible part. Thus, we also +penalize violation degree V iophm in the loss function to force +the predicted values with big V iophm close to the square’s +feasible half space for smaller constraint violations. +Although the proposed NN with elaborate loss function has +high training accuracy, it is still difficult to guarantee the gen- +eralization of the NN OPF solver to the whole input space with +conventional random sampling. Therefore, it is indispensable +and challenging to obtain a representative training dataset with +moderate size to train the proposed NN, which is the focus of +the following subsection. +C. Worth-learning data generation +As shown in Fig. 3, we adopt an iterative process to identify +the worth-learning data. For an NN trained in the previous +iteration, we utilize its output V iophm to help identify new +data samples that are not yet properly generalized. Specifically, +if an input SL* is feasible for the original OPF problem while +the current NN outputs a large violation degree V io∗ +phm, the +contradiction means the NN has a large fitting error at SL*. +Input feasible set module +Fig. 7. The input feasible set module. +This is probably because sample SL* was not included in +the previous training set and was not generalized by the NN. +Hence, this sample SL* can be regarded as a worth-learning +sample. Including the sample in the training dataset in the next +iteration helps enhance the generalization of the NN. +The key to the proposed worth-learning data generation +method is to identify worth-learning samples efficiently. In- +stead of traversing all of the possible inputs, we maximize +V iophm for a given NN to identify the input with a large vio- +lation degree. However, the inputs identified in the maximizing +process should be feasible for the original OPF problem. +Otherwise, the found inputs might be infeasible and useless +for the representation of the training data. +1) Input feasible set module: To keep the inputs identified +in the maximizing process feasible for the original OPF +problem, we formulate the input feasible set module to restrict +power loads SL to their feasible set. The feasible set is +composed of box constraints, current limits, and KCL&KVL +constraints, which are transformed from the feasible set of the +OPF problem defined in Eq. (1). The partial formulations of +the input feasible set are as follows, where the subscript “ifs” +denotes the input feasible set module: +SG +ifs = dRe +� +S′G +ifs, SG, S +G� +, S′G +ifs ∈ Rn, +(11a) +V ifs = dRe +� +V ′ +ifs, V , V +� +, V ′ +ifs ∈ Rn, +(11b) +SL +ifs = SG +ifs − [V ifs]Y∗V ∗ +ifs, +(11c) +Iifs = YbV ifs, +(11d) +where S′G +ifs and V ′ifs are auxiliary n × 1 vectors in Rn and +have no physical meaning. Symbols SG +ifs and V ifs are restricted +in their box constraints in Eqs. (11a) and (11b). Then the +KCL&KVL correlations of SL +ifs, SG +ifs, and V ifs are described +by Eq. (11c). Symbol Iifs in Eq. (11d) denotes the currents at +all branches. +The other formulations of the input feasible set aim to calcu- +late V ioifs, the AC-OPF’s constraint violations corresponding +to SL +ifs and Iifs, as follows: +V ioS +ifs = ReLU(SL +ifs − S +L) + ReLU(SL − SL +ifs), +(12a) +V ioI +ifs = ReLU(|Iifs| − I), +(12b) +V ioifs = (V ioS +ifs +V ioI +ifs)⊤, +(12c) +where V ioS +ifs denotes the violation of the upper or lower limit +of SL +phm, and V ioI +ifs denotes the violation of branch current. +Remark 2. This module takes S′G +ifs and V ′ifs as the inputs, +and then outputs SL +ifs and V ioifs, as shown in Fig. 7. When +V ioifs = 0, the corresponding SL +ifs lies in the feasible set of + +6 +Conventional +NN module +Physical +model module +Input +feasible set +module +Updated +variables +Fig. 8. +The novel NN for max violation backpropagation by integrating +physical-model-integrated NN with the input feasible set module. +the AC-OPF problem. To identify feasible SL +ifs in the process of +maximizing V iophm, this module backpropagate the ∂V iophm +∂SL +ifs +with V ioifs ≤ ζ (ζ is a small positive tolerance), and then +it updates S′G +ifs and V ′ifs. As a result, the corresponding SL +ifs +is always feasible. Furthermore, because S′G +ifs and V ′ifs are +not bounded, changing them can theoretically find any feasible +SL +ifs. +2) Max violation backpropagation: +To identify worth- +learning data, a novel NN is created by inputting SL +ifs into +the physical-model-integrated NN (see Fig. 8). This NN has +two outputs, i.e., V iophm and V ioifs. The former measures +the constraint violation degree of the OPF solution SG*; the +latter indicates the feasibility of the OPF input SL +ifs. If SL +ifs +is a feasible input, i.e., V ioifs ≤ ζ, but the optimal solution +SG* is infeasible, i.e., V iophm ≥ ξ (ξ is a threshold), this +means the corresponding input is worth learning (i.e., it is +not learned or generalized by the current NN). Based on this +analysis, we design the loss function lossmax for max violation +backpropagation, as follows: +lossmax = V iophm − λ × V ioifs, +(13) +where λ is a large, constant weight parameter. When maxi- +mizing this loss function, the algorithm tends to find a worth- +learning SL +ifs that has small V ioifs but large V iophm. +During the max violation backpropagation, the proposed +algorithm maximizes lossmax to update the variables S′G +ifs +and V ′ifs by gradient backpropagation until lossmax converges +to the local maximum. After the process, the corresponding +SL +ifs is also found. Because the maximizing process can be +processed in parallel by the deep learning module PyTorch, +the worth-learning samples are found in batch, where the +max violation backpropagation uses the previous training set +as initial points to identify the new data. Further, the auto- +differentiation technique in PyTorch can accelerate the process +of parallel computation. Based on these techniques, massive +worth-learning data samples are identified efficiently. +D. Overall training process +The overall training process is presented in Algorithm 1, +which first takes an initial training dataset Dt (obtained by any +conventional sampling method) as input. The learning rate η is +equal to 10−3, the loss difference tolerance ϵ is equal to 10−2, +the added dataset A is empty, and the loss difference ∆L is +equal to infinity at initialization. The training is performed for +a fixed number of epochs (lines 2–5). Then the max violation +backpropagation starts to identify worth-learning data (lines +6 and 7) by using the training data as the initial points (line +8) and updating S′G +ifs and V ′ifs until ∆L is less than ϵ (lines +9–12), which indicates lossmax has converged to the terminal. +Algorithm +1 +Training +process +of +the +physical-model- +integrated NN OPF solver with worth-learning data generation. +Input: Dt = +� ˆ +SL, ˆV , ˆ +SG� +Initialization : η ← 10−3, ϵ ← 10−2, A ← ∅, ∆L ← ∞ +1: repeat +2: +for epoch k = 0, 1, ... do +3: +Train the NN with loss Eq. (10): +4: +w ← w − η∇loss. +5: +end for +6: +while ∆L ≥ ϵ do +7: +Identify data with lossmax Eq. (13): +8: +S′G +ifs, V ′ +ifs ← SG +ifs, V ifs ← ˆ +SG, ˆV +9: +S′G +ifs ← S′G +ifs + η∇lossmax +10: +V ′ +ifs ← V ′ +ifs + η∇lossmax +11: +∆L ← | lossmax,i − lossmax,i−100 | +12: +end while +13: +{V iophm,N} ← ffilter(V iophm,N ≥ ξ) +14: +Collect {SL +ifs} corresponding to {V iophm,N} based on the +novel NN in Fig. 8 +15: +Calculate { ˆV , ˆ +SG} corresponding to {SG +ifs} using numerical +solvers +16: +A ← {SL +ifs, ˆV , ˆ +SG} +17: +Dt ← Dt ∪ A +18: until A is ∅ +After the max violation backpropagation, a series of com- +mands are designed to add proper data to the training set. +First, a filter function ffilter is employed to eliminate data with +terminal violation V iophm,N less than a given threshold ξ (the +value depends on the acceptable violation settings). Second, +{ ˆV , ˆ +SG} is calculated by numerical solvers corresponding to +SL +ifs with large violation degree (lines 14 and 15). They consist +of added set A (line 16). Third, the training set Dt is expanded +with A (line 17). The loop is repeated until the added set A is +empty (line 18), meaning no worth-learning data are identified. +E. Efficiency and convergence of the proposed method +Unlike general training processes for conventional NNs, the +proposed physical-model-integrated NN with worth-learning +data generation adopts an iterative training process. It iter- +atively checks the NN’s generalization to the input’s feasi- +ble space by identifying worth-learning data, as shown in +Fig. 3 and Algorithm 1. This difference introduces two critical +questions. 1) Efficiency: is the process of identifying worth- +learning data computationally efficient? 2) Convergence: is +the training set representative of the whole input space after +iterations? In terms of the computational efficiency of the +proposed method, the theoretical analysis (detailed in the +Appendix A) shows it takes no more than 0.08 s to find +one sample, which brings little computational burden into +the training process. According to the experiment results, the +average consumption time for finding one sample is 0.056 s. In +terms of the convergence, we prove that the training set would +gradually represent the whole input space in the Appendix +B, because the number of worth-learning samples identified +would converge to zero after a finite number of iterations. + +7 +The number of times the sequence codes are +repeated in the data generation loop (/100) +The violation degree (MW) +Fig. 9. Time consumption of the worth-learning data codes in three different +iterations. The number of times the sequence codes are repeated in the data +generation loop (x-axis) represents the time consumed in one data generation +loop; the violation degrees (y-axis) quickly converge to the terminal stage. +IV. NUMERICAL EXPERIMENTS +The proposed method is evaluated using the IEEE 12-bus, +14-bus, 30-bus, 57-bus, and 118-bus systems. The ground truth +datasets are constructed using PANDAPOWER based on a +prime-dual interior points algorithm. +A. The efficiency of worth-learning data generation +As shown in Algorithm 1, the proposed worth-learning data +generation (lines 6–12) is the second loop in one iteration +(lines 1–18), and the number of initial data points for the +generation varies with iterations (lines 8, 15–17). To evalu- +ate the efficiency of the worth-learning data generation, we +conduct an experiment on the IEEE 57-bus system in three +different iterations to quantitatively measure how much time +it takes to finish one worth-learning data generation loop. The +time consumption of the data-generation loops in the three +different iterations is illustrated in Fig. 9. The x-axis is the +number of times the codes are repeated (lines 6–12) divided +by 100, which represents the time consumed in one data +generation loop; the y-axis is the violation degree. The three +lines converge to the terminal stage within 4000 times. The +trends are similar: they increase very quickly at first (with 100 +epochs) and then approach the local maximum slowly (with +2900–3900 epochs). The inflection points on the three lines +are (1, 7228), (1, 9065), and (1, 5841). +In the three iterations, 300, 500, and 800 new data samples +are identified. Each data-generation loop in iterations takes +30 s on average to run 3000–4000 times. Hence, one worth- +learning data sample costs (30×3)/(300+500+800) ≈ 0.056 +s, which introduces little computational burden into the train- +ing process compared to the other steps in Algorithm 1. For ex- +ample, each label value calculated by numerical solvers costs +around 1 s (line 14), and the NN training on a dataset with +1100 samples costs around 600 s (lines 2–5). In conclusion, +the numerical experiment verifies that the worth-learning data +generation brings little computational burden to the training +process. +Furthermore, we list the time consumption comparison of +the conventional and proposed training processes in Table I, +where the conventional training process uses simple random +sampling in place of the data generation loop (lines 6–12) +in Algorithm 1. By comparing the time consumption of the +TABLE I +TRAINING TIME BASED ON THE CONVENTIONAL SIMPLE RANDOM +SAMPLING AND PROPOSED WORTH-LEARNING DATA GENERATION +Cases +Conventional (min.) +Proposed (min.) +30-bus +27.9 +30.1 +57-bus +79.8 +85.5 +118-bus +174.1 +181.2 +two methods, we can conclude that the training time of the +proposed method only increases by 4%–8%. Hence, these +experiments validate that the proposed worth-learning data +generation is computationally efficient. +B. Reliability and optimality of the proposed solver +To validate the superiority of the proposed NN OPF solver +(denoted by Proposed NN), we compare it with two bench- +marks: 1) B1 NN, which adopts the conventional loss function +and NN model (MLP) with a training dataset generated +by simple random sampling; 2) B2 NN, which adopts the +proposed loss function and physical-model-integrated model +with a training dataset generated by simple random sampling. +A particular test set different from the training datasets +above is created to examine the effect of these models fairly. +The test set has 600 samples that are produced by uniformly +sampling 200 points in [80%, 120%] of the nominal value +of one load three times. The other loads in the three times +are fixed at light (80% × nominal value), nominal (100% × +nominal value), and heavy (120%×nominal value) load con- +ditions. The load sampled has the largest nominal value to +cover a big region of the input space. Based on these settings, +the test set includes much “unseen” data for those models. +The reliability of the NN OPF solvers is evaluated by the +constraint violation degrees on all test data. The optimality loss +is evaluated by the relative error between predicted results and +label values. For a fair comparison, the three methods all stop +their training processes when the value of || ˆV − V NN||1 is +less than 2 × 10−4. In view of the iterative training process, +the performance of the three solvers is studied with increasing +training data, and the initial NNs are identical because they +are trained on an initial dataset with N samples. +The results are statistically analyzed by creating box plots +displayed in Fig. 10. The violation degrees and optimality +losses of the results of the NNs from the three methods con- +verge to the terminal stages gradually. The rate of convergence +of Proposed NN is the largest, that of B2 NN is in the middle, +and that of B1 NN is the smallest. +In Figs. 10(a) to 10(c), the comparison of the last violation +degree gives notable results in the three cases. Specifically, +the median values in three cases are 7, 15, and 75 for B1 +NN; 6, 12.5, and 60 for B2 NN; and 3.2, 6.1, and 25 for +Proposed NN, respectively. The novel loss function brings a +19% reduction of violation degree on average by comparing +B1 NN and B2 NN. The proposed training data generation +method introduces a 50% reduction of violation degree on +average according to the comparison of B2 NN and Proposed +NN. Moreover, the height of the last boxes in each subfigure +suggests the robustness of the three solvers, and Proposed NN + +10000 +8000 +6000 +4000 +Data at 1st iteration +2000 +Data at 2nd iteration +Data at 3rd iteration +10 +20 +30 +40 +0 +The number of epoches （/1008 +(a) +(b) +(c) +(d) +(e) +(f) +Proposed NN +B1 NN +B2 NN +Fig. 10. The violation degree and optimality loss of the results of the NNs +trained by three methods change with the number of training data in different +cases: (a), (d) IEEE 30-bus; (b), (e) IEEE 57-bus; (c), (f) IEEE 118-bus. +has the smallest height in all three cases, which indicates the +worth-learning data generation can improve the reliability in +encountering “unseen” data from the feasible region. +The comparison of optimality losses is similar to that of +violation degrees, as illustrated in Figs. 10(d) to 10(f). The +proposed NN method has the best results in the three cases, +and the final median values of optimality losses are 0.6%, +0.5%, and 0.3% in the three different cases, respectively. The +optimality losses of B2 NN and B1 NN increase by 150%, +66%, and 360% and 142%, 167%, and 460% compared to +those of the proposed NN method in the three cases. +In conclusion, the proposed physical-model-integrated NN +OPF solver with worth-learning data generation can improve +the generalization of NN models compared to the conventional +NN solvers. Specifically, the proposed method introduces an +over 50% reduction of constraint violations and optimality +losses in the results on average. +C. Comparison with numerical solvers +To further evaluate the capability of the proposed method, +the next experiment focuses on the comparison with the +results of the classical AC-OPF solver based on the prime- +dual interior points algorithm and the classical DC-OPF solver +with a linear approximation of the power flow equations. The +classical AC-OPF solver produces the optimal solutions as +the ground truth values, and the DC-OPF solver is a widely +used approximation in the power industry. The test set is the +same as that in Section IV-B. The performance of the three +methods is evaluated by the following metrics: 1) the average +consumption time to solve an OPF problem; 2) the average +constraint violation degree V iophm, which is calculated by +Eqs. (8) and (9) for the two numerical solvers; and 3) the +average relative error of dispatch costs. These three metrics are +denoted as Time (ms), Vio.(MW), and Opt.(%), respectively. +The results are tabulated in Table II. The bottom row of the +table shows the average results over the three cases. As shown, +the proposed method achieves high computational efficiency, +which is at least three orders of magnitude faster than the +DC-OPF solver and four orders of magnitude faster than the +AC-OPF solver. Furthermore, the method also has much lower +constraint violations and optimality losses compared with the +DC OPF solver. The average Vio. (MW) and Opt. (%) of the +proposed solver are only 10.882 and 0.462, which are 44% +and 18% of those of the DC-OPF solver, respectively. +D. Interpretation of worth-learning data generation +This subsection interprets why the worth-learning data gen- +erated by the proposed method improve the representativeness +of the training dataset. The proposed worth-learning data +generation method is compared with the conventional simple +random sampling method. Without loss of generality, the +experiment is conducted on the 14-bus system. Beginning +with an identical initial dataset, the conventional and proposed +methods generate 100 samples in every step, and there are 8 +steps for both. To visualize the representativeness, we draw the +distribution of these high-dimensional training samples based +on the t-distributed Stochastic Neighbor Embedding algorithm +[29], [30], which is a statistical method for visualizing high- +dimensional data by giving each data point a location in a two- +or three-dimensional map. +The reduced-dimensional data distributions of the conven- +tional and proposed methods are shown in Fig.11. In Fig. +11(a), the data are produced by the simple random sampling +method, and their distribution is almost in a “�” region, +which means the possibility of sampling in this region is high. +Furthermore, the new data added in each step overlap with +existing data or fill in the intervals. The new data overlapping +with existing data are redundant in terms of NN training. +The data filling in the intervals may be also redundant when +the blanks are generalized well by the trained NN model. In +contrast, as shown in Fig. 11(b), the new data generated by + +(MW) +X101 +8 +10 +12 +16 +2 +The number of the training data (320 + x X 64)(MW +12 +16 +The number of the training data (320 + x X 64)× 102 +(MW +Vio. +10 +12 +16 +The number of the training data (320 + x X 64)% +Optimality loss( +2 +8 +10 +12 +16 +3 +4 +6 +The number of the training data (320 + x X 64)% +Optimality loss( +2 +8 +10 +12 +3 +6 +16 +The number of the training data (320 + x X 64)（%） +2 +3 +4 +8 +10 +12 +16 +6 +The number of the training data (320 + x X 64)9 +TABLE II +PERFORMANCE COMPARISON OF NUMERICAL SOLVERS AND THE PROPOSED SOLVER +Test +cases +AC-OPF solver +DC-OPF solver +Proposed NN solver +Time (ms) +Vio. (MW) +Opt. (%) +Time (ms) +Vio. (MW) +Opt. (%) +Time (ms) +Vio. (MW) +Opt. (%) +30-bus +530.3 +0 +0 +14.8 +5.340 +0.908 +0.110 +4.415 +0.603 +57-bus +991.6 +0 +0 +36.2 +15.611 +1.758 +0.113 +7.226 +0.499 +118-bus +1606.7 +0 +0 +78.5 +52.199 +4.762 +0.116 +21.004 +0.285 +Avg. +1024.9 +0 +0 +129.5 +24.383 +2.476 +0.113 +10.882 +0.462 +(a) Training dataset generated by simple random sampling +(b) Training dataset generated by worth-learning data generation method +Fig. 11. Reduced-dimensional distributions of the training datasets generated +by two different methods. +the proposed method in each step hardly overlap with existing +data and are usually outside the region covered by the initial +data. These new data increase the area covered by the training +set so that the training set can have better representativeness +of the input feasible region. This explains the effectiveness of +the proposed worth-learning data generation method. +V. CONCLUSION +This study proposes an AC-OPF solver based on a physical- +model-integrated NN with worth-learning data generation to +produce reliable solutions efficiently. To the best of our knowl- +edge, this is the first study that has addressed the generalization +problem of NN OPF solvers regarding the representativeness +of training datasets. The physical-model-integrated NN is +designed by integrating an MLP and an OPF-model module. +This specific structure outputs not only the optimal decision +variables of the OPF problem but also the constraint violation +degree. Based on this NN, the worth-learning data generation +method can identify feasible training samples that are not well +generalized by the NN. Accordingly, by iteratively applying +this method and including the newly identified worth-learning +data samples in the training set, the representativeness of the +training set can be significantly enhanced. +The theoretical analysis shows that the method brings little +computational burden into the training process and can make +the models generalize over the feasible region. Experimen- +tal results show that the proposed method leads to over a +50% reduction of both constraint violations and optimality +loss compared to conventional NN solvers. Furthermore, the +computation speed of the proposed method is three orders of +magnitude faster than that of the DC-OPF solver. +APPENDIX A +COMPUTATIONAL EFFICIENCY OF WORTH-LEARNING +DATA GENERATION +To analyze the computational complexity of the proposed +NN model with worth-learning data generation, we adopt a +widely used measure—the number of floating-point operations +(FLOPs) during the NN model’s forward-backward propaga- +tion. The total FLOPs of one single layer of a fully-connected +NN model can be calculated as follows: +Forward : +FLOPs = (2I − 1) × O, +(14a) +Backward : +FLOPs = (2I − 1) × O, +(14b) +where I is the dimension of the layer’s input, and O is the +dimension of its output. +To approximate an OPF mapping based on a 57-bus system, +the proposed NN model uses the following structure: 84 × +1000×2560×2560×5120×2000×114. According to Eq. (14), +the total FLOPs of the NN per forward-backward process is +around 1×108. The GPU used in the experiment is the Quadro +P6000, and its performance is 12.2 TFLOP/s (1 TFLOP/s = +1012 FLOP/s). Using the GPU, we can perform the forward- +backward process 1.22 × 105 times per second. +For the worth-learning data generation in Algorithm 1, the +forward process is to calculate V ioifs and V iophm, and the +backward process is to update S′G +ifs and V ′ifs by the gradients. +We concatenate S′G +ifs and V ′ifs as a vector x, and we suppose +the range of each item in x is [0, 10], and x changes 10−3 +in each update step. Varying from 0 to 10, it costs 104 times +the forward-backward processes. In other words, the algorithm +can at least update 1.22 × 105/104 ≈ 12 samples in 1 s, so +finding one sample costs no longer than 0.08 s. +In practice, there is a slight error between the actual speed +in experiments and the theoretical analysis. According to the +numerical experiments in Section IV-A, an average of 533 +samples are found in 30 s. The average consumption time for +identifying one sample is 0.056 s. + +2nd step +Initial data +step +80 +60 +40 +20 +D +0 +-20 +-40 +-60 +-80 +-100 +8th step +6th +Overall data +step +80 +60 +40 +20 +D +0 +-20 +-40 +-60 +-80 +-100 +-50 +50 +100 +-50 +-100 +0 +-100 +50 +100 +-100 +-50 +50 +100 +0 +D2 +D2 +D2Initial data +2nd step +4th step +80 +60 +40 +20 +D +0 +-20 +-40 +-60 +-80 +-100 +6th step +8th step +Overall data +80 +60 +40 +20 +D +0 +-20 +-40 +-60 +-80 +-100 +-100 +-50 +0 +50 +100 +-100 +-50 +0 +50 +100 +-100 +-50 +0 +50 +100 +D2 +D2 +D2 +Initial data +Identified data in previous steps +New data10 +Fig. 12. +Illustration of the covered region Scover expanding its area by the +generalized region Sadd. +From the analysis presented above, we can conclude that the +proposed worth-learning data generation method brings little +computational burden into the training process. +APPENDIX B +CONVERGENCE OF WORTH-LEARNING DATA GENERATION +This section verifies that the proposed NN with worth- +learning data generation can generalize to the whole feasible +set. NN models are continuous functions because both linear +layers and activation functions are continuous. We define a +critical violation value ϵ that divides the input space into +two types: the covered region (the V iophm values of all of +the points are less or equal to ϵ) and the uncovered region +(the V iophm values of all of the points are greater than +ϵ). The boundaries of the two regions consist of the points +whose V iophm values are approximately equal to ϵ. Using +these points as initial points, we can identify points with the +local maximum in the uncovered region by max violation +backpropagation. +Next, these new points {x1} (the red points) are added to +the training set. After training, the neighborhood of these new +points {x1} would be covered. 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Commun., vol. 10, no. 1, pp. 1–12, 2019. + +C \ No newline at end of file diff --git a/9dE2T4oBgHgl3EQfQAa_/content/tmp_files/load_file.txt b/9dE2T4oBgHgl3EQfQAa_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a15d0a7beb2ae76f0dfe41d835e8c233861d926e --- /dev/null +++ b/9dE2T4oBgHgl3EQfQAa_/content/tmp_files/load_file.txt @@ -0,0 +1,920 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf,len=919 +page_content='1 Optimal Power Flow Based on Physical-Model-Integrated Neural Network with Worth-Learning Data Generation Zuntao Hu, Graduate Student Member, IEEE, and Hongcai Zhang, Member, IEEE Abstract—Fast and reliable solvers for optimal power flow (OPF) problems are attracting surging research interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As surrogates of physical-model-based OPF solvers, neural network (NN) solvers can accelerate the solving process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' However, they may be unreliable for “unseen” inputs when the training dataset is unrepresentative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Enhancing the representativeness of the training dataset for NN solvers is indispensable but is not well studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To tackle this challenge, we propose an OPF solver based on a physical-model-integrated NN with worth- learning data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The designed NN is a combination of a conventional multi-layer perceptron (MLP) and an OPF- model module, which outputs not only the optimal decision variables of the OPF problem but also the constraints violation degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Based on this NN, the worth-learning data generation method can identify feasible samples that are not well generalized by the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' By iteratively applying this method and including the newly identified worth-learning samples in the training set, the representativeness of the training set can be significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Therefore, the solution reliability of the NN solver can be remarkably improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Experimental results show that the proposed method leads to an over 50% reduction of constraint violations and optimality loss compared to conventional NN solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Index Terms—Optimal power flow, physical-model-integrated neural network, worth-learning data generation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' INTRODUCTION O PTIMAL power flow (OPF) is a fundamental but chal- lenging problem for power systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A typical OPF problem usually involves determining the optimal power dis- patch with an objective, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', minimizing total generation costs or power loss, while satisfying nonlinear power flow equations and other physical or engineering constraints [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Due to the nonlinear interrelation of nodal power injections and voltages, OPF is non-convex, NP-hard, and cannot be solved efficiently [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' With the increasing integration of renewable generation and flexible demands, uncertainty and volatility have been rising on both the demand and supply sides of modern power systems [4], which requires OPF to be solved more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Thus, fast and reliable OPF solvers have become indispensable to ensure effective operations of modern power systems and have attracted surging interest in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' There is a dilemma between the solving efficiency and solution reliability of OPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Conventionally, OPF is solved by iterative algorithms, such as interior point algorithms, based on explicit physical models [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' However, these methods may converge to locally optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Recently, some researchers have made great progress in designing conic relaxation models for OPF, which are convex and can be efficiently solved [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Nevertheless, the exactness of these relaxations may not hold in practical scenarios, and they may obtain infeasible solutions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In addition, the scalability of the conic relaxation of alternating current optimal power flow (AC-OPF) may still be a challenge, particularly in online, combinatorial, and stochastic settings [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To overcome the limitation of the aforementioned physical- model-based solvers, some researchers propose surrogate OPF solvers based on neural networks (NNs) [11]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' These solvers use NNs to approximate the functional mapping from the operational parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', profiles of renewable gen- eration and power demands) to the decision variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', power dispatch) of OPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Compared to iterative algorithms, they can introduce significant speedup because an NN is only composed of simple fundamental functions in sequence [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' However, one of the critical problems of NN solvers is that they may be unreliable if not properly trained, especially for “unseen” inputs in feasible regions due to NNs’ mystery generalization mechanism [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The generalization of NNs is mainly influenced by their structures, loss functions, and training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Most published papers propose to enhance the generalization of NN OPF solvers by adjusting the structures and loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Various advanced NN structures rather than conventional fully con- nected networks are employed to imitate AC-OPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For exam- ple, Owerko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [15] use graph NNs to approximate a given optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [16] employ a deep belief network to fit the generator’s power in OPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [17] construct a convex NN solving DC-OPF to guarantee the generalization of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Jeyaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [18] employ a Bayesian regularized deep NN to solve the OPF in DC microgrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Some researchers design elaborate loss functions that penalize the constraints violation, combine Karush-Kuhn-Tucker conditions, or include derivatives of decision variables to operational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For example, Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [11] introduce a penalty term related to the inequality constraints into the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This approach can speed up the computation by up to two orders of magnitude compared to the Gurobi solver, but 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='3% of its solutions are infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Ferdinando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [12] include a Lagrange item in the loss function of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Their method’s prediction errors are as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2%, and its solving speed is faster than DC- OPF by at least two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Manish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [10] include sensitivity information in the training of NN so that only using about 10% to 25% of training data can attain the same approximation accuracy as methods without sensitivity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Nellikkath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [19] apply physics-informed arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='03766v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='LG] 10 Jan 2023 2 NNs to OPF problems, and their results have higher accuracy than conventional NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The above-mentioned studies have made significant progress in designing elaborate network structures and loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' However, little attention has been paid to the training set generation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Specifically, they all adopt conventional probability sampling methods to produce datasets for training and testing, such as simple random sampling [10]–[13], [15], [17], [20], Monte Carlo simulation [18], or Latin hypercube sampling [16], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' These probability sampling methods can- not provide a theoretical guarantee that a generated training set can represent the input space of the OPF problem prop- erly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As a result, probability sampling methods may generate insufficient and unrepresentative training sets, so the trained NN solvers may provide unreliable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' It is important to create a sufficiently representative dataset for training an NN OPF solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A training set’s representative- ness depends on its size and distribution in its feasible region [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Taking a medium-scale OPF problem as an example, millions of data samples may still be sparse given the high dimension of the NN’s inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', operational parameters of the OPF problem: renewable generation and power demands at all buses);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' in addition, because the OPF problem is non- convex, the feasible region of the NN’s inputs is a complicated irregular space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Thus, generating a representative training set to cover all the feasible regions of the inputs with an acceptable size is quite challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Without a representative training set, it is difficult to guarantee that the NN OPF solver’s outputs are reliable, especially given “unseen” inputs in the inference process, as discussed in [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To address the above challenge, this study proposes a physical-model-integrated deep NN method with worth- learning data generation to solve AC-OPF problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To the best of our knowledge, this is the first study that has addressed the representativeness problem of the training dataset for NN OPF solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The major contributions of this study are twofold: 1) A novel physical-model-integrated NN is designed for solving the AC-OPF problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This NN is constructed by a conventional MLP integrating an OPF-model module, which outputs not only the optimal decision variables of the OPF problem but also the violation degree of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' By penalizing the latter in the loss function during training, the NN can generate more reliable decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2) Based on the designed NN, a novel generation method for worth-learning training data is proposed, which can identify samples in the input feasible region that are not well generalized by the previous NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' By iteratively applying this method during the training process, the trained NN gradually generalizes to the whole feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As a result, the generalization and reliability of the proposed NN solver can be significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Furthermore, comprehensive numerical experiments are con- ducted, which prove that the proposed method is effective in terms of both reliability and optimality for solving AC-OPF problems with high computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The remainder of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Section II provides preliminary models and the motivations behind this Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The 3-bus system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Section III introduces the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Section IV details the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Section V concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' ANALYSIS OF APPROXIMATING OPF PROBLEMS BY NN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' AC-OPF problem The AC-OPF problem aims to determine the optimal power dispatch (usually for generators) given specific operating con- ditions of a power system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', power loads and renewable generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A typical AC-OPF model can be formulated as min V , SG C � SG� (1a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' : [V ] Y∗ busV ∗ = SG − SL, (1b) SG ≤ SG ≤ S G, (1c) V ≤ V ≤ V, (1d) |YbV | ≤ ¯I, (1e) where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (1a) is the objective, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', minimizing total genera- tion costs, and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (1b) to (1e) denote constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbols SG and SL are n × 1 vectors representing complex bus injections from generators and loads, respectively, where n is the number of buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbol V is an n×1 vector denoting node voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbol [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='] denotes an operator that transforms a vector into a diagonal matrix with the vector elements on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbol Ybus is a complex n × n bus admittance matrix written as Y at other sections for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbol Yb is a complex nb × n branch admittance matrix, and nb is the number of branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The upper and lower bounds of any variable x are represented by ¯x and x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Vector ¯I denotes the current flow limit of branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' AC-OPF mapping from loads to optimal dispatch An NN model describes an input-output mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Specifi- cally, for an NN model solving the AC-OPF problem shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (1), the input is the power demand SL, and the output is the optimal generation SG *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Hence, an NN OPF solver describes the mapping SG * = f OPF(SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A well-trained NN should be able to accurately approximate this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' We provide a basic example of a 3-bus system, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1, to illustrate how NN works for OPF problems and explain the corresponding challenge for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For simplicity, we assume there is no reactive power in the system and set r31 = r12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='01 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' P i = 0 , P i = 4, for i ∈ {1, 3}, P2 ∈ [−7, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' V i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='95 and V i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='05, for i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Load P2E[-7,0]3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Examples of an NN fitting the OPF of the 3-bus system based on (a) simple random sampling, and (b) worth-learning data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Then, the OPF model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (1) is reduced to the following quadratic programming: min V, PG P1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='5 × P3 (2a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' : P1 = V1 (V1 − V2) /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='01 + V1 (V1 − V3) /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='01, (2b) P2 = V2 (V2 − V1) /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='01, (2c) P3 = V3 (V3 − V1) /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='01, (2d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='95 ≤ V3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='95 ≤ V2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='05, (2e) 0 ≤ P1 ≤ 4, 0 ≤ P3 ≤ 4, V1 = 1, (2f) where V is [V1 V2 V3]⊤, and P G is [P1 P3]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Given that P2 ranges from -7 to 0, the 3-bus OPF model can be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The closed-form solution of [P ∗ 1 P ∗ 3 ] = f OPF 3-bus(P2), is formulated as follows: P ∗ 1 = � 50 − 50√0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='04P2 + 1, c1, 4, c2, (3) P ∗ 3 = � � � 0, c1, 213 � 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='34√0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='04P2 + 1 �2 +50√0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='04P2 + 1 − 146 , c2, (4) where c1 denotes condition 1: −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='84 ≤ P2 ≤ 0, and c2 denotes condition 2: −7 ≤ P2 < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To further analyze the mapping f OPF 3-bus, we draw the [P ∗ 1 P ∗ 3 ]–P2 curve according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (3) and (4), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Both the P ∗ 1 –P2 and P ∗ 3 –P2 curves are piecewise nonlinear functions, in which two oblique lines are nonlinear because of the quadratic equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The reason why the two curves above are piecewise is that the active inequalities change the [P ∗ 1 P ∗ 3 ]–P2 relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' From an optimization perspective, each active inequality will add a unique equality constraint to the relationship, so the pieces in f OPF 3-bus are determined by the sets of active inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In this example, the two pieces in each curve correspond to two sets of active inequalities: P1 ≤ 4 and 0 ≤ P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Moreover, the two intersection points are the critical points where these inequalities are just satisfied as equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For a general AC-OPF problem, its input is usually high- dimensional (commonly determined by the number of buses), and its feasible space is partitioned into some distinct regions by different sets of active inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' From an optimization perspective, a set of active constraints uniquely characterizes the relationship SG * = f OPF(SL), and the number of pieces theoretically increases with the number of inequality constraints by exponential order [24]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' There- fore, there are massive regions, and each region corresponds to a unique mapping relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', a piece of mapping function f OPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Challenges of fitting OPF mapping by NN As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2(a), to fit the two-dimensional piecewise nonlinear curve of f OPF 3-bus, we first adopt four data samples by simple random sampling and then use an NN to learn the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Obviously, there are significant fitting errors between the fitting and the original lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Because the training set lacks the samples near the intersections in the curve (where p2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='384 in this case), the NN cannot accurately approximate the mapping in the neighboring region of the intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A training set representing the whole input space is a prereq- uisite for an NN approximating the curve properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' However, it is nontrivial to generate a representative training set by probability sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2(a), the intersections of f OPF are key points for the representativeness, and the number of intersections increases exponentially with that of the inequality constraints, as analyzed in II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' When each sample is selected with a small possibility ρ, the generation of a dataset containing all the intersection points are in a low possibility event whose probability is equal to ρm, where m is the number of intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In practice, the only way to collect sufficient data representing the input space by probability sampling is to expand the dataset as much as possible [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This is impractical for large power networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Therefore, the conventional probability sampling in the literature can hardly produce a representative dataset with a moderate size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2(b), if we are able to identify the two intersections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', (P2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='384, P1 = 4) and (P2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='384, P3 = 0), and include them as new samples in the training dataset, the corresponding large fitting errors of the NN can be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' These samples are termed as the worth- learning data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The focus of this study is to propose a worth-learning data generation method that can help identify worth-learning data samples and overcome the aforementioned disadvantage of conventional probability sampling (detailed in the following section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A PHYSICAL-MODEL-INTEGRATED NN WITH WORTH-LEARNING DATA GENERATION This section proposes a physical-model-integrated NN with a worth-learning data generation method to solve AC-OPF problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The proposed NN is a combination of a fully- connected network and a transformed OPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' It outputs not only the optimal decision variables of the OPF problem but also the violation degree of constraints, which provides guid- ance for identifying worth-learning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The worth-learning data generation method creates representative training sets to enhance the generalization of the NN solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' P* Data from simple random sampling Fitting line Fitting error Worth-learning data4 START Initialize a training set by randomly sampling Train the NN on the current training set Identify worth-learning data for the current NN Worth-learning data are identified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Output the current NN END Y N Add identified data to the training set Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Framework of the proposed training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Framework of the proposed method The proposed data generation method has an iterative pro- cess, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' First, a training set is initialized by random sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' second, the physical-model-integrated NN is trained on the training set, where an elaborate loss function is utilized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' third, worth-learning data for the current NN are identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' fourth, if worth-learning data are identified, these data are added to the training set and returns to the second step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' otherwise, the current NN is output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The above training process converges until no worth- learning data are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This means that the training set is sufficiently representative of the input space of the OPF problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As a result, the NN trained based on this dataset can generalize to the input feasible set well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The following subsections introduce the proposed method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Physical-model-integrated NN In the second step of the proposed method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 3), the NN is trained to fit the mapping SG * = f OPF(SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To obtain better results, we design a physical-model-integrated NN structure consisting of a conventional NN module and a physical-model module, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The former is a conventional MLP, while the latter is a computational graph transformed from the OPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1) Conventional NN module: This module first adopts a conventional MLP with learnable parameters to fit the mapping from the SL to the optimal decision variable V NN [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The V NN has its box constraint defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To ensure that the output V NN satisfies this constraint, we design a function dRe() to adjust any infeasible output V NN into its feasible region, which is formulated as follows: x ← dRe(x, x, x) = ReLU(x − x) − ReLU(x − x) + x, (5) where ReLU(x) = max(x, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' x is the input of the function, and its lower and upper bounds are x and x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The diagram of this function is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Input … … Physical model module Conventional NN module Output Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The physical-model-integrated NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Applying dRe() as the activation function of the last layer of the conventional MLP, the mathematical model of this conventional NN module is formulated as follows: V NN = MLP(SL), (6) V NN ← dRe(V NN, V , V ), (7) where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (6) describes the conventional model of MLP and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (7) adjusts the output of the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2) Physical model module: This module receives V NN from the previous module, and then it outputs the optimal power generation SG phm and the corresponding constraints violation V iophm, where the subscript “phm” denotes the physical model module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The first output SG phm is the optimal decision variable of the AC-OPF problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' It can be calculated by V NN and SL, as follows: SG phm = [V NN]Y∗V ∗ NN + SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (8) The second output V iophm (termed as violation degree) measures the quality of SG phm and is the key metric to guide the proposed worth-learning data generation (see details in the following subsection III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Given V NN and SG phm, the violations of inequality constraints of the AC-OPF problem V iophm are calculated as follows: V ioS phm = ReLU(SG phm − S G) + ReLU(SG − SG phm), (9a) V ioI phm = ReLU(|YfV NN| − ¯I), (9b) V iophm = (V ioS phm V ioI phm)⊤, (9c) where V ioS phm denotes the violation of the upper or lower limit of SG phm, and V ioI phm represents the violation of branch currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The physical-model-integrated NN is formed by combining the conventional NN module and the physical model module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' It inputs SL and outputs SG phm and V iophm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Its function is the same as conventional OPF numerical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In addition, it is convenient for users Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The dRe() function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 5 Feasible region Label value Region with tiny predicted error Predicted value Effect of the proposed loss function Point 1 Point 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Illustration of the effectiveness of the three terms in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' to directly determine whether the result of the NN OPF solver is acceptable or not based on the violation degree V iophm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In contrast, most NN OPF solvers in the literature are incapable of outputting the violation degree directly [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 3) Loss function: To enhance the training accuracy of the physical-model-integrated NN, we design an elaborate loss function, which consists of V NN from the conventional NN module, and SG phm and V iophm from the physical model module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The formula is as follows: loss = || ˆV − V NN||1 + || ˆ SG − SG phm||1 + V iophm, (10) where ˆV and ˆ SG are label values from the training set, which is a ground truth dataset from numerical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Combining the three terms in the loss function can help en- hance fitting precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 6, if the loss function only has the first two items || ˆV − V NN||1+ || ˆ SG − SG phm||1 to penalize conventional fitting errors, the predicted value will be in a tiny square space (the red square in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 6) around the label value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' From the optimization perspective, the optimal label value is usually on the edge of its feasible region (the blue polyhedron in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This edge through the label value splits the square into two parts: the feasible (blue) part and the infeasible (white) part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Intuitively, we would prefer the predicted values to be in the feasible part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Thus, we also penalize violation degree V iophm in the loss function to force the predicted values with big V iophm close to the square’s feasible half space for smaller constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Although the proposed NN with elaborate loss function has high training accuracy, it is still difficult to guarantee the gen- eralization of the NN OPF solver to the whole input space with conventional random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Therefore, it is indispensable and challenging to obtain a representative training dataset with moderate size to train the proposed NN, which is the focus of the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Worth-learning data generation As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 3, we adopt an iterative process to identify the worth-learning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For an NN trained in the previous iteration, we utilize its output V iophm to help identify new data samples that are not yet properly generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Specifically, if an input SL* is feasible for the original OPF problem while the current NN outputs a large violation degree V io∗ phm, the contradiction means the NN has a large fitting error at SL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Input feasible set module Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The input feasible set module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This is probably because sample SL* was not included in the previous training set and was not generalized by the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Hence, this sample SL* can be regarded as a worth-learning sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Including the sample in the training dataset in the next iteration helps enhance the generalization of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The key to the proposed worth-learning data generation method is to identify worth-learning samples efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In- stead of traversing all of the possible inputs, we maximize V iophm for a given NN to identify the input with a large vio- lation degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' However, the inputs identified in the maximizing process should be feasible for the original OPF problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Otherwise, the found inputs might be infeasible and useless for the representation of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1) Input feasible set module: To keep the inputs identified in the maximizing process feasible for the original OPF problem, we formulate the input feasible set module to restrict power loads SL to their feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The feasible set is composed of box constraints, current limits, and KCL&KVL constraints, which are transformed from the feasible set of the OPF problem defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The partial formulations of the input feasible set are as follows, where the subscript “ifs” denotes the input feasible set module: SG ifs = dRe � S′G ifs, SG, S G� , S′G ifs ∈ Rn, (11a) V ifs = dRe � V ′ ifs, V , V � , V ′ ifs ∈ Rn, (11b) SL ifs = SG ifs − [V ifs]Y∗V ∗ ifs, (11c) Iifs = YbV ifs, (11d) where S′G ifs and V ′ifs are auxiliary n × 1 vectors in Rn and have no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbols SG ifs and V ifs are restricted in their box constraints in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (11a) and (11b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Then the KCL&KVL correlations of SL ifs, SG ifs, and V ifs are described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (11c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Symbol Iifs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (11d) denotes the currents at all branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The other formulations of the input feasible set aim to calcu- late V ioifs, the AC-OPF’s constraint violations corresponding to SL ifs and Iifs, as follows: V ioS ifs = ReLU(SL ifs − S L) + ReLU(SL − SL ifs), (12a) V ioI ifs = ReLU(|Iifs| − I), (12b) V ioifs = (V ioS ifs V ioI ifs)⊤, (12c) where V ioS ifs denotes the violation of the upper or lower limit of SL phm, and V ioI ifs denotes the violation of branch current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This module takes S′G ifs and V ′ifs as the inputs, and then outputs SL ifs and V ioifs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' When V ioifs = 0, the corresponding SL ifs lies in the feasible set of 6 Conventional NN module Physical model module Input feasible set module Updated variables Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The novel NN for max violation backpropagation by integrating physical-model-integrated NN with the input feasible set module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' the AC-OPF problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To identify feasible SL ifs in the process of maximizing V iophm, this module backpropagate the ∂V iophm ∂SL ifs with V ioifs ≤ ζ (ζ is a small positive tolerance), and then it updates S′G ifs and V ′ifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As a result, the corresponding SL ifs is always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Furthermore, because S′G ifs and V ′ifs are not bounded, changing them can theoretically find any feasible SL ifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2) Max violation backpropagation: To identify worth- learning data, a novel NN is created by inputting SL ifs into the physical-model-integrated NN (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This NN has two outputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', V iophm and V ioifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The former measures the constraint violation degree of the OPF solution SG*;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' the latter indicates the feasibility of the OPF input SL ifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' If SL ifs is a feasible input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', V ioifs ≤ ζ, but the optimal solution SG* is infeasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', V iophm ≥ ξ (ξ is a threshold), this means the corresponding input is worth learning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=', it is not learned or generalized by the current NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Based on this analysis, we design the loss function lossmax for max violation backpropagation, as follows: lossmax = V iophm − λ × V ioifs, (13) where λ is a large, constant weight parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' When maxi- mizing this loss function, the algorithm tends to find a worth- learning SL ifs that has small V ioifs but large V iophm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' During the max violation backpropagation, the proposed algorithm maximizes lossmax to update the variables S′G ifs and V ′ifs by gradient backpropagation until lossmax converges to the local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' After the process, the corresponding SL ifs is also found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Because the maximizing process can be processed in parallel by the deep learning module PyTorch, the worth-learning samples are found in batch, where the max violation backpropagation uses the previous training set as initial points to identify the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Further, the auto- differentiation technique in PyTorch can accelerate the process of parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Based on these techniques, massive worth-learning data samples are identified efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Overall training process The overall training process is presented in Algorithm 1, which first takes an initial training dataset Dt (obtained by any conventional sampling method) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The learning rate η is equal to 10−3, the loss difference tolerance ϵ is equal to 10−2, the added dataset A is empty, and the loss difference ∆L is equal to infinity at initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The training is performed for a fixed number of epochs (lines 2–5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Then the max violation backpropagation starts to identify worth-learning data (lines 6 and 7) by using the training data as the initial points (line 8) and updating S′G ifs and V ′ifs until ∆L is less than ϵ (lines 9–12), which indicates lossmax has converged to the terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Algorithm 1 Training process of the physical-model- integrated NN OPF solver with worth-learning data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Input: Dt = � ˆ SL, ˆV , ˆ SG� Initialization : η ← 10−3, ϵ ← 10−2, A ← ∅, ∆L ← ∞ 1: repeat 2: for epoch k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' do 3: Train the NN with loss Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (10): 4: w ← w − η∇loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 5: end for 6: while ∆L ≥ ϵ do 7: Identify data with lossmax Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (13): 8: S′G ifs, V ′ ifs ← SG ifs, V ifs ← ˆ SG, ˆV 9: S′G ifs ← S′G ifs + η∇lossmax 10: V ′ ifs ← V ′ ifs + η∇lossmax 11: ∆L ← | lossmax,i − lossmax,i−100 | 12: end while 13: {V iophm,N} ← ffilter(V iophm,N ≥ ξ) 14: Collect {SL ifs} corresponding to {V iophm,N} based on the novel NN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 8 15: Calculate { ˆV , ˆ SG} corresponding to {SG ifs} using numerical solvers 16: A ← {SL ifs, ˆV , ˆ SG} 17: Dt ← Dt ∪ A 18: until A is ∅ After the max violation backpropagation, a series of com- mands are designed to add proper data to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' First, a filter function ffilter is employed to eliminate data with terminal violation V iophm,N less than a given threshold ξ (the value depends on the acceptable violation settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Second, { ˆV , ˆ SG} is calculated by numerical solvers corresponding to SL ifs with large violation degree (lines 14 and 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' They consist of added set A (line 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Third, the training set Dt is expanded with A (line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The loop is repeated until the added set A is empty (line 18), meaning no worth-learning data are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Efficiency and convergence of the proposed method Unlike general training processes for conventional NNs, the proposed physical-model-integrated NN with worth-learning data generation adopts an iterative training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' It iter- atively checks the NN’s generalization to the input’s feasi- ble space by identifying worth-learning data, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 3 and Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This difference introduces two critical questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1) Efficiency: is the process of identifying worth- learning data computationally efficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2) Convergence: is the training set representative of the whole input space after iterations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In terms of the computational efficiency of the proposed method, the theoretical analysis (detailed in the Appendix A) shows it takes no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='08 s to find one sample, which brings little computational burden into the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' According to the experiment results, the average consumption time for finding one sample is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='056 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In terms of the convergence, we prove that the training set would gradually represent the whole input space in the Appendix B, because the number of worth-learning samples identified would converge to zero after a finite number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 7 The number of times the sequence codes are repeated in the data generation loop (/100) The violation degree (MW) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Time consumption of the worth-learning data codes in three different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The number of times the sequence codes are repeated in the data generation loop (x-axis) represents the time consumed in one data generation loop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' the violation degrees (y-axis) quickly converge to the terminal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' NUMERICAL EXPERIMENTS The proposed method is evaluated using the IEEE 12-bus, 14-bus, 30-bus, 57-bus, and 118-bus systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The ground truth datasets are constructed using PANDAPOWER based on a prime-dual interior points algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The efficiency of worth-learning data generation As shown in Algorithm 1, the proposed worth-learning data generation (lines 6–12) is the second loop in one iteration (lines 1–18), and the number of initial data points for the generation varies with iterations (lines 8, 15–17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To evalu- ate the efficiency of the worth-learning data generation, we conduct an experiment on the IEEE 57-bus system in three different iterations to quantitatively measure how much time it takes to finish one worth-learning data generation loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The time consumption of the data-generation loops in the three different iterations is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The x-axis is the number of times the codes are repeated (lines 6–12) divided by 100, which represents the time consumed in one data generation loop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' the y-axis is the violation degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The three lines converge to the terminal stage within 4000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The trends are similar: they increase very quickly at first (with 100 epochs) and then approach the local maximum slowly (with 2900–3900 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The inflection points on the three lines are (1, 7228), (1, 9065), and (1, 5841).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In the three iterations, 300, 500, and 800 new data samples are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Each data-generation loop in iterations takes 30 s on average to run 3000–4000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Hence, one worth- learning data sample costs (30×3)/(300+500+800) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='056 s, which introduces little computational burden into the train- ing process compared to the other steps in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For ex- ample, each label value calculated by numerical solvers costs around 1 s (line 14), and the NN training on a dataset with 1100 samples costs around 600 s (lines 2–5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In conclusion, the numerical experiment verifies that the worth-learning data generation brings little computational burden to the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Furthermore, we list the time consumption comparison of the conventional and proposed training processes in Table I, where the conventional training process uses simple random sampling in place of the data generation loop (lines 6–12) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' By comparing the time consumption of the TABLE I TRAINING TIME BASED ON THE CONVENTIONAL SIMPLE RANDOM SAMPLING AND PROPOSED WORTH-LEARNING DATA GENERATION Cases Conventional (min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=') Proposed (min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=') 30-bus 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='1 57-bus 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='5 118-bus 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='1 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2 two methods, we can conclude that the training time of the proposed method only increases by 4%–8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Hence, these experiments validate that the proposed worth-learning data generation is computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Reliability and optimality of the proposed solver To validate the superiority of the proposed NN OPF solver (denoted by Proposed NN), we compare it with two bench- marks: 1) B1 NN, which adopts the conventional loss function and NN model (MLP) with a training dataset generated by simple random sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2) B2 NN, which adopts the proposed loss function and physical-model-integrated model with a training dataset generated by simple random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A particular test set different from the training datasets above is created to examine the effect of these models fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The test set has 600 samples that are produced by uniformly sampling 200 points in [80%, 120%] of the nominal value of one load three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The other loads in the three times are fixed at light (80% × nominal value), nominal (100% × nominal value), and heavy (120%×nominal value) load con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The load sampled has the largest nominal value to cover a big region of the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Based on these settings, the test set includes much “unseen” data for those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The reliability of the NN OPF solvers is evaluated by the constraint violation degrees on all test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The optimality loss is evaluated by the relative error between predicted results and label values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For a fair comparison, the three methods all stop their training processes when the value of || ˆV − V NN||1 is less than 2 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In view of the iterative training process, the performance of the three solvers is studied with increasing training data, and the initial NNs are identical because they are trained on an initial dataset with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The results are statistically analyzed by creating box plots displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The violation degrees and optimality losses of the results of the NNs from the three methods con- verge to the terminal stages gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The rate of convergence of Proposed NN is the largest, that of B2 NN is in the middle, and that of B1 NN is the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 10(a) to 10(c), the comparison of the last violation degree gives notable results in the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Specifically, the median values in three cases are 7, 15, and 75 for B1 NN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 6, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='5, and 60 for B2 NN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='1, and 25 for Proposed NN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The novel loss function brings a 19% reduction of violation degree on average by comparing B1 NN and B2 NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The proposed training data generation method introduces a 50% reduction of violation degree on average according to the comparison of B2 NN and Proposed NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Moreover, the height of the last boxes in each subfigure suggests the robustness of the three solvers, and Proposed NN 10000 8000 6000 4000 Data at 1st iteration 2000 Data at 2nd iteration Data at 3rd iteration 10 20 30 40 0 The number of epoches （/1008 (a) (b) (c) (d) (e) (f) Proposed NN B1 NN B2 NN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The violation degree and optimality loss of the results of the NNs trained by three methods change with the number of training data in different cases: (a), (d) IEEE 30-bus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (b), (e) IEEE 57-bus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (c), (f) IEEE 118-bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' has the smallest height in all three cases, which indicates the worth-learning data generation can improve the reliability in encountering “unseen” data from the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The comparison of optimality losses is similar to that of violation degrees, as illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 10(d) to 10(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The proposed NN method has the best results in the three cases, and the final median values of optimality losses are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='6%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='5%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='3% in the three different cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The optimality losses of B2 NN and B1 NN increase by 150%, 66%, and 360% and 142%, 167%, and 460% compared to those of the proposed NN method in the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In conclusion, the proposed physical-model-integrated NN OPF solver with worth-learning data generation can improve the generalization of NN models compared to the conventional NN solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Specifically, the proposed method introduces an over 50% reduction of constraint violations and optimality losses in the results on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Comparison with numerical solvers To further evaluate the capability of the proposed method, the next experiment focuses on the comparison with the results of the classical AC-OPF solver based on the prime- dual interior points algorithm and the classical DC-OPF solver with a linear approximation of the power flow equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The classical AC-OPF solver produces the optimal solutions as the ground truth values, and the DC-OPF solver is a widely used approximation in the power industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The test set is the same as that in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The performance of the three methods is evaluated by the following metrics: 1) the average consumption time to solve an OPF problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 2) the average constraint violation degree V iophm, which is calculated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (8) and (9) for the two numerical solvers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' and 3) the average relative error of dispatch costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' These three metrics are denoted as Time (ms), Vio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (MW), and Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (%), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The results are tabulated in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The bottom row of the table shows the average results over the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' As shown, the proposed method achieves high computational efficiency, which is at least three orders of magnitude faster than the DC-OPF solver and four orders of magnitude faster than the AC-OPF solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Furthermore, the method also has much lower constraint violations and optimality losses compared with the DC OPF solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The average Vio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (MW) and Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (%) of the proposed solver are only 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='882 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='462, which are 44% and 18% of those of the DC-OPF solver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Interpretation of worth-learning data generation This subsection interprets why the worth-learning data gen- erated by the proposed method improve the representativeness of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The proposed worth-learning data generation method is compared with the conventional simple random sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Without loss of generality, the experiment is conducted on the 14-bus system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Beginning with an identical initial dataset, the conventional and proposed methods generate 100 samples in every step, and there are 8 steps for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To visualize the representativeness, we draw the distribution of these high-dimensional training samples based on the t-distributed Stochastic Neighbor Embedding algorithm [29], [30], which is a statistical method for visualizing high- dimensional data by giving each data point a location in a two- or three-dimensional map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The reduced-dimensional data distributions of the conven- tional and proposed methods are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 11(a), the data are produced by the simple random sampling method, and their distribution is almost in a “�” region, which means the possibility of sampling in this region is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Furthermore, the new data added in each step overlap with existing data or fill in the intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The new data overlapping with existing data are redundant in terms of NN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The data filling in the intervals may be also redundant when the blanks are generalized well by the trained NN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In contrast, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 11(b), the new data generated by (MW) X101 8 10 12 16 2 The number of the training data (320 + x X 64)(MW 12 16 The number of the training data (320 + x X 64)× 102 (MW Vio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 10 12 16 The number of the training data (320 + x X 64)% Optimality loss( 2 8 10 12 16 3 4 6 The number of the training data (320 + x X 64)% Optimality loss( 2 8 10 12 3 6 16 The number of the training data (320 + x X 64)（%） 2 3 4 8 10 12 16 6 The number of the training data (320 + x X 64)9 TABLE II PERFORMANCE COMPARISON OF NUMERICAL SOLVERS AND THE PROPOSED SOLVER Test cases AC-OPF solver DC-OPF solver Proposed NN solver Time (ms) Vio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (MW) Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (%) Time (ms) Vio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (MW) Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (%) Time (ms) Vio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (MW) Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (%) 30-bus 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='3 0 0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='110 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='603 57-bus 991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='6 0 0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='611 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='113 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='499 118-bus 1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='7 0 0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='199 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='116 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='285 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='9 0 0 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='383 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='113 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='462 (a) Training dataset generated by simple random sampling (b) Training dataset generated by worth-learning data generation method Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Reduced-dimensional distributions of the training datasets generated by two different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' the proposed method in each step hardly overlap with existing data and are usually outside the region covered by the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' These new data increase the area covered by the training set so that the training set can have better representativeness of the input feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This explains the effectiveness of the proposed worth-learning data generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' CONCLUSION This study proposes an AC-OPF solver based on a physical- model-integrated NN with worth-learning data generation to produce reliable solutions efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To the best of our knowl- edge, this is the first study that has addressed the generalization problem of NN OPF solvers regarding the representativeness of training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The physical-model-integrated NN is designed by integrating an MLP and an OPF-model module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' This specific structure outputs not only the optimal decision variables of the OPF problem but also the constraint violation degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Based on this NN, the worth-learning data generation method can identify feasible training samples that are not well generalized by the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Accordingly, by iteratively applying this method and including the newly identified worth-learning data samples in the training set, the representativeness of the training set can be significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The theoretical analysis shows that the method brings little computational burden into the training process and can make the models generalize over the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Experimen- tal results show that the proposed method leads to over a 50% reduction of both constraint violations and optimality loss compared to conventional NN solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Furthermore, the computation speed of the proposed method is three orders of magnitude faster than that of the DC-OPF solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' APPENDIX A COMPUTATIONAL EFFICIENCY OF WORTH-LEARNING DATA GENERATION To analyze the computational complexity of the proposed NN model with worth-learning data generation, we adopt a widely used measure—the number of floating-point operations (FLOPs) during the NN model’s forward-backward propaga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The total FLOPs of one single layer of a fully-connected NN model can be calculated as follows: Forward : FLOPs = (2I − 1) × O, (14a) Backward : FLOPs = (2I − 1) × O, (14b) where I is the dimension of the layer’s input, and O is the dimension of its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' To approximate an OPF mapping based on a 57-bus system, the proposed NN model uses the following structure: 84 × 1000×2560×2560×5120×2000×114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' (14), the total FLOPs of the NN per forward-backward process is around 1×108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The GPU used in the experiment is the Quadro P6000, and its performance is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2 TFLOP/s (1 TFLOP/s = 1012 FLOP/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Using the GPU, we can perform the forward- backward process 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='22 × 105 times per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' For the worth-learning data generation in Algorithm 1, the forward process is to calculate V ioifs and V iophm, and the backward process is to update S′G ifs and V ′ifs by the gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' We concatenate S′G ifs and V ′ifs as a vector x, and we suppose the range of each item in x is [0, 10], and x changes 10−3 in each update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Varying from 0 to 10, it costs 104 times the forward-backward processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In other words, the algorithm can at least update 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='22 × 105/104 ≈ 12 samples in 1 s, so finding one sample costs no longer than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='08 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In practice, there is a slight error between the actual speed in experiments and the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' According to the numerical experiments in Section IV-A, an average of 533 samples are found in 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The average consumption time for identifying one sample is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='056 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2nd step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='Initial data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='60 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='8th step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='6th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='Overall data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='80 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D2Initial data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='2nd step ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='Overall data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='Initial data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='Identified data in previous steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='New data10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Illustration of the covered region Scover expanding its area by the generalized region Sadd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' From the analysis presented above, we can conclude that the proposed worth-learning data generation method brings little computational burden into the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' APPENDIX B CONVERGENCE OF WORTH-LEARNING DATA GENERATION This section verifies that the proposed NN with worth- learning data generation can generalize to the whole feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' NN models are continuous functions because both linear layers and activation functions are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' We define a critical violation value ϵ that divides the input space into two types: the covered region (the V iophm values of all of the points are less or equal to ϵ) and the uncovered region (the V iophm values of all of the points are greater than ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' The boundaries of the two regions consist of the points whose V iophm values are approximately equal to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Using these points as initial points, we can identify points with the local maximum in the uncovered region by max violation backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Next, these new points {x1} (the red points) are added to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' After training, the neighborhood of these new points {x1} would be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Due to the generalization of NNs, most points in the area Sadd = {x|a × xini 0 + (1 − a) × x1, 0 ≤ a ≤ 1} would also be covered, where {xini 0 } are the initial points on the boundaries (the black points), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Therefore, the area Sadd is subtracted from the uncovered region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Through iterations, the uncovered region is emptied, and the number of added samples converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' In practice, we choose the training set instead of the boundary points as initial points for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Although some samples in the training set are not at boundaries, they are eliminated by the filter function, as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Therefore, the replacement of the boundary points has no impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Taylor, Convex optimization of power systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' Cambridge University Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' [2] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' 1–12, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} +page_content=' C' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQfQAa_/content/2301.03766v1.pdf'} diff --git a/9tE2T4oBgHgl3EQfQQYW/content/tmp_files/2301.03767v1.pdf.txt b/9tE2T4oBgHgl3EQfQQYW/content/tmp_files/2301.03767v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eedc6c800b48a7a47155adb693508f43938ec124 --- /dev/null +++ b/9tE2T4oBgHgl3EQfQQYW/content/tmp_files/2301.03767v1.pdf.txt @@ -0,0 +1,1506 @@ +Online Backfilling with No Regret for Large-Scale Image Retrieval +Seonguk Seo1,† +Mustafa Gokhan Uzunbas3 +Bohyung Han1,2 +Sara Cao3 +Joena Zhang3 +Taipeng Tian3 +Ser-Nam Lim3 +1ECE & 1,2IPAI, Seoul National University +3Meta AI +{seonguk, bhhan}@snu.ac.kr +{gokhanuzunbas, joenazhang, xuefeicao01, ttp, sernamlim}@meta.com +Abstract +Backfilling is the process of re-extracting all gallery em- +beddings from upgraded models in image retrieval systems. +It inevitably requires a prohibitively large amount of com- +putational cost and even entails the downtime of the ser- +vice. Although backward-compatible learning sidesteps this +challenge by tackling query-side representations, this leads +to suboptimal solutions in principle because gallery em- +beddings cannot benefit from model upgrades. We address +this dilemma by introducing an online backfilling algorithm, +which enables us to achieve a progressive performance im- +provement during the backfilling process while not sacri- +ficing the final performance of new model after the com- +pletion of backfilling. To this end, we first propose a sim- +ple distance rank merge technique for online backfilling. +Then, we incorporate a reverse transformation module for +more effective and efficient merging, which is further en- +hanced by adopting a metric-compatible contrastive learn- +ing approach. These two components help to make the dis- +tances of old and new models compatible, resulting in de- +sirable merge results during backfilling with no extra com- +putational overhead. Extensive experiments show the effec- +tiveness of our framework on four standard benchmarks in +various settings. +1. Introduction +Image retrieval models [5, 10, 21, 23] have achieved re- +markable performance by adopting deep neural networks +for representing images. Yet, all models need to be up- +graded at times to take advantage of improvements in train- +ing datasets, network architectures, and training techniques. +This unavoidably leads to the need for re-extracting the fea- +tures from millions or even billions of gallery images using +the upgraded new model. This process, called backfilling +† This work was mostly done during an internship at Meta AI. +or re-indexing, needs to be completed before the retrieval +system can benefit from the new model, which may take +months in practice. +To sidestep this bottleneck, several backfilling-free ap- +proaches based on backward-compatible learning [4,13,19, +20,22] have been proposed. They learn a new model while +ensuring that its feature space is still compatible with the old +one, thus avoiding the need for updating old gallery embed- +dings. Although these approaches have achieved substantial +performance gains without backfilling, they achieve feature +compatibility at the expense of feature discriminability and +their performance is suboptimal. We argue that backward- +compatible learning is not a fundamental solution and back- +filling is still essential to accomplish state-of-the-art perfor- +mance without performance sacrifices. +To resolve this compatibility-discriminability dilemma, +we relax the backfill-free constraint and propose a novel +online backfilling algorithm equipped with three technical +components. We posit that an online backfilling technique +needs to satisfy three essential conditions: 1) immediate de- +ployment after the completion of model upgrade, 2) pro- +gressive and non-trivial performance gains in the middle +of backfilling, and 3) no degradation of final performance +compared to offline backfilling. To this end, we first pro- +pose a distance rank merge framework to make online back- +filling feasible, which retrieves images from both the old +and new galleries separately and merge their results to ob- +tain the final retrieval outputs even when backfilling is still +ongoing. While this approach provides a monotonic perfor- +mance increase with the progress of backfilling regardless +of the gallery of interest and network architectures, it re- +quires feature computations twice, once from the old model +and another from the new one at the inference stage of a +query. To overcome this limitation, we introduce a reverse +transformation module, which is a lightweight mapping net- +work between the old and new embeddings. The reverse +transformation module allows us to obtain the query repre- +sentations compatible with both the old and new galleries +arXiv:2301.03767v1 [cs.CV] 10 Jan 2023 + +using only a single feature extraction. On the other hand, +however, we notice that the scales of distance in the embed- +ding spaces of the two models could be significantly dif- +ferent. We resolve the limitation with a metric compatible +learning technique, which calibrates the distances of two +models via contrastive learning, further enhancing perfor- +mance of rank merge. +The main contributions of our work are summarized as +follows. +• We propose an online backfilling approach, a funda- +mental solution for model upgrades in image retrieval +systems, based on distance rank merge to overcome +the compatibility-discriminability dilemma in existing +compatible learning methods. +• We incorporate a reverse query transform module to +make it compatible with both the old and new galleries +while computing the feature extraction of query only +once in the middle of the backfilling process. +• We adopt a metric-compatible learning technique to +make the merge process robust by calibrating distances +in the feature embedding spaces given by the old and +new models. +• The proposed approach outperforms all existing meth- +ods by significant margins on four standard benchmark +datasets under various scenarios. +The rest of this paper is organized as follows. Section 2 +reviews the related works. We present the main framework +of online backfilling in Section 3, and discuss the techni- +cal components for improvement in Section 4 and 5. We +demonstrate the effectiveness of the proposed framework in +Section 6 and conclude this paper in Section 7. +2. Related Work +Backward compatible learning +Backward compatibility +refers to the property to support older versions in hardware +or software systems. It has been recently used in model +upgrade scenarios in image retrieval systems. Since the fea- +ture spaces given by the models relying on training datasets +in different regimes are not compatible [11, 24], model up- +grades require re-extraction of all gallery images from new +models, which takes a huge amount of computational cost. +To prevent this time-consuming backfilling cost, backward +compatible training (BCT) [1, 13, 15, 19, 22, 26] has been +proposed to learn better feature representations while be- +ing compatible with old embeddings, which makes the new +model backfill-free. Shen et al. [19] employ the influence +loss that utilizes the old classifier as a regularizer when +training the new model. LCE [13] introduces an alignment +loss to align the class centers between old and new mod- +els and a boundary loss that restricts more compact intra- +class distributions for the new model. Bai et al. [1] pro- +pose a joint prototype transfer with structural regularization +to align two embedding features. UniBCT [26] presents a +structural prototype refinement algorithm that first refines +noisy old features with graph transition and then conducts +backward compatible training. Although these approaches +improved compatible performance without backfilling, they +clearly sacrifice feature discriminability to achieve feature +compatibility with non-ideal old gallery embeddings. +Compatible learning with backfilling +To overcome the +inherent limitation of backward compatible learning, sev- +eral approaches [17, 20, 25] have been proposed to uti- +lize backfilling but efficiently. Forward compatible train- +ing (FCT) [17] learn a lightweight transformation mod- +ule that updates old gallery embeddings to be compati- +ble with new embeddings. Although it gives better com- +patible performance than BCT, it requires an additional +side-information [2] to map from old to new embeddings, +which limits its practicality. Moreover, FCT still suffers +from computational bottleneck until all old gallery embed- +dings are transformed, especially when the side-information +needs to be extracted. +On the other hand, RACT [25] +and BiCT [20] alleviate this bottleneck issue by backfilling +the gallery embeddings in an online manner. RACT first +trains a backward-compatible new model with regression- +alleviating loss, then backfills the old gallery embeddings +with the new model. +Because the new feature space is +compatible with the old one, the new model can be de- +ployed right away while backfilling is carried out in the +background. +BiCT further reduces the backfilling cost +by transforming the old gallery embeddings with forward- +compatible training [17]. Although both approaches can +utilize online backfilling, they still sacrifice the final perfor- +mance because the final new embeddings are constrained by +the old ones. Unlike these methods, our framework enables +online backfilling while fully exploiting the final new model +performance without any degradation. +3. Image Retrieval by Rank Merge +This section discusses our baseline image retrieval al- +gorithm that makes online backfilling feasible. +We first +present our motivation and then describe technical details +with empirical observations. +3.1. Overview +Our goal is to develop a fundamental solution via online +backfilling to overcome the compatibility-discriminability +trade-off in compatible model upgrade. +This strategy +removes inherent limitations of backfill-free backward- +compatible learning—the inability to use state-of-the- + +Figure 1. Image retrieval with the proposed distance rank merge technique. In the middle of backfilling, we retrieve images independently +using two separate models and their galleries, and merge the retrieval results based on their distances. Note that the total number of gallery +embeddings are fixed throughout the backfilling process, i.e., |G| = |Gnew| + |Gold|. +art representations of gallery images through model +upgrades—while avoiding prohibitive costs, including the +situation that we cannot benefit from model upgrade of the +offline backfilling process, until backfilling is completed. +To be specific, the proposed image retrieval system with +online backfilling should satisfy the following three condi- +tions: +1. The system can be deployed immediately as soon as +model upgrade is complete. +2. The performance should monotonically increase with- +out negative flips1 as backfill progresses. +3. The final performance should not be sacrificed com- +pared to the algorithm relying on offline backfilling. +We present a distance rank merge approach for image re- +trieval, which enables online backfilling in arbitrary model +upgrade scenarios. Our method maintains two separate re- +trieval pipelines corresponding to the old and new models +and merges the retrieval results from the two models based +on distances from a query embedding. This allows us to run +the retrieval system without a warm-up period and achieve +surprisingly good results during the backfill process. Note +that the old and new models are not required to be com- +patible at this moment but we will make them so to further +improve performance in the subsequent sections. +3.2. Formulation +Let q ∈ Q be a query image and G = {g1, ..., gN} be +a gallery composed of N images. An embedding network +φ(·) projects an image onto a learned feature embedding +space. To retrieve the closest gallery image given a query, +we find arg ming∈G dist (φ(q), φ(g)), where dist(·, ·) is a +distance metric. Following [19], we define the retrieval per- +formance as +M(φ(Q), φ(G)), +(1) +1The “negative flip” refers to performance degradation caused by in- +correct retrievals of samples by the new model, which were correctly rec- +ognized by the old model. +where M(·, ·) is an evaluation metric such as mean aver- +age precision (mAP) or cumulative matching characteristics +(CMC), and φ(·) indicates embedding models for query and +gallery, respectively. +Backward compatibility +Denote the old and new embed- +ding networks by φold(·) and φnew(·) respectively. If φnew(·) +is backward compatible with φold(·), then we can perform +search on a set of old gallery embeddings using a new +query embedding, i.e., arg ming∈G dist(φnew(q), φold(g)). +As stated in [19], the backward compatibility is achieved +when the following criterion is satisfied: +M(φnew(Q), φold(G)) > M(φold(Q), φold(G)). +(2) +From now, we refer to a pair of embedding networks for +query and gallery as a retrieval system, e.g., {φ(·), φ(·)}. +Rank merge +Assume that the first M out of a total of +N images are backfilled, i.e., Gnew = {g1, ..., gM} and +Gold = {gM+1, ..., gN}. Note that the total number of +stored gallery embeddings is fixed to N during the back- +filling process, i.e., Gold = G − Gnew. Then, we first con- +duct image retrieval using the individual retrieval systems, +{φold, φold} and {φnew, φnew}, independently as +gm = arg min +gi∈Gold dist +� +φold(q), φold(gi) +� +, +(3) +gn = arg min +gj∈Gnew dist (φnew(q), φnew(gj)) . +(4) +Figure 1 illustrates the retrieval process. For each query +image q, we finally select gm if dist(φold(q), φold(gm)) < +dist(φnew(q), φnew(gn)) and gn otherwise. +The retrieval +performance after rank merge during backfilling is given by +Mt := +(5) +M({φold(Q), φnew(Q)}, {φold(Gold +t ), φnew(Gnew +t +)}), +where t ∈ [0, 1] indicates the rate of backfilling completion, +i.e., |Gnew +t +| = t|G| and |Gold +t | = (1 − t)|G|. The criteria + +Old retrieval system +retrieval +Plo +dold (Gold) +Query +Gallery (G) +(q) +retrieval +IG| = |Gnew| + |Gold] +backfilling +New retrieval system anew0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.62 +0.64 +0.66 +0.68 +0.70 +0.72 +0.74 +0.76 +mAP +New (0.773) +Merge (0.693) +Old (0.627) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.84 +0.86 +0.88 +0.90 +CMC (Top1 Acc.) +New (0.909) +Merge (0.871) +Old (0.827) +(a) ImageNet-1K +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.25 +0.30 +0.35 +0.40 +0.45 +mAP +New (0.474) +Merge (0.308) +Old (0.216) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +CMC (Top1 Acc.) +New (0.626) +Merge (0.490) +Old (0.343) +(b) CIFAR-100 +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.17 +0.18 +0.19 +0.20 +0.21 +0.22 +0.23 +mAP +New (0.234) +Merge (0.195) +Old (0.165) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.32 +0.34 +0.36 +0.38 +CMC (Top1 Acc.) +New (0.391) +Merge (0.358) +Old (0.307) +(c) Places-365 +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.35 +0.40 +0.45 +0.50 +mAP +New (0.513) +Merge (0.400) +Old (0.312) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.50 +0.55 +0.60 +0.65 +0.70 +CMC (Top1 Acc.) +New (0.703) +Merge (0.639) +Old (0.497) +(d) Market-1501 +Figure 2. mAP and CMC results on the standard benchmarks using ResNet-18. Old and New denote the performance without backfilling +and with offline backfilling, respectively. The proposed distance rank merging of the old and new models, denoted by Merge, exhibits +desirable results; the performance monotonically increases as backfill progresses without negative flips for all datasets and our algorithm +based on online backfilling achieves competitive final performances with offline backfilling. The numbers in the legend indicate either +AUCmAP or AUCCMC scores. +discussed in Section 3.1 are formally defined as +M0 ≥ M(φold(Q), φold(G)), +(6) +M1 ≥ M(φnew(Q), φnew(G)), +(7) +Mt1 ≥ Mt2 if t1 ≥ t2. +(8) +Comprehensive evaluation +To measure both backfilling +cost and model performance comprehensively during online +backfilling, we utilize the following metrics that calculate +the area under mAP or CMC curves as +AUCmAP = +� 1 +0 +mAPtdt and AUCCMC = +� 1 +0 +CMCtdt. +3.3. Merge Results +We present the results from the rank merge strategy on +two standard benchmarks, including ImageNet-1K [18] and +Places-365 [28], in Figure 2. Our rank merging approach +yields strong and robust results for all datasets; both mAP +and CMC monotonically increase without negative flips as +backfill progresses even though the old and new models are +not compatible each other. Also, it takes full advantage of +the new model until the end of backfilling without suffering +from performance degradation. This validates that our rank +merge technique satisfies the criteria for online backfilling +discussed in Section 3.1 and 3.2. Please refer to Section 6.1 +for the experimental detail. +Figure 3. Reverse query transform module, ψ(·), learns a mapping +from new to old feature spaces. We only update the parameters of +the module ψ(·) (in red rectangle) during training. +4. Reverse Query Transform +Our baseline image retrieval method is model-agnostic, +free from extra training, and effective for performance im- +provement. However, one may argue that the proposed ap- +proach is computationally expensive at inference time be- +cause we need to conduct feature extraction twice per query +for both the old and new models. This section discusses how +to alleviate this limitation by introducing a small network, +called the reverse query transform module. +4.1. Basic Formulation +To reduce the computational cost incurred by comput- +ing query embeddings twice at inference stage, we compute +the embedding using the new model and transform it to the +version compatible with the old model through the reverse + +old +tnew +(.) +new +revFigure 4. Image retrieval merging with reverse query transform module. Backward retrieval system consists of reversely transformed new +query and old gallery, {φrev, φold}. The final image retrieval results are given by merging the outputs from {φrev, φold} and {φnew, φnew}. +query transform module as illustrated in Figure 3. To estab- +lish such a mechanism, we fix the parameters of the old and +new models {φold, φnew} after training them independently, +and train a lightweight network, ψ(·), which transforms the +embedding in the new model to the one in the old model. +For each training example x, our objective is minimizing +the following loss: +LRQT(x) := dist +� +ψ (φnew(x)) , φold(x) +� +, +(9) +where dist(·, ·) is a distance metric such as ℓ2 or cosine dis- +tances. Because we only update the parameters in ψ(·), not +the ones in φnew(·) or φold(·), we can still access the repre- +sentations given by the new model at no cost even after the +optimization of ψ(·). Note that this reverse query transform +module differs from FCT [17] mainly in terms of transfor- +mation direction and requirement of side information. FCT +performs a transformation from the old representation to the +new, while the opposite is true for our proposed approach. +Since the embedding quality of a new model is highly likely +to be better than that of an old one, our reverse transforma- +tion module performs well even without additional side in- +formation and, consequently, is more practical and efficient. +4.2. Integration into Baseline Retrieval System +Figure 4 illustrates the distance rank merge process to- +gether with the proposed reverse transformation module. +The whole procedure consists of two retrieval systems de- +fined by a pair of query and gallery representations, back- +ward retrieval system {φrev, φold} and new retrieval system +{φnew, φnew}, where φrev := ψ(φnew). Note that we obtain +both the new and compatible query embeddings, φnew(q) +and φrev(q) = ψ(φnew(q)), using a shared feature extrac- +tion network, φnew(·). +The entire image retrieval pipeline consists of two parts: +1) feature extraction of a query image and 2) search for the +nearest image in a gallery from the query. Compared to the +image retrieval based on a single model, the computational +cost of the proposed model with rank merge requires negli- +gible additional cost, which corresponds to feature transfor- +mation ψ(·) in the first part. Note that the number of total +gallery embeddings is fixed, i.e., |Gnew| + |Gold| = |G|, so +the cost of the second part is always the same in both cases. +5. Distance Calibration +While the proposed rank merge technique with the ba- +sic reverse transformation module works well, there ex- +ists room for improvement in calibrating feature embedding +spaces of both systems. This section discusses the issues in +details and presents how we figure them out. +5.1. Cross-Model Contrastive Learning +The objective in (9) cares about the positive pairs φold +and φrev with no consideration of negative pairs, which can +sometimes lead to misranked position. To handle this issue, +we employ a supervised contrastive learning loss [7, 14] to +consider both positive and negative pairs as follows: +LCL(xi, yi) = − log +� +yk=yi sold +ik +� +yk=yi sold +ik + � +yk̸=yi sold +ik +, +(10) +where sold +ij += exp +� +−dist +� +φrev(xi), φold(xj) +�� +and yi de- +notes the class membership of the ith sample. For more ro- +bust contrastive training, we perform hard example mining +for both the positive and negative pairs2. Such a contrastive +learning approach facilitates distance calibration and im- +proves feature discrimination because it promotes separa- +tion of the positive and negative examples. +Now, although the distances within the backward re- +trieval system {φrev, φold} become more comparable, they +are still not properly calibrated in terms of the distances +in the new retrieval system {φnew, φnew}. Considering dis- +tances in both retrieval systems jointly when we train the +reverse transformation module, we can obtain more com- +parable distances and consequently achieve more reliable +rank merge results. From this perspective, we propose a +2For each anchor, we select the half of the examples in each of positive +and negative labels based on the distances from the anchor. + +Backward retrieval system +pold (.) +retrieval +(.) +grev(q) +dold (Gold) +Gallery +(G) +Query +retrieval +(q) +(q) +backfilling +New retrieval system [@new +,dnew1Figure 5. Illustration of cross-model contrastive learning loss with +backward retrieval system {φold, φrev} and new retrieval system +{φnew, φnew}. +Two boxes with dotted lines corresponds to two +terms in (11). For each retrieval system, the distances between +positive pairs are learned to be both smaller than those of negative +pairs in the two retrieval systems. +cross-model contrastive learning loss as +LCMCL(xi, yi) = +(11) +− log +� +yk=yi sold +ik +� +yk=yi sold +ik + � +yk̸=yi sold +ik + � +yk̸=yi snew +ik +− log +� +yk=yi snew +ik +� +yk=yi snew +ik + � +yk̸=yi snew +ik + � +yk̸=yi sold +ik +, +where snew +ij += exp(−dist +� +φnew(xi), φnew(xj) +� +) and sold +ij = +exp(−dist +� +φrev(xi), φold(xj) +� +). +Figure 5 illustrates the +concept of the loss function. The positive pairs from the +backward retrieval system {φrev, φold} are trained to locate +closer to the anchor than not only the negative pairs from +the same system but also the ones from the new system +{φnew, φnew}, and vice versa. We finally replace (9) with +(11) for training the reverse transformation module. Com- +pared to (10), additional heterogeneous negative terms in +the denominator of (11) play a role as a regularizer to make +the distances from one model directly comparable to those +from other one, which is desirable for our rank merge strat- +egy. +5.2. Training New Feature Embedding +Until now, we do not jointly train the reverse transfor- +mation module ψ(·) and the new feature extraction module +φnew(·) as illustrated in Figure 3. This hampers the compat- +ibility between the backward and new retrieval systems be- +cause the backward retrieval system {φrev, φold} is the only +part to be optimized while the new system {φnew, φnew} is +fixed. To provide more flexibility, we add another transfor- +mation module ρ(·) on top of the new model as shown in +Figure 6, where ρnew = ρ(φnew) and ρrev = ψ(ρ(φnew)). In +this setting, we use ρnew as the final new model instead of +φnew, and our rank merge process employs {ρrev, φold} and +Figure 6. +Compatible training with learnable new embedding. +Compared to Figure 3, another transformation module ρ(·) is in- +corporated on top of the new model to learn new embedding fa- +vorable to our rank merging. The retrieval results are now merged +from {ρrev, φold} and {ρnew, ρnew}. +{ρnew, ρnew} eventually. This strategy helps to achieve a bet- +ter compatibility by allowing both systems to be trainable. +The final loss function to train the reverse transformation +module has the identical form to LCMCL in (11) except for +the definitions of snew +ij +and sold +ij , which are given by +snew +ij += exp (−dist (ρnew(xi), ρnew(xj))) +(12) +sold +ij = exp +� +−dist +� +ρrev(xi), φold(xj) +�� +. +(13) +Note that this extension does not result in computational +overhead at inference stage but yet improves the perfor- +mance even further. +6. Experiments +We present our experiment setting, the performance of +the proposed approach, and results from the analysis of al- +gorithm characteristics. +6.1. Dataset and Evaluation Protocol +We employ four standard benchmarks, which includes +ImageNet-1K [18], +CIFAR-100 [9], +Places-365 [28], +Market-1501 [27]. As in previous works [17,19], we adopt +the extended-class setting in model upgrade; the old model +is trained with examples from a half of all classes while the +new model is trained with all samples. For example, on the +ImageNet-1K dataset, the old model is trained with the first +500 classes and the new model is trained with the whole +1,000 classes. +Following the previous works [17, 20, 25], we measure +mean average precision (mAP) and cumulative matching +characteristics (CMC)3. We also report our comprehensive +results in terms of AUCmAP and AUCCMC at 10 backfill time +slices, i.e., t ∈ {0.0, 0.1, ..., 1.0} in (5). +6.2. Implementation Details +We employ ResNet-18 [6], ResNet-50 [6], and ViT- +B/32 [3] as our backbone architectures for either old or new +3CMC corresponds to top-k accuracy, and we report top-1 accuracy in +all tables and graphs. + +old +rev +new +D +anchor +positive +negativeold +(.) +p() +() +new +rev +0Table 1. Comparison with existing compatible learning methods on four standard benchmarks in homogeneous model upgrades. Gain +denotes relative gain that each method achieves from old model in terms of AUCmAP, compared to the gain of new model. The proposed +framework, dubbed as RM, consistently outperforms all other models with significantly large margins for all datasets. Note that RMna¨ıve +indicates the basic version of distance rank merge described in Sec. 3.2 and that Old and New denote embedding models of gallery images. +ImageNet-1K +CIFAR-100 +Places-365 +Market-1501 +AUCmAP +AUCCMC +Gain +AUCmAP +AUCCMC +Gain +AUCmAP +AUCCMC +Gain +AUCmAP +AUCCMC +Gain +Old +31.2 +49.7 +0% +21.6 +34.3 +0% +16.5 +30.7 +0% +62.7 +82.7 +0% +New +51.3 +70.3 +100% +47.4 +62.6 +100% +23.4 +39.1 +100% +77.3 +90.9 +100% +RMna¨ıve (Ours) +40.0 +63.9 +44% +30.8 +49.1 +36% +19.5 +35.8 +43% +69.2 +87.0 +45% +BCT [19] +32.0 +46.3 +4% +26.4 +43.5 +19% +17.5 +37.0 +14% +66.6 +84.3 +27% +FCT [17] +36.9 +58.7 +28% +27.1 +49.4 +21% +22.5 +37.3 +87% +66.4 +84.2 +25% +FCT (w/ side-info) [17] +43.6 +65.0 +62% +37.0 +53.9 +60% +23.7 +38.3 +104% +66.4 +84.4 +25% +BiCT [20] +35.1 +59.7 +19% +29.0 +48.3 +29% +19.0 +34.9 +36% +65.0 +82.4 +16% +RM (Ours) +53.4 +68.1 +110% +41.4 +60.7 +78% +28.2 +41.7 +170% +70.7 +87.6 +55% +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +mAP +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.50 +0.55 +0.60 +0.65 +0.70 +CMC (Top1 Acc.) +(a) ImageNet-1K +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +mAP +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +CMC (Top1 Acc.) +(b) CIFAR-100 +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.175 +0.200 +0.225 +0.250 +0.275 +0.300 +mAP +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.32 +0.34 +0.36 +0.38 +0.40 +0.42 +CMC (Top1 Acc.) +(c) Places-365 +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.62 +0.64 +0.66 +0.68 +0.70 +0.72 +0.74 +0.76 +mAP +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.80 +0.82 +0.84 +0.86 +0.88 +0.90 +CMC (Top1 Acc.) +(d) Market-1501 +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +mAP +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.50 +0.55 +0.60 +0.65 +0.70 +Top1 Acc. +Old +New +BCT +FCT* +FCT(w/ side-info)* +BiCT +RM_naïve (Ours) +RM (Ours) +Figure 7. mAP and CMC (Top-1 Acc.) results of our full framework in comparison to existing approaches. The numbers in the legend +indicate either AUCmAP or AUCCMC scores. +models. All transformation modules, ψ(·) and ρ(·), con- +sist of 1 to 5 linear layer blocks, where each block is com- +posed of a sequence of operations, (Linear → BatchNorm +→ ReLU), except for the last block that only has a Lin- +ear layer. Our algorithm does not use any side-information. +Our modules are trained with the Adam optimizer [8] for 50 +epoch, where the learning rate is 1 × 10−4 at the beginning +and decayed using cosine annealing [12]. Our frameworks +are implemented with the Pytorch [16] library and we plan +to release the source codes of our work. +6.3. Results +Homogeneous model upgrade +We present the quantita- +tive results in the homogeneous model upgrade scenario, +where old and new models have the same architecture. We +employ ResNet-50 for ImageNet and ResNet-18 for other +datasets. Table 1 and Figure 7 compare the proposed frame- +work, referred to as RM (Rank Merge), with existing com- +patible learning approaches, including BCT [19], FCT [17], +and BiCT [20]. As shown in the table, RM consistently out- +performs all the existing compatible learning methods by +remarkably significant margins in all datasets. BCT [19] +learns backward compatible feature representations, which +is backfill-free, but its performance gain is not impressive. + +FCT [17] achieves meaningful performance improvement +by transforming old gallery features, but most of the gains +come from side-information [2]. +For example, if side- +information is not available, the performance gain of FCT +drops from 62% to 28% on the ImageNet dataset. Also, +such side-information is not useful for the re-identification +dataset, Market-1501, mainly because the model for the +side-information is trained for image classification using the +ImageNet dataset, which shows its limited generalizability. +On the other hand, although BiCT [20] takes advantage of +online backfilling with less backfilling cost, it suffers from +degraded final performance and negative flips in the mid- +dle of backfilling. Note that RMna¨ıve, our na¨ıve rank merg- +ing between old and new models, is already competitive to +other approaches. +Heterogeneous model upgrade +We evaluate our frame- +work in more challenging scenarios and present the results +in Figure 8, where the old and new models have different +architectures, e.g., ResNet-18 → ResNet-50 or ResNet-18 +→ ViT-B/32. In this figure, RMRQT (green line) denotes +our ablative model trained with (9). Even in this setting, +where both embedding spaces are more incompatible, our +rank merge results from the old and new models still man- +age to achieve a monotonous performance growth curve and +RM improves the overall performance significantly further, +which validates the robustness of our frameworks. +Ablation study +We analyze the results from the abla- +tions of models for our cross-model contrastive learning. +For compatible training, CL-S employs contrastive learn- +ing within the backward system only as in (10) while our +CMCL considers distance metrics from both backward and +new retrieval systems simultaneously as in (11). For a more +thorough ablation study, we also design and test another +metric learning objective, called CL-M, which is given by +LCL-M(xi, yi) = − log +� +yk=yi sold +ik +� +yk=yi sold +ik + � +yk̸=yi sold +ik +− log +� +yk=yi snew +ik +� +yk=yi snew +ik + � +yk̸=yi snew +ik +, (14) +which conducts contrastive learning for both backward and +new retrieval systems separately. Figure 9 visualizes the re- +sults from the ablation studies, where CMCL consistently +outperforms both CL-S and CL-M in various datasets and +architectures. CL-M generally gives better merge results +than CL-S because it calibrates the distances of new re- +trieval system additionally. +However, CL-M still suffers +from negative flips because the distance metrics of both re- +trieval systems are calibrated independently and not learned +to be directly comparable to each other. +On the other +hand, CMCL improves overall performance curves con- +sistently without negative flips. +This validates that con- +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.3 +0.4 +0.5 +0.6 +mAP +Old (0.223) +New (0.513) +RM (0.509) +RM_RQT (0.372) +RM_naïve (0.365) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +Top1 Acc. +Old (0.436) +New (0.703) +RM (0.673) +RM_RQT (0.631) +RM_naïve (0.615) +(a) ImageNet (ResNet-18 → ResNet-50) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +mAP +Old (0.216) +New (0.448) +RM (0.420) +RM_RQT (0.364) +RM_naïve (0.309) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +Top1 Acc. +Old (0.343) +New (0.626) +RM (0.611) +RM_RQT (0.568) +RM_naïve (0.514) +(b) CIFAR-100 (ResNet-18 → ViT-B/32) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.65 +0.70 +0.75 +0.80 +mAP +MCT (Ours) (0.747) +Direct Alignment (0.709) +Merge [Old+New] (0.722) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.82 +0.84 +0.86 +0.88 +0.90 +0.92 +Top1 Acc. +MCT (Ours) (0.900) +Direct Alignment (0.871) +Merge [Old+New] (0.883) +(c) Market-1501 (ResNet-18 → ResNet-50) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.175 +0.200 +0.225 +0.250 +0.275 +0.300 +0.325 +mAP +Old (0.164) +New (0.249) +RM (0.292) +RM_RQT (0.217) +RM_naïve (0.208) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.32 +0.34 +0.36 +0.38 +0.40 +0.42 +Top1 Acc. +Old (0.309) +New (0.398) +RM (0.427) +RM_RQT (0.375) +RM_naïve (0.366) +(d) Places-365 (ResNet-18 → ResNet-50) +Figure 8. +Experimental results with heterogeneous model up- +grades. Our na¨ıve rank merge between different architectures still +achieves promising performance curves in various settings, and +our full algorithm exhibits significantly better results. +sidering the distance metrics of both systems simultane- +ously helps to achieve better metric compatibility and con- +sequently stronger merge results. +7. Conclusion +We presented a novel compatible training framework for +effective and efficient online backfilling. We first addressed + +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.45 +0.50 +0.55 +0.60 +mAP +CMCL (Ours) (0.534) +CL-M (0.487) +CL-S (0.461) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.60 +0.62 +0.64 +0.66 +0.68 +0.70 +Top1 Acc. +CMCL (Ours) (0.681) +CL-M (0.648) +CL-S (0.637) +(a) ImageNet (ResNet-50 → ResNet-50) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.350 +0.375 +0.400 +0.425 +0.450 +0.475 +0.500 +mAP +CMCL (Ours) (0.433) +CL-M (0.411) +CL-S (0.400) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.54 +0.56 +0.58 +0.60 +0.62 +Top1 Acc. +CMCL (Ours) (0.617) +CL-M (0.572) +CL-S (0.594) +(b) CIFAR-100 (ViT-B/32 → ViT-B/32) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.175 +0.200 +0.225 +0.250 +0.275 +0.300 +0.325 +mAP +CMCL (Ours) (0.282) +CL-M (0.228) +CL-S (0.243) +0 +20 +40 +60 +80 +100 +Backfill progress (%) +0.30 +0.32 +0.34 +0.36 +0.38 +0.40 +0.42 +Top1 Acc. +CMCL (Ours) (0.417) +CL-M (0.383) +CL-S (0.385) +(c) Places-365 (ResNet-18 → ResNet-18) +Figure 9. Ablation study of the cross-model contrastive learning +loss on several datasets. CMCL outperforms other ablative mod- +els, CL-M and CL-S, which validates that the distance calibration +plays a crucial role for effective rank merging. +the inherent trade-off between compatibility and discrim- +inability, and proposed a practical alternative, online back- +filling, to handle this dilemma. Our distance rank merge +framework elegantly sidesteps this issue by bridging the gap +between old and new models, and our metric-compatible +learning further enhances the merge results with distance +calibration. Our framework was validated via extensive ex- +periments with significant improvement. We believe our +work will provide a fundamental and practical foundation +for promoting new directions in this line of research. +References +[1] Yan Bai, Jile Jiao, Shengsen Wu, Yihang Lou, Jun Liu, Xue- +tao Feng, and Ling-Yu Duan. 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NIPS, 2014. 4, 6 + diff --git a/9tE2T4oBgHgl3EQfQQYW/content/tmp_files/load_file.txt b/9tE2T4oBgHgl3EQfQQYW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a2e9a49ab202da6aeb0bd3000c1c550f122e342 --- /dev/null +++ b/9tE2T4oBgHgl3EQfQQYW/content/tmp_files/load_file.txt @@ -0,0 +1,718 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf,len=717 +page_content='Online Backfilling with No Regret for Large-Scale Image Retrieval Seonguk Seo1,† Mustafa Gokhan Uzunbas3 Bohyung Han1,2 Sara Cao3 Joena Zhang3 Taipeng Tian3 Ser-Nam Lim3 1ECE & 1,2IPAI, Seoul National University 3Meta AI {seonguk, bhhan}@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='kr {gokhanuzunbas, joenazhang, xuefeicao01, ttp, sernamlim}@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='com Abstract Backfilling is the process of re-extracting all gallery em- beddings from upgraded models in image retrieval systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' It inevitably requires a prohibitively large amount of com- putational cost and even entails the downtime of the ser- vice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Although backward-compatible learning sidesteps this challenge by tackling query-side representations, this leads to suboptimal solutions in principle because gallery em- beddings cannot benefit from model upgrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We address this dilemma by introducing an online backfilling algorithm, which enables us to achieve a progressive performance im- provement during the backfilling process while not sacri- ficing the final performance of new model after the com- pletion of backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To this end, we first propose a sim- ple distance rank merge technique for online backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Then, we incorporate a reverse transformation module for more effective and efficient merging, which is further en- hanced by adopting a metric-compatible contrastive learn- ing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' These two components help to make the dis- tances of old and new models compatible, resulting in de- sirable merge results during backfilling with no extra com- putational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Extensive experiments show the effec- tiveness of our framework on four standard benchmarks in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Introduction Image retrieval models [5, 10, 21, 23] have achieved re- markable performance by adopting deep neural networks for representing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Yet, all models need to be up- graded at times to take advantage of improvements in train- ing datasets, network architectures, and training techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This unavoidably leads to the need for re-extracting the fea- tures from millions or even billions of gallery images using the upgraded new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This process, called backfilling † This work was mostly done during an internship at Meta AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' or re-indexing, needs to be completed before the retrieval system can benefit from the new model, which may take months in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To sidestep this bottleneck, several backfilling-free ap- proaches based on backward-compatible learning [4,13,19, 20,22] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' They learn a new model while ensuring that its feature space is still compatible with the old one, thus avoiding the need for updating old gallery embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Although these approaches have achieved substantial performance gains without backfilling, they achieve feature compatibility at the expense of feature discriminability and their performance is suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We argue that backward- compatible learning is not a fundamental solution and back- filling is still essential to accomplish state-of-the-art perfor- mance without performance sacrifices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To resolve this compatibility-discriminability dilemma, we relax the backfill-free constraint and propose a novel online backfilling algorithm equipped with three technical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We posit that an online backfilling technique needs to satisfy three essential conditions: 1) immediate de- ployment after the completion of model upgrade, 2) pro- gressive and non-trivial performance gains in the middle of backfilling, and 3) no degradation of final performance compared to offline backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To this end, we first pro- pose a distance rank merge framework to make online back- filling feasible, which retrieves images from both the old and new galleries separately and merge their results to ob- tain the final retrieval outputs even when backfilling is still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' While this approach provides a monotonic perfor- mance increase with the progress of backfilling regardless of the gallery of interest and network architectures, it re- quires feature computations twice, once from the old model and another from the new one at the inference stage of a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To overcome this limitation, we introduce a reverse transformation module, which is a lightweight mapping net- work between the old and new embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The reverse transformation module allows us to obtain the query repre- sentations compatible with both the old and new galleries arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='03767v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='CV] 10 Jan 2023 using only a single feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' On the other hand, however, we notice that the scales of distance in the embed- ding spaces of the two models could be significantly dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We resolve the limitation with a metric compatible learning technique, which calibrates the distances of two models via contrastive learning, further enhancing perfor- mance of rank merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The main contributions of our work are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We propose an online backfilling approach, a funda- mental solution for model upgrades in image retrieval systems, based on distance rank merge to overcome the compatibility-discriminability dilemma in existing compatible learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We incorporate a reverse query transform module to make it compatible with both the old and new galleries while computing the feature extraction of query only once in the middle of the backfilling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We adopt a metric-compatible learning technique to make the merge process robust by calibrating distances in the feature embedding spaces given by the old and new models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The proposed approach outperforms all existing meth- ods by significant margins on four standard benchmark datasets under various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Section 2 reviews the related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We present the main framework of online backfilling in Section 3, and discuss the techni- cal components for improvement in Section 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We demonstrate the effectiveness of the proposed framework in Section 6 and conclude this paper in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Related Work Backward compatible learning Backward compatibility refers to the property to support older versions in hardware or software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' It has been recently used in model upgrade scenarios in image retrieval systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Since the fea- ture spaces given by the models relying on training datasets in different regimes are not compatible [11, 24], model up- grades require re-extraction of all gallery images from new models, which takes a huge amount of computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To prevent this time-consuming backfilling cost, backward compatible training (BCT) [1, 13, 15, 19, 22, 26] has been proposed to learn better feature representations while be- ing compatible with old embeddings, which makes the new model backfill-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' [19] employ the influence loss that utilizes the old classifier as a regularizer when training the new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' LCE [13] introduces an alignment loss to align the class centers between old and new mod- els and a boundary loss that restricts more compact intra- class distributions for the new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' [1] pro- pose a joint prototype transfer with structural regularization to align two embedding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' UniBCT [26] presents a structural prototype refinement algorithm that first refines noisy old features with graph transition and then conducts backward compatible training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Although these approaches improved compatible performance without backfilling, they clearly sacrifice feature discriminability to achieve feature compatibility with non-ideal old gallery embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Compatible learning with backfilling To overcome the inherent limitation of backward compatible learning, sev- eral approaches [17, 20, 25] have been proposed to uti- lize backfilling but efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Forward compatible train- ing (FCT) [17] learn a lightweight transformation mod- ule that updates old gallery embeddings to be compati- ble with new embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Although it gives better com- patible performance than BCT, it requires an additional side-information [2] to map from old to new embeddings, which limits its practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Moreover, FCT still suffers from computational bottleneck until all old gallery embed- dings are transformed, especially when the side-information needs to be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' On the other hand, RACT [25] and BiCT [20] alleviate this bottleneck issue by backfilling the gallery embeddings in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' RACT first trains a backward-compatible new model with regression- alleviating loss, then backfills the old gallery embeddings with the new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Because the new feature space is compatible with the old one, the new model can be de- ployed right away while backfilling is carried out in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' BiCT further reduces the backfilling cost by transforming the old gallery embeddings with forward- compatible training [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Although both approaches can utilize online backfilling, they still sacrifice the final perfor- mance because the final new embeddings are constrained by the old ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Unlike these methods, our framework enables online backfilling while fully exploiting the final new model performance without any degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Image Retrieval by Rank Merge This section discusses our baseline image retrieval al- gorithm that makes online backfilling feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We first present our motivation and then describe technical details with empirical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Overview Our goal is to develop a fundamental solution via online backfilling to overcome the compatibility-discriminability trade-off in compatible model upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This strategy removes inherent limitations of backfill-free backward- compatible learning—the inability to use state-of-the- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Image retrieval with the proposed distance rank merge technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' In the middle of backfilling, we retrieve images independently using two separate models and their galleries, and merge the retrieval results based on their distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that the total number of gallery embeddings are fixed throughout the backfilling process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', |G| = |Gnew| + |Gold|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' art representations of gallery images through model upgrades—while avoiding prohibitive costs, including the situation that we cannot benefit from model upgrade of the offline backfilling process, until backfilling is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To be specific, the proposed image retrieval system with online backfilling should satisfy the following three condi- tions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The system can be deployed immediately as soon as model upgrade is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The performance should monotonically increase with- out negative flips1 as backfill progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The final performance should not be sacrificed com- pared to the algorithm relying on offline backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We present a distance rank merge approach for image re- trieval, which enables online backfilling in arbitrary model upgrade scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our method maintains two separate re- trieval pipelines corresponding to the old and new models and merges the retrieval results from the two models based on distances from a query embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This allows us to run the retrieval system without a warm-up period and achieve surprisingly good results during the backfill process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that the old and new models are not required to be com- patible at this moment but we will make them so to further improve performance in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Formulation Let q ∈ Q be a query image and G = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', gN} be a gallery composed of N images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' An embedding network φ(·) projects an image onto a learned feature embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To retrieve the closest gallery image given a query, we find arg ming∈G dist (φ(q), φ(g)), where dist(·, ·) is a distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Following [19], we define the retrieval per- formance as M(φ(Q), φ(G)), (1) 1The “negative flip” refers to performance degradation caused by in- correct retrievals of samples by the new model, which were correctly rec- ognized by the old model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' where M(·, ·) is an evaluation metric such as mean aver- age precision (mAP) or cumulative matching characteristics (CMC), and φ(·) indicates embedding models for query and gallery, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Backward compatibility Denote the old and new embed- ding networks by φold(·) and φnew(·) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' If φnew(·) is backward compatible with φold(·), then we can perform search on a set of old gallery embeddings using a new query embedding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', arg ming∈G dist(φnew(q), φold(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' As stated in [19], the backward compatibility is achieved when the following criterion is satisfied: M(φnew(Q), φold(G)) > M(φold(Q), φold(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' (2) From now, we refer to a pair of embedding networks for query and gallery as a retrieval system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', {φ(·), φ(·)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Rank merge Assume that the first M out of a total of N images are backfilled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', Gnew = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', gM} and Gold = {gM+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', gN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that the total number of stored gallery embeddings is fixed to N during the back- filling process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', Gold = G − Gnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Then, we first con- duct image retrieval using the individual retrieval systems, {φold, φold} and {φnew, φnew}, independently as gm = arg min gi∈Gold dist � φold(q), φold(gi) � , (3) gn = arg min gj∈Gnew dist (φnew(q), φnew(gj)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' (4) Figure 1 illustrates the retrieval process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For each query image q, we finally select gm if dist(φold(q), φold(gm)) < dist(φnew(q), φnew(gn)) and gn otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The retrieval performance after rank merge during backfilling is given by Mt := (5) M({φold(Q), φnew(Q)}, {φold(Gold t ), φnew(Gnew t )}), where t ∈ [0, 1] indicates the rate of backfilling completion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', |Gnew t | = t|G| and |Gold t | = (1 − t)|G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The criteria Old retrieval system retrieval Plo dold (Gold) Query Gallery (G) (q) retrieval IG| = |Gnew| + |Gold] backfilling New retrieval system anew0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='76 mAP New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='773) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='693) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='627) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='90 CMC (Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='909) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='871) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='827) (a) ImageNet-1K 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 mAP New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='474) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='308) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='216) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 CMC (Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='626) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='490) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='343) (b) CIFAR-100 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='23 mAP New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='234) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='195) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='165) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='38 CMC (Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='391) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='358) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='307) (c) Places-365 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 mAP New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='513) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='400) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='312) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 CMC (Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='703) Merge (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='639) Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='497) (d) Market-1501 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' mAP and CMC results on the standard benchmarks using ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Old and New denote the performance without backfilling and with offline backfilling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The proposed distance rank merging of the old and new models, denoted by Merge, exhibits desirable results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' the performance monotonically increases as backfill progresses without negative flips for all datasets and our algorithm based on online backfilling achieves competitive final performances with offline backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The numbers in the legend indicate either AUCmAP or AUCCMC scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1 are formally defined as M0 ≥ M(φold(Q), φold(G)), (6) M1 ≥ M(φnew(Q), φnew(G)), (7) Mt1 ≥ Mt2 if t1 ≥ t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' (8) Comprehensive evaluation To measure both backfilling cost and model performance comprehensively during online backfilling, we utilize the following metrics that calculate the area under mAP or CMC curves as AUCmAP = � 1 0 mAPtdt and AUCCMC = � 1 0 CMCtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Merge Results We present the results from the rank merge strategy on two standard benchmarks, including ImageNet-1K [18] and Places-365 [28], in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our rank merging approach yields strong and robust results for all datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' both mAP and CMC monotonically increase without negative flips as backfill progresses even though the old and new models are not compatible each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Also, it takes full advantage of the new model until the end of backfilling without suffering from performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This validates that our rank merge technique satisfies the criteria for online backfilling discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Please refer to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1 for the experimental detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Reverse query transform module, ψ(·), learns a mapping from new to old feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We only update the parameters of the module ψ(·) (in red rectangle) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Reverse Query Transform Our baseline image retrieval method is model-agnostic, free from extra training, and effective for performance im- provement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' However, one may argue that the proposed ap- proach is computationally expensive at inference time be- cause we need to conduct feature extraction twice per query for both the old and new models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This section discusses how to alleviate this limitation by introducing a small network, called the reverse query transform module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Basic Formulation To reduce the computational cost incurred by comput- ing query embeddings twice at inference stage, we compute the embedding using the new model and transform it to the version compatible with the old model through the reverse old tnew (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') new revFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Image retrieval merging with reverse query transform module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Backward retrieval system consists of reversely transformed new query and old gallery, {φrev, φold}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The final image retrieval results are given by merging the outputs from {φrev, φold} and {φnew, φnew}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' query transform module as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To estab- lish such a mechanism, we fix the parameters of the old and new models {φold, φnew} after training them independently, and train a lightweight network, ψ(·), which transforms the embedding in the new model to the one in the old model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For each training example x, our objective is minimizing the following loss: LRQT(x) := dist � ψ (φnew(x)) , φold(x) � , (9) where dist(·, ·) is a distance metric such as ℓ2 or cosine dis- tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Because we only update the parameters in ψ(·), not the ones in φnew(·) or φold(·), we can still access the repre- sentations given by the new model at no cost even after the optimization of ψ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that this reverse query transform module differs from FCT [17] mainly in terms of transfor- mation direction and requirement of side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' FCT performs a transformation from the old representation to the new, while the opposite is true for our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Since the embedding quality of a new model is highly likely to be better than that of an old one, our reverse transforma- tion module performs well even without additional side in- formation and, consequently, is more practical and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Integration into Baseline Retrieval System Figure 4 illustrates the distance rank merge process to- gether with the proposed reverse transformation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The whole procedure consists of two retrieval systems de- fined by a pair of query and gallery representations, back- ward retrieval system {φrev, φold} and new retrieval system {φnew, φnew}, where φrev := ψ(φnew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that we obtain both the new and compatible query embeddings, φnew(q) and φrev(q) = ψ(φnew(q)), using a shared feature extrac- tion network, φnew(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The entire image retrieval pipeline consists of two parts: 1) feature extraction of a query image and 2) search for the nearest image in a gallery from the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Compared to the image retrieval based on a single model, the computational cost of the proposed model with rank merge requires negli- gible additional cost, which corresponds to feature transfor- mation ψ(·) in the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that the number of total gallery embeddings is fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', |Gnew| + |Gold| = |G|, so the cost of the second part is always the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Distance Calibration While the proposed rank merge technique with the ba- sic reverse transformation module works well, there ex- ists room for improvement in calibrating feature embedding spaces of both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This section discusses the issues in details and presents how we figure them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Cross-Model Contrastive Learning The objective in (9) cares about the positive pairs φold and φrev with no consideration of negative pairs, which can sometimes lead to misranked position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To handle this issue, we employ a supervised contrastive learning loss [7, 14] to consider both positive and negative pairs as follows: LCL(xi, yi) = − log � yk=yi sold ik � yk=yi sold ik + � yk̸=yi sold ik , (10) where sold ij = exp � −dist � φrev(xi), φold(xj) �� and yi de- notes the class membership of the ith sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For more ro- bust contrastive training, we perform hard example mining for both the positive and negative pairs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Such a contrastive learning approach facilitates distance calibration and im- proves feature discrimination because it promotes separa- tion of the positive and negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Now, although the distances within the backward re- trieval system {φrev, φold} become more comparable, they are still not properly calibrated in terms of the distances in the new retrieval system {φnew, φnew}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Considering dis- tances in both retrieval systems jointly when we train the reverse transformation module, we can obtain more com- parable distances and consequently achieve more reliable rank merge results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' From this perspective, we propose a 2For each anchor, we select the half of the examples in each of positive and negative labels based on the distances from the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Backward retrieval system pold (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') retrieval (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') grev(q) dold (Gold) Gallery (G) Query retrieval (q) (q) backfilling New retrieval system [@new ,dnew1Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Illustration of cross-model contrastive learning loss with backward retrieval system {φold, φrev} and new retrieval system {φnew, φnew}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Two boxes with dotted lines corresponds to two terms in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For each retrieval system, the distances between positive pairs are learned to be both smaller than those of negative pairs in the two retrieval systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' cross-model contrastive learning loss as LCMCL(xi, yi) = (11) − log � yk=yi sold ik � yk=yi sold ik + � yk̸=yi sold ik + � yk̸=yi snew ik − log � yk=yi snew ik � yk=yi snew ik + � yk̸=yi snew ik + � yk̸=yi sold ik , where snew ij = exp(−dist � φnew(xi), φnew(xj) � ) and sold ij = exp(−dist � φrev(xi), φold(xj) � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Figure 5 illustrates the concept of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The positive pairs from the backward retrieval system {φrev, φold} are trained to locate closer to the anchor than not only the negative pairs from the same system but also the ones from the new system {φnew, φnew}, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We finally replace (9) with (11) for training the reverse transformation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Com- pared to (10), additional heterogeneous negative terms in the denominator of (11) play a role as a regularizer to make the distances from one model directly comparable to those from other one, which is desirable for our rank merge strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Training New Feature Embedding Until now, we do not jointly train the reverse transfor- mation module ψ(·) and the new feature extraction module φnew(·) as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This hampers the compat- ibility between the backward and new retrieval systems be- cause the backward retrieval system {φrev, φold} is the only part to be optimized while the new system {φnew, φnew} is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To provide more flexibility, we add another transfor- mation module ρ(·) on top of the new model as shown in Figure 6, where ρnew = ρ(φnew) and ρrev = ψ(ρ(φnew)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' In this setting, we use ρnew as the final new model instead of φnew, and our rank merge process employs {ρrev, φold} and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Compatible training with learnable new embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Compared to Figure 3, another transformation module ρ(·) is in- corporated on top of the new model to learn new embedding fa- vorable to our rank merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The retrieval results are now merged from {ρrev, φold} and {ρnew, ρnew}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' {ρnew, ρnew} eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This strategy helps to achieve a bet- ter compatibility by allowing both systems to be trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The final loss function to train the reverse transformation module has the identical form to LCMCL in (11) except for the definitions of snew ij and sold ij , which are given by snew ij = exp (−dist (ρnew(xi), ρnew(xj))) (12) sold ij = exp � −dist � ρrev(xi), φold(xj) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' (13) Note that this extension does not result in computational overhead at inference stage but yet improves the perfor- mance even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Experiments We present our experiment setting, the performance of the proposed approach, and results from the analysis of al- gorithm characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Dataset and Evaluation Protocol We employ four standard benchmarks, which includes ImageNet-1K [18], CIFAR-100 [9], Places-365 [28], Market-1501 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' As in previous works [17,19], we adopt the extended-class setting in model upgrade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' the old model is trained with examples from a half of all classes while the new model is trained with all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For example, on the ImageNet-1K dataset, the old model is trained with the first 500 classes and the new model is trained with the whole 1,000 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Following the previous works [17, 20, 25], we measure mean average precision (mAP) and cumulative matching characteristics (CMC)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We also report our comprehensive results in terms of AUCmAP and AUCCMC at 10 backfill time slices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', t ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='0} in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Implementation Details We employ ResNet-18 [6], ResNet-50 [6], and ViT- B/32 [3] as our backbone architectures for either old or new 3CMC corresponds to top-k accuracy, and we report top-1 accuracy in all tables and graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' old rev new D anchor positive negativeold (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') p() () new rev 0Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Comparison with existing compatible learning methods on four standard benchmarks in homogeneous model upgrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Gain denotes relative gain that each method achieves from old model in terms of AUCmAP, compared to the gain of new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The proposed framework, dubbed as RM, consistently outperforms all other models with significantly large margins for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that RMna¨ıve indicates the basic version of distance rank merge described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='2 and that Old and New denote embedding models of gallery images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' ImageNet-1K CIFAR-100 Places-365 Market-1501 AUCmAP AUCCMC Gain AUCmAP AUCCMC Gain AUCmAP AUCCMC Gain AUCmAP AUCCMC Gain Old 31.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='300 mAP 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='42 CMC (Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') (c) Places-365 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='76 mAP 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='90 CMC (Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') (d) Market-1501 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 mAP 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Old New BCT FCT* FCT(w/ side-info)* BiCT RM_naïve (Ours) RM (Ours) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' mAP and CMC (Top-1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=') results of our full framework in comparison to existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' The numbers in the legend indicate either AUCmAP or AUCCMC scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' All transformation modules, ψ(·) and ρ(·), con- sist of 1 to 5 linear layer blocks, where each block is com- posed of a sequence of operations, (Linear → BatchNorm → ReLU), except for the last block that only has a Lin- ear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our algorithm does not use any side-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our modules are trained with the Adam optimizer [8] for 50 epoch, where the learning rate is 1 × 10−4 at the beginning and decayed using cosine annealing [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our frameworks are implemented with the Pytorch [16] library and we plan to release the source codes of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Results Homogeneous model upgrade We present the quantita- tive results in the homogeneous model upgrade scenario, where old and new models have the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We employ ResNet-50 for ImageNet and ResNet-18 for other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Table 1 and Figure 7 compare the proposed frame- work, referred to as RM (Rank Merge), with existing com- patible learning approaches, including BCT [19], FCT [17], and BiCT [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' As shown in the table, RM consistently out- performs all the existing compatible learning methods by remarkably significant margins in all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' BCT [19] learns backward compatible feature representations, which is backfill-free, but its performance gain is not impressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' FCT [17] achieves meaningful performance improvement by transforming old gallery features, but most of the gains come from side-information [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For example, if side- information is not available, the performance gain of FCT drops from 62% to 28% on the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Also, such side-information is not useful for the re-identification dataset, Market-1501, mainly because the model for the side-information is trained for image classification using the ImageNet dataset, which shows its limited generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' On the other hand, although BiCT [20] takes advantage of online backfilling with less backfilling cost, it suffers from degraded final performance and negative flips in the mid- dle of backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Note that RMna¨ıve, our na¨ıve rank merg- ing between old and new models, is already competitive to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Heterogeneous model upgrade We evaluate our frame- work in more challenging scenarios and present the results in Figure 8, where the old and new models have different architectures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=', ResNet-18 → ResNet-50 or ResNet-18 → ViT-B/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' In this figure, RMRQT (green line) denotes our ablative model trained with (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Even in this setting, where both embedding spaces are more incompatible, our rank merge results from the old and new models still man- age to achieve a monotonous performance growth curve and RM improves the overall performance significantly further, which validates the robustness of our frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Ablation study We analyze the results from the abla- tions of models for our cross-model contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For compatible training, CL-S employs contrastive learn- ing within the backward system only as in (10) while our CMCL considers distance metrics from both backward and new retrieval systems simultaneously as in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' For a more thorough ablation study, we also design and test another metric learning objective, called CL-M, which is given by LCL-M(xi, yi) = − log � yk=yi sold ik � yk=yi sold ik + � yk̸=yi sold ik − log � yk=yi snew ik � yk=yi snew ik + � yk̸=yi snew ik , (14) which conducts contrastive learning for both backward and new retrieval systems separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Figure 9 visualizes the re- sults from the ablation studies, where CMCL consistently outperforms both CL-S and CL-M in various datasets and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' CL-M generally gives better merge results than CL-S because it calibrates the distances of new re- trieval system additionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' However, CL-M still suffers from negative flips because the distance metrics of both re- trieval systems are calibrated independently and not learned to be directly comparable to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' On the other hand, CMCL improves overall performance curves con- sistently without negative flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' This validates that con- 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='6 mAP Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='223) New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='513) RM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='509) RM_RQT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='372) RM_naïve (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='365) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='436) New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='703) RM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='673) RM_RQT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='631) RM_naïve (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='615) (a) ImageNet (ResNet-18 → ResNet-50) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 mAP Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='216) New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='448) RM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='420) RM_RQT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='364) RM_naïve (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='309) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='343) New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='626) RM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='611) RM_RQT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='568) RM_naïve (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='514) (b) CIFAR-100 (ResNet-18 → ViT-B/32) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='80 mAP MCT (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='747) Direct Alignment (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='709) Merge [Old+New] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='722) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='92 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' MCT (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='900) Direct Alignment (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='871) Merge [Old+New] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='883) (c) Market-1501 (ResNet-18 → ResNet-50) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='325 mAP Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='164) New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='249) RM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='292) RM_RQT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='217) RM_naïve (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='208) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='42 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Old (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='309) New (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='398) RM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='427) RM_RQT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='375) RM_naïve (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='366) (d) Places-365 (ResNet-18 → ResNet-50) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Experimental results with heterogeneous model up- grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our na¨ıve rank merge between different architectures still achieves promising performance curves in various settings, and our full algorithm exhibits significantly better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' sidering the distance metrics of both systems simultane- ously helps to achieve better metric compatibility and con- sequently stronger merge results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Conclusion We presented a novel compatible training framework for effective and efficient online backfilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We first addressed 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 mAP CMCL (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='534) CL-M (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='487) CL-S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='461) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='70 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' CMCL (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='681) CL-M (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='648) CL-S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='637) (a) ImageNet (ResNet-50 → ResNet-50) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='500 mAP CMCL (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='433) CL-M (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='411) CL-S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='400) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='62 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' CMCL (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='617) CL-M (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='572) CL-S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='594) (b) CIFAR-100 (ViT-B/32 → ViT-B/32) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='325 mAP CMCL (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='282) CL-M (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='228) CL-S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='243) 0 20 40 60 80 100 Backfill progress (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='42 Top1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' CMCL (Ours) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='417) CL-M (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='383) CL-S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='385) (c) Places-365 (ResNet-18 → ResNet-18) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Ablation study of the cross-model contrastive learning loss on several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' CMCL outperforms other ablative mod- els, CL-M and CL-S, which validates that the distance calibration plays a crucial role for effective rank merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' the inherent trade-off between compatibility and discrim- inability, and proposed a practical alternative, online back- filling, to handle this dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our distance rank merge framework elegantly sidesteps this issue by bridging the gap between old and new models, and our metric-compatible learning further enhances the merge results with distance calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Our framework was validated via extensive ex- periments with significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' We believe our work will provide a fundamental and practical foundation for promoting new directions in this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' References [1] Yan Bai, Jile Jiao, Shengsen Wu, Yihang Lou, Jun Liu, Xue- tao Feng, and Ling-Yu Duan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Dual-tuning: Joint prototype transfer and structure regularization for compatible feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' arXiv preprint arXiv:2108.' metadata={'source': 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Hot-refresh model upgrades with regression-free compatible training in image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' In ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 2, 6 [26] Binjie Zhang, Yixiao Ge, Yantao Shen, Shupeng Su, Chun Yuan, Xuyuan Xu, Yexin Wang, and Ying Shan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' To- wards universal backward-compatible representation learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content='01583, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 2 [27] Liang Zheng, Liyue Shen, Lu Tian, Shengjin Wang, Jing- dong Wang, and Qi Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Scalable person re-identification: A benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' In ICCV, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 6 [28] Bolei Zhou, Agata Lapedriza, Jianxiong Xiao, Antonio Tor- ralba, and Aude Oliva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' Learning deep features for scene recognition using places database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' NIPS, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} +page_content=' 4, 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE2T4oBgHgl3EQfQQYW/content/2301.03767v1.pdf'} diff --git a/A9E2T4oBgHgl3EQf8Qn_/vector_store/index.faiss b/A9E2T4oBgHgl3EQf8Qn_/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..bf03581fb302326639d470ddbe7552cc57d4fca4 --- /dev/null +++ b/A9E2T4oBgHgl3EQf8Qn_/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e5d45a06089696787439d3d13625ee9decdecc3a65f754e7a60d4b1b48f2433 +size 4784173 diff --git a/A9E2T4oBgHgl3EQf8Qn_/vector_store/index.pkl b/A9E2T4oBgHgl3EQf8Qn_/vector_store/index.pkl new file mode 100644 index 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a/BNE1T4oBgHgl3EQfDgMw/content/tmp_files/2301.02877v1.pdf.txt b/BNE1T4oBgHgl3EQfDgMw/content/tmp_files/2301.02877v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5dd106021a71ef56de78c1d38305c603650952f2 --- /dev/null +++ b/BNE1T4oBgHgl3EQfDgMw/content/tmp_files/2301.02877v1.pdf.txt @@ -0,0 +1,1950 @@ +Deep Learning for Mean Field Games with +non-separable Hamiltonians +Mouhcine Assoulia, Badr Missaouib +aModeling, Simulation and Data Analysis Lab, Lot 660, Ben Guerir, 43150, Morocco +bModeling,Simulation and Data Analysis Lab, Lot 660, Ben Guerir, 43150, Morocco +Abstract +This paper introduces a new method based on Deep Galerkin Methods (DGMs) +for solving high-dimensional stochastic Mean Field Games (MFGs). +We +achieve this by using two neural networks to approximate the unknown so- +lutions of the MFG system and forward-backward conditions. Our method +is efficient, even with a small number of iterations, and is capable of han- +dling up to 300 dimensions with a single layer, which makes it faster than +other approaches. +In contrast, methods based on Generative Adversarial +Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We +demonstrate the effectiveness of our approach by applying it to a traffic flow +problem, which was previously solved using the Newton iteration method +only in the deterministic case. We compare the results of our method to +analytical solutions and previous approaches, showing its efficiency. We also +prove the convergence of our neural network approximation with a single +hidden layer using the universal approximation theorem. +Keywords: +Mean Field Games, Deep Learning, Deep Galerkin Method, +Traffic Flow, Non-Separable Hamiltonian +1. Introduction +Mean Field Games (MFGs) are a widely studied topic that can model +a variety of phenomena, including autonomous vehicles [1, 2], finance [3, 4], +economics [5, 6, 7], industrial engineering [8, 9, 10], and data science [11, 12]. +MFGs are dynamic, symmetric games where the agents are indistinguishable +but rational, meaning that their actions can affect the mean of the popu- +lation. In the optimal case, the MFG system reaches a Nash equilibrium +January 10, 2023 +arXiv:2301.02877v1 [cs.LG] 7 Jan 2023 + +(NE), in which no agent can further improve their objective. MFGs are de- +scribed by a system of coupled partial differential equations (PDEs) known +as equation +� +� +� +−∂tφ − ν∆φ + H(x, ρ, ∇φ) = 0, in +E1, +∂tρ − ν∆ρ − div (ρ∇pH(x, ρ, ∇φ)) = 0, in +E2, +ρ(0, x) = ρ0(x), +φ(T, x) = g(x, ρ(T, x)), in +Ω, +(1) +where, E1 = (0, T] × Ω, E2 = [0, T) × Ω, Ω ⊂ Rd and g denotes the terminal +cost. The Hamiltonian H with separable structure is defined as +H(x, ρ, p) = infv{−p.v + L0(x, v)} − f0(x, ρ) = H0(x, p) − f0(x, ρ), +(2) +consisting of a forward-time Fokker-Planck equation (FP) and a backward- +time Hamilton-Jacobi-Bellman equation (HJB), which describe the evolution +of the population density (ρ) and the cost value (φ), respectively. The PDEs +are defined in the domain E1 = (0, T]×Ω and E2 = [0, T). The Hamiltonian +H has a separable structure and is defined as the infimum of the Lagrangian +function L0, which is the Legendre transform of the Hamiltonian, minus the +interaction function f0 between the population of agents. The MFG system +also includes boundary conditions, with the initial density ρ(0, x) given by +ρ0(x) and the terminal cost φ(T, x) given by g(x, ρ(T, x)). These boundary +conditions apply in the domain Ω ⊂ Rd. +One of the main challenges of MFGs is the viscosity problem, in addi- +tion to the complexity of the PDEs and forward-backward conditions. Many +methods for solving MFGs are limited to the deterministic setting (ν = 0). +For example, the Newton iteration method has been applied to the prob- +lem of traffic flow in [1], where a flexible machine learning framework was +provided for the numerical solution of potential MFGs. +While numerical +methods do exist for solving the system of PDEs (1) [13, 14, 15, 16], they +are not always effective due to computational complexity, especially in high +dimensional problems. Deep learning methods, such as Generative Adver- +sarial Networks (GANs) [17, 18], have been used to address this issue by +reformulating MFGs as a primal-dual problem [19, 20, 14]. This approach +uses the Hopf formula in density space [21] to establish a connection between +MFGs and GANs. However, this method requires the Hamiltonian H to be +separable in ρ and p. In cases where the Hamiltonian is non-separable, such +as in traffic flow [1], it is not possible to reformulate MFGs as a primal-dual +2 + +problem. Recently, [22] proposed a policy iteration algorithm for MFGs with +non-separable Hamiltonians using the contraction fixed point method. +Contributions In this work, we present a new method based on DGM for +solving stochastic MFG with a non-separable Hamiltonian. Inspired by the +work [23, 24, 25], we approximate the unknown solutions of the system (1) +by two neural networks trained simultaneously to satisfy each equation of the +MFGs system and forward-backward conditions. While the GAN-based tech- +niques are limited to problems with separable Hamiltonians, our algorithm, +called New-Method, can solve any MFG system. Moreover, we prove the +convergence of the neural network approximation with a single layer using a +fundamental result of the universal approximation theorem. Then, we test +the effectiveness of our New-Method through several numerical experiments, +where we compare our results of New-Method with previous approaches to +assess their reliability. At last, our approach is applied to solve the MFG +system of traffic flow accounting for the stochastic case. +Contents The structure of the rest of the paper is as follows: in Section 2, +we introduce the main description of our approach. Section 3 examines the +convergence of our neural network approximation with a single hidden layer. +In Section 4, we present a review of prior methods. Section 5 investigates +the numerical performance of our proposed algorithms. +We evaluate our +method using a simple analytical solution in Section 5.1 and compare it +to the previous approach in Section 5.2. We also apply our method to the +traffic flow problem in Section 5.3. Finally, we conclude the paper and discuss +potential future work in Section 6. +2. Methodology +Our method involves using two neural networks, Nθ and Nω, to approx- +imate the unknown variables ρ and φ, respectively. The weights for these +networks are θ and ω. Each iteration of our method involves updating ρ +and φ with the approximations from Nθ and Nω. To optimize the accuracy +of these approximations, we use a loss function based on the residual of the +first equation (HJB) to update the parameters of the neural networks. We +repeat this process using the second equation (FP) and new parameters; see +Figure 1. Both neural networks are simultaneously trained on the first equa- +tion, and the results are then checked in the second equation, where they are +3 + +Figure 1: The learning mechanism of our method. +fine-tuned until an equilibrium is reached. This equilibrium represents the +convergence of the two neural networks and, therefore, the solution to both +the Hamilton Jacobi Bellman equations and the Fokker-Planck equation. +We have developed a solution for the problem of MFG systems 1 that +does not rely on the Hamiltonian structure. Our approach involves using a +combination of physics-informed deep learning [24] and deep hidden physics +models [25] to train our model to solve high-dimensional PDEs that adhere to +specified differential operators, initial conditions, and boundary conditions. +Our model is also designed to adhere to general nonlinear partial differential +equations that describe physical laws. To train our model, we define a loss +function that minimizes the residual of the equation at randomly chosen +points in time and space within the domain Ω. +We initialize the neural networks as a solution to our system. We let: +φω(t, x) = Nω(t, x), +ρθ(t, x) = Nθ(t, x). +(3) +Our training strategy starts by solving (HJB). We compute the loss (4) at +randomly sampled points {(tb1, xb1)}B1 +b1=1 from E1, and {xs1}S1 +s1=1 from Ω. +Loss(HJB) +total += Loss(HJB) + Loss(HJB) +cond , +(4) +4 + +Update +Input +HJB +On, Wn +On+1,Wn+1 +On+2, Wn+2 +On+1, Wn+1 +FP +Input +Updatewhere +Loss(HJB) = 1 +B1 +B1 +� +b1=1 +���∂tφω(tb1, xb1) + ν∆φω(tb1, xb1) +− H(xb1, ρθ(tb1, xb1), ∇φω(tb1, xb1)) +��� +2 +, +and +Loss(HJB) +cond += 1 +S1 +S1 +� +s1=1 +���φω(T, xs1) − g(xs1, ρθ(T, xs1)) +��� +2 +. +We then update the weights of φω and ρθ by back-propagating the loss (4). +We do the same to (FP) with the updated weights. We compute (5) at ran- +domly sampled points {(tb2, xb2)}B2 +b2=1 from E2, and {xs2}S2 +s2=1 from Ω. +Loss(FP) +total = Loss(FP) + Loss(FP) +cond , +(5) +where +Loss(FP) = 1 +B2 +B2 +� +b2=1 +���∂tρθ(tb2, xb2) − ν∆ρθ(tb2, xb2) +− div (ρθ(tb2, xb2)∇pH(xb2, ρθ(tb2, xb2), ∇φω(tb2, xb2))) +��� +2 +, +and +Loss(FP) +cond = 1 +S2 +S2 +� +s2=1 +���ρθ(0, xs2) − ρ0(xs2) +��� +2 +. +Finally, we update the weights of φω and ρθ by back-propagating the loss (5); +see Algorithm [1]. +3. Convergence +Following the steps of [23], this section presents theoretical results that +guarantee the existence of a single layer feedforward neural networks ρθ and +φω which can universally approximate the solutions of (1). Denote +L1(ρθ, φω) = +���H1(ρθ, φω) +��� +2 +L2(E1) + +���φω(T, x) − φ(T, x) +��� +2 +L2(Ω), +(6) +5 + +Algorithm 1 New-Method +Require: H Hamiltonian, ν diffusion parameter, g terminal cost. +Require: Initialize neural networks Nω0 and Nθ0 +Train +for n=0,1,2...,K-2 do +Sample batch {(tb1, xb1)}B1 +b1=1 from E1, and {xs1}S1 +s1=1 from Ω +L(HJB) ← +1 +B1 +�B1 +b1=1 +���∂tφωn(tb1, xb1) + ν∆φωn(tb1, xb1) +−H(xb1, ρθn(tb1, xb1), ∇φωn(tb1, xb1)) +��� +2 +. +L(HJB) +cond +← +1 +S1 +�S1 +s1=1 +���φωn(T, xs1) − g(xs1, ρθn(T, xs1)) +��� +2 +. +Backpropagate Loss(HJB) +total +to ωn+1, θn+1 weights. +Sample batch {(tb2, xb2)}B2 +b2=1 from E2, and {xs2}S2 +s2=1 from Ω. +L(FP) ← +1 +B2 +�B2 +b2=1 +���∂tρθn+1(tb2, xb2) − ν∆ρθn+1(tb2, xb2) +− div(∇pH(xb2, ρθn+1(tb2, xb2), ∇φωn+1(tb2, xb2)) +×ρθn+1(tb2, xb2)) +��� +2 +. +Lcond(FP) ← +1 +S2 +�S2 +s2=1 +���ρθn+1(0, xs2) − ρ0(xs2) +��� +2 +. +Backpropagate Loss(FP) +total to ωn+2 θn+2 weights. +return θK, ωK +where +H1(ρθ, φω) = ∂tφω(t, x) + ν∆φω(t, x) − H(x, ρθ(x, t), ∇φω(t, x)). +L2(ρθ, φω) = +���H2(ρθ, φω) +��� +2 +L2(E2) + +���ρθ(0, x) − ρ0(x) +��� +2 +L2(Ω), +(7) +and +H2(ρθ, φω) = ∂tρθ(t, x)−ν∆ρθ(t, x)−div (ρθ(t, x)∇pH(x, ρθ(t, x), ∇φω(t, x))) . +Denote ||f(x)||L2(E) = +�� +E |f(x)|2dµ(x) +� 1 +2 the norm on L2 and µ is a positive +probability density on E. The aim of our approach is to identify a set of +6 + +parameters θ and ω such that the functions ρθ(x, t) and φω(x, t) minimizes +the error L1(ρθ, φω) and L2(ρθ, φω). If L1(ρθ, φω) = 0 and L2(ρθ, φω) = 0, +then ρθ(t, x) and φω(t, x) are solutions to (1). To prove the convergence of +the neural networks, we use the results [26] on the universal approximation +of functions and their derivatives. Define the class of neural networks with a +single hidden layer and n hidden units, +N n(σ) = +� +Φ(t, x) : R1+d �→ R : Φ(t, x) = +n +� +i=1 +βiσ +� +α1,it + +d +� +j=1 +αj,ixj + cj +� � +, +Where +θ = (β1, · · · , βn, α1,1, · · · , αd,n, c1, c1, · · · , cn) ∈ R2n+n(1+d), +the vector of the parameter to be learned. The set of all functions imple- +mented by such a network with a single hidden layer and n hidden units +is +N(σ) = +� +n≥1 +N n(σ), +(8) +We consider E a compact subset of Rd+1, from [26, Th 3]. we know that if +σ ∈ C2 � +Rd+1� +is non constant and bounded, then N(σ) is uniformly 2-dense +on E. This means by [26, Th 2] that for all u ∈ C1,2 � +[0, T] × Rd� +and ϵ > 0, +there is fθ ∈ N(σ) such that: +sup +(t,x)∈E +|∂tu(t, x) − ∂tfθ(t, x)| + max +|a|≤2 sup +(t,x)∈E +��∂(a) +x u(t, x) − ∂(a) +x fθ(t, x) +�� < ϵ. (9) +To prove the convergence of our algorithm, we make the following assump- +tions, +• (H1): E1, E2 are compacts and consider the measures µ1, µ2, µ3, and µ4 +whose support is contained in E1, Ω, E2, and Ω respectively. +• (H2): System (1) has a unique solution (φ, ρ) ∈ X × X such that: +X = +� +u(t, x) ∈ C +� +¯ +[0, T] × Ω +� � +C1+η/2,2+η ([0, T] × Ω) +with η ∈ (0, 1)and that +sup +(t,x)∈[0,T]×Ω +2 +� +k=1 +��∇(k) +x u(t, x) +�� < ∞ +� +. +7 + +• (H3): H, ∇pH, ∇ppH, ∇ρpH are locally Lipschitz continuous in (ρ, p) +with Lipschitz constant that can have at most polynomial growth in ρ +and p, uniformly with respect to t, x. +Remark 3.1. It is important to note that the nonlinear term of L2 can be +simplified as follows, +div(ρ∇pH(x, ρ, ∇φ)) = ∇pH(x, ρ, ∇φ)∇ρ + ∇pρH(x, ρ, ∇φ)∇ρ.ρ ++ +� +i,j +∇pipjH(x, ρ, ∇φ)(∂xjxiφ)ρ. +Theorem 3.1. Let consider N(σ) where σ is C2 � +Rd+1� +, non constant and +bounded. +Suppose (H1), (H2), (H3) hold. +Then for every ϵ1, ϵ2 > 0, +there exists two positives constant C1, C2 > 0 and there exists two functions +(ρθ, φω) ∈ N(σ) × N(σ), such that, +Li(ρθ, φω) ≤ Ci(ϵ1 + ϵ2), +for +i = {1, 2}. +The proof of this theorem is in Appendix A. +Now we have L1(ρn +θ, φn +ω) �→ 0, and L2(ρn +θ, φn +ω) �→ 0 as n �→ ∞ but it does +not necessarily imply that (ρn +θ, φn +ω) �→ (ρ, ω) is the unique solution. +We now prove, under stronger conditions, the convergence of the neural net- +work, (ρn +θ, φn +w) to the solution (ρ, φ) of the system 1 as n → ∞. +To avoid some difficulties, we add homogeneous boundary conditions that +assume the solution is vanishing on the boundary. The MFG system (1) +writes +� +� +� +� +� +� +� +−∂tφ − ν div (a1(∇φ)) + γ(ρ, ∇φ) = 0, in +ΩT, +∂tρ − ν div (a2(∇ρ)) − div (a3(ρ, ∇φ)) = 0, in +ΩT, +ρ(0, x) = ρ0(x), +φ(T, x) = g(x, ρ(T, x)), in +Ω, +ρ(t, x) = φ(t, x) = 0, in +Γ, +(10) +where, ΩT = (0, T) × Ω, Γ = (0, T) × ∂Ω and +a1(t, x, ∇φ) = ∇φ, +a2(t, x, ∇ρ) = ∇ρ, +a3(t, x, ρ, ∇φ) = ρ∇pH(x, ρ, ∇φ), +γ(t, x, ρ, ∇φ) = H(x, ρ, ∇φ), +8 + +a1 : ΩT × RN → RN, a2 : ΩT × RN × RN → RN, a3 : ΩT × R × RN → RN +and γ : ΩT × R × RN → R are Caratheodory functions. +Then we introduce the approximate problem of the system (10) as +� +� +� +� +� +� +� +−∂tφn +ω − ν div (a1(∇φn +ω)) + γ(ρn +θ, ∇φn +ω) = 0, in +ΩT, +∂tρn +θ − ν div (a2(∇ρn +θ)) − div (a3(ρn +θ, ∇φn +ω) = 0, in +ΩT, +ρn +θ(0, x) = ρ0(x), +φn +ω(T, x) = g(x, ρn +θ(T, x)), in +Ω, +ρn +θ(t, x) = φn +ω(t, x) = 0. in +Γ, +(11) +Let us first introduce some definitions. +Let r ≥ 1. In the sequel we denote by Lr � +0, T; W 1,r +0 (Ω) +� +the set of functions +u such that u ∈ Lr (ΩT), u(t, ·) ∈ W 1,r +0 (Ω). The space Lr � +0, T; W 1,r +0 (Ω) +� +is +equipped with the norm +∥u∥Lr(0,T;W 1,r +0 +(Ω)) := +�� T +0 +� +Ω +|∇u(x, t)|rdxdt +� 1 +r +, +is a Banach space. For s, r ≥ 1, the space V s,r +0 +(ΩT) := L∞ (0, T; Ls(Ω)) ∩ +Lr � +0, T; W 1,r +0 (Ω) +� +endowed with the norm +∥ϕ∥V s,r +0 +(ΩT ) := ess sup +0≤t≤T +∥ϕ(., t)∥Ls(Ω) + ∥ϕ∥Lr(0,T;W 1,r +0 +(Ω)), +is also a Banach space. +For this convergence, we make the following set of assumptions, +• (H4): There is a constant µ > 0 and positive functions κ(t, x), λ(t, x) +such that for all (t, x) ∈ ΩT, we have +∥a3(t, x, ρ, p)∥ ≤ µ(κ(t, x) + ∥p∥), and |γ(t, x, ρ, p)| ≤ λ(t, x)∥p∥, +with κ ∈ L2 (ΩT) , λ ∈ Ld+2 (ΩT) . +• (H5): a3(t, x, ρ, p) and γ(t, x, ρ, p) are Lipschitz continuous in (t, x, ρ, p) ∈ +ΩT×R×Rd uniformly on compacts of the form +� +(t, x) ∈ ¯ΩT, |ρ| ≤ C, |p| ≤ C +� +. +• (H6): There is a positive constant α > 0 such that +a3(t, x, ρ, p)p ≥ α|p|2. +9 + +• (H7): For every n ∈ N, ρn +θ, φn +ω ∈ C1,2 �¯ΩT +� +. In addition, (ρn +θ)n∈N , (φn +ω)n∈N ∈ +L2 (ΩT) . +Theorem 3.2. Under previous assumptions (H4)-(H7), if we assume that +(10) has a unique bounded solution (φ, ρ) ∈ V 2,2 +0 +×V 2,2 +0 +, then (φn +ω, ρn +θ) converge +to (φ, ρ) strongly in Lp (ΩT) × Lp (ΩT) for every p < 2. +The proof of this theorem is in Appendix B. Related Works +4. Related Works +GANs: Generative adversarial networks, or GANs, are a class of ma- +chine learning introduced in 2014 [27] that have been successful in generat- +ing images and processing data [28, 29, 30]. In recent years, there has been +increasing interest in using GANs for financial modeling as well [31]. GANs +consist of two neural networks, a generator network, and a discriminator +network, that work against each other in order to generate samples from a +specific distribution. As described in various sources [27, 32, 33], the goal is +to reach equilibrium for the following problem, +min +G max +D +� +Ex∼Pdata(x)[log(D(x)] + Ez∼Pg(z)[log(1 − D(G(z))] +� +, +(12) +where Pdata(x) is the original data and Pg(z) is the noise data. In (12), the +goal is to minimize the generator’s output (G) and maximize the discrimina- +tor’s output (D). This is achieved by comparing the probability of the original +data Pdata(x) being correctly identified by the discriminator D with the prob- +ability of the generated data G produced by the generator using noise data +Pg(z) being incorrectly identified as real by the discriminator 1 − D(G(z)). +Essentially, the discriminator is trying to accurately distinguish between real +and fake data, while the generator is attempting to create fake data that can +deceive the discriminator. +APAC-Net: In [17], the authors present a method (APAC-Net) based +on GANs for solving high-dimensional MFGs in the stochastic case. They use +of the Hopf formula in density space to reformulate the MFGs as a saddle- +point problem given by, +inf +ρ(x,t) sup +φ(x,t) +� +Ez∼P(z),t∼Unif[0,T][∂tφ(ρ(t, z), t) + ν∆φ(ρ(t, z), t) +− H(ρ(t, z), ∇φ)] + Ez∼P(z)φ(0, ρ(0, z)) − Ex∼ρT φ(T, x) +� +, +(13) +10 + +where +H(x, p) = infv{−p.v + L(x, v)}. +In this case, we have a connection between the GANs and MFGs, since +(13) allows them to reach the Kantorovich-Rubenstein dual formulation of +Wasserstein GANs [33] given by, +min +G max +D {Ex∼Pdata(x)[(D(x)] − Ez∼Pg(z)[(D(G(z))]}, +s.t. ||∇D|| ≤ 1. +(14) +Finally, we can use an algorithm similar to GANs to solve the problems of +MFGs. Unfortunately, we notice that the Hamiltonian in this situation has a +separable structure. Due to this, we cannot solve the MFG-LWR system (to +be detailed in section 5.3). In general, we cannot solve the MFGs problems, +where its Hamiltonian is non-separable, since we cannot reformulate MFGs +as 13. +MFGANs: In [18, 17], the connection between GANs and MFGs is +demonstrated by the fact that equation (13) allows them to both reach the +Kantorovich-Rubinstein dual formulation of Wasserstein GANs, as described +in reference [33]. This is shown in equation (12), which can be solved using +an algorithm similar to those used for GANs. However, it is not possible to +solve MFGs problems with non-separable Hamiltonians, as they cannot be +reformulated as in equation (13). This is because the Hamiltonian in these +cases has a separable structure, which prevents the solution of the MFG- +LWR system (to be discussed in section 5.3). +DGM-MFG: In [34], section 4 discusses the adaptation of the DGM al- +gorithm to solve mean field games, referred to as DGM-MFG. This method +is highly versatile and can effectively solve a wide range of partial differential +equations due to its lack of reliance on the specific structure of the problem. +Our own work is similar to DGM-MFG in that we also utilize neural net- +works to approximate unknown functions and adjust parameters to minimize +a loss function based on the PDE residual, as seen in [34] and [18]. However, +our approach, referred to as New-Method, differs in the way it is trained. +Instead of using the sum of PDE residuals as the loss function and SGD for +optimization, we define a separate loss function for each equation and use +ADAM for training, following the approach in [18]. This modification allows +11 + +for faster and more accurate convergence. +Policy iteration Method: To the best of our knowledge, [22] was the +first to successfully solve systems of mean field game partial differential equa- +tions with non-separable Hamiltonians. They proposed two algorithms based +on policy iteration, which involve iteratively updating the population distri- +bution, value function, and control. These algorithms only require the solu- +tion of two decoupled, linear PDEs at each iteration due to the fixed control. +This approach reduces the complexity of the equations, but it is limited to +low-dimensional problems due to the computationally intensive nature of the +method. In contrast, our method utilizes neural networks to solve the HJB +and FP equations at each iteration, allowing for updates to the population +distribution and value function in each equation without the limitations of +[22]. +5. Numerical Experiments +To evaluate the effectiveness of the proposed algorithm [1], we use the +example provided in [17], as it has an explicitly defined solution structure +that allows for easy numerical comparison. We compare the performance of +New-Method, APAC-Net’s MFGAN, and DGM-MFG on the same data to +assess their reliability. Additionally, we apply New-Method to the traffic flow +problem [19], which is characterized by its non-separable Hamiltonian [20], +to determine its ability to solve this type of problem in a stochastic case. +5.1. Analytic Comparison +We test our method by comparing it to a simple example of the analytic +solution used to test the effectiveness of Apac-Net [17]. +For the sake of +simplicity, we take the spatial domain Ω = [−2, 2]d, the final time T = 1, +and without congestion (γ = 0). For +H0(x, p) = ||p||2 +2 +− β ||x||2 +2 , +f0(x, ρ) = γln(ρ), +g(x) = α ||x||2 +2 +− (νdα + γ d +2ln α +2πν), +(15) +and ν = β = 1, where +α = −γ + +� +γ2 + 4ν2β +2ν += 1. +12 + +The corresponding MFG system is: +� +� +� +� +� +� +� +� +� +−∂tφ − ∆φ + ||∇φ||2 +2 +− ||x||2 +2 += 0, +∂tρ − ∆ρ − div (ρ∇φ) = 0, +ρ(0, x) = ( 1 +2π) +d +2e− ||x||2 +2 , +φ(T, x) = x2 +2 − d, +(16) +and the explicit formula is given by +φ(t, x) = ||x||2 +2 +− d.t, +ρ(t, x) = ( 1 +2π) +d +2e− ||x||2 +2 . +(17) +Test 1: We consider the system of PDEs [16] in one dimension (d = +1). +To obtain results, we run Algorithm [1] for 5.103 iterations, using a +minibatch of 50 samples at each iteration. The neural networks employed +have three hidden layers with 100 neurons each, and utilize the Softplus +activation function for Nω and the Tanh activation function for Nθ. Both +networks use ADAM with a learning rate of 10−4 and a weight decay of 10−3. +We employ ResNet as the architecture of the neural networks, with a skip +connection weight of 0.5. The numerical results are shown in Figure 2, which +compares the approximate solutions obtained by New-Method to the exact +solutions at different time states. +To evaluate the performance of New-Method, we compute the relative error +between the model predictions and the exact solutions on a 100 × 100 grid +within the domain [0, 1] × [−2, 2]. Additionally, we plot the HJB and FP +residual loss, as defined in Algorithm [1], to monitor the convergence of our +method (see Figure 3). +Test 2: In this experiment, we use a single hidden layer with vary- +ing numbers of hidden units (nU) for both neural networks. As previously +shown in section 2, the number of hidden units can affect the convergence of +the model. To verify this, we repeat the previous test using the same hyper- +parameters and a single hidden layer but with different numbers of hidden +units. The relative error between the model predictions and the exact solu- +tions is then calculated on a 100×100 grid within the domain [0, 1]×[−2, 2], +as shown in Figure 4. +Test 3: We solve the MFG system [16] for dimensions 2, 50, and 100. +Figure 5 shows the residuals of the HJB and FP equations over 5.104 iter- +ations. A minibatch of 1024, 512, and 128 samples were used for d=100, +d=50, and d=2, respectively. The neural networks had three hidden layers +13 + +Figure 2: The exact solution and prediction calculated by New-Method in dimension one +at t=(0.25, 0.5, 0.75 ). +Figure 3: The relative error for ρ, φ for the figure on the left. On the right, the HJB, FP +Loss. +14 + +t=0.25 +175 +pexact +150 +p new method +125 + exact +@ new method +1D0 +0.75 +0.50 +0.25 +0.00 +0.25 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1D +15 +2Dt=0.5 +150 +pexact +125 +p new method +LDO - + exact +@ new method +0.75 +0.50 +0.25 +0.0 +0.25 +0.50 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1D +15 +2Dt=0.75 +125 +pexact +1D0 +p new method +0.75 + exact +@ new method +0.50 +0.25 +0.0 +0.25 +0.50 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1D +15 +2DThe relative errorof p and +10 +Relative errar +0 +2400 +ODE +50D0 +iberationsResiduals for Fp and HjB cguation +loss FP +10° +loss HJB +10-1 +Losg +10- +10 +10 +0 +2400 +ODE +400 +50D0 +IeratiorsFigure 4: The relative error for ρ , φ in 1-dimension for nU=(2, 5, 10, 20, 50). +with 100 neurons each and utilized the Softplus activation function for Nω +and the Tanh activation function for Nθ. Both networks used ADAM with +a learning rate of 10−4, weight decay of 10−3, and employed ResNet as their +architecture with a skip connection weight of 0.5. The results were obtained +by recording the residuals every 100 iterations and using a rolling average +over 5 points to smooth out the curves. +Figure 5: The loss HJB and FP equation for d=(2,10,100) +Test 4: In this test, we use the same setup as before, but with a single +layer of 100 neurons instead of multiple layers. We keep all other neural +network hyperparameters unchanged. +This test is meant to demonstrate +that a single layer can perform better than multiple layers, even when the +dimension increases, as seen in section 2. Figure 6 shows improved results +compared to the previous test, even with few iterations, which allows for +faster computation times. +15 + +Residuals for Fp equation +10-2 1 +.- d=2 +- d=50 +E-OT +. d=100 +10+ +lenprs +10 + 10- +107 +10-8 +10- +0 +14000 +24000 +ODIDE +4D00 +50000 +iberationa cf AdamThe relative error phi +100 +nU=2 +10-1 +nU=5 +nU=10 +nU=20 +nU=50 +0 +2400 +30 +400 +50D0 +iberatiors of AdamlThe relative eror rha +nU=2 +nU=5 += nU=10 +100 +nU=20 +nU=50 +XYYY +101 +0 +2400 +30D0 +400 +50D0 +iberatior of AdamlResiduals for HjB eguation +- d=2 +103 +-- d=50 +... d=100 +107 +10 +100 +10-1 +10~2 +0 +1ADO0 +24000 +ODIDE +4D00 +50000 +iberationa cf AdamFigure 6: The loss HJB and FP equation with a minibatch of 128, 512, and 1024 samples +for d=2, d=50, and d=(100,200,300), respectively. +5.2. Comparison +In previous sections, we introduced and discussed four methods for solv- +ing MFGs: APAC-Net, MFGAN, DGM-MFG, and New-Method. Here, we +compare these approaches to assess their performance. For APAC-Net, it is +only possible to compare the cost values φ due to the unavailability of the +density function. In APAC-Net, the generator neural network represents ρ, +which generates the distribution. In order to compare the results, we need +to use kernel density estimation to transform the distribution into a density, +which is only an estimate. We use the simple example from the analytic so- +lution with d = 1 and T = 1 for this comparison. The two neural networks in +this comparison have three hidden layers with 100 neurons each, and utilize +ResNet as their architecture with a skip connection weight of 0.5. They also +use the Softplus activation function for Nω and the Tanh activation function +for Nθ. For training APAC-Net, MFGAN, and New-Method, we use ADAM +with a learning rate of 10−4 and a weight decay of 10−3 for both networks. +For training DGM-MFG, we use SGD initialized with a value of 10−3 and a +weight decay of 10−3 for both networks. +We run the four algorithms for 5.103 iterations, using a minibatch of 50 sam- +ples at each iteration. The relative error between the model predictions and +the exact solutions is then calculated on a 100 × 100 grid within the domain +[0, 1] × [−2, 2], as shown in Figure 7. +5.3. Application (Traffic Flow): +In a study published in [1], the authors focused on the longitudinal speed +control of autonomous vehicles. They developed a mathematical model called +16 + +Residual for HjB eguation +... d=2 +104 +- d=50 +- d=100 +- d=200 +102 +— d=300 +Bso1 lenpgad +100 +102 , +0 +2400 +14000 +keratiorsResidual for Fp eguation +101 +10-2 +10-3 +10-4 +d=2 +101 +d=50 +d=100 +10-0 +d=200 +d=300 +0 +2400 +410 +6+DO +14000 +keratiorsFigure 7: comparison between APAC-Net, MFGAN, DGM-MFG, and New-Method. +a Mean Field Game (MFG) to solve a traffic flow problem for autonomous +vehicles and demonstrated that the traditional Lighthill-Whitham-Richards +(LWR) model can be used as a solution to the MFG-LWR model described +by the following system of equations: +MFG − LWR +� +� +� +� +� +� +� +Vt + U(ρ)Vx − 1 +2V 2 +x = 0, +ρt + (ρu)x = 0, +u = U(ρ) − Vx, +VT = g(·, ρT), +ρ(·, 0) = ρ0. +(18) +Here, ρ, V , and u represent the density, optimal cost, and speed function, +respectively, and the Greenshields density-speed relation is given by U(ρ) = +umax(1 − ρ/ρjam), where ρjam is the jam density and umax is the maximum +speed. By setting ρjam = 1 and umax = 1, the authors generalized the MFG- +LWR model to include a viscosity term µ > 0, resulting in the following +system: +MFG − LWR +� +� +� +Vt + ν∆V − H(x, p, ρ) = 0, +ρt − ν∆ρ − div(∇pH(x, p, ρ)ρ) = 0, +VT = g(·, ρT), +ρ(·, 0) = ρ0. +(19) +In this model, ρ and V represent the density and optimal cost function, +respectively, and H is the Hamiltonian with a non-separable structure given +by +H(x, p, ρ) = 1 +2||p||2 − (1 − ρ)p, +with p = Vx, +(20) +where p = Vx. +The authors solved the system in (19) using the New- +ton Iteration method for the deterministic case (ν = 0) with a numerical +17 + +Comparison of p +- New-Method +100 +MFGAN +DGM-MFG +Relative errar +10-1 +0 +2400 +ODE +50D0 +iberationsComparison of +..- New-Method +MFGAN +DGM-MFG +100 +APAC net +Relative errar +10-1 +0 +24D0 +ODE +5400 +iberationsmethod that considers only a finite number of discretization points to re- +duce computational complexity. +In this work, we propose a new method +using a neural network to approximate the unknown and solve the prob- +lem in the stochastic case, while also avoiding the computational complexity +of the previous method. To evaluate the performance of the new method, +we consider the traffic flow problem defined by the MFG-LWR model in 19 +with a non-separable Hamiltonian in (20) on the spatial domain Ω = [0, 1] +with dimension d = 1 and final time T = 1. +The terminal cost g is +set to zero and the initial density ρ0 is given by a Gaussian distribution, +ρ0(x) = 0.2 − 0.6 exp +� +−1 +2 +� x−0.5 +0.1 +�2� +. The aim is to investigate the perfor- +mance of the new method, called the ”New-Method,” in solving this traffic +flow problem. +The corresponding MFG system is, +� +� +� +� +� +� +� +Vt + ν∆V − 1 +2||Vx||2 + (1 − ρ)Vx = 0 +ρt − ν∆ρ − div((Vx − (1 − ρ))ρ) = 0 +ρ(x, 0) = 0.2 − 0.6 exp( −1 +2 ( x−0.5 +0.1 )2), +φ(x, T) = 0. +(21) +We study the deterministic case (ν = 0) and stochastic case (ν = 0.5). We +represent the unknown solutions by two neural networks Nω and Nθ, which +have a single hidden layer of 50 neurons. We use the ResNet architecture +with a skip connection weight of 0.5. We employ ADAM with learning rate +4 × 10−4 for Nω and 5 × 10−4 for Nθ and weight decay of 10−4 for both +networks, batch size 100, in both cases ν = 0 and ν = 0.5 we use the +activation function Softmax and Relu for Nω and Nθ respectively. In Figure +8 we plot over different times the density function, the optimal cost, and the +speed which is calculated according to the density and the optimal cost [1] +by the following formula, +u = umax(1 − ρ/ρjam) − Vx +where, we take the jam density ρjam = 1 and the maximum speed umax = 1 +and 104 iterations. +++ In Figure (9), we plot the HJB, FP residual loss for ν = 0 and ν = 0.5, +which helps us monitor the convergence of our method. Unfortunately, we +do not have the exact solution to compute the error. To validate the results +of Figure (8), we use the fundamental traffic flow diagram, an essential tool +to comprehend classic traffic flow models. Precisely, this is a graphic that +18 + +t=0 +t=0.5 +t=1 +Figure 8: The solution of the problem MFG-LWR by New-Method for (ν = 0) and (ν = +0.5) at t=(0,0.5,1). +Figure 9: The loss HJB and FP equation for (ν = 0) and (ν = 0.5). +19 + +v=o +10-1 , +.- loss FP +loss HJB +10-3 +i +10- +10~7 +10-5 +1022, +1013 +0 +2400 +400 +14000 +keratiorsv=05 +101 +- +loss FP +loss HJB +102 +10-3 +Bso1 lenpgad +10 +100 +10~7 +10-8 +0 +2400 +6+DO +14000 +keratiorsdensity +speed +Optimal costdensity +speed +Optimal costdensity +speed +Optimal costdensity +speed +Optimal costdensity +speed +Optimal costdensity +speed +Optimal cost0.8 +0.7 +0.6 +0.5 +density +0.4 +speed +EO +Optimal cost +0.2 +0.1 +0.0 +0'0 +0.2 +0.4 +0.6 +0.8 +10V=0.5 +0.8 +0.7 +0.6 +0.5 +density +0.4 +speed +Optimalcost +0.3 +0.2 +0.1 +0.0 +00 +0.2 +0.4 +0.6 +0.8 +10 +xV=O +0.8 +0.7 +0.6 +0.5 +density +0.4 : +speed +EO +Optimal cost +0.2 +0.1 +0.0 +00 +0.2 +0.4 +0.6 +0.8 +1.0 +x0.8 +0.7 +0.6 +0.5 +density +0.4 : +speed +EO +Optimalcost +0.2 +0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +8'0 +1.0 +x0.8 +0.7 +0.6 +0.5 +density +0.4 +speed +Optimalcost +EO +0.2 +0.1 +0.0 +00 +0.2 +0.4 +0.6 +0.8 +1.00.8 +0.7 +0.6 +0.5 +density +0.4 +speed +E0 +Optimal cost +0.2 +0.1 +0.0 +0'0 +0.2 +0.4 +9:0 +0.8 +10displays a link between road traffic flux (vehicles/hour) and the traffic density +(vehicles/km) [35, 36, 37]. We can find this diagram numerically [1] such as +its function q is given by, +q(t, x) = ρ(t, x)u(t, x). +Figure (10) shows the fundamental diagram of our results. +t=0 +t=0.5 +t=1 +Figure 10: Fundamental diagram for ν = (0, 0.5) at t = (0, 0.5, 1). +6. Conclusion +• We present a new method based on the deep galerkin method (DGM) +for solving high-dimensional stochastic mean field games (MFGs). The +key idea of our algorithm is to approximate the unknown solutions by +two neural networks that were simultaneously trained to satisfy each +equation of the MFGs system and forward-backward conditions. +• Consequently, our method shows better results even in a small number +of iterations because of its learning mechanism. Moreover, it shows the +20 + +densitydenstbydensitydensitydansitydensityV=0.5 +0.24 +0.22 +flow +0.20 +0.18 +0.16 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +density0.24 +0.22 +MOU +0.20 +0.18 +0.16 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +density0.168 +0.167 +0.166 +10 +0.165 +0.164 +0.163 +0.162 +0.204 +0.206 +0.208 +0.210 +0.212 +0.214 +densityV=O +0.24 +0.22 +0.18 +0.16 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +density0.24 +0.22 +0.20 +0.18 +0.16 +0.20 +0.25 +0.30 +SEO +0.40 +density0.180 +0.175 +0.170 +0.165 +0.160 +0.20 +0.21 +0.22 +0.23 +0.24 +densitypotential of up to 300 dimensions with a single layer, which gives more +speed to our method. +• we proved that as the number of hidden units increases, the neural +networks converge to the MFG solution. +• Comparison with the previous methods shows the efficiency of our ap- +proach even with multilayer neural networks. +• Test on traffic flow problem in the deterministic case gives results sim- +ilar to the newton iteration method, showing that it can solve this +problem in the stochastic case. +To address the issue of high dimensions in the problem, we used a neural +network but found that it took a significant amount of time. +While our +approach has helped to reduce the time required, it is still not fast enough. +Therefore, we are seeking an alternative to neural networks in future research +to improve efficiency. +Appendix A. Proof of Theorem 3.1. +Denote N(σ) the space of all functions implemented by such a network +with a single hidden layer and n hidden units, where σ in C2 � +Rd+1� +non- +constant, and bonded. By (H1) we have that for all ρ, φ ∈ C1,2 � +[0, T] × Rd� +and ε1, ε2 > 0, There is ρθ, φω ∈ N(σ) such That, +sup +(t,x)∈E1 +|∂tφ(t, x) − ∂tφω(t, x)| ++ max +|a|≤2 +sup +(t,x)∈E1 +��∂(a) +x φ(t, x) − ∂(a) +x φω(t, x) +�� < ϵ1 +(A.1) +sup +(t,x)∈E2 +|∂tρ(t, x) − ∂tρθ(t, x)| ++ max +|a|≤2 +sup +(t,x)∈E2 +��∂(a) +x ρ(t, x) − ∂(a) +x ρθ(t, x) +�� < ϵ2 +(A.2) +From (H3) we have that (ρ, p) �→ H(x, ρ, p) is locally Lipschitz continuous +in (ρ, p), with Lipschitz constant that can have at most polynomial growth +in ρ and p, uniformly with respect to t, x. This means that +|H(x, ρ, p) − H(x, γ, s)| ≤ +� +|ρ|q1/2 + |p|q2/2 + |γ|q3/2 + |s|q4/2� +× (|ρ − γ| + |p − s|). +21 + +with some constants 0 ≤ q1, q2, q3, q4 < ∞. As a result, we get using H¨older +inequality with exponents r1, r2, +� +E1 +|H (x, ρθ, ∇xφω) − H (x, ρ, ∇φ)|2 dµ1(t, x) +≤ +� +E1 +(|ρθ(t, x)|q1 + |∇φω(t, x)|q2 + |ρ(t, x)|q3 + |∇φ(t, x)|q4) +× +� +|ρθ(t, x) − ρ(t, x)|2 + |∇φω(t, x) − ∇φ(t, x)|2� +dµ1(t, x) +≤ +� � +E1 +(|ρθ(t, x)|q1 + |∇φω(t, x)|q2 + |ρ(t, x)|q3 + |∇φ(t, x)|q4)r1dµ1(t, x) +�1/r1 +× +� � +E1 +(|ρθ(t, x) − ρ(t, x)|2 + |∇φω(t, x) − ∇φ(t, x)|2)r2dµ1(t, x) +�1/r2 +≤ C1 +� � +E1 +(|ρθ(t, x) − ρ(t, x)|q1 + |∇φω(t, x) − ∇φ(t, x)|q2 ++ |ρ(t, x)|q1∨q3 + |∇φ(t, x)|q2∨q4)r1dµ1(t, x) +�1/r1 +× +� � +E1 +(|ρθ(t, x) − ρ(t, x)|2 + |∇φω(t, x) − ∇φ(t, x)|2)r2dµ1(t, x) +�1/r2 +≤ C1 +� +ϵq1 +1 + ϵq2 +2 + sup +E1 +|ρ|q1∨q3 + sup +E1 +|∇φ|q2∨q4 +� +(ϵ2 +1 + ϵ2 +2) +≤ C1(ϵ2 +1 + ϵ2 +2), +where the constant C1 < ∞ may change from line to line and qi ∨ qj = +max{qi, qj}. In the two last steps we used A.1, A.2 and (H2). We recall +that, +H1(ρθ, φω) = ∂tφω(t, x) + ν∆φω(t, x) − H(x, ρθ(t, x), ∇φω(t, x)). +22 + +Note that H1(ρ, φ) = 0 for ρ, θ that solves the system of PDEs, +L1(ρθ, φω) = +���H1(ρθ, φω) +��� +2 +L2(E1) + +���φω(T, x) − φ(T, x) +��� +2 +L2(Ω) += +���H1(ρθ, φω) − H1(ρ, φ) +��� +2 +L2(E1) + +���φω(x, T) − g(x, ρθ(x, T)) +��� +2 +L2(Ω) +≤ +� +E1 +|∂tφω(t, x) − ∂tφ(t, x)|2 dµ1(t, x) ++ |ν| +� +E1 +|∆φω(t, x) − ∆φ(t, x)|2 dµ1(t, x) ++ +� +E1 +|H (x, ρθ, ∇φω) − H (x, ρ, ∇φ)|2 dµ1(t, x) ++ +� +Ω +|φω(T, x) − φ(T, x)|2dµ2(t, x) +≤C1(ϵ2 +1 + ϵ2 +2) +for an appropriate constant C1 < ∞. In the last step, we use A.1, A.2 and +the previous result. +For L2 we use remark 3.1 to simplified the nonlinear term, +div(ρ∇pH(x, ρ, ∇φ)) = α1(x, ρ, ∇φ) + α2(x, ρ, ∇φ) + α3(x, ρ, ∇φ), +where, +α1(x, ρ, ∇φ) = ∇pH(x, ρ, ∇φ)∇ρ, +α2(x, ρ, ∇φ) = ∇pρH(x, ρ, ∇φ)∇ρ.ρ, +α3(x, ρ, ∇φ) = +� +i,j +∇pipjH(x, ρ, ∇φ)(∂xjxiφ)ρ. +In addition, from (H3) we have also ∇pH(x, ρ, p), ∇pρH(x, ρ, p), and ∇ppH(x, ρ, p) +are locally Lipschitz continuous in (ρ, p). Then, we have after an application +of Holder inequality, for some constant C2 < ∞ that may change from line +23 + +to line, +� +E2 +|α1 (x, ρθ, ∇φω) − α1(x, ρ, ∇φ)|2 dµ3(t, x) += +� +E2 +|∇pωH (x, ρθ, ∇φω) ∇ρθ − ∇pH(x, ρ, ∇φ)∇ρ|2 dµ3(t, x) +≤ +� +E2 +��� +� +∇pωH (x, ρθ, ∇φω) − ∇pH(x, ρ, ∇φ) +� +∇ρ +��� +2 +dµ3(t, x) ++ +� +E2 +���∇pωH (x, ρθ, ∇φω) (∇ρθ − ∇ρ) +��� +2 +dµ3(t, x) +≤ C2 +�� +E2 +���∇pωH (x, ρθ, ∇φω) − ∇pH(x, ρ, ∇φ) +��� +2r1dµ3 (t, x) +�1/r1 +× +� � +E2 +|∇ρ|2r2dµ3(t, x) +�1/r2 + C2 +�� +E2 +���∇pωH (x, ρθ, φω) +��� +2s1dµ3(t, x) +�1/s1 +× +�� +E2 +|∇ρθ − ∇ρ|2s2 dµ3(t, x) +�1/s2 +≤ C2 +� � +E2 +|∇ρ|2r2dµ3(t, x) +�1/r2 +× +� � +E2 +(|ρθ(t, x) − ρ(t, x)|q1 + |∇φω(t, x) − ∇φ(t, x)|q2 ++ |ρ(t, x)|q1∨q3 + |∇φ(t, x)|q2∨q4)v1r1dµ3(t, x) +�1/v1r1 +× +� � +E2 +(|ρθ(t, x) − ρ(t, x)|2 + |∇xφω(t, x) − ∇xφ(t, x)|2)v2r2dµ3(t, x) +�1/v2r2 ++ C2 +�� +E2 +���∇pωH (x, ρθ, φω) +��� +2s1dµ3(t, x) +�1/s1 +× +�� +E2 +|∇ρθ − ∇ρ|2s2 dµ3(t, x) +�1/s2 +≤ C2(ϵ2 +1 + ϵ2 +2) +where in the last steps, we followed the computations previously. We do +same for α2(x, ρ, ∇φ) and α3(x, ρ, ∇φ), we obtain for a C2 < ∞, +� +E2 +��� div(ρθ∇pωH(x, ρθ, ∇φω)) − div(ρ∇pH(x, ρ, ∇φ)) +��� +2 +dµ3(t, x) +≤ C2(ϵ2 +1 + ϵ2 +2). +24 + +We recall that, +H2(ρθ, φω) = ∂tρθ(t, x)−ν∆ρθ(t, x)−div (ρθ(t, x)∇pH(x, ρθ(t, x), ∇φω(t, x))) +Note that H2(ρ, φ) = 0 for ρ, θ that solves the system of PDEs, then we have, +L2(ρθ, φω) = +���H2(ρθ, φω) +��� +2 +L2(E2) + +���ρθ(0, x) − ρ0(x) +��� +2 +L2(Ω) += +���H2(ρθ, φω) − H2(ρ, φ) +��� +2 +L2(E2) + +���ρθ(0, x) − ρ0(x) +��� +2 +L2(Ω) +≤ +� +E2 +|∂tρθ(t, x) − ∂tρ(t, x)|2 dµ3(t, x) ++ |ν| +� +E2 +|∆ρθ(t, x) − ∆ρ(t, x)|2 dµ3(t, x) ++ +� +E2 +��� div(ρθ∇pωH(x, ρθ, ∇φω)) − div(ρ∇pH(x, ρ, ∇φ)) +��� +2 +dµ3(t, x) ++ +� +Ω +|ρθ(0, x) − ρ0(x)|2dµ4(t, x) +≤C2(ϵ2 +1 + ϵ2 +2) +for an appropriate constant C2 < ∞. The proof of theorem 3.1 is complete +after rescaling ϵ1 and ϵ2 +Appendix B. Proof of Theorem 3.2. +We follow the method used in [23] for a single PDE. (See also section 4 +in [38] for a coupled system). Let us denote the solution of problem 11 by. +� +ˆρn +θ, ˆφn +ω +� +∈ V = V 2,2 +0 +× V 2,2 +0 +. Due to Conditions (H4) − (H6) and by using +lemma 1.4 [39] on each equation then, there exist, C1, C2 such that: +∥ˆρn +θ∥V 2,2 +0 +≤ C1 +∥ˆφn +ω∥V 2,2 +0 +≤ C2 +These applies and gives that the both sequence {ˆρn +θ}n∈N, {ˆφn +ω}n∈N are uni- +formly bounded with respect to n in at least V . These uniform energy bounds +25 + +imply the existence of two subsequences, (still denoted in the same way) +{ˆρn +θ}n∈N, {ˆφn +ω}n∈N and two functions ρ, φ in L2 � +0, T; W 1,2 +0 (Ω) +� +such that, +ˆρn +θ → ρ weakly in L2 � +0, T : W 1,2 +0 (Ω) +� +ˆφn +ω → φ weakly in L2 � +0, T : W 1,2 +0 (Ω) +� +Next let us set q = 1 + +d +d+4 ∈ (1, 2) and note that for conjugates, r1, r2 > 1 +such that 1/r1 + 1/r2 = 1 +� +ΩT +���γ +� +t, x, ˆρn +θ, ∇ˆφn +ω +���� +q +≤ +� +ΩT +|λ|q ���∇ˆφn +ω +��� +q +≤ +�� +ΩT +|λ|r1q +�1/r1 �� +ΩT +���∇ˆφn +ω +��� +r2q�1/r2 +Let us choose r2 = 2/q > 1. Then we calculate r1 = +r2 +r2−1 = +2 +2−q. Hence, +we have that r1q = d + 2. Recalling the assumption λ ∈ Ld+2 (ΩT) and the +uniform bound on the ∇ˆφn +ω we subsequently obtain that for q = 1 + +d +d+4, +there is a constant C < ∞ such that +� +ΩT +���γ +� +t, x, ˆρn +θ, ∇ˆφn +ω +���� +q +≤ C +On the other hand, it is obvious that a1 is bounded uniformly then, according +to the HJB equation of 11, we have +� +∂t ˆφn +ω +� +n∈N is bounded uniformly with +respect to n in L2 (0, T; W −1,2(Ω)). Then we can extract a subsequence, (still +denoted in the same way) +� +∂t ˆφn +ω +� +n∈N such that +∂t ˆφn +θ → ∂tφ weakly in L2 � +0, T; W −1,2(Ω) +� +Also, it will be shown that +∂tˆρn +θ → ∂tρ weakly in L2 � +0, T; W −1,2(Ω) +� +Since the problem is nonlinear, the weak convergence of ˆφn +ω and ˆρn +θ in the +space L2 � +0, T; W 1,2 +0 (Ω) +� +is not enough in order to prove that φ and ρ are a +solution of problem 10. To do this, we need the almost everywhere conver- +gence of the gradients for a subsequence of the approximating solutions ˆφn +ω +and ˆρn +θ. +26 + +However, the uniform boundedness of {ˆφn +ω}n∈N and {ˆρn +θ}n∈N in L2 � +0, T; W 1,2 +0 (Ω) +� +and their weak convergence to φ and ρ respectively in that space, allows us +to conclude, by using Theorem 3.3 of [40] on each equation, that +∇ˆφn +ω → ∇φ almost everywhere in ΩT. +∇ˆρn +θ → ∇ρ almost everywhere in ΩT. +Hence, we obtain that {ˆφn +ω}n∈N and {ˆρn +θ}n∈N converges respectively to φ and +ρ strongly in Lp � +0, T; W 1,p +0 (Ω) +� +for every p < 2. It remains to discuss the +convergence of φn +ω − ˆφn +ω and ρn +θ − ˆρn +θ to zero. By last step of proof theorem 7.3 +[23] we get +� +φn +ω − ˆφn +ω +� +n∈N and {ρn +θ − ˆρn +θ}n∈N goes to zero strongly in Lp (ΩT) +for every p < 2. Finally we conclude the proof of the convergence in Lp (ΩT) +for every p < 2 +References +[1] K. Huang, X. Di, Q. Du, X. 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Orsina, Nonlinear parabolic +equations with measure data, journal of functional analysis 147 (1) +(1997) 237–258. +31 + diff --git a/BNE1T4oBgHgl3EQfDgMw/content/tmp_files/load_file.txt b/BNE1T4oBgHgl3EQfDgMw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c05332274db46cf2ac3e44f6554b359c80cd8fe --- /dev/null +++ b/BNE1T4oBgHgl3EQfDgMw/content/tmp_files/load_file.txt @@ -0,0 +1,907 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf,len=906 +page_content='Deep Learning for Mean Field Games with non-separable Hamiltonians Mouhcine Assoulia, Badr Missaouib aModeling, Simulation and Data Analysis Lab, Lot 660, Ben Guerir, 43150, Morocco bModeling,Simulation and Data Analysis Lab, Lot 660, Ben Guerir, 43150, Morocco Abstract This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We achieve this by using two neural networks to approximate the unknown so- lutions of the MFG system and forward-backward conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Our method is efficient, even with a small number of iterations, and is capable of han- dling up to 300 dimensions with a single layer, which makes it faster than other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We compare the results of our method to analytical solutions and previous approaches, showing its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Keywords: Mean Field Games, Deep Learning, Deep Galerkin Method, Traffic Flow, Non-Separable Hamiltonian 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Introduction Mean Field Games (MFGs) are a widely studied topic that can model a variety of phenomena, including autonomous vehicles [1, 2], finance [3, 4], economics [5, 6, 7], industrial engineering [8, 9, 10], and data science [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' MFGs are dynamic, symmetric games where the agents are indistinguishable but rational, meaning that their actions can affect the mean of the popu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In the optimal case, the MFG system reaches a Nash equilibrium January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='02877v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='LG] 7 Jan 2023 (NE), in which no agent can further improve their objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' MFGs are de- scribed by a system of coupled partial differential equations (PDEs) known as equation � � � −∂tφ − ν∆φ + H(x, ρ, ∇φ) = 0, in E1, ∂tρ − ν∆ρ − div (ρ∇pH(x, ρ, ∇φ)) = 0, in E2, ρ(0, x) = ρ0(x), φ(T, x) = g(x, ρ(T, x)), in Ω, (1) where, E1 = (0, T] × Ω, E2 = [0, T) × Ω, Ω ⊂ Rd and g denotes the terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The Hamiltonian H with separable structure is defined as H(x, ρ, p) = infv{−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='v + L0(x, v)} − f0(x, ρ) = H0(x, p) − f0(x, ρ), (2) consisting of a forward-time Fokker-Planck equation (FP) and a backward- time Hamilton-Jacobi-Bellman equation (HJB), which describe the evolution of the population density (ρ) and the cost value (φ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The PDEs are defined in the domain E1 = (0, T]×Ω and E2 = [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The Hamiltonian H has a separable structure and is defined as the infimum of the Lagrangian function L0, which is the Legendre transform of the Hamiltonian, minus the interaction function f0 between the population of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The MFG system also includes boundary conditions, with the initial density ρ(0, x) given by ρ0(x) and the terminal cost φ(T, x) given by g(x, ρ(T, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' These boundary conditions apply in the domain Ω ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' One of the main challenges of MFGs is the viscosity problem, in addi- tion to the complexity of the PDEs and forward-backward conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Many methods for solving MFGs are limited to the deterministic setting (ν = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For example, the Newton iteration method has been applied to the prob- lem of traffic flow in [1], where a flexible machine learning framework was provided for the numerical solution of potential MFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' While numerical methods do exist for solving the system of PDEs (1) [13, 14, 15, 16], they are not always effective due to computational complexity, especially in high dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Deep learning methods, such as Generative Adver- sarial Networks (GANs) [17, 18], have been used to address this issue by reformulating MFGs as a primal-dual problem [19, 20, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This approach uses the Hopf formula in density space [21] to establish a connection between MFGs and GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' However, this method requires the Hamiltonian H to be separable in ρ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In cases where the Hamiltonian is non-separable, such as in traffic flow [1], it is not possible to reformulate MFGs as a primal-dual 2 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Recently, [22] proposed a policy iteration algorithm for MFGs with non-separable Hamiltonians using the contraction fixed point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Contributions In this work, we present a new method based on DGM for solving stochastic MFG with a non-separable Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Inspired by the work [23, 24, 25], we approximate the unknown solutions of the system (1) by two neural networks trained simultaneously to satisfy each equation of the MFGs system and forward-backward conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' While the GAN-based tech- niques are limited to problems with separable Hamiltonians, our algorithm, called New-Method, can solve any MFG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Moreover, we prove the convergence of the neural network approximation with a single layer using a fundamental result of the universal approximation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Then, we test the effectiveness of our New-Method through several numerical experiments, where we compare our results of New-Method with previous approaches to assess their reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' At last, our approach is applied to solve the MFG system of traffic flow accounting for the stochastic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Contents The structure of the rest of the paper is as follows: in Section 2, we introduce the main description of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Section 3 examines the convergence of our neural network approximation with a single hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In Section 4, we present a review of prior methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Section 5 investigates the numerical performance of our proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We evaluate our method using a simple analytical solution in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1 and compare it to the previous approach in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We also apply our method to the traffic flow problem in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Finally, we conclude the paper and discuss potential future work in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Methodology Our method involves using two neural networks, Nθ and Nω, to approx- imate the unknown variables ρ and φ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The weights for these networks are θ and ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Each iteration of our method involves updating ρ and φ with the approximations from Nθ and Nω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To optimize the accuracy of these approximations, we use a loss function based on the residual of the first equation (HJB) to update the parameters of the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We repeat this process using the second equation (FP) and new parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Both neural networks are simultaneously trained on the first equa- tion, and the results are then checked in the second equation, where they are 3 Figure 1: The learning mechanism of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' fine-tuned until an equilibrium is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This equilibrium represents the convergence of the two neural networks and, therefore, the solution to both the Hamilton Jacobi Bellman equations and the Fokker-Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We have developed a solution for the problem of MFG systems 1 that does not rely on the Hamiltonian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Our approach involves using a combination of physics-informed deep learning [24] and deep hidden physics models [25] to train our model to solve high-dimensional PDEs that adhere to specified differential operators, initial conditions, and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Our model is also designed to adhere to general nonlinear partial differential equations that describe physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To train our model, we define a loss function that minimizes the residual of the equation at randomly chosen points in time and space within the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We initialize the neural networks as a solution to our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We let: φω(t, x) = Nω(t, x), ρθ(t, x) = Nθ(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (3) Our training strategy starts by solving (HJB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We compute the loss (4) at randomly sampled points {(tb1, xb1)}B1 b1=1 from E1, and {xs1}S1 s1=1 from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Loss(HJB) total = Loss(HJB) + Loss(HJB) cond , (4) 4 Update Input HJB On, Wn On+1,Wn+1 On+2, Wn+2 On+1, Wn+1 FP Input Updatewhere Loss(HJB) = 1 B1 B1 � b1=1 ���∂tφω(tb1, xb1) + ν∆φω(tb1, xb1) − H(xb1, ρθ(tb1, xb1), ∇φω(tb1, xb1)) ��� 2 , and Loss(HJB) cond = 1 S1 S1 � s1=1 ���φω(T, xs1) − g(xs1, ρθ(T, xs1)) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We then update the weights of φω and ρθ by back-propagating the loss (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We do the same to (FP) with the updated weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We compute (5) at ran- domly sampled points {(tb2, xb2)}B2 b2=1 from E2, and {xs2}S2 s2=1 from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Loss(FP) total = Loss(FP) + Loss(FP) cond , (5) where Loss(FP) = 1 B2 B2 � b2=1 ���∂tρθ(tb2, xb2) − ν∆ρθ(tb2, xb2) − div (ρθ(tb2, xb2)∇pH(xb2, ρθ(tb2, xb2), ∇φω(tb2, xb2))) ��� 2 , and Loss(FP) cond = 1 S2 S2 � s2=1 ���ρθ(0, xs2) − ρ0(xs2) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Finally, we update the weights of φω and ρθ by back-propagating the loss (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' see Algorithm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Convergence Following the steps of [23], this section presents theoretical results that guarantee the existence of a single layer feedforward neural networks ρθ and φω which can universally approximate the solutions of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Denote L1(ρθ, φω) = ���H1(ρθ, φω) ��� 2 L2(E1) + ���φω(T, x) − φ(T, x) ��� 2 L2(Ω), (6) 5 Algorithm 1 New-Method Require: H Hamiltonian, ν diffusion parameter, g terminal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Require: Initialize neural networks Nω0 and Nθ0 Train for n=0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=',K-2 do Sample batch {(tb1, xb1)}B1 b1=1 from E1, and {xs1}S1 s1=1 from Ω L(HJB) ← 1 B1 �B1 b1=1 ���∂tφωn(tb1, xb1) + ν∆φωn(tb1, xb1) −H(xb1, ρθn(tb1, xb1), ∇φωn(tb1, xb1)) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' L(HJB) cond ← 1 S1 �S1 s1=1 ���φωn(T, xs1) − g(xs1, ρθn(T, xs1)) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Backpropagate Loss(HJB) total to ωn+1, θn+1 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Sample batch {(tb2, xb2)}B2 b2=1 from E2, and {xs2}S2 s2=1 from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' L(FP) ← 1 B2 �B2 b2=1 ���∂tρθn+1(tb2, xb2) − ν∆ρθn+1(tb2, xb2) − div(∇pH(xb2, ρθn+1(tb2, xb2), ∇φωn+1(tb2, xb2)) ×ρθn+1(tb2, xb2)) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Lcond(FP) ← 1 S2 �S2 s2=1 ���ρθn+1(0, xs2) − ρ0(xs2) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Backpropagate Loss(FP) total to ωn+2 θn+2 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' return θK, ωK where H1(ρθ, φω) = ∂tφω(t, x) + ν∆φω(t, x) − H(x, ρθ(x, t), ∇φω(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' L2(ρθ, φω) = ���H2(ρθ, φω) ��� 2 L2(E2) + ���ρθ(0, x) − ρ0(x) ��� 2 L2(Ω), (7) and H2(ρθ, φω) = ∂tρθ(t, x)−ν∆ρθ(t, x)−div (ρθ(t, x)∇pH(x, ρθ(t, x), ∇φω(t, x))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Denote ||f(x)||L2(E) = �� E |f(x)|2dµ(x) � 1 2 the norm on L2 and µ is a positive probability density on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The aim of our approach is to identify a set of 6 parameters θ and ω such that the functions ρθ(x, t) and φω(x, t) minimizes the error L1(ρθ, φω) and L2(ρθ, φω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' If L1(ρθ, φω) = 0 and L2(ρθ, φω) = 0, then ρθ(t, x) and φω(t, x) are solutions to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To prove the convergence of the neural networks, we use the results [26] on the universal approximation of functions and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Define the class of neural networks with a single hidden layer and n hidden units, N n(σ) = � Φ(t, x) : R1+d �→ R : Φ(t, x) = n � i=1 βiσ � α1,it + d � j=1 αj,ixj + cj � � , Where θ = (β1, · · · , βn, α1,1, · · · , αd,n, c1, c1, · · · , cn) ∈ R2n+n(1+d), the vector of the parameter to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The set of all functions imple- mented by such a network with a single hidden layer and n hidden units is N(σ) = � n≥1 N n(σ), (8) We consider E a compact subset of Rd+1, from [26, Th 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' we know that if σ ∈ C2 � Rd+1� is non constant and bounded, then N(σ) is uniformly 2-dense on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This means by [26, Th 2] that for all u ∈ C1,2 � [0, T] × Rd� and ϵ > 0, there is fθ ∈ N(σ) such that: sup (t,x)∈E |∂tu(t, x) − ∂tfθ(t, x)| + max |a|≤2 sup (t,x)∈E ��∂(a) x u(t, x) − ∂(a) x fθ(t, x) �� < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (9) To prove the convergence of our algorithm, we make the following assump- tions, (H1): E1, E2 are compacts and consider the measures µ1, µ2, µ3, and µ4 whose support is contained in E1, Ω, E2, and Ω respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (H2): System (1) has a unique solution (φ, ρ) ∈ X × X such that: X = � u(t, x) ∈ C � ¯ [0, T] × Ω � � C1+η/2,2+η ([0, T] × Ω) with η ∈ (0, 1)and that sup (t,x)∈[0,T]×Ω 2 � k=1 ��∇(k) x u(t, x) �� < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 7 (H3): H, ∇pH, ∇ppH, ∇ρpH are locally Lipschitz continuous in (ρ, p) with Lipschitz constant that can have at most polynomial growth in ρ and p, uniformly with respect to t, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' It is important to note that the nonlinear term of L2 can be simplified as follows, div(ρ∇pH(x, ρ, ∇φ)) = ∇pH(x, ρ, ∇φ)∇ρ + ∇pρH(x, ρ, ∇φ)∇ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='ρ + � i,j ∇pipjH(x, ρ, ∇φ)(∂xjxiφ)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Let consider N(σ) where σ is C2 � Rd+1� , non constant and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Suppose (H1), (H2), (H3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Then for every ϵ1, ϵ2 > 0, there exists two positives constant C1, C2 > 0 and there exists two functions (ρθ, φω) ∈ N(σ) × N(σ), such that, Li(ρθ, φω) ≤ Ci(ϵ1 + ϵ2), for i = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The proof of this theorem is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Now we have L1(ρn θ, φn ω) �→ 0, and L2(ρn θ, φn ω) �→ 0 as n �→ ∞ but it does not necessarily imply that (ρn θ, φn ω) �→ (ρ, ω) is the unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We now prove, under stronger conditions, the convergence of the neural net- work, (ρn θ, φn w) to the solution (ρ, φ) of the system 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To avoid some difficulties, we add homogeneous boundary conditions that assume the solution is vanishing on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The MFG system (1) writes � � � � � � � −∂tφ − ν div (a1(∇φ)) + γ(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' in ΩT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∂tρ − ν div (a2(∇ρ)) − div (a3(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ)) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' in ΩT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) = ρ0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φ(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) = g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) = φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' in Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (10) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ΩT = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' T) × Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Γ = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' T) × ∂Ω and a1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ) = ∇φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' a2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇ρ) = ∇ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' a3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ) = ρ∇pH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' γ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ) = H(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 8 a1 : ΩT × RN → RN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' a2 : ΩT × RN × RN → RN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' a3 : ΩT × R × RN → RN and γ : ΩT × R × RN → R are Caratheodory functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Then we introduce the approximate problem of the system (10) as � � � � � � � −∂tφn ω − ν div (a1(∇φn ω)) + γ(ρn θ, ∇φn ω) = 0, in ΩT, ∂tρn θ − ν div (a2(∇ρn θ)) − div (a3(ρn θ, ∇φn ω) = 0, in ΩT, ρn θ(0, x) = ρ0(x), φn ω(T, x) = g(x, ρn θ(T, x)), in Ω, ρn θ(t, x) = φn ω(t, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' in Γ, (11) Let us first introduce some definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Let r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In the sequel we denote by Lr � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,r 0 (Ω) � the set of functions u such that u ∈ Lr (ΩT), u(t, ·) ∈ W 1,r 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The space Lr � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,r 0 (Ω) � is equipped with the norm ∥u∥Lr(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='W 1,r 0 (Ω)) := �� T 0 � Ω |∇u(x, t)|rdxdt � 1 r , is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For s, r ≥ 1, the space V s,r 0 (ΩT) := L∞ (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Ls(Ω)) ∩ Lr � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,r 0 (Ω) � endowed with the norm ∥ϕ∥V s,r 0 (ΩT ) := ess sup 0≤t≤T ∥ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=', t)∥Ls(Ω) + ∥ϕ∥Lr(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='W 1,r 0 (Ω)), is also a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For this convergence, we make the following set of assumptions, (H4): There is a constant µ > 0 and positive functions κ(t, x), λ(t, x) such that for all (t, x) ∈ ΩT, we have ∥a3(t, x, ρ, p)∥ ≤ µ(κ(t, x) + ∥p∥), and |γ(t, x, ρ, p)| ≤ λ(t, x)∥p∥, with κ ∈ L2 (ΩT) , λ ∈ Ld+2 (ΩT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (H5): a3(t, x, ρ, p) and γ(t, x, ρ, p) are Lipschitz continuous in (t, x, ρ, p) ∈ ΩT×R×Rd uniformly on compacts of the form � (t, x) ∈ ¯ΩT, |ρ| ≤ C, |p| ≤ C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (H6): There is a positive constant α > 0 such that a3(t, x, ρ, p)p ≥ α|p|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 9 (H7): For every n ∈ N, ρn θ, φn ω ∈ C1,2 �¯ΩT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In addition, (ρn θ)n∈N , (φn ω)n∈N ∈ L2 (ΩT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Under previous assumptions (H4)-(H7), if we assume that (10) has a unique bounded solution (φ, ρ) ∈ V 2,2 0 ×V 2,2 0 , then (φn ω, ρn θ) converge to (φ, ρ) strongly in Lp (ΩT) × Lp (ΩT) for every p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The proof of this theorem is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Related Works 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Related Works GANs: Generative adversarial networks, or GANs, are a class of ma- chine learning introduced in 2014 [27] that have been successful in generat- ing images and processing data [28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In recent years, there has been increasing interest in using GANs for financial modeling as well [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' GANs consist of two neural networks, a generator network, and a discriminator network, that work against each other in order to generate samples from a specific distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' As described in various sources [27, 32, 33], the goal is to reach equilibrium for the following problem, min G max D � Ex∼Pdata(x)[log(D(x)] + Ez∼Pg(z)[log(1 − D(G(z))] � , (12) where Pdata(x) is the original data and Pg(z) is the noise data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In (12), the goal is to minimize the generator’s output (G) and maximize the discrimina- tor’s output (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This is achieved by comparing the probability of the original data Pdata(x) being correctly identified by the discriminator D with the prob- ability of the generated data G produced by the generator using noise data Pg(z) being incorrectly identified as real by the discriminator 1 − D(G(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Essentially, the discriminator is trying to accurately distinguish between real and fake data, while the generator is attempting to create fake data that can deceive the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' APAC-Net: In [17], the authors present a method (APAC-Net) based on GANs for solving high-dimensional MFGs in the stochastic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' They use of the Hopf formula in density space to reformulate the MFGs as a saddle- point problem given by, inf ρ(x,t) sup φ(x,t) � Ez∼P(z),t∼Unif[0,T][∂tφ(ρ(t, z), t) + ν∆φ(ρ(t, z), t) − H(ρ(t, z), ∇φ)] + Ez∼P(z)φ(0, ρ(0, z)) − Ex∼ρT φ(T, x) � , (13) 10 where H(x, p) = infv{−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='v + L(x, v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In this case, we have a connection between the GANs and MFGs, since (13) allows them to reach the Kantorovich-Rubenstein dual formulation of Wasserstein GANs [33] given by, min G max D {Ex∼Pdata(x)[(D(x)] − Ez∼Pg(z)[(D(G(z))]}, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ||∇D|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (14) Finally, we can use an algorithm similar to GANs to solve the problems of MFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Unfortunately, we notice that the Hamiltonian in this situation has a separable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Due to this, we cannot solve the MFG-LWR system (to be detailed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In general, we cannot solve the MFGs problems, where its Hamiltonian is non-separable, since we cannot reformulate MFGs as 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' MFGANs: In [18, 17], the connection between GANs and MFGs is demonstrated by the fact that equation (13) allows them to both reach the Kantorovich-Rubinstein dual formulation of Wasserstein GANs, as described in reference [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This is shown in equation (12), which can be solved using an algorithm similar to those used for GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' However, it is not possible to solve MFGs problems with non-separable Hamiltonians, as they cannot be reformulated as in equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This is because the Hamiltonian in these cases has a separable structure, which prevents the solution of the MFG- LWR system (to be discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' DGM-MFG: In [34], section 4 discusses the adaptation of the DGM al- gorithm to solve mean field games, referred to as DGM-MFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This method is highly versatile and can effectively solve a wide range of partial differential equations due to its lack of reliance on the specific structure of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Our own work is similar to DGM-MFG in that we also utilize neural net- works to approximate unknown functions and adjust parameters to minimize a loss function based on the PDE residual, as seen in [34] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' However, our approach, referred to as New-Method, differs in the way it is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Instead of using the sum of PDE residuals as the loss function and SGD for optimization, we define a separate loss function for each equation and use ADAM for training, following the approach in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This modification allows 11 for faster and more accurate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Policy iteration Method: To the best of our knowledge, [22] was the first to successfully solve systems of mean field game partial differential equa- tions with non-separable Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' They proposed two algorithms based on policy iteration, which involve iteratively updating the population distri- bution, value function, and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' These algorithms only require the solu- tion of two decoupled, linear PDEs at each iteration due to the fixed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This approach reduces the complexity of the equations, but it is limited to low-dimensional problems due to the computationally intensive nature of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In contrast, our method utilizes neural networks to solve the HJB and FP equations at each iteration, allowing for updates to the population distribution and value function in each equation without the limitations of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Numerical Experiments To evaluate the effectiveness of the proposed algorithm [1], we use the example provided in [17], as it has an explicitly defined solution structure that allows for easy numerical comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We compare the performance of New-Method, APAC-Net’s MFGAN, and DGM-MFG on the same data to assess their reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Additionally, we apply New-Method to the traffic flow problem [19], which is characterized by its non-separable Hamiltonian [20], to determine its ability to solve this type of problem in a stochastic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Analytic Comparison We test our method by comparing it to a simple example of the analytic solution used to test the effectiveness of Apac-Net [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For the sake of simplicity, we take the spatial domain Ω = [−2, 2]d, the final time T = 1, and without congestion (γ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For H0(x, p) = ||p||2 2 − β ||x||2 2 , f0(x, ρ) = γln(ρ), g(x) = α ||x||2 2 − (νdα + γ d 2ln α 2πν), (15) and ν = β = 1, where α = −γ + � γ2 + 4ν2β 2ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 12 The corresponding MFG system is: � � � � � � � � � −∂tφ − ∆φ + ||∇φ||2 2 − ||x||2 2 = 0, ∂tρ − ∆ρ − div (ρ∇φ) = 0, ρ(0, x) = ( 1 2π) d 2e− ||x||2 2 , φ(T, x) = x2 2 − d, (16) and the explicit formula is given by φ(t, x) = ||x||2 2 − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='t, ρ(t, x) = ( 1 2π) d 2e− ||x||2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (17) Test 1: We consider the system of PDEs [16] in one dimension (d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To obtain results, we run Algorithm [1] for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='103 iterations, using a minibatch of 50 samples at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The neural networks employed have three hidden layers with 100 neurons each, and utilize the Softplus activation function for Nω and the Tanh activation function for Nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Both networks use ADAM with a learning rate of 10−4 and a weight decay of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We employ ResNet as the architecture of the neural networks, with a skip connection weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The numerical results are shown in Figure 2, which compares the approximate solutions obtained by New-Method to the exact solutions at different time states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To evaluate the performance of New-Method, we compute the relative error between the model predictions and the exact solutions on a 100 × 100 grid within the domain [0, 1] × [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Additionally, we plot the HJB and FP residual loss, as defined in Algorithm [1], to monitor the convergence of our method (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Test 2: In this experiment, we use a single hidden layer with vary- ing numbers of hidden units (nU) for both neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' As previously shown in section 2, the number of hidden units can affect the convergence of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To verify this, we repeat the previous test using the same hyper- parameters and a single hidden layer but with different numbers of hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The relative error between the model predictions and the exact solu- tions is then calculated on a 100×100 grid within the domain [0, 1]×[−2, 2], as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Test 3: We solve the MFG system [16] for dimensions 2, 50, and 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Figure 5 shows the residuals of the HJB and FP equations over 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='104 iter- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' A minibatch of 1024, 512, and 128 samples were used for d=100, d=50, and d=2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The neural networks had three hidden layers 13 Figure 2: The exact solution and prediction calculated by New-Method in dimension one at t=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='75 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Figure 3: The relative error for ρ, φ for the figure on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' On the right, the HJB, FP Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 14 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 175 pexact 150 p new method 125 exact @ new method 1D0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 1D 15 2Dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 150 pexact 125 p new method LDO - exact @ new method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 1D 15 2Dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='75 125 pexact 1D0 p new method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='75 exact @ new method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 1D 15 2DThe relative errorof p and 10 Relative errar 0 2400 ODE 50D0 iberationsResiduals for Fp and HjB cguation loss FP 10° loss HJB 10-1 Losg 10- 10 10 0 2400 ODE 400 50D0 IeratiorsFigure 4: The relative error for ρ , φ in 1-dimension for nU=(2, 5, 10, 20, 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' with 100 neurons each and utilized the Softplus activation function for Nω and the Tanh activation function for Nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Both networks used ADAM with a learning rate of 10−4, weight decay of 10−3, and employed ResNet as their architecture with a skip connection weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The results were obtained by recording the residuals every 100 iterations and using a rolling average over 5 points to smooth out the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Figure 5: The loss HJB and FP equation for d=(2,10,100) Test 4: In this test, we use the same setup as before, but with a single layer of 100 neurons instead of multiple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We keep all other neural network hyperparameters unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This test is meant to demonstrate that a single layer can perform better than multiple layers, even when the dimension increases, as seen in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Figure 6 shows improved results compared to the previous test, even with few iterations, which allows for faster computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 15 Residuals for Fp equation 10-2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='- d=2 d=50 E-OT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' d=100 10+ lenprs 10 10- 107 10-8 10- 0 14000 24000 ODIDE 4D00 50000 iberationa cf AdamThe relative error phi 100 nU=2 10-1 nU=5 nU=10 nU=20 nU=50 0 2400 30 400 50D0 iberatiors of AdamlThe relative eror rha nU=2 nU=5 = nU=10 100 nU=20 nU=50 XYYY 101 0 2400 30D0 400 50D0 iberatior of AdamlResiduals for HjB eguation d=2 103 -- d=50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' d=100 107 10 100 10-1 10~2 0 1ADO0 24000 ODIDE 4D00 50000 iberationa cf AdamFigure 6: The loss HJB and FP equation with a minibatch of 128, 512, and 1024 samples for d=2, d=50, and d=(100,200,300), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Comparison In previous sections, we introduced and discussed four methods for solv- ing MFGs: APAC-Net, MFGAN, DGM-MFG, and New-Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Here, we compare these approaches to assess their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For APAC-Net, it is only possible to compare the cost values φ due to the unavailability of the density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In APAC-Net, the generator neural network represents ρ, which generates the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In order to compare the results, we need to use kernel density estimation to transform the distribution into a density, which is only an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We use the simple example from the analytic so- lution with d = 1 and T = 1 for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The two neural networks in this comparison have three hidden layers with 100 neurons each, and utilize ResNet as their architecture with a skip connection weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' They also use the Softplus activation function for Nω and the Tanh activation function for Nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For training APAC-Net, MFGAN, and New-Method, we use ADAM with a learning rate of 10−4 and a weight decay of 10−3 for both networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For training DGM-MFG, we use SGD initialized with a value of 10−3 and a weight decay of 10−3 for both networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We run the four algorithms for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='103 iterations, using a minibatch of 50 sam- ples at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The relative error between the model predictions and the exact solutions is then calculated on a 100 × 100 grid within the domain [0, 1] × [−2, 2], as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Application (Traffic Flow): In a study published in [1], the authors focused on the longitudinal speed control of autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' They developed a mathematical model called 16 Residual for HjB eguation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' d=2 104 d=50 d=100 d=200 102 — d=300 Bso1 lenpgad 100 102 , 0 2400 14000 keratiorsResidual for Fp eguation 101 10-2 10-3 10-4 d=2 101 d=50 d=100 10-0 d=200 d=300 0 2400 410 6+DO 14000 keratiorsFigure 7: comparison between APAC-Net, MFGAN, DGM-MFG, and New-Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' a Mean Field Game (MFG) to solve a traffic flow problem for autonomous vehicles and demonstrated that the traditional Lighthill-Whitham-Richards (LWR) model can be used as a solution to the MFG-LWR model described by the following system of equations: MFG − LWR � � � � � � � Vt + U(ρ)Vx − 1 2V 2 x = 0, ρt + (ρu)x = 0, u = U(ρ) − Vx, VT = g(·, ρT), ρ(·, 0) = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (18) Here, ρ, V , and u represent the density, optimal cost, and speed function, respectively, and the Greenshields density-speed relation is given by U(ρ) = umax(1 − ρ/ρjam), where ρjam is the jam density and umax is the maximum speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' By setting ρjam = 1 and umax = 1, the authors generalized the MFG- LWR model to include a viscosity term µ > 0, resulting in the following system: MFG − LWR � � � Vt + ν∆V − H(x, p, ρ) = 0, ρt − ν∆ρ − div(∇pH(x, p, ρ)ρ) = 0, VT = g(·, ρT), ρ(·, 0) = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (19) In this model, ρ and V represent the density and optimal cost function, respectively, and H is the Hamiltonian with a non-separable structure given by H(x, p, ρ) = 1 2||p||2 − (1 − ρ)p, with p = Vx, (20) where p = Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The authors solved the system in (19) using the New- ton Iteration method for the deterministic case (ν = 0) with a numerical 17 Comparison of p New-Method 100 MFGAN DGM-MFG Relative errar 10-1 0 2400 ODE 50D0 iberationsComparison of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='.- New-Method MFGAN DGM-MFG 100 APAC net Relative errar 10-1 0 24D0 ODE 5400 iberationsmethod that considers only a finite number of discretization points to re- duce computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In this work, we propose a new method using a neural network to approximate the unknown and solve the prob- lem in the stochastic case, while also avoiding the computational complexity of the previous method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To evaluate the performance of the new method, we consider the traffic flow problem defined by the MFG-LWR model in 19 with a non-separable Hamiltonian in (20) on the spatial domain Ω = [0, 1] with dimension d = 1 and final time T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The terminal cost g is set to zero and the initial density ρ0 is given by a Gaussian distribution, ρ0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='6 exp � −1 2 � x−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The aim is to investigate the perfor- mance of the new method, called the ”New-Method,” in solving this traffic flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The corresponding MFG system is, � � � � � � � Vt + ν∆V − 1 2||Vx||2 + (1 − ρ)Vx = 0 ρt − ν∆ρ − div((Vx − (1 − ρ))ρ) = 0 ρ(x, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='6 exp( −1 2 ( x−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1 )2), φ(x, T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (21) We study the deterministic case (ν = 0) and stochastic case (ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We represent the unknown solutions by two neural networks Nω and Nθ, which have a single hidden layer of 50 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We use the ResNet architecture with a skip connection weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We employ ADAM with learning rate 4 × 10−4 for Nω and 5 × 10−4 for Nθ and weight decay of 10−4 for both networks, batch size 100, in both cases ν = 0 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 we use the activation function Softmax and Relu for Nω and Nθ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In Figure 8 we plot over different times the density function, the optimal cost, and the speed which is calculated according to the density and the optimal cost [1] by the following formula, u = umax(1 − ρ/ρjam) − Vx where, we take the jam density ρjam = 1 and the maximum speed umax = 1 and 104 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ++ In Figure (9), we plot the HJB, FP residual loss for ν = 0 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5, which helps us monitor the convergence of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Unfortunately, we do not have the exact solution to compute the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To validate the results of Figure (8), we use the fundamental traffic flow diagram, an essential tool to comprehend classic traffic flow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Precisely, this is a graphic that 18 t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 t=1 Figure 8: The solution of the problem MFG-LWR by New-Method for (ν = 0) and (ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5) at t=(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Figure 9: The loss HJB and FP equation for (ν = 0) and (ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 19 v=o 10-1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='- loss FP loss HJB 10-3 i 10- 10~7 10-5 1022, 1013 0 2400 400 14000 keratiorsv=05 101 loss FP loss HJB 102 10-3 Bso1 lenpgad 10 100 10~7 10-8 0 2400 6+DO 14000 keratiorsdensity speed Optimal costdensity speed Optimal costdensity speed Optimal costdensity speed Optimal costdensity speed Optimal costdensity speed Optimal cost0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='4 speed EO Optimal cost 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='8 10displays a link between road traffic flux (vehicles/hour) and the traffic density (vehicles/km) [35, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We can find this diagram numerically [1] such as its function q is given by, q(t, x) = ρ(t, x)u(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Figure (10) shows the fundamental diagram of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 t=1 Figure 10: Fundamental diagram for ν = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5) at t = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Conclusion We present a new method based on the deep galerkin method (DGM) for solving high-dimensional stochastic mean field games (MFGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The key idea of our algorithm is to approximate the unknown solutions by two neural networks that were simultaneously trained to satisfy each equation of the MFGs system and forward-backward conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Consequently, our method shows better results even in a small number of iterations because of its learning mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Moreover, it shows the 20 densitydenstbydensitydensitydansitydensityV=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='22 flow 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='24 densitypotential of up to 300 dimensions with a single layer, which gives more speed to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' we proved that as the number of hidden units increases, the neural networks converge to the MFG solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Comparison with the previous methods shows the efficiency of our ap- proach even with multilayer neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Test on traffic flow problem in the deterministic case gives results sim- ilar to the newton iteration method, showing that it can solve this problem in the stochastic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To address the issue of high dimensions in the problem, we used a neural network but found that it took a significant amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' While our approach has helped to reduce the time required, it is still not fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Therefore, we are seeking an alternative to neural networks in future research to improve efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Denote N(σ) the space of all functions implemented by such a network with a single hidden layer and n hidden units, where σ in C2 � Rd+1� non- constant, and bonded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' By (H1) we have that for all ρ, φ ∈ C1,2 � [0, T] × Rd� and ε1, ε2 > 0, There is ρθ, φω ∈ N(σ) such That, sup (t,x)∈E1 |∂tφ(t, x) − ∂tφω(t, x)| + max |a|≤2 sup (t,x)∈E1 ��∂(a) x φ(t, x) − ∂(a) x φω(t, x) �� < ϵ1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1) sup (t,x)∈E2 |∂tρ(t, x) − ∂tρθ(t, x)| + max |a|≤2 sup (t,x)∈E2 ��∂(a) x ρ(t, x) − ∂(a) x ρθ(t, x) �� < ϵ2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2) From (H3) we have that (ρ, p) �→ H(x, ρ, p) is locally Lipschitz continuous in (ρ, p), with Lipschitz constant that can have at most polynomial growth in ρ and p, uniformly with respect to t, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' This means that |H(x, ρ, p) − H(x, γ, s)| ≤ � |ρ|q1/2 + |p|q2/2 + |γ|q3/2 + |s|q4/2� × (|ρ − γ| + |p − s|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 21 with some constants 0 ≤ q1, q2, q3, q4 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' we get using H¨older inequality with exponents r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' � E1 |H (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇xφω) − H (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ)|2 dµ1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) ≤ � E1 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q1 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q2 + |ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q3 + |∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q4) × � |ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2� dµ1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) ≤ � � E1 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q1 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q2 + |ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q3 + |∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q4)r1dµ1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r1 × � � E1 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2)r2dµ1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r2 ≤ C1 � � E1 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q1 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q2 + |ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q1∨q3 + |∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q2∨q4)r1dµ1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r1 × � � E1 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2)r2dµ1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r2 ≤ C1 � ϵq1 1 + ϵq2 2 + sup E1 |ρ|q1∨q3 + sup E1 |∇φ|q2∨q4 � (ϵ2 1 + ϵ2 2) ≤ C1(ϵ2 1 + ϵ2 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' where the constant C1 < ∞ may change from line to line and qi ∨ qj = max{qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' qj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In the two last steps we used A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2 and (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We recall that, H1(ρθ, φω) = ∂tφω(t, x) + ν∆φω(t, x) − H(x, ρθ(t, x), ∇φω(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 22 Note that H1(ρ, φ) = 0 for ρ, θ that solves the system of PDEs, L1(ρθ, φω) = ���H1(ρθ, φω) ��� 2 L2(E1) + ���φω(T, x) − φ(T, x) ��� 2 L2(Ω) = ���H1(ρθ, φω) − H1(ρ, φ) ��� 2 L2(E1) + ���φω(x, T) − g(x, ρθ(x, T)) ��� 2 L2(Ω) ≤ � E1 |∂tφω(t, x) − ∂tφ(t, x)|2 dµ1(t, x) + |ν| � E1 |∆φω(t, x) − ∆φ(t, x)|2 dµ1(t, x) + � E1 |H (x, ρθ, ∇φω) − H (x, ρ, ∇φ)|2 dµ1(t, x) + � Ω |φω(T, x) − φ(T, x)|2dµ2(t, x) ≤C1(ϵ2 1 + ϵ2 2) for an appropriate constant C1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In the last step, we use A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2 and the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' For L2 we use remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1 to simplified the nonlinear term, div(ρ∇pH(x, ρ, ∇φ)) = α1(x, ρ, ∇φ) + α2(x, ρ, ∇φ) + α3(x, ρ, ∇φ), where, α1(x, ρ, ∇φ) = ∇pH(x, ρ, ∇φ)∇ρ, α2(x, ρ, ∇φ) = ∇pρH(x, ρ, ∇φ)∇ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='ρ, α3(x, ρ, ∇φ) = � i,j ∇pipjH(x, ρ, ∇φ)(∂xjxiφ)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' In addition, from (H3) we have also ∇pH(x, ρ, p), ∇pρH(x, ρ, p), and ∇ppH(x, ρ, p) are locally Lipschitz continuous in (ρ, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' we have after an application of Holder inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' for some constant C2 < ∞ that may change from line 23 to line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' � E2 |α1 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω) − α1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ)|2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) = � E2 |∇pωH (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω) ∇ρθ − ∇pH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ)∇ρ|2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) ≤ � E2 ��� � ∇pωH (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω) − ∇pH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ) � ∇ρ ��� 2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) + � E2 ���∇pωH (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω) (∇ρθ − ∇ρ) ��� 2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) ≤ C2 �� E2 ���∇pωH (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω) − ∇pH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ) ��� 2r1dµ3 (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r1 × � � E2 |∇ρ|2r2dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r2 + C2 �� E2 ���∇pωH (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φω) ��� 2s1dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/s1 × �� E2 |∇ρθ − ∇ρ|2s2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/s2 ≤ C2 � � E2 |∇ρ|2r2dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/r2 × � � E2 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q1 + |∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q2 + |ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q1∨q3 + |∇φ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|q2∨q4)v1r1dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/v1r1 × � � E2 (|ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2 + |∇xφω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∇xφ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2)v2r2dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/v2r2 + C2 �� E2 ���∇pωH (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φω) ��� 2s1dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/s1 × �� E2 |∇ρθ − ∇ρ|2s2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) �1/s2 ≤ C2(ϵ2 1 + ϵ2 2) where in the last steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' we followed the computations previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We do same for α2(x, ρ, ∇φ) and α3(x, ρ, ∇φ), we obtain for a C2 < ∞, � E2 ��� div(ρθ∇pωH(x, ρθ, ∇φω)) − div(ρ∇pH(x, ρ, ∇φ)) ��� 2 dµ3(t, x) ≤ C2(ϵ2 1 + ϵ2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 24 We recall that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' H2(ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φω) = ∂tρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)−ν∆ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)−div (ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)∇pH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x))) Note that H2(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φ) = 0 for ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' θ that solves the system of PDEs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' then we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' L2(ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φω) = ���H2(ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φω) ��� 2 L2(E2) + ���ρθ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ0(x) ��� 2 L2(Ω) = ���H2(ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φω) − H2(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' φ) ��� 2 L2(E2) + ���ρθ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ0(x) ��� 2 L2(Ω) ≤ � E2 |∂tρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∂tρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) + |ν| � E2 |∆ρθ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ∆ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x)|2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) + � E2 ��� div(ρθ∇pωH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φω)) − div(ρ∇pH(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇φ)) ��� 2 dµ3(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) + � Ω |ρθ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) − ρ0(x)|2dµ4(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' x) ≤C2(ϵ2 1 + ϵ2 2) for an appropriate constant C2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' The proof of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='1 is complete after rescaling ϵ1 and ϵ2 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' We follow the method used in [23] for a single PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' (See also section 4 in [38] for a coupled system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Let us denote the solution of problem 11 by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' � ˆρn θ, ˆφn ω � ∈ V = V 2,2 0 × V 2,2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Due to Conditions (H4) − (H6) and by using lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='4 [39] on each equation then, there exist, C1, C2 such that: ∥ˆρn θ∥V 2,2 0 ≤ C1 ∥ˆφn ω∥V 2,2 0 ≤ C2 These applies and gives that the both sequence {ˆρn θ}n∈N, {ˆφn ω}n∈N are uni- formly bounded with respect to n in at least V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' These uniform energy bounds 25 imply the existence of two subsequences, (still denoted in the same way) {ˆρn θ}n∈N, {ˆφn ω}n∈N and two functions ρ, φ in L2 � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,2 0 (Ω) � such that, ˆρn θ → ρ weakly in L2 � 0, T : W 1,2 0 (Ω) � ˆφn ω → φ weakly in L2 � 0, T : W 1,2 0 (Ω) � Next let us set q = 1 + d d+4 ∈ (1, 2) and note that for conjugates, r1, r2 > 1 such that 1/r1 + 1/r2 = 1 � ΩT ���γ � t, x, ˆρn θ, ∇ˆφn ω ���� q ≤ � ΩT |λ|q ���∇ˆφn ω ��� q ≤ �� ΩT |λ|r1q �1/r1 �� ΩT ���∇ˆφn ω ��� r2q�1/r2 Let us choose r2 = 2/q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Then we calculate r1 = r2 r2−1 = 2 2−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Hence, we have that r1q = d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Recalling the assumption λ ∈ Ld+2 (ΩT) and the uniform bound on the ∇ˆφn ω we subsequently obtain that for q = 1 + d d+4, there is a constant C < ∞ such that � ΩT ���γ � t, x, ˆρn θ, ∇ˆφn ω ���� q ≤ C On the other hand, it is obvious that a1 is bounded uniformly then, according to the HJB equation of 11, we have � ∂t ˆφn ω � n∈N is bounded uniformly with respect to n in L2 (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W −1,2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Then we can extract a subsequence, (still denoted in the same way) � ∂t ˆφn ω � n∈N such that ∂t ˆφn θ → ∂tφ weakly in L2 � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W −1,2(Ω) � Also, it will be shown that ∂tˆρn θ → ∂tρ weakly in L2 � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W −1,2(Ω) � Since the problem is nonlinear, the weak convergence of ˆφn ω and ˆρn θ in the space L2 � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,2 0 (Ω) � is not enough in order to prove that φ and ρ are a solution of problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' To do this, we need the almost everywhere conver- gence of the gradients for a subsequence of the approximating solutions ˆφn ω and ˆρn θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' 26 However, the uniform boundedness of {ˆφn ω}n∈N and {ˆρn θ}n∈N in L2 � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,2 0 (Ω) � and their weak convergence to φ and ρ respectively in that space, allows us to conclude, by using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='3 of [40] on each equation, that ∇ˆφn ω → ∇φ almost everywhere in ΩT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' ∇ˆρn θ → ∇ρ almost everywhere in ΩT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' Hence, we obtain that {ˆφn ω}n∈N and {ˆρn θ}n∈N converges respectively to φ and ρ strongly in Lp � 0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' W 1,p 0 (Ω) � for every p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' It remains to discuss the convergence of φn ω − ˆφn ω and ρn θ − ˆρn θ to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content=' By last step of proof theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQfDgMw/content/2301.02877v1.pdf'} +page_content='3 [23] we get � φn ω − ˆφn ω � n∈N and {ρn θ − ˆρn θ}n∈N goes to zero strongly in Lp (ΩT) for every p < 2.' metadata={'source': 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a/E9FRT4oBgHgl3EQfyzhp/content/tmp_files/2301.13647v1.pdf.txt b/E9FRT4oBgHgl3EQfyzhp/content/tmp_files/2301.13647v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed14ba8ad7bca92e08306fe0482265b69ca39abb --- /dev/null +++ b/E9FRT4oBgHgl3EQfyzhp/content/tmp_files/2301.13647v1.pdf.txt @@ -0,0 +1,1420 @@ +arXiv:2301.13647v1 [physics.data-an] 31 Jan 2023 +Bayesian estimation of information-theoretic metrics for sparsely sampled distributions +Angelo Piga,∗ Lluc Font-Pomarol,† Marta Sales-Pardo,‡ and Roger Guimer`a§ +(Dated: February 1, 2023) +Estimating the Shannon entropy of a discrete distribution from which we have only observed a small sample is +challenging. Estimating other information-theoretic metrics, such as the Kullback-Leibler divergence between +two sparsely sampled discrete distributions, is even harder. Existing approaches to address these problems +have shortcomings: they are biased, heuristic, work only for some distributions, and/or cannot be applied to all +information-theoretic metrics. Here, we propose a fast, semi-analytical estimator for sparsely sampled distribu- +tions that is efficient, precise, and general. Its derivation is grounded in probabilistic considerations and uses a +hierarchical Bayesian approach to extract as much information as possible from the few observations available. +Our approach provides estimates of the Shannon entropy with precision at least comparable to the state of the +art, and most often better. It can also be used to obtain accurate estimates of any other information-theoretic +metric, including the notoriously challenging Kullback-Leibler divergence. Here, again, our approach performs +consistently better than existing estimators. +I. +INTRODUCTION +Information theory is gaining momentum as a methodolog- +ical framework to study complex systems. In network sci- +ence, information theory provides rigorous tools to predict +unobserved links [1] and to infer community structure [2]. +In neuroscience, Shannon entropy of spike train distributions +characterizes brain activity from neural responses [3], while +mutual information identifies correlations between brain stim- +uli and responses [4]. Recently, the Kullback-Leibler diver- +gence [5] and its regularized version, the Jensen-Shannon dis- +tance, have also been successfully used in a wide variety of +contexts: in cognitive science as a measure of “surprise,” to +quantify and predict how human attention is oriented between +changing screen images [6]; in quantitative social science, +in combination with topic models, to track the propagation +of political and social discourses [7, 8] or to understand the +emergence of social disruption from the analysis of judicial +decisions [9]; and in machine learning, at the intersection be- +tween the statistical physics of diffusive processes, probabilis- +tic models and deep neural networks [10]. +Information theoretical metrics are measured on distribu- +tions. In practice, a distribution ρ over the possible states +of a system, as well as functions F(ρ) of this distribution +(such as Shannon entropy or other metrics), have to be in- +ferred from experimental observations. However, this infer- +ence process is difficult for many real complex systems since, +due to experimental limitations, the observations are often +sparse, and statistical estimates of the distribution ρ and its +functions can be severely biased. Here, we focus on the par- +ticular yet important case of discrete (or categorical) distribu- +tions ρi, i = 1, . . . , K, where K is the number of possible +∗ angelo.piga@gmail.com; Department of Chemical Engineering, Universi- +tat Rovira i Virgili, Tarragona 43007, Catalonia. +† lluc.font@urv.cat; +Department of Chemical Engineering, +Universitat +Rovira i Virgili, Tarragona 43007, Catalonia. +‡ marta.sales@urv.cat; Department of Chemical Engineering, Universitat +Rovira i Virgili, Tarragona 43007, Catalonia. +§ roger.guimera@urv.cat; Department of Chemical Engineering, Universitat +Rovira i Virgili, Tarragona 43007, Catalonia.; ICREA, Barcelona 08010, +Catalonia. +states (or categories), which is known and fixed. Inferences +about ρ and any function must be based on ni, the number +of observations in the i-th state (with N = � +i ni the sam- +ple size) and, in the undersampled regime we are interested +in, N ≲ K. The challenge is thus, from the sparse observa- +tions {ni}, to infer the probability ρi of each category i and +estimate metrics F(ρ). +A theoretically well-founded approach to tackle this prob- +lem is provided by the principles of conditional probability, +encapsulated in Bayes’ theorem [11]. This framework is in +general preferable because of its transparency—it requires +that all assumptions of the underlying generative model for the +data are made explicit, expressed via the choice of a likelihood +function and a prior distribution that reflects the knowledge +about the system before observing any data. In probabilistic +reasoning, the combination of observations and prior distri- +bution provides an updated (posterior) probability distribution +of the quantity under study. Other estimation strategies make +implicit assumptions and often provide only point estimates, +as opposed to full distributions. +A class of expressive generative models for categorical dis- +tributions amenable to a Bayesian framework is the well- +studied family of Dirichlet distributions. However, as Ne- +menmann, Shafee, and Bialek (henceforth NSB) pointed out +in [12], when sample sizes are small (N ≲ K), the inferred +Shannon entropy is tightly determined by the specific parame- +ters one chooses for the Dirichlet model; therefore, inaccurate +choices result in severe biases of the Shannon entropy esti- +mates. To overcome this problem, they introduced a mixture +of Dirichlet models, which results in a very precise estimator +of the Shannon entropy that works for a wide variety of distri- +butions, even in the sparse sampling regime N ≲ K [12, 13]. +Although, in terms of precision, NSB can be considered the +state of the art for estimating the Shannon entropy, it does not +provide estimates for the distribution ρ. For this reason, its ap- +plicability is limited to estimating the Shannon entropy (and +related information theoretic quantities like mutual informa- +tion and Jensen-Shannon distance, which can be expressed in +terms of entropies). By contrast, it cannot be used to estimate +the Kullback-Leibler divergence. To cover this gap, Hausser +and Strimmer derived a James-Stein-type shrinkage estimator +for ρ [14] (henceforth HS), which has the advantage of being + +2 +analytical and applicable to any information-theoretic metric, +but at the price of making implicit ad hoc assumptions, and +of being less precise than NSB for the Shannon entropy, and +lacking error estimation. +Here, we propose an alternative fast, semi-analytical es- +timator for distributions that is efficient, precise, and gen- +eral. Its derivation is grounded in probabilistic considerations, +without any ad hoc assumptions. We consider Dirichlet gen- +erative models and use a hierarchical Bayesian approach to +extract as much information as possible from the few observa- +tions at hand. In the case of Shannon entropy, we can estimate +the expected value and higher order moments with precision +at least comparable to the NSB estimator, and most often bet- +ter. Additionally, because our method provides estimates of +the probability distribution, it can be used to obtain accurate +estimations of the Kullback-Leibler divergence. In this case +our approach also performs equally or better than existing es- +timators. +II. +BACKGROUND +Let us consider a system with K possible output states +whose observations follow an unknown discrete distribu- +tion ρ = {ρi; i = 1, . . . , K} with � +i ρi = 1. +The vector +n = {ni; i = 1, . . . , K} represents the number of times each +state was observed in a set of � +i ni = N independent obser- +vations of the system. We also consider a function F(ρ) of ρ, +such as, for example, the Shannon entropy +S(ρ) = − +K +� +i=1 +ρi log ρi , +(1) +which we want to estimate from the set of observations. +The posterior distribution over the values of the function F +given the observed counts n is +p (F|n) = +� +dρ δ (F − F(ρ)) p(ρ|n) , +(2) +where p(ρ|n) is the posterior of the distribution ρ given the +counts n. We further assume that the prior over distributions +depends on a parameter β, which becomes a hyperparameter +of our generative model. Then, using the laws of conditional +probability, we can write the posterior p(ρ|n, β) as +p(ρ|n, β) = p(n|ρ, β) p(ρ|β) +p(n|β) +, +(3) +where p(n|ρ, β) is the likelihood, p(ρ|β) is the prior over +distributions, and p(n|β) = +� +dρ p(n|ρ) p(ρ|β) is the evi- +dence and acts as normalization factor. The likelihood is the +probability of the empirical observations n given ρ; for in- +dependent multinomial samples, the probability of observing +an event of type i is ρi, and the full likelihood is the prod- +uct p(n|ρ, β) = p(n|ρ) = N! �K +i ρni +i /ni! and, given ρ, it +is independent of the hyperparameter β. The prior p(ρ|β) +expresses the probability of each distribution ρ prior to ob- +serving any data, and plays a crucial role in the discussion be- +low. Symmetric Dirichlet distributions are convenient priors +because they are a generative model for a broad class of dis- +crete distributions. Additionally, they have been widely used +in this setting [15], and are parametrized as follows +p(ρ|β) = +1 +BK(β) +K +� +i=1 +ρβ−1 +i +, +BK(β) = Γ(β)K +Γ(βK) , +(4) +where Γ is the gamma function, while the hyperparameter β is +a real, positive number known as the concentration parameter. +In the first row of Fig. 1, examples of categorical distributions +sampled from symmetric Dirichlet priors are shown. +Besides being very expressive, Dirichlet priors are conju- +gate distributions of categorical likelihoods, meaning that the +posterior is still a Dirichlet distribution, a property that often +makes the inference via Eqs. (3) and (2) analytically tractable. +For example, when F(ρ) = ρ, Dirichlet priors lead to ex- +pected posterior probabilities ⟨ρi⟩ given by the widely-used +generalized Laplace’s formula +⟨ρi⟩ = +ni + β +N + Kβ . +(5) +It is worth noting the improvement of Eq. (5) with respect to +the maximum likelihood (or frequency) estimator ρi = ni/N, +which is recovered by the former in the limit β → 0. In par- +ticular, Laplace’s formula assigns non zero probability to non +observed states, a desirable property whose advantage will be- +come evident later, when estimating Kullback-Leibler diver- +gences. This example also illustrates how non-Bayesian ap- +proaches to inference make implicit and non-trivial assump- +tions, in this case assuming β → 0 amounts to assuming that +infinitely concentrated distributions ρ are a priori much more +plausible than more homogeneous ones. +Going back to the estimation of F from the observations +n, and given Eq. (5), one may be tempted to directly plug +the value of ⟨ρi⟩ in the explicit expression of F(ρ) to get a +point estimate. However, this is just an approximation; the +exact procedure consists in finding and using the whole pos- +terior p(F|n). Specifically, the expected value of this pos- +terior ⟨F⟩ = +� +dF F p(F|n) minimizes the mean-squared +error [16], and its mode is a consistent estimator, meaning +that it converges to the true value of F(ρ) when the num- +ber of observations increases, regardless of the prior and, in +particular, regardless of the hyperparameter β. Wolpert and +Wolf in Refs. [16, 17] provided analytical formulas for all +the moments of p(F|n) when F is the Shannon entropy and +for Dirichlet priors (we report the formula for the mean in +Eq. (15) and for the second moment in Appendix B). +However, an unbiased estimation of F is not guaranteed for +small samples. This is often the case for Dirichlet priors, es- +pecially when the parameter β is unknown. Several options +for the value of β have been proposed in literature, each one +suitable to some specific case but deficient in others (for a dis- +cussion, refer to Refs. [12, 14]). In [12], NSB suggested that, +when samples are scarce, any attempt to find a single universal +β is hopeless; the fundamental reason being that categorical +distributions generated by a Dirichlet have a Shannon entropy +that is narrowly determined by, and monotonically dependent + +3 +on, β. In other words, for small samples, the posterior distri- +bution (2) is dominated by the prior. To overcome this prob- +lem, Refs. [12, 13] proposed, as the prior pNSB(ρ), an infinite +mixture of Dirichlet priors +pNSB(ρ) ∝ +� +dβ pNSB(β) p(ρ|β) , +(6) +where the weights pNSB(β) were set so as to obtain a flat prior +over entropies S, and have the functional form +pNSB(β) ∝ d E[S|ni = 0, β] +dβ += Kψ1(Kβ +1)−ψ1(β +1) , +(7) +where E[S|n, β] is the expected entropy given the observa- +tions n, and then E[S|ni = 0, β] is the expected entropy of the +distributions ρ generated from a symmetric Dirichlet priors +(that is if there are no observations), with fixed β and K, and +ψm(x) = +� d +dx +�m+1 log Γ(x) are the polygamma functions. +The NSB prior leads to very accurate estimates of the Shannon +entropy, and can be considered the state of the art. Even if best +suited for situations in which the number of states K is known +and fixed, it is quite versatile and has been later extended for +countable infinite number of states [18] and further optimized +for binary states [19] and long tail distributions [18]. Other es- +timators, for example, the Chao-Shen estimator [20], perform +at most as well as the NSB (or its derivatives), but never better +(see [14] for a comprehensive review). Additionally, given an +estimator of S, a number of other quantities can be indirectly +estimated. For example, the mutual information M between +two distributions ρ and σ is M(ρ ; σ) = S(ρ)+S(σ)−S(π), +where π is the joint distribution of ρ and σ [21]. Similar re- +lations can be derived for Jensen-Shannon distance and other +information-theoretic quantities [8] [22]. +However, consider the estimation of the Kullback-Leibler +divergence (DKL) between two distributions ρ and σ with the +same dimension K +DKL(ρ∥σ) = +K +� +i=1 +ρi log2 +ρi +σi +. +(8) +To estimate DKL from samples n = {ni; i = 1, . . . , K} from +ρ, and m = {mi; i = 1, . . . , K} from σ, one cannot use the +NSB approach. First, DKL is not a combination of the Shan- +non entropies of the two underlying distributions ρ and σ. +Second, DKL is unbounded, and any attempt to find a hyper- +prior in the spirit of Eq. (7) results in improper hyperpriors. +Finally, with the NSB prior one renounces to any estimation +of β and, in turn, to a good a point estimation of DKL by +means of Laplace’s formula. +III. +HIERARCHICAL BAYES POINT ESTIMATE FOR β +Here, we address these limitations of the NSB estimator +while maintaining and even improving its performance. We +posit that the success of the NSB approach stems, not from +mixing infinitely many values of the concentration parameter +β, but rather from the flexibility to accommodate for any par- +ticular value of β. Indeed, we surmise that, in general, only +a narrow interval of β values are compatible with a given ob- +servation n and therefore contribute to the mixture, whereas +most others do not contribute. Motivated by this, we propose +an approach that aims to directly estimate the value of β that +most contributes to the posterior given the data n. +First, we observe that the posterior p(ρ|n) can be written as +p(ρ|n) = +� +dβ p(ρ|n, β) p(β|n) += +� +dβ p(n|ρ) p(ρ|β) +p(n|β) +p(β|n) , +(9) +where we have applied Bayes’ rule, and the fact that n condi- +tioned on ρ is independent of β, so that p(n|ρ, β) = p(n|ρ). +Then, we assume that the conditional distribution p(β|n) is +very peaked around a given value β⋆ , so that the posterior +p(ρ|n) can be approximated as +p(ρ|n) ≈ p(n|ρ) p(ρ|β⋆) +p(n|β⋆) +. +(10) +This approximation, sometimes referred to as empirical +Bayes, is a point estimate for the fully hierarchical probabilis- +tic model given by p(n|ρ) and p(ρ|β). Eq. (10) is identical to +Eq. (3), with the difference that the concentration parameter +is now the most likely value of β given the observed counts n, +that is, +β⋆ = argmax +β +p(β|n) = argmax +β +p(n|β) p(β) +p(n) +, +(11) +where p(n|β) = +� +dρ p(n|β, ρ)p(ρ|β). For Dirichlet priors +(Eq. (4)), β∗ satisfies (see Appendix A) +K +� +i=1 +ni−1 +� +m=0 +1 +m + β⋆ − +N−1 +� +m=0 +K +m + Kβ⋆ + +1 +p(β⋆) +d p(β) +d β +��� +β⋆ = 0 , +(12) +which is the key analytical result of this paper. +The hyperprior p(β) reflects our prior knowledge about the +shape of the distribution of the hyperparameter. To be com- +pletely agnostic in this regard, we can use a uniform hyper- +prior +pU(β) = +1 +∆β = const. , +∆β = βmax − βmin , +(13) +with cut-offs 0 < βmin < βmax < ∞. In this case, the deriva- +tive term in Eq. (12) disappears. The NSB hyperprior (7) is a +valid alternative; in this case, the last term in Eq. (12) is (see +appendix A for details) +1 +pNSB(β∗) +d pNSB(β) +d β +���� +β∗ = K2ψ2(kβ⋆ + 1) − ψ2(β⋆ + 1) +Kψ1(kβ⋆ + 1) − ψ1(β⋆ + 1) . +(14) +Despite the complex appearance of Eq. (12), β∗ is not +hard to obtain numerically, giving a computational improve- +ment with respect the NSB estimator, whose algorithm is +involved and has higher computational costs [23]. +The +source code of the implementations in Python is available at +https://github.com/angelopiga/info-metric-estimation/. + +4 +1 +200 +400 +600 +800 +1000 +i +0.1 +0.2 +0.3 +ρ +i +Dirichlet +: + β += +0.01, S += +0.394 +1 +200 +400 +600 +800 +1000 +i +0.0025 +0.0050 +0.0075 +Dirichlet +: + β += +1, S += +0.936 +1 +200 +400 +600 +800 +1000 +i +0.001 +0.002 +Dirichlet +: + β += +10, S += +0.993 +1 +200 +400 +600 +800 +1000 +i +0.005 +0.010 +ρ +i +half empty Dirichlet +: + β += +1, S += +0.838 +1 +200 +400 +600 +800 +1000 +i +10 +−4 +10 +−3 +10 +−2 +10 +−1 +Zipf +: + a += +1.001, S += +0.751 +1 +200 +400 +600 +800 +1000 +i +0.0025 +0.0050 +0.0075 +Bimodal +: + S += +0.854 +FIG. 1. Examples of target distributions. First row: three categorical distributions sampled from uniform Dirichlet with β = 0.01, 1, 10, +respectively. Second row: a categorical distribution sampled from a uniform Dirichlet, β = 1, but where half bins are set to zero; Zipf’s dis- +tribution with exponent a = 1.001; binomial distribution: two gaussians with {mean, standard deviation} respectively {10, 20} and {100, 5}, +are concatenated and then discretized over a histogram of 1000 categories. +IV. +RESULTS +We test our method in a variety of scenarios and com- +pare the results with the main alternative available estima- +tors, the NSB [12, 13] and the Hausser-Strimmer (HS) [14]. +In our experiments, we generate synthetic target distributions +and sample multinomial counts {ni} from those distributions. +We fix K = 1000 and generate samples of increasing size +N = 20, . . ., 10000. After calculating β⋆ from (12), we es- +timate the Shannon entropy S and the Kullback-Leibler di- +vergence DKL. For each case, we repeat this procedure 1000 +times; we always report averages over these repetitions [24]. +As target distributions (see Fig. 1 as reference) we consider +categorical distributions that are both typical in the Dirich- +let prior (that is, they are generated by a symmetric Dirich- +let prior; we use several values of concentration parameter +β = 0.01, 1, 10) and atypical in the Dirichlet prior (that is, +they cannot be attributed to or have a negligible probability of +being generated from a symmetric Dirichlet prior). Among +the latter, we consider: (i) distributions with added struc- +tural zeroes (that is, we sample from a symmetric Dirich- +let prior with a given β, but half of the categories are then +forced to have zero probability) [25]; (ii) Bimodal distribu- +tions, which represent, for example, the degree distributions +of core-periphery complex networks [26]; (iii) Zipf’s distri- +bution, ubiquitous in nature, in biological as well as social +systems [27], characterized by probabilities ρi ∝ i−a, with a +exponent a ≥ 1 [28]. +A. +Shannon entropy +To estimate the posterior p(S|n) of the Shannon entropy we +use the exact formulas of its moments, derived in Refs. [16, +17] (later refined in Ref. [18]). The first moment is given by +E[S|n, β] = +� +dρ S(ρ|β) p(ρ|n) += ψ0(N + Kβ + 1) +− +K +� +i=1 +ni + β +N + Kβ ψ0(ni + β + 1) . +(15) +In Appendix B we also show the expression of the standard +deviation. +In practice, given a dataset n we calculate the most prob- +able β⋆ from Eq. (12) by assuming either a flat hyperprior, +Eq. (13), or the NSB hyperprior, Eq. (7). Then, we compute +the required moments of the Shannon entropy; we indicate +the estimated values of the Shannon entropy as S(β⋆ +flat) and +S(β⋆ +NSB), respectively. In Figs. 2 and 3, we show that our +estimator with a flat hyperprior is the most accurate estima- +tor overall. In particular, S(β⋆ +flat) is consistently more accu- +rate than the NSB estimator, except in the deep sparse regime +N < 30 of two of the distributions atypical in the Dirichlet +prior, where it is comparable but slightly less accurate. The +Bayesian estimators also behave better than the HS estimator +SHS except for very uniform distributions sampled from the +Dirichlet prior with β = 10. Overall, the S(β⋆ +flat) has little +bias often even in the very sparse regime and for distributions +atypical in the Dirichlet prior. It is also interesting to note +that both S(β⋆ +flat) and S(β⋆ +NSB) have a more regular scaling +behavior, in the convergence toward the true values as N in- +creases, in particular when compared with NSB and HS for +Zipf’s distribution. +We also analyze the variability of the Shannon entropy es- +timates, as measured by the root mean squared error (insets +in Figs. 2 and 3). This analysis reveals that, besides having +less bias, the S(β⋆ +flat) estimator has a variability that is typi- +cally comparable to or smaller than the other estimators. It is + +5 +10 +2 +10 +3 +10 +4 +N +0.0 +0.2 +0.4 +ΔS +rel +Dirichlet : K += +1000, β += +0.01, S +true += +0.422 +S +true +β +⋆ +flat +β +⋆ +NSB +NSB +HS +10 +2 +10 +3 +10 +4 +N +10 +−2 +10 +−1 +RMSE +10 +2 +10 +3 +10 +4 +N +−0.15 +−0.10 +−0.05 +0.00 +0.05 +ΔS +rel +Dirichlet : K += +1000, β += +1, S +true += +0.939 +10 +2 +10 +3 +10 +4 +N +10 +−2 +10 +−1 +RMSE +10 +2 +10 +3 +10 +4 +N +−0.20 +−0.15 +−0.10 +−0.05 +0.00 +ΔS +rel +Dirichlet : K += +1000, β += +10, S +true += +0.993 +10 +2 +10 +3 +10 +4 +N +10 +−3 +10 +−2 +10 +−1 +RMSE +FIG. 2. +Shannon entropy estimation for distributions typical in +a Dirichlet prior, for β += +0.01, 1, 10 and sample size N += +25, . . . , 10000. +Each point corresponds to an average over 1000 +samples. The Strue in the titles serves as a reference and indicates +the average over the entropies of the runs. Main plots: relative er- +rors of entropies ∆Srel = (Sest − Strue)/Strue. Insets: roots +mean-squared errors (note the logarithmic scale in both axes). Black +squares: our estimator with β⋆ from a flat hyperprior. Cyan pluses: +our estimator but with β⋆ from NSB hyperprior. Pink upper triangle: +NSB estimator. Red crosses: Hausser-Strimmer plug-in estimator. +Here and in the rest of figures, the standard-errors bars of the main +plots are smaller then symbols and are not shown. +also worth noting that, differently from Bayesian estimators, +for which all the moments can be estimated also from a single +sample, the HS estimator is limited to a point estimate of the +mean value of Shannon entropy. +Note that, contrary to what one may expect, SNSB differs +from our estimate S(β⋆ +NSB) in that the latter is always smaller +for small samples. +This happens because the NSB hyper- +prior (7) is a positive monotonically-decreasing function that +assigns higher probabilities to smaller β’s, while the Shannon +entropy of distributions sampled from a symmetric Dirichlet +is a monotonically-increasing function of β. However, it is +not the same estimating β⋆ with the NSB hyperprior and then +plug it in (15) or directly estimating the Shannon entropy with +the NSB prior (6) and the latter in fact provides better results. +On the other side, S(β⋆ +flat) and SNSB should not substantially +differ, being based on the same first principles of estimation. +The differences are attributable to the numerical and compu- +tational difficulties in implementing the NSB approach that +required both a fine discretization over β and solving as many +equations (15) as β’s, which have to be finally integrated with +weights given by the hyperprior (7), in contrast with our ap- +proach, which needs solving just Eq. (12) and Eq. (15) once. +10 +2 +10 +3 +10 +4 +N +−0.1 +0.0 +0.1 +ΔS +rel +Half empty Dirichlet : K += +1000, β += +1, S +true += +0.839 +10 +2 +10 +3 +10 +4 +N +10 +−2 +10 +−1 +RMSE +10 +2 +10 +3 +10 +4 N +−0.3 +−0.2 +−0.1 +0.0 +0.1 +ΔSrel +Zipf : K = 1000, a = 1.001, Strue = 0.751 +10 +2 +10 +3 +10 +4 +N +10 +−2 +10 +−1 +RMSE +10 +2 +10 +3 +10 +4 +N +−0.1 +0.0 +0.1 +ΔS +rel +Bimodal : K += +1000, S=0.857 +10 +2 +10 +3 +10 +4 +N +10 +−2 +10 +−1 +RMSE +FIG. 3. Shannon entropy estimation for atypical distributions in a +Dirichlet prior (same legend as in Fig. 2): Dirichlet with β = 1 +but half bins are set to zeros; Zipf’s distribution with exponent a = +1.001; bimodal distribution. + +6 +B. +Kullback-Leibler divergence +Regarding the Kullback-Leibler divergence DKL, there are +no exact formulas for the moments of the posterior distribu- +tion p(DKL|n). Therefore, we have to rely on a point estimate +of the mean by first estimating the distributions via Laplace’s +formula (5) with the inferred β⋆ and then plugging these val- +ues into expression (8). The flat hyperprior in Eq. (13) is the +only reasonable one to estimate β⋆ in this case, since the NSB +prior (Eq. (7)) can only be justified for the Shannon entropy. +10 +2 +10 +3 +10 +4 N +0 +1 +2 +3 +DKL +K = 1000, β = 0.01 +β⋆ +plugin +HS +βplugin = ⋆ +10 +2 +10 +3 +10 +4 N +0.00 +0.25 +0.50 +0.75 +1.00 +DKL +K = 1000, β = 1 +10 +2 +10 +3 +10 +4 N +0.0 +0.2 +0.4 +0.6 +DKL +K = 1000, β = 10 +FIG. 4. +Kullback-Leibler estimation for distributions typical in +a Dirichlet prior, for β = 0.01, 1, 101 and sample size N += +25 . . . 10000. Each point corresponds to an average over 1000 sam- +ples. Black squares: our plug-in estimator, that is Laplace’s formula +with β⋆ estimated from a flat hyperprior. Red crosses: Hausser- +Strimmer plug-in estimator. Purple circles: Laplace’s estimator for +uniform prior β = 1. +We compare the results with Laplace’s estimator (5) with +β = 1 and with the HS estimator, since both have the same +desirable property of assigning non-null probabilities to un- +observed states (ni = 0) and are suitable estimators for com- +puting DKL. Indeed, β = 1 in Laplace’s formula is a com- +mon choice and amounts to assigning the same probability +to all possible distributions. We test the estimators in a sce- +nario typical in machine learning and variational inference, +in which one wants to minimize the DKL between a complex, +target distribution and some model approximation. Here, after +generating a synthetic discrete distribution ρ, we measure the +DKL(ρ; ˆρ), where ˆρ is the distribution estimated from counts; +hence a good estimator should make DKL as small as possi- +ble. +10 +2 +10 +3 +10 +4 +N +0.0 +0.5 +1.0 +D +KL +Half empty Dirichlet : K += +1000, β += +1 +10 +2 +10 +3 +10 +4 +N +0.0 +0.5 +1.0 +D +KL +Zipf : K += +1000, a += +1.001 +10 +2 +10 +3 +10 +4 +N +0.0 +0.5 +1.0 +D +KL +Bimodal +: + K += +1000 +FIG. 5. Kullback-Leibler divergence estimation for atypical distribu- +tions in a Dirichlet prior (same legend as in Fig. 4): Dirichlet with +β = 1 but half bins set to zero; Zipf’s distribution with exponent +a = 1, 001; bimodal distribution. +In Figs. 4 and 5, we show that our estimator and the HS +estimator provide similar results, although DKL(β⋆) is more + +7 +accurate in the very sparse regime N < 50, and when the tar- +get distributions are atypical in the Dirichlet priors, especially +in the important case of Zipf’s distributions. The estimator +based on Laplace’s formula wih β = 1 performs generally +worse, unless in the trivial case when the target distribution +itself was also generated just from a Dirichlet with β = 1. +Importantly, in this case in which β = 1 is optimal, our ap- +proach provides virtually identical results. +V. +CONCLUSIONS +Inferring the shape of discrete distributions and their infor- +mation content from experimental data is a fundamental task +in fields that spread from machine learning to computational +social science and neuroscience. However, it is common in +experiments to have a very low number of observations that +hinder a correct estimation. In this paper, we have proposed +a new method for the solution of this problem that applies +to discrete distributions with a known number of states. It +is pinned on the laws of conditional probability, in the form +of Bayes’ rules, with the explicit assumption of a Dirichlet +prior distribution as the mechanism behind the generation of +data. In particular, we are able to provide a semi-analytical +formula (Eq. (12)), easily solvable with moderated computa- +tion efforts, to find the concentration parameter characterizing +the Dirichlet distribution. This result is a step forward with re- +spect to many previous works that share the same background +but ultimately focused on constructing an infinite mixture of +Dirichlet priors, which weights were chosen to optimize the +estimation of the Shannon entropy only [12, 13, 18, 19]. Be- +sides their precision and success, these other approaches are +computationally involved and ignore any estimation of the +probability distribution, which could be necessary, in partic- +ular, for the estimation of the Kullback-Leibler divergence. +Our approach allows the reconstruction of the posterior distri- +bution of Shannon entropy for a broad variety of data types, by +using the exact formulas in Ref. [16], with a precision compa- +rable to or better than other estimators developed for the same +purposes. In the case of Kullback-Leibler divergence, on the +contrary, we were not able to estimate its full posterior dis- +tribution, but we obtained a good point-wise estimation of its +mean value, by estimating the two involved probability dis- +tributions and then plugging them into the explicit expression +of the Kulback-Leibler divergence. In regard to this point and +for future studies, it is in general desirable having some ana- +lytical expression for the posterior distribution (conditioned to +observations) of the Kullback-Leibler divergence in the same +spirit as the Shannon entropy. Further efforts should be de- +voted to extending the same approach to more specific pri- +ors than Dirichlet, for example for data that follow power +law distributions, including Zipf’s laws, for binary distribu- +tions [19], or when the number of states is unknown, as in +Refs. [18, 20, 29, 30]. +VI. +ACKNOWLEDGEMENTS +This research was funded by the Social Observatory of the +“la Caixa” Foundation as part of the project LCF / PR / SR19 +/ 52540009, by MCIN / AEI / 10.13039 / 501100011033 +(Project No. PID2019–106811GB-C31) and by the Govern- +ment of Catalonia (Project No. 2017SGR-896). +[1] Roger Guimer`a and Marta Sales-Pardo. Missing and spurious +interactions and the reconstruction of complex networks. 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Entropy, 15(5):1738–1755, 2013. +[22] As observed in [21] and [30], mutual information can be ex- +pressed in terms of different combinations of the Shannon en- +tropy of the two distributions. But its estimations in general dif- +fer. The expression M(ρ ; σ) = S(ρ) + S(σ) − S(π) seems +to be the less biased, however, in the absence of a unique con- +sistent prior over the joint distribution, it is not guaranteed it +minimizes the mean-squared error. +[23] Although we have not proved that the solution β⋆ is unique, it +seems reasonable that it is and, indeed, our simulations suggest +that, even for N ≪ K, if a finite β⋆ exists, it is unique. +[24] Averaging on multiple runs is preferable in order to highlight +the scaling behaviors of the estimators while mitigating the +effects of outliers (for example, very singular distributions or +samples). +[25] This scenario corresponds to an experiment in which some +states are not observable. +[26] Xiao Zhang, Travis Martin, and Mark EJ Newman. Identifica- +tion of core-periphery structure in networks. Physical Review +E, 91(3):032803, 2015. +[27] Mark EJ Newman. Power laws, pareto distributions and zipf’s +law. Contemporary physics, 46(5):323–351, 2005. +[28] In Refs. [12, 13] a rigorous definition of atypicity is provided, +related to the shape of the tails of a Zipf’s distribution. +[29] Gregory Valiant and Paul Valiant. Estimating the unseen: im- +proved estimators for entropy and other properties. Journal of +the ACM (JACM), 64(6):1–41, 2017. +[30] David H Wolpert and Simon DeDeo. Estimating functions of +distributions defined over spaces of unknown size. +Entropy, +15(11):4668–4699, 2013. +Appendix A: Derivation of results (Eq. (12) in main text) +Let us suppose that we have K different categories (or types of random events) and that we observe N independent random +events distributed in the K categories n = {ni; i = 1, . . . , K}, with � +i ni = N. We also assume that the probabilities of +observing counts in each category ρi are distributed according to a Dirichlet prior with the same hyper-parameters β for all +ρ = {ρi; i = 1, . . . , K}, so that +p(ρ|β) = +1 +BK(β) +K +� +i=1 +ρβ−1 +i +, +BK(β) = Γ(β)K +Γ(βK). +(A1) +Our goal is to compute the most likely value of β given the observed counts {ni}. To that end, we need to compute the +conditional probability p(β|n). We can do this by marginalizing over the possible combinations of ρ = {ρi} as follows: +p(β|n) = p(β) +p(n)p(n|β) , +p(n|β) = +� +dρ p(n|β, ρ)p(ρ|β) . +(A2) +Since the probability of observing an event in category i is ρi, the probability of observing ni events of type i is ρni +i . Therefore, +for the integral in Eq. (A2) we have that +p(n|β, ρ) = +K +� +i=1 +ρni +i +, +(A3) +so that +p(n|β) = +1 +BK(β) +� +dρ +K +� +i=1 +ρni+β−1 +i +, +(A4) +where we have used Eq. (A1) for p(ρ|β) and the integral is over the simplex that satisfies the condition � +i=1 ρi = 1. +To perform the integrals above we first evaluate the normalization condition for ρk = 1 − R(K − 1) with RK−1 = �K−1 +i=1 ρi +so that for ρk−1 we have the following integral: +IK−1 = +� 1−RK−2 +0 +dρK−1 ρnK−1+β−1 +K−1 +(1 − ρk−1 − RK−2)nK+β−1 . +(A5) + +9 +To evaluate this integral we use the fact that +� (1−R) +0 +dx xa(1 − x − R)b = Γ(a + 1)Γ(b + 1) +Γ(a + b + 2) +(1 − R)a+b+1 +if +Re(R) < 1 +and +Im(R) = 0 +(A6) +so that +IK−1 = Γ(nK−1 + β)Γ(nK + β) +Γ(nk + nK−1 + 2β) +(1 − RK−2)nK+nK−1+2β−1 +(A7) +Which gives for ρK−2 the following integral: +IK−2 = +� 1−RK−3 +0 +dρK−2 ρnK−2+β−1 +K−2 +(1 − ρK−2 − RK−3)nK+NK−1+2β−1 +(A8) += Γ(nK−2 + β)Γ(nK + nK−1 + 2β) +Γ(nk + nK−1 + nk−2 + 3β) +(1 − RK−3)nK+nK−1+nK−2+3β−1 +(A9) +which have evaluated using Eq. (A6). If we do this for all ρ we end up having +� +dρ +� +i +ρni+β−1 +i += +K +� +i=1 +Ii = +�K +i=1 Γ(ni + β) +Γ(N + Kβ) +. +(A10) +Thus, we obtain the following expression for p(n|β) +p(n|β) = +1 +BK(β) +� +i Γ(ni + β) +Γ(N + Kβ) = Γ(Kβ) +Γ(β)K +� +i Γ(ni + β) +Γ(N + Kβ) +(A11) +Our goal is to find β⋆ that maximizes p(β|n) = p(β) +p(n)p(n|β). To that end we take the derivative of log p(β|n), +log p(β|n) = log Γ(Kβ) − K log Γ(β) + +� +i +log Γ(ni + β) − log Γ(N + Kβ) + log p(β) − log p(n) +(A12) +so that β⋆ is the one that satisfies the condition: +d log p(β|n) +dβ +���� +β=β⋆ = 0 . +(A13) +To evaluate this equation we use the following definitions and properties of the log Gamma function: +1. +� d +dx +�m+1 log Γ(x) = ψm(x) +(A14) +2. +ψ0(x + n) = �n−1 +m=0 +1 +x+m + ψ(x) . +(A15) +Using the expressions above and the consideration that p(β) = const. we obtain that: +d log p(β|n) +dβ += Kψ0(Kβ) − Kψ0(β) + +� +i +ψ0(ni + β) − Kψ0(N + Kβ) +(A16) += +K +� +i=1 +ni−1 +� +m=0 +1 +m + β − +N−1 +� +m=0 +K +m + Kβ +(A17) +Therefore the condition that gives β⋆ is +K +� +i=1 +ni−1 +� +m=0 +1 +m + β⋆ − +N−1 +� +m=0 +K +m + Kβ⋆ = 0 , +(A18) +that is, the Eq. (12) in main text for uniform hyperprior (13). If instead we consider a prior for beta that results in a close-to- +uniform distribution of Shannon entropy such as in Nemenman et al. [12, 13] then +pNSB(β) = dS +dβ , +(A19) + +10 +with S = E[S|ni = 0, β] = ψ0(Kβ + 1) − ψ0(β + 1), the average entropy of the distributions generated from a Dirichlet +prior p(ρ|β). Note that this prior is already normalized since +� ∞ +0 +dS/dβdβ = S(∞; K) − S(0; K) = 1. The derivative of the +logarithm of this prior with respect to β is then +d log pNSB(β) +dβ += +1 +pNSB(β) +dpNSB(β) +dβ += 1 +dS +dβ +d2S +dβ2 = K2ψ2(kβ + 1) − ψ2(β + 1) +Kψ1(kβ + 1) − ψ1(β + 1) , +which is the formula (12) in main text. The condition of the β⋆ that maximizes p(β|n) is in this case: +d log p(β|n) +dβ += Kψ0(Kβ) − Kψ0(β) + +� +i +ψ0(ni + β) − Kψ0(N + Kβ) + 1 +dS +dβ +d2S +dβ2 = += +K +� +i=1 +ni−1 +� +m=0 +1 +m + β⋆ − +N−1 +� +m=0 +K +m + Kβ⋆ + K2ψ2(kβ⋆ + 1) − ψ2(β⋆ + 1) +Kψ1(kβ⋆ + 1) − ψ1(β⋆ + 1) = 0 . +Appendix B: Analytical moments of the Shannon entropy posterior +In the specific case of S(ρ), instead of solving p (F|n) = +� +dρ δ (F − F(ρ)) p(ρ|n) (Eq. (2) in main text) directly, it is +possible to obtain closed form expression for all the moments of the posterior [16–18]. Here we report the first two, the mean +E[S|n, β] = +� +dρ S(ρ|β) p(ρ|n) = ψ0(N + Kβ + 1) − +K +� +i=1 +ni + β +N + Kβ ψ0(ni + β + 1) , +(B1) +and the second moment +E[S2|n, β] = +� +dρ S(ρ|β)2 p(ρ|n) = +K +� +i̸=j +(ni + β) (nj + β) +(N + Kβ + 1) (N + Kβ) Ii,j + +K +� +i=1 +(ni + β + 1) (ni + β) +(N + Kβ + 1) (N + Kβ) Ji , +(B2) +with +Ii,j = +� +ψ0(ni + β + 1) − ψ0(N + Kβ + 2) +� +· +� +ψ0(nj + β + 1) − ψ0(N + Kβ + 2) +� +− ψ1(N + Kβ + 2) ; +Ji = +� +ψ0(ni + β + 2) − ψ0(N + Kβ + 2) +�2 ++ ψ1(ni + β + 2) − ψ1(N + Kβ + 2) ; +(B3) +from which the standard deviation is in turn calculated as the square root of the variance Var(S|n, β) = E[S2|n, β]−E[S|n, β]2. + diff --git a/E9FRT4oBgHgl3EQfyzhp/content/tmp_files/load_file.txt b/E9FRT4oBgHgl3EQfyzhp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..968d668489fd4ed166bcb2f98dc006fb4f9a82f2 --- /dev/null +++ b/E9FRT4oBgHgl3EQfyzhp/content/tmp_files/load_file.txt @@ -0,0 +1,531 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf,len=530 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='13647v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='data-an] 31 Jan 2023 Bayesian estimation of information-theoretic metrics for sparsely sampled distributions Angelo Piga,∗ Lluc Font-Pomarol,† Marta Sales-Pardo,‡ and Roger Guimer`a§ (Dated: February 1, 2023) Estimating the Shannon entropy of a discrete distribution from which we have only observed a small sample is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Estimating other information-theoretic metrics, such as the Kullback-Leibler divergence between two sparsely sampled discrete distributions, is even harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Existing approaches to address these problems have shortcomings: they are biased, heuristic, work only for some distributions, and/or cannot be applied to all information-theoretic metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here, we propose a fast, semi-analytical estimator for sparsely sampled distribu- tions that is efficient, precise, and general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Its derivation is grounded in probabilistic considerations and uses a hierarchical Bayesian approach to extract as much information as possible from the few observations available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Our approach provides estimates of the Shannon entropy with precision at least comparable to the state of the art, and most often better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' It can also be used to obtain accurate estimates of any other information-theoretic metric, including the notoriously challenging Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here, again, our approach performs consistently better than existing estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' INTRODUCTION Information theory is gaining momentum as a methodolog- ical framework to study complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In network sci- ence, information theory provides rigorous tools to predict unobserved links [1] and to infer community structure [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In neuroscience, Shannon entropy of spike train distributions characterizes brain activity from neural responses [3], while mutual information identifies correlations between brain stim- uli and responses [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Recently, the Kullback-Leibler diver- gence [5] and its regularized version, the Jensen-Shannon dis- tance, have also been successfully used in a wide variety of contexts: in cognitive science as a measure of “surprise,” to quantify and predict how human attention is oriented between changing screen images [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' in quantitative social science, in combination with topic models, to track the propagation of political and social discourses [7, 8] or to understand the emergence of social disruption from the analysis of judicial decisions [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' and in machine learning, at the intersection be- tween the statistical physics of diffusive processes, probabilis- tic models and deep neural networks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Information theoretical metrics are measured on distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In practice, a distribution ρ over the possible states of a system, as well as functions F(ρ) of this distribution (such as Shannon entropy or other metrics), have to be in- ferred from experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, this infer- ence process is difficult for many real complex systems since, due to experimental limitations, the observations are often sparse, and statistical estimates of the distribution ρ and its functions can be severely biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here, we focus on the par- ticular yet important case of discrete (or categorical) distribu- tions ρi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K, where K is the number of possible ∗ angelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='piga@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Department of Chemical Engineering, Universi- tat Rovira i Virgili, Tarragona 43007, Catalonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' † lluc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='font@urv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='cat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona 43007, Catalonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ‡ marta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='sales@urv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='cat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona 43007, Catalonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' § roger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='guimera@urv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='cat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona 43007, Catalonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ICREA, Barcelona 08010, Catalonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' states (or categories), which is known and fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Inferences about ρ and any function must be based on ni, the number of observations in the i-th state (with N = � i ni the sam- ple size) and, in the undersampled regime we are interested in, N ≲ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The challenge is thus, from the sparse observa- tions {ni}, to infer the probability ρi of each category i and estimate metrics F(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' A theoretically well-founded approach to tackle this prob- lem is provided by the principles of conditional probability, encapsulated in Bayes’ theorem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' This framework is in general preferable because of its transparency—it requires that all assumptions of the underlying generative model for the data are made explicit, expressed via the choice of a likelihood function and a prior distribution that reflects the knowledge about the system before observing any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In probabilistic reasoning, the combination of observations and prior distri- bution provides an updated (posterior) probability distribution of the quantity under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Other estimation strategies make implicit assumptions and often provide only point estimates, as opposed to full distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' A class of expressive generative models for categorical dis- tributions amenable to a Bayesian framework is the well- studied family of Dirichlet distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, as Ne- menmann, Shafee, and Bialek (henceforth NSB) pointed out in [12], when sample sizes are small (N ≲ K), the inferred Shannon entropy is tightly determined by the specific parame- ters one chooses for the Dirichlet model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' therefore, inaccurate choices result in severe biases of the Shannon entropy esti- mates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To overcome this problem, they introduced a mixture of Dirichlet models, which results in a very precise estimator of the Shannon entropy that works for a wide variety of distri- butions, even in the sparse sampling regime N ≲ K [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Although, in terms of precision, NSB can be considered the state of the art for estimating the Shannon entropy, it does not provide estimates for the distribution ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' For this reason, its ap- plicability is limited to estimating the Shannon entropy (and related information theoretic quantities like mutual informa- tion and Jensen-Shannon distance, which can be expressed in terms of entropies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' By contrast, it cannot be used to estimate the Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To cover this gap, Hausser and Strimmer derived a James-Stein-type shrinkage estimator for ρ [14] (henceforth HS), which has the advantage of being 2 analytical and applicable to any information-theoretic metric, but at the price of making implicit ad hoc assumptions, and of being less precise than NSB for the Shannon entropy, and lacking error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here, we propose an alternative fast, semi-analytical es- timator for distributions that is efficient, precise, and gen- eral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Its derivation is grounded in probabilistic considerations, without any ad hoc assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We consider Dirichlet gen- erative models and use a hierarchical Bayesian approach to extract as much information as possible from the few observa- tions at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In the case of Shannon entropy, we can estimate the expected value and higher order moments with precision at least comparable to the NSB estimator, and most often bet- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Additionally, because our method provides estimates of the probability distribution, it can be used to obtain accurate estimations of the Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In this case our approach also performs equally or better than existing es- timators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' BACKGROUND Let us consider a system with K possible output states whose observations follow an unknown discrete distribu- tion ρ = {ρi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K} with � i ρi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The vector n = {ni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K} represents the number of times each state was observed in a set of � i ni = N independent obser- vations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We also consider a function F(ρ) of ρ, such as, for example, the Shannon entropy S(ρ) = − K � i=1 ρi log ρi , (1) which we want to estimate from the set of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The posterior distribution over the values of the function F given the observed counts n is p (F|n) = � dρ δ (F − F(ρ)) p(ρ|n) , (2) where p(ρ|n) is the posterior of the distribution ρ given the counts n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We further assume that the prior over distributions depends on a parameter β, which becomes a hyperparameter of our generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Then, using the laws of conditional probability, we can write the posterior p(ρ|n, β) as p(ρ|n, β) = p(n|ρ, β) p(ρ|β) p(n|β) , (3) where p(n|ρ, β) is the likelihood, p(ρ|β) is the prior over distributions, and p(n|β) = � dρ p(n|ρ) p(ρ|β) is the evi- dence and acts as normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The likelihood is the probability of the empirical observations n given ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' for in- dependent multinomial samples, the probability of observing an event of type i is ρi, and the full likelihood is the prod- uct p(n|ρ, β) = p(n|ρ) = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' �K i ρni i /ni!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' and, given ρ, it is independent of the hyperparameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The prior p(ρ|β) expresses the probability of each distribution ρ prior to ob- serving any data, and plays a crucial role in the discussion be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Symmetric Dirichlet distributions are convenient priors because they are a generative model for a broad class of dis- crete distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Additionally, they have been widely used in this setting [15], and are parametrized as follows p(ρ|β) = 1 BK(β) K � i=1 ρβ−1 i , BK(β) = Γ(β)K Γ(βK) , (4) where Γ is the gamma function, while the hyperparameter β is a real, positive number known as the concentration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 1, examples of categorical distributions sampled from symmetric Dirichlet priors are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Besides being very expressive, Dirichlet priors are conju- gate distributions of categorical likelihoods, meaning that the posterior is still a Dirichlet distribution, a property that often makes the inference via Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (3) and (2) analytically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' For example, when F(ρ) = ρ, Dirichlet priors lead to ex- pected posterior probabilities ⟨ρi⟩ given by the widely-used generalized Laplace’s formula ⟨ρi⟩ = ni + β N + Kβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (5) It is worth noting the improvement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (5) with respect to the maximum likelihood (or frequency) estimator ρi = ni/N, which is recovered by the former in the limit β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In par- ticular, Laplace’s formula assigns non zero probability to non observed states, a desirable property whose advantage will be- come evident later, when estimating Kullback-Leibler diver- gences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' This example also illustrates how non-Bayesian ap- proaches to inference make implicit and non-trivial assump- tions, in this case assuming β → 0 amounts to assuming that infinitely concentrated distributions ρ are a priori much more plausible than more homogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Going back to the estimation of F from the observations n, and given Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (5), one may be tempted to directly plug the value of ⟨ρi⟩ in the explicit expression of F(ρ) to get a point estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, this is just an approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' the exact procedure consists in finding and using the whole pos- terior p(F|n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Specifically, the expected value of this pos- terior ⟨F⟩ = � dF F p(F|n) minimizes the mean-squared error [16], and its mode is a consistent estimator, meaning that it converges to the true value of F(ρ) when the num- ber of observations increases, regardless of the prior and, in particular, regardless of the hyperparameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Wolpert and Wolf in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [16, 17] provided analytical formulas for all the moments of p(F|n) when F is the Shannon entropy and for Dirichlet priors (we report the formula for the mean in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (15) and for the second moment in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, an unbiased estimation of F is not guaranteed for small samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' This is often the case for Dirichlet priors, es- pecially when the parameter β is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Several options for the value of β have been proposed in literature, each one suitable to some specific case but deficient in others (for a dis- cussion, refer to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [12, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In [12], NSB suggested that, when samples are scarce, any attempt to find a single universal β is hopeless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' the fundamental reason being that categorical distributions generated by a Dirichlet have a Shannon entropy that is narrowly determined by, and monotonically dependent 3 on, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In other words, for small samples, the posterior distri- bution (2) is dominated by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To overcome this prob- lem, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 13] proposed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' as the prior pNSB(ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' an infinite mixture of Dirichlet priors pNSB(ρ) ∝ � dβ pNSB(β) p(ρ|β) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (6) where the weights pNSB(β) were set so as to obtain a flat prior over entropies S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' and have the functional form pNSB(β) ∝ d E[S|ni = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' β] dβ = Kψ1(Kβ +1)−ψ1(β +1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (7) where E[S|n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' β] is the expected entropy given the observa- tions n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' and then E[S|ni = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' β] is the expected entropy of the distributions ρ generated from a symmetric Dirichlet priors (that is if there are no observations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' with fixed β and K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' and ψm(x) = � d dx �m+1 log Γ(x) are the polygamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The NSB prior leads to very accurate estimates of the Shannon entropy, and can be considered the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Even if best suited for situations in which the number of states K is known and fixed, it is quite versatile and has been later extended for countable infinite number of states [18] and further optimized for binary states [19] and long tail distributions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Other es- timators, for example, the Chao-Shen estimator [20], perform at most as well as the NSB (or its derivatives), but never better (see [14] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Additionally, given an estimator of S, a number of other quantities can be indirectly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' For example, the mutual information M between two distributions ρ and σ is M(ρ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' σ) = S(ρ)+S(σ)−S(π), where π is the joint distribution of ρ and σ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Similar re- lations can be derived for Jensen-Shannon distance and other information-theoretic quantities [8] [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, consider the estimation of the Kullback-Leibler divergence (DKL) between two distributions ρ and σ with the same dimension K DKL(ρ∥σ) = K � i=1 ρi log2 ρi σi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (8) To estimate DKL from samples n = {ni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K} from ρ, and m = {mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K} from σ, one cannot use the NSB approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' First, DKL is not a combination of the Shan- non entropies of the two underlying distributions ρ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Second, DKL is unbounded, and any attempt to find a hyper- prior in the spirit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (7) results in improper hyperpriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Finally, with the NSB prior one renounces to any estimation of β and, in turn, to a good a point estimation of DKL by means of Laplace’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' HIERARCHICAL BAYES POINT ESTIMATE FOR β Here, we address these limitations of the NSB estimator while maintaining and even improving its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We posit that the success of the NSB approach stems, not from mixing infinitely many values of the concentration parameter β, but rather from the flexibility to accommodate for any par- ticular value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Indeed, we surmise that, in general, only a narrow interval of β values are compatible with a given ob- servation n and therefore contribute to the mixture, whereas most others do not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Motivated by this, we propose an approach that aims to directly estimate the value of β that most contributes to the posterior given the data n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' First, we observe that the posterior p(ρ|n) can be written as p(ρ|n) = � dβ p(ρ|n, β) p(β|n) = � dβ p(n|ρ) p(ρ|β) p(n|β) p(β|n) , (9) where we have applied Bayes’ rule, and the fact that n condi- tioned on ρ is independent of β, so that p(n|ρ, β) = p(n|ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Then, we assume that the conditional distribution p(β|n) is very peaked around a given value β⋆ , so that the posterior p(ρ|n) can be approximated as p(ρ|n) ≈ p(n|ρ) p(ρ|β⋆) p(n|β⋆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (10) This approximation, sometimes referred to as empirical Bayes, is a point estimate for the fully hierarchical probabilis- tic model given by p(n|ρ) and p(ρ|β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (10) is identical to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (3), with the difference that the concentration parameter is now the most likely value of β given the observed counts n, that is, β⋆ = argmax β p(β|n) = argmax β p(n|β) p(β) p(n) , (11) where p(n|β) = � dρ p(n|β, ρ)p(ρ|β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' For Dirichlet priors (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (4)), β∗ satisfies (see Appendix A) K � i=1 ni−1 � m=0 1 m + β⋆ − N−1 � m=0 K m + Kβ⋆ + 1 p(β⋆) d p(β) d β ��� β⋆ = 0 , (12) which is the key analytical result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The hyperprior p(β) reflects our prior knowledge about the shape of the distribution of the hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To be com- pletely agnostic in this regard, we can use a uniform hyper- prior pU(β) = 1 ∆β = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , ∆β = βmax − βmin , (13) with cut-offs 0 < βmin < βmax < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In this case, the deriva- tive term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12) disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The NSB hyperprior (7) is a valid alternative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' in this case, the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12) is (see appendix A for details) 1 pNSB(β∗) d pNSB(β) d β ���� β∗ = K2ψ2(kβ⋆ + 1) − ψ2(β⋆ + 1) Kψ1(kβ⋆ + 1) − ψ1(β⋆ + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (14) Despite the complex appearance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12), β∗ is not hard to obtain numerically, giving a computational improve- ment with respect the NSB estimator, whose algorithm is involved and has higher computational costs [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The source code of the implementations in Python is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='com/angelopiga/info-metric-estimation/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 4 1 200 400 600 800 1000 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='3 ρ i Dirichlet : β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='394 1 200 400 600 800 1000 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0075 Dirichlet : β = 1, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='936 1 200 400 600 800 1000 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='002 Dirichlet : β = 10, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='993 1 200 400 600 800 1000 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='010 ρ i half empty Dirichlet : β = 1, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='838 1 200 400 600 800 1000 i 10 −4 10 −3 10 −2 10 −1 Zipf : a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='001, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='751 1 200 400 600 800 1000 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0075 Bimodal : S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='854 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Examples of target distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' First row: three categorical distributions sampled from uniform Dirichlet with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01, 1, 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Second row: a categorical distribution sampled from a uniform Dirichlet, β = 1, but where half bins are set to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Zipf’s dis- tribution with exponent a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' binomial distribution: two gaussians with {mean, standard deviation} respectively {10, 20} and {100, 5}, are concatenated and then discretized over a histogram of 1000 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' RESULTS We test our method in a variety of scenarios and com- pare the results with the main alternative available estima- tors, the NSB [12, 13] and the Hausser-Strimmer (HS) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In our experiments, we generate synthetic target distributions and sample multinomial counts {ni} from those distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We fix K = 1000 and generate samples of increasing size N = 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=', 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' After calculating β⋆ from (12), we es- timate the Shannon entropy S and the Kullback-Leibler di- vergence DKL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' For each case, we repeat this procedure 1000 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' we always report averages over these repetitions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' As target distributions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 1 as reference) we consider categorical distributions that are both typical in the Dirich- let prior (that is, they are generated by a symmetric Dirich- let prior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' we use several values of concentration parameter β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01, 1, 10) and atypical in the Dirichlet prior (that is, they cannot be attributed to or have a negligible probability of being generated from a symmetric Dirichlet prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Among the latter, we consider: (i) distributions with added struc- tural zeroes (that is, we sample from a symmetric Dirich- let prior with a given β, but half of the categories are then forced to have zero probability) [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (ii) Bimodal distribu- tions, which represent, for example, the degree distributions of core-periphery complex networks [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (iii) Zipf’s distri- bution, ubiquitous in nature, in biological as well as social systems [27], characterized by probabilities ρi ∝ i−a, with a exponent a ≥ 1 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Shannon entropy To estimate the posterior p(S|n) of the Shannon entropy we use the exact formulas of its moments, derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [16, 17] (later refined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The first moment is given by E[S|n, β] = � dρ S(ρ|β) p(ρ|n) = ψ0(N + Kβ + 1) − K � i=1 ni + β N + Kβ ψ0(ni + β + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (15) In Appendix B we also show the expression of the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In practice, given a dataset n we calculate the most prob- able β⋆ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12) by assuming either a flat hyperprior, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (13), or the NSB hyperprior, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Then, we compute the required moments of the Shannon entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' we indicate the estimated values of the Shannon entropy as S(β⋆ flat) and S(β⋆ NSB), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 2 and 3, we show that our estimator with a flat hyperprior is the most accurate estima- tor overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In particular, S(β⋆ flat) is consistently more accu- rate than the NSB estimator, except in the deep sparse regime N < 30 of two of the distributions atypical in the Dirichlet prior, where it is comparable but slightly less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The Bayesian estimators also behave better than the HS estimator SHS except for very uniform distributions sampled from the Dirichlet prior with β = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Overall, the S(β⋆ flat) has little bias often even in the very sparse regime and for distributions atypical in the Dirichlet prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' It is also interesting to note that both S(β⋆ flat) and S(β⋆ NSB) have a more regular scaling behavior, in the convergence toward the true values as N in- creases, in particular when compared with NSB and HS for Zipf’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We also analyze the variability of the Shannon entropy es- timates, as measured by the root mean squared error (insets in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' This analysis reveals that, besides having less bias, the S(β⋆ flat) estimator has a variability that is typi- cally comparable to or smaller than the other estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' It is 5 10 2 10 3 10 4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='4 ΔS rel Dirichlet : K = 1000, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01, S true = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='422 S true β ⋆ flat β ⋆ NSB NSB HS 10 2 10 3 10 4 N 10 −2 10 −1 RMSE 10 2 10 3 10 4 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='05 ΔS rel Dirichlet : K = 1000, β = 1, S true = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='939 10 2 10 3 10 4 N 10 −2 10 −1 RMSE 10 2 10 3 10 4 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='00 ΔS rel Dirichlet : K = 1000, β = 10, S true = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='993 10 2 10 3 10 4 N 10 −3 10 −2 10 −1 RMSE FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Shannon entropy estimation for distributions typical in a Dirichlet prior, for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01, 1, 10 and sample size N = 25, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Each point corresponds to an average over 1000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The Strue in the titles serves as a reference and indicates the average over the entropies of the runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Main plots: relative er- rors of entropies ∆Srel = (Sest − Strue)/Strue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Insets: roots mean-squared errors (note the logarithmic scale in both axes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Black squares: our estimator with β⋆ from a flat hyperprior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Cyan pluses: our estimator but with β⋆ from NSB hyperprior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Pink upper triangle: NSB estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Red crosses: Hausser-Strimmer plug-in estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here and in the rest of figures, the standard-errors bars of the main plots are smaller then symbols and are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' also worth noting that, differently from Bayesian estimators, for which all the moments can be estimated also from a single sample, the HS estimator is limited to a point estimate of the mean value of Shannon entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Note that, contrary to what one may expect, SNSB differs from our estimate S(β⋆ NSB) in that the latter is always smaller for small samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' This happens because the NSB hyper- prior (7) is a positive monotonically-decreasing function that assigns higher probabilities to smaller β’s, while the Shannon entropy of distributions sampled from a symmetric Dirichlet is a monotonically-increasing function of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, it is not the same estimating β⋆ with the NSB hyperprior and then plug it in (15) or directly estimating the Shannon entropy with the NSB prior (6) and the latter in fact provides better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' On the other side, S(β⋆ flat) and SNSB should not substantially differ, being based on the same first principles of estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The differences are attributable to the numerical and compu- tational difficulties in implementing the NSB approach that required both a fine discretization over β and solving as many equations (15) as β’s, which have to be finally integrated with weights given by the hyperprior (7), in contrast with our ap- proach, which needs solving just Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (15) once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 10 2 10 3 10 4 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 ΔS rel Half empty Dirichlet : K = 1000, β = 1, S true = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='839 10 2 10 3 10 4 N 10 −2 10 −1 RMSE 10 2 10 3 10 4 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 ΔSrel Zipf : K = 1000, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='001, Strue = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='751 10 2 10 3 10 4 N 10 −2 10 −1 RMSE 10 2 10 3 10 4 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='1 ΔS rel Bimodal : K = 1000, S=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='857 10 2 10 3 10 4 N 10 −2 10 −1 RMSE FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Shannon entropy estimation for atypical distributions in a Dirichlet prior (same legend as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 2): Dirichlet with β = 1 but half bins are set to zeros;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Zipf’s distribution with exponent a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' bimodal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Kullback-Leibler divergence Regarding the Kullback-Leibler divergence DKL, there are no exact formulas for the moments of the posterior distribu- tion p(DKL|n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Therefore, we have to rely on a point estimate of the mean by first estimating the distributions via Laplace’s formula (5) with the inferred β⋆ and then plugging these val- ues into expression (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The flat hyperprior in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (13) is the only reasonable one to estimate β⋆ in this case, since the NSB prior (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (7)) can only be justified for the Shannon entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 10 2 10 3 10 4 N 0 1 2 3 DKL K = 1000, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01 β⋆ plugin HS βplugin = ⋆ 10 2 10 3 10 4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='00 DKL K = 1000, β = 1 10 2 10 3 10 4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='6 DKL K = 1000, β = 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Kullback-Leibler estimation for distributions typical in a Dirichlet prior, for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='01, 1, 101 and sample size N = 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Each point corresponds to an average over 1000 sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Black squares: our plug-in estimator, that is Laplace’s formula with β⋆ estimated from a flat hyperprior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Red crosses: Hausser- Strimmer plug-in estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Purple circles: Laplace’s estimator for uniform prior β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We compare the results with Laplace’s estimator (5) with β = 1 and with the HS estimator, since both have the same desirable property of assigning non-null probabilities to un- observed states (ni = 0) and are suitable estimators for com- puting DKL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Indeed, β = 1 in Laplace’s formula is a com- mon choice and amounts to assigning the same probability to all possible distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We test the estimators in a sce- nario typical in machine learning and variational inference, in which one wants to minimize the DKL between a complex, target distribution and some model approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here, after generating a synthetic discrete distribution ρ, we measure the DKL(ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ˆρ), where ˆρ is the distribution estimated from counts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' hence a good estimator should make DKL as small as possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 10 2 10 3 10 4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 D KL Half empty Dirichlet : K = 1000, β = 1 10 2 10 3 10 4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 D KL Zipf : K = 1000, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='001 10 2 10 3 10 4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 D KL Bimodal : K = 1000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Kullback-Leibler divergence estimation for atypical distribu- tions in a Dirichlet prior (same legend as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 4): Dirichlet with β = 1 but half bins set to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Zipf’s distribution with exponent a = 1, 001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' bimodal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 4 and 5, we show that our estimator and the HS estimator provide similar results, although DKL(β⋆) is more 7 accurate in the very sparse regime N < 50, and when the tar- get distributions are atypical in the Dirichlet priors, especially in the important case of Zipf’s distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The estimator based on Laplace’s formula wih β = 1 performs generally worse, unless in the trivial case when the target distribution itself was also generated just from a Dirichlet with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Importantly, in this case in which β = 1 is optimal, our ap- proach provides virtually identical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' CONCLUSIONS Inferring the shape of discrete distributions and their infor- mation content from experimental data is a fundamental task in fields that spread from machine learning to computational social science and neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' However, it is common in experiments to have a very low number of observations that hinder a correct estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In this paper, we have proposed a new method for the solution of this problem that applies to discrete distributions with a known number of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' It is pinned on the laws of conditional probability, in the form of Bayes’ rules, with the explicit assumption of a Dirichlet prior distribution as the mechanism behind the generation of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In particular, we are able to provide a semi-analytical formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12)), easily solvable with moderated computa- tion efforts, to find the concentration parameter characterizing the Dirichlet distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' This result is a step forward with re- spect to many previous works that share the same background but ultimately focused on constructing an infinite mixture of Dirichlet priors, which weights were chosen to optimize the estimation of the Shannon entropy only [12, 13, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Be- sides their precision and success, these other approaches are computationally involved and ignore any estimation of the probability distribution, which could be necessary, in partic- ular, for the estimation of the Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Our approach allows the reconstruction of the posterior distri- bution of Shannon entropy for a broad variety of data types, by using the exact formulas in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [16], with a precision compa- rable to or better than other estimators developed for the same purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In the case of Kullback-Leibler divergence, on the contrary, we were not able to estimate its full posterior dis- tribution, but we obtained a good point-wise estimation of its mean value, by estimating the two involved probability dis- tributions and then plugging them into the explicit expression of the Kulback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' In regard to this point and for future studies, it is in general desirable having some ana- lytical expression for the posterior distribution (conditioned to observations) of the Kullback-Leibler divergence in the same spirit as the Shannon entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Further efforts should be de- voted to extending the same approach to more specific pri- ors than Dirichlet, for example for data that follow power law distributions, including Zipf’s laws, for binary distribu- tions [19], or when the number of states is unknown, as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [18, 20, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This research was funded by the Social Observatory of the “la Caixa” Foundation as part of the project LCF / PR / SR19 / 52540009, by MCIN / AEI / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='13039 / 501100011033 (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' PID2019–106811GB-C31) and by the Govern- ment of Catalonia (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' 2017SGR-896).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [1] Roger Guimer`a and Marta Sales-Pardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Missing and spurious interactions and the reconstruction of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Pro- ceedings of the National Academy of Sciences, 106(52):22073– 22078, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [2] Tiago P Peixoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Entropy of stochastic blockmodel ensembles.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [21] Evan W Archer, Il Memming Park, and Jonathan W Pillow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Bayesian and quasi-bayesian estimators for mutual information from discrete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Entropy, 15(5):1738–1755, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [22] As observed in [21] and [30], mutual information can be ex- pressed in terms of different combinations of the Shannon en- tropy of the two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' But its estimations in general dif- fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The expression M(ρ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' σ) = S(ρ) + S(σ) − S(π) seems to be the less biased, however, in the absence of a unique con- sistent prior over the joint distribution, it is not guaranteed it minimizes the mean-squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [23] Although we have not proved that the solution β⋆ is unique, it seems reasonable that it is and, indeed, our simulations suggest that, even for N ≪ K, if a finite β⋆ exists, it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [24] Averaging on multiple runs is preferable in order to highlight the scaling behaviors of the estimators while mitigating the effects of outliers (for example, very singular distributions or samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [25] This scenario corresponds to an experiment in which some states are not observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [26] Xiao Zhang, Travis Martin, and Mark EJ Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Identifica- tion of core-periphery structure in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Physical Review E, 91(3):032803, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [27] Mark EJ Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Power laws, pareto distributions and zipf’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Contemporary physics, 46(5):323–351, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [28] In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [12, 13] a rigorous definition of atypicity is provided, related to the shape of the tails of a Zipf’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [29] Gregory Valiant and Paul Valiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Estimating the unseen: im- proved estimators for entropy and other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Journal of the ACM (JACM), 64(6):1–41, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [30] David H Wolpert and Simon DeDeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Estimating functions of distributions defined over spaces of unknown size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Entropy, 15(11):4668–4699, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Appendix A: Derivation of results (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12) in main text) Let us suppose that we have K different categories (or types of random events) and that we observe N independent random events distributed in the K categories n = {ni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K}, with � i ni = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We also assume that the probabilities of observing counts in each category ρi are distributed according to a Dirichlet prior with the same hyper-parameters β for all ρ = {ρi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' , K}, so that p(ρ|β) = 1 BK(β) K � i=1 ρβ−1 i , BK(β) = Γ(β)K Γ(βK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A1) Our goal is to compute the most likely value of β given the observed counts {ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To that end, we need to compute the conditional probability p(β|n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' We can do this by marginalizing over the possible combinations of ρ = {ρi} as follows: p(β|n) = p(β) p(n)p(n|β) , p(n|β) = � dρ p(n|β, ρ)p(ρ|β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A2) Since the probability of observing an event in category i is ρi, the probability of observing ni events of type i is ρni i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Therefore, for the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A2) we have that p(n|β, ρ) = K � i=1 ρni i , (A3) so that p(n|β) = 1 BK(β) � dρ K � i=1 ρni+β−1 i , (A4) where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A1) for p(ρ|β) and the integral is over the simplex that satisfies the condition � i=1 ρi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To perform the integrals above we first evaluate the normalization condition for ρk = 1 − R(K − 1) with RK−1 = �K−1 i=1 ρi so that for ρk−1 we have the following integral: IK−1 = � 1−RK−2 0 dρK−1 ρnK−1+β−1 K−1 (1 − ρk−1 − RK−2)nK+β−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(A5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='To evaluate this integral we use the fact that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='� (1−R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='dx xa(1 − x − R)b = Γ(a + 1)Γ(b + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='Γ(a + b + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(1 − R)a+b+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='Re(R) < 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='Im(R) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(A6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='so that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='IK−1 = Γ(nK−1 + β)Γ(nK + β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='Γ(nk + nK−1 + 2β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(1 − RK−2)nK+nK−1+2β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(A7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='Which gives for ρK−2 the following integral: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='IK−2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='� 1−RK−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='dρK−2 ρnK−2+β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='K−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(1 − ρK−2 − RK−3)nK+NK−1+2β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(A8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='= Γ(nK−2 + β)Γ(nK + nK−1 + 2β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='Γ(nk + nK−1 + nk−2 + 3β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(1 − RK−3)nK+nK−1+nK−2+3β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='(A9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content='which have evaluated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' If we do this for all ρ we end up having � dρ � i ρni+β−1 i = K � i=1 Ii = �K i=1 Γ(ni + β) Γ(N + Kβ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A10) Thus, we obtain the following expression for p(n|β) p(n|β) = 1 BK(β) � i Γ(ni + β) Γ(N + Kβ) = Γ(Kβ) Γ(β)K � i Γ(ni + β) Γ(N + Kβ) (A11) Our goal is to find β⋆ that maximizes p(β|n) = p(β) p(n)p(n|β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' To that end we take the derivative of log p(β|n), log p(β|n) = log Γ(Kβ) − K log Γ(β) + � i log Γ(ni + β) − log Γ(N + Kβ) + log p(β) − log p(n) (A12) so that β⋆ is the one that satisfies the condition: d log p(β|n) dβ ���� β=β⋆ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A13) To evaluate this equation we use the following definitions and properties of the log Gamma function: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' � d dx �m+1 log Γ(x) = ψm(x) (A14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' ψ0(x + n) = �n−1 m=0 1 x+m + ψ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (A15) Using the expressions above and the consideration that p(β) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' we obtain that: d log p(β|n) dβ = Kψ0(Kβ) − Kψ0(β) + � i ψ0(ni + β) − Kψ0(N + Kβ) (A16) = K � i=1 ni−1 � m=0 1 m + β − N−1 � m=0 K m + Kβ (A17) Therefore the condition that gives β⋆ is K � i=1 ni−1 � m=0 1 m + β⋆ − N−1 � m=0 K m + Kβ⋆ = 0 , (A18) that is, the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (12) in main text for uniform hyperprior (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' If instead we consider a prior for beta that results in a close-to- uniform distribution of Shannon entropy such as in Nemenman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' [12, 13] then pNSB(β) = dS dβ , (A19) 10 with S = E[S|ni = 0, β] = ψ0(Kβ + 1) − ψ0(β + 1), the average entropy of the distributions generated from a Dirichlet prior p(ρ|β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Note that this prior is already normalized since � ∞ 0 dS/dβdβ = S(∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' K) − S(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The derivative of the logarithm of this prior with respect to β is then d log pNSB(β) dβ = 1 pNSB(β) dpNSB(β) dβ = 1 dS dβ d2S dβ2 = K2ψ2(kβ + 1) − ψ2(β + 1) Kψ1(kβ + 1) − ψ1(β + 1) , which is the formula (12) in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' The condition of the β⋆ that maximizes p(β|n) is in this case: d log p(β|n) dβ = Kψ0(Kβ) − Kψ0(β) + � i ψ0(ni + β) − Kψ0(N + Kβ) + 1 dS dβ d2S dβ2 = = K � i=1 ni−1 � m=0 1 m + β⋆ − N−1 � m=0 K m + Kβ⋆ + K2ψ2(kβ⋆ + 1) − ψ2(β⋆ + 1) Kψ1(kβ⋆ + 1) − ψ1(β⋆ + 1) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Appendix B: Analytical moments of the Shannon entropy posterior In the specific case of S(ρ), instead of solving p (F|n) = � dρ δ (F − F(ρ)) p(ρ|n) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (2) in main text) directly, it is possible to obtain closed form expression for all the moments of the posterior [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Here we report the first two, the mean E[S|n, β] = � dρ S(ρ|β) p(ρ|n) = ψ0(N + Kβ + 1) − K � i=1 ni + β N + Kβ ψ0(ni + β + 1) , (B1) and the second moment E[S2|n, β] = � dρ S(ρ|β)2 p(ρ|n) = K � i̸=j (ni + β) (nj + β) (N + Kβ + 1) (N + Kβ) Ii,j + K � i=1 (ni + β + 1) (ni + β) (N + Kβ + 1) (N + Kβ) Ji , (B2) with Ii,j = � ψ0(ni + β + 1) − ψ0(N + Kβ + 2) � � ψ0(nj + β + 1) − ψ0(N + Kβ + 2) � − ψ1(N + Kβ + 2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' Ji = � ψ0(ni + β + 2) − ψ0(N + Kβ + 2) �2 + ψ1(ni + β + 2) − ψ1(N + Kβ + 2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} +page_content=' (B3) from which the standard deviation is in turn calculated as the square root of the variance Var(S|n, β) = E[S2|n, β]−E[S|n, β]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfyzhp/content/2301.13647v1.pdf'} diff --git a/EdFKT4oBgHgl3EQfZy6F/content/2301.11805v1.pdf b/EdFKT4oBgHgl3EQfZy6F/content/2301.11805v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2d0cd3e03ab7932dbf3c1985bc546e841e5e1504 --- /dev/null +++ b/EdFKT4oBgHgl3EQfZy6F/content/2301.11805v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6b5681c31a49d2a4771f7b99ff1f94563b5217dd4ac58cb338e141ca9327b6a +size 203530 diff --git a/FtAzT4oBgHgl3EQfxP5f/vector_store/index.faiss b/FtAzT4oBgHgl3EQfxP5f/vector_store/index.faiss new file mode 100644 index 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Florida, +Gainesville, 32611, FL, USA +cComputational Mechanics and Materials Department, Idaho National +Laboratory, Idaho Falls, ID 83415, United States +∗Corresponding author. E-mail addresses: +chatterjee.s@ufl.edu (S. Chatterjee), daniel.schwen@inl.gov (D. Schwen), +nele.moelans@kuleuven.be (N. Moelans) +1 +arXiv:2301.01747v1 [cond-mat.mtrl-sci] 4 Jan 2023 + +Abstract +Engineering alloys generally exhibit multi-phase microstructures. For simu- +lating their microstructure evolution during solid-state phase transformation, +CALPHAD-guided multi-phase-field models coupled with micro-mechanics +have proven to be a reliable simulation tool. Nevertheless, their efficiency +and accuracy still depend on the homogenization scheme used to interpolate +the elastic properties in the interfacial regions. In this paper, we present a +phase-field model for multi-phase and multi-component solids using a partial +rank-one homogenization scheme that enforces static and kinematic compat- +ibilities in the interfacial regions. To this end, we first extend the rank-one +homogenization scheme to multi-phase systems. Moreover, for computational +efficiency, we analytically solve the static compatibility equations for linear +elastic three-phase solids. For quantitative accuracy, a coupling technique is +used to extract the prerequisite thermodynamic and kinetic properties from +CALPHAD databases. The model is solved numerically in an open source +finite-element framework. As numerical applications, the microstructure of +two elastically stressed intermetallic-containing three-phase alloys: Ni-Al and +Al-Cr-Ni, are simulated. The accuracy of the model is verified against analyt- +ically obtained solutions for planar and concentric ring interfaces. We show +that the simulation results remain unaltered with varying interface width. +Except for one simulation, all cases show better or nearly equal convergence +using the partial rank-one scheme compared to the Voigt-Taylor scheme. +2 + +Keywords: chemo-mechanical processes; microstructure; phase transforma- +tion; inhomogeneous material; homogenization +1 +Introduction +Engineering alloys, such as Ni-base superalloys, steels, etc., generally com- +prise multiple chemical constituents and phases. +Their physical and me- +chanical properties are strongly related to the microstructure formed during +interdiffusion processes at elevated temperatures. However, predicting the +kinetics of microstructure evolution during diffusive transformations, espe- +cially when elastic stresses are included, is difficult since this requires solving +a free-boundary problem, which is seldom analytically soluble [1–3]. Thus, +reliable and efficient computational approaches are often needed to gain a +quantitative understanding of microstructure evolution in elastically stressed +multiphase and multicomponent alloys. +The phase-field method has emerged as a useful tool to predict microstruc- +ture evolution in engineering alloys [4–8]. Its well-known advantage is that +the interface or interphase separating either the grains or phases is implic- +itly represented by a phase-field variable that varies smoothly across a finite +region of thickness, referred to as the interface width. Further, for simula- +tions to be well-resolved, the interface width has to be at least five times +the grid spacing [9]. Therefore, simulations using this method are particu- +larly difficult when the desired microstructural length scale is in the micro +to millimeter range due to a stringent limit on interface width [10], [11]. +In addition, this limit typically varies with the bulk alloy properties [12]. +3 + +To overcome this limitation, it is thus essential that the interface width in +a phase-field model can be independently controlled without affecting the +accuracy of the simulation. +This requirement has led to the development of alloy phase-field models +in which the interface width is treated as a simulation parameter that can be +selected depending on numerical convenience [13], [10]. This is because the +bulk and interfacial properties in such models are independent, even when +the interface width is artificially enlarged [10], [14]. Nevertheless, the gener- +alization of such alloy phase-field models to problems that require coupling +with mechanics is not straightforward due to the dependence of bulk prop- +erties on elastic fields. More precisely, in a mechanically coupled phase-field +model, the scheme of interpolation or homogenization of elastic fields in the +interfacial regions may affect this desired separation of bulk from interfa- +cial properties due to an interfacial excess elastic energy contribution that +depends on the homogenization scheme [15], [16]. +So far, two types of mechanically uncoupled phase-field models for alloys +have been proposed that allow the interface width to be selected arbitrarily +[10]. As pointed out by Plapp [10], the first derives the evolution equations +starting from a Helmholtz functional [13], while the second derives it from +a grand-potential functional [10], [17]. Moreover, the former approach re- +quires thermochemical properties as functions of composition(s), while the +latter requires them as functions of diffusion potential(s) [10]. Although both +models are equivalent [10], the latter offers possible computational gains as it +requires solving (n − 1)(p − 1) less equations for a n-component and p-phase +alloy system [18]. It is worth noting that this is strictly true assuming ei- +4 + +ther non-dilute or non-ideal or non-quadratic free energies since only then the +equal diffusion potential or “quasiequilibrium” conditions have to be numeri- +cally solved at each grid point and time step [19] [20]. Further, to decrease the +computational costs in such simulations, some studies have developed sim- +plified approaches to solving these conditions [19], [21], [22]. Nevertheless, +the appropriate homogenization approach for coupling these alloy phase-field +models with mechanics is still debatable. +Specifically, the coupling of the above-mentioned models with small-strain +elasticity theory has been considered by many workers; either based on a +Helmholtz functional, e.g., [23], [24], [25], [15], [26] or a grand-potential +functional [27–30]. Nevertheless, the accuracy of such coupled models still +depends on the homogenization assumptions with regard to the elastic fields +[24], [15], [31], [16], [32]. To be precise, depending on the scheme of homog- +enization, these mechanically coupled models can be subdivided into two +categories. +The models in the first category follow those homogenization +schemes that are either statically or kinematically compatible. For instance, +Khachaturyan [23], [27], Reuss/Sachs [33], [25], and Voigt/Taylor [24], [29]. +On the other hand, models in the second category follow those schemes that +enforce both static and kinematic compatibilities: by either using a mixed +scheme that is a combination of Reuss/Sachs and Voigt/Taylor, e.g., [15], +[26], [28], or a partial rank-one scheme [30]. Moreover, it has been argued +that models in the first category are less accurate compared to models in the +second category due to an interfacial excess energy contribution coming from +the interpolated elastic strain energy [15], [26], [16]. +Nevertheless, from the standpoint of computational efficiency, the partic- +5 + +ular scheme used for enforcing the static and kinematic compatibilities is also +a topic of relevance. For example, the mixed scheme that is a combination +of Reuss/Sachs and Voigt/Taylor proposed in the works of Durga et al. [15], +[34] and Schneider et al. [16], [35] requires a coordinate transformation of +elastic fields in order to formulate the interfacial elastic driving force con- +tribution as a function of only continuous elastic fields. As shown in [15], +[16], this is needed because then this interfacial excess contribution vanishes +in the model. Consequently, their approaches are computationally intensive +[36]. Naturally, this limits the application of their model to simple systems. +Specifically, Durga’s model has been so far applied to simulate an elasti- +cally anisotropic four-phase Cu-Sn alloy having only planar interfaces [34], +while Schneider’s model has been limited to elastically isotropic two-phase +[28] and multi-phase [37] binary alloys. These works, however, assume only +small-strain deformations. For sake of completeness, it is worth mentioning +that Schneider et al. and Hermann et al. have also proposed a numerical +approach to enforce the static compatibility equations for multiphase solids +undergoing finite-strain and small-strain inelastic deformations, respectively. +Svendsen et al. [32] independently proposed a more unified framework that +extends Helmholtz-based models, such as [15], to multiphase multicomponent +solids undergoing finite-strain and inelastic deformations. +Contrary to the mixed scheme, Mosler et al. +[31] proposed a partial +rank-one homogenization scheme to enforce static and kinematic compati- +bilities for two-phase solids undergoing finite deformations. The advantage +of this scheme over the mixed scheme is that it does not require coordi- +nate transformation. Keifer et al. showed improved convergence using this +6 + +scheme compared to schemes that ensure either static or kinematic com- +patibility for two-phase solids undergoing small-strain deformations. Subse- +quently, Bartels et al. [38] applied this scheme to couple mechanics with a +WBM (Wheeler-Boettinger-McFadden) type chemical model [39]. Unfortu- +nately, unlike the previously discussed mechanically uncoupled models, the +interface width in this model cannot be controlled due to an interfacial ex- +cess energy contribution coming from bulk chemical free energies [10], [17]. +Later, Bartel’s model was improved by the present authors by combining a +grand-potential model with the partial rank-one scheme for two-phase solids +undergoing small-strain deformations [30]. Using this model, we also found +that the rank-one scheme offered improved numerical convergence compared +to either static or kinematically compatible schemes [30]. +Despite these advantages, the partial rank-one scheme has so far not been +extended to multiphase and multicomponent solids undergoing linear elastic +deformations. To our knowledge, the only published work that extends the +rank-one scheme to multi-phase solids is by Sarhil et al. [40]. However, there +are two limitations to this model. The first limitation is that it has not been +coupled with diffusion equations and hence cannot be applied to simulate +diffusive transformations. The second limitation is that it takes an interpo- +lation function that is equal to the phase-field variable, i.e., hθ(φ) = φθ, to +interpolate elastic properties. As noted by Moelans [9], this assumption may +shift the local minima of the free energies and may cause inaccuracies. Hence, +this paper aims to fill these gaps by formulating a multi-phase-field model +based on a partial rank-one homogenization scheme starting from a grand- +potential functional, thereby ensuring that the interfacial excess contribution +7 + +due to bulk properties vanishes. To this end, we present an analytical ap- +proach to solving the static compatibility equations for a three-phase linear +elastic solid. For quantitative accuracy, we use a coupling method devel- +oped in [18] that allows incorporating thermodynamic and kinetic properties +obtained from CALPHAD databases into a grand-potential-based model. +The paper is organized as follows. The phase-field formulation with the +rank-one homogenization scheme is introduced in Section 2. In Section 3, +the prerequisite chemical properties and the elastic properties for two—a +Ni-Al and an Al-Ni-Cr—three-phase alloys are given. To demonstrate the +application of our model, four numerical simulations are performed, and the +results are discussed in Section 4. The accuracy of our numerical results is +tested by comparing the phase-field simulations with analytically obtained +solutions. Finally, the conclusions of the paper are discussed in Section 5. +2 +Formulation +2.1 +Notations +In this paper, we assume an isothermal system consisting of n diffusing com- +ponent and p phases. We denote a set of scalar fields with boldface letters. +For example, the set of (n − 1) independent diffusion potential fields is de- +noted as ˜µ = +� +˜µk=1...(n−1) +� +. Similarly, the set of p phase-field variables is +shown as φ = {φθ=α...p}. Vector and tensors are also represented with bold- +face letters, e.g., the displacement is written as u = uiei, where ui=1,2,3 are +the components of u relative to a chosen orthonormal basis {ei}. Following +8 + +standard notations, the Einstein summation convention is used throughout +the paper to indicate summation over spatial dimensions. The dot, outer +and inner products between two vectors, say a and b, are written as a · b, +a ⊗ b and a : b, respectively. The norm, divergence, gradient and laplacian +of a physical quantity, say Φ, are written as ∥Φ∥, div Φ, grad Φ, and ∆Φ, +respectively. +2.2 +Definitions of field variables and jump +As mentioned before, since the diffusion potentials are the independent vari- +ables in a grand-potential-based model, any prerequisite property in the +model should be expressed as functions of diffusion potentials. +Precisely, +the diffusion potential of a diffusing component, say k, is defined as the +difference between its chemical potential and the chemical potential of the +dependent component, i.e., ˜µk = (µk − µn), and it has units of J/mol. Fur- +ther, an arbitrary phase in the system, say θ, at any given spatial point x +and time t is indicated by the phase-field variable, φθ(x, t), such that the +bulk regions occupied by this phase are when φθ = 1. Moreover, the jump of +a field or property, �Φ�αβ = Φα − Φβ, at an interface, say α/β, is defined as +the difference between its bulk values within the two phases. +2.3 +Partial rank-one scheme for multi-phase systems +Starting from the two-phase approach of Mosler et al. [31], we assume that +the total strain, ϵ(u), in the interfacial regions is a smooth function of the +9 + +phase strains assigned to each phase in the system. Precisely, +ϵij(u) = +p +� +θ=1 +ϵθ +ijhθ(φ), +(1) +where ϵ(u) is the total strain as a function of the displacement u, ϵθ and +hθ(φ) are the (total) phase strain and interpolation function attributed to +phase θ. Moreover, the total strain at a point is calculated using the linear +strain-displacement relations +ϵij(u) = (1/2) [grad u + (grad u)T]. +(2) +The choice of the interpolation function, h(φ), is such that in the bulk re- +gions: hθ = 1 for (φθ = 1, φσ̸=θ = 0) and hθ = 0 for (φθ = 0, φσ̸=θ = 1), +while in the interfacial regions: 0 < hθ < 1 for 0 < φθ < 1. Further, as noted +in [41], [9], the function h(φ) must satisfy two additional requirements: i) +�p +θ=1 hθ = 1, and ii) dhθ/dφθ [φθ = 1, φσ̸=θ = 0] = 0. Thus, similar to the +function proposed by Moelans [9], three different interpolation functions that +fulfill these requirements have been formulated by Schneider and co-workers +[35, 42, 43]. However, for the sake of convenience, in this work we chose the +function first proposed by Moelans [9]: +hθ(φ) = +φ2 +θ +�p +θ=1 φ2 +θ +for +θ = {α, β . . . , p}. +(3) +As discussed in the Introduction section, it should be noted that Sarhil et +al. [40] proposed hθ(φ) = φθ which does not satisfy the above-mentioned +requirements and may lead to inaccuracies. Another noteworthy difference +10 + +between our model and the models proposed by Sarhil et al. [40] and Schnei- +der et al. [35, 42, 43] is that our model does not require a constraint that +the sum of phase-fields should add up to 1, i.e., �p +θ=1 φθ = 1. +It is worth noting that the phase strains introduced in Eq. (1) are phys- +ically meaningful only in the bulk regions of a phase (hθ = 1) but not in the +interfacial regions. It is because, in the bulk regions, they become equal to +the total strain, which is a physically measurable quantity that depends on +the stiffness tensor and boundary conditions. But in the interfacial regions, +the variation of phase strains depends on the interface width; a numerical +parameter selected arbitrarily. In section 2.4, we will show that the phase +strains also depend on the homogenization scheme in the interfacial regions. +Similar to phase concentrations introduced in solidification studies [44], their +primary purpose is to separate the bulk and interfacial contributions in the +total energy for artificially enlarged interfaces. +Following the two-phase approach [31], to ensure kinematic compatibility, +the phase strains must satisfy the Hadamard jump conditions. Consequently, +the p unknown phase strains, {ϵα, ϵβ, ϵγ . . . ϵp}, in Eq. (1) must satisfy the +following (p − 1) Hadamard jump conditions +�ϵij�αβ = ϵα +ij − ϵβ +ij = sym(aαβ +i nαβ +j ), +�ϵij�βγ = ϵβ +ij − ϵγ +ij = sym(aβγ +i nβγ +j ), +... +�ϵij�(p−1),p = ϵp−1 +ij +− ϵp +ij = sym +� +a(p−1),p +i +n(p−1),p +j +� +, +(4) +where {aαβ, aβγ, . . . , a(p−1),p} and {nαβ, nβγ, . . . , and n(p−1),p} are the +11 + +jump vectors and unit normals at the {α/β, β/γ, . . . , (p − 1)/p} interfaces, +respectively. +Here, the notation �ϵ�(p−1),p denotes the strain jump at the +interface between phases (p − 1) and p, and is equal to the outer product +between the jump vector and unit normal at that interface. It should be noted +that the jump vector, aαβ, at the α/β interface is symmetric with respect to +the superscripts αβ [42], i.e., aαβ = aβα, since by definition �ϵ�αβ = −�ϵ�βα +and nαβ = −nβα. +Moreover, following our previous work [30], we define the unit normal at +an interface as [45], [42] +nθσ,σ̸=θ = −grad φθ/ ∥grad φθ∥ , +θ = {α, β . . . (p − 1)} & σ = {β, γ, . . . , p}, +(5) +where ∥∇φθ∥ is the norm of the gradient of the phase-field variable φθ. We +note that although a multi-phase-field version of Eq. (5) exists (see [46] & +[47]), we have not used that definition in this paper for the sake of simplicity. +Moreover, Schneider et al. [42] has noted that the solution of elastic fields is +not significantly dependent on the definition of the unit normal vector. +Eqs. (1) and (4) form a system of p equations that can be analytically +solved to explicitly determine the p-phase strains: {ϵα, ϵβ, ϵγ . . . ϵp}, as func- +tions of the total strain ϵ(u), p interpolation functions hθ=α,β...p(φ) and (p−1) +strain jumps: {�ϵ�αβ, �ϵ�βγ, . . . , �ϵ�(p−1),p}. In Appendix A, we show how to +analytically calculate the phase strains for a multiphase system as functions +of the total strain, interpolation functions and strain jumps. +Next, we calculate the unknown jump vectors: {aαβ, aβγ, . . . , a(p−1),p} in +12 + +order to determine the (p − 1) strain jumps. Similar to previous works, e.g., +[31], [30], [42], we also calculate the jump vector at an interface by solving +the static compatibility equation. To be precise, the following (p − 1) static +compatibility equations must be solved to determine the same number of +unknown jump vectors +�σij�αβnαβ +j += +� +σα +ij − σβ +ij +� +nαβ +j += 0i, +�σij�βγnβγ +j += +� +σβ +ij − σγ +ij +� +nβγ +j += 0i, +... +�σij�(p−1),pn(p−1),p +j += +� +σp−1 +ij +− σp +ij +� +n(p−1),p +j += 0i, +(6) +where σθ is the elastic stress associated with phase θ. In this paper, we +have introduced static compatibility equations as a means to calculate the +jump vectors. Alternatively, these equations can be derived by minimizing +the total elastic strain energy with respect to the jump vectors, as pointed +out by Mosler et al. [31] and Sarhil et al. [40] . +Using linear elastic theory, the elastic phase stresses, {σα, σβ, σγ . . . σp}, +are related to the phase strains by the generalized Hooke’s law +σθ +ij = Cθ +ijkl +� +ϵθ +kl − ϵ⋆θ +kl +� +for +θ = {α, β . . . , p}, +(7) +where Cθ and ϵ⋆θ denote the fourth-rank stiffness tensor and eigenstrain +belonging to a particular phase θ, respectively. +It follows from the set of Eqs. (4) & (6) that the rank-one scheme ensures +both kinematic and static compatibilities at (p − 1) interfacial regions of a +13 + +p-phase system. It is worth pointing out that a system consisting of p-phases +may have p(p − 1)/2 two-phase junctions. However, Eqs. (4) & (6) only +ensure static and kinematic compatibilities at (p − 1) of these junctions. It +can be shown that mechanical compatibilities at remaining (p − 1)(p − 2)/2 +junctions are implicitly ensured. For instance, by adding Eqs. (4) & (6), we +obtain the following set of compatibility equations +�ϵij�α,p = ϵα +ij − ϵp +ij = {aαβ +i nαβ +j ++ aβγ +i nβγ +j ++ . . . + a(p−1),p +i +}, +(8) +�σij�α,pnα,p +j += +� +σα +ij − σp +ij +� +nα,p +j += 0i. +(9) +From Eqs. (8) and (9), it follows that both static and kinematic compat- +ibilities are ensured at the interface between phases α and p. +Finally, it should be emphasized that, depending on the constitutive equa- +tions, the jump vectors can be solved either analytically or numerically. As +discussed in the Introduction section, for non-linear elastic solids, the jump +vectors can be obtained only by numerically solving the set of static compat- +ibility equations at each grid point and time step (see [42],[43]), i.e., Eqs. (6). +However, for linear elastic solids, the jump vectors can be determined either +analytically or numerically. Although restricted to two phases, the analyt- +ical approach was followed in [48] & [30], while a Newton-Raphson scheme +was used in [49]. Nevertheless, to our knowledge, analytical expressions for +the jump vectors in a multi-phase-field setting do not exist. Since analytical +approaches may offer computational gains over numerical solutions, partic- +ularly when linear constitutive equations are assumed, we follow the former +approach. +14 + +However, deriving a general analytical expression for the jump vectors +for a p-phase system is not straightforward as it requires explicit analytical +expressions for phase strains. Since the analytical expressions for the phase +strains become increasingly complicated as the number of phases increases +(see Appendix A), we therefore take a special case to illustrate how to derive +the jump vectors in a multi-phase-field context. For the sake of convenience, +we chose a three-phase system for this derivation. +2.4 +Partial rank-one scheme for three-phase systems +For a three-phase system, the phase strains belonging to phases, say α, β +and γ, may be written as (see Appendix A) +ϵα +ij +� +ϵ, φ, �ϵ�αβ, �ϵ�βγ� += ϵij(u) + [hβ(φ) + hγ(φ)] �ϵij�αβ + hγ(φ)�ϵij�βγ, +(10) +ϵβ +ij +� +ϵ, φ, �ϵ�αβ, �ϵ�βγ� += ϵij(u) − hα(φ)�ϵij�αβ + hγ(φ)�ϵij�βγ, +(11) +ϵγ +ij +� +ϵ, φ, �ϵ�αβ, �ϵ�βγ� += ϵij(u) − hα(φ)�ϵij�αβ − [hβ(φ) + hα(φ)] �ϵij�βγ. +(12) +It follows from Eqs. (10)-(12) that the phase strains are always equal to +the total strain ϵ(u) in the bulk regions of the phases. +However, in the +interfacial regions, they differ depending on the definition of strain jumps, +which in turn depends on the homogenization assumption. Concretely, the +strain jumps vanishes, i.e., �ϵ�αβ = �ϵ�βγ = 0, for the case of Voigt-Taylor +homogenization scheme (henceforth referred to as the VT scheme) or the +Khachaturyan scheme [15], while for the partial rank-one scheme (henceforth +15 + +referred to as PR scheme) the strain jumps are given by (see Eqs. 4) +�ϵ�αβ = aαβ ⊗ nαβ, +(13) +�ϵ�βγ = aβγ ⊗ nβγ. +(14) +As previously discussed, the jump vectors aαβ and aβγ in Eqs. (13) & +(14) are obtained by solving the static compatibility equations. Precisely, +the set of Eqs. (6) for a three-phase system reduced to +� +σα +ij − σβ +ij +� +nαβ +j += 0i, +(15) +� +σβ +ij − σγ +ij +� +nβγ +j += 0i. +(16) +Next, it follows from Eq. (7) that the elastic phase stresses in Eqs. (15) & +(16) are related to the phase strains by +σα +ij = Cα +ijkl [ϵα +kl − ϵ⋆α +kl ] , +(17) +σβ +ij = Cβ +ijkl +� +ϵβ +kl − ϵ⋆β +kl +� +, +(18) +σγ +ij = Cγ +ijkl [ϵγ +kl − ϵ⋆γ +kl ] . +(19) +Now, substituting Eqs. (17)-(19) in Eqs. (15)-(16) yields +�� +Cα +ijkl − Cβ +ijkl +� +ϵkl + λ1 +ijkl�ϵkl�1 + λ2 +ijkl�ϵkl�2� +n1 +j = Z1 +i , +(20) +�� +Cβ +ijkl − Cγ +ijkl +� +ϵkl + M1 +ijkl�ϵkl�1 + M2 +ijkl�ϵkl�2� +n2 +j = Z2 +i , +(21) +where, we have denoted the superscripts αβ and βγ by 1 & 2, respectively, +16 + +and +λ1 +ijkl(φ) = hβ(φ)Cα +ijkl + hα(φ)Cβ +ijkl + hγ(φ)Cα +ijkl, +λ2 +ijkl(φ) = hγ(φ)Cα +ijkl − hγ(φ)Cβ +ijkl, +M1 +ijkl(φ) = hα(φ)Cγ +ijkl − hα(φ)Cβ +ijkl, +M2 +ijkl(φ) = hγ(φ)Cβ +ijkl + hα(φ)Cγ +ijkl + hβ(φ)Cγ +ijkl, +Z1 +i (n1) = +� +Cα +ijklϵ⋆α +kl − Cβ +ijklϵ⋆β +kl +� +n1 +j, +Z2 +i (n2) = +� +Cβ +ijklϵ⋆β +kl − Cγ +ijklϵ⋆γ +kl +� +n2 +j, +(22) +Then, substituting Eqs. (13) & (14) in Eqs. (20) & (21) yields +� +mα1 +i +− mβ1 +i +� ++ λ# +ika1 +k + λ⋆ +ika2 +k = Z1 +i , +(23) +� +ψβ2 +i +− ψγ2 +i +� ++ L# +ika1 +k + L⋆ +ika2 +k = Z2 +i , +(24) +where +mα1 +i +� +ϵ, n1� += Cα +ijklϵkln1 +j, +mβ1 +i +� +ϵ, n1� += Cβ +ijklϵkln1 +j, +ψβ2 +i +� +ϵ, n2� += Cβ +ijklϵkln2 +j, +ψγ2 +i +� +ϵ, n2� += Cγ +ijklϵkln2 +j, +λ# +ki(φ, n1) = n1 +l λ1 +lkij(φ)n1 +j, +λ⋆ +ki(φ, n1, n2) = n2 +l λ2 +lkij(φ)n1 +j, +L# +ki(φ, n1, n2) = n1 +l M1 +lkij(φ)n2 +j, +L⋆ +ki(φ, n2) = n2 +l M2 +lkij(φ)n2 +j. +(25) +17 + +Rearranging Eq. (24) and solving for a2 yields +a2 +j(φ, ϵ, n1, n2) = Sji(φ, , n2)bi, +(26) +where Sji = +� +L⋆ +ij +�−1 and bi = Z2 +i − +� +ψβ2 +i +− ψγ2 +i ++ L# +ika1 +k +� +. Next, substituting +a2 +j in Eq. (23) yields +λ# +pqa1 +q + λ⋆ +pq (Sqibi) = Z1 +i − +� +mα1 +p − mβ1 +p +� +(27) +Using the expression for b and then solving for a1 using Eq.(27) finally yields +a1 +k(ϵ, φ, n1, n2) = (Dpk)−1 � +Z1 +p − +� +mα1 +p − mβ1 +p +� +− λ⋆ +prSri +� +Z2 +i − +� +ψβ2 +i +− ψγ2 +i +��� +(28) +where Dpk = λ# +pk − λ⋆ +prSriL# +ik. +For a three-phase-field model, Eqs. +(26) +and (28) are the most general expressions for the jump vectors a2 and a1, +respectively. +To further simplify these expressions, we assume that the two second- +rank tensors, λ⋆ and L#, are zero. Because from Eq. (25) we see that these +tensors depend on both the unit vectors, n1 and n2, which are simultaneously +non-zero only at the triple points. Perhaps not surprisingly, by making this +assumption we have strictly restricted the definition of jump vectors to the +two-phase regions. Stated differently, we have enforced static compatibility +only at the two-phase junctions. Our assumption is justified since the set +of Eqs. (6) is strictly valid at the two-phase junctions only where the unit +normal vector to the interface is uniquely defined. As a consequence, Eqs. +18 + +(26) and (28) simplifies to +a2 +j = − +� +L⋆ +ij +�−1 �� +Cβ +ikpq − Cγ +ikpq +� +ϵpq − +� +Cβ +ikpqϵ⋆β +pq − Cγ +ikpqϵ⋆γ +pq +�� +n2 +k, +(29) +a1 +j = − +� +λ# +ij +�−1 �� +Cα +ikpq − Cβ +ikpq +� +ϵpq − +� +Cα +ikpqϵ⋆α +pq − Cβ +ikpqϵ⋆β +pq +�� +n1 +k, +(30) +where +L⋆ +ij(φ, n2) = n2 +l +� +hγ(φ)Cβ +lijr + hα(φ)Cγ +lijr + hβ(φ)Cγ +lijr +� +n2 +r, +(31) +λ# +ij(φ, n2) = n1 +l +� +hβ(φ)Cα +lijr + hα(φ)Cβ +lijr + hγ(φ)Cα +lijr +� +n1 +r. +(32) +Expectedly, we see that the analytically derived expressions for jump +vectors, i.e., Eqs. (29) & (30), are similar to the expression of jump vector +derived in a two-phase setting (cf. Eq. (9) in [30]). As noted in a previous +work [30], we find that the magnitude of the jump vector at an interface, say +α/β, is proportional to two elastic properties: i) the jump in stiffness tensors +of the bulk phases, and ii) the eigenstrains in the bulk phases. +2.5 +Functional, overall molar density and elastic stresses +Here, starting from a grand-potential functional we derive expressions for +the overall molar density of a diffusing component and elastic stresses. As +discussed in the Introduction section, we follow the grand-potential approach +[10], [17] in this work because we don’t need to explicitly solve for the +quasiequilibrium conditions [19], that may lead to computational gains. +The grand-potential functional, Ω[φ, ˜µ, u], of the system for an elastically +19 + +stressed multiphase multicomponent alloy is given by +Ω [φ, ˜µ, u] = +� +V +[ωbulk (φ, ˜µ, ϵ) + ωint (φ, ∇φ)] dv, +(33) +where the bulk contribution to the total grand-potential density is denoted by +ωbulk (φ, ˜µ, ϵ); the interfacial energy contribution to the total grand-potential +density is indicated by ωint(φ, ∇φ); and V is the total volume. Further, the +bulk contribution to the total functional, i.e., ωbulk [J/m3], is defined as +ωbulk(φ, ˜µ, ϵ) = +p +� +θ=1 +hθ(φ)ωθ +bulk +�˜µ, ϵθ� +, +(34) +where hθ(φ) is the interpolation function, which is defined at Eq. (3) and +ωθ +bulk is the grand-potential density of phase θ expressed as functions of dif- +fusion potentials ˜µ and phase strains ϵθ. Under the assumption that each +phase is represented by a single grain orientation, the interfacial energy con- +tribution to the total energy may be written as [9] +ωint(φ, ∇φ) = +p +� +θ=1 +(1/2)κ ∥grad φθ∥2 + mg(φ), +(35) +where the two constant parameters κ [J/m] and m +� +J/m3� +are related to the +interfacial energy σαβ and interface width lαβ by [9] +κ = (3.0/4.0) σαβlαβ, +m = 6.0 (σαβ/lαβ) , +(36) +assuming uniform interface properties and a multi-well function g(φ) of the +20 + +form [9] +g(φ) = +p +� +θ=1 +� +(1/4) φ4 +θ − (1/2) φ2 +θ +� ++ (3/4) +p +� +θ=1 +p +� +σ=1 +σ>θ +φ2 +θφ2 +σ + (1/4). +(37) +Following our previous work [30], the bulk grand-potential density ωθ +bulk(˜µ, ϵθ) +in Eq. (34) is written as +ωθ +bulk(ϵθ, ˜µ) = ωθ +chem(˜µ) + (1/2)Cθ +ijkl(˜µ) +� +ϵθ +kl − ϵ⋆θ +kl(˜µ) +� � +ϵθ +ij − ϵ⋆θ +ij (˜µ) +� +. +(38) +The first and second terms in Eq. (38) are the chemical and elastic energy +contributions to the bulk grand-potential density of a phase θ, respectively. +Precisely, ωθ +chem is defined as Ωθ +m/Vm, where Ωθ +m is the molar grand-potential +and Vm is the molar volume, which is assumed to be constant. Moreover, Ωθ +m +can be analytically calculated by assuming either parabolic or dilute or ideal +free energies [10]. This was the approach taken in our previous study [30]. +However, it is difficult to extend this approach to multi-phase and multi- +component alloy systems. Thus, in this work we take a numerical approach +to calculate the chemical grand-potential from CALPHAD databases using +the method developed in [18]. It should be noted that, similar to our previous +study [30], here we have assumed that the stiffness tensor and the eigenstrains +are functions of diffusion potentials to account for composition-engendered +stresses in the model. +Next, we derive an expression for the overall molar density. Thus, differ- +21 + +entiating Eq. (34) with respect to the diffusion potential gives +cr(φ, ˜µ, ϵθ) = −∂ωbulk +∂µr += +p +� +θ=1 +hθ(φ)cθ +r +� +˜µ, ϵθ� +, +(39) +where cr and cθ +r are the overall and phase molar densities of a diffusing +component r, and have units of mol/m3. More precisely, using Eq. (38) the +phase molar density may be explicitly written as [30] +cθ +r +� +ϵθ, ˜µ +� += −∂ωθ +bulk +∂µr += Xθ +r (˜µ) +Vm +− 1 +2 +∂Cθ +ijkl +∂˜µr +� +ϵθ +kl − ϵ⋆θ +kl +� � +ϵθ +ij − ϵ⋆θ +ij +� ++ ∂ϵ⋆θ +ij +∂˜µr +σθ +ij, +(40) +where Xθ +r (˜µ) = −∂Ωθ +m/∂µr [18] is the phase mole fraction of component +r in phase θ. It follows from Eq. (40) that if the stiffness tensor and the +eigenstrains are assumed to be uniform throughout the system, then Eq. (40) +simplifies to +cθ +r (˜µ) = Xθ +r (˜µ) +Vm +. +(41) +Since we will assume uniform elastic properties in this paper, we will use +Eq. (41) to define the phase molar densities. Moreover, the prerequisite +phase mole fractions, Xθ +r , can be calculated either analytically assuming ei- +ther parabolic or dilute or ideal free energies [10], or can be directly obtained +from CALPHAD databases [18]. In this work, we will follow the latter ap- +proach since simplistic free energies may cause inaccuracies, particularly for +multiphase and multicomponent alloys. Finally, we derive an expression for +22 + +the overall stress. Thus, differentiating Eq. (34) with respect to total strain, +it can be shown that (Appendix C) +σij(φ, ˜µ, ϵθ) = ∂ωbulk +∂ϵij += +p +� +θ=1 +hθ(φ)σθ +ij(˜µ, ϵθ), +(42) +where σij and σθ +ij = ∂ωθ +bulk/∂ϵθ +ij are the overall and phase elastic stresses. +The latter is defined at Eq. (7) +2.6 +Governing equations +Taking the first variation of Eq. (33) and using Eqs. (39) and (42) yields: +p +� +θ=1 +hθ(φ)cθ +k(˜µ) − ck = 0 +∀ k = 1 . . . (n − 1), +(43) +div +� +p +� +θ=1 +hθ(φ)σθ +ij +� += 0, +(44) +∂φθ +∂t + Lφ +� +m∂g (φ) +∂φθ +− κ∆φθ + ∂ωbulk +∂φθ +� += 0 +∀ θ = 1 . . . p, +(45) +where Lφ is the Allen-Cahn mobility and is assumed to be uniform in this +work. It should be noted that the standard diffusion equations do not nat- +urally come out of the variational derivative in the case of grand-potential- +based models. Thus, to ensure mass conservation, the evolution of overall +molar density, ck in Eq. (43), is given by [18] +∂ck(x, t) +∂t +− div +�n−1 +� +j=1 +Ln +kj (˜µ, φ) +Vm +grad ˜µj +� += 0 +∀ k = 1 . . . (n − 1), +(46) +23 + +where the components of the overall Onsager matrix Ln +kj (˜µ, φ) are interpo- +lated as [18] +Ln +kj (˜µ, φ) = +p +� +θ=1 +hθ (φ) Lnθ +kj (˜µ) . +(47) +Here, the notation Lnθ +kj (˜µ) represents the components of the Onsager matrix +specific to a particular phase θ expressed as a function of diffusion potentials. +Again, this term can also be either directly obtained as functions of diffusion +potentials from CALPHAD databases [18] or can be assumed to be uniform. +It should be noted that we do not follow grand-potential-based models, +e.g., [10], [17], [27, 37, 50–52], that requires formulating a diffusion potential +rate equation by first taking the time derivative of Eq. (43) and then substi- +tuting Eq. (46). Instead, we calculate the diffusion potential by iteratively +solving Eq. (43). Consequently, this approach requires calculating a Jaco- +bian matrix, that can be evaluated by differentiating Eq. (43) with respect +to the diffusion potential [30]. This yields +p +� +θ=1 +hθ(φ)χθ +jr (˜µ) − ∂cj +∂˜µr += 0, +(48) +where χθ +jr(˜µ) = ∂cθ +j/∂µr are the coefficients of the susceptibility matrix ex- +pressed as a function of diffusion potentials. Further, these coefficients can +be determined either by analytical approaches assuming parabolic or dilute +or ideal free energies [10] or numerically from CALPHAD databases [18]. +Moreover, as previously discussed, the scheme of homogenization may in- +fluence the independence of bulk and interfacial properties. More specifically, +24 + +this independence is achieved provided that the last term in Eq. (45) van- +ishes at equilibrium [10]. However, this may not be evident in mechanically +coupled alloy phase-field models. Concretely, consider a three-phase system +consisting of phases—α, β and γ; then the last term for a specific phase-field +variable, say φα, may be explicitly written as (Appendix D) +∂ωbulk +∂φα += − ∂hβ +∂φα +�� +ωα +bulk − ωβ +bulk +� +− +� +p +� +θ=1 +hθσθ +ij +� +�ϵij�αβ +� +− ∂hγ +∂φα +� +(ωα +bulk − ωγ +bulk) − +� +p +� +θ=1 +hθσθ +ij +� +�ϵij�αγ +� +. +(49) +It follows from Eq. (49) that the terms within the large curly braces depend +on the strain jumps, �ϵ�αβ and �ϵ�αγ, which are in turn dependent on the +scheme of homogenization. +For instance, if Voigt/Taylor or Khacturayan +scheme is followed, then the strain jumps vanish and consequently these +terms are proportional to the jump in the grand potentials, i.e., �ωbulk�αβ +& �ωbulk�αγ. Further, since the bulk grand-potentials are functions of both +continuous and discontinuous (total) strain components (see Eq. (38)), these +terms would not necessarily vanish at equilibrium. On the other hand, in +the case of the partial rank-one scheme the strain jumps are non-zero and +it can be shown that these terms reduce to the sharp interfacial chemical +equilibrium conditions for coherently stressed two-phase solids (see Eq. (7.31) +in [53]). For sake of completeness, we have also provided the derivatives with +respect to φβ and φγ in Appendix D, which are similar to Eq. (49). +Moreover, it must be noted that in writing Eq. +(45) we have tacitly +assumed that the variational contribution to the driving force is negligible. +25 + +Precisely, the variational term may be written as: +div +� +∂ωbulk +∂ (grad φθ) +� += div +�� +p +� +θ=1 +hθσθ +ij +� +∂ϵθ +ij +∂ (grad φθ) +� +. +(50) +Note that due to the dependence of phase strains on the unit vectors: nαβ +and nβγ, the term within the curly braces in Eq. (50) is nonzero in case of the +rank-one scheme. However, based on our previous study [30], we found that +this term does not significantly affect the temporal variation of the interface +for cases with a small difference in stiffness tensors [30]. This is because the +term is proportional to the magnitude of jump vectors, aαβ and aβγ, and are +consequently proportional to the difference in stiffness tensors (see Eqs. (29) +& (30)). Thus, we have neglected this term in our calculations which renders +our formulation non-variational. +Finally, the Allen-Cahn mobility is calculated using [9] +Lφ = 4m/(3κζ), +(51) +where m and κ are defined at Eq. (36) and the parameter ζ = �n−1 +k=1(Xθ,eq +k +− +Xσ,eq +k +) �n−1 +j=1 +� +Lnθ,eq +kj +Vm +�−1 +(Xθ,eq +j +−Xσ,eq +j +), is obtained assuming infinite inter- +face kinetics [54]. This choice of ζ ensures that local equilibrium is maintained +near the interface and the growth is diffusion-controlled [9]. +3 +Coupling with CALPHAD databases +As discussed before, our model requires thermodynamic properties and mo- +bilities as functions of diffusion potentials. Specifically, four properties are +26 + +needed for any given phase [18]. First, the molar grand-potential, Ωθ +m, of an +individual phase to calculate the chemical contribution to the bulk grand- +potential density in Eq. (38). Second, the phase mole fractions to calculate +the phase molar densities using Eq. (41). Third, the susceptibility matrix to +evaluate Eq. (48). Finally, the Onsager matrix pertaining to each individ- +ual phase is also required to evaluate Eq. (47). Moreover, for non-dilute and +non-ideal solid solutions, these properties cannot be analytically expressed as +functions of diffusion potentials. Thus, we numerically evaluated these prop- +erties using the MATLAB-ThermoCalc interface by minimising the prereq- +uisite properties with respect to a discretized range of diffusion potential(s). +This discretized range was predetermined based on the phase diagram [18]. +Concretely, we chose two three-phase alloys: a binary Ni-Al and a ternary +Ni-Al-Cr, to illustrate the coupling procedure. For all phases except the bi- +nary and ternary B2 phases, the above-mentioned properties were extracted +as functions of diffusion potentials from the TCNi8 and MOBNi4 databases +using the TC-Toolbox for MATLAB. Specifically, in the case of Ni-Al, we +evaluated the thermodynamic properties and mobilities as discretized func- +tions of Al diffusion potential in the interval of [−2e5, 2e5] J/mol. Similarly, +for the Al-Cr-Ni simulations, we obtained the discretized properties by vary- +ing the Cr and Al diffusion potentials from −1e5 J/mol to 1e5 J/mol. These +limits were selected to ensure that the Al and Cr mole fractions of an ar- +bitrary phase are very close to the limits of 0 and 1 (see Appendix C in +Ref. [18], for details). Following this, the properties assigned to a given +phase were non-dimensionalized and stored in a tabulated format and then +supplied as an input to MOOSE (Multiphysics Object-Oriented Simulation +27 + +Environment) [55] for phase-field simulations. Fig.1 shows the coupling pro- +cedure schematically. The details of the non-dimenionalization are given in +Appendix E. +For the binary and ternary B2 phases, we could obtain only the thermody- +namic properties as discretized functions of diffusion potentials. The Onsager +coefficients were assumed to be constants. Specifically, the mobilities were +obtained from ThermoCalc at the equilibrium mole fractions. Following this, +these mobilities were used to evaluate the ζ parameter in Eq. (51), which is +needed to calculate the Allen-Cahn mobility. The mobilities, the equilibrium +mole fractions, the parameter ζ, and the simulation temperatures are listed +in Table 1. +Fig. 1. Schematic showing the coupling procedure between CALPHAD databases +and MOOSE in case of a grand-potential-based model. +28 + +Table 1 +Constant material parameters for the Ni-Al and Al-Cr-Ni alloy systems. The equi- +librium mole fractions and the Onsager mobilities were obtained from ThermoCalc. +Ni-Al +Al-Cr-Ni +T [K] +1000 +1473 +σ [J/m2] +0.5 +0.5 +Vm [m3/mol] +7.5e−5 +7.5e−5 +Xα,eq +B +0.27457 +0.2209 +Xβ,eq +B +0.40646 +0.2912 +Xα,eq +C +- +0.07575 +Xβ,eq +C +- +0.06756 +Lβ,eq [mol m2/Js] +1.7534e-17 +�0.8238 +0.0552 +0.0552 +0.2684 +� +× 1e−17 +ζ [Js/m5] +1.3228e19 +8.8423e18 +4 +Results and discussion +As previously discussed, we have considered two three-phase alloys, an Al- +Ni alloy and an Al-Cr-Ni alloy, to demonstrate the application of our model. +Further, we have considered two interface geometries per alloy system. Specif- +ically, the first two cases assume planar interfaces, while the remaining two +cases assume concentric ring interfaces. We have employed both the par- +tial rank-one (hereafter referred to as PR) and the Voigt-Taylor (hereafter +referred to as VT) homogenization schemes to simulate all four cases. As +noted earlier, this was achieved by controlling the jump in phase strains, i.e., +�ϵ�, in Eqs. (10)-(12). +For sake of clarity, Table 2 provides the mechanical boundary conditions +and the eigenstrains for each considered case. From Table 2, we note that the +eigenstrains in the binary and ternary γ′ phases are identical. Although in +real alloys, the strength of the eigenstrain depends on the alloy composition, +29 + +we made this simplifying assumption due to the lack of any experimental +data in the literature. +Moreover, the assumed elastic constants for each +simulated case are listed in Table 3. Except for case II, we have assumed +isotropic elastic constants for all considered cases (Table 3). Finally, to verify +the accuracy of our model, we have compared the simulated elastic fields in +each of these cases against the analytically obtained solution. The analytical +solutions are provided in Appendix F. Here, it is worth emphasizing that +the analytical solutions depend on the interface positions, which have been +calculated numerically by tracking the phase-field variables (φθ=α,β,γ = 0.5). +Table 2 +Summary of eigenstrains and mechanical boundary conditions for all cases. The x- +and y-components of displacement u are denoted by ux & uy, respectively. Here, +lc = 0.033 µm denotes a characteristic length scale used for non-dimensionalization. +Simulation +Eigenstrains [Phase] +Boundary conditions +Planar Al-Ni +ϵ⋆ [γ′] = −0.3%1 +u at left boundary = 0 +(Case I) +u at right boundary = 0 +ϵ⋆ [γ] = ϵ⋆ [B2] = 0 +u is periodic along y-direction +u at left boundary = 0 +Planar Al-Cr-Ni +ϵ⋆ [γ′] = −0.3%1 +ux/lc at right boundary = 5 +(Cases II) +uy/lc at right boundary = −5 +ϵ⋆ [γ] = ϵ⋆ [B2] = 0 +u is periodic along y-direction +Non-planar Al-Ni +ϵ⋆ [γ′] = −0.3%1 +ux at left boundary = 0 +(Case III) +uy at bottom boundary = 0 +ϵ⋆ [γ] = 0 +traction is zero at outer boundary +Non-planar Al-Cr-Ni +ϵ⋆ [γ′] = 0 +ux at left boundary = 0 +(Case IV) +uy at bottom boundary = 0 +ϵ⋆ [γ] = ϵ⋆ [B2] = 0 +ux at outer boundary = 0.1%x +uy at outer boundary = 0.1%y +30 + +Table 3 +Summary of elastic constants for all simulated cases. Here, the left/inner label +refers to the leftmost or the innermost phase in the simulations depending on +the planar or concentric interface case. Likewise, the right/outer label refers to +the rightmost or outermost phase, and the centre label refers to the intermediate +phase. +Simulation +Left/Inner +Centre +Right/Outer +Refs. +Case I, +E = 158 GPa +E = 147 GPa +G = 76.6 GPa +[56], [57] +ν = 0.3 +ν = 0.3 +ν = 0.3387 +II & III +Case II +C11 = 188.3 GPa +C11 = 194.37 GPa +G = 76.6 GPa +[58], [57] +C12 = 143.54 GPa +C12 = 140.82 GPa +ν = 0.3387 +C44 = 80.734 GPa +C44 = 84.04 GPa +4.1 +Planar three-phase Ni-Al simulation +First, we simulated a coherently stressed planar fcc−γ/γ′−Ni3Al/NiAl alloy +that is mechanically constrained at the left and right boundaries (Fig. 2a). +We have assumed periodic boundary conditions for the phase field, compo- +sition and displacement variables at the top and bottom boundaries. While +homogeneous Neumann boundary conditions are applied at the left and right +boundaries for the phase-field and composition variables, viz. +grad φ · nΓ(x = ±Lx/2, y, t) = 0, +(52) +grad ˜µAl · nΓ(x = ±Lx/2, y, t) = 0, +(53) +where ˜µAl is the Al-diffusion potential; Lx is the length of the simulation +domain, and nΓ is the unit normal at the left and right boundaries. The +31 + +displacement boundary conditions at these boundaries are (Table 2): +ux(x = ±Lx/2, y, t) = 0, +(54) +uy(x = ±Lx/2, y, t) = 0. +(55) +Since the three phases cannot coexist, the intermediate γ′−Ni3Al phase grows +at the expense of γ and NiAl phases. Fig. 2b shows the Al mole fraction +field at time t = 37 s. +Moreover, we find that the thickness of γ′ phase +increases linearly as a function of the square root of simulation time (Fig. +2c), thus indicating parabolic growth kinetics. This thickness is numerically +determined by locating the γ/γ′ and γ′/NiAl interface positions as a func- +tion of time using the phase-field variables. To further test the influence of +interface width on kinetics, we vary the interfacial parameters: κ, m and Lφ, +using Eqs. (36) & (51), for three different interface widths. We find that +the thickness of the Ni3Al phase remains relatively unaltered with varying +interface width using both schemes (Fig. 2c). Expectedly, for both PR and +VT schemes, the CPU time decreases with increasing interface width; since +the grid spacing, ∆x = lw/6.0, is directly proportional to interface width +lw (Fig. 2d). However, we find that the PR scheme shows comparatively +better convergence compared to the VT scheme (Fig. 2d). This shows that +the proposed PR scheme is computationally efficient compared to the VT +scheme for a longer simulation time. +To further verify the spatial accuracy, we sample the spatial variation of +the composition field and the elastic quantities across a line normal to the +interface at time t = 37 s. Fig. 3a compares the Al mole fraction profile +32 + +for three different interface widths using the PR scheme. We find that the +simulated Al mole fraction profile remains independent of interface width in +the bulk regions. Nevertheless, we find marginal deviations in the interfacial +regions since the composition is interpolated in this region. We also find this +deviation in the x-component of the displacement field near the interfaces. +Specifically, we find that the displacement fields using interface widths of 1.2 +µm and 1.5 µm are in agreement with the analytically obtained solutions +(Fig. 3b). It should be noted that the interface positions required in this +analytical solution are obtained assuming an interface width of 1.2 µm. Con- +sequently, the simulated solution using an interface width of 0.9 µm shows +deviation from this analytical solution near the interfaces (Fig. 3b). This +is expected because the analytical solution depends on the accuracy of the +numerically determined interface positions (see Appendix F), which slightly +depends on interface width (see the thickness variation in Fig. 2c). To ver- +ify this, we re-compare the simulated displacement field having an interface +width of 0.9 µm against an analytical solution that uses the interface posi- +tions determined from the same simulation. We then find good quantitative +agreement between the two results (Fig.3b). It should be noted that we will +obtain similar quantitative agreement between the analytical and simulated +elastic fields using the VT scheme. This is because the interface positions as a +function of time are relatively independent of the scheme of homogenization. +Moreover, from Fig. 3b, we see that the maximum displacement is at the +γ/γ′ and γ′/NiAl interfaces. This is because of the assumed eigenstrain in the +γ′ phase. As shown in Appendix F, since the displacement field varies linearly +as a function of distance, the total strain and stress normal to the interface are +33 + +spatially constant within the three phases (Figs. 3c and Fig. 3d). Moreover, +due to the deviation in the simulated displacement field having an interface +width of 0.9 µm from the analytical solution, we find similar disagreement +in the total strain and stress normal to the interface from this analytical +solution. However, by comparing this case against the analytical solution +having an interface width of 0.9 µm (shown as a dotted magenta coloured +line in Figs. 3c and 3d), we find good quantitative agreement. +4.2 +Planar three-phase Al-Cr-Ni simulation +Secondly, we considered a planar ternary Al-Cr-Ni fcc−γ/γ′/B2 alloy having +a similar geometry compared to the previous case (Fig. 4a). Moreover, the +boundary conditions at the top, left, and bottom boundaries are identical +to the previous case. However, the mechanical displacements at the right +boundary are +ux(x = Lx/2, y, t) = uR +x , +(56) +uy(x = Lx/2, y, t) = uR +y , +(57) +where uR +x and uR +y are the imposed mechanical displacements (Table 2). +Unlike the previous case, the three phases, in this case, may coexist in +equilibrium because the system is ternary. However, the initial conditions are +set such that the system is out of equilibrium. Consequently, we find that γ′ +phase shrinks while the γ and B2 phases grow. The simulated Al and Cr mole +fraction fields using the PR scheme at time t = 100 s are shown in Figs. 4b +and 4c. Moreover, we find that the thickness of the ternary γ′ phase decreases +34 + +(a) +15 µm +t = 0 +FCC-γ +γ′ +NiAl +Mole fraction Al +0.125 +0.266 +0.406 +(b) +0 +1 +2 +3 +4 +5 +6 +7 +8 +√ +simulation time [√s] +30 +40 +50 +60 +70 +80 +90 +Thickness [µm] +0.9 µm (PR) +1.2 µm (PR) +1.5 µm (PR) +0.9 µm (VT) +1.2 µm (VT) +1.5 µm (VT) +(c) +0 +10 +20 +30 +40 +50 +60 +Simulation time [s] +0 +2 +4 +6 +8 +10 +12 +14 +CPU time [hrs.] +0.9 µm (PR) +1.2 µm (PR) +1.5 µm (PR) +0.9 µm (VT) +1.2 µm (VT) +1.5 µm (VT) +(d) +Fig. 2. For a Ni-Al fcc-γ/Ni3Al−γ′/NiAl coherently stressed planar diffusion cou- +ple: a) schematic of the simulation domain, eigenstrains and mechanical boundary +conditions; b) simulated Al-mole fraction field at time t = 37 s. For three different +interface widths, the temporal variation in Ni3Al thickness as a function of the +square root of simulation time using the partial rank-one (PR) and Voigt-Taylor +(VT) homogenization schemes (c); and the CPU time as a function of simulation +time for both these schemes (d). +linearly as a function of the square root of simulation time using the PR +scheme (Fig. 4d). Moreover, we find that this variation remains independent +of interface width using the partial rank-one (PR) scheme (Fig. 4d). This +is, however, not true for simulations using the VT scheme. Specifically, we +35 + +−150 +−100 +−50 +0 +50 +100 +150 +Distance [µm] +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Mole fraction Al +FCC-γ +Ni3Al-γ′ +NiAl +lw = 0.9 µm +lw = 1.2 µm +lw = 1.5 µm +(a) +−150 +−100 +−50 +0 +50 +100 +150 +Distance [µm] +−0.10 +−0.05 +0.00 +0.05 +0.10 +x-component of displacement [µm] +lw = 0.9 µm +lw = 1.2 µm +lw = 1.5 µm +Exact for 0.9 µm +Exact for 1.2 µm +(b) +−150 +−100 +−50 +0 +50 +100 +150 +Distance [µm] +-3e-03 +-2e-03 +-1e-03 +0e+00 +1e-03 +Total strain in x-direction [ϵxx] +lw = 0.9 µm +lw = 1.2 µm +lw = 1.5 µm +Exact for 0.9 µm +Exact for 1.2 µm +(c) +−150 +−100 +−50 +0 +50 +100 +150 +Distance [µm] +0.1 +0.2 +0.3 +0.4 +0.5 +Stress components σxx and σxy [GPa] +σxy +σxx +σxx +lw = 0.9 µm +lw = 1.2 µm +lw = 1.5 µm +Exact for 0.9 µm +Exact for 1.2 µm +(d) +Fig. 3. +For the Ni-Al fcc-γ/Ni3Al−γ′/NiAl planar diffusion couple case using +the partial rank-one scheme and three different interface widths: a) Al-mole frac- +tion profiles; b) x-component of displacement field; c) total normal strain; and d) +normal and shear stresses as functions of distance perpendicular to the interface. +The superimposed black and magenta dotted lines are the analytically calculated +elastic fields using interface width values of 1.2 µm and 0.9 µm, respectively. +find that as the interface width is increased from 0.4 µm to 0.6 µm, the VT +scheme shows deviation from the expected parabolic growth kinetics (Fig. +4d). Because this behaviour is a consequence of the increase in the interface +width, this deviation from the parabolic kinetics may be attributed to the +excess interfacial energy contribution arising in the case of the VT scheme. +Surprisingly, we find that the CPU time is higher using both PR and VT +36 + +schemes for the simulations with interface widths of 0.6 µm compared to cases +having interface widths of 0.4 µm and 0.5 µm (Fig. 4e). Nevertheless, we +find that the convergence of the PR scheme is significantly faster compared +to the VT scheme for interface width values of 0.4 µm and 0.5 µm (Fig. 4e). +To verify the spatial accuracy, we calculate the composition and elastic +fields along a line parallel to the interface normal at time t = 100 s. We +find that the spatial distribution of the simulated Al and Cr mole fraction +fields normal to the interface is independent of the interface width (Fig. 5a). +Moreover, the simulated x-component of the displacement field normal to the +interface shows good quantitative agreement with the analytical solution, in- +dependent of the choice of interface width (Fig.5b). +Due to the applied +mechanical displacement at the right boundary, the y-component of the dis- +placement field is also non-zero in this case. Fig. 5c shows that the simulated +and analytically obtained solutions for the y-component displacement field +are also in quantitative agreement in the bulk domains. Likewise, this agree- +ment is not a function of the interface width. Expectedly, we find that the +total strain normal to the interface is constant within the bulk phases and +is in agreement with the analytical solution (Fig. 5d). Since the system is +elastically anisotropic, the shear strains are non-zero and constant within the +bulk phases (Fig. 5e). Finally, the non-zero elastic stresses as a function of +distance normal to the interface are shown in Fig. 5f. +37 + +4.3 +Non-planar three-phase Ni-Al alloy +Thirdly, we simulated a fcc−γ/γ′−Ni3Al/NiAl alloy having concentric ring +interfaces (Fig. 6a). We have assumed homogeneous Neumann boundary +conditions along the left, bottom and outer boundaries for the composition +and phase-field variables. +Further, we have imposed symmetry boundary +conditions on displacements along the bottom and left boundaries, and zero +traction boundary conditions at the outer surface, viz. +ux(x = 0, y, t) = 0, +(58) +uy(x, y = 0, t) = 0, +(59) +σnΓ(x, y, t) = 0 +on +x2 + y2 = R2 +0, +(60) +where R0 is the radius of the domain. +Similar to our first case, the three phases cannot coexist in equilibrium. +Thus, we find that the intermediate γ′ phase grows while the innermost γ +and outermost NiAl phases shrink. We run this simulation until the γ/γ′ +interface vanishes. Figs. 6a and 6b show the simulated contour map of the +Al mole fraction and the radial displacement fields at time t = 1 s for an +interface thickness of 0.15 µm using the PR scheme. The variation in the γ′ +thickness as a function of the square root of simulation time using the PR +and VT schemes are shown in Fig.6c. As expected, we find parabolic growth +kinetics using both these schemes. To check the influence of this result on +interface width, we vary the interface width from 0.10 µm to 0.30 µm. +We find that the interface kinetics remains unaffected for interface width +38 + +values of 0.10 µm and 0.30 µm using both PR and VT schemes (Fig. 6c). +However, for the case with interface width value of 0.60 µm, we find that +the calculated thickness of Ni3Al is slightly lower in both schemes (Fig. 6c). +Nevertheless, we find that the kinetics is still parabolic. +Moreover, for a +given simulation time, CPU time in the case of the PR scheme is always +lower compared to the VT scheme for interface width values of 0.10 µm and +0.30 µm (Fig. 6d). The difference in CPU time is however negligible for an +interface width of 0.60 µm. This suggests that the PR scheme converges at +a faster or nearly equal rate compared to the VT scheme for this system. +To verify the spatial accuracy of the simulated solution, we calculated the +composition and elastic fields along the radius at time t = 100 s (Fig.6b). We +find that the radial distribution of the Al mole fraction field within the bulk +domains remains independent of interface width for values between 0.10 µm +and 0.30 µm (Fig. 7a). Likewise, the radial displacement within the bulk +phases remains unaltered with varying interface width (Fig. 7b). Also, note +that the tangential displacement is negligible within the bulk phases (Fig. +7b). We also find excellent quantitative agreement between the simulated and +analytically obtained radial displacement in the bulk γ and Ni3Al phases. It +should be emphasized that the analytical solution uses the interface positions +calculated from the simulation with an interface width of 0.15 µm (Fig. 7b). +However, for the simulation with an interface width of 0.30 µm, we see +that the radial displacement near the Ni3Al/NiAl interface deviates marginally +from this analytical solution. It should be noted that a similar observation +was made for the planar Ni-Al case. Furthermore, we explained this devia- +tion by accounting for the inaccuracy caused by numerically determining the +39 + +interface positions. As discussed before, the analytical solution is sensitive +to the calculated interface positions, which are, in turn, dependent on the +interface width. Further, we have verified this assertion by matching the +simulated field for this case with an analytical solution where the interface +positions are calculated using the same interface thickness. +We also find quantitative agreement between the simulated and the ana- +lytically obtained radial and hoop strains (Figs. 7c and 7d). As expected, the +radial and hoop strains are equal and constant in the bulk γ phase. However, +the radial and hoop strains are dependent on the radius in the γ′-Ni3Al and +NiAl phases. Likewise, for the radial and hoop stress fields, we also obtained +a good match between the analytical and simulated fields (Figs. 7e and 7f). +4.4 +Non-planar three-phase Al-Cr-Ni alloy +Lastly, we simulated a ternary Al-Cr-Ni fcc−γ/γ′/B2 alloy having concentric +interfaces (Fig. 8a). As listed in Table 2, compared to the previous case, the +mechanical displacements at the outer boundary are different in this case. +Specifically, +ux(x, y, t) = ϵg +Rx +on +x2 + y2 = R2 +0, +(61) +uy(x, y, t) = ϵg +Ry +on +x2 + y2 = R2 +0, +(62) +where ϵg +R = 0.1% is the imposed hoop strain. For sake of completeness, we +note that the boundary conditions at the remaining boundaries are identical +to the previous case. Moreover, we have assumed that the eigenstrains in +the bulk phases are zero. Because of this, VT and Khachaturayan schemes +40 + +become identical for this special case [30]. Since there are no eigenstrains, +the mechanical stresses are simply due to the imposed boundary conditions. +As noted previously for the planar case, the three phases may coexist +since the overall alloy composition lies in the three-phase region. However, +the initial conditions are set such that the γ′ grows at the expense of the +ternary γ and B2 phases. Fig. 8a shows the spatial variation in the Al-mole +fraction field at time t = 100 s. As shown in Fig. 8b, the radial displacement +field is symmetric due to the imposed boundary conditions and isotropic +elastic properties. +Fig. 8c shows the variation in the thickness of the γ′ phase as a function +of the square root of simulation time. +Unlike the previous case, we find +that the γ′ phase first grows parabolically as a function of time. Eventually, +its growth slows down as the system reaches towards the equilibrium state. +This parabolic growth behaviour of the γ′ phase is due to the fact that the +process is diffusion-controlled. Moreover, we find that the accuracy of the +temporal variation in the γ′ phase thickness is independent of the interface +width choice and the homogenization scheme (Fig. 8c). However, in contrast +to the previous three simulations, we find that the convergence of the VT +scheme is marginally faster in this case compared to the PR scheme for two +interface width values of 0.15 µm and 0.30 µm (Fig. 8d). We think this is +possibly due to the absence of eigenstrains in this simulation compared to +all other previous cases. Nevertheless, we find that the PR scheme converges +faster compared to the VT scheme only for the case with an interface width +of 0.10 µm. +To test the accuracy of our simulations, we calculate the spatial variation +41 + +of elastic and composition fields along the radial direction based on the PR +scheme (Fig. 8b). In the bulk phases, we find that the spatial variation in the +Al and Cr mole fraction fields along the radius is independent of the choice +of interface width (Fig. 9a). Moreover, our simulated radial displacement +field is consistent with the analytically obtained solution (Fig. 9b). We also +find that the accuracy remains unaltered for three different interface widths +(Fig. 9b). As shown in Fig. 9b, the tangential displacement is negligible for +this case. Figs. 9c and 9d show the variation in the total radial and hoop +strains as functions of radial distance. Similar to the previous case, notice +that the radial and hoop strains in the γ phase are constant and equal. +However, the radial and hoop strains in the γ′ and B2 phases depend on the +radius. Moreover, our simulated radial and hoop stresses in the bulk phases +are also consistent with the analytical solution (Figs. 9e and 9f). Finally, we +emphasize that the simulated stress and strain fields are independent of the +choice of interface width. +5 +Conclusions +This paper first generalizes the partial rank-one homogenization scheme to +multi-phase systems. Subsequently, it implements this scheme for a three- +phase system by analytically solving the static compatibility equations, thereby +ensuring both static and kinematic compatibilities in the interfacial regions. +Following this, a multi-phase-field grand-potential-based model is formulated +using the rank-one scheme for solids undergoing small-strain deformations. +To demonstrate its application for real alloys, a coupling technique is utilized +42 + +to extract the prerequisite properties directly from CALPHAD databases. +Specifically, we test the model for two three-phase Ni-based alloys having +either planar or concentric ring interfaces. We verify the accuracy of the +simulated elastic fields against analytical solutions for all simulated cases. +Our results show that the simulation accuracy using the rank-one scheme +remains independent of the choice of interface width. Except for one case, we +find that the rank-one scheme shows improved or nearly equal convergence +compared to the Voigt-Taylor homogenization scheme, which ensures only +kinematic compatibility. +Nevertheless, the current implementation is still +limited to linear elastic deformation, and in the future numerical approaches +to solving the static compatibility equations, as demonstrated in [35], will be +explored. +6 +CRediT authorship contribution statement +Sourav Chatterjee: Conceptualization, Methodology, Software, Valida- +tion, Writing - Original Draft, Writing - Review & Editing, Visualization. +Daniel Schwen: Software, Writing - Review & Editing. Nele Moelans: +Conceptualization, Methodology, Resources, Writing - Review & Editing, +Supervision, Funding acquisition. +7 +Acknowledgement +This work was supported by the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovation program (INTER- +43 + +DIFFUSION, grant agreement no. 714754). The computational resources +and services used in this work were provided by the VSC (Flemish Super- +computer Center), funded by the Research Foundation - Flanders (FWO) +and the Flemish Government - department EWI. +44 + +(a) +10 µm +t = 0 +FCC-γ +γ′ +B2 +Mole fraction Al +0.156 +0.229 +0.301 +(b) +10 µm +t = 0 +FCC-γ +γ′ +B2 +Mole fraction Cr +0.052 +0.106 +0.160 +(c) +0 +10 +20 +30 +40 +√ +simulation time [√s] +0 +5 +10 +15 +20 +25 +30 +35 +Thickness [µm] +0.4 µm (PR) +0.5 µm (PR) +0.6 µm (PR) +0.4 µm (VT) +0.5 µm (VT) +0.6 µm (VT) +(d) +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Simulation time [s] +0 +10 +20 +30 +40 +50 +CPU time [hrs.] +0.4 µm (PR) +0.5 µm (PR) +0.6 µm (PR) +0.4 µm (VT) +0.5 µm (VT) +0.6 µm (VT) +(e) +Fig. 4. For an Al-Cr-Ni fcc-γ/γ′/B2 coherently stressed planar diffusion couple: +schematic of the simulation domain, eigenstrains and mechanical boundary condi- +tions (a); the simulated Al-mole fraction field (b) and Cr-mole fraction field (c) at +time t = 100 s. For three different interface widths, the temporal variation in the +ternary γ′ thickness as a function of the square root of simulation time using the +partial rank-one (PR) and Voigt-Taylor (VT) homogenization schemes (d); and +change in CPU time with simulation time for both these schemes (e). +45 + +−100 +−75 +−50 +−25 +0 +25 +50 +75 +100 +Distance [µm] +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Al and Cr mole fractions [xAl and xCr] +xCr +xAl +0.4 µm +0.5 µm +0.6 µm +(a) +−100 +−50 +0 +50 +100 +Distance [µm] +−0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +x-component of displacement [µm] +0.4 µm +0.5 µm +0.6 µm +Analytical +(b) +−100 +−50 +0 +50 +100 +Distance [µm] +−0.175 +−0.150 +−0.125 +−0.100 +−0.075 +−0.050 +−0.025 +0.000 +y-component of displacement [µm] +0.4 µm +0.5 µm +0.6 µm +Analytical (uα +y) +Analytical (uβ +y) +Analytical (uγ +y) +−75 +−50 +−25 +−0.04 +−0.02 +(c) +−100 +−50 +0 +50 +100 +Distance [µm] +-3e-03 +-2e-03 +-1e-03 +0e+00 +1e-03 +2e-03 +Total strain in x-direction +0.4 µm +0.5 µm +0.6 µm +Analytical +(d) +−100 +−50 +0 +50 +100 +Distance [µm] +−0.000430 +−0.000425 +−0.000420 +−0.000415 +−0.000410 +−0.000405 +−0.000400 +−0.000395 +Shear strain [ϵxy] +0.4 µm +0.5 µm +0.6 µm +Analytical +(e) +−100 +−75 +−50 +−25 +0 +25 +50 +75 +100 +Distance [µm] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Stress components [σxx, σxy and σyy] +σxx +σxy +σyy +0.4 µm +0.5 µm +0.6 µm +Analytical +(f) +Fig. 5. For an Al-Cr-Ni fcc-γ/γ′/B2 coherently stressed planar diffusion couple, +the spatial distribution of Al and Cr-mole fraction profiles (a); x and y-components +of displacement field (b) and (c); total normal and shear strain (d) and (e); normal +and shear stresses (f), as functions of distance perpendicular to the interface at time +t = 100 s. Simulations using different interface widths are also superimposed on +these figures. The superimposed dotted black lines indicate the analytical solution. +For sake of clarity, the analytical solution for the y-component of displacement field +within the bulk regions are denoted by different colours in Fig. 5b. +46 + +3.0 µm +FCC-γ +γ′ +NiAl-B2 +Mole fraction Al +ux = 0 +uy = 0 +t=0 +0.124 +0.153 +0.181 +0.210 +0.238 +0.267 +0.295 +0.324 +0.353 +0.381 +(a) +3.0 µm +[µm] +Radial displacement at t = 1 s +-0.0360 +-0.0320 +-0.0280 +-0.0240 +-0.0200 +-0.0160 +-0.0120 +-0.0080 +-0.0040 +0.0000 +(b) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +√ +simulation time [√s] +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Thickness [µm] +0.10 µm (PR) +0.30 µm (PR) +0.60 µm (PR) +0.10 µm (VT) +0.30 µm (VT) +0.60 µm (VT) +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Simulation time [s] +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +CPU time [hrs.] +0.10 µm (PR) +0.30 µm (PR) +0.60 µm (PR) +0.10 µm (VT) +0.30 µm (VT) +0.60 µm (VT) +(d) +Fig. 6. +For a non-planar Ni-Al fcc-γ/Ni3Al/NiAl coherently stressed diffusion +couple: simulation domain, eigenstrains, mechanical boundary conditions and the +simulated Al-mole fraction field at time t = 1 s (a); the simulated radial displace- +ment field at the same time (b). For three different interface widths, variation in +Ni3Al thickness as a function of the square root of simulation time using the partial +rank-one (PR) and Voigt-Taylor (VT) homogenization schemes (c); the CPU time +as a function of simulation time for both these schemes (d). +47 + +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Mole fraction Al +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +(a) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +−0.040 +−0.035 +−0.030 +−0.025 +−0.020 +−0.015 +−0.010 +−0.005 +0.000 +Radial and tangential displacements [µm] +ut +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(b) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +−0.004 +−0.003 +−0.002 +−0.001 +0.000 +0.001 +0.002 +Radial strain +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(c) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +−0.00200 +−0.00175 +−0.00150 +−0.00125 +−0.00100 +−0.00075 +−0.00050 +−0.00025 +Hoop strain +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(d) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +−0.05 +0.00 +0.05 +0.10 +0.15 +Radial stress [GPa] +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(e) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Hoop stress [GPa] +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(f) +Fig. 7. +For a non-planar Ni-Al fcc-γ/Ni3Al/NiAl diffusion couple, the spatial +variation in a) Al-mole fraction; b) radial and tangential displacements; c) radial +strain; d) hoop strain; e) radial stress; and f) hoop stress as functions of radial +distance at time t = 1 s. The plots show this variation for three different interface +widths lw using the partial rank-one scheme. The dotted and discontinuous black +lines are the analytically obtained solutions. +48 + +3.0 µm +FCC-γ +γ′ +B2 +Mole fraction Al +ux = 0 +uy = 0 +t=0 +0.163 +0.176 +0.190 +0.203 +0.216 +0.229 +0.242 +0.256 +0.269 +0.282 +(a) +3.0 µm +[µm] +Radial displacement at t = 100 s +0.0000 +0.0032 +0.0063 +0.0095 +0.0126 +0.0158 +0.0189 +0.0221 +0.0252 +0.0284 +(b) +0 +5 +10 +15 +20 +25 +30 +35 +40 +√ +simulation time [√s] +2.0 +2.5 +3.0 +3.5 +Thickness [µm] +0.10 µm (PR) +0.15 µm (PR) +0.30 µm (PR) +0.10 µm (VT) +0.15 µm (VT) +0.30 µm (VT) +(c) +0 +200 +400 +600 +800 +1000 +Simulation time [s] +0 +5 +10 +15 +20 +CPU time [hrs.] +0.10 µm (PR) +0.15 µm (PR) +0.30 µm (PR) +0.10 µm (VT) +0.15 µm (VT) +0.30 µm (VT) +(d) +Fig. 8. For a non-planar Al-Cr-Ni fcc-γ/γ′/B2 coherently stressed diffusion cou- +ple: simulation domain, mechanical boundary conditions and the simulated Al- +mole fraction field at time t = 100 s (a); the simulated radial displacement field at +the same time (b). For three different interface widths, variation in γ′ thickness as +a function of the square root of simulation time using the partial rank-one (PR) +and Voigt-Taylor (VT) homogenization schemes (c); the CPU time as a function +of simulation time for both these schemes (d). +49 + +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.10 +0.15 +0.20 +0.25 +0.30 +Al and Cr mole fractions [xAl and xCr] +xCr +xAl +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +(a) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +Radial and tangential displacements [µm] +ut +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(b) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.0006 +0.0008 +0.0010 +0.0012 +0.0014 +Radial strain +lw = 0.15 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(c) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.00100 +0.00105 +0.00110 +0.00115 +0.00120 +0.00125 +Hoop strain +lw = 0.15 µm +lw = 0.20 µm +lw = 0.30 µm +Exact for 0.15 µm +(d) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.375 +0.380 +0.385 +0.390 +0.395 +0.400 +0.405 +0.410 +Radial stress [GPa] +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(e) +0 +5 +10 +15 +20 +25 +30 +Radial distance [µm] +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +0.48 +Hoop stress [GPa] +lw = 0.10 µm +lw = 0.15 µm +lw = 0.30 µm +Exact for 0.15 µm +(f) +Fig. 9. For a non-planar Al-Cr-Ni fcc-γ/γ′/B2 diffusion couple, the spatial vari- +ation in a) Al and Cr mole fraction fields; b) radial and tangential displacements; +c) radial strain; d) hoop strain; e) radial stress; and f) hoop stress as functions +of radial distance at time t = 100 s. +The plots show this variation for three +different interface widths lw using the partial rank-one scheme. The dotted and +discontinuous black lines are the analytically obtained solutions. +50 + +Appendix A +Calculation of phase strains +In this section, we provide an analytical approach to calculate the phase +strains for a p-phase system as functions of the total strain ϵ(u), interpola- +tions functions h(φ) and strain jumps. To this end, we first begin by deriving +the phase strains for two-phase and three-phase systems and then extend it +to multi-phase systems. +A.1 +Phase strains for a two-phase system +For a two-phase α/β system, Eqs. (1) and (4) reduces to +ϵ(u) = ϵαhα + ϵβhβ +(A.1) +ϵα − ϵβ = �ϵ�αβ +(A.2) +Multiplying Eq. (A.2) with hβ and then adding Eq. (A.1) yields +ϵα (hα + hβ) = ϵ + hβ�ϵ�αβ +(A.3) +Since for a two-phase system hα + hβ = 1, Eq. (A.3) simplifies to +ϵα = ϵ + hβ�ϵ�αβ +(A.4) +51 + +Since hα = 1 − hβ, it follows from Eqs. (A.2) and (A.4) that +ϵβ = ϵ − hα�ϵ�αβ +(A.5) +We simply note that Eqs. (A.4) and (A.5) are completely equivalent to Eqs. +(A.1) and (A.2). Next, we attempt to use the two-phase relations to extend +the model to three-phase systems. +A.2 +Phase strains for a three-phase system +For a system consisting of three phases, say α, β & γ, Eqs. (1) and (4) +reduces to +ϵ = ϵαhα + ϵβhβ + ϵγhγ, +(A.6) +ϵα − ϵβ = �ϵ�αβ, +(A.7) +ϵβ − ϵγ = �ϵ�βγ. +(A.8) +By defining ϵ′ = ϵ − ϵγhγ, Eq. (A.6) may be written as +ϵ′ = ϵαhα + ϵβhβ. +(A.9) +Notice that Eqs. (A.9) and (A.7) are similar to Eqs. (A.1) and (A.2). Be- +cause of this similarity, we can directly use Eq. (A.3) to write +ϵα (hα + hβ) = ϵ′ + hβ�ϵ�αβ = ϵ − ϵγhγ + hβ�ϵ�αβ. +(A.10) +52 + +It should be noticed from the right-hand side of Eq. (A.10) that (hα + hβ) ̸= +1 since this is a three-phase system. Consequently, adding Eqs. (A.7) & +(A.8) and substituting: ϵγ = ϵα − �ϵ�αβ − �ϵ�βγ, in Eq. (A.10) gives +ϵα(hα + hβ + hγ) = ϵ + hβ�ϵ�αβ + hγ +� +�ϵ�αβ + �ϵ�βγ� +. +(A.11) +Now, since (hα + hβ + hγ) = 1 for a three-phase system, Eq. (A.11) reduces +to +ϵα = ϵ + hβ�ϵ�αβ + hγ +� +�ϵ�αβ + �ϵ�βγ� +. +(A.12) +Next, substituting Eq. (A.12) in Eq. (A.7) and then using (1−hβ −hγ) = hα +gives +ϵβ = ϵ − hα�ϵ�αβ + hγ�ϵ�βγ. +(A.13) +Finally, substituting Eq. (A.13) in Eq. (A.8) yields +ϵγ = ϵ − hα�ϵ�αβ − (hα + hβ) �ϵ�βγ. +(A.14) +Thus, we have obtained the phase strains as functions of the total strain, +interpolation functions and strain jumps for a three-phase system. We again +note that Eqs. (A.12), (A.13) & (A.14) are completely equivalent to Eqs. +(A.6), (A.7) & (A.8). +53 + +A.3 +Phase strains for a multi-phase system +Based on the previous two derivations, it is worth noting that once the phase +strain pertaining to a particular phase, say α, is determined, the phase strains +of the remaining (p−1) phases may be obtained using the (p−1) compatibility +equations, i.e., Eqs. (4). For sake of concretness, if phase strain pertaining +to α-phase is known, then the phase strains in β, γ, . . . , (p − 1), p phases are +ϵβ = ϵα − �ϵ�αβ, +ϵγ = ϵβ − �ϵ�βγ, +ϵδ = ϵγ − �ϵ�γδ, +... +ϵp = ϵp−1 − �ϵ�(p−1),p. +(A.15) +Therefore, if an analytical expression for the α-phase strain in a system +consisting of p phases is known, all remaining phase strains can be calculated. +It can be observed from Eqs. (A.4) & (A.12) that the α-phase strain +for a three-phase system differs from a two-phase system by just one term. +Specifically, this term is equal to the product of the interpolation function +associated with the new phase and the sum of all jump vectors in that system, +i.e., hγ +� +�ϵ�αβ + �ϵ�βγ� +. Consequently, ϵα for a multi-phase system may be +written as +ϵα = ϵ(u) + hβ�ϵ�αβ + hγ +� +�ϵ�αβ + �ϵ�βγ� ++ . . . + hp +� +� +(p−1),p +� +i=αβ +�ϵ�i +� +� , (A.16) +54 + +where +(p−1),p +� +i=αβ +�ϵ�i = �ϵ�αβ + �ϵ�βγ + �ϵ�γδ + . . . + �ϵ�(p−1),p. +(A.17) +By substituting Eq. (A.16) in the first of the set of Eqs. (A.15) and using +the relation 1 − (hβ + hγ + . . . + hp) = hα, it follows that +ϵβ = ϵ(u) − hα�ϵ�αβ + hγ�ϵ�βγ + . . . + hp +� +�ϵ�βγ + �ϵ�γδ + . . . + �ϵ�(p−1),p� +. +(A.18) +Similarly, by substituting Eq. (A.18) in the second of the set of Eqs. (A.15) +and using 1 − (hγ + hδ + . . . + hp) = (hα + hβ), it follows that +ϵγ = ϵ(u) − hα�ϵ�αβ − (hα + hβ) �ϵ�βγ + . . . + hp +� +�ϵ�γδ + . . . + �ϵ�(p−1),p� +. +(A.19) +Thus, by using Eqs. (A.15) and (A.16) we can calculate the phase strains +for an arbitrary multi-phase system. +55 + +Appendix B +Some useful relations +B.1 +Derivatives with respect to phase-field variables +Since hα + hβ + hγ = 1, it follows that +∂hα +∂φα += − +�∂hβ +∂φα ++ ∂hγ +∂φα +� +(B.1) +∂hγ +∂φα += − +�∂hβ +∂φα ++ ∂hα +∂φα +� +(B.2) +Differentiating Eqs. (10), (11) and (12) with respect to φα yields +∂ϵα +ij +∂φα += +�∂hβ +∂φα ++ ∂hγ +∂∂φα +� +�ϵij�αβ + (hβ + hγ)∂�ϵij�αβ +∂φα ++ ∂hγ +∂φα +�ϵij�βγ + hγ +∂�ϵij�βγ +∂φα +(B.3) +∂ϵβ +ij +∂φα += −∂hα +∂φα +�ϵij�αβ − hα +∂�ϵij�αβ +∂φα ++ ∂hγ +∂φα +�ϵij�β +γ + hγ +∂�ϵij�βγ +∂φα +(B.4) +∂ϵγ +ij +∂φα += −∂hα +∂φα +�ϵij�αβ − hα +∂�ϵij�αβ +∂φα +− +�∂hβ +∂φα ++ ∂hα +∂φα +� +�ϵij�βγ − (hβ + hα)∂�ϵij�βγ +∂φα +(B.5) +Multiplying Eqs. (B.3), (B.4) and (B.5) with hασα +ij, hβσβ +ij and hγσγ +ij, respec- +tively, and then setting the terms premultiplied by hγhα to zero, and using +56 + +Eqs. (B.1) and (B.2) yields +hασα +ij +∂ϵα +ij +∂φα += −hα +∂hα +∂φα +σα +ij�ϵij�α +β + hαhβσα +ij +∂�ϵij�α +β +∂φα ++ hα +∂hγ +∂φα +σα +ij�ϵij�β +γ +(B.6) +hβσβ +ij +∂ϵβ +ij +∂∂φα += −hβ +∂hα +∂φα +σβ +ij�ϵij�α +β − hαhβσβ +ij +∂�ϵij�α +β +∂φα ++ hβ +∂hγ +∂φα +σβ +ij�ϵij�β +γ + hβhγσβ +ij +∂�ϵij�β +γ +∂φα +(B.7) +hγσγ +ij +∂ϵγ +ij +∂φα += −hγ +∂hα +∂φα +σγ +ij�ϵij�α +β + hγ +∂hγ +∂φα +σγ +ij�ϵij�β +γ − hγhβσγ +ij +∂�ϵij�β +γ +∂φα +(B.8) +Adding Eqs. (B.6), (B.7) and (B.8) and using Eqs. (13) and (14) yields +γ +� +θ=α +hθ(φ)σθ +ij +∂ϵθ +ij +∂φα += hαhβ +� +σα +ij − σβ +ij +� +nαβ +j +∂aαβ +i +∂φα ++ hβhγ +� +σβ +ij − σγ +ij +� +nβγ +j +∂aβγ +i +∂φα +− ∂hα +∂φα +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φα +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(B.9) +It follows from Eqs. (15) and (16), that the first two terms on the right hand +side of Eq. (B.9) are zero. Thus, Eq. (B.9) reduces to +γ +� +θ=α +hθ(φ)σθ +ij +∂ϵθ +ij +∂φα += −∂hα +∂φα +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φα +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(B.10) +57 + +Following a similar procedure, it can be shown that +γ +� +θ=α +hθ(φ)σθ +ij +∂ϵθ +ij +∂φβ += −∂hα +∂φβ +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φβ +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(B.11) +γ +� +θ=α +hθ(φ)σθ +ij +∂ϵθ +ij +∂φγ += −∂hα +∂φγ +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φγ +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(B.12) +Now, we obtain another similar relation by multiplying Eqs. (B.3), (B.4) +and (B.5) with hαCα +klij, hβCβ +klij and hγCγ +klij, respectively, and setting the terms +premultiplied by hγhα to zero yields +hαCα +klij +∂ϵα +ij +∂φα += hα +�∂hβ +∂φα ++ ∂hγ +∂φα +� +Cα +klij�ϵij�α +β + hαhβCα +klij +∂�ϵij�α +β +∂φα ++ hα +∂hγ +∂φα +Cα +klij�ϵij�β +γ +(B.13) +hβCβ +klij +∂ϵβ +ij +∂φα += −hβ +∂hα +∂φα +Cβ +klij�ϵij�α +β − hβhαCβ +klij +∂�ϵij�α +β +∂φα ++ hβ +∂hγ +∂φα +Cβ +klij�ϵij�β +γ + hβhγCβ +klij +∂�ϵij�β +γ +∂φα +(B.14) +hγCγ +klij +∂ϵγ +ij +∂φα += −hγ +∂hα +∂φα +Cγ +klij�ϵij�α +β − hγ +�∂hβ +∂φα ++ ∂hα +∂φα +� +Cγ +klij�ϵij�β +γ − hβhγCγ +klij +∂�ϵij�β +γ +∂φα +(B.15) +58 + +Substituting Eqs. (B.1) and (B.2) in Eqs. (B.13) and (B.15) gives +hαCα +klij +∂ϵα +ij +∂φα += −hα +∂hα +∂φα +Cα +klij�ϵij�α +β + hαhβCα +klij +∂�ϵij�α +β +∂φα ++ hα +∂hγ +∂φα +Cα +klij�ϵij�β +γ +(B.16) +hβCβ +klij +∂ϵβ +ij +∂φα += −hβ +∂hα +∂φα +Cβ +klij�ϵij�α +β − hβhαCβ +klij +∂�ϵij�α +β +∂φα ++ hβ +∂hγ +∂φα +Cβ +klij�ϵij�β +γ + hβhγCβ +klij +∂�ϵij�β +γ +∂φα +(B.17) +hγCγ +klij +∂ϵγ +ij +∂φα += −hγ +∂hα +∂φα +Cγ +klij�ϵij�α +β + hγ +∂hγ +∂φα +Cγ +klij�ϵij�β +γ − hβhγCγ +klij +∂�ϵij�β +γ +∂φα +(B.18) +Adding Eqs. (B.16), (B.17), and (B.18) yields +γ +� +θ=α +hθCθ +klij +∂ϵθ +ij +∂φα += hαhβ +� +Cα +klij − Cβ +klij +� ∂�ϵij�α +β +∂φα ++ hβhγ +� +Cβ +klij − Cγ +klij +� ∂�ϵij�β +γ +∂φα +−∂hα +∂φα +� γ +� +θ=α +hθCθ +klij +� +�ϵij�α +β + ∂hγ +∂φα +� γ +� +θ=α +hθCθ +klij +� +�ϵij�β +γ +(B.19) +By replacing ∂φα with ∂φβ and ∂φγ with ∂φγ equivalent expressions for φβ +and φγ can be easily obtained. +59 + +B.2 +Derivatives with respect to total strain +Differentiating Eqs. (10), (11) and (12) with respect to total strain ϵ and +then multiplying with hασα +ij, hβσβ +ij and hγσγ +ij, respectively, yields +hασα +ij +∂ϵα +ij +∂ϵmn += hασα +mn + hα (hβ + hγ) σα +ij +∂�ϵij�α +β +∂ϵmn ++ hγhασα +ij +∂�ϵij�β +γ +∂ϵmn +(B.20) +hβσβ +ij +∂ϵβ +ij +∂ϵmn += hβσβ +mn − hαhβσβ +ij +∂�ϵij�α +β +∂ϵmn ++ hγhβσβ +ij +∂�ϵij�β +γ +∂ϵmn +(B.21) +hγσγ +ij +∂ϵγ +ij +∂ϵmn += hγσγ +mn − hαhγσγ +ij +∂�ϵij�α +β +∂ϵmn +− hγ (hβ + hα) σγ +ij +∂�ϵij�β +γ +∂ϵmn +(B.22) +Now, we note that hγhα is non-zero only near the γ/α interface boundary and +the terms �ϵ�α +β and �ϵ�β +γ are also non-zero only within the interfacial regions +of β/γ and α/β boundaries. We therefore set all terms premultiplied by hγhα +to zero in Eqs. (B.20), (B.21) and (B.22). Now adding these equations yields +γ +� +θ=α +hθσθ +ij +∂ϵθ +ij +∂ϵmn += +γ +� +θ=α +hθσθ +mn + hαhβ +� +σα +ij − σβ +ij +� +nαβ +j +∂aαβ +i +∂ϵmn ++ hγhβ +� +σβ +ij − σγ +ij +� +nβγ +j +∂aβγ +i +∂ϵmn +(B.23) +Due to Eqs. (15) and (16), the last two terms in Eq. (B.23) must be zero. +This gives +γ +� +θ=α +hθσθ +ij +∂ϵθ +ij +∂ϵmn += +γ +� +θ=α +hθσθ +mn +(B.24) +Next, differentiating Eqs. (10), (11) and (12) with respect to total strain ϵ +and multiplying with hαCα +ijkl, hβCβ +mnkl and hγCγ +mnkl, yields +60 + +hαCα +mnkl +∂ϵα +kl +∂ϵrs += hαCα +mnrs + hα (hβ + hγ) Cα +mnkl +∂�ϵkl�α +β +∂ϵrs ++ hγhαCα +mnkl +∂�ϵkl�β +γ +∂ϵrs +(B.25) +hβCβ +mnkl +∂ϵβ +kl +∂ϵrs += hβCβ +mnrs − hαhβCβ +mnkl +∂�ϵkl�α +β +∂ϵrs ++ hγhβCβ +mnkl +∂�ϵkl�β +γ +∂ϵrs +(B.26) +hγCγ +mnkl +∂ϵγ +kl +∂ϵrs += hγCγ +mnrs − hαhγCγ +mnkl +∂�ϵkl�α +β +∂ϵrs +− hγ (hβ + hα) Cγ +mnkl +∂�ϵkl�β +γ +∂ϵrs +(B.27) +Again, we set the four terms in Eqs. (B.25), (B.26) and (B.27) which are +premultiplied by hγhα to zero. Next adding these equations, we see that +Jmnrs = +γ +� +θ=α +hθ(φ)Cθ +mnkl +∂ϵθ +kl +∂ϵrs += +γ +� +θ=α +hθ(φ)Cθ +mnkl + hαhβ +� +Cα +mnkl − Cβ +mnkl +� ∂�ϵkl�α +β +∂ϵrs ++ hγhβ +� +Cβ +mnkl − Cγ +mnkl +� ∂�ϵkl�β +γ +∂ϵrs +(B.28) +Appendix C +Derivation of stress and its derivatives +Differentiating Eq. (34) with respect to total strain yields +∂ωbulk +∂ϵmn += +γ +� +θ=α +hθ(φ)∂ωθ +bulk +∂ϵθ +ij +∂ϵθ +ij +∂ϵmn +(C.1) +61 + +Using the definition of the phase stress tensor, we can replace ∂ωθ +bulk/∂ϵij +with σθ +ij. Using the relation (B.24) , Eq.(D.1) can be written as +∂ωbulk +∂ϵij += +γ +� +θ=α +hθσθ +ij +∂ϵθ +ij +∂ϵmn += +γ +� +θ=α +hθ(φ)σθ +mn +(C.2) +Appendix D +Derivation of driving force and its derivatives +Differentiating Eq. (34) with respect to phase-field variable φθ yields +∂ωbulk +∂φθ += +γ +� +σ=α +∂hσ +∂φθ +ωσ + +γ +� +σ=α +hσ(φ)∂ωσ +∂ϵσ +ij +∂ϵσ +ij +∂φθ += +γ +� +σ=α +∂hσ +∂φθ +ωσ + +γ +� +σ=α +hσ(φ)σσ +ij +∂ϵσ +ij +∂φθ +(D.1) +For θ = α, substituting Eqs. (B.1), (B.2) and (B.10) in Eq. (D.1) yields +∂ωbulk +∂φα += ∂hβ +∂φα +� +ωβ − ωα� ++ ∂hγ +∂φα +(ωγ − ωα) +− ∂hα +∂φα +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φα +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(D.2) +62 + +Similarly, one can derive the bulk driving force for phase-field variables φβ +and φγ by using Eqs. (B.11) and (B.12) +∂ωbulk +∂φβ += ∂hα +∂φβ +� +ωα − ωβ� ++ ∂hγ +∂φβ +� +ωγ − ωβ� +− ∂hα +∂φβ +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φβ +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(D.3) +∂ωbulk +∂φγ += ∂hα +∂φγ +(ωα − ωγ) + ∂hβ +∂φγ +� +ωβ − ωγ� +− ∂hα +∂φγ +�� +θ=α +hθσθ +ij +� +�ϵij�α +β + ∂hγ +∂φγ +� γ +� +θ=α +hθσθ +ij +� +�ϵij�β +γ +(D.4) +Appendix E +Non-dimensionalization +Eqs. (43)-(46) were solved in the MOOSE (Multiphysics Object-Oriented +Simulation Environment) finite-element framework [55]. +To ensure good +convergence, we formed a non-dimensional form of these equations. In this +section, we provide the dimensionless form of the governing equations. +We will denote dimensionless quantities using the symbol, (·). Let lc and +tc denote characteristic length and time scales. Then, the non-dimensional +position and time may be written as: x = x/lc and t = t/tc. The dimen- +sionless form of the displacement field is defined as: u = u/lc. Similarly, we +define the dimensionless form of the set of diffusion potentials as: ˜µ = ˜µ/RT, +where R is gas constant and T is simulation temperature. After change of +63 + +variables and using the relation ck = Xk/Vm, it can be shown that the di- +mensionless form of Eqs. (43)-(46) may be written as +p +� +θ=1 +hθ(φ)Xθ +k(˜µ) − Xk = 0, +∀ k = 1 . . . (n − 1), +(E.1) +div +� +p +� +θ=1 +hθ(φ)σθ +ij +� += 0, +(E.2) +∂φθ +∂t + Lφ +�∂g (φ) +∂φθ +− κ∆φθ + λ1 +∂ωchem +∂φθ ++ λ2 +∂ωmech +∂φθ +� += 0 +∀ θ = 1 . . . p, +(E.3) +∂Xk +∂t − ∇ +�n−1 +� +j=1 +Ln +kj +� +˜µ,φ +� +∇˜µj +� +x, t +� +� += 0 +∀ k = 1 . . . (n − 1), +(E.4) +64 + +where +σ = σ/µel, +(E.5) +Lφ = tcLφm, +(E.6) +κ = κ/ +� +l2 +cm +� +, +(E.7) +λ1 = RT/(mVm), +(E.8) +λ2 = µel/m, +(E.9) +Ln +kj = Ln +kjtcRT/l2 +c, +(E.10) +∂ωchem +∂φθ += +� 1 +RT +� � +p +� +σ=1 +∂hσ +∂φθ +ωσ +chem +� +, +(E.11) +∂ωmech +∂φθ += +� 1 +µel +� � +p +� +σ=1 +∂hσ +∂φθ +ωσ +elastic + +p +� +σ=1 +hσ +∂ωσ +elastic +∂φθ +� +, +(E.12) +ωθ +elastic = (1/2)Cθ +ijkl +� +ϵθ +kl − ϵ⋆θ +kl +� � +ϵθ +ij − ϵ⋆θ +ij +� +(E.13) +Appendix F +Analytical solutions +To test the simulation accuracy, we have compared our simulated results +with analytically obtained solution. However, it bears emphasis that these +analytical solutions require prior knowledge of the domain size, and thus of +the interface positions. Therefore, we have first performed numerical sim- +ulations to calculate the position of these interfaces. Once these positions +65 + +were calculated, they were used as input in the analytical solutions to make +comparisons with simulated solutions. Moreover, unless stated otherwise, we +have assumed zero flux boundary conditions at all boundaries. For the two +planar simulations, in order to compare with analytical solutions we have +taken all fields to be periodic in the top and bottom boundaries. Moreover, +to reduce the computational costs by taking advantage of the domain sym- +metry, we have used symmetry boundary conditions at the left and bottom +boundaries for two non-planar simulations (see cases III and IV). +In this section, we provide the analytical solutions to the four set of three- +phase simulations performed in this paper. It must be emphasized that to +analytically solve the mechanical equilibrium equations, the instantaneous +positions of the two two-phase interphases are required. These prerequisite +positions are therefore numerically obtained based on the phase-field results +and then compared against analytical solutions. +F.1 +Solution for the planar Ni-Al case +Fig. F.1 shows the system geometry and boundary conditions for the planar +Ni-Al case. For the sake of generality, we will refer to the leftmost (fcc−γ), +center (Ni3Al-γ′) and rightmost (NiAl) phases as α, β and γ, respectively. +Let x1(t) and x2(t) represent the positions of the α/β and β/γ interphases +at time t. As mentioned before, we numerically determine these positions at +any given instant from the phase-field simulations and thus the accuracy of +the solution depends on the interface positions. Moreover, assuming plane +stress conditions and neglecting externally applied body forces, the mechan- +66 + +Fig. F.1. A schematic showing the phases, eigenstrains and mechanical boundary +conditions for the planar Ni-Al case. +ical equilibrium equations in Cartesian frame within the bulk regions of a +phase θ = {α, β, γ} reduces to: +∂σθ +x +∂x + ∂σθ +xy +∂y += 0, +(F.1) +∂σθ +xy +∂y + ∂σθ +yy +∂y += 0. +(F.2) +As depicted in Fig. F.1, we have assumed that origin of the Cartesian frame +lies at the center of the domain. As shown in Table 3, we have assumed +the elastic constants to be isotropic but spatially heterogeneous. Thus, the +stress-strain relation within the bulk phases may be written as [59]: +σθ +ij = λθδij +� +ϵθ +kk − ϵ⋆,θ +kk +� ++ 2µθ(ϵθ +ij − ϵ⋆,θ +ij ), +(F.3) +where λθ, µθ and ϵ⋆,θ are Lame’s constant, shear modulus and eigenstrain of +phase θ, respectively. As show in Fig. F.1, the mechanical displacements at +67 + +the left and right boundaries may be written as: +ux(x = ±Lx/2, y, t) = 0, +(F.4) +uy(x = ±Lx/2, y, t) = 0. +(F.5) +On the other hand, the mechanical displacements are assumed to be peri- +odic along the y-direction. Due to these boundary conditions, only the x- +component of displacement, ux(x, t), and the normal strain along x-direction, +ϵθ +x = duθ +x/dx, are nonzero in the bulk regions. +Using Eq. (F.3), it follows that the nonzero mechanical stresses within +the bulk phases are: +σx(x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(λα + 2µα) ϵα +x +−Lx/2 < x < x1, +� +λβ + 2µβ� +ϵβ +x − 2(λβ + µβ)ϵ⋆ +x1 < x < x2, +(λγ + 2µγ) ϵγ +x +x2 < x < Lx/2, +(F.6) +σy(x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +λαϵα +x +−Lx/2 < x < x1, +λβϵβ +x − 2(λβ + µβ)ϵ⋆ +x1 < x < x2, +λγϵγ +x +x2 < x < Lx/2. +(F.7) +It should be noticed from Eqs. (F.6)-(F.7) that the β stress components +are different compared to the α (FCC) and γ (NiAl) phases because we have +assumed a two-dimensional eigenstrain ϵ⋆ = ( ϵ⋆ 0 +0 ϵ⋆ ) only within the Ni3Al +phase. Next, by substituting Eq. (F.6) in Eq. (F.1) and using the strain- +68 + +displacement relations, we see that +� +λθ + 2µθ� d2uθ +x +dx2 = 0. +(F.8) +By integrating Eq. (F.8), it follows that the x-component of displacement +field must vary linearly with distance within the bulk phases. More precisely, +ux(x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Aαx + Bα +−Lx/2 < x < x1, +Aβx + Bβ +x1 < x < x2, +Aγx + Bγ +x2 < x ≤ Lx/2, +(F.9) +where Aα, Bα, Aβ, Bβ, Aγ and Bγ are unknown constants. Moreover, these +six constants can be determined using the two imposed boundary conditions +(Eqs. F.4 & F.5) and four interfacial conditions. Two of these interfacial +conditions arise due to the continuity of x-component of displacement at the +two interfaces, �ux� = 0, and the remaining two are a result of continuity of +normal stresses along x, �σx� = 0. Specifically, +uα +x|x1 = uβ +x +�� +x1 +(F.10) +uβ +x +�� +x2 = uγ +x|x2 +(F.11) +σα +x|x1 = σβ +x +�� +x1 +(F.12) +σβ +x +�� +x2 = σγ +x|x2 +(F.13) +69 + +Next, substituting the expressions in Eq. (F.9) in Eqs. (F.4) and (F.5) yields +−AαLx/2 + Bα = 0 +(F.14) +AγLx/2 + Bγ = 0 +(F.15) +Then using Eq. (F.9), Eqs. (F.10)-(F.13) may be written as +Aαx1 + Bα − +� +Aβx1 + Bβ� += 0, +(F.16) +Aβx2 + Bβ − (Aγx2 + Bγ) = 0, +(F.17) +(λα + 2µα)Aα − (λβ + 2µβ)Aβ + 2 +� +λβ + µβ� +ϵ⋆ = 0, +(F.18) +(λβ + 2µβ)Aβ − 2 +� +λβ + µβ� +ϵ⋆ − (λγ + 2µγ)Aγ = 0 +(F.19) +Eqs. (F.14)-(F.19) form a set of six equations that can be solved to determine +the six unknowns. This was performed using the Python library for symbolic +mathematics, SymPy [60]. A python script for solving these equations is +available (see the python script threephase planar analytical.py). +F.2 +Solution for the planar Ni-Al-Cr case +Fig.F.2 shows the system geometry and boundary condition for the planar +Ni-Al-Cr case. +In contrast to the previous case, the leftmost (fcc-γ) and center (γ′) phases +are elastically anisotropic (see Table 3). Consequently, the stress-strain rela- +70 + +Fig. F.2. A schematic showing the phases, eigenstrains and mechanical boundary +conditions for the planar Ni-Al-Cr case. +tions within these phases may be written as [59]: +σθ +ij = λθδij +� +ϵθ +kk − ϵ⋆,θ +kk +� ++ 2µθ(ϵθ +ij − ϵ⋆,θ +ij ) + µ′θδijkl +� +ϵθ +ij − ϵ⋆,θ +ij +� +, +(F.20) +where λθ = Cθ +12, µθ = Cθ +44, µ′θ = Cθ +11 − Cθ +12 − 2Cθ +44 and δijkl is zero except for +δ1111 = δ2222 = 1. +As shown in Fig. F.2, the imposed mechanical boundary conditions at +the left and right boundaries yields: +ux(x = −Lx/2, y, t) = 0, +(F.21) +uy(x = −Lx/2, y, t) = 0, +(F.22) +ux(x = Lx/2, y, t) = uR +x , +(F.23) +uy(x = Lx/2, y, t) = uR +y , +(F.24) +where uR +x and uR +y are the x and y components of the imposed mechanical +displacement at the right boundary. Consequently, unlike the previous case, +both x and y components of the displacement field, i.e., ux(x, t) and uy(x, t), +71 + +are nonzero within the bulk phases. Precisely, +ux(x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Aα +xx + Bα +x +−Lx/2 < x < x1, +Aβ +xx + Bβ +x +x1 < x < x2, +Aγ +xx + Bγ +x +x2 < x ≤ Lx/2, +(F.25) +uy(x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Aα +yx + Bα +y +−Lx/2 < x < x1, +Aβ +yx + Bβ +y +x1 < x < x2, +Aγ +yx + Bγ +y +x2 < x ≤ Lx/2, +(F.26) +where {Aθ=α,β,γ +x +}, {Aθ=α,β,γ +y +}, {Bθ=α,β,γ +x +} and {Bθ=α,β,γ +y +} are the 12 unknown +constants. +Next, to determine the unknown constants, we first solve the x-component +of displacement field. This requires calculating the six unknowns: {Aθ=α,β,γ +x +} +and {Bθ=α,β,γ +x +}. After substituting the expressions in Eq. (F.25) in Eqs. +(F.21) & (F.23), we get +−Aα +xLx/2 + Bα +x = 0 +(F.27) +uR +x − (Aγ +xLx/2 + Bγ +x) = 0 +(F.28) +The remaining four unknowns can be determined by solving continuity +of x-component of displacement field and normal stress along x. Thus, using +72 + +Eqs. (F.3), (F.20) and (F.25) in Eqs. (F.10)-(F.13), it follows that: +Aα +xx1 + Bα +x − +� +Aβ +xx1 + Bβ +x +� += 0, +(F.29) +Aβ +xx2 + Bβ +x − (Aγ +xx2 + Bγ +x) = 0, +(F.30) +(λα + 2µα)Aα +x − (λβ + 2µβ)Aβ +x + µ′αAα +x − µ′βAβ +x + ζβϵ⋆ = 0, +(F.31) +(λβ + 2µβ)Aβ +x − (λγ + 2µγ)Aγ +x + µ′βAβ +x − µ′γAγ +x − ζβϵ⋆ = 0, +(F.32) +where λθ=α,β = Cθ +12, µθ=α,β = Cθ +44, µ′θ=α,β = Cθ +11 − Cθ +12 − 2Cθ +44 and ζβ = +2(λβ + µβ) + µ′β. By solving Eqs. (F.27)-(F.32) we can obtain six of the 12 +unknown constants. This is achieved symbolically using SymPy [60] and the +python script, threephase aniso planar analytical.py, is provided with +this paper. +Following this, the remaining six constants can be obtained by solving the +y-component of displacement field. Specifically, we need another set of six +equations to determine the unknown constants: {Aθ=α,β,γ +y +} and {Bθ=α,β,γ +y +}. +To this end, substituting Eq. (F.26) in Eqs. (F.22) & (F.24) yields the first +two of these equations: +−Aα +yLx/2 + Bα +y = 0 +(F.33) +uR +y − +� +Aγ +yLx/2 + Bγ +y +� += 0 +(F.34) +Since the y-component of displacement field must be continuous at the +73 + +two interfaces, it follows that +uα +y +�� +x1 = uβ +y +�� +x1 =⇒ Aα +yx1 + Bα +y − +� +Aβ +yx1 + Bβ +y +� += 0, +(F.35) +uβ +y +�� +x2 = uγ +y +�� +x2 =⇒ Aβ +yx2 + Bβ +y − +� +Aγ +yx2 + Bγ +y +� += 0. +(F.36) +Additionally, the shear stress must be continuous at the two interfaces. +This yields +σα +xy +�� +x1 = σβ +xy +�� +x1 +(F.37) +σβ +xy +�� +x2 = σγ +xy +�� +x2 +(F.38) +Using constitutive Eqs. (F.3) and (F.20) in Eqs. (F.37) & (F.38) yields: +2µαAα +y − 2µβAβ +y = 0, +(F.39) +2µβAα +y − 2µγAγ +y = 0. +(F.40) +Thus, by solving Eqs. (F.33)-(F.40) the remaining six unknowns: {Aθ=α,β,γ +y +} +and {Bθ=α,β,γ +y +} can be determined. These equations were also solved sym- +bolically. The python script, threephase aniso shear components.py, is +provided with this paper. +F.3 +Solution for the non-planar Ni-Al case +Fig. F.3 shows the system geometry and boundary conditions for the three- +phase Ni-Al case with concentric interfaces. +As shown in Fig. +F.3, the +innermost (fcc-γ), center (Ni3Al) and outermost (NiAl) phases are hereafter +74 + +referred to as α, β and γ, respectively. Moreover, due to the concentric ring +geometry of the system, we analytically solve the mechanical equilibrium +equations in polar coordinates, (r, φ), even though the simulation was per- +formed in a Cartesian frame, (x, y). It should be noted that to compare the +analytically obtained solution against the simulated solution we transform +the simulated elastic fields from the Cartesian frame to polar coordinates. +For instance, the displacement field in polar coordinates may be calculated +Fig. F.3. A schematic showing the phases, eigenstrains and mechanical boundary +conditions for the concentric interface Ni-Al case. +from Cartesian frame using +� +� +� +� +� +ur +ut +� +� +� +� +� += +� +�� +cos ζ +sin ζ +− sin ζ +cos ζ +� +�� +� +� +� +� +� +ux +uy +� +� +� +� +� +, +(F.41) +where ζ = tan−1(y/x) is the angle of rotation between the two frames (Fig. +F.3). For this case, the displacement field within the bulk phase θ takes the +75 + +form +uθ(r, t) = uθ +r(r, t)er + uθ +φ(r, t)eφ. +(F.42) +As shown in Fig. F.3, due to the imposed boundary conditions, the radial +displacement is zero at the origin and the radial stress at the outer surface +is zero. This yields +uα +r (r = 0, t) = 0 +(F.43) +σγ +r (r = R, t) = 0 +(F.44) +Note that the superscripts on the mechanical fields identify the phases in the +system. Because of these imposed boundary conditions, it can be assumed +that the φ component of displacement field is zero throughout the system, +i.e., uα +φ = uβ +φ = uγ +φ = 0. Consequently, the strain-displacement relation in +polar coordinates within the bulk domains simplifies to +ϵθ +r(r) = duθ +r(r)/dr, +(F.45) +ϵθ +φ(r) = uθ +r(r)/r, +(F.46) +ϵθ +rφ(r) = 0 +(F.47) +Further, assuming plane stress conditions, the mechanical equilibrium equa- +tions within the bulk domains in polar coordinates simplifies to +∂σθ +r +∂r + σθ +r − σθ +φ +r += 0 +(F.48) +76 + +Because we have assumed isotropic elastic properties, it follows from Eqs. +(F.3) & (F.45)-(F.47) that the nonzero stresses in polar coordinates are +σr(r, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(λα + 2µα) ϵα +r + λαϵα +φ +0 < r < r1, +� +λβ + 2µβ� +ϵβ +r + λβϵβ +φ − 2(λβ + µβ)ϵ⋆ +r1 < r < r2, +(λγ + 2µγ) ϵγ +r + λγϵγ +φ +r2 < r < R, +(F.49) +σφ(r, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(λα + 2µα) ϵα +φ + λαϵα +r +0 < r < r1, +� +λβ + 2µβ� +ϵβ +φ + λβϵβ +r − 2(λβ + µβ)ϵ⋆ +r1 < r < r2, +(λγ + 2µγ) ϵγ +φ + λγϵγ +r +r2 < r < R. +(F.50) +Here r1(t) and r2(t) represent the numerically obtained interface positions at +the α/β and β/γ interfaces at time t (see Fig. F.3). Next, by substituting +Eqs. (F.49) & (F.50) in Eq. (F.48) it can be shown that in a bulk phase θ +the mechanical equilibrium equation reduces to +� +λθ + 2µθ� �d2uθ +r +dr2 + 1 +r +duθ +r +dr − uθ +r +r2 +� += 0 ⇔ d +dr +�1 +r +d +dr +� +uθ +rr +�� += 0. +(F.51) +Integrating Eq. (F.51) yields the radial displacement within the bulk phases +yields +ur(r) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +uα +r := Aαr + Bα/r +0 < r < r1, +uβ +r := Aβr + Bβ/r +r1 < r < r2, +uγ +r := Aγr + Bγ/r +r2 < r ≤ R, +(F.52) +Now, the problem reduces to finding the solution to the six unknown con- +77 + +stants: {Aθ=α,β,γ} and {Bθ=α,β,γ}. +Since the displacement field must be +bounded as r → 0, the constant Bα must be zero. This ensures that the +radial displacement is zero at the origin (see Eq. (F.43)). +Using the strain-displacement relations, i.e., Eqs.(F.45)-(F.47), and Eq.(F.52), +it can be shown that the nonzero strains within the bulk phases are +ϵr(r) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ϵα +r := Aα +0 < r < r1, +ϵβ +r := Aβ − Bβ/r2 +r1 < r < r2, +ϵγ +r := Aγ − Bγ/r2 +r2 < r ≤ R, +(F.53) +ϵφ(r) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ϵα +r := Aα +0 < r < r1, +ϵβ +r := Aβ + Bβ/r2 +r1 < r < r2, +ϵγ +r := Aγ + Bγ/r2 +r2 < r ≤ R, +(F.54) +Note that we have set Bα to be zero in Eqs.(F.53) & (F.54). To determine +the remaining five unknowns, we need five equations. The first equation is +a consequence of boundary condition at the outer surface, i.e., Eq. (F.44). +Thus, substituting Eqs. (F.53) and (F.54) in Eq. (F.49) and setting r = R +yields +(λγ + 2µγ) +� +Aγ − Bγ/R2� ++ λγ � +Aγ + Bγ/R2� += 0 +(F.55) +The four remaining equations are obtained as a consequence of the interfacial +conditions, specifically the continuity of radial displacement and radial stress. +78 + +The continuity of displacement field yields: +uα +r |r1 = uβ +r +�� +r1 +(F.56) +uβ +r +�� +r2 = uγ +r|r2 +(F.57) +Similarly, stress continuity implies: +σα +r |r1 = σβ +r +�� +r1 +(F.58) +σβ +r +�� +r2 = σγ +r |r2 +(F.59) +Substituting Eq. (F.52) in Eqs. (F.56) and (F.57) yields two of the required +equations +(Aα − Aβ) r1 − Bβ/r1 = 0 +(F.60) +(Aβ − Aγ) r2 + (Bβ − Bγ) /r1 = 0 +(F.61) +Similarly, using Eqs. (F.49), (F.52) & (F.53) in Eqs. (F.58) & (F.59) yields +the remaining two equations: +[(λα + 2µα)Aα + λαAα] − +� +(λβ + 2µβ) +� +Aβ − Bβ/r2 +1 +�� +− λβ � +Aβ + Bβ/r2 +1 +� ++ 2(λβ + µβ)ϵ⋆ = 0 +(F.62) +� +(λβ + 2µβ) +� +Aβ − Bβ/r2 +2 +�� ++ λβ � +Aβ + Bβ/r2 +2 +� +− 2(λβ + µβ)ϵ⋆ +− +� +(λγ + 2µγ) +� +Aγ − Bγ/r2 +2 +�� +− λγ � +Aγ + Bγ/r2 +2 +� += 0 +(F.63) +79 + +By solving Eq. (F.55) and Eqs. (F.60)-(F.63) yields the five unknown con- +stants. A python script, threephase nonplanar analytical.py, to solve +these equations symbolically is provided with this paper. +F.4 +Solution for the non-planar Ni-Al-Cr case +Fig. F.4 shows the simulation domain and boundary conditions for the con- +centric ring Ni-Al-Cr case. Despite the similarities, there are two important +differences that affects the analytical solution. First, we have assumed that +there are no eigenstrains in the system; and second, we have imposed a hoop +strain at the outer boundary. The outer boundary condition may be written +as: +ϵγ +φ(r = R, t) = ϵg +R, +(F.64) +where ϵg +R is the assumed hoop strain. In the simulation, this hoop strain is +imposed by assuming that the Cartesian displacements at the outer boundary +are: +ux(r = R, t) = ϵg +Rx +uy(r = R, t) = ϵg +Ry +(F.65) +Using Eqs. (F.41), (F.46) and (F.65), it can be shown that the hoop strain +at the outer boundary is equal to ϵg +R. Moreover, due to fact that the geome- +try of the system is similar to the previous case, it can be assumed that the +displacement fields within the bulk regions are given by Eq. (F.52). Further- +more, since the boundary conditions at the left and bottom boundaries are +80 + +Fig. F.4. A schematic showing the phases, eigenstrains and mechanical boundary +conditions for the concentric interface Ni-Al-Cr case. +identical to the previous case, it follows that the radial displacement at the +origin must be zero, and consequently Bα = 0. Therefore, we need to solve +for only the five unknown constants in Eq. (F.52). +The first of these conditions is obtained by solving the outer boundary +condition. Thus, substituting Eq. (F.52) in Eq. (F.54) and using Eq. (F.64) +yields +Aγ + Bγ/R2 = ϵg +r +(F.66) +Similar to the previous case, the remaining four equations arise from the +interfacial conditions. Moreover, the equations resulting from continuity of +displacement field are identical to the previous case, i.e., Eqs. +(F.60) & +(F.61). The remaining two equations are obtained assuming that the radial +81 + +stress is continuous at the two interfaces. Specifically, +[(λα + 2µα)Aα + λαAα] − +� +(λβ + 2µβ) +� +Aβ − Bβ/r2 +1 +�� +− λβ � +Aβ + Bβ/r2 +1 +� += 0 +(F.67) +� +(λβ + 2µβ) +� +Aβ − Bβ/r2 +2 +�� ++ λβ � +Aβ + Bβ/r2 +2 +� +− +� +(λγ + 2µγ) +� +Aγ − Bγ/r2 +2 +�� +− λγ � +Aγ + Bγ/r2 +2 +� += 0 +(F.68) +Thus, by solving Eqs. (F.66), (F.67), (F.68), (F.60) and (F.61) we can +obtain the five unknowns in Eq. (F.52. This was achieved using the python +package SymPy [60]. The python script, threephase iso nonplanar- +applied strain.py, is also available with this paper. +Data Availability +The processed data required to reproduce the figures are available from the +corresponding author on request. The simulation software required to repro- +duce the results is available to download from https://github.com/souravmat- +git/gibbs. The MOOSE input files required to run the simulations are avail- +able to download from the folder stressed multiphase. The MATLAB scripts +required to reproduce the precomputed input thermodynamic and kinetic +properties are available to download from the folder Precomputed properties +[61]. Finally, the python scripts required to symbolically calculate the con- +stants in the analytical solutions are available to download from the folder +symbolic python. +82 + +References +[1] Eliot Fried and Morton E Gurtin. 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Mendeley Data, v1, 2021. +92 + diff --git a/I9AzT4oBgHgl3EQfx_6k/content/tmp_files/load_file.txt b/I9AzT4oBgHgl3EQfx_6k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6eb5e336c8bc11b3900b3650398472ff240f535a --- /dev/null +++ b/I9AzT4oBgHgl3EQfx_6k/content/tmp_files/load_file.txt @@ -0,0 +1,2059 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf,len=2058 +page_content='A computationally efficient and mechanically compatible multi-phase-field model applied to coherently stressed three-phase solids Sourav Chatterjeea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='b⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Daniel Schwenc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nele Moelansa aDepartment of Materials Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' KU Leuven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Kasteelpark Arenberg 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Leuven BE-3001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Belgium bDepartment of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' University of Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Gainesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 32611,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' FL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' USA cComputational Mechanics and Materials Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Idaho National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Idaho Falls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' ID 83415,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' United States ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' E-mail addresses: chatterjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='s@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='edu (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Chatterjee), daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='schwen@inl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='gov (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Schwen), nele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='moelans@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='be (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moelans) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='01747v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='mtrl-sci] 4 Jan 2023 Abstract Engineering alloys generally exhibit multi-phase microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For simu- lating their microstructure evolution during solid-state phase transformation, CALPHAD-guided multi-phase-field models coupled with micro-mechanics have proven to be a reliable simulation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, their efficiency and accuracy still depend on the homogenization scheme used to interpolate the elastic properties in the interfacial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In this paper, we present a phase-field model for multi-phase and multi-component solids using a partial rank-one homogenization scheme that enforces static and kinematic compat- ibilities in the interfacial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To this end, we first extend the rank-one homogenization scheme to multi-phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, for computational efficiency, we analytically solve the static compatibility equations for linear elastic three-phase solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For quantitative accuracy, a coupling technique is used to extract the prerequisite thermodynamic and kinetic properties from CALPHAD databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The model is solved numerically in an open source finite-element framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As numerical applications, the microstructure of two elastically stressed intermetallic-containing three-phase alloys: Ni-Al and Al-Cr-Ni, are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The accuracy of the model is verified against analyt- ically obtained solutions for planar and concentric ring interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We show that the simulation results remain unaltered with varying interface width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Except for one simulation, all cases show better or nearly equal convergence using the partial rank-one scheme compared to the Voigt-Taylor scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2 Keywords: chemo-mechanical processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' microstructure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' phase transforma- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' inhomogeneous material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' homogenization 1 Introduction Engineering alloys, such as Ni-base superalloys, steels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', generally com- prise multiple chemical constituents and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Their physical and me- chanical properties are strongly related to the microstructure formed during interdiffusion processes at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, predicting the kinetics of microstructure evolution during diffusive transformations, espe- cially when elastic stresses are included, is difficult since this requires solving a free-boundary problem, which is seldom analytically soluble [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, reliable and efficient computational approaches are often needed to gain a quantitative understanding of microstructure evolution in elastically stressed multiphase and multicomponent alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The phase-field method has emerged as a useful tool to predict microstruc- ture evolution in engineering alloys [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Its well-known advantage is that the interface or interphase separating either the grains or phases is implic- itly represented by a phase-field variable that varies smoothly across a finite region of thickness, referred to as the interface width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, for simula- tions to be well-resolved, the interface width has to be at least five times the grid spacing [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Therefore, simulations using this method are particu- larly difficult when the desired microstructural length scale is in the micro to millimeter range due to a stringent limit on interface width [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In addition, this limit typically varies with the bulk alloy properties [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3 To overcome this limitation, it is thus essential that the interface width in a phase-field model can be independently controlled without affecting the accuracy of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This requirement has led to the development of alloy phase-field models in which the interface width is treated as a simulation parameter that can be selected depending on numerical convenience [13], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is because the bulk and interfacial properties in such models are independent, even when the interface width is artificially enlarged [10], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, the gener- alization of such alloy phase-field models to problems that require coupling with mechanics is not straightforward due to the dependence of bulk prop- erties on elastic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' More precisely, in a mechanically coupled phase-field model, the scheme of interpolation or homogenization of elastic fields in the interfacial regions may affect this desired separation of bulk from interfa- cial properties due to an interfacial excess elastic energy contribution that depends on the homogenization scheme [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' So far, two types of mechanically uncoupled phase-field models for alloys have been proposed that allow the interface width to be selected arbitrarily [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As pointed out by Plapp [10], the first derives the evolution equations starting from a Helmholtz functional [13], while the second derives it from a grand-potential functional [10], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the former approach re- quires thermochemical properties as functions of composition(s), while the latter requires them as functions of diffusion potential(s) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Although both models are equivalent [10], the latter offers possible computational gains as it requires solving (n − 1)(p − 1) less equations for a n-component and p-phase alloy system [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It is worth noting that this is strictly true assuming ei- 4 ther non-dilute or non-ideal or non-quadratic free energies since only then the equal diffusion potential or “quasiequilibrium” conditions have to be numeri- cally solved at each grid point and time step [19] [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, to decrease the computational costs in such simulations, some studies have developed sim- plified approaches to solving these conditions [19], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, the appropriate homogenization approach for coupling these alloy phase-field models with mechanics is still debatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, the coupling of the above-mentioned models with small-strain elasticity theory has been considered by many workers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' either based on a Helmholtz functional, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', [23], [24], [25], [15], [26] or a grand-potential functional [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, the accuracy of such coupled models still depends on the homogenization assumptions with regard to the elastic fields [24], [15], [31], [16], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To be precise, depending on the scheme of homog- enization, these mechanically coupled models can be subdivided into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The models in the first category follow those homogenization schemes that are either statically or kinematically compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For instance, Khachaturyan [23], [27], Reuss/Sachs [33], [25], and Voigt/Taylor [24], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' On the other hand, models in the second category follow those schemes that enforce both static and kinematic compatibilities: by either using a mixed scheme that is a combination of Reuss/Sachs and Voigt/Taylor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', [15], [26], [28], or a partial rank-one scheme [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, it has been argued that models in the first category are less accurate compared to models in the second category due to an interfacial excess energy contribution coming from the interpolated elastic strain energy [15], [26], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, from the standpoint of computational efficiency, the partic- 5 ular scheme used for enforcing the static and kinematic compatibilities is also a topic of relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For example, the mixed scheme that is a combination of Reuss/Sachs and Voigt/Taylor proposed in the works of Durga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [15], [34] and Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [16], [35] requires a coordinate transformation of elastic fields in order to formulate the interfacial elastic driving force con- tribution as a function of only continuous elastic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in [15], [16], this is needed because then this interfacial excess contribution vanishes in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, their approaches are computationally intensive [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Naturally, this limits the application of their model to simple systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, Durga’s model has been so far applied to simulate an elasti- cally anisotropic four-phase Cu-Sn alloy having only planar interfaces [34], while Schneider’s model has been limited to elastically isotropic two-phase [28] and multi-phase [37] binary alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' These works, however, assume only small-strain deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For sake of completeness, it is worth mentioning that Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and Hermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' have also proposed a numerical approach to enforce the static compatibility equations for multiphase solids undergoing finite-strain and small-strain inelastic deformations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Svendsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [32] independently proposed a more unified framework that extends Helmholtz-based models, such as [15], to multiphase multicomponent solids undergoing finite-strain and inelastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Contrary to the mixed scheme, Mosler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [31] proposed a partial rank-one homogenization scheme to enforce static and kinematic compati- bilities for two-phase solids undergoing finite deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The advantage of this scheme over the mixed scheme is that it does not require coordi- nate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Keifer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' showed improved convergence using this 6 scheme compared to schemes that ensure either static or kinematic com- patibility for two-phase solids undergoing small-strain deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Subse- quently, Bartels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [38] applied this scheme to couple mechanics with a WBM (Wheeler-Boettinger-McFadden) type chemical model [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Unfortu- nately, unlike the previously discussed mechanically uncoupled models, the interface width in this model cannot be controlled due to an interfacial ex- cess energy contribution coming from bulk chemical free energies [10], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Later, Bartel’s model was improved by the present authors by combining a grand-potential model with the partial rank-one scheme for two-phase solids undergoing small-strain deformations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Using this model, we also found that the rank-one scheme offered improved numerical convergence compared to either static or kinematically compatible schemes [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Despite these advantages, the partial rank-one scheme has so far not been extended to multiphase and multicomponent solids undergoing linear elastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To our knowledge, the only published work that extends the rank-one scheme to multi-phase solids is by Sarhil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, there are two limitations to this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The first limitation is that it has not been coupled with diffusion equations and hence cannot be applied to simulate diffusive transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The second limitation is that it takes an interpo- lation function that is equal to the phase-field variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', hθ(φ) = φθ, to interpolate elastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As noted by Moelans [9], this assumption may shift the local minima of the free energies and may cause inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Hence, this paper aims to fill these gaps by formulating a multi-phase-field model based on a partial rank-one homogenization scheme starting from a grand- potential functional, thereby ensuring that the interfacial excess contribution 7 due to bulk properties vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To this end, we present an analytical ap- proach to solving the static compatibility equations for a three-phase linear elastic solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For quantitative accuracy, we use a coupling method devel- oped in [18] that allows incorporating thermodynamic and kinetic properties obtained from CALPHAD databases into a grand-potential-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The phase-field formulation with the rank-one homogenization scheme is introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In Section 3, the prerequisite chemical properties and the elastic properties for two—a Ni-Al and an Al-Ni-Cr—three-phase alloys are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To demonstrate the application of our model, four numerical simulations are performed, and the results are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The accuracy of our numerical results is tested by comparing the phase-field simulations with analytically obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, the conclusions of the paper are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2 Formulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 Notations In this paper, we assume an isothermal system consisting of n diffusing com- ponent and p phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We denote a set of scalar fields with boldface letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For example, the set of (n − 1) independent diffusion potential fields is de- noted as ˜µ = � ˜µk=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='(n−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similarly, the set of p phase-field variables is shown as φ = {φθ=α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Vector and tensors are also represented with bold- face letters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', the displacement is written as u = uiei, where ui=1,2,3 are the components of u relative to a chosen orthonormal basis {ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Following 8 standard notations, the Einstein summation convention is used throughout the paper to indicate summation over spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The dot, outer and inner products between two vectors, say a and b, are written as a · b, a ⊗ b and a : b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The norm, divergence, gradient and laplacian of a physical quantity, say Φ, are written as ∥Φ∥, div Φ, grad Φ, and ∆Φ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 Definitions of field variables and jump As mentioned before, since the diffusion potentials are the independent vari- ables in a grand-potential-based model, any prerequisite property in the model should be expressed as functions of diffusion potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Precisely, the diffusion potential of a diffusing component, say k, is defined as the difference between its chemical potential and the chemical potential of the dependent component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', ˜µk = (µk − µn), and it has units of J/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fur- ther, an arbitrary phase in the system, say θ, at any given spatial point x and time t is indicated by the phase-field variable, φθ(x, t), such that the bulk regions occupied by this phase are when φθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the jump of a field or property, �Φ�αβ = Φα − Φβ, at an interface, say α/β, is defined as the difference between its bulk values within the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 Partial rank-one scheme for multi-phase systems Starting from the two-phase approach of Mosler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [31], we assume that the total strain, ϵ(u), in the interfacial regions is a smooth function of the 9 phase strains assigned to each phase in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Precisely, ϵij(u) = p � θ=1 ϵθ ijhθ(φ), (1) where ϵ(u) is the total strain as a function of the displacement u, ϵθ and hθ(φ) are the (total) phase strain and interpolation function attributed to phase θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the total strain at a point is calculated using the linear strain-displacement relations ϵij(u) = (1/2) [grad u + (grad u)T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (2) The choice of the interpolation function, h(φ), is such that in the bulk re- gions: hθ = 1 for (φθ = 1, φσ̸=θ = 0) and hθ = 0 for (φθ = 0, φσ̸=θ = 1), while in the interfacial regions: 0 < hθ < 1 for 0 < φθ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, as noted in [41], [9], the function h(φ) must satisfy two additional requirements: i) �p θ=1 hθ = 1, and ii) dhθ/dφθ [φθ = 1, φσ̸=θ = 0] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, similar to the function proposed by Moelans [9], three different interpolation functions that fulfill these requirements have been formulated by Schneider and co-workers [35, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, for the sake of convenience, in this work we chose the function first proposed by Moelans [9]: hθ(φ) = φ2 θ �p θ=1 φ2 θ for θ = {α, β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (3) As discussed in the Introduction section, it should be noted that Sarhil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [40] proposed hθ(φ) = φθ which does not satisfy the above-mentioned requirements and may lead to inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Another noteworthy difference 10 between our model and the models proposed by Sarhil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [40] and Schnei- der et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [35, 42, 43] is that our model does not require a constraint that the sum of phase-fields should add up to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', �p θ=1 φθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It is worth noting that the phase strains introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (1) are phys- ically meaningful only in the bulk regions of a phase (hθ = 1) but not in the interfacial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It is because, in the bulk regions, they become equal to the total strain, which is a physically measurable quantity that depends on the stiffness tensor and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' But in the interfacial regions, the variation of phase strains depends on the interface width;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' a numerical parameter selected arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4, we will show that the phase strains also depend on the homogenization scheme in the interfacial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similar to phase concentrations introduced in solidification studies [44], their primary purpose is to separate the bulk and interfacial contributions in the total energy for artificially enlarged interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Following the two-phase approach [31], to ensure kinematic compatibility, the phase strains must satisfy the Hadamard jump conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, the p unknown phase strains, {ϵα, ϵβ, ϵγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' ϵp}, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (1) must satisfy the following (p − 1) Hadamard jump conditions �ϵij�αβ = ϵα ij − ϵβ ij = sym(aαβ i nαβ j ), �ϵij�βγ = ϵβ ij − ϵγ ij = sym(aβγ i nβγ j ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' �ϵij�(p−1),p = ϵp−1 ij − ϵp ij = sym � a(p−1),p i n(p−1),p j � , (4) where {aαβ, aβγ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , a(p−1),p} and {nαβ, nβγ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , and n(p−1),p} are the 11 jump vectors and unit normals at the {α/β, β/γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , (p − 1)/p} interfaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Here, the notation �ϵ�(p−1),p denotes the strain jump at the interface between phases (p − 1) and p, and is equal to the outer product between the jump vector and unit normal at that interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that the jump vector, aαβ, at the α/β interface is symmetric with respect to the superscripts αβ [42], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', aαβ = aβα, since by definition �ϵ�αβ = −�ϵ�βα and nαβ = −nβα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, following our previous work [30], we define the unit normal at an interface as [45], [42] nθσ,σ̸=θ = −grad φθ/ ∥grad φθ∥ , θ = {α, β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (p − 1)} & σ = {β, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , p}, (5) where ∥∇φθ∥ is the norm of the gradient of the phase-field variable φθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We note that although a multi-phase-field version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (5) exists (see [46] & [47]), we have not used that definition in this paper for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [42] has noted that the solution of elastic fields is not significantly dependent on the definition of the unit normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (1) and (4) form a system of p equations that can be analytically solved to explicitly determine the p-phase strains: {ϵα, ϵβ, ϵγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' ϵp}, as func- tions of the total strain ϵ(u), p interpolation functions hθ=α,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='p(φ) and (p−1) strain jumps: {�ϵ�αβ, �ϵ�βγ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , �ϵ�(p−1),p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In Appendix A, we show how to analytically calculate the phase strains for a multiphase system as functions of the total strain, interpolation functions and strain jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, we calculate the unknown jump vectors: {aαβ, aβγ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , a(p−1),p} in 12 order to determine the (p − 1) strain jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similar to previous works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', [31], [30], [42], we also calculate the jump vector at an interface by solving the static compatibility equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To be precise, the following (p − 1) static compatibility equations must be solved to determine the same number of unknown jump vectors �σij�αβnαβ j = � σα ij − σβ ij � nαβ j = 0i, �σij�βγnβγ j = � σβ ij − σγ ij � nβγ j = 0i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' �σij�(p−1),pn(p−1),p j = � σp−1 ij − σp ij � n(p−1),p j = 0i, (6) where σθ is the elastic stress associated with phase θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In this paper, we have introduced static compatibility equations as a means to calculate the jump vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Alternatively, these equations can be derived by minimizing the total elastic strain energy with respect to the jump vectors, as pointed out by Mosler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [31] and Sarhil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [40] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Using linear elastic theory, the elastic phase stresses, {σα, σβ, σγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' σp}, are related to the phase strains by the generalized Hooke’s law σθ ij = Cθ ijkl � ϵθ kl − ϵ⋆θ kl � for θ = {α, β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , p}, (7) where Cθ and ϵ⋆θ denote the fourth-rank stiffness tensor and eigenstrain belonging to a particular phase θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It follows from the set of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (4) & (6) that the rank-one scheme ensures both kinematic and static compatibilities at (p − 1) interfacial regions of a 13 p-phase system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It is worth pointing out that a system consisting of p-phases may have p(p − 1)/2 two-phase junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (4) & (6) only ensure static and kinematic compatibilities at (p − 1) of these junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It can be shown that mechanical compatibilities at remaining (p − 1)(p − 2)/2 junctions are implicitly ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For instance, by adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (4) & (6), we obtain the following set of compatibility equations �ϵij�α,p = ϵα ij − ϵp ij = {aαβ i nαβ j + aβγ i nβγ j + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + a(p−1),p i }, (8) �σij�α,pnα,p j = � σα ij − σp ij � nα,p j = 0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (9) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (8) and (9), it follows that both static and kinematic compat- ibilities are ensured at the interface between phases α and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, it should be emphasized that, depending on the constitutive equa- tions, the jump vectors can be solved either analytically or numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As discussed in the Introduction section, for non-linear elastic solids, the jump vectors can be obtained only by numerically solving the set of static compat- ibility equations at each grid point and time step (see [42],[43]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, for linear elastic solids, the jump vectors can be determined either analytically or numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Although restricted to two phases, the analyt- ical approach was followed in [48] & [30], while a Newton-Raphson scheme was used in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, to our knowledge, analytical expressions for the jump vectors in a multi-phase-field setting do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Since analytical approaches may offer computational gains over numerical solutions, partic- ularly when linear constitutive equations are assumed, we follow the former approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 14 However, deriving a general analytical expression for the jump vectors for a p-phase system is not straightforward as it requires explicit analytical expressions for phase strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Since the analytical expressions for the phase strains become increasingly complicated as the number of phases increases (see Appendix A), we therefore take a special case to illustrate how to derive the jump vectors in a multi-phase-field context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For the sake of convenience, we chose a three-phase system for this derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 Partial rank-one scheme for three-phase systems For a three-phase system, the phase strains belonging to phases, say α, β and γ, may be written as (see Appendix A) ϵα ij � ϵ, φ, �ϵ�αβ, �ϵ�βγ� = ϵij(u) + [hβ(φ) + hγ(φ)] �ϵij�αβ + hγ(φ)�ϵij�βγ, (10) ϵβ ij � ϵ, φ, �ϵ�αβ, �ϵ�βγ� = ϵij(u) − hα(φ)�ϵij�αβ + hγ(φ)�ϵij�βγ, (11) ϵγ ij � ϵ, φ, �ϵ�αβ, �ϵ�βγ� = ϵij(u) − hα(φ)�ϵij�αβ − [hβ(φ) + hα(φ)] �ϵij�βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (12) It follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (10)-(12) that the phase strains are always equal to the total strain ϵ(u) in the bulk regions of the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, in the interfacial regions, they differ depending on the definition of strain jumps, which in turn depends on the homogenization assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Concretely, the strain jumps vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', �ϵ�αβ = �ϵ�βγ = 0, for the case of Voigt-Taylor homogenization scheme (henceforth referred to as the VT scheme) or the Khachaturyan scheme [15], while for the partial rank-one scheme (henceforth 15 referred to as PR scheme) the strain jumps are given by (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4) �ϵ�αβ = aαβ ⊗ nαβ, (13) �ϵ�βγ = aβγ ⊗ nβγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (14) As previously discussed, the jump vectors aαβ and aβγ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (13) & (14) are obtained by solving the static compatibility equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Precisely, the set of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (6) for a three-phase system reduced to � σα ij − σβ ij � nαβ j = 0i, (15) � σβ ij − σγ ij � nβγ j = 0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (16) Next, it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (7) that the elastic phase stresses in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (15) & (16) are related to the phase strains by σα ij = Cα ijkl [ϵα kl − ϵ⋆α kl ] , (17) σβ ij = Cβ ijkl � ϵβ kl − ϵ⋆β kl � , (18) σγ ij = Cγ ijkl [ϵγ kl − ϵ⋆γ kl ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (19) Now, substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (17)-(19) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (15)-(16) yields �� Cα ijkl − Cβ ijkl � ϵkl + λ1 ijkl�ϵkl�1 + λ2 ijkl�ϵkl�2� n1 j = Z1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (20) �� Cβ ijkl − Cγ ijkl � ϵkl + M1 ijkl�ϵkl�1 + M2 ijkl�ϵkl�2� n2 j = Z2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (21) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' we have denoted the superscripts αβ and βγ by 1 & 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 16 and λ1 ijkl(φ) = hβ(φ)Cα ijkl + hα(φ)Cβ ijkl + hγ(φ)Cα ijkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' λ2 ijkl(φ) = hγ(φ)Cα ijkl − hγ(φ)Cβ ijkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' M1 ijkl(φ) = hα(φ)Cγ ijkl − hα(φ)Cβ ijkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' M2 ijkl(φ) = hγ(φ)Cβ ijkl + hα(φ)Cγ ijkl + hβ(φ)Cγ ijkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Z1 i (n1) = � Cα ijklϵ⋆α kl − Cβ ijklϵ⋆β kl � n1 j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Z2 i (n2) = � Cβ ijklϵ⋆β kl − Cγ ijklϵ⋆γ kl � n2 j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (22) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (13) & (14) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (20) & (21) yields � mα1 i − mβ1 i � + λ# ika1 k + λ⋆ ika2 k = Z1 i , (23) � ψβ2 i − ψγ2 i � + L# ika1 k + L⋆ ika2 k = Z2 i , (24) where mα1 i � ϵ, n1� = Cα ijklϵkln1 j, mβ1 i � ϵ, n1� = Cβ ijklϵkln1 j, ψβ2 i � ϵ, n2� = Cβ ijklϵkln2 j, ψγ2 i � ϵ, n2� = Cγ ijklϵkln2 j, λ# ki(φ, n1) = n1 l λ1 lkij(φ)n1 j, λ⋆ ki(φ, n1, n2) = n2 l λ2 lkij(φ)n1 j, L# ki(φ, n1, n2) = n1 l M1 lkij(φ)n2 j, L⋆ ki(φ, n2) = n2 l M2 lkij(φ)n2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (25) 17 Rearranging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (24) and solving for a2 yields a2 j(φ, ϵ, n1, n2) = Sji(φ, , n2)bi, (26) where Sji = � L⋆ ij �−1 and bi = Z2 i − � ψβ2 i − ψγ2 i + L# ika1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, substituting a2 j in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (23) yields λ# pqa1 q + λ⋆ pq (Sqibi) = Z1 i − � mα1 p − mβ1 p � (27) Using the expression for b and then solving for a1 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (27) finally yields a1 k(ϵ, φ, n1, n2) = (Dpk)−1 � Z1 p − � mα1 p − mβ1 p � − λ⋆ prSri � Z2 i − � ψβ2 i − ψγ2 i ��� (28) where Dpk = λ# pk − λ⋆ prSriL# ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For a three-phase-field model, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (26) and (28) are the most general expressions for the jump vectors a2 and a1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To further simplify these expressions, we assume that the two second- rank tensors, λ⋆ and L#, are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Because from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (25) we see that these tensors depend on both the unit vectors, n1 and n2, which are simultaneously non-zero only at the triple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Perhaps not surprisingly, by making this assumption we have strictly restricted the definition of jump vectors to the two-phase regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Stated differently, we have enforced static compatibility only at the two-phase junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Our assumption is justified since the set of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (6) is strictly valid at the two-phase junctions only where the unit normal vector to the interface is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As a consequence, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 18 (26) and (28) simplifies to a2 j = − � L⋆ ij �−1 �� Cβ ikpq − Cγ ikpq � ϵpq − � Cβ ikpqϵ⋆β pq − Cγ ikpqϵ⋆γ pq �� n2 k, (29) a1 j = − � λ# ij �−1 �� Cα ikpq − Cβ ikpq � ϵpq − � Cα ikpqϵ⋆α pq − Cβ ikpqϵ⋆β pq �� n1 k, (30) where L⋆ ij(φ, n2) = n2 l � hγ(φ)Cβ lijr + hα(φ)Cγ lijr + hβ(φ)Cγ lijr � n2 r, (31) λ# ij(φ, n2) = n1 l � hβ(φ)Cα lijr + hα(φ)Cβ lijr + hγ(φ)Cα lijr � n1 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (32) Expectedly, we see that the analytically derived expressions for jump vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (29) & (30), are similar to the expression of jump vector derived in a two-phase setting (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (9) in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As noted in a previous work [30], we find that the magnitude of the jump vector at an interface, say α/β, is proportional to two elastic properties: i) the jump in stiffness tensors of the bulk phases, and ii) the eigenstrains in the bulk phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 Functional, overall molar density and elastic stresses Here, starting from a grand-potential functional we derive expressions for the overall molar density of a diffusing component and elastic stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As discussed in the Introduction section, we follow the grand-potential approach [10], [17] in this work because we don’t need to explicitly solve for the quasiequilibrium conditions [19], that may lead to computational gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The grand-potential functional, Ω[φ, ˜µ, u], of the system for an elastically 19 stressed multiphase multicomponent alloy is given by Ω [φ, ˜µ, u] = � V [ωbulk (φ, ˜µ, ϵ) + ωint (φ, ∇φ)] dv, (33) where the bulk contribution to the total grand-potential density is denoted by ωbulk (φ, ˜µ, ϵ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' the interfacial energy contribution to the total grand-potential density is indicated by ωint(φ, ∇φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and V is the total volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, the bulk contribution to the total functional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', ωbulk [J/m3], is defined as ωbulk(φ, ˜µ, ϵ) = p � θ=1 hθ(φ)ωθ bulk �˜µ, ϵθ� , (34) where hθ(φ) is the interpolation function, which is defined at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (3) and ωθ bulk is the grand-potential density of phase θ expressed as functions of dif- fusion potentials ˜µ and phase strains ϵθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Under the assumption that each phase is represented by a single grain orientation, the interfacial energy con- tribution to the total energy may be written as [9] ωint(φ, ∇φ) = p � θ=1 (1/2)κ ∥grad φθ∥2 + mg(φ), (35) where the two constant parameters κ [J/m] and m � J/m3� are related to the interfacial energy σαβ and interface width lαβ by [9] κ = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0) σαβlαβ, m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 (σαβ/lαβ) , (36) assuming uniform interface properties and a multi-well function g(φ) of the 20 form [9] g(φ) = p � θ=1 � (1/4) φ4 θ − (1/2) φ2 θ � + (3/4) p � θ=1 p � σ=1 σ>θ φ2 θφ2 σ + (1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (37) Following our previous work [30], the bulk grand-potential density ωθ bulk(˜µ, ϵθ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (34) is written as ωθ bulk(ϵθ, ˜µ) = ωθ chem(˜µ) + (1/2)Cθ ijkl(˜µ) � ϵθ kl − ϵ⋆θ kl(˜µ) � � ϵθ ij − ϵ⋆θ ij (˜µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (38) The first and second terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (38) are the chemical and elastic energy contributions to the bulk grand-potential density of a phase θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Precisely, ωθ chem is defined as Ωθ m/Vm, where Ωθ m is the molar grand-potential and Vm is the molar volume, which is assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, Ωθ m can be analytically calculated by assuming either parabolic or dilute or ideal free energies [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This was the approach taken in our previous study [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, it is difficult to extend this approach to multi-phase and multi- component alloy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, in this work we take a numerical approach to calculate the chemical grand-potential from CALPHAD databases using the method developed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that, similar to our previous study [30], here we have assumed that the stiffness tensor and the eigenstrains are functions of diffusion potentials to account for composition-engendered stresses in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, we derive an expression for the overall molar density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, differ- 21 entiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (34) with respect to the diffusion potential gives cr(φ, ˜µ, ϵθ) = −∂ωbulk ∂µr = p � θ=1 hθ(φ)cθ r � ˜µ, ϵθ� , (39) where cr and cθ r are the overall and phase molar densities of a diffusing component r, and have units of mol/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' More precisely, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (38) the phase molar density may be explicitly written as [30] cθ r � ϵθ, ˜µ � = −∂ωθ bulk ∂µr = Xθ r (˜µ) Vm − 1 2 ∂Cθ ijkl ∂˜µr � ϵθ kl − ϵ⋆θ kl � � ϵθ ij − ϵ⋆θ ij � + ∂ϵ⋆θ ij ∂˜µr σθ ij, (40) where Xθ r (˜µ) = −∂Ωθ m/∂µr [18] is the phase mole fraction of component r in phase θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (40) that if the stiffness tensor and the eigenstrains are assumed to be uniform throughout the system, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (40) simplifies to cθ r (˜µ) = Xθ r (˜µ) Vm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (41) Since we will assume uniform elastic properties in this paper, we will use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (41) to define the phase molar densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the prerequisite phase mole fractions, Xθ r , can be calculated either analytically assuming ei- ther parabolic or dilute or ideal free energies [10], or can be directly obtained from CALPHAD databases [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In this work, we will follow the latter ap- proach since simplistic free energies may cause inaccuracies, particularly for multiphase and multicomponent alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, we derive an expression for 22 the overall stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (34) with respect to total strain, it can be shown that (Appendix C) σij(φ, ˜µ, ϵθ) = ∂ωbulk ∂ϵij = p � θ=1 hθ(φ)σθ ij(˜µ, ϵθ), (42) where σij and σθ ij = ∂ωθ bulk/∂ϵθ ij are the overall and phase elastic stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The latter is defined at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 Governing equations Taking the first variation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (33) and using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (39) and (42) yields: p � θ=1 hθ(φ)cθ k(˜µ) − ck = 0 ∀ k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (n − 1), (43) div � p � θ=1 hθ(φ)σθ ij � = 0, (44) ∂φθ ∂t + Lφ � m∂g (φ) ∂φθ − κ∆φθ + ∂ωbulk ∂φθ � = 0 ∀ θ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' p, (45) where Lφ is the Allen-Cahn mobility and is assumed to be uniform in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that the standard diffusion equations do not nat- urally come out of the variational derivative in the case of grand-potential- based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, to ensure mass conservation, the evolution of overall molar density, ck in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (43), is given by [18] ∂ck(x, t) ∂t − div �n−1 � j=1 Ln kj (˜µ, φ) Vm grad ˜µj � = 0 ∀ k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (n − 1), (46) 23 where the components of the overall Onsager matrix Ln kj (˜µ, φ) are interpo- lated as [18] Ln kj (˜µ, φ) = p � θ=1 hθ (φ) Lnθ kj (˜µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (47) Here, the notation Lnθ kj (˜µ) represents the components of the Onsager matrix specific to a particular phase θ expressed as a function of diffusion potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Again, this term can also be either directly obtained as functions of diffusion potentials from CALPHAD databases [18] or can be assumed to be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that we do not follow grand-potential-based models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', [10], [17], [27, 37, 50–52], that requires formulating a diffusion potential rate equation by first taking the time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (43) and then substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Instead, we calculate the diffusion potential by iteratively solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, this approach requires calculating a Jaco- bian matrix, that can be evaluated by differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (43) with respect to the diffusion potential [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This yields p � θ=1 hθ(φ)χθ jr (˜µ) − ∂cj ∂˜µr = 0, (48) where χθ jr(˜µ) = ∂cθ j/∂µr are the coefficients of the susceptibility matrix ex- pressed as a function of diffusion potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, these coefficients can be determined either by analytical approaches assuming parabolic or dilute or ideal free energies [10] or numerically from CALPHAD databases [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, as previously discussed, the scheme of homogenization may in- fluence the independence of bulk and interfacial properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' More specifically, 24 this independence is achieved provided that the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (45) van- ishes at equilibrium [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, this may not be evident in mechanically coupled alloy phase-field models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Concretely, consider a three-phase system consisting of phases—α, β and γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' then the last term for a specific phase-field variable, say φα, may be explicitly written as (Appendix D) ∂ωbulk ∂φα = − ∂hβ ∂φα �� ωα bulk − ωβ bulk � − � p � θ=1 hθσθ ij � �ϵij�αβ � − ∂hγ ∂φα � (ωα bulk − ωγ bulk) − � p � θ=1 hθσθ ij � �ϵij�αγ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (49) It follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (49) that the terms within the large curly braces depend on the strain jumps, �ϵ�αβ and �ϵ�αγ, which are in turn dependent on the scheme of homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For instance, if Voigt/Taylor or Khacturayan scheme is followed, then the strain jumps vanish and consequently these terms are proportional to the jump in the grand potentials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', �ωbulk�αβ & �ωbulk�αγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, since the bulk grand-potentials are functions of both continuous and discontinuous (total) strain components (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (38)), these terms would not necessarily vanish at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' On the other hand, in the case of the partial rank-one scheme the strain jumps are non-zero and it can be shown that these terms reduce to the sharp interfacial chemical equilibrium conditions for coherently stressed two-phase solids (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='31) in [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For sake of completeness, we have also provided the derivatives with respect to φβ and φγ in Appendix D, which are similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, it must be noted that in writing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (45) we have tacitly assumed that the variational contribution to the driving force is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 25 Precisely, the variational term may be written as: div � ∂ωbulk ∂ (grad φθ) � = div �� p � θ=1 hθσθ ij � ∂ϵθ ij ∂ (grad φθ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (50) Note that due to the dependence of phase strains on the unit vectors: nαβ and nβγ, the term within the curly braces in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (50) is nonzero in case of the rank-one scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, based on our previous study [30], we found that this term does not significantly affect the temporal variation of the interface for cases with a small difference in stiffness tensors [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is because the term is proportional to the magnitude of jump vectors, aαβ and aβγ, and are consequently proportional to the difference in stiffness tensors (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (29) & (30)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, we have neglected this term in our calculations which renders our formulation non-variational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, the Allen-Cahn mobility is calculated using [9] Lφ = 4m/(3κζ), (51) where m and κ are defined at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (36) and the parameter ζ = �n−1 k=1(Xθ,eq k − Xσ,eq k ) �n−1 j=1 � Lnθ,eq kj Vm �−1 (Xθ,eq j −Xσ,eq j ), is obtained assuming infinite inter- face kinetics [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This choice of ζ ensures that local equilibrium is maintained near the interface and the growth is diffusion-controlled [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3 Coupling with CALPHAD databases As discussed before, our model requires thermodynamic properties and mo- bilities as functions of diffusion potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, four properties are 26 needed for any given phase [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' First, the molar grand-potential, Ωθ m, of an individual phase to calculate the chemical contribution to the bulk grand- potential density in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Second, the phase mole fractions to calculate the phase molar densities using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Third, the susceptibility matrix to evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, the Onsager matrix pertaining to each individ- ual phase is also required to evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, for non-dilute and non-ideal solid solutions, these properties cannot be analytically expressed as functions of diffusion potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, we numerically evaluated these prop- erties using the MATLAB-ThermoCalc interface by minimising the prereq- uisite properties with respect to a discretized range of diffusion potential(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This discretized range was predetermined based on the phase diagram [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Concretely, we chose two three-phase alloys: a binary Ni-Al and a ternary Ni-Al-Cr, to illustrate the coupling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For all phases except the bi- nary and ternary B2 phases, the above-mentioned properties were extracted as functions of diffusion potentials from the TCNi8 and MOBNi4 databases using the TC-Toolbox for MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, in the case of Ni-Al, we evaluated the thermodynamic properties and mobilities as discretized func- tions of Al diffusion potential in the interval of [−2e5, 2e5] J/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similarly, for the Al-Cr-Ni simulations, we obtained the discretized properties by vary- ing the Cr and Al diffusion potentials from −1e5 J/mol to 1e5 J/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' These limits were selected to ensure that the Al and Cr mole fractions of an ar- bitrary phase are very close to the limits of 0 and 1 (see Appendix C in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' [18], for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Following this, the properties assigned to a given phase were non-dimensionalized and stored in a tabulated format and then supplied as an input to MOOSE (Multiphysics Object-Oriented Simulation 27 Environment) [55] for phase-field simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 shows the coupling pro- cedure schematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The details of the non-dimenionalization are given in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For the binary and ternary B2 phases, we could obtain only the thermody- namic properties as discretized functions of diffusion potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The Onsager coefficients were assumed to be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, the mobilities were obtained from ThermoCalc at the equilibrium mole fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Following this, these mobilities were used to evaluate the ζ parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (51), which is needed to calculate the Allen-Cahn mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The mobilities, the equilibrium mole fractions, the parameter ζ, and the simulation temperatures are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Schematic showing the coupling procedure between CALPHAD databases and MOOSE in case of a grand-potential-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 28 Table 1 Constant material parameters for the Ni-Al and Al-Cr-Ni alloy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The equi- librium mole fractions and the Onsager mobilities were obtained from ThermoCalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Ni-Al Al-Cr-Ni T [K] 1000 1473 σ [J/m2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 Vm [m3/mol] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5e−5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5e−5 Xα,eq B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='27457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2209 Xβ,eq B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='40646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2912 Xα,eq C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='07575 Xβ,eq C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='06756 Lβ,eq [mol m2/Js] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7534e-17 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2684 � × 1e−17 ζ [Js/m5] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3228e19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8423e18 4 Results and discussion As previously discussed, we have considered two three-phase alloys, an Al- Ni alloy and an Al-Cr-Ni alloy, to demonstrate the application of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, we have considered two interface geometries per alloy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specif- ically, the first two cases assume planar interfaces, while the remaining two cases assume concentric ring interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We have employed both the par- tial rank-one (hereafter referred to as PR) and the Voigt-Taylor (hereafter referred to as VT) homogenization schemes to simulate all four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As noted earlier, this was achieved by controlling the jump in phase strains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', �ϵ�, in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (10)-(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For sake of clarity, Table 2 provides the mechanical boundary conditions and the eigenstrains for each considered case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' From Table 2, we note that the eigenstrains in the binary and ternary γ′ phases are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Although in real alloys, the strength of the eigenstrain depends on the alloy composition, 29 we made this simplifying assumption due to the lack of any experimental data in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the assumed elastic constants for each simulated case are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Except for case II, we have assumed isotropic elastic constants for all considered cases (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, to verify the accuracy of our model, we have compared the simulated elastic fields in each of these cases against the analytically obtained solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The analytical solutions are provided in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Here, it is worth emphasizing that the analytical solutions depend on the interface positions, which have been calculated numerically by tracking the phase-field variables (φθ=α,β,γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Table 2 Summary of eigenstrains and mechanical boundary conditions for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The x- and y-components of displacement u are denoted by ux & uy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Here, lc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='033 µm denotes a characteristic length scale used for non-dimensionalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Simulation Eigenstrains [Phase] Boundary conditions Planar Al-Ni ϵ⋆ [γ′] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3%1 u at left boundary = 0 (Case I) u at right boundary = 0 ϵ⋆ [γ] = ϵ⋆ [B2] = 0 u is periodic along y-direction u at left boundary = 0 Planar Al-Cr-Ni ϵ⋆ [γ′] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3%1 ux/lc at right boundary = 5 (Cases II) uy/lc at right boundary = −5 ϵ⋆ [γ] = ϵ⋆ [B2] = 0 u is periodic along y-direction Non-planar Al-Ni ϵ⋆ [γ′] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3%1 ux at left boundary = 0 (Case III) uy at bottom boundary = 0 ϵ⋆ [γ] = 0 traction is zero at outer boundary Non-planar Al-Cr-Ni ϵ⋆ [γ′] = 0 ux at left boundary = 0 (Case IV) uy at bottom boundary = 0 ϵ⋆ [γ] = ϵ⋆ [B2] = 0 ux at outer boundary = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1%x uy at outer boundary = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1%y 30 Table 3 Summary of elastic constants for all simulated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Here, the left/inner label refers to the leftmost or the innermost phase in the simulations depending on the planar or concentric interface case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Likewise, the right/outer label refers to the rightmost or outermost phase, and the centre label refers to the intermediate phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Simulation Left/Inner Centre Right/Outer Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Case I, E = 158 GPa E = 147 GPa G = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 GPa [56], [57] ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3387 II & III Case II C11 = 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 GPa C11 = 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='37 GPa G = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 GPa [58], [57] C12 = 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='54 GPa C12 = 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='82 GPa ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3387 C44 = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='734 GPa C44 = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='04 GPa 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 Planar three-phase Ni-Al simulation First, we simulated a coherently stressed planar fcc−γ/γ′−Ni3Al/NiAl alloy that is mechanically constrained at the left and right boundaries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We have assumed periodic boundary conditions for the phase field, compo- sition and displacement variables at the top and bottom boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' While homogeneous Neumann boundary conditions are applied at the left and right boundaries for the phase-field and composition variables, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' grad φ · nΓ(x = ±Lx/2, y, t) = 0, (52) grad ˜µAl · nΓ(x = ±Lx/2, y, t) = 0, (53) where ˜µAl is the Al-diffusion potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Lx is the length of the simulation domain, and nΓ is the unit normal at the left and right boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The 31 displacement boundary conditions at these boundaries are (Table 2): ux(x = ±Lx/2, y, t) = 0, (54) uy(x = ±Lx/2, y, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (55) Since the three phases cannot coexist, the intermediate γ′−Ni3Al phase grows at the expense of γ and NiAl phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2b shows the Al mole fraction field at time t = 37 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, we find that the thickness of γ′ phase increases linearly as a function of the square root of simulation time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2c), thus indicating parabolic growth kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This thickness is numerically determined by locating the γ/γ′ and γ′/NiAl interface positions as a func- tion of time using the phase-field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To further test the influence of interface width on kinetics, we vary the interfacial parameters: κ, m and Lφ, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (36) & (51), for three different interface widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We find that the thickness of the Ni3Al phase remains relatively unaltered with varying interface width using both schemes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Expectedly, for both PR and VT schemes, the CPU time decreases with increasing interface width;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' since the grid spacing, ∆x = lw/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0, is directly proportional to interface width lw (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, we find that the PR scheme shows comparatively better convergence compared to the VT scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This shows that the proposed PR scheme is computationally efficient compared to the VT scheme for a longer simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To further verify the spatial accuracy, we sample the spatial variation of the composition field and the elastic quantities across a line normal to the interface at time t = 37 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3a compares the Al mole fraction profile 32 for three different interface widths using the PR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We find that the simulated Al mole fraction profile remains independent of interface width in the bulk regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, we find marginal deviations in the interfacial regions since the composition is interpolated in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We also find this deviation in the x-component of the displacement field near the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, we find that the displacement fields using interface widths of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm are in agreement with the analytically obtained solutions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that the interface positions required in this analytical solution are obtained assuming an interface width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Con- sequently, the simulated solution using an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm shows deviation from this analytical solution near the interfaces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is expected because the analytical solution depends on the accuracy of the numerically determined interface positions (see Appendix F), which slightly depends on interface width (see the thickness variation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To ver- ify this, we re-compare the simulated displacement field having an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm against an analytical solution that uses the interface posi- tions determined from the same simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We then find good quantitative agreement between the two results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that we will obtain similar quantitative agreement between the analytical and simulated elastic fields using the VT scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is because the interface positions as a function of time are relatively independent of the scheme of homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3b, we see that the maximum displacement is at the γ/γ′ and γ′/NiAl interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is because of the assumed eigenstrain in the γ′ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in Appendix F, since the displacement field varies linearly as a function of distance, the total strain and stress normal to the interface are 33 spatially constant within the three phases (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, due to the deviation in the simulated displacement field having an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm from the analytical solution, we find similar disagreement in the total strain and stress normal to the interface from this analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, by comparing this case against the analytical solution having an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm (shown as a dotted magenta coloured line in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3c and 3d), we find good quantitative agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 Planar three-phase Al-Cr-Ni simulation Secondly, we considered a planar ternary Al-Cr-Ni fcc−γ/γ′/B2 alloy having a similar geometry compared to the previous case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the boundary conditions at the top, left, and bottom boundaries are identical to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, the mechanical displacements at the right boundary are ux(x = Lx/2, y, t) = uR x , (56) uy(x = Lx/2, y, t) = uR y , (57) where uR x and uR y are the imposed mechanical displacements (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Unlike the previous case, the three phases, in this case, may coexist in equilibrium because the system is ternary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, the initial conditions are set such that the system is out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, we find that γ′ phase shrinks while the γ and B2 phases grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The simulated Al and Cr mole fraction fields using the PR scheme at time t = 100 s are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4b and 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, we find that the thickness of the ternary γ′ phase decreases 34 (a) 15 µm t = 0 FCC-γ γ′ NiAl Mole fraction Al 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='406 (b) 0 1 2 3 4 5 6 7 8 √ simulation time [√s] 30 40 50 60 70 80 90 Thickness [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm (PR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (PR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm (VT) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (VT) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (VT) (c) 0 10 20 30 40 50 60 Simulation time [s] 0 2 4 6 8 10 12 14 CPU time [hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm (PR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (PR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm (VT) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (VT) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (VT) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For a Ni-Al fcc-γ/Ni3Al−γ′/NiAl coherently stressed planar diffusion cou- ple: a) schematic of the simulation domain, eigenstrains and mechanical boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' b) simulated Al-mole fraction field at time t = 37 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For three different interface widths, the temporal variation in Ni3Al thickness as a function of the square root of simulation time using the partial rank-one (PR) and Voigt-Taylor (VT) homogenization schemes (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and the CPU time as a function of simulation time for both these schemes (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' linearly as a function of the square root of simulation time using the PR scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, we find that this variation remains independent of interface width using the partial rank-one (PR) scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is, however, not true for simulations using the VT scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, we 35 −150 −100 −50 0 50 100 150 Distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='40 Mole fraction Al FCC-γ Ni3Al-γ′ NiAl lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (a) −150 −100 −50 0 50 100 150 Distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 x-component of displacement [µm] lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm Exact for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (b) −150 −100 −50 0 50 100 150 Distance [µm] 3e-03 2e-03 1e-03 0e+00 1e-03 Total strain in x-direction [ϵxx] lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm Exact for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (c) −150 −100 −50 0 50 100 150 Distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 Stress components σxx and σxy [GPa] σxy σxx σxx lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm lw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm Exact for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For the Ni-Al fcc-γ/Ni3Al−γ′/NiAl planar diffusion couple case using the partial rank-one scheme and three different interface widths: a) Al-mole frac- tion profiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' b) x-component of displacement field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' c) total normal strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and d) normal and shear stresses as functions of distance perpendicular to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The superimposed black and magenta dotted lines are the analytically calculated elastic fields using interface width values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' find that as the interface width is increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm, the VT scheme shows deviation from the expected parabolic growth kinetics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Because this behaviour is a consequence of the increase in the interface width, this deviation from the parabolic kinetics may be attributed to the excess interfacial energy contribution arising in the case of the VT scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Surprisingly, we find that the CPU time is higher using both PR and VT 36 schemes for the simulations with interface widths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm compared to cases having interface widths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, we find that the convergence of the PR scheme is significantly faster compared to the VT scheme for interface width values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To verify the spatial accuracy, we calculate the composition and elastic fields along a line parallel to the interface normal at time t = 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We find that the spatial distribution of the simulated Al and Cr mole fraction fields normal to the interface is independent of the interface width (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the simulated x-component of the displacement field normal to the interface shows good quantitative agreement with the analytical solution, in- dependent of the choice of interface width (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Due to the applied mechanical displacement at the right boundary, the y-component of the dis- placement field is also non-zero in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5c shows that the simulated and analytically obtained solutions for the y-component displacement field are also in quantitative agreement in the bulk domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Likewise, this agree- ment is not a function of the interface width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Expectedly, we find that the total strain normal to the interface is constant within the bulk phases and is in agreement with the analytical solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Since the system is elastically anisotropic, the shear strains are non-zero and constant within the bulk phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, the non-zero elastic stresses as a function of distance normal to the interface are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 Non-planar three-phase Ni-Al alloy Thirdly, we simulated a fcc−γ/γ′−Ni3Al/NiAl alloy having concentric ring interfaces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We have assumed homogeneous Neumann boundary conditions along the left, bottom and outer boundaries for the composition and phase-field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, we have imposed symmetry boundary conditions on displacements along the bottom and left boundaries, and zero traction boundary conditions at the outer surface, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' ux(x = 0, y, t) = 0, (58) uy(x, y = 0, t) = 0, (59) σnΓ(x, y, t) = 0 on x2 + y2 = R2 0, (60) where R0 is the radius of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similar to our first case, the three phases cannot coexist in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, we find that the intermediate γ′ phase grows while the innermost γ and outermost NiAl phases shrink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We run this simulation until the γ/γ′ interface vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6a and 6b show the simulated contour map of the Al mole fraction and the radial displacement fields at time t = 1 s for an interface thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm using the PR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The variation in the γ′ thickness as a function of the square root of simulation time using the PR and VT schemes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As expected, we find parabolic growth kinetics using both these schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To check the influence of this result on interface width, we vary the interface width from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We find that the interface kinetics remains unaffected for interface width 38 values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm using both PR and VT schemes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, for the case with interface width value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60 µm, we find that the calculated thickness of Ni3Al is slightly lower in both schemes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, we find that the kinetics is still parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, for a given simulation time, CPU time in the case of the PR scheme is always lower compared to the VT scheme for interface width values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The difference in CPU time is however negligible for an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This suggests that the PR scheme converges at a faster or nearly equal rate compared to the VT scheme for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To verify the spatial accuracy of the simulated solution, we calculated the composition and elastic fields along the radius at time t = 100 s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We find that the radial distribution of the Al mole fraction field within the bulk domains remains independent of interface width for values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Likewise, the radial displacement within the bulk phases remains unaltered with varying interface width (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Also, note that the tangential displacement is negligible within the bulk phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We also find excellent quantitative agreement between the simulated and analytically obtained radial displacement in the bulk γ and Ni3Al phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be emphasized that the analytical solution uses the interface positions calculated from the simulation with an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, for the simulation with an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm, we see that the radial displacement near the Ni3Al/NiAl interface deviates marginally from this analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that a similar observation was made for the planar Ni-Al case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Furthermore, we explained this devia- tion by accounting for the inaccuracy caused by numerically determining the 39 interface positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As discussed before, the analytical solution is sensitive to the calculated interface positions, which are, in turn, dependent on the interface width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further, we have verified this assertion by matching the simulated field for this case with an analytical solution where the interface positions are calculated using the same interface thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We also find quantitative agreement between the simulated and the ana- lytically obtained radial and hoop strains (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7c and 7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As expected, the radial and hoop strains are equal and constant in the bulk γ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, the radial and hoop strains are dependent on the radius in the γ′-Ni3Al and NiAl phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Likewise, for the radial and hoop stress fields, we also obtained a good match between the analytical and simulated fields (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7e and 7f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 Non-planar three-phase Al-Cr-Ni alloy Lastly, we simulated a ternary Al-Cr-Ni fcc−γ/γ′/B2 alloy having concentric interfaces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As listed in Table 2, compared to the previous case, the mechanical displacements at the outer boundary are different in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, ux(x, y, t) = ϵg Rx on x2 + y2 = R2 0, (61) uy(x, y, t) = ϵg Ry on x2 + y2 = R2 0, (62) where ϵg R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1% is the imposed hoop strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For sake of completeness, we note that the boundary conditions at the remaining boundaries are identical to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, we have assumed that the eigenstrains in the bulk phases are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Because of this, VT and Khachaturayan schemes 40 become identical for this special case [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Since there are no eigenstrains, the mechanical stresses are simply due to the imposed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As noted previously for the planar case, the three phases may coexist since the overall alloy composition lies in the three-phase region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, the initial conditions are set such that the γ′ grows at the expense of the ternary γ and B2 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8a shows the spatial variation in the Al-mole fraction field at time t = 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8b, the radial displacement field is symmetric due to the imposed boundary conditions and isotropic elastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8c shows the variation in the thickness of the γ′ phase as a function of the square root of simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Unlike the previous case, we find that the γ′ phase first grows parabolically as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Eventually, its growth slows down as the system reaches towards the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This parabolic growth behaviour of the γ′ phase is due to the fact that the process is diffusion-controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, we find that the accuracy of the temporal variation in the γ′ phase thickness is independent of the interface width choice and the homogenization scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, in contrast to the previous three simulations, we find that the convergence of the VT scheme is marginally faster in this case compared to the PR scheme for two interface width values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We think this is possibly due to the absence of eigenstrains in this simulation compared to all other previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, we find that the PR scheme converges faster compared to the VT scheme only for the case with an interface width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To test the accuracy of our simulations, we calculate the spatial variation 41 of elastic and composition fields along the radial direction based on the PR scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In the bulk phases, we find that the spatial variation in the Al and Cr mole fraction fields along the radius is independent of the choice of interface width (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, our simulated radial displacement field is consistent with the analytically obtained solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We also find that the accuracy remains unaltered for three different interface widths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9b, the tangential displacement is negligible for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9c and 9d show the variation in the total radial and hoop strains as functions of radial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similar to the previous case, notice that the radial and hoop strains in the γ phase are constant and equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, the radial and hoop strains in the γ′ and B2 phases depend on the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, our simulated radial and hoop stresses in the bulk phases are also consistent with the analytical solution (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9e and 9f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, we emphasize that the simulated stress and strain fields are independent of the choice of interface width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5 Conclusions This paper first generalizes the partial rank-one homogenization scheme to multi-phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Subsequently, it implements this scheme for a three- phase system by analytically solving the static compatibility equations, thereby ensuring both static and kinematic compatibilities in the interfacial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Following this, a multi-phase-field grand-potential-based model is formulated using the rank-one scheme for solids undergoing small-strain deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To demonstrate its application for real alloys, a coupling technique is utilized 42 to extract the prerequisite properties directly from CALPHAD databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, we test the model for two three-phase Ni-based alloys having either planar or concentric ring interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We verify the accuracy of the simulated elastic fields against analytical solutions for all simulated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Our results show that the simulation accuracy using the rank-one scheme remains independent of the choice of interface width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Except for one case, we find that the rank-one scheme shows improved or nearly equal convergence compared to the Voigt-Taylor homogenization scheme, which ensures only kinematic compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nevertheless, the current implementation is still limited to linear elastic deformation, and in the future numerical approaches to solving the static compatibility equations, as demonstrated in [35], will be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6 CRediT authorship contribution statement Sourav Chatterjee: Conceptualization, Methodology, Software, Valida- tion, Writing - Original Draft, Writing - Review & Editing, Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Daniel Schwen: Software, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Nele Moelans: Conceptualization, Methodology, Resources, Writing - Review & Editing, Supervision, Funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7 Acknowledgement This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (INTER- 43 DIFFUSION, grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 714754).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The computational resources and services used in this work were provided by the VSC (Flemish Super- computer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government - department EWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 44 (a) 10 µm t = 0 FCC-γ γ′ B2 Mole fraction Al 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='301 (b) 10 µm t = 0 FCC-γ γ′ B2 Mole fraction Cr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='160 (c) 0 10 20 30 40 √ simulation time [√s] 0 5 10 15 20 25 30 35 Thickness [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm (VT) (d) 0 250 500 750 1000 1250 1500 1750 2000 Simulation time [s] 0 10 20 30 40 50 CPU time [hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm (VT) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For an Al-Cr-Ni fcc-γ/γ′/B2 coherently stressed planar diffusion couple: schematic of the simulation domain, eigenstrains and mechanical boundary condi- tions (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' the simulated Al-mole fraction field (b) and Cr-mole fraction field (c) at time t = 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For three different interface widths, the temporal variation in the ternary γ′ thickness as a function of the square root of simulation time using the partial rank-one (PR) and Voigt-Taylor (VT) homogenization schemes (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and change in CPU time with simulation time for both these schemes (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 45 −100 −75 −50 −25 0 25 50 75 100 Distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 Al and Cr mole fractions [xAl and xCr] xCr xAl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm (a) −100 −50 0 50 100 Distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='175 x-component of displacement [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm Analytical (b) −100 −50 0 50 100 Distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='175 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='150 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='125 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='075 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000 y-component of displacement [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm Analytical (uα y) Analytical (uβ y) Analytical (uγ y) −75 −50 −25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='02 (c) −100 −50 0 50 100 Distance [µm] 3e-03 2e-03 1e-03 0e+00 1e-03 2e-03 Total strain in x-direction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm Analytical (d) −100 −50 0 50 100 Distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000430 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000425 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000420 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000415 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000410 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000405 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000400 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000395 Shear strain [ϵxy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm Analytical (e) −100 −75 −50 −25 0 25 50 75 100 Distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 Stress components [σxx, σxy and σyy] σxx σxy σyy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6 µm Analytical (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For an Al-Cr-Ni fcc-γ/γ′/B2 coherently stressed planar diffusion couple, the spatial distribution of Al and Cr-mole fraction profiles (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' x and y-components of displacement field (b) and (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' total normal and shear strain (d) and (e);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' normal and shear stresses (f), as functions of distance perpendicular to the interface at time t = 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Simulations using different interface widths are also superimposed on these figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The superimposed dotted black lines indicate the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For sake of clarity, the analytical solution for the y-component of displacement field within the bulk regions are denoted by different colours in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 µm FCC-γ γ′ NiAl-B2 Mole fraction Al ux = 0 uy = 0 t=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='381 (a) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 µm [µm] Radial displacement at t = 1 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0000 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='75 √ simulation time [√s] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 Thickness [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60 µm (VT) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 Simulation time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 CPU time [hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60 µm (VT) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For a non-planar Ni-Al fcc-γ/Ni3Al/NiAl coherently stressed diffusion couple: simulation domain, eigenstrains, mechanical boundary conditions and the simulated Al-mole fraction field at time t = 1 s (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' the simulated radial displace- ment field at the same time (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For three different interface widths, variation in Ni3Al thickness as a function of the square root of simulation time using the partial rank-one (PR) and Voigt-Taylor (VT) homogenization schemes (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' the CPU time as a function of simulation time for both these schemes (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 47 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='40 Mole fraction Al lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (a) 0 5 10 15 20 25 30 Radial distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='040 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='030 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='025 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='020 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000 Radial and tangential displacements [µm] ut lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (b) 0 5 10 15 20 25 30 Radial distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='002 Radial strain lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (c) 0 5 10 15 20 25 30 Radial distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00200 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00175 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00150 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00125 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00075 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00025 Hoop strain lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (d) 0 5 10 15 20 25 30 Radial distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 Radial stress [GPa] lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (e) 0 5 10 15 20 25 30 Radial distance [µm] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 Hoop stress [GPa] lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For a non-planar Ni-Al fcc-γ/Ni3Al/NiAl diffusion couple, the spatial variation in a) Al-mole fraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' b) radial and tangential displacements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' c) radial strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' d) hoop strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' e) radial stress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and f) hoop stress as functions of radial distance at time t = 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The plots show this variation for three different interface widths lw using the partial rank-one scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The dotted and discontinuous black lines are the analytically obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 µm FCC-γ γ′ B2 Mole fraction Al ux = 0 uy = 0 t=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='282 (a) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 µm [µm] Radial displacement at t = 100 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0284 (b) 0 5 10 15 20 25 30 35 40 √ simulation time [√s] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5 Thickness [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (VT) (c) 0 200 400 600 800 1000 Simulation time [s] 0 5 10 15 20 CPU time [hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (PR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (VT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (VT) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For a non-planar Al-Cr-Ni fcc-γ/γ′/B2 coherently stressed diffusion cou- ple: simulation domain, mechanical boundary conditions and the simulated Al- mole fraction field at time t = 100 s (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' the simulated radial displacement field at the same time (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For three different interface widths, variation in γ′ thickness as a function of the square root of simulation time using the partial rank-one (PR) and Voigt-Taylor (VT) homogenization schemes (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' the CPU time as a function of simulation time for both these schemes (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 49 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 Al and Cr mole fractions [xAl and xCr] xCr xAl lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm (a) 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='030 Radial and tangential displacements [µm] ut lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (b) 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='0014 Radial strain lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (c) 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='00125 Hoop strain lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (d) 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='410 Radial stress [GPa] lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (e) 0 5 10 15 20 25 30 Radial distance [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='48 Hoop stress [GPa] lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm lw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30 µm Exact for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15 µm (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For a non-planar Al-Cr-Ni fcc-γ/γ′/B2 diffusion couple, the spatial vari- ation in a) Al and Cr mole fraction fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' b) radial and tangential displacements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' c) radial strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' d) hoop strain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' e) radial stress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and f) hoop stress as functions of radial distance at time t = 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The plots show this variation for three different interface widths lw using the partial rank-one scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The dotted and discontinuous black lines are the analytically obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 50 Appendix A Calculation of phase strains In this section, we provide an analytical approach to calculate the phase strains for a p-phase system as functions of the total strain ϵ(u), interpola- tions functions h(φ) and strain jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To this end, we first begin by deriving the phase strains for two-phase and three-phase systems and then extend it to multi-phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 Phase strains for a two-phase system For a two-phase α/β system, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (1) and (4) reduces to ϵ(u) = ϵαhα + ϵβhβ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) ϵα − ϵβ = �ϵ�αβ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) Multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) with hβ and then adding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) yields ϵα (hα + hβ) = ϵ + hβ�ϵ�αβ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) Since for a two-phase system hα + hβ = 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) simplifies to ϵα = ϵ + hβ�ϵ�αβ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) 51 Since hα = 1 − hβ, it follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) that ϵβ = ϵ − hα�ϵ�αβ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) We simply note that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) are completely equivalent to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, we attempt to use the two-phase relations to extend the model to three-phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 Phase strains for a three-phase system For a system consisting of three phases, say α, β & γ, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (1) and (4) reduces to ϵ = ϵαhα + ϵβhβ + ϵγhγ, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6) ϵα − ϵβ = �ϵ�αβ, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) ϵβ − ϵγ = �ϵ�βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) By defining ϵ′ = ϵ − ϵγhγ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6) may be written as ϵ′ = ϵαhα + ϵβhβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) Notice that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) are similar to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Be- cause of this similarity, we can directly use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) to write ϵα (hα + hβ) = ϵ′ + hβ�ϵ�αβ = ϵ − ϵγhγ + hβ�ϵ�αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) 52 It should be noticed from the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) that (hα + hβ) ̸= 1 since this is a three-phase system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) & (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) and substituting: ϵγ = ϵα − �ϵ�αβ − �ϵ�βγ, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) gives ϵα(hα + hβ + hγ) = ϵ + hβ�ϵ�αβ + hγ � �ϵ�αβ + �ϵ�βγ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='11) Now, since (hα + hβ + hγ) = 1 for a three-phase system, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='11) reduces to ϵα = ϵ + hβ�ϵ�αβ + hγ � �ϵ�αβ + �ϵ�βγ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) Next, substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) and then using (1−hβ −hγ) = hα gives ϵβ = ϵ − hα�ϵ�αβ + hγ�ϵ�βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) Finally, substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) yields ϵγ = ϵ − hα�ϵ�αβ − (hα + hβ) �ϵ�βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='14) Thus, we have obtained the phase strains as functions of the total strain, interpolation functions and strain jumps for a three-phase system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We again note that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) & (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='14) are completely equivalent to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) & (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 53 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 Phase strains for a multi-phase system Based on the previous two derivations, it is worth noting that once the phase strain pertaining to a particular phase, say α, is determined, the phase strains of the remaining (p−1) phases may be obtained using the (p−1) compatibility equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For sake of concretness, if phase strain pertaining to α-phase is known, then the phase strains in β, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' , (p − 1), p phases are ϵβ = ϵα − �ϵ�αβ, ϵγ = ϵβ − �ϵ�βγ, ϵδ = ϵγ − �ϵ�γδ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' ϵp = ϵp−1 − �ϵ�(p−1),p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) Therefore, if an analytical expression for the α-phase strain in a system consisting of p phases is known, all remaining phase strains can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It can be observed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) & (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) that the α-phase strain for a three-phase system differs from a two-phase system by just one term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, this term is equal to the product of the interpolation function associated with the new phase and the sum of all jump vectors in that system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', hγ � �ϵ�αβ + �ϵ�βγ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, ϵα for a multi-phase system may be written as ϵα = ϵ(u) + hβ�ϵ�αβ + hγ � �ϵ�αβ + �ϵ�βγ� + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + hp � � (p−1),p � i=αβ �ϵ�i � � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='16) 54 where (p−1),p � i=αβ �ϵ�i = �ϵ�αβ + �ϵ�βγ + �ϵ�γδ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + �ϵ�(p−1),p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='17) By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='16) in the first of the set of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) and using the relation 1 − (hβ + hγ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + hp) = hα, it follows that ϵβ = ϵ(u) − hα�ϵ�αβ + hγ�ϵ�βγ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + hp � �ϵ�βγ + �ϵ�γδ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + �ϵ�(p−1),p� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='18) Similarly, by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='18) in the second of the set of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) and using 1 − (hγ + hδ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + hp) = (hα + hβ), it follows that ϵγ = ϵ(u) − hα�ϵ�αβ − (hα + hβ) �ϵ�βγ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + hp � �ϵ�γδ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' + �ϵ�(p−1),p� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='19) Thus, by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='16) we can calculate the phase strains for an arbitrary multi-phase system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 55 Appendix B Some useful relations B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 Derivatives with respect to phase-field variables Since hα + hβ + hγ = 1, it follows that ∂hα ∂φα = − �∂hβ ∂φα + ∂hγ ∂φα � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) ∂hγ ∂φα = − �∂hβ ∂φα + ∂hα ∂φα � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) Differentiating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (10), (11) and (12) with respect to φα yields ∂ϵα ij ∂φα = �∂hβ ∂φα + ∂hγ ∂∂φα � �ϵij�αβ + (hβ + hγ)∂�ϵij�αβ ∂φα + ∂hγ ∂φα �ϵij�βγ + hγ ∂�ϵij�βγ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) ∂ϵβ ij ∂φα = −∂hα ∂φα �ϵij�αβ − hα ∂�ϵij�αβ ∂φα + ∂hγ ∂φα �ϵij�β γ + hγ ∂�ϵij�βγ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) ∂ϵγ ij ∂φα = −∂hα ∂φα �ϵij�αβ − hα ∂�ϵij�αβ ∂φα − �∂hβ ∂φα + ∂hα ∂φα � �ϵij�βγ − (hβ + hα)∂�ϵij�βγ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) Multiplying Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) with hασα ij, hβσβ ij and hγσγ ij, respec- tively, and then setting the terms premultiplied by hγhα to zero, and using 56 Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) yields hασα ij ∂ϵα ij ∂φα = −hα ∂hα ∂φα σα ij�ϵij�α β + hαhβσα ij ∂�ϵij�α β ∂φα + hα ∂hγ ∂φα σα ij�ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6) hβσβ ij ∂ϵβ ij ∂∂φα = −hβ ∂hα ∂φα σβ ij�ϵij�α β − hαhβσβ ij ∂�ϵij�α β ∂φα + hβ ∂hγ ∂φα σβ ij�ϵij�β γ + hβhγσβ ij ∂�ϵij�β γ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) hγσγ ij ∂ϵγ ij ∂φα = −hγ ∂hα ∂φα σγ ij�ϵij�α β + hγ ∂hγ ∂φα σγ ij�ϵij�β γ − hγhβσγ ij ∂�ϵij�β γ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) Adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) and using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (13) and (14) yields γ � θ=α hθ(φ)σθ ij ∂ϵθ ij ∂φα = hαhβ � σα ij − σβ ij � nαβ j ∂aαβ i ∂φα + hβhγ � σβ ij − σγ ij � nβγ j ∂aβγ i ∂φα − ∂hα ∂φα �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φα � γ � θ=α hθσθ ij � �ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) It follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (15) and (16), that the first two terms on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) reduces to γ � θ=α hθ(φ)σθ ij ∂ϵθ ij ∂φα = −∂hα ∂φα �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φα � γ � θ=α hθσθ ij � �ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) 57 Following a similar procedure, it can be shown that γ � θ=α hθ(φ)σθ ij ∂ϵθ ij ∂φβ = −∂hα ∂φβ �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φβ � γ � θ=α hθσθ ij � �ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='11) γ � θ=α hθ(φ)σθ ij ∂ϵθ ij ∂φγ = −∂hα ∂φγ �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φγ � γ � θ=α hθσθ ij � �ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) Now, we obtain another similar relation by multiplying Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) with hαCα klij, hβCβ klij and hγCγ klij, respectively, and setting the terms premultiplied by hγhα to zero yields hαCα klij ∂ϵα ij ∂φα = hα �∂hβ ∂φα + ∂hγ ∂φα � Cα klij�ϵij�α β + hαhβCα klij ∂�ϵij�α β ∂φα + hα ∂hγ ∂φα Cα klij�ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) hβCβ klij ∂ϵβ ij ∂φα = −hβ ∂hα ∂φα Cβ klij�ϵij�α β − hβhαCβ klij ∂�ϵij�α β ∂φα + hβ ∂hγ ∂φα Cβ klij�ϵij�β γ + hβhγCβ klij ∂�ϵij�β γ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='14) hγCγ klij ∂ϵγ ij ∂φα = −hγ ∂hα ∂φα Cγ klij�ϵij�α β − hγ �∂hβ ∂φα + ∂hα ∂φα � Cγ klij�ϵij�β γ − hβhγCγ klij ∂�ϵij�β γ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) 58 Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) gives hαCα klij ∂ϵα ij ∂φα = −hα ∂hα ∂φα Cα klij�ϵij�α β + hαhβCα klij ∂�ϵij�α β ∂φα + hα ∂hγ ∂φα Cα klij�ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='16) hβCβ klij ∂ϵβ ij ∂φα = −hβ ∂hα ∂φα Cβ klij�ϵij�α β − hβhαCβ klij ∂�ϵij�α β ∂φα + hβ ∂hγ ∂φα Cβ klij�ϵij�β γ + hβhγCβ klij ∂�ϵij�β γ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='17) hγCγ klij ∂ϵγ ij ∂φα = −hγ ∂hα ∂φα Cγ klij�ϵij�α β + hγ ∂hγ ∂φα Cγ klij�ϵij�β γ − hβhγCγ klij ∂�ϵij�β γ ∂φα (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='18) Adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='16), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='17), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='18) yields γ � θ=α hθCθ klij ∂ϵθ ij ∂φα = hαhβ � Cα klij − Cβ klij � ∂�ϵij�α β ∂φα + hβhγ � Cβ klij − Cγ klij � ∂�ϵij�β γ ∂φα −∂hα ∂φα � γ � θ=α hθCθ klij � �ϵij�α β + ∂hγ ∂φα � γ � θ=α hθCθ klij � �ϵij�β γ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='19) By replacing ∂φα with ∂φβ and ∂φγ with ∂φγ equivalent expressions for φβ and φγ can be easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 59 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 Derivatives with respect to total strain Differentiating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (10), (11) and (12) with respect to total strain ϵ and then multiplying with hασα ij, hβσβ ij and hγσγ ij, respectively, yields hασα ij ∂ϵα ij ∂ϵmn = hασα mn + hα (hβ + hγ) σα ij ∂�ϵij�α β ∂ϵmn + hγhασα ij ∂�ϵij�β γ ∂ϵmn (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20) hβσβ ij ∂ϵβ ij ∂ϵmn = hβσβ mn − hαhβσβ ij ∂�ϵij�α β ∂ϵmn + hγhβσβ ij ∂�ϵij�β γ ∂ϵmn (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='21) hγσγ ij ∂ϵγ ij ∂ϵmn = hγσγ mn − hαhγσγ ij ∂�ϵij�α β ∂ϵmn − hγ (hβ + hα) σγ ij ∂�ϵij�β γ ∂ϵmn (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='22) Now, we note that hγhα is non-zero only near the γ/α interface boundary and the terms �ϵ�α β and �ϵ�β γ are also non-zero only within the interfacial regions of β/γ and α/β boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We therefore set all terms premultiplied by hγhα to zero in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='21) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Now adding these equations yields γ � θ=α hθσθ ij ∂ϵθ ij ∂ϵmn = γ � θ=α hθσθ mn + hαhβ � σα ij − σβ ij � nαβ j ∂aαβ i ∂ϵmn + hγhβ � σβ ij − σγ ij � nβγ j ∂aβγ i ∂ϵmn (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='23) Due to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (15) and (16), the last two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='23) must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This gives γ � θ=α hθσθ ij ∂ϵθ ij ∂ϵmn = γ � θ=α hθσθ mn (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='24) Next, differentiating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (10), (11) and (12) with respect to total strain ϵ and multiplying with hαCα ijkl, hβCβ mnkl and hγCγ mnkl, yields 60 hαCα mnkl ∂ϵα kl ∂ϵrs = hαCα mnrs + hα (hβ + hγ) Cα mnkl ∂�ϵkl�α β ∂ϵrs + hγhαCα mnkl ∂�ϵkl�β γ ∂ϵrs (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25) hβCβ mnkl ∂ϵβ kl ∂ϵrs = hβCβ mnrs − hαhβCβ mnkl ∂�ϵkl�α β ∂ϵrs + hγhβCβ mnkl ∂�ϵkl�β γ ∂ϵrs (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='26) hγCγ mnkl ∂ϵγ kl ∂ϵrs = hγCγ mnrs − hαhγCγ mnkl ∂�ϵkl�α β ∂ϵrs − hγ (hβ + hα) Cγ mnkl ∂�ϵkl�β γ ∂ϵrs (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='27) Again, we set the four terms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='26) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='27) which are premultiplied by hγhα to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next adding these equations, we see that Jmnrs = γ � θ=α hθ(φ)Cθ mnkl ∂ϵθ kl ∂ϵrs = γ � θ=α hθ(φ)Cθ mnkl + hαhβ � Cα mnkl − Cβ mnkl � ∂�ϵkl�α β ∂ϵrs + hγhβ � Cβ mnkl − Cγ mnkl � ∂�ϵkl�β γ ∂ϵrs (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='28) Appendix C Derivation of stress and its derivatives Differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (34) with respect to total strain yields ∂ωbulk ∂ϵmn = γ � θ=α hθ(φ)∂ωθ bulk ∂ϵθ ij ∂ϵθ ij ∂ϵmn (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) 61 Using the definition of the phase stress tensor, we can replace ∂ωθ bulk/∂ϵij with σθ ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Using the relation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='24) , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) can be written as ∂ωbulk ∂ϵij = γ � θ=α hθσθ ij ∂ϵθ ij ∂ϵmn = γ � θ=α hθ(φ)σθ mn (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) Appendix D Derivation of driving force and its derivatives Differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (34) with respect to phase-field variable φθ yields ∂ωbulk ∂φθ = γ � σ=α ∂hσ ∂φθ ωσ + γ � σ=α hσ(φ)∂ωσ ∂ϵσ ij ∂ϵσ ij ∂φθ = γ � σ=α ∂hσ ∂φθ ωσ + γ � σ=α hσ(φ)σσ ij ∂ϵσ ij ∂φθ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) For θ = α, substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) yields ∂ωbulk ∂φα = ∂hβ ∂φα � ωβ − ωα� + ∂hγ ∂φα (ωγ − ωα) − ∂hα ∂φα �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φα � γ � θ=α hθσθ ij � �ϵij�β γ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) 62 Similarly, one can derive the bulk driving force for phase-field variables φβ and φγ by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='11) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) ∂ωbulk ∂φβ = ∂hα ∂φβ � ωα − ωβ� + ∂hγ ∂φβ � ωγ − ωβ� − ∂hα ∂φβ �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φβ � γ � θ=α hθσθ ij � �ϵij�β γ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) ∂ωbulk ∂φγ = ∂hα ∂φγ (ωα − ωγ) + ∂hβ ∂φγ � ωβ − ωγ� − ∂hα ∂φγ �� θ=α hθσθ ij � �ϵij�α β + ∂hγ ∂φγ � γ � θ=α hθσθ ij � �ϵij�β γ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) Appendix E Non-dimensionalization Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (43)-(46) were solved in the MOOSE (Multiphysics Object-Oriented Simulation Environment) finite-element framework [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To ensure good convergence, we formed a non-dimensional form of these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In this section, we provide the dimensionless form of the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' We will denote dimensionless quantities using the symbol, (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Let lc and tc denote characteristic length and time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Then, the non-dimensional position and time may be written as: x = x/lc and t = t/tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The dimen- sionless form of the displacement field is defined as: u = u/lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Similarly, we define the dimensionless form of the set of diffusion potentials as: ˜µ = ˜µ/RT, where R is gas constant and T is simulation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' After change of 63 variables and using the relation ck = Xk/Vm, it can be shown that the di- mensionless form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (43)-(46) may be written as p � θ=1 hθ(φ)Xθ k(˜µ) − Xk = 0, ∀ k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (n − 1), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) div � p � θ=1 hθ(φ)σθ ij � = 0, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) ∂φθ ∂t + Lφ �∂g (φ) ∂φθ − κ∆φθ + λ1 ∂ωchem ∂φθ + λ2 ∂ωmech ∂φθ � = 0 ∀ θ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' p, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) ∂Xk ∂t − ∇ �n−1 � j=1 Ln kj � ˜µ,φ � ∇˜µj � x, t � � = 0 ∀ k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (n − 1), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) 64 where σ = σ/µel, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) Lφ = tcLφm, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6) κ = κ/ � l2 cm � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) λ1 = RT/(mVm), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) λ2 = µel/m, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) Ln kj = Ln kjtcRT/l2 c, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) ∂ωchem ∂φθ = � 1 RT � � p � σ=1 ∂hσ ∂φθ ωσ chem � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='11) ∂ωmech ∂φθ = � 1 µel � � p � σ=1 ∂hσ ∂φθ ωσ elastic + p � σ=1 hσ ∂ωσ elastic ∂φθ � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) ωθ elastic = (1/2)Cθ ijkl � ϵθ kl − ϵ⋆θ kl � � ϵθ ij − ϵ⋆θ ij � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) Appendix F Analytical solutions To test the simulation accuracy, we have compared our simulated results with analytically obtained solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' However, it bears emphasis that these analytical solutions require prior knowledge of the domain size, and thus of the interface positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Therefore, we have first performed numerical sim- ulations to calculate the position of these interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Once these positions 65 were calculated, they were used as input in the analytical solutions to make comparisons with simulated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, unless stated otherwise, we have assumed zero flux boundary conditions at all boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For the two planar simulations, in order to compare with analytical solutions we have taken all fields to be periodic in the top and bottom boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, to reduce the computational costs by taking advantage of the domain sym- metry, we have used symmetry boundary conditions at the left and bottom boundaries for two non-planar simulations (see cases III and IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In this section, we provide the analytical solutions to the four set of three- phase simulations performed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It must be emphasized that to analytically solve the mechanical equilibrium equations, the instantaneous positions of the two two-phase interphases are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' These prerequisite positions are therefore numerically obtained based on the phase-field results and then compared against analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 Solution for the planar Ni-Al case Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1 shows the system geometry and boundary conditions for the planar Ni-Al case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For the sake of generality, we will refer to the leftmost (fcc−γ), center (Ni3Al-γ′) and rightmost (NiAl) phases as α, β and γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Let x1(t) and x2(t) represent the positions of the α/β and β/γ interphases at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As mentioned before, we numerically determine these positions at any given instant from the phase-field simulations and thus the accuracy of the solution depends on the interface positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, assuming plane stress conditions and neglecting externally applied body forces, the mechan- 66 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A schematic showing the phases, eigenstrains and mechanical boundary conditions for the planar Ni-Al case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' ical equilibrium equations in Cartesian frame within the bulk regions of a phase θ = {α, β, γ} reduces to: ∂σθ x ∂x + ∂σθ xy ∂y = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) ∂σθ xy ∂y + ∂σθ yy ∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2) As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1, we have assumed that origin of the Cartesian frame lies at the center of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in Table 3, we have assumed the elastic constants to be isotropic but spatially heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, the stress-strain relation within the bulk phases may be written as [59]: σθ ij = λθδij � ϵθ kk − ϵ⋆,θ kk � + 2µθ(ϵθ ij − ϵ⋆,θ ij ), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) where λθ, µθ and ϵ⋆,θ are Lame’s constant, shear modulus and eigenstrain of phase θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1, the mechanical displacements at 67 the left and right boundaries may be written as: ux(x = ±Lx/2, y, t) = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) uy(x = ±Lx/2, y, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) On the other hand, the mechanical displacements are assumed to be peri- odic along the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Due to these boundary conditions, only the x- component of displacement, ux(x, t), and the normal strain along x-direction, ϵθ x = duθ x/dx, are nonzero in the bulk regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3), it follows that the nonzero mechanical stresses within the bulk phases are: σx(x, t) = � � � � � � � � � � � � � � � (λα + 2µα) ϵα x −Lx/2 < x < x1, � λβ + 2µβ� ϵβ x − 2(λβ + µβ)ϵ⋆ x1 < x < x2, (λγ + 2µγ) ϵγ x x2 < x < Lx/2, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6) σy(x, t) = � � � � � � � � � � � � � � � λαϵα x −Lx/2 < x < x1, λβϵβ x − 2(λβ + µβ)ϵ⋆ x1 < x < x2, λγϵγ x x2 < x < Lx/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) It should be noticed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='7) that the β stress components are different compared to the α (FCC) and γ (NiAl) phases because we have assumed a two-dimensional eigenstrain ϵ⋆ = ( ϵ⋆ 0 0 ϵ⋆ ) only within the Ni3Al phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='6) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='1) and using the strain- 68 displacement relations, we see that � λθ + 2µθ� d2uθ x dx2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8) By integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='8), it follows that the x-component of displacement field must vary linearly with distance within the bulk phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' More precisely, ux(x, t) = � � � � � � � � � � � � � � � Aαx + Bα −Lx/2 < x < x1, Aβx + Bβ x1 < x < x2, Aγx + Bγ x2 < x ≤ Lx/2, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) where Aα, Bα, Aβ, Bβ, Aγ and Bγ are unknown constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, these six constants can be determined using the two imposed boundary conditions (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 & F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) and four interfacial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Two of these interfacial conditions arise due to the continuity of x-component of displacement at the two interfaces, �ux� = 0, and the remaining two are a result of continuity of normal stresses along x, �σx� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, uα x|x1 = uβ x �� x1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10) uβ x �� x2 = uγ x|x2 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='11) σα x|x1 = σβ x �� x1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='12) σβ x �� x2 = σγ x|x2 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) 69 Next, substituting the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='5) yields −AαLx/2 + Bα = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='14) AγLx/2 + Bγ = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='15) Then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='9), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13) may be written as Aαx1 + Bα − � Aβx1 + Bβ� = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='16) Aβx2 + Bβ − (Aγx2 + Bγ) = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='17) (λα + 2µα)Aα − (λβ + 2µβ)Aβ + 2 � λβ + µβ� ϵ⋆ = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='18) (λβ + 2µβ)Aβ − 2 � λβ + µβ� ϵ⋆ − (λγ + 2µγ)Aγ = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='19) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='14)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='19) form a set of six equations that can be solved to determine the six unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This was performed using the Python library for symbolic mathematics, SymPy [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A python script for solving these equations is available (see the python script threephase planar analytical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='py).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 Solution for the planar Ni-Al-Cr case Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2 shows the system geometry and boundary condition for the planar Ni-Al-Cr case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In contrast to the previous case, the leftmost (fcc-γ) and center (γ′) phases are elastically anisotropic (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, the stress-strain rela- 70 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A schematic showing the phases, eigenstrains and mechanical boundary conditions for the planar Ni-Al-Cr case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' tions within these phases may be written as [59]: σθ ij = λθδij � ϵθ kk − ϵ⋆,θ kk � + 2µθ(ϵθ ij − ϵ⋆,θ ij ) + µ′θδijkl � ϵθ ij − ϵ⋆,θ ij � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20) where λθ = Cθ 12, µθ = Cθ 44, µ′θ = Cθ 11 − Cθ 12 − 2Cθ 44 and δijkl is zero except for δ1111 = δ2222 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='2, the imposed mechanical boundary conditions at the left and right boundaries yields: ux(x = −Lx/2, y, t) = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='21) uy(x = −Lx/2, y, t) = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='22) ux(x = Lx/2, y, t) = uR x , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='23) uy(x = Lx/2, y, t) = uR y , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='24) where uR x and uR y are the x and y components of the imposed mechanical displacement at the right boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, unlike the previous case, both x and y components of the displacement field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', ux(x, t) and uy(x, t), 71 are nonzero within the bulk phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Precisely, ux(x, t) = � � � � � � � � � � � � � � � Aα xx + Bα x −Lx/2 < x < x1, Aβ xx + Bβ x x1 < x < x2, Aγ xx + Bγ x x2 < x ≤ Lx/2, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25) uy(x, t) = � � � � � � � � � � � � � � � Aα yx + Bα y −Lx/2 < x < x1, Aβ yx + Bβ y x1 < x < x2, Aγ yx + Bγ y x2 < x ≤ Lx/2, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='26) where {Aθ=α,β,γ x }, {Aθ=α,β,γ y }, {Bθ=α,β,γ x } and {Bθ=α,β,γ y } are the 12 unknown constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, to determine the unknown constants, we first solve the x-component of displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This requires calculating the six unknowns: {Aθ=α,β,γ x } and {Bθ=α,β,γ x }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' After substituting the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='21) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='23), we get −Aα xLx/2 + Bα x = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='27) uR x − (Aγ xLx/2 + Bγ x) = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='28) The remaining four unknowns can be determined by solving continuity of x-component of displacement field and normal stress along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, using 72 Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='25) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='10)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='13), it follows that: Aα xx1 + Bα x − � Aβ xx1 + Bβ x � = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='29) Aβ xx2 + Bβ x − (Aγ xx2 + Bγ x) = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='30) (λα + 2µα)Aα x − (λβ + 2µβ)Aβ x + µ′αAα x − µ′βAβ x + ζβϵ⋆ = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='31) (λβ + 2µβ)Aβ x − (λγ + 2µγ)Aγ x + µ′βAβ x − µ′γAγ x − ζβϵ⋆ = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='32) where λθ=α,β = Cθ 12, µθ=α,β = Cθ 44, µ′θ=α,β = Cθ 11 − Cθ 12 − 2Cθ 44 and ζβ = 2(λβ + µβ) + µ′β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' By solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='27)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='32) we can obtain six of the 12 unknown constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This is achieved symbolically using SymPy [60] and the python script, threephase aniso planar analytical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='py, is provided with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Following this, the remaining six constants can be obtained by solving the y-component of displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, we need another set of six equations to determine the unknown constants: {Aθ=α,β,γ y } and {Bθ=α,β,γ y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To this end, substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='26) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='22) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='24) yields the first two of these equations: −Aα yLx/2 + Bα y = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='33) uR y − � Aγ yLx/2 + Bγ y � = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='34) Since the y-component of displacement field must be continuous at the 73 two interfaces, it follows that uα y �� x1 = uβ y �� x1 =⇒ Aα yx1 + Bα y − � Aβ yx1 + Bβ y � = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='35) uβ y �� x2 = uγ y �� x2 =⇒ Aβ yx2 + Bβ y − � Aγ yx2 + Bγ y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='36) Additionally, the shear stress must be continuous at the two interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This yields σα xy �� x1 = σβ xy �� x1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='37) σβ xy �� x2 = σγ xy �� x2 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='38) Using constitutive Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='20) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='37) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='38) yields: 2µαAα y − 2µβAβ y = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='39) 2µβAα y − 2µγAγ y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='40) Thus, by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='33)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='40) the remaining six unknowns: {Aθ=α,β,γ y } and {Bθ=α,β,γ y } can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' These equations were also solved sym- bolically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The python script, threephase aniso shear components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='py, is provided with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 Solution for the non-planar Ni-Al case Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3 shows the system geometry and boundary conditions for the three- phase Ni-Al case with concentric interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3, the innermost (fcc-γ), center (Ni3Al) and outermost (NiAl) phases are hereafter 74 referred to as α, β and γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, due to the concentric ring geometry of the system, we analytically solve the mechanical equilibrium equations in polar coordinates, (r, φ), even though the simulation was per- formed in a Cartesian frame, (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' It should be noted that to compare the analytically obtained solution against the simulated solution we transform the simulated elastic fields from the Cartesian frame to polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For instance, the displacement field in polar coordinates may be calculated Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A schematic showing the phases, eigenstrains and mechanical boundary conditions for the concentric interface Ni-Al case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' from Cartesian frame using � � � � � ur ut � � � � � = � �� cos ζ sin ζ − sin ζ cos ζ � �� � � � � � ux uy � � � � � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='41) where ζ = tan−1(y/x) is the angle of rotation between the two frames (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' For this case, the displacement field within the bulk phase θ takes the 75 form uθ(r, t) = uθ r(r, t)er + uθ φ(r, t)eφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='42) As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3, due to the imposed boundary conditions, the radial displacement is zero at the origin and the radial stress at the outer surface is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This yields uα r (r = 0, t) = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='43) σγ r (r = R, t) = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='44) Note that the superscripts on the mechanical fields identify the phases in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Because of these imposed boundary conditions, it can be assumed that the φ component of displacement field is zero throughout the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', uα φ = uβ φ = uγ φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Consequently, the strain-displacement relation in polar coordinates within the bulk domains simplifies to ϵθ r(r) = duθ r(r)/dr, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='45) ϵθ φ(r) = uθ r(r)/r, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='46) ϵθ rφ(r) = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='47) Further, assuming plane stress conditions, the mechanical equilibrium equa- tions within the bulk domains in polar coordinates simplifies to ∂σθ r ∂r + σθ r − σθ φ r = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='48) 76 Because we have assumed isotropic elastic properties, it follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='45)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='47) that the nonzero stresses in polar coordinates are σr(r, t) = � � � � � � � � � � � � � � � (λα + 2µα) ϵα r + λαϵα φ 0 < r < r1, � λβ + 2µβ� ϵβ r + λβϵβ φ − 2(λβ + µβ)ϵ⋆ r1 < r < r2, (λγ + 2µγ) ϵγ r + λγϵγ φ r2 < r < R, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='49) σφ(r, t) = � � � � � � � � � � � � � � � (λα + 2µα) ϵα φ + λαϵα r 0 < r < r1, � λβ + 2µβ� ϵβ φ + λβϵβ r − 2(λβ + µβ)ϵ⋆ r1 < r < r2, (λγ + 2µγ) ϵγ φ + λγϵγ r r2 < r < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='50) Here r1(t) and r2(t) represent the numerically obtained interface positions at the α/β and β/γ interfaces at time t (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Next, by substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='49) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='50) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='48) it can be shown that in a bulk phase θ the mechanical equilibrium equation reduces to � λθ + 2µθ� �d2uθ r dr2 + 1 r duθ r dr − uθ r r2 � = 0 ⇔ d dr �1 r d dr � uθ rr �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='51) Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='51) yields the radial displacement within the bulk phases yields ur(r) = � � � � � � � � � � � � � � � uα r := Aαr + Bα/r 0 < r < r1, uβ r := Aβr + Bβ/r r1 < r < r2, uγ r := Aγr + Bγ/r r2 < r ≤ R, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52) Now, the problem reduces to finding the solution to the six unknown con- 77 stants: {Aθ=α,β,γ} and {Bθ=α,β,γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Since the displacement field must be bounded as r → 0, the constant Bα must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This ensures that the radial displacement is zero at the origin (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='43)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Using the strain-displacement relations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='45)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='47), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52), it can be shown that the nonzero strains within the bulk phases are ϵr(r) = � � � � � � � � � � � � � � � ϵα r := Aα 0 < r < r1, ϵβ r := Aβ − Bβ/r2 r1 < r < r2, ϵγ r := Aγ − Bγ/r2 r2 < r ≤ R, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='53) ϵφ(r) = � � � � � � � � � � � � � � � ϵα r := Aα 0 < r < r1, ϵβ r := Aβ + Bβ/r2 r1 < r < r2, ϵγ r := Aγ + Bγ/r2 r2 < r ≤ R, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='54) Note that we have set Bα to be zero in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='53) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' To determine the remaining five unknowns, we need five equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The first equation is a consequence of boundary condition at the outer surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='53) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='54) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='49) and setting r = R yields (λγ + 2µγ) � Aγ − Bγ/R2� + λγ � Aγ + Bγ/R2� = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='55) The four remaining equations are obtained as a consequence of the interfacial conditions, specifically the continuity of radial displacement and radial stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 78 The continuity of displacement field yields: uα r |r1 = uβ r �� r1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='56) uβ r �� r2 = uγ r|r2 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='57) Similarly, stress continuity implies: σα r |r1 = σβ r �� r1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='58) σβ r �� r2 = σγ r |r2 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='59) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='56) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='57) yields two of the required equations (Aα − Aβ) r1 − Bβ/r1 = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60) (Aβ − Aγ) r2 + (Bβ − Bγ) /r1 = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='61) Similarly, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='49), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='53) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='58) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='59) yields the remaining two equations: [(λα + 2µα)Aα + λαAα] − � (λβ + 2µβ) � Aβ − Bβ/r2 1 �� − λβ � Aβ + Bβ/r2 1 � + 2(λβ + µβ)ϵ⋆ = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='62) � (λβ + 2µβ) � Aβ − Bβ/r2 2 �� + λβ � Aβ + Bβ/r2 2 � − 2(λβ + µβ)ϵ⋆ − � (λγ + 2µγ) � Aγ − Bγ/r2 2 �� − λγ � Aγ + Bγ/r2 2 � = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='63) 79 By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='55) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='63) yields the five unknown con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A python script, threephase nonplanar analytical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='py, to solve these equations symbolically is provided with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 Solution for the non-planar Ni-Al-Cr case Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4 shows the simulation domain and boundary conditions for the con- centric ring Ni-Al-Cr case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Despite the similarities, there are two important differences that affects the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' First, we have assumed that there are no eigenstrains in the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' and second, we have imposed a hoop strain at the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The outer boundary condition may be written as: ϵγ φ(r = R, t) = ϵg R, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='64) where ϵg R is the assumed hoop strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' In the simulation, this hoop strain is imposed by assuming that the Cartesian displacements at the outer boundary are: ux(r = R, t) = ϵg Rx uy(r = R, t) = ϵg Ry (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='65) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='41), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='46) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='65), it can be shown that the hoop strain at the outer boundary is equal to ϵg R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, due to fact that the geome- try of the system is similar to the previous case, it can be assumed that the displacement fields within the bulk regions are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Further- more, since the boundary conditions at the left and bottom boundaries are 80 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' A schematic showing the phases, eigenstrains and mechanical boundary conditions for the concentric interface Ni-Al-Cr case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' identical to the previous case, it follows that the radial displacement at the origin must be zero, and consequently Bα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Therefore, we need to solve for only the five unknown constants in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The first of these conditions is obtained by solving the outer boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Thus, substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='54) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='64) yields Aγ + Bγ/R2 = ϵg r (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='66) Similar to the previous case, the remaining four equations arise from the interfacial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Moreover, the equations resulting from continuity of displacement field are identical to the previous case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60) & (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The remaining two equations are obtained assuming that the radial 81 stress is continuous at the two interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Specifically, [(λα + 2µα)Aα + λαAα] − � (λβ + 2µβ) � Aβ − Bβ/r2 1 �� − λβ � Aβ + Bβ/r2 1 � = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='67) � (λβ + 2µβ) � Aβ − Bβ/r2 2 �� + λβ � Aβ + Bβ/r2 2 � − � (λγ + 2µγ) � Aγ − Bγ/r2 2 �� − λγ � Aγ + Bγ/r2 2 � = 0 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='68) Thus, by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='66), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='67), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='68), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='60) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='61) we can obtain the five unknowns in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' This was achieved using the python package SymPy [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The python script, threephase iso nonplanar- applied strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='py, is also available with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Data Availability The processed data required to reproduce the figures are available from the corresponding author on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The simulation software required to repro- duce the results is available to download from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content='com/souravmat- git/gibbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The MOOSE input files required to run the simulations are avail- able to download from the folder stressed multiphase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' The MATLAB scripts required to reproduce the precomputed input thermodynamic and kinetic properties are available to download from the folder Precomputed properties [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Finally, the python scripts required to symbolically calculate the con- stants in the analytical solutions are available to download from the folder symbolic python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 82 References [1] Eliot Fried and Morton E Gurtin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Coherent solid-state phase transi- tions with atomic diffusion: a thermomechanical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Journal of statistical physics, 95(5):1361–1427, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} 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simulating growth of intermetallic phases in multi- component alloy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' Mendeley Data, v1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} +page_content=' 92' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf'} diff --git a/J9E2T4oBgHgl3EQfAQbB/content/tmp_files/2301.03590v1.pdf.txt b/J9E2T4oBgHgl3EQfAQbB/content/tmp_files/2301.03590v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..17619483383150db22669478d8d4ed6f900a8fd8 --- /dev/null +++ b/J9E2T4oBgHgl3EQfAQbB/content/tmp_files/2301.03590v1.pdf.txt @@ -0,0 +1,1417 @@ +Avoidance of singularity during the gravitational collapse with string T-duality effects +Kimet Jusufi1, ∗ +1Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, North Macedonia +In this paper, we explore the gravitational collapse of matter (dust) under the effect of zero-point length l0. +During the gravitational collapse, we have neglected the backreaction effect of the pre-Hawking radiation (in +the sense that it is small effect and cannot prevent the formation of apparent horizon), then we recast the internal +metric of a collapsing star as a closed FRW universe for any spherically symmetric case and, finally, we obtain +the minimal value for the scale factor meaning that the particles never hit the singularity. We argue that the object +emerging at the end of the gravitational collapse can be interpreted as Planck stars (black hole core) hidden inside +the event horizon of the black hole, with radius proportional to (GMl2 +0/c2)1/3. Quite interestingly, we found +the same result for radius of the Planck star using a free falling observer point of view. In addition, we pointed +out a correspondence between the modified Friedmann’s equations in loop quantum gravity and the modified +Friedmann’s equation in string T-duality. In the end, we discuss two possibilities regarding the final stage of the +black hole. The first possibility is that we end up with a Planck-size black hole remnants. The second possibility +is that the inner core can be unstable and, due to the quantum tunnelling effect, the spacetime can undergo a +black hole-to-white hole transition (a bouncing Planck star). +I. +INTRODUCTION +Using classical general relativity, in was shown by Oppen- +heimer and Snyder [1], that the ultimate fate of a spherically +symmetric collapsing star must be a black hole. According +to general relativity, classical black hole solutions have singu- +larities arising during the gravitational collapse. In particular, +Penrose showed that even deviations from spherical symme- +try cannot prevent space-time singularities from arising [2]. +On the other hand, Hawking [3], used quantum field theory +in strong gravitational field and found that there must be a +thermal flux of particle production, known as Hawking ra- +diation. This means that a static observer located far away +from the black hole should detect temperature. Such temper- +ature is very small and, as of today, it has not been measured. +It is widely believed that such spacetime singularities can be +cured within a quantum theory of gravity. Regular black holes +attracted a lot of attention (see different regular black hole so- +lutions [4–9]), including a recent review [10], and possible +constraints to the regular black holes with the Event Horizon +Telescope image of Sagittarius A∗ and S2 star [11, 12]. Dif- +ferent concerns have been raised about the stability of regu- +lar black holes. Specifically, it was argued that regular black +holes can be generically unstable because of the phenomenon +known as the “mass inflation” which can destabilize the inner +horizon and the role of Hawking radiation to cure this insta- +bility [13, 14], and the problem with such a claim see [15]. +In the present paper, we shall peruse a different scenario, +namely it was argued how ideas from T-duality can regular- +ize the gravitational potential [16–19] and how this can play +an important role to resolve the black hole singularity [17]. +Using T-duality, it is possible to show that the description of +string theory below the length ls = α′ is the same as its de- +scription above ls = α′. In this framework, the physics of +four dimensions can be obtained by compactifying the other +dimensions. For a single compact dimension with radius R, +∗ kimet.jusufi@unite.edu.mk +one can use the boundary conditions by writing +X4(τ, σ + 2π) = X4(τ, σ) + 2πwR, +(1) +where w is known as the winding number. Furthermore, for +the mass spectrum for such a system, we have +m2 = +1 +2α′ +� +n2 α′ +R2 + w2 R2 +α′ +� ++ . . . , +(2) +here n is the known as the Kaluza-Klein excitation level. The +main idea behind T-duality is that the above spectrum does not +change if we exchange winding number w and Kaluza-Klein +excitation level n, namely we can write [17], +w → n,R → α +′2/R. +(3) +On physical grounds, this also suggests that the description +of string theory below a certain length is equivalent to its de- +scription above it [16]. Furthermore, by means of T-duality, +one can show that Green’s function is invariant under w → +n, R → α +′2/R. In particular, for the Green function in the +momentum space, it was found the following [16–19], +G(k) = −2πR +√ +k2 K1(2πR +√ +k2), +(4) +in which K1(k) is a modified Bessel function of second kind. +The zero point length (l0 = 2πR) is therefore produced by +means of compactified extra dimension of radius R, and can- +not be probed below this length. Taking the limit l0k2 → 0,we +get the standard relation for the Green’s function is obtained, +i.e., G(k) = −k−2. In such a limit, the stringy effects are +very small and can be totally neglected. Using the modified +Green’s function it was also shown that the point-like source +distribution is replaced by a smearing matter density. Specif- +ically, using the regularized potential due to the zero-point +length l0, and by solving the Poisson equation one can obtain +the energy density and the stress-energy tensor describing the +smearing matter distribution [17]. The static and spherically +symmetric metric that solves the Einstein field equations with +arXiv:2301.03590v1 [gr-qc] 9 Jan 2023 + +2 +stringy effect is given by [17] +ds2 = − +� +1 − +2Mr2 +(r2 + l2 +0)3/2 +� +dt2 + +dr2 +1 − +2Mr2 +(r2+l2 +0)3/2 ++r2dΩ2, +(5) +where M denotes the Komar mass. This is a very impor- +tant solution since it is a non-pertubative solution that de- +scribes a static and spherically symmetric black hole geom- +etry. For M > 3 +√ +3 l0/4, there exist two roots: the inner +and outer horizon, r− and r+, respectively. We can also say +that such a metric describes two possible phases of matter, +the particle sector (M < 3 +√ +3 l0/4) and the black hole sector +(M > 3 +√ +3 l0/4). For very large mass, the solution is ef- +fectively the Schwarzschild black hole. Recently, such non- +pertubative modification were used to find a charged black +hole solution in T-duality [21], regular black holes in 3 di- +mensions [22], charged black holes in 4D Einstein-Gauss- +Bonnet gravity [23], entropic corrections to Friedmann equa- +tions [24], regular black strings and torus-like black holes +[25]. By considering the Hawking evaporation, it was argued +that black remnants should occur due to stringy theoretical +effects [26]. In the present work, we would like to study in +more details the gravitational collapse and the final stage of +the collapse using such stringy corrections. +This paper is outlined as follows. In Section II, we study the +gravitational collapse of the interior of star. In Section III, we +discuss the Planck star radius using an infalling observer point +of view. In Section IV, we discuss the Hawking evaporation +and the final Planck-size remnants. In Section V, we study the +bouncing Planck star scenario. We comment on our results in +Section VI. +II. +GRAVITATIONAL COLLAPSE AND PLANCK STARS +IN T-DUALITY +Let us start by considering a spherically-symmetric star +composed by matter (dust) with vanishing pressure which un- +dergoes a gravitational collapse. In general, the stress energy +tensor of the collapsing matter is given by +T µν = (ρ + p)uµuν + pgµν, +(6) +in which ρ is the energy density, uµ is the fluid 4-velocity, and +p is the pressure which in our case vanishes. For such a fluid +we have to consider the energy conservation, i.e., ∇αT αβ = +0, along with the Einstein equations. For the interior region, +having a spherically symmetry star, we can write in general +[39] +ds2 +int = −e2φ(r,τ)dτ 2 + eλ(r,τ)dr2 + R(r, τ)2dΩ2, +(7) +in which R(r, τ) is the area radius. +In addition, we take +φ′(r, τ) = 0, which follows from the Einstein equations for +the case of homogeneous dust [39]. To study the gravitational +collapse we are going to use the well known Tolman-Bondi +spacetime by introducing following function [35–38, 40, 41] +eλ(r,τ) = +[R(r, τ)]2 +,r +1 − K(r) , +(8) +which leads to +ds2 +int = −dτ 2 + +[R(r, τ)]2 +,r +1 − K(r) dr2 + R(r, τ)2dΩ2. +(9) +For the exterior metric, we shall use the modified vacuum +solution due to the stringy effects given by Eq. (5) and rewrit- +ten as +ds2 +ext = − +� +1 − +2Mr2 +(r2 + l2 +0)3/2 +� +dt2+ +dr2 +1 − +2Mr2 +(r2+l2 +0)3/2 ++r2dΩ2. +(10) +At this point, we will utilize the Misner-Sharp mass func- +tion which is defined by using the area radius, which at fixed +R, reads +gµν(∇µR)(∇νR) = 1 − +2MR2 +(R2 + l2 +0)3/2 , +(11) +from this equation it follows that +[R(r, τ)]2 +,τ = +2MR2 +(R2 + l2 +0)3/2 − K(r). +(12) +A very important result that follows from the Tolman-Bondi +spacetime is that one can obtain the Friedmann’s equations as +a special case. To see this, we need to introduce the following +relations +R(r, τ) = a(τ)r and +K(r) = k r2. +(13) +It can be easily seen how the FRW universe metric is obtained +ds2 = −dτ 2 + a(τ)2 +� +dr2 +1 − k r2 + r2dΩ2 +� +. +(14) +Here k denotes the curvature of space with k = 0, 1, −1 cor- +responding to flat, closed, and open universes, respectively. +Using this equivalence we can model and study the interior +spherically symmetric homogeneous stars. Using Eqs. (12) +and (13) we can find +� ˙a +a +�2 ++ k +a2 = 8πρ +3 +� +1 + l2 +0 +R2 +�−3/2 +. +(15) +It is important to note here that we shall neglect the back- +reaction effect of the pre-Hawking radiation during the gravi- +tational collapse. In particular, it has been shown that such an +effect is small and cannot prevent the formation of apparent +horizon (see for example [42]). The dynamical apparent hori- +zon, a marginally trapped surface with vanishing expansion, +is determined by the relation +hµν (∂µR) (∂νR) = 0, +(16) +where the two dimensional metric reads +hµν = diag(−1, +a2 +1 − kr2 ). +(17) +It is a simple calculation to find out the relation for the appar- +ent horizon radius of the FRW universe +R = a r = +1 +�� ˙a +a +�2 + k +a2 +. +(18) + +3 +We are going to simplify the work, since l0 is a very small +number, we can consider a series expansion around l0 via +� +1 + +l2 +0 +r2a2 +�−3/2 += 1 − 3 +2 +l2 +0 +r2a2 + ... +(19) +then using the Friedmann’s equation (15) we obtain in leading +order terms +� ˙a +a +�2 ++ k +a2 = 8πρ +3 +� +1 − 3l2 +0 +2 +�� ˙a +a +�2 ++ k +a2 +�� +. (20) +The last equation can be further written as +� ˙a +a +�2 ++ k +a2 = 8πρ +3 +[1 − Γρ] , +(21) +where Γ is a constant defined as +Γ ≡ 4l2 +0π +3 +. +(22) +This result is nothing but the corrected Fredmann equation +reported recently in Ref. [24] using a different approach (Ver- +linde’s entropic force scenario). In fact, it coincides with [24] +by taking ω = 0 (dust). It is quite remarkable that we found +a bridge between two different and competing directions in +quantum gravity. In one hand, by considering string T-duality +effects we found modified Friedmann equations (21) which +coincides with the conclusions obtained from the loop quan- +tum gravity approach [45] +� ˙a +a +�2 ++ k +a2 = 8πρ +3 +� +1 − ρ +ρc +� +, +(23) +where ρc is the critical energy density +ρc ≡ +3 +8πγ2λ2 , +(24) +where λ ∼ 5.2l2 +P l [45] is area gap that sets the discreteness +scale of loop quantum gravity and γ is the Immirzi parameter. +The correspondence is achieved by identifying Γ = ρ−1 +c . A +direct computation yields γ = 0.310086 l0/lP l. Using l0 = +23/4/33/4lP l = 0.73778lP l [17], we get γ = 0.2287783, +which is in perfect agreement with the value proposed in loop +quantum gravity γ = 0.2375. Once the gravitational collapse +takes place, we can now use the modified Friedmann equa- +tions and explore the possibility that the collapse stops at some +point due to the stringy corrections. To do so, we have to use +the condition +˙a = 0|(a=amin,ρ=ρcrit.), +(25) +along with k = 1. From this condition, we can get the criti- +cal density, in fact we obtain two branches of solution for the +critical density +ρcrit. = 1 +2Γ +� +1 ± +� +1 − +3Γ +2πa2 +min +� +. +(26) +From this result it follows that +1 − +3Γ +2πa2 +min +≥ 0, +(27) +which basically allows us to find the minimal quantity for the +scale factor +amin = +� +3Γ +2π = +√ +2 l0. +(28) +Again, this is in perfect agreement with what has been found +in Ref. [24]. Such a critical density is thus inversely propor- +tional to the minimal length +ρcrit. ∼ 1 +l2 +0 +. +(29) +The above arguments show that during the gravitational col- +lapsing phase, the singularity is never reached and the interior +solution of the black hole (black hole core) is a kind of very +dense star. This possibility that a very dense star or Planck +star exists inside the black hole was proposed in Ref. [27]. +The radius of such a star was conjectured to be proportional +to the collapsed mass [27]. It is very interesting, as we shall +see, such Planck stars hidden inside the stringy corrected reg- +ular black holes can naturally appear in our analyses. In what +follows, we are going to compute the radius of such a star. +First, we need to rewrite the FRW metric in a simple form. +Let us define the proper time τ using +τ = +� +a(η)dη, +(30) +where η is the conformal time, along with radial coordinate +defined as +r(τ) = a(τ) sin χ. +(31) +From these equations we obtain the FRW metric as +ds2 +int = −dτ 2 + a2(τ) +� +dχ2 + sin2 χdr2� += a2(η) +� +−dη2 + dχ2 + sin2 χdr2� +. +(32) +Choosing a surface Σ, with fixed χ = χ0, by matching the +metrics, we can obtain the first equation +R(τ) = a(τ) sin χ0, +(33) +along with the second equation +− +� +1 − +2MR2 +(R2 + l2 +0)3/2 +� � dt +dτ +�2 ++ +� dR +dτ +�2 +1 − +2MR2 +(R2+l2 +0)3/2 += −1. +(34) +From the last equation it is not difficult to show that +dt +dτ = ± +� +˙R2 + 1 − +2MR2 +(R2+l2 +0)3/2 +1 − +2MR2 +(R2+l2 +0)3/2 +. +(35) +Although we use a rather simple and idealized model of col- +lapse, it highlights the main features of the interior dynamics + +4 +of interior of the star. Using the matching procedure of the +interior and exterior metrics at the surface of the star it is pos- +sible to study the motion of the star’s surface. In what follows +we shall show some important results; first, we are going to +approximate the last equation as +dt ≃ ± +dR +1 − +2MR2 +(R2+l2 +0)3/2 +, +(36) +where plus/minus sign corresponds to the case of expansion +or collapse. Since we are interested in the collapse, we chose +the minus sign (R decreases with time) and by doing further +simplifications we get +R(t) ≃ 2M − exp +� +−4tM + 8M 2 + 8M 2Ξ + 3l2 +0Ξ +8M 2 + 3l2 +0 +� +, +(37) +where +Ξ = LabertW +� +�−4Me +− 4M(t+2M) +8M2+3l2 +0 +8M 2 + 3l2 +0 +� +� . +(38) +We see that from the point of view of the outside observer, +it takes infinite amount of time t → ∞ to see the formation +of the black hole horizon R → 2M. The whole process is +viewed in “very slow motion”. However, from the point of +view from the inside, it takes a finite proper time for particles +to reach the minimal distance. In the Oppenheimer-Snyder +model [1], the surface of a gravitationally collapsing spheri- +cally symmetric star made up of dust with radius Rs, can be +obtained via Eq. (33). At this point, let us define the following +constant quantity +a0 ≡ 8πρa3 +3 +|(τ=0) = const. +(39) +where a0 = a(τ = 0) is the scale factor in the initial moment +of collapse. We can see that this quantity is constant simply by +taking ρ = ρ0a−3, for the dust matter. At the initial time we +also have τ = η = 0, along with radius of the star Rs(0) = +R0. Furthermore one can show that +a0 = +� +R3 +0 +2M , sin χ0 = +� +2M +R0 +. +(40) +From the corrected Fredmann’s equation [setting k = 1], we +obtain +˙a(τ) + 1 = +a0 +a(τ) +� +1 − +a0l2 +0 +2a(τ)3 +� +(41) +or, in terms of η, we get +˙a(η) + a(η)2 = a0a(η) +� +1 − +a0l2 +0 +2a(η)3 +� +. +(42) +Solving the last two equation exactly is not an easy task. +One simple guess is to try and generalize the parametric form +a(η) = a0(1+cos η)/2 which is a solution when l0 = 0, then +by using the following equation +a(η) = a0 +2 (1 + ξ(η)) +(43) +we get +ξ(η) = η ± +� +(1 + a(η))a0da(η) +� +(1 − a(η))(1 + a(η))3 − 8l2 +0 ++ C. +(44) +There are two branches in this solution which can describe the +contraction and expansion, respectively. Again, finding an ex- +act solution in closed form is outside the scope of the present +work. Since the mass is conserved during the gravitational +collapse (having in mind the Hawking radiation is very small) +we must also have +a(τmax) ≡ 8πρa3 +3 +|(τmax) = const. +(45) +once the Planck star is formed. At the surface of the star when +the gravitational collapse stops, we also have +a(τmax) = +� +R3τmax +2M , sin χτmax = +� +2M +Rτmax +, +(46) +note here that a(τmax) = amin = +√ +2 l0. This means that we +can obtain the proportionality +ρ0a2 +0 = ρ(τmax)a2 +min, +(47) +where we can identify ρ(τmax) = ρcrit. Put in other words, +during the gravitational collapse, the scale factor decreases, +but the density per unit volume increases. Another way of +stating this result is to say the mass of the collapsing matter is +constant +ρ0R3 +0 = ρcritR2 +τmax. +(48) +When the gravitational collapse stops, we can find the ra- +dius of the Planck star using Rs|τmax,amin = a(τmax) sin χτmax, +namely we get +Rs|τmax,amin ∼ amin(2M)1/2R−1/2 +τmax. +(49) +Using Rs = Rs|τmax,amin = Rτmax, we estimate the radius of +Planck star as follows +Rs ∼ 22/3 M 1/3 l2/3 +0 +. +(50) +The above value for the radius is in good agreement with Ref. +[27], having set n = 1/3. The phenomenological aspects of +Planck stars have been studied in Refs. [28, 29, 31, 32]. For +a stellar mass black hole with mass M ∼ 10 × Msun and +l0 ∼ 10−34 m we can obtain the radius of the Planck star +[by restoring the constants G and c] Rs ≃ [GMl2 +0/c2]1/3 ∼ +10−22 m. Although a small value, this shows that the radius of +such a star is many order of magnitudes greater compared to +l0. Such a star is hidden inside the event horizon of the black +hole with the geometry described by the metric (5). +III. +A FREE FALLING OBSERVER AND PLANCK STAR +RADIUS +Let us now study the whole process as seen from a free +falling observer. +To do this, we can use the Painlev´e– +Gullstrand coordinates through the definition of a new time + +5 +coordinate as +dtp = dt + +� +1 − f(r) +f(r) +dr +(51) +for some arbitary function f(r), along with the new metric +ds2 = −f(r)dt2 +p + 2 +� +1 − f(r)dtpdr + dr2 + r2dΩ2 . (52) +We see that there is no coordinate singularity at the hori- +zon. The time coordinate of the Painlev´e–Gullstrand metric +is the same as the proper time of a freely-falling observer who +starts from infinity at zero velocity. We denote the Painlev´e– +Gullstrand coordinates as (tp, rp) and the Schwarzschild co- +ordinates as (t, r). One can use the Jacobian to relate these +coordinates given by [43] +∂(tp, rp) +∂(t, r) += +� +∂tp +∂t +∂tp +∂r +∂rp +∂t +∂rp +∂r +� += +� +1 +√ +1−f(r) +f(r) +0 +1 +� +, +along with the inverse of the transformation matrix +∂(t, r) +∂(tp, rp) = +� +∂t +∂tp +∂t +∂rp +∂r +∂tp +∂r +∂rp +� += +� +1 − +√ +1−f(r) +f(r) +0 +1 +� +. +From the point of view of a static observer located far away +from the black hole, the total energy momentum is the sum +of the energy momentum of the black hole core or the Planck +star energy density and the renormalized stress energy tensor +is we add the effect of Hawking radiation. For example, one +can choose the Unruh vacuum stat [see, [43]]. Hence we can +write +T tot. +µν = T Core +µν ++ T RSET +µν +(53) +For a freely-falling observer we can write the components +in Painlev´e–Gullstrand coordinates by means of a coordinate +transformation +T GP +αβ = ∂xµ +∂xα +∂xν +∂xβ T tot +µν . +(54) +As we did in the last section, we shall neglect here too the +Hawking radiation effect as perceived by a freely-falling ob- +server, and focus only on T Core. For simplicity we work in +1 + 1 dimensions, this yields +Ttptp = f(r)ρ(r) +(55) +Ttprp = − +� +1 − f(r)ρ(r), +(56) +Trprp = −ρ, +(57) +with the components of the energy-momentum for the black +hole core in Schwarzschild coordinates given by +T Coreµ +ν = (−ρ, Pr) +(58) +For a freely-falling observer, the velocity in Painlev´e– +Gullstrand coordinates is given by +V a = +� +1, − +� +1 − f(r) +� +. +(59) +Using this velocity, we find that the energy density as mea- +sured by such an observer is given by +ρGP = TabV aV b = ρ . +(60) +In other words, the energy density stays invariant quantity. At +this point, we use the condition +V aVa = −1 =⇒ f(r)′|r=rmin = 0, +(61) +and after solving this equation we get the minimal value at +rmin = +√ +2 l0. +(62) +This is in perfect agreement with the minimal scale factor +found in the last section. We can now compute the total time +measured by such a free falling observer using +� T +0 +dtp = − +� rmin +r+ +dr +� +1 − f(r) +(63) +where we approximate r+ ≃ 2M. After solving this integral +we obtain +T ≃ 4 +3M + 21/4l2/3 +0 +12 +√ +M +− 3l2 +0 +4M + .... +(64) +The proper time of a particle is therefore finite. We can +show that the time for light reaching the minimal distance, +say from event horizon is also finite. In this case, one can use +ds2, to find the radial equations, then using the integrating the +equation we obtain +� T +0 +dtp = − +� rmin +rmax +dr +1 + +� +1 − f(r) +(65) +where again we can use the approximation rmax ≃ 2M. After +solving this integral we obtain a finite amount of time +T ≃ 2M ln M + 6M ln 2 − 2M + 2 +√ +2M +� +l0 +√ +2 +− +√ +2l0 − 4M ln( +√ +2M +� +l0 +√ +2) +(66) +Let us now use Eq. (51) to find the time t measured by an +observer located far away from the black hole +t = T − +� � +1 − f(r) +f(r) +dr +(67) +which yields +t ≃ T + 2 +√ +2M +�√ +2M arctan−1 +�� r +2M +� +− √r +� ++ C +(68) +where C is an integration constant. In the limit r → 2M, +we obtain t → ∞, meaning that from this observer point of +view, it takes an infinite amount of time to see the collapsing +of matter. Due to the quantum gravitational effect, or the zero +point length effect, we found that the particles never reach +the singularity, but this also implies the existence of Plank + +6 +stars. This can be seen from Einstein field equations and using +ρ(r) = ρcrit., we must have +R ∼ 8πρcrit.(r) ∼ 3 +l2 +0 +. +(69) +This shows that there is no singularity in the expression for +the Ricci scalar, provided l0 > 0. One can calculate one more +scalar invariant, known as the Kretschmann scalar given by +the following result K ∼ l−4 +0 . To estimate the radius of the +Planck star, here we recall that the Ricci scalar (R) for the +above black hole given is found +R = 2M +� +2r4 − 11l2 +0r2 + 2l2 +0 +� +(l2 +0 + r2)7/2 +. +(70) +From these two equations we obtain +2M +� +2r4 − 11l2 +0r2 + 2l2 +0 +� +(l2 +0 + r2)7/2 +− 3 +l2 +0 += 0, +(71) +considering a series expansion around l0, and by setting the +radial coordinate to be the Planck star radius [we call it r = +Rs] we get +4M +R3s +− 3 +l2 +0 += 0. +(72) +Solving for the Rs we obtain +Rs ∼ 22/33−1/3M 1/3l2/3 +0 +, +(73) +which is in perfect agreement with Eq. (50) found in the last +section in leading order terms. +IV. +PLANCK-SIZE REMNANTS +In this last section, we would like to speculate about the +final state of the Planck star hidden inside the black hole. As- +suming that black hole has been formed along with a Planck +star inside it, due to the presence of the horizon, we can now +take into the account the Hawking radiation and its back reac- +tion effect. Viewed from the outside region, we have the outer +horizon [17] +r+ ≃ 2M − 3l2 +0 +4M , +(74) +and the inner horizon +r− ≃ l0 +√ +2 +� l0 +M +�1/2 +, +(75) +respectively. The Hawking radiation is computed via [17] +TH = f ′(r) +4π |r=r+ = +1 +4πr+ +� +1 − +3l2 +0 +l2 +0 + r2 ++ +� +. +(76) +Due to the backreaction effect of the Hawking evaporation +the mass of the black hole decreases M(t), this means that we +have a slowly shrinking outer horizon, but in the same time the +inner horizon increases [as can be seen from Eqs. (74)-(75)]. +For instance, we can compute the evaporation time viewed +from the outside using +− dM(t) +dt +∼ A σ T 4 +H +(77) +where A = 4πr2 ++ is the area of the black hole horizon and σ +is the Stefan–Boltzmann constant. For the evaporation time it +is not difficult to show that +tevapo. ≃ A +� +M 3 − M 3 +ext +� ++ l2 +0B (M − Mext) . +(78) +where A and B are two constants of proportionality. +The +stringy effects are small and the evaporation time will be very +long. It was shown that for some extremal configuration with +M = M ext = 3 +√ +3l0/4 the outer and inner horizon coincide +r− = r+ = +√ +2l0 (see, for details [17]). This is interest- +ing since it coincides exactly with the minimal scale factor +obtained in the present work. There is a significant differ- +ence compare to the classical Schwarzschild black hole case, +namely instead of getting increasingly hotter and eventually +with a final explosion, due to the stringy effect, here it cools +down and eventually vanishes (TH = 0) at the extremal con- +figuration. This offers a possibility that the final state - which +is a result of a very long time, to be stable remnant with +Rext +s +∼ r− = r+ = +√ +2 l0, +(79) +This small mass of the remnant is nothing but a particle, and +it has been speculated to be a candidate for the dark matter. +V. +BOUNCING PLANCK STAR: BLACK HOLE TO WHITE +HOLE TRANSITION +There is another possibility, perhaps a more interesting one, +in which, the Planck star bounces instead of decreasing it’s +radius. This is due to the fact that that the inner core (Planck +star) solution may not be a stable state after all. Mathemati- +cally, the bouncing at the critical point can be stated using the +conditions: amin > 0, H|a=amin = 0, along with the condition +¨a|a=amin > 0. This shows that there is a great level of sim- +ilarity between the physics that describes the cosmic bounce +and the possible bounce inside black holes. One can use the +second modified Friedmann equation that describes the dy- +namical evolution reported in Ref. [24] +¨a +a = − +�4π +3 +� +(ρcrit + 3pcrit) +� +1 − 3 +2 +l2 +0 +a2 +min(ω) + ... +� +, +(80) +with the minimal scale factor given by [24] +amin(ω) = +√ +2 +� +1 + 3ω +1 + ω l0. +(81) +We see that in general, if we have matter with non-vanishing +pressure then the scale factor can be a function of ω. Imposing +the condition amin > 0, we get the interval ω ∈ (−∞, −1) ∪ +(−1/3, ∞). On the other hand, if we use the equation of state +via pcrit = ωρcrit, along with ρcrit = 1/(2Γ), we obtain +¨a +a = − 1 +l2 +0 +1 + 9ω +8 +, +(82) + +7 +which is further rewritten as +¨a(τ) − ζ2 a(τ) = 0, +(83) +with +ζ2 = − 1 +l2 +0 +1 + 9ω +8 +> 0, +(84) +provided ω < −1/9. But we must also have in mind that +amin > 0, therefore, we are left with the allowed interval +−1/3 < ω < −1/9. The general solution in this interval +is given by +a(τ) = B1 exp (ζτ) + B2 exp (−ζτ), +(85) +where we can take the interval amin ≤ a(τ) ≤ amax. At the +initial moment τ = 0, one has a(τ = 0) = amin = B1, +hence we can fix the constant B2 = 0, which yields a(τ) = +amin exp(ζτ). The interior metric now reads +ds2 +in = −dτ 2 + a2 +mine2ζτ � +dχ2 + sin2 χdr2� +(86) +As was argued in [24], this metric can describe the bounc- +ing universe. For reasons we elaborated above, we need a spe- +cial form of matter with a specific interval for EoS parameter +ω in order to justify the bouncing effect. Coming back to our +case, where we studied the collapsing of matter (dust) with +zero pressure i.e., pcrit = 0, along with ω = 0, this means that +the above bouncing condition is not satisfied. At this point +a natural question arises: even if we have a collapsing dust +which clearly does not satisfy the above bouncing condition, +can we still say that the final state of the internal core of the +black hole will be eternally stable? Of course, we don’t know +the answer to this question, but from a quantum mechanical +point of view, we may speculate that the bouncing effect can +be also a consequence of the black hole-to white hole transi- +tion (BHWH). In other words, instead of the bouncing con- +dition given by Eq. (84) which is classical effect, we can +have a purely quantum mechanical bounce due to the quan- +tum tunneling effect. The idea behind the BHWH transition +is not new, for example in [33], authors have tried to compute +the probability amplitude between two configurations, say h− +and h+, with the corresponding hypersurfaces Σ− and Σ+. In +particular, the probability amplitude for the BHWH transition +can be computed from [33] +PBH→W H(M, ∆0) = +� ∆0 +0 +| ⟨WH|BH⟩M,∆′ +0 |2 d∆′ +0, +(87) +where ∆0 is a parameter measuring the width of the interpo- +lating region. Furthermore, it was estimated for the BHWH +transition probability a exponential decay law [33] +PBH→W H(M, ∆0) ≃ 1 − e−M∆0, +(88) +with a mean lifetime τ ≤ 1/2M. There are other other ar- +guments about the black hole-to-white hole transition. For +instance, the probability increases with time if we take into +account the Hawking radiation (see, [29, 30]). This can be +explained from the fact that as the mass of the Planck star de- +creases with time and the bouncing mass will be smaller com- +pared to the initial Planck mass, then in accordance with the +semiclassical standard tunnelling factor ∼ e−SE/ℏ [29, 30], +the probability for the black hole-to-white hole transition in- +creases as the mass decreases. Here we note that SE is the +Euclidian action, where SE = M 2. The tunnelling probabil- +ity per unit time can also be written [here we restore ℏ for a +moment] PBH→W H ∼ e−M 2/ℏ/M [29, 30]. Now let us con- +sider again the Painlev´e–Gullstrand coordinates that relate the +time measured by an outside observer and the time measured +by an outgoing observer given by +dtp = dt − +� +1 − f(r) +f(r) +dr, +(89) +along with the white hole metric +ds2 +W H = −f(r)dt2 +p − 2 +� +1 − f(r)dtpdr + dr2 + r2dΩ2 . +(90) +Due to the bouncing effect, the black hole becomes essentially +a white hole with an explicit time-reversal symmetry. In that +sense, a white hole is a solution in general relativity, with a +spacetime region to which cannot be entered from the outside. +From the point of view of an outside observer measuring in +Schwarzschild coordinates, the time-reversed solution or the +white hole geometry is the same as black hole. As we saw, it +takes a finite proper time to form a Planck star from the grav- +itational collapse and yet from the point of view of outside +observer, due to the strong redshift effect, the gravitational +collapse appears ’frozen’ in time due to the formation of the +horizon. The same can be shown for the bouncing process. +An outside observer sees the collapse/bouncing in “very slow +motion”, and the entire process takes a long time. To see this, +let us consider a white hole region with a time-reversed solu- +tion, i.e., t → −t in Eq. (89), then the time measured outside +the white hole is given by +t = +� +dtp + +� � +1 − f(r) +f(r) +dr. +(91) +The first terms is finite proper time measured by the outgoing +observer and can be computed via +T = +� T +0 +dtp = +� rmax +rmin +dr +� +1 − f(r) +, +(92) +and the result is similar to Eq. (64). For the time measured by +the outside observer we get +t ≃ T + 2 +√ +2M +�√ +2M arctan−1 +�� r +2M +� +− √r +� ++ C, +(93) +meaning that a particle to reach the event horizon r ∼ 2M, +we need t → ∞. Put in other words, the bouncing effect of +the star appears in “very slow motion” when observed from +the outside. We can basically deduce the same conclusion +using the matching of the interior and exterior metrics. To do +such a computation we need of course the explicit form of the +scale factor. Let us take just for fun the exponential law i.e., +a(τ) ∼ exp(ζτ), then we get +R(τ) = amin exp(ζτ) +� +2M +Rs +, +(94) + +8 +where ζ = C/l0, here C is some constant. In the initial time +of expansion τ = 0, we must have R(τ = 0) = Rs, i.e. R +should coincide with the radius of the Planck star. Now as- +suming that during the expansion we reach the classical hori- +zon radius with R → 2M, we get the total proper time +τ ≃ l0 +2C ln +�2MRs +a2 +min +� +. +(95) +Since we have an expansion in this case, we need to take +the plus sign in the right hand side of Eq. (36), along with the +time-reversed condition t → −t. In doing so, we get +dt ≃ +dR +2MR2 +(R2+l2 +0)3/2 − 1 +≃ +dR +2M +R − 1. +(96) +Solving this equation for the time leads to the following +result +t = −2M ln +� +M +√ +2 − e +Cτ +l0 amin +� +M +Rs +� +− e +Cτ +l0 amin +� +2M +Rs +. +(97) +If we replace the expression for the proper time τ, we finally +get +t = −2M lim +x→0 ln(x) − 2M. +(98) +The first term goes like limx→0 ln(x) → −∞ and, again, +this confirms the fact that that the time measured from the +outside observer will be very large, i.e., t → ∞, which +is in agreement with Eq. (93). At this point, one can ask +whether white holes can be stable remnants? +Authors in +[30], argued that such a unitary process may not violate +any known physics. +This question is outside the scope of +the present work, but if there is surrounding matter, most +probably the white holes are unstable objects too and collapse +again to black holes. According to [30], there is a difference +in the lifetime between the black holes and white holes. The +former are described by the law τBH ∼ M 3, and the latter +τW H ∼ M 4. If such a spacetime bounce happens there is a +possibility that strings can increase the size. This is similar to +the so-called fuzzball structure to the black hole, speculated +in Ref. [44]. +VI. +CONCLUSIONS +In this paper, we studied the gravitational collapse of mat- +ter (dust) under the effect of zero-point length l0. Initially, +we have neglected the backreaction effect of the pre-Hawking +radiation, then we found that the internal metric of a collaps- +ing star is precisely modeled as a closed FRW universe. Us- +ing the modified Friedmann equations and by matching the +interior and exterior metrics, we studied the dynamics of the +collapsing star and found that the gravitational collapse stops +at some minimal scale factor meaning that the particles never +hit the singularity. We argue that such an object emerging at +the end of the gravitational collapse are Planck stars hidden +inside the event horizon of the black hole with radius propor- +tional to Rs ∼ (GMl2 +0/c2)1/3. To enhance this conclusion, +we found the same result for radius for the Planck star using a +free falling observer point of view. +In the final part of this work we have speculated about the +final stage and pointed out two possibilities: (i) First possi- +bility is that black holes (and the Planck stars inside the core +of black hole), due to the backreaction effect of the Hawk- +ing evaporation decrease their mass to a specific value where +there exists an extremal configuration, at this point, the Hawk- +ing temperature vanishes, and the resulting object is a Planck- +size remnant (a particle). (ii) Second possibility is that the +inner core (Planck star), might be unstable. In particular, due +to the quantum tunnelling effect, the spacetime can undergo a +black hole-to-white hole transition (a bouncing Planck star). +We also showed that, from the outside point of view, the col- +lapse/bounce are viewed in a very slow motion due to the +strong redshift effect. 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Ashtekar and D. Sloan, , Gen. Rel. Grav. 43, 3619 (2011). + diff --git a/J9E2T4oBgHgl3EQfAQbB/content/tmp_files/load_file.txt b/J9E2T4oBgHgl3EQfAQbB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c50c2c7f055653b24daf0ccea698c8ea890485f8 --- /dev/null +++ b/J9E2T4oBgHgl3EQfAQbB/content/tmp_files/load_file.txt @@ -0,0 +1,541 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf,len=540 +page_content='Avoidance of singularity during the gravitational collapse with string T-duality effects Kimet Jusufi1, ∗ 1Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, North Macedonia In this paper, we explore the gravitational collapse of matter (dust) under the effect of zero-point length l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' During the gravitational collapse, we have neglected the backreaction effect of the pre-Hawking radiation (in the sense that it is small effect and cannot prevent the formation of apparent horizon), then we recast the internal metric of a collapsing star as a closed FRW universe for any spherically symmetric case and, finally, we obtain the minimal value for the scale factor meaning that the particles never hit the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We argue that the object emerging at the end of the gravitational collapse can be interpreted as Planck stars (black hole core) hidden inside the event horizon of the black hole, with radius proportional to (GMl2 0/c2)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Quite interestingly, we found the same result for radius of the Planck star using a free falling observer point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In addition, we pointed out a correspondence between the modified Friedmann’s equations in loop quantum gravity and the modified Friedmann’s equation in string T-duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the end, we discuss two possibilities regarding the final stage of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The first possibility is that we end up with a Planck-size black hole remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The second possibility is that the inner core can be unstable and, due to the quantum tunnelling effect, the spacetime can undergo a black hole-to-white hole transition (a bouncing Planck star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' INTRODUCTION Using classical general relativity, in was shown by Oppen- heimer and Snyder [1], that the ultimate fate of a spherically symmetric collapsing star must be a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' According to general relativity, classical black hole solutions have singu- larities arising during the gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In particular, Penrose showed that even deviations from spherical symme- try cannot prevent space-time singularities from arising [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' On the other hand, Hawking [3], used quantum field theory in strong gravitational field and found that there must be a thermal flux of particle production, known as Hawking ra- diation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This means that a static observer located far away from the black hole should detect temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Such temper- ature is very small and, as of today, it has not been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' It is widely believed that such spacetime singularities can be cured within a quantum theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Regular black holes attracted a lot of attention (see different regular black hole so- lutions [4–9]), including a recent review [10], and possible constraints to the regular black holes with the Event Horizon Telescope image of Sagittarius A∗ and S2 star [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Dif- ferent concerns have been raised about the stability of regu- lar black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Specifically, it was argued that regular black holes can be generically unstable because of the phenomenon known as the “mass inflation” which can destabilize the inner horizon and the role of Hawking radiation to cure this insta- bility [13, 14], and the problem with such a claim see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the present paper, we shall peruse a different scenario, namely it was argued how ideas from T-duality can regular- ize the gravitational potential [16–19] and how this can play an important role to resolve the black hole singularity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Using T-duality, it is possible to show that the description of string theory below the length ls = α′ is the same as its de- scription above ls = α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In this framework, the physics of four dimensions can be obtained by compactifying the other dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For a single compact dimension with radius R, ∗ kimet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='jusufi@unite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='mk one can use the boundary conditions by writing X4(τ, σ + 2π) = X4(τ, σ) + 2πwR, (1) where w is known as the winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Furthermore, for the mass spectrum for such a system, we have m2 = 1 2α′ � n2 α′ R2 + w2 R2 α′ � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' , (2) here n is the known as the Kaluza-Klein excitation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The main idea behind T-duality is that the above spectrum does not change if we exchange winding number w and Kaluza-Klein excitation level n, namely we can write [17], w → n,R → α ′2/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (3) On physical grounds, this also suggests that the description of string theory below a certain length is equivalent to its de- scription above it [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Furthermore, by means of T-duality, one can show that Green’s function is invariant under w → n, R → α ′2/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In particular, for the Green function in the momentum space, it was found the following [16–19], G(k) = −2πR √ k2 K1(2πR √ k2), (4) in which K1(k) is a modified Bessel function of second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The zero point length (l0 = 2πR) is therefore produced by means of compactified extra dimension of radius R, and can- not be probed below this length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Taking the limit l0k2 → 0,we get the standard relation for the Green’s function is obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', G(k) = −k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In such a limit, the stringy effects are very small and can be totally neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Using the modified Green’s function it was also shown that the point-like source distribution is replaced by a smearing matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Specif- ically, using the regularized potential due to the zero-point length l0, and by solving the Poisson equation one can obtain the energy density and the stress-energy tensor describing the smearing matter distribution [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The static and spherically symmetric metric that solves the Einstein field equations with arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='03590v1 [gr-qc] 9 Jan 2023 2 stringy effect is given by [17] ds2 = − � 1 − 2Mr2 (r2 + l2 0)3/2 � dt2 + dr2 1 − 2Mr2 (r2+l2 0)3/2 +r2dΩ2, (5) where M denotes the Komar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This is a very impor- tant solution since it is a non-pertubative solution that de- scribes a static and spherically symmetric black hole geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For M > 3 √ 3 l0/4, there exist two roots: the inner and outer horizon, r− and r+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We can also say that such a metric describes two possible phases of matter, the particle sector (M < 3 √ 3 l0/4) and the black hole sector (M > 3 √ 3 l0/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For very large mass, the solution is ef- fectively the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Recently, such non- pertubative modification were used to find a charged black hole solution in T-duality [21], regular black holes in 3 di- mensions [22], charged black holes in 4D Einstein-Gauss- Bonnet gravity [23], entropic corrections to Friedmann equa- tions [24], regular black strings and torus-like black holes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' By considering the Hawking evaporation, it was argued that black remnants should occur due to stringy theoretical effects [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the present work, we would like to study in more details the gravitational collapse and the final stage of the collapse using such stringy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This paper is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In Section II, we study the gravitational collapse of the interior of star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In Section III, we discuss the Planck star radius using an infalling observer point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In Section IV, we discuss the Hawking evaporation and the final Planck-size remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In Section V, we study the bouncing Planck star scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We comment on our results in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' GRAVITATIONAL COLLAPSE AND PLANCK STARS IN T-DUALITY Let us start by considering a spherically-symmetric star composed by matter (dust) with vanishing pressure which un- dergoes a gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In general, the stress energy tensor of the collapsing matter is given by T µν = (ρ + p)uµuν + pgµν, (6) in which ρ is the energy density, uµ is the fluid 4-velocity, and p is the pressure which in our case vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For such a fluid we have to consider the energy conservation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', ∇αT αβ = 0, along with the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For the interior region, having a spherically symmetry star, we can write in general [39] ds2 int = −e2φ(r,τ)dτ 2 + eλ(r,τ)dr2 + R(r, τ)2dΩ2, (7) in which R(r, τ) is the area radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In addition, we take φ′(r, τ) = 0, which follows from the Einstein equations for the case of homogeneous dust [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To study the gravitational collapse we are going to use the well known Tolman-Bondi spacetime by introducing following function [35–38, 40, 41] eλ(r,τ) = [R(r, τ)]2 ,r 1 − K(r) , (8) which leads to ds2 int = −dτ 2 + [R(r, τ)]2 ,r 1 − K(r) dr2 + R(r, τ)2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (9) For the exterior metric, we shall use the modified vacuum solution due to the stringy effects given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (5) and rewrit- ten as ds2 ext = − � 1 − 2Mr2 (r2 + l2 0)3/2 � dt2+ dr2 1 − 2Mr2 (r2+l2 0)3/2 +r2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (10) At this point, we will utilize the Misner-Sharp mass func- tion which is defined by using the area radius, which at fixed R, reads gµν(∇µR)(∇νR) = 1 − 2MR2 (R2 + l2 0)3/2 , (11) from this equation it follows that [R(r, τ)]2 ,τ = 2MR2 (R2 + l2 0)3/2 − K(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (12) A very important result that follows from the Tolman-Bondi spacetime is that one can obtain the Friedmann’s equations as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To see this, we need to introduce the following relations R(r, τ) = a(τ)r and K(r) = k r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (13) It can be easily seen how the FRW universe metric is obtained ds2 = −dτ 2 + a(τ)2 � dr2 1 − k r2 + r2dΩ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (14) Here k denotes the curvature of space with k = 0, 1, −1 cor- responding to flat, closed, and open universes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Using this equivalence we can model and study the interior spherically symmetric homogeneous stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (12) and (13) we can find � ˙a a �2 + k a2 = 8πρ 3 � 1 + l2 0 R2 �−3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (15) It is important to note here that we shall neglect the back- reaction effect of the pre-Hawking radiation during the gravi- tational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In particular, it has been shown that such an effect is small and cannot prevent the formation of apparent horizon (see for example [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The dynamical apparent hori- zon, a marginally trapped surface with vanishing expansion, is determined by the relation hµν (∂µR) (∂νR) = 0, (16) where the two dimensional metric reads hµν = diag(−1, a2 1 − kr2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (17) It is a simple calculation to find out the relation for the appar- ent horizon radius of the FRW universe R = a r = 1 �� ˙a a �2 + k a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (18) 3 We are going to simplify the work, since l0 is a very small number, we can consider a series expansion around l0 via � 1 + l2 0 r2a2 �−3/2 = 1 − 3 2 l2 0 r2a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (19) then using the Friedmann’s equation (15) we obtain in leading order terms � ˙a a �2 + k a2 = 8πρ 3 � 1 − 3l2 0 2 �� ˙a a �2 + k a2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (20) The last equation can be further written as � ˙a a �2 + k a2 = 8πρ 3 [1 − Γρ] , (21) where Γ is a constant defined as Γ ≡ 4l2 0π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (22) This result is nothing but the corrected Fredmann equation reported recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [24] using a different approach (Ver- linde’s entropic force scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In fact, it coincides with [24] by taking ω = 0 (dust).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' It is quite remarkable that we found a bridge between two different and competing directions in quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In one hand, by considering string T-duality effects we found modified Friedmann equations (21) which coincides with the conclusions obtained from the loop quan- tum gravity approach [45] � ˙a a �2 + k a2 = 8πρ 3 � 1 − ρ ρc � , (23) where ρc is the critical energy density ρc ≡ 3 8πγ2λ2 , (24) where λ ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='2l2 P l [45] is area gap that sets the discreteness scale of loop quantum gravity and γ is the Immirzi parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The correspondence is achieved by identifying Γ = ρ−1 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' A direct computation yields γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='310086 l0/lP l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Using l0 = 23/4/33/4lP l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='73778lP l [17], we get γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='2287783, which is in perfect agreement with the value proposed in loop quantum gravity γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='2375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Once the gravitational collapse takes place, we can now use the modified Friedmann equa- tions and explore the possibility that the collapse stops at some point due to the stringy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To do so, we have to use the condition ˙a = 0|(a=amin,ρ=ρcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' ), (25) along with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' From this condition, we can get the criti- cal density, in fact we obtain two branches of solution for the critical density ρcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' = 1 2Γ � 1 ± � 1 − 3Γ 2πa2 min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (26) From this result it follows that 1 − 3Γ 2πa2 min ≥ 0, (27) which basically allows us to find the minimal quantity for the scale factor amin = � 3Γ 2π = √ 2 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (28) Again, this is in perfect agreement with what has been found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Such a critical density is thus inversely propor- tional to the minimal length ρcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' ∼ 1 l2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (29) The above arguments show that during the gravitational col- lapsing phase, the singularity is never reached and the interior solution of the black hole (black hole core) is a kind of very dense star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This possibility that a very dense star or Planck star exists inside the black hole was proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The radius of such a star was conjectured to be proportional to the collapsed mass [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' It is very interesting, as we shall see, such Planck stars hidden inside the stringy corrected reg- ular black holes can naturally appear in our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In what follows, we are going to compute the radius of such a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' First, we need to rewrite the FRW metric in a simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Let us define the proper time τ using τ = � a(η)dη, (30) where η is the conformal time, along with radial coordinate defined as r(τ) = a(τ) sin χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (31) From these equations we obtain the FRW metric as ds2 int = −dτ 2 + a2(τ) � dχ2 + sin2 χdr2� = a2(η) � −dη2 + dχ2 + sin2 χdr2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (32) Choosing a surface Σ, with fixed χ = χ0, by matching the metrics, we can obtain the first equation R(τ) = a(τ) sin χ0, (33) along with the second equation − � 1 − 2MR2 (R2 + l2 0)3/2 � � dt dτ �2 + � dR dτ �2 1 − 2MR2 (R2+l2 0)3/2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (34) From the last equation it is not difficult to show that dt dτ = ± � ˙R2 + 1 − 2MR2 (R2+l2 0)3/2 1 − 2MR2 (R2+l2 0)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (35) Although we use a rather simple and idealized model of col- lapse, it highlights the main features of the interior dynamics 4 of interior of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Using the matching procedure of the interior and exterior metrics at the surface of the star it is pos- sible to study the motion of the star’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In what follows we shall show some important results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' first, we are going to approximate the last equation as dt ≃ ± dR 1 − 2MR2 (R2+l2 0)3/2 , (36) where plus/minus sign corresponds to the case of expansion or collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Since we are interested in the collapse, we chose the minus sign (R decreases with time) and by doing further simplifications we get R(t) ≃ 2M − exp � −4tM + 8M 2 + 8M 2Ξ + 3l2 0Ξ 8M 2 + 3l2 0 � , (37) where Ξ = LabertW � �−4Me − 4M(t+2M) 8M2+3l2 0 8M 2 + 3l2 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (38) We see that from the point of view of the outside observer, it takes infinite amount of time t → ∞ to see the formation of the black hole horizon R → 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The whole process is viewed in “very slow motion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' However, from the point of view from the inside, it takes a finite proper time for particles to reach the minimal distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the Oppenheimer-Snyder model [1], the surface of a gravitationally collapsing spheri- cally symmetric star made up of dust with radius Rs, can be obtained via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At this point, let us define the following constant quantity a0 ≡ 8πρa3 3 |(τ=0) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (39) where a0 = a(τ = 0) is the scale factor in the initial moment of collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We can see that this quantity is constant simply by taking ρ = ρ0a−3, for the dust matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At the initial time we also have τ = η = 0, along with radius of the star Rs(0) = R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Furthermore one can show that a0 = � R3 0 2M , sin χ0 = � 2M R0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (40) From the corrected Fredmann’s equation [setting k = 1], we obtain ˙a(τ) + 1 = a0 a(τ) � 1 − a0l2 0 2a(τ)3 � (41) or, in terms of η, we get ˙a(η) + a(η)2 = a0a(η) � 1 − a0l2 0 2a(η)3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (42) Solving the last two equation exactly is not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' One simple guess is to try and generalize the parametric form a(η) = a0(1+cos η)/2 which is a solution when l0 = 0, then by using the following equation a(η) = a0 2 (1 + ξ(η)) (43) we get ξ(η) = η ± � (1 + a(η))a0da(η) � (1 − a(η))(1 + a(η))3 − 8l2 0 + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (44) There are two branches in this solution which can describe the contraction and expansion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Again, finding an ex- act solution in closed form is outside the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Since the mass is conserved during the gravitational collapse (having in mind the Hawking radiation is very small) we must also have a(τmax) ≡ 8πρa3 3 |(τmax) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (45) once the Planck star is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At the surface of the star when the gravitational collapse stops, we also have a(τmax) = � R3τmax 2M , sin χτmax = � 2M Rτmax , (46) note here that a(τmax) = amin = √ 2 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This means that we can obtain the proportionality ρ0a2 0 = ρ(τmax)a2 min, (47) where we can identify ρ(τmax) = ρcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Put in other words, during the gravitational collapse, the scale factor decreases, but the density per unit volume increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Another way of stating this result is to say the mass of the collapsing matter is constant ρ0R3 0 = ρcritR2 τmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (48) When the gravitational collapse stops, we can find the ra- dius of the Planck star using Rs|τmax,amin = a(τmax) sin χτmax, namely we get Rs|τmax,amin ∼ amin(2M)1/2R−1/2 τmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (49) Using Rs = Rs|τmax,amin = Rτmax, we estimate the radius of Planck star as follows Rs ∼ 22/3 M 1/3 l2/3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (50) The above value for the radius is in good agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [27], having set n = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The phenomenological aspects of Planck stars have been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [28, 29, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For a stellar mass black hole with mass M ∼ 10 × Msun and l0 ∼ 10−34 m we can obtain the radius of the Planck star [by restoring the constants G and c] Rs ≃ [GMl2 0/c2]1/3 ∼ 10−22 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Although a small value, this shows that the radius of such a star is many order of magnitudes greater compared to l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Such a star is hidden inside the event horizon of the black hole with the geometry described by the metric (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' A FREE FALLING OBSERVER AND PLANCK STAR RADIUS Let us now study the whole process as seen from a free falling observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To do this, we can use the Painlev´e– Gullstrand coordinates through the definition of a new time 5 coordinate as dtp = dt + � 1 − f(r) f(r) dr (51) for some arbitary function f(r), along with the new metric ds2 = −f(r)dt2 p + 2 � 1 − f(r)dtpdr + dr2 + r2dΩ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (52) We see that there is no coordinate singularity at the hori- zon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The time coordinate of the Painlev´e–Gullstrand metric is the same as the proper time of a freely-falling observer who starts from infinity at zero velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We denote the Painlev´e– Gullstrand coordinates as (tp, rp) and the Schwarzschild co- ordinates as (t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' One can use the Jacobian to relate these coordinates given by [43] ∂(tp, rp) ∂(t, r) = � ∂tp ∂t ∂tp ∂r ∂rp ∂t ∂rp ∂r � = � 1 √ 1−f(r) f(r) 0 1 � , along with the inverse of the transformation matrix ∂(t, r) ∂(tp, rp) = � ∂t ∂tp ∂t ∂rp ∂r ∂tp ∂r ∂rp � = � 1 − √ 1−f(r) f(r) 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' From the point of view of a static observer located far away from the black hole, the total energy momentum is the sum of the energy momentum of the black hole core or the Planck star energy density and the renormalized stress energy tensor is we add the effect of Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For example, one can choose the Unruh vacuum stat [see, [43]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Hence we can write T tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' µν = T Core µν + T RSET µν (53) For a freely-falling observer we can write the components in Painlev´e–Gullstrand coordinates by means of a coordinate transformation T GP αβ = ∂xµ ∂xα ∂xν ∂xβ T tot µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (54) As we did in the last section, we shall neglect here too the Hawking radiation effect as perceived by a freely-falling ob- server, and focus only on T Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For simplicity we work in 1 + 1 dimensions, this yields Ttptp = f(r)ρ(r) (55) Ttprp = − � 1 − f(r)ρ(r), (56) Trprp = −ρ, (57) with the components of the energy-momentum for the black hole core in Schwarzschild coordinates given by T Coreµ ν = (−ρ, Pr) (58) For a freely-falling observer, the velocity in Painlev´e– Gullstrand coordinates is given by V a = � 1, − � 1 − f(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (59) Using this velocity, we find that the energy density as mea- sured by such an observer is given by ρGP = TabV aV b = ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (60) In other words, the energy density stays invariant quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At this point, we use the condition V aVa = −1 =⇒ f(r)′|r=rmin = 0, (61) and after solving this equation we get the minimal value at rmin = √ 2 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (62) This is in perfect agreement with the minimal scale factor found in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We can now compute the total time measured by such a free falling observer using � T 0 dtp = − � rmin r+ dr � 1 − f(r) (63) where we approximate r+ ≃ 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' After solving this integral we obtain T ≃ 4 3M + 21/4l2/3 0 12 √ M − 3l2 0 4M + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='. (64) The proper time of a particle is therefore finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We can show that the time for light reaching the minimal distance, say from event horizon is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In this case, one can use ds2, to find the radial equations, then using the integrating the equation we obtain � T 0 dtp = − � rmin rmax dr 1 + � 1 − f(r) (65) where again we can use the approximation rmax ≃ 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' After solving this integral we obtain a finite amount of time T ≃ 2M ln M + 6M ln 2 − 2M + 2 √ 2M � l0 √ 2 − √ 2l0 − 4M ln( √ 2M � l0 √ 2) (66) Let us now use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (51) to find the time t measured by an observer located far away from the black hole t = T − � � 1 − f(r) f(r) dr (67) which yields t ≃ T + 2 √ 2M �√ 2M arctan−1 �� r 2M � − √r � + C (68) where C is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the limit r → 2M, we obtain t → ∞, meaning that from this observer point of view, it takes an infinite amount of time to see the collapsing of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Due to the quantum gravitational effect, or the zero point length effect, we found that the particles never reach the singularity, but this also implies the existence of Plank 6 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This can be seen from Einstein field equations and using ρ(r) = ρcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', we must have R ∼ 8πρcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (r) ∼ 3 l2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (69) This shows that there is no singularity in the expression for the Ricci scalar, provided l0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' One can calculate one more scalar invariant, known as the Kretschmann scalar given by the following result K ∼ l−4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To estimate the radius of the Planck star, here we recall that the Ricci scalar (R) for the above black hole given is found R = 2M � 2r4 − 11l2 0r2 + 2l2 0 � (l2 0 + r2)7/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (70) From these two equations we obtain 2M � 2r4 − 11l2 0r2 + 2l2 0 � (l2 0 + r2)7/2 − 3 l2 0 = 0, (71) considering a series expansion around l0, and by setting the radial coordinate to be the Planck star radius [we call it r = Rs] we get 4M R3s − 3 l2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (72) Solving for the Rs we obtain Rs ∼ 22/33−1/3M 1/3l2/3 0 , (73) which is in perfect agreement with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (50) found in the last section in leading order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' PLANCK-SIZE REMNANTS In this last section, we would like to speculate about the final state of the Planck star hidden inside the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' As- suming that black hole has been formed along with a Planck star inside it, due to the presence of the horizon, we can now take into the account the Hawking radiation and its back reac- tion effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Viewed from the outside region, we have the outer horizon [17] r+ ≃ 2M − 3l2 0 4M , (74) and the inner horizon r− ≃ l0 √ 2 � l0 M �1/2 , (75) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The Hawking radiation is computed via [17] TH = f ′(r) 4π |r=r+ = 1 4πr+ � 1 − 3l2 0 l2 0 + r2 + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (76) Due to the backreaction effect of the Hawking evaporation the mass of the black hole decreases M(t), this means that we have a slowly shrinking outer horizon, but in the same time the inner horizon increases [as can be seen from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (74)-(75)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For instance, we can compute the evaporation time viewed from the outside using − dM(t) dt ∼ A σ T 4 H (77) where A = 4πr2 + is the area of the black hole horizon and σ is the Stefan–Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For the evaporation time it is not difficult to show that tevapo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' ≃ A � M 3 − M 3 ext � + l2 0B (M − Mext) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (78) where A and B are two constants of proportionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The stringy effects are small and the evaporation time will be very long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' It was shown that for some extremal configuration with M = M ext = 3 √ 3l0/4 the outer and inner horizon coincide r− = r+ = √ 2l0 (see, for details [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This is interest- ing since it coincides exactly with the minimal scale factor obtained in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' There is a significant differ- ence compare to the classical Schwarzschild black hole case, namely instead of getting increasingly hotter and eventually with a final explosion, due to the stringy effect, here it cools down and eventually vanishes (TH = 0) at the extremal con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This offers a possibility that the final state - which is a result of a very long time, to be stable remnant with Rext s ∼ r− = r+ = √ 2 l0, (79) This small mass of the remnant is nothing but a particle, and it has been speculated to be a candidate for the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' BOUNCING PLANCK STAR: BLACK HOLE TO WHITE HOLE TRANSITION There is another possibility, perhaps a more interesting one, in which, the Planck star bounces instead of decreasing it’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This is due to the fact that that the inner core (Planck star) solution may not be a stable state after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Mathemati- cally, the bouncing at the critical point can be stated using the conditions: amin > 0, H|a=amin = 0, along with the condition ¨a|a=amin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This shows that there is a great level of sim- ilarity between the physics that describes the cosmic bounce and the possible bounce inside black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' One can use the second modified Friedmann equation that describes the dy- namical evolution reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [24] ¨a a = − �4π 3 � (ρcrit + 3pcrit) � 1 − 3 2 l2 0 a2 min(ω) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' � , (80) with the minimal scale factor given by [24] amin(ω) = √ 2 � 1 + 3ω 1 + ω l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (81) We see that in general, if we have matter with non-vanishing pressure then the scale factor can be a function of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Imposing the condition amin > 0, we get the interval ω ∈ (−∞, −1) ∪ (−1/3, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' On the other hand, if we use the equation of state via pcrit = ωρcrit, along with ρcrit = 1/(2Γ), we obtain ¨a a = − 1 l2 0 1 + 9ω 8 , (82) 7 which is further rewritten as ¨a(τ) − ζ2 a(τ) = 0, (83) with ζ2 = − 1 l2 0 1 + 9ω 8 > 0, (84) provided ω < −1/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' But we must also have in mind that amin > 0, therefore, we are left with the allowed interval −1/3 < ω < −1/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The general solution in this interval is given by a(τ) = B1 exp (ζτ) + B2 exp (−ζτ), (85) where we can take the interval amin ≤ a(τ) ≤ amax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At the initial moment τ = 0, one has a(τ = 0) = amin = B1, hence we can fix the constant B2 = 0, which yields a(τ) = amin exp(ζτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The interior metric now reads ds2 in = −dτ 2 + a2 mine2ζτ � dχ2 + sin2 χdr2� (86) As was argued in [24], this metric can describe the bounc- ing universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For reasons we elaborated above, we need a spe- cial form of matter with a specific interval for EoS parameter ω in order to justify the bouncing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Coming back to our case, where we studied the collapsing of matter (dust) with zero pressure i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', pcrit = 0, along with ω = 0, this means that the above bouncing condition is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At this point a natural question arises: even if we have a collapsing dust which clearly does not satisfy the above bouncing condition, can we still say that the final state of the internal core of the black hole will be eternally stable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Of course, we don’t know the answer to this question, but from a quantum mechanical point of view, we may speculate that the bouncing effect can be also a consequence of the black hole-to white hole transi- tion (BHWH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In other words, instead of the bouncing con- dition given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (84) which is classical effect, we can have a purely quantum mechanical bounce due to the quan- tum tunneling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The idea behind the BHWH transition is not new, for example in [33], authors have tried to compute the probability amplitude between two configurations, say h− and h+, with the corresponding hypersurfaces Σ− and Σ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In particular, the probability amplitude for the BHWH transition can be computed from [33] PBH→W H(M, ∆0) = � ∆0 0 | ⟨WH|BH⟩M,∆′ 0 |2 d∆′ 0, (87) where ∆0 is a parameter measuring the width of the interpo- lating region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Furthermore, it was estimated for the BHWH transition probability a exponential decay law [33] PBH→W H(M, ∆0) ≃ 1 − e−M∆0, (88) with a mean lifetime τ ≤ 1/2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' There are other other ar- guments about the black hole-to-white hole transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For instance, the probability increases with time if we take into account the Hawking radiation (see, [29, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This can be explained from the fact that as the mass of the Planck star de- creases with time and the bouncing mass will be smaller com- pared to the initial Planck mass, then in accordance with the semiclassical standard tunnelling factor ∼ e−SE/ℏ [29, 30], the probability for the black hole-to-white hole transition in- creases as the mass decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Here we note that SE is the Euclidian action, where SE = M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The tunnelling probabil- ity per unit time can also be written [here we restore ℏ for a moment] PBH→W H ∼ e−M 2/ℏ/M [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Now let us con- sider again the Painlev´e–Gullstrand coordinates that relate the time measured by an outside observer and the time measured by an outgoing observer given by dtp = dt − � 1 − f(r) f(r) dr, (89) along with the white hole metric ds2 W H = −f(r)dt2 p − 2 � 1 − f(r)dtpdr + dr2 + r2dΩ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (90) Due to the bouncing effect, the black hole becomes essentially a white hole with an explicit time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In that sense, a white hole is a solution in general relativity, with a spacetime region to which cannot be entered from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' From the point of view of an outside observer measuring in Schwarzschild coordinates, the time-reversed solution or the white hole geometry is the same as black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' As we saw, it takes a finite proper time to form a Planck star from the grav- itational collapse and yet from the point of view of outside observer, due to the strong redshift effect, the gravitational collapse appears ’frozen’ in time due to the formation of the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The same can be shown for the bouncing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' An outside observer sees the collapse/bouncing in “very slow motion”, and the entire process takes a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To see this, let us consider a white hole region with a time-reversed solu- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', t → −t in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (89), then the time measured outside the white hole is given by t = � dtp + � � 1 − f(r) f(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (91) The first terms is finite proper time measured by the outgoing observer and can be computed via T = � T 0 dtp = � rmax rmin dr � 1 − f(r) , (92) and the result is similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' For the time measured by the outside observer we get t ≃ T + 2 √ 2M �√ 2M arctan−1 �� r 2M � − √r � + C, (93) meaning that a particle to reach the event horizon r ∼ 2M, we need t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Put in other words, the bouncing effect of the star appears in “very slow motion” when observed from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We can basically deduce the same conclusion using the matching of the interior and exterior metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To do such a computation we need of course the explicit form of the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Let us take just for fun the exponential law i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', a(τ) ∼ exp(ζτ), then we get R(τ) = amin exp(ζτ) � 2M Rs , (94) 8 where ζ = C/l0, here C is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the initial time of expansion τ = 0, we must have R(τ = 0) = Rs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' R should coincide with the radius of the Planck star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Now as- suming that during the expansion we reach the classical hori- zon radius with R → 2M, we get the total proper time τ ≃ l0 2C ln �2MRs a2 min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (95) Since we have an expansion in this case, we need to take the plus sign in the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (36), along with the time-reversed condition t → −t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In doing so, we get dt ≃ dR 2MR2 (R2+l2 0)3/2 − 1 ≃ dR 2M R − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (96) Solving this equation for the time leads to the following result t = −2M ln � M √ 2 − e Cτ l0 amin � M Rs � − e Cτ l0 amin � 2M Rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (97) If we replace the expression for the proper time τ, we finally get t = −2M lim x→0 ln(x) − 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (98) The first term goes like limx→0 ln(x) → −∞ and, again, this confirms the fact that that the time measured from the outside observer will be very large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=', t → ∞, which is in agreement with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' At this point, one can ask whether white holes can be stable remnants?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Authors in [30], argued that such a unitary process may not violate any known physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This question is outside the scope of the present work, but if there is surrounding matter, most probably the white holes are unstable objects too and collapse again to black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' According to [30], there is a difference in the lifetime between the black holes and white holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' The former are described by the law τBH ∼ M 3, and the latter τW H ∼ M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' If such a spacetime bounce happens there is a possibility that strings can increase the size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' This is similar to the so-called fuzzball structure to the black hole, speculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' CONCLUSIONS In this paper, we studied the gravitational collapse of mat- ter (dust) under the effect of zero-point length l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Initially, we have neglected the backreaction effect of the pre-Hawking radiation, then we found that the internal metric of a collaps- ing star is precisely modeled as a closed FRW universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' Us- ing the modified Friedmann equations and by matching the interior and exterior metrics, we studied the dynamics of the collapsing star and found that the gravitational collapse stops at some minimal scale factor meaning that the particles never hit the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We argue that such an object emerging at the end of the gravitational collapse are Planck stars hidden inside the event horizon of the black hole with radius propor- tional to Rs ∼ (GMl2 0/c2)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' To enhance this conclusion, we found the same result for radius for the Planck star using a free falling observer point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In the final part of this work we have speculated about the final stage and pointed out two possibilities: (i) First possi- bility is that black holes (and the Planck stars inside the core of black hole), due to the backreaction effect of the Hawk- ing evaporation decrease their mass to a specific value where there exists an extremal configuration, at this point, the Hawk- ing temperature vanishes, and the resulting object is a Planck- size remnant (a particle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' (ii) Second possibility is that the inner core (Planck star), might be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' In particular, due to the quantum tunnelling effect, the spacetime can undergo a black hole-to-white hole transition (a bouncing Planck star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfAQbB/content/2301.03590v1.pdf'} +page_content=' We also showed that, from the outside point of view, the col- lapse/bounce are viewed in a very slow motion due to the strong redshift effect.' metadata={'source': 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+South East Technological University +Carlow, Ireland +rejwanul.haque@adaptcentre.ie +Andy Way +ADAPT Centre +School of Computing +Dublin City University +Dublin, Ireland +andy.way@adaptcentre.ie +Abstract +Consistency is a key requirement of high- +quality translation. It is especially important +to adhere to pre-approved terminology and +corrected +translations +in +domain-specific +projects. +Machine translation (MT) has +achieved significant progress in the area +of domain adaptation. +However, real-time +adaptation remains challenging. Large-scale +language +models +(LLMs) +have +recently +shown interesting capabilities of in-context +learning, where they learn to replicate certain +input-output text generation patterns, without +further fine-tuning. By feeding an LLM with +a prompt that consists of a list of translation +pairs, it can then simulate the domain and +style characteristics at inference time. This +work aims to investigate how we can utilize +in-context learning to improve real-time +adaptive MT. Our extensive experiments +show promising results at translation time. +For example, GPT-3.5 can adapt to a set of +in-domain sentence pairs and/or terminology +while translating a new sentence. We observe +that the translation quality with few-shot +in-context learning can surpass that of strong +encoder-decoder MT systems, +especially +for high-resource languages. Moreover, we +investigate whether we can combine MT +from strong encoder-decoder models with +fuzzy matches, which can further improve +the translation, especially for less supported +languages. +We conduct our experiments +across +five +diverse +languages, +namely +English-to-Arabic +(EN-AR), +English- +to-Chinese +(EN-ZH), +English-to-French +(EN-FR), +English-to-Kinyarwanda +(EN- +RW), +and +English-to-Spanish +(EN-ES) +language pairs. +© 2023 The authors. This article is licensed under a Creative +Commons 4.0 licence, no derivative works, attribution, CC- +BY-ND. +Figure 1: Evaluation results for GPT-3 zero-shot, and few-shot translation with +random context or fuzzy matches. Average scores across EN-AR, EN-ES, EN- +FR, and EN-ZH language pairs. While using a random context outperforms +zero-shot translation, using fuzzy matches reveals the best results. +1 +Introduction +Adaptive MT is a type of machine translation that +utilizes feedback from users to improve the qual- +ity of the translations over time. Feedback usually +includes corrections to previous translations, ter- +minology and style guides, as well as ratings of +the quality of the translations. This can be partic- +ularly useful for domain-specific scenarios, where +baseline MT systems may have insufficient rele- +vant data to accurately translate certain terms or +phrases. There are still several challenges to ef- +fectively incorporate user feedback into the trans- +lation process, especially at inference time. In this +work, we use a relatively wide definition of adap- +tive MT to refer to learning from similar transla- +tions (fuzzy matches) found in approved transla- +tion memories (TMs) on the fly (Farajian et al., +arXiv:2301.13294v1 [cs.CL] 30 Jan 2023 + +zero-shot +random 2-shot +fuzzy 2-shot +fuzzy 5-shot +SpBLEU +72.96 +chrF++ +COMET +70.75 +63.64 +63.52 +62.23 +58.73 +57.98 +55.91 +50.61 +48.51 +42.09 +38.882017; Wuebker et al., 2018; Peris and Casacuberta, +2019; Etchegoyhen et al., 2021), as well as real- +time terminology-constrained MT (Hokamp and +Liu, 2017; Post and Vilar, 2018; Dinu et al., 2019). +Autoregressive decoder-only LLMs, such as +GPT-3 (Brown et al., 2020; Ouyang et al., 2022), +GPT-J (Wang and Komatsuzaki, 2021), BLOOM +(Le Scao et al., 2022), and PaLM (Chowdhery et +al., 2022) are trained to predict the next word given +the previous context. +During unsupervised pre- +training, a language model develops a broad set of +pattern recognition abilities. It then uses these abil- +ities at inference time to rapidly adapt to or recog- +nize the desired task. In their experiments, Brown +et al. (2020) use the term “in-context learning” to +describe the inner loop of this process, which oc- +curs within the forward-pass upon each sequence. +In this sense, in-context learning is a scenario +where a pre-trained language model at inference +time learns to replicate certain input-output text +generation patterns without further fine-tuning. +They show that autoregressive LLMs such as GPT- +3 can perform well on diverse tasks, through zero- +shot, one-shot, and few-shot in-context learning +without weight updates. Previous researchers in- +vestigated using neural language models for MT +through few-shot in-context learning (Vilar et al., +2022) and even in zero-shot settings (Wang et al., +2021). Other researchers proposed using LLMs for +generating synthetic domain-specific data for MT +domain adaptation (Moslem et al., 2022). +The main contribution of this paper is investi- +gating the capabilities of LLMs such as GPT-3 for +real-time adaptive MT through in-context learning. +In particular, we would like to understand the qual- +ity with which such models can perform the fol- +lowing tasks, without any further training: +• Adapting new translations to match the termi- +nology and style of previously approved TM +fuzzy matches, at inference time; +• Matching or outperforming the quality of +translations generated by encoder-decoder +MT models across a number of languages; +• Fixing translations from stronger encoder- +decoder MT systems using fuzzy matches, +which is especially useful for low-resource +languages; and +• Terminology-constrained MT, by first defin- +ing terminology in the relevant sentences or +dataset, and then forcing new translations to +use these terms. +2 +Experimental Setup +In all our experiments, we use GPT-3.5 text- +davinci-003 model via its official API.1 For param- +eters, we use top-p 1, with temperature 0.3 for the +three translation tasks, and 0 for the terminology +extraction task. To avoid generating new lines in +the translation tasks, the option stop can be set. For +the maximum length of tokens, we observe that +French and Spanish tokens can be 3–4 times the +number of English source words, while other lan- +guages can be longer. Hence, we roughly choose +a length multiplier value, which we set to 8 for +Arabic, 5 for Chinese and Kinyarwanda, and 4 for +French and Spanish. We used bach requests with a +batch size of 20 segments.2 +For the test dataset, we use TICO-19 (Anasta- +sopoulos et al., 2020), which includes 3070 unique +segments. We target a range of languages with di- +verse scripts and amounts of resources, namely En- +glish as the source language, and Arabic, Chinese, +French, Kinyarwanda, and Spanish as the target +languages. +3 +Adaptive MT with Fuzzy Matches +In translation environments, similar approved seg- +ments are usually referred to as “fuzzy matches”, +and are stored in translation memories (TMs). Re- +searchers have investigated the possibilities of im- +proving MT quality and consistency with fuzzy +matches (Knowles et al., 2018; Bulte and Tez- +can, 2019; Xu et al., 2020). Incorporating fuzzy +matches into the MT process can help the system +generate more accurate translations, and try to en- +sure adherence to pre-approved terminology and +preferred style requirements. +In this set of experiments, we first extract sen- +tence pairs similar to each segment in the test +dataset, TICO-19. +To this end, we use the +paraphrase mining module from the Sentence- +Transformers library (Reimers and Gurevych, +2019). Paraphrase mining is the task of finding +texts with a similar meaning in a large corpus of +sentences. We use the all-MiniLM-L6-v2 model +because of its high accuracy and efficiency.3 For +each sentence, we retrieve up to top k other sen- +tences. +We experiment with diverse values of +1https://openai.com/api/ +2For higher values of few-shot prediction with Arabic, we had +to decrease the batch size. +3https://www.sbert.net/docs/pretrained_ +models.html + +Lang +GPT-3 Context +spBLEU ↑ +chrF++ ↑ +TER ↓ +COMET ↑ +EN-AR +zero-shot +27.6 +48.36 +70.6 +41.28 +random 2-shot +28.94 +49.35 +70.55 +43.32 +fuzzy1-shot +36.38 +55.08 +63.99 +55.1 +fuzzy 2-shot +38.41 +56.57 +62.31 +57.36 +fuzzy 3-shot +39.75 +57.52 +61.12 +59.68 +fuzzy 4-shot +40.84 +58.27 +60.39 +62.16 +fuzzy 5-shot +41.33 +58.64 +59.95 +62.65 +fuzzy 7-shot +41.81 +59.1 +59.38 +64.01 +EN-ES +zero-shot +50.63 +69.16 +40.44 +75.1 +random 2-shot +54.78 +73.12 +36.09 +85.25 +fuzzy 2-shot +59.64 +75.83 +32.56 +90.37 +fuzzy 5-shot +61.24 +76.73 +31.32 +91.51 +fuzzy 10-shot +61.77 +77.05 +30.9 +92.0 +EN-FR +zero-shot +44.87 +65.29 +50.34 +58.67 +random 2-shot +45.91 +65.4 +49.92 +57.6 +fuzzy 1-shot +48.39 +66.58 +48.18 +59.49 +fuzzy 2-shot +49.79 +67.41 +46.79 +61.38 +fuzzy 3-shot +50.96 +68.06 +45.85 +61.97 +fuzzy 4-shot +51.89 +68.5 +44.94 +62.7 +fuzzy 5-shot +51.94 +68.43 +45.09 +62.81 +fuzzy 10-shot +53.72 +69.39 +43.82 +63.57 +EN-RW +zero-shot +2.84 +22.37 +142.08 +N/A +random 2-shot +3.8 +25.19 +129.88 +N/A +fuzzy 2-shot +12.23 +36.66 +105.54 +N/A +fuzzy 5-shot +14.96 +39.84 +100.11 +N/A +fuzzy 10-shot +17.87 +41.44 +92.84 +N/A +EN-ZH +zero-shot +32.41 +40.82 +99.45 +59.87 +random 2-shot +38.72 +44.06 +87.56 +68.39 +fuzzy 2-shot +46.18 +49.12 +69.0 +73.9 +fuzzy 5-shot +47.94 +50.28 +64.96 +74.86 +fuzzy 10-shot +49.11 +51.22 +63.14 +75.3 +Table 1: Adaptive MT with fuzzy matches for GPT-3 few-shot in-context learn- +ing outperforms using random sentence pairs. Increasing the number of fuzzy +matches can improve the translation quality further. The table shows consistent +results for EN-AR, EN-ES, EN-FR, EN-RW, and EN-ZH language pairs. +1 to 10 sentence(s). We arrange fuzzy matches to +make higher matches closer to the segment to be +translated.4 +Table 2 shows numbers of matches +based on fuzzy match threshold in a 2-shot and 5- +shot scenarios. +Fuzzy +Threshold +Segment Statistics +fuzzy 2-shot +fuzzy 5-shot +>90% +167 +2.7% +168 +1.1% +89-80% +751 +12.2% +1,103 +7.2% +79-70% +1,593 +25.9% +3,143 +20.5% +69-60% +1,825 +29.7% +4,661 +30.4% +<60% +1,804 +29.4% +6,275 +40.9% +Total +6,140 = 3,070*2 +15,350 = 3,070*5 +Table 2: Numbers and percentages of segments based on +fuzzy threshold, for 2-shot and 5-shot experiments using +fuzzy matches for in-context learning. The English source +is used to calculate similarity across the 5 language pairs. +The following example shows an English- +to-Arabic prompt that incorporates two fuzzy +matches (2-shot) into the translation request. +4We experimented with reversing the order, and there was no +significant difference according to the evaluation results. +English: +Arabic: +English: +Arabic: +English: +Arabic: +Results illustrated by Figure 1 show that few- +shot translation with GPT-3 using fuzzy matches +as context outperforms few-shot translation with +random examples, although using random sen- +tence pairs outperforms zero-shot translation. As +demonstrated by Table 1 across five language +pairs, adding more fuzzy matches improves the +translation quality further. At some point, there +might be diminishing returns of adding more sim- +ilar sentences as their similarity score decreases. +In other words, increasing the number of fuzzy +matches from 2 sentences to 3, 4, 5, and 10 sen- +tences incrementally improves translation quality, +but with smaller quality gains. +4 +GPT-3 vs Encoder-Decoder MT Models +In this section, we aim to compare evaluation +results we obtained from various MT encoder- +decoder Transformer-based systems (Vaswani et +al., 2017) with those from GPT-3. To this end, +we translate our test dataset with a range of open- +source and commercial MT models, including +DeepL Translate API,5 Google Cloud Translation +API, OPUS (Tiedemann, 2020),6 and NLLB-200 +(NLLB Team et al., 2022). We converted OPUS +and NLLB models to the CTranslate2 format with +int8 quantization for efficiency. Inference parame- +ters include beam size 4 and max batch size 2024, +on a GPU A100-SXM4-40GB (Google Colab Pro). +For tokenization, we used SentencePiece (Kudo +and Richardson, 2018) with the source and tar- +get sub-wording models provided for each OPUS +model, and the multilingual model provided by +NLLB for tokenization.7 +We observe that for high-resource languages, +adaptive MT with fuzzy matches using GPT-3 +few-shot in-context learning (cf. Section 3) can +outperform strong encoder-decoder MT systems. +5DeepL supports French, Spanish, and Chinese, but not Ara- +bic and Kinyarwanda. +6We use OPUS models from Tatoeba-Challenge, specifically +the models augmented with back-translation, and trained with +Transformer-Big. +7flores200 sacrebleu tokenizer spm.model is used for +both tokenization for NLLB and also for spBLEU (Goyal et +al., 2022) in sacreBLEU. + +Figure 2: Evaluation results for GPT-3 few-shot translation with 5 or 10 fuzzy matches compared to encoder-decoder MT models (DeepL, Google, OPUS, and +NLLB). Specifically, for EN-ES, EN-FR, and EN-ZH language pairs, few-shot translation with GPT-3 outperforms conventional systems. +For the English-to-French and English-to-Spanish +language pairs, few-shot translation with GPT-3 +incorporating only 5 fuzzy matches outperforms +strong encoder-decoder MT models, as demon- +strated by Figure 2. For English-to-Chinese trans- +lation, only when we used 10 fuzzy matches could +we achieve better results. However, for English-to- +Arabic and English-to-Kinyarwanda translations, +results were not on par with the other three lan- +guage pairs, most likely due to the limited support +of these languages.8 The results are detailed in Ta- +ble 3. +5 +Incorporating Encoder-Decoder MT +As we demonstrated in the previous section, +encoder-decoder MT models have achieved high +translation quality for several language pairs. +Nevertheless, adaptive MT with LLM few-shot in- +context learning can surpass such quality, espe- +cially for high-resource languages. In this section, +we investigate whether we can utilize encoder- +decoder MT models to further improve adaptive +translation with GPT-3. In the next subsections, +we study two scenarios: +1. appending fuzzy matches with MT from an +8For English-to-Arabic, we could only test up to 7 matches +(not 10 matches) because the GPT-3.5 tokenizer generates +many more tokens for some Unicode languages, which can +easily hit the max length of 4000 tokens. The way we under- +stand it, this is a bug in the GPT-3 tokenizer. +encoder-decoder model to enhance in-context +learning. Here is an example for a few-shot +English-to-Chinese prompt. +English: +Chinese: +English: +Chinese: +English: +MT: +Chinese: +2. translating the source side of fuzzy matches, +and using these translations for few-shot in- +context learning along with the original trans- +lation. Here is an example for an English-to- +Chinese prompt: +English: +MT: +Chinese: +English: +MT: +Chinese: +English: +MT: +Chinese: +5.1 +Fuzzy matches + new segment MT +Incorporating a translation from an encoder- +decoder MT model with fuzzy matches, we could + +spBLEU +Lang +OPUS (bt-big) +DeepL APl +NLLB3.3B +Google APl +GPT-3 fuzzy 5-shot +GPT-3 fuzzy 10-shot +chrF++ +91.51 +92.00 +COMET +85.68 +86.86 +86.62 +83.69 +76.73 +77.05 +72.66 +72.87 +74.60 +75.17 +58.98 +61.24 +61.77 +EN-ES +55.39 +57.47 +54.99 +68.43 +69.39 +65.08 +66.45 +66.89 +66.34 +61.01 +60.91 +62.81 +63.57 +EN-FR +56.29 +59.01 +51.94 +53.72 +46.05 +47.38 +47.27 +46.81 +73.62 +74.86 +75.30 +69.92 +EN-ZH +53.89 +50.40 +52.02 +50.28 +51.22 +47.67 +48.58 +47.94 +49.11 +40.72 +37.51 +37.79 +39.08 +31.35achieve substantial improvements over the base- +line MT. For example, although OPUS English-to- +Arabic translation quality outperforms GPT-3 few- +shot translation with 5 fuzzy matches, appending +these fuzzy matches with OPUS translation out- +performs both OPUS translation only and GPT- +3 translation with fuzzy matches only. Similarly, +adding Google English-to-Chinese translation to 5 +fuzzy matches outperforms both baselines. Even +for the very low-resource English-to-Kinyarwanda +language pair, we relatively notice a similar be- +haviour, using MT outputs of OPUS or NLLB +models. Table 4 illustrates the results. +Figure 3: The English-to-Arabic translation by OPUS is better than the GPT-3 +few-shot translation. +When we incorporate both fuzzy matches and OPUS +translation for the new segment into GPT-3 few-shot in-context learning, the +generated translation outperforms both baseline translations. +However, we observe that if the translation with +only fuzzy matches is significantly better than the +encoder-decoder MT baseline, we may not achieve +further gains. +For example, the GPT-3 transla- +tions with 5 fuzzy matches are already much better +than the OPUS translation for English-to-French +or Google translation for English-to-Spanish. That +is why incorporating the MT output from OPUS +or Google did not enhance the GPT-3 translation +quality for these language pairs. +5.2 +Fuzzy matches + all segments MT +In Section 5.1, we added MT of the new segment +from an encoder-decoder model to fuzzy matches, +in order to enhance GPT-3 in-context learning. In +this set of experiments, we include MT for all +fuzzy matches and also for the new source segment +to be translated. For the English-to-Spanish lan- +guage pair, this reveals slightly better results than +including MT for only the new source segment to +be translated. +Lang +System +spBLEU ↑ +chrF++ ↑ +TER ↓ +COMET ↑ +EN-AR +OPUS (bt-big) +43.11 +60.79 +57.24 +63.64 +NLLB 600M +35.66 +54.6 +62.07 +54.53 +NLLB 1.2B +41.1 +58.51 +57.15 +63.85 +NLLB 3.3B +43.42 +60.11 +55.58 +66.8 +Google API +43.56 +61.58 +57.79 +65.5 +GPT-3 fuzzy 5-shot +41.33 +58.64 +59.95 +62.65 +GPT-3 fuzzy 7-shot +41.81 +59.1 +59.38 +64.01 +EN-ES +OPUS (bt-big) +54.99 +72.66 +36.26 +83.69 +NLLB 600M +53.31 +72.19 +37.13 +83.09 +NLLB 1.2B +56.1 +73.85 +34.96 +85.91 +NLLB 3.3B +57.47 +74.6 +33.99 +86.86 +DeepL API +55.39 +72.87 +36.21 +85.68 +Google API +58.98 +75.17 +32.46 +86.62 +GPT-3 fuzzy 5-shot +61.24 +76.73 +31.32 +91.51 +GPT-3 fuzzy 10-shot +61.77 +77.05 +30.9 +92.0 +EN-FR +OPUS (bt-big) +46.05 +65.08 +49.8 +56.29 +NLLB 600M +43.25 +64.17 +51.28 +56.16 +NLLB 1.2B +46.3 +66.25 +48.68 +59.76 +NLLB 3.3B +47.27 +66.89 +48.19 +60.91 +DeepL API +47.38 +66.45 +48.47 +61.01 +Google API +46.81 +66.34 +47.01 +59.01 +GPT-3 fuzzy 5-shot +51.94 +68.43 +45.09 +62.81 +GPT-3 fuzzy 10-shot +53.72 +69.39 +43.82 +63.57 +EN-RW +OPUS (Tatoeba 2021) +1.38 +15.32 +153.58 +N/A +OPUS (2020) +5.58 +27.05 +101.25 +N/A +NLLB 600M +19.46 +47.61 +80.01 +N/A +NLLB 1.2B +23.6 +50.73 +74.53 +N/A +NLLB 3.3B +25.17 +52.59 +73.06 +N/A +Google API +20.63 +48.37 +73.54 +N/A +GPT-3 fuzzy 5-shot +14.96 +39.84 +100.11 +N/A +GPT-3 fuzzy 10-shot +17.87 +41.44 +92.84 +N/A +EN-ZH +OPUS (bt-big) +37.51 +40.72 +121.49 +50.4 +NLLB 600M +24.9 +33.87 +109.37 +39.28 +NLLB 1.2B +29.02 +37.45 +110.22 +50.05 +NLLB 3.3B +31.35 +39.08 +109.52 +53.89 +DeepL API +37.79 +47.67 +100.83 +69.92 +Google API +48.58 +52.02 +70.87 +73.62 +GPT-3 fuzzy 5-shot +47.94 +50.28 +64.96 +74.86 +GPT-3 fuzzy 10-shot +49.11 +51.22 +63.14 +75.3 +Table 3: Comparing GPT-3.5 few-shot translation using fuzzy matches with +encoder-decoder MT systems, DeepL Translate API, Google Cloud Translation +API, OPUS (Tatoeba-Challenge, with back-translation and Transformer-Big), +and NLLB-200 (600M, 1.2B & 3.3B parameters). +6 +Bilingual Terminology Extraction +Terminology extraction is the task of automatically +defining domain-specific terms in a dataset. Ex- +tracted terms are naturally used for building glos- +saries to help translators. Furthermore, it is pos- +sible to improve MT performance through finding +sentences that include these terms and fine-tuning +the system with them (Hu et al., 2019; Haque et +al., 2020). +In this set of experiments, we use GPT-3 to auto- +matically extract 5 bilingual terms from each sen- +tence pair in the test dataset. For parameters, we +use temperature 0 and top p 1. The prompt we use +for bilingual terminology extraction is as follows. +: +: +Extract terms from the above sentence pair. +Type each term and its +equivalent in one line, separated by ’’. +1. + +MT (OPUS) +GPT-3 fuzzy 5-shot +GPT-3 fuzzy 5-shot + MT +SpBLEU +chrF++ +67.74 +COMET +63.64 +62.65 +62.90 +60.79 +58.64 +45.90 +43.11 +41.33Lang +System +spBLEU ↑ +chrF++ ↑ +TER ↓ +COMET ↑ +EN-AR +MT (OPUS) +43.11 +60.79 +57.24 +63.64 +GPT-3 fuzzy 5-shot +41.33 +58.64 +59.95 +62.65 +GPT-3 fuzzy 5-shot + 1-MT +45.9 +62.9 +55.14 +67.74 +EN-ES +MT (Google) +58.98 +75.17 +32.46 +86.62 +GPT-3 fuzzy 2-shot +59.64 +75.83 +32.56 +90.37 +GPT-3 fuzzy 2-shot + 1-MT +59.82 +75.73 +32.16 +89.0 +GPT-3 fuzzy 2-shot + all-MT +60.2 +76.06 +32.32 +92.0 +GPT-3 fuzzy 5-shot +61.24 +76.73 +31.32 +91.51 +GPT-3 fuzzy 5-shot + 1-MT +60.49 +76.16 +31.49 +89.55 +GPT-3 fuzzy 5-shot + all-MT +61.1 +76.52 +31.8 +92.07 +EN-FR +MT (OPUS) +46.05 +65.08 +49.8 +56.29 +GPT-3 fuzzy 5-shot +51.94 +68.43 +45.09 +62.81 +GPT-3 fuzzy 5-shot + 1-MT +47.95 +66.72 +48.34 +59.69 +EN-RW +MT #1 (Google) +20.63 +48.37 +73.54 +N/A +GPT-3 fuzzy 5-shot +14.96 +39.84 +100.11 +N/A +GPT-3 fuzzy 5-shot + 1-MT #1 +22.51 +49.69 +72.97 +N/A +MT #2 (NLLB 3.3B) +25.17 +52.59 +73.06 +N/A +GPT-3 fuzzy 5-shot + 1-MT #2 +25.59 +53.12 +72.73 +N/A +EN-ZH +MT (Google) +48.58 +52.02 +70.87 +73.62 +GPT-3 fuzzy 5-shot +47.94 +50.28 +64.96 +74.86 +GPT-3 fuzzy 5-shot + 1-MT +49.45 +52.4 +67.81 +74.61 +Table 4: Combining fuzzy matches with high-quality MT can improve translation with GPT-3 few-shot in-context learning, especially for low-resource and medium- +resource languages. 1-MT refers to appending fuzzy matches with the MT of the segment to be translated, while all-MT refers to additionally adding MT for each +segment of the fuzzy matches along with its approved translation. +Human evaluation was performed for Arabic, +French, and Spanish. We provided the evaluators +with a random sample of 500 sentences and their +extracted terms. They were asked to use a 0-1 scale +to determine whether each source and target term +were equivalent, and whether the extracted terms +were actually in the sentence pair (relevant inflex- +ions are acceptable). In several cases where the +evaluators marked the extracted term pair with 0, +the model had made up either the source, target, +or both; although it might be correct, it was not in +the provided sentence pair. In other cases, the ex- +tracted term was partial, sometimes due to reach- +ing the maximum length of tokens. Nevertheless, +as Table 5 illustrates, the majority of the terms in +the provided sample were accurately extracted by +the model. +Lang +Sentences +Terms +Correct +% +EN-AR +500 +2,500 +2,427 +97.08 +EN-ES +500 +2,500 +2,397 +95.88 +EN-FR +500 +2,500 +2,382 +95.28 +Table 5: Human evaluation results for the terminology extrac- +tion task for English-to-Arabic (EN-AR), English-to-Spanish +(EN-ES), and English-to-French (EN-FR) language pairs. +7 +Terminology-Constrained MT +As we saw in Section 3, adding more fuzzy +matches enhances in-context learning and hence +improves the translation quality. However, early in +the translation project, we might not have so many +fuzzy matches. By incorporating domain-specific +terminology, the system can produce translations +that are more accurate and consistent with the +terminology used in that field. +In this section, +we investigate integrating terms in the process +when there are N fuzzy matches. For example, +if we have only two fuzzy matches, we either ex- +tract terms from these similar sentences or from a +glossary, and use those that match up to 5-gram +phrases in the source sentence to be translated. In +this work, we use the terminology extraction pro- +cess elaborated in Section 6. Obviously, if a pre- +approved glossary is available, it can be used in- +stead. We investigate three scenarios: +1. Few-shot translation with 2 fuzzy matches +and their terms. +As we do not have terms +for the segment to be translated, we use +terms from the 2 fuzzy matches if they are +found in a set of n-grams (1-5) of the seg- +ment to be translated. Integrating terms into +two-shot prediction, i.e. +using both terms +and two fuzzy matches for in-context learn- + +(a) Zero-shot translation with max 5 glossary terms found in the source +(b) Few-shot translation with both fuzzy matches and glossary terms +Figure 4: Terminology-constrained MT with GPT-3. Evaluation results across EN-AR, EN-ES, EN-FR, and EN-ZH language pairs. Integration of terms from +a pre-approved glossary improves translation for both zero-shot and 2-shot scenarios, although gains from zero-shot prediction are significantly higher. +ing, outperforms using fuzzy matches only. +This is an example for an English-to-Spanish +prompt: +Terms: +English: +Spanish: +Terms: +English: +Spanish: +Terms: +English: +Spanish: +2. We automatically compile a glossary includ- +ing all terms from the dataset, with 2+ fre- +quency, and up to 5-grams. If there are multi- +ple targets for the same source, the term pair +with the highest frequency is selected. Stop +words and terms with empty source or tar- +get sides are excluded. The list is sorted by +n-gram length, so terms with longer n-grams +are prioritized. As illustrated by Table 6, in- +tegrating terms from a glossary outperforms +adding terms from only two fuzzy matches, +most likely due to the diversity that this op- +tion offers. In prompts, we experiment with +adding maximum 5 terms and maximum 10 +terms, which does not show a huge difference +in performance; in some cases only a smaller +number of terms is available in the glossary. +This is an example for an English-to-Spanish +prompt: +Terms: +English: +Spanish: +Terms: +English: +Spanish: +Terms: +English: +Spanish: +3. Zero-shot translation, i.e. without any fuzzy +matches. This is similar to the previous sce- +nario, except that we only use terms from +the glossary. In zero-shot prediction, adding +terms from the glossary improves the transla- +tion quality. As shown in Table 6, improve- +ments are significant across all 5 language +pairs. An English-to-Spanish prompt might +look like this: +Terms: = - += ... = +Engligh: +Spanish: +8 +Conclusion +In this work, we conducted several experiments to +assess the performance of GPT-3.5 across multi- +ple translation tasks, namely adaptive MT using +fuzzy matches (cf. Section 3), MT post-editing (cf. +Section 5), terminology extraction (cf. Section 6), +and terminology-constrained MT (cf. Section 7). + +Lang +zero-shot +zero-shot +terms +SpBLEU 个 +chrF++ 个 +COMET +54.53 +54.91 +EN-AR +48.36 +41.28 +35.38 +27.60 +87.21 +75.10 +74.18 +69.16 +55.99 +EN-ES +50.63 +65.29 +66.01 +58.67 +59.78 +EN-FR +44.87 +45.94 +68.60 +59.87 +EN-ZH +44.72 +40.82 +36.31 +32.41Lang +fuzzy 2-shot +fuzzy 2-shot + terms +SpBLEU 个 +chrF++ 个 +COMET +58.84 +62.17 +56.57 +57.36 +EN-AR +38.41 +41.27 +75.83 +90.37 +76.55 +91.05 +59.64 +60.50 +EN-ES +67.41 +67.74 +61.38 +59.90 +EN-FR +49.79 +50.63 +73.90 +73.88 +EN-ZH +49.12 +46.60 +49.51 +46.18Lang +System +spBLEU ↑ +chrF++ ↑ +TER ↓ +COMET ↑ +EN-AR +zero-shot +27.6 +48.36 +70.6 +41.28 +zero-shot + max 5 terms (glossary) +35.38 +54.53 +65.36 +54.91 +fuzzy 2-shot +38.41 +56.57 +62.31 +57.36 +fuzzy 2-shot + terms (fuzzy) +39.38 +57.22 +62.01 +59.36 +fuzzy 2-shot + max 5 terms (glossary) +41.27 +58.84 +60.09 +62.17 +fuzzy 2-shot + max 10 terms (glossary) +41.95 +59.34 +59.45 +62.48 +EN-ES +zero-shot +50.63 +69.16 +40.44 +75.1 +zero-shot + max 5 terms (glossary) +55.99 +74.18 +35.3 +87.21 +fuzzy 2-shot +59.64 +75.83 +32.56 +90.37 +fuzzy 2-shot + terms (fuzzy) +59.66 +75.91 +32.53 +90.04 +fuzzy 2-shot + max 5 terms (glossary) +60.5 +76.55 +31.93 +91.05 +fuzzy 2-shot + max 10 terms (glossary) +60.54 +76.58 +32.02 +91.05 +EN-FR +zero-shot +44.87 +65.29 +50.34 +58.67 +zero-shot + max 5 terms (glossary) +45.94 +66.01 +49.22 +59.78 +fuzzy 2-shot +49.79 +67.41 +46.79 +61.38 +fuzzy 2-shot + terms (fuzzy) +50.58 +67.93 +45.81 +62.04 +fuzzy 2-shot + max 3 terms (glossary) +50.46 +67.69 +46.22 +68.94 +fuzzy 2-shot + max 5 terms (glossary) +50.63 +67.74 +46.15 +59.9 +fuzzy 2-shot + max 10 terms (glossary) +49.64 +66.86 +47.34 +58.57 +EN-RW +zero-shot +2.84 +22.37 +142.08 +N/A +zero-shot + max 5 terms (glossary) +7.26 +30.83 +115.44 +N/A +fuzzy 2-shot +12.23 +36.66 +105.54 +N/A +fuzzy 2-shot + terms (fuzzy) +12.43 +36.48 +102.22 +N/A +fuzzy 2-shot + max 5 terms (glossary) +15.34 +39.96 +96.09 +N/A +fuzzy 2-shot + max 10 terms (glossary) +15.49 +40.53 +96.0 +N/A +EN-ZH +zero-shot +32.41 +40.82 +99.45 +59.87 +zero-shot + max 5 terms (glossary) +36.31 +44.72 +96.45 +68.6 +zero-shot + max 10 terms (glossary) +36.64 +45.06 +96.24 +68.94 +fuzzy 2-shot +46.18 +49.12 +69.0 +73.9 +fuzzy 2-shot + terms (fuzzy) +46.16 +49.11 +68.79 +73.41 +fuzzy 2-shot + max 5 terms (glossary) +46.6 +49.51 +69.46 +73.88 +fuzzy 2-shot + max 10 terms (glossary) +46.31 +49.25 +69.39 +73.57 +Table 6: Terminology-constrained MT with GPT 3.5 outperforms both zero-shot and 2-shot translation with fuzzy matches, although gains are much higher for +zero-shot translation. For zero-shot translation, we experimented with adding terms from a glossary. For 2-shot translation with fuzzy matches, we compared +adding terms from these 2 fuzzy matches to adding terms from a glossary. The latter revealed better results. +Moreover, we compared its translation quality with +strong encoder-decoder MT systems. +Generally +speaking, results obtained from these experiments +are very promising. +While some high-resource +languages such as English-to-French, English-to- +Spanish and even English-to-Chinese show excel- +lent results, other languages have lower support +either because they are low-resource languages +such as English-to-Kinyarwanda or because of +issues in the GPT-3 tokenizer such as English-to- +Arabic. Nevertheless, when we used GPT-3.5 for +MT post-editing of the English-to-Arabic trans- +lation obtained from OPUS, the quality signifi- +cantly surpassed that obtained from both OPUS +and Google Translation API. This means that dif- +ferent pipelines can be adopted in production for +different language pairs, based on the level of sup- +port of these languages by an LLM. +In the future, we would like to conduct simi- +lar experiments with open-source LLMs such as +BLOOM and GPT-J. We observe that OpenAI ser- +vices, including GPT-3, do not cover several coun- +tries. Accordingly, open-source works are not only +important for their quality equivalence or cost- +effectiveness, but perhaps most importantly for +better accessibility. +For terminology extraction, we would like to try +“phrases” instead of “terms”. This would gener- +ate longer strings. We would like to see the effect +of using such longer phrases, especially for low- +resource languages. +This work mainly aims at understanding the +quality and level of support that LLMs like GPT-3 +can offer (out of the box) for translation across a +range of diverse language pairs. In the future, we +might consider starting with fine-tuning the model, +and then conducting similar experiments. This can +be especially useful for low-resource languages + +and rare domains, and can help enhance quality +and efficiency. +Finally, we intend to extend human evaluation +to cover not only terminology extraction, but also +other aspects, especially terminology-constrained +MT, to see to what extent the model adheres to +the required terms, and how this affects the overall +translation quality, from the perspective of profes- +sional linguists. +Acknowledgements +This work is supported by the Science Foun- +dation Ireland Centre for Research Training in +Digitally-Enhanced Reality (d-real) under Grant +No. 18/CRT/6224, the ADAPT Centre for Digital +Content Technology which is funded under the +Science Foundation Ireland (SFI) Research Cen- +tres Programme (Grant No. 13/RC/2106) and is +co-funded under the European Regional Develop- +ment Fund, and Microsoft Research. +We would like to extend our sincere thanks to +Julie Locquet, Senior Linguist; Philippe Locquet, +Senior Linguist and Academic Program Manager +at Wordfast; and Dr Muhammed Yaman Muhaisen, +Ophthalmologist and Linguist, for conducting the +evaluation of the terminology extraction task. +References +[Anastasopoulos et al.2020] Anastasopoulos, +Antonios, +Alessandro Cattelan, +Zi-Yi Dou, +Marcello Federico, +Christian +Federmann, +Dmitriy +Genzel, +Franscisco +Guzm´an, Junjie Hu, Macduff Hughes, Philipp Koehn, +Rosie Lazar, Will Lewis, Graham Neubig, Mengmeng +Niu, Alp ¨Oktem, Eric Paquin, Grace Tang, and Sylwia Tur. +2020. 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Association for Computational +Linguistics. + diff --git a/K9FQT4oBgHgl3EQfTzY7/content/tmp_files/load_file.txt b/K9FQT4oBgHgl3EQfTzY7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e94cd847ca863b11728f4b52de2522c05e7e12c3 --- /dev/null +++ b/K9FQT4oBgHgl3EQfTzY7/content/tmp_files/load_file.txt @@ -0,0 +1,1153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf,len=1152 +page_content='Adaptive Machine Translation with Large Language Models Yasmin Moslem ADAPT Centre School of Computing Dublin City University Dublin, Ireland yasmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='moslem@adaptcentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='ie Rejwanul Haque ADAPT Centre Computing Department South East Technological University Carlow, Ireland rejwanul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='haque@adaptcentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='ie Andy Way ADAPT Centre School of Computing Dublin City University Dublin, Ireland andy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='way@adaptcentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='ie Abstract Consistency is a key requirement of high- quality translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' It is especially important to adhere to pre-approved terminology and corrected translations in domain-specific projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Machine translation (MT) has achieved significant progress in the area of domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' However, real-time adaptation remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Large-scale language models (LLMs) have recently shown interesting capabilities of in-context learning, where they learn to replicate certain input-output text generation patterns, without further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' By feeding an LLM with a prompt that consists of a list of translation pairs, it can then simulate the domain and style characteristics at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This work aims to investigate how we can utilize in-context learning to improve real-time adaptive MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Our extensive experiments show promising results at translation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For example, GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 can adapt to a set of in-domain sentence pairs and/or terminology while translating a new sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We observe that the translation quality with few-shot in-context learning can surpass that of strong encoder-decoder MT systems, especially for high-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Moreover, we investigate whether we can combine MT from strong encoder-decoder models with fuzzy matches, which can further improve the translation, especially for less supported languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We conduct our experiments across five diverse languages, namely English-to-Arabic (EN-AR), English- to-Chinese (EN-ZH), English-to-French (EN-FR), English-to-Kinyarwanda (EN- RW), and English-to-Spanish (EN-ES) language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' © 2023 The authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This article is licensed under a Creative Commons 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='0 licence, no derivative works, attribution, CC- BY-ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Figure 1: Evaluation results for GPT-3 zero-shot, and few-shot translation with random context or fuzzy matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Average scores across EN-AR, EN-ES, EN- FR, and EN-ZH language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' While using a random context outperforms zero-shot translation, using fuzzy matches reveals the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 1 Introduction Adaptive MT is a type of machine translation that utilizes feedback from users to improve the qual- ity of the translations over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Feedback usually includes corrections to previous translations, ter- minology and style guides, as well as ratings of the quality of the translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This can be partic- ularly useful for domain-specific scenarios, where baseline MT systems may have insufficient rele- vant data to accurately translate certain terms or phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' There are still several challenges to ef- fectively incorporate user feedback into the trans- lation process, especially at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this work, we use a relatively wide definition of adap- tive MT to refer to learning from similar transla- tions (fuzzy matches) found in approved transla- tion memories (TMs) on the fly (Farajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='13294v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='CL] 30 Jan 2023 zero-shot random 2-shot fuzzy 2-shot fuzzy 5-shot SpBLEU 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 chrF++ COMET 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='75 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='52 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='23 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='98 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='91 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='61 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='09 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='882017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Wuebker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Peris and Casacuberta, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Etchegoyhen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2021), as well as real- time terminology-constrained MT (Hokamp and Liu, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Post and Vilar, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Dinu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Autoregressive decoder-only LLMs, such as GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022), GPT-J (Wang and Komatsuzaki, 2021), BLOOM (Le Scao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022), and PaLM (Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022) are trained to predict the next word given the previous context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' During unsupervised pre- training, a language model develops a broad set of pattern recognition abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' It then uses these abil- ities at inference time to rapidly adapt to or recog- nize the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In their experiments, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' (2020) use the term “in-context learning” to describe the inner loop of this process, which oc- curs within the forward-pass upon each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this sense, in-context learning is a scenario where a pre-trained language model at inference time learns to replicate certain input-output text generation patterns without further fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' They show that autoregressive LLMs such as GPT- 3 can perform well on diverse tasks, through zero- shot, one-shot, and few-shot in-context learning without weight updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Previous researchers in- vestigated using neural language models for MT through few-shot in-context learning (Vilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022) and even in zero-shot settings (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Other researchers proposed using LLMs for generating synthetic domain-specific data for MT domain adaptation (Moslem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The main contribution of this paper is investi- gating the capabilities of LLMs such as GPT-3 for real-time adaptive MT through in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In particular, we would like to understand the qual- ity with which such models can perform the fol- lowing tasks, without any further training: Adapting new translations to match the termi- nology and style of previously approved TM fuzzy matches, at inference time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Matching or outperforming the quality of translations generated by encoder-decoder MT models across a number of languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Fixing translations from stronger encoder- decoder MT systems using fuzzy matches, which is especially useful for low-resource languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' and Terminology-constrained MT, by first defin- ing terminology in the relevant sentences or dataset, and then forcing new translations to use these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 2 Experimental Setup In all our experiments, we use GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 text- davinci-003 model via its official API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 For param- eters, we use top-p 1, with temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3 for the three translation tasks, and 0 for the terminology extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' To avoid generating new lines in the translation tasks, the option stop can be set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For the maximum length of tokens, we observe that French and Spanish tokens can be 3–4 times the number of English source words, while other lan- guages can be longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Hence, we roughly choose a length multiplier value, which we set to 8 for Arabic, 5 for Chinese and Kinyarwanda, and 4 for French and Spanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We used bach requests with a batch size of 20 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2 For the test dataset, we use TICO-19 (Anasta- sopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2020), which includes 3070 unique segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We target a range of languages with di- verse scripts and amounts of resources, namely En- glish as the source language, and Arabic, Chinese, French, Kinyarwanda, and Spanish as the target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 3 Adaptive MT with Fuzzy Matches In translation environments, similar approved seg- ments are usually referred to as “fuzzy matches”, and are stored in translation memories (TMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Re- searchers have investigated the possibilities of im- proving MT quality and consistency with fuzzy matches (Knowles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Bulte and Tez- can, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Incorporating fuzzy matches into the MT process can help the system generate more accurate translations, and try to en- sure adherence to pre-approved terminology and preferred style requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this set of experiments, we first extract sen- tence pairs similar to each segment in the test dataset, TICO-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' To this end, we use the paraphrase mining module from the Sentence- Transformers library (Reimers and Gurevych, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Paraphrase mining is the task of finding texts with a similar meaning in a large corpus of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We use the all-MiniLM-L6-v2 model because of its high accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3 For each sentence, we retrieve up to top k other sen- tences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We experiment with diverse values of 1https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='com/api/ 2For higher values of few-shot prediction with Arabic, we had to decrease the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='sbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='net/docs/pretrained_ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='html Lang GPT-3 Context spBLEU ↑ chrF++ ↑ TER ↓ COMET ↑ EN-AR zero-shot 27.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='31 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='36 fuzzy 3-shot 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='75 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='52 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='12 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='68 fuzzy 4-shot 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='27 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='16 fuzzy 5-shot 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='33 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='95 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='65 fuzzy 7-shot 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='01 EN-ES zero-shot 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='63 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='16 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='44 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 random 2-shot 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='78 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='12 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='09 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='25 fuzzy 2-shot 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='83 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='56 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='37 fuzzy 5-shot 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='32 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 fuzzy 10-shot 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='77 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='05 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='0 EN-FR zero-shot 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='29 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='34 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='67 random 2-shot 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='91 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='92 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='6 fuzzy 1-shot 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='58 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='18 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='49 fuzzy 2-shot 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='41 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 fuzzy 3-shot 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='06 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='85 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='97 fuzzy 4-shot 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='89 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='7 fuzzy 5-shot 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='43 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 fuzzy 10-shot 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='72 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='82 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='57 EN-RW zero-shot 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='37 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='08 N/A random 2-shot 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='19 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='88 N/A fuzzy 2-shot 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='23 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='66 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='54 N/A fuzzy 5-shot 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 N/A fuzzy 10-shot 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='44 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 N/A EN-ZH zero-shot 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='41 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='82 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='45 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 random 2-shot 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='72 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='06 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='56 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 fuzzy 2-shot 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='18 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='12 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9 fuzzy 5-shot 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='86 fuzzy 10-shot 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='22 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='14 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3 Table 1: Adaptive MT with fuzzy matches for GPT-3 few-shot in-context learn- ing outperforms using random sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Increasing the number of fuzzy matches can improve the translation quality further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The table shows consistent results for EN-AR, EN-ES, EN-FR, EN-RW, and EN-ZH language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 1 to 10 sentence(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We arrange fuzzy matches to make higher matches closer to the segment to be translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='4 Table 2 shows numbers of matches based on fuzzy match threshold in a 2-shot and 5- shot scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Fuzzy Threshold Segment Statistics fuzzy 2-shot fuzzy 5-shot >90% 167 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='7% 168 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1% 89-80% 751 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2% 1,103 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2% 79-70% 1,593 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9% 3,143 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5% 69-60% 1,825 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='7% 4,661 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='4% <60% 1,804 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='4% 6,275 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9% Total 6,140 = 3,070*2 15,350 = 3,070*5 Table 2: Numbers and percentages of segments based on fuzzy threshold, for 2-shot and 5-shot experiments using fuzzy matches for in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The English source is used to calculate similarity across the 5 language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The following example shows an English- to-Arabic prompt that incorporates two fuzzy matches (2-shot) into the translation request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 4We experimented with reversing the order, and there was no significant difference according to the evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' English: Arabic: English: Arabic: English: Arabic: Results illustrated by Figure 1 show that few- shot translation with GPT-3 using fuzzy matches as context outperforms few-shot translation with random examples, although using random sen- tence pairs outperforms zero-shot translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' As demonstrated by Table 1 across five language pairs, adding more fuzzy matches improves the translation quality further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' At some point, there might be diminishing returns of adding more sim- ilar sentences as their similarity score decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In other words, increasing the number of fuzzy matches from 2 sentences to 3, 4, 5, and 10 sen- tences incrementally improves translation quality, but with smaller quality gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 4 GPT-3 vs Encoder-Decoder MT Models In this section, we aim to compare evaluation results we obtained from various MT encoder- decoder Transformer-based systems (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2017) with those from GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' To this end, we translate our test dataset with a range of open- source and commercial MT models, including DeepL Translate API,5 Google Cloud Translation API, OPUS (Tiedemann, 2020),6 and NLLB-200 (NLLB Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We converted OPUS and NLLB models to the CTranslate2 format with int8 quantization for efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Inference parame- ters include beam size 4 and max batch size 2024, on a GPU A100-SXM4-40GB (Google Colab Pro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For tokenization, we used SentencePiece (Kudo and Richardson, 2018) with the source and tar- get sub-wording models provided for each OPUS model, and the multilingual model provided by NLLB for tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='7 We observe that for high-resource languages, adaptive MT with fuzzy matches using GPT-3 few-shot in-context learning (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Section 3) can outperform strong encoder-decoder MT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 5DeepL supports French, Spanish, and Chinese, but not Ara- bic and Kinyarwanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 6We use OPUS models from Tatoeba-Challenge, specifically the models augmented with back-translation, and trained with Transformer-Big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 7flores200 sacrebleu tokenizer spm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='model is used for both tokenization for NLLB and also for spBLEU (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2022) in sacreBLEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Figure 2: Evaluation results for GPT-3 few-shot translation with 5 or 10 fuzzy matches compared to encoder-decoder MT models (DeepL, Google, OPUS, and NLLB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Specifically, for EN-ES, EN-FR, and EN-ZH language pairs, few-shot translation with GPT-3 outperforms conventional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For the English-to-French and English-to-Spanish language pairs, few-shot translation with GPT-3 incorporating only 5 fuzzy matches outperforms strong encoder-decoder MT models, as demon- strated by Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For English-to-Chinese trans- lation, only when we used 10 fuzzy matches could we achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' However, for English-to- Arabic and English-to-Kinyarwanda translations, results were not on par with the other three lan- guage pairs, most likely due to the limited support of these languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='8 The results are detailed in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 5 Incorporating Encoder-Decoder MT As we demonstrated in the previous section, encoder-decoder MT models have achieved high translation quality for several language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Nevertheless, adaptive MT with LLM few-shot in- context learning can surpass such quality, espe- cially for high-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this section, we investigate whether we can utilize encoder- decoder MT models to further improve adaptive translation with GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In the next subsections, we study two scenarios: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' appending fuzzy matches with MT from an 8For English-to-Arabic, we could only test up to 7 matches (not 10 matches) because the GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 tokenizer generates many more tokens for some Unicode languages, which can easily hit the max length of 4000 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The way we under- stand it, this is a bug in the GPT-3 tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' encoder-decoder model to enhance in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Here is an example for a few-shot English-to-Chinese prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' English: Chinese: English: Chinese: English: MT: Chinese: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' translating the source side of fuzzy matches, and using these translations for few-shot in- context learning along with the original trans- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Here is an example for an English-to- Chinese prompt: English: MT: Chinese: English: MT: Chinese: English: MT: Chinese: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 Fuzzy matches + new segment MT Incorporating a translation from an encoder- decoder MT model with fuzzy matches, we could spBLEU Lang OPUS (bt-big) DeepL APl NLLB3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3B Google APl GPT-3 fuzzy 5-shot GPT-3 fuzzy 10-shot chrF++ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='00 COMET 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='68 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='86 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='62 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='69 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='05 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='66 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='60 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='17 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='98 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='77 EN-ES 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='47 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='99 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='43 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='08 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='45 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='89 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='34 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='01 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='91 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='57 EN-FR 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='29 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='01 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='72 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='05 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='27 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='62 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='86 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='30 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='92 EN-ZH 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='89 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='40 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='02 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='22 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='67 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='58 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='72 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='08 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='35achieve substantial improvements over the base- line MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For example, although OPUS English-to- Arabic translation quality outperforms GPT-3 few- shot translation with 5 fuzzy matches, appending these fuzzy matches with OPUS translation out- performs both OPUS translation only and GPT- 3 translation with fuzzy matches only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Similarly, adding Google English-to-Chinese translation to 5 fuzzy matches outperforms both baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Even for the very low-resource English-to-Kinyarwanda language pair, we relatively notice a similar be- haviour, using MT outputs of OPUS or NLLB models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Table 4 illustrates the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Figure 3: The English-to-Arabic translation by OPUS is better than the GPT-3 few-shot translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' When we incorporate both fuzzy matches and OPUS translation for the new segment into GPT-3 few-shot in-context learning, the generated translation outperforms both baseline translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' However, we observe that if the translation with only fuzzy matches is significantly better than the encoder-decoder MT baseline, we may not achieve further gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For example, the GPT-3 transla- tions with 5 fuzzy matches are already much better than the OPUS translation for English-to-French or Google translation for English-to-Spanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' That is why incorporating the MT output from OPUS or Google did not enhance the GPT-3 translation quality for these language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2 Fuzzy matches + all segments MT In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1, we added MT of the new segment from an encoder-decoder model to fuzzy matches, in order to enhance GPT-3 in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this set of experiments, we include MT for all fuzzy matches and also for the new source segment to be translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For the English-to-Spanish lan- guage pair, this reveals slightly better results than including MT for only the new source segment to be translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Lang System spBLEU ↑ chrF++ ↑ TER ↓ COMET ↑ EN-AR OPUS (bt-big) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 NLLB 600M 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='66 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='07 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='53 NLLB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2B 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='15 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='85 NLLB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3B 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='42 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='58 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='8 Google API 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='56 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='58 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 GPT-3 fuzzy 5-shot 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='33 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='95 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='65 GPT-3 fuzzy 7-shot 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='01 EN-ES OPUS (bt-big) 54.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='68 Google API 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='98 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='17 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='46 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='62 GPT-3 fuzzy 5-shot 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='32 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 GPT-3 fuzzy 10-shot 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='77 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='05 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='0 EN-FR OPUS (bt-big) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} 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+page_content='72 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='49 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='4 NLLB 600M 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='37 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 NLLB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} 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+page_content='08 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='52 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='89 DeepL API 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='67 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='83 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='92 Google API 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='58 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='02 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='62 GPT-3 fuzzy 5-shot 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='86 GPT-3 fuzzy 10-shot 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='22 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='14 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3 Table 3: Comparing GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 few-shot translation using fuzzy matches with encoder-decoder MT systems, DeepL Translate API, Google Cloud Translation API, OPUS (Tatoeba-Challenge, with back-translation and Transformer-Big), and NLLB-200 (600M, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2B & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3B parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 6 Bilingual Terminology Extraction Terminology extraction is the task of automatically defining domain-specific terms in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Ex- tracted terms are naturally used for building glos- saries to help translators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Furthermore, it is pos- sible to improve MT performance through finding sentences that include these terms and fine-tuning the system with them (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Haque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this set of experiments, we use GPT-3 to auto- matically extract 5 bilingual terms from each sen- tence pair in the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For parameters, we use temperature 0 and top p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The prompt we use for bilingual terminology extraction is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' : : Extract terms from the above sentence pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Type each term and its equivalent in one line, separated by ’’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' MT (OPUS) GPT-3 fuzzy 5-shot GPT-3 fuzzy 5-shot + MT SpBLEU chrF++ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='74 COMET 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='65 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='90 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='90 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='33Lang System spBLEU ↑ chrF++ ↑ TER ↓ COMET ↑ EN-AR MT (OPUS) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 GPT-3 fuzzy 5-shot 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='33 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='95 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='65 GPT-3 fuzzy 5-shot + 1-MT 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='9 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='14 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='74 EN-ES MT (Google) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='98 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='17 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='46 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='62 GPT-3 fuzzy 2-shot 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='83 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='56 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='37 GPT-3 fuzzy 2-shot + 1-MT 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='82 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='16 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='0 GPT-3 fuzzy 2-shot + all-MT 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='06 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='32 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='0 GPT-3 fuzzy 5-shot 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='32 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 GPT-3 fuzzy 5-shot + 1-MT 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='49 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='16 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='49 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='55 GPT-3 fuzzy 5-shot + all-MT 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='52 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='07 EN-FR MT (OPUS) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='05 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='08 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='29 GPT-3 fuzzy 5-shot 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='43 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 GPT-3 fuzzy 5-shot + 1-MT 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='95 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='72 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='34 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='69 EN-RW MT #1 (Google) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='63 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='37 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='54 N/A GPT-3 fuzzy 5-shot 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='11 N/A GPT-3 fuzzy 5-shot + 1-MT #1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='69 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='97 N/A MT #2 (NLLB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3B) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='17 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='59 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='06 N/A GPT-3 fuzzy 5-shot + 1-MT #2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='59 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='12 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='73 N/A EN-ZH MT (Google) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='58 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='02 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='62 GPT-3 fuzzy 5-shot 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='96 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='86 GPT-3 fuzzy 5-shot + 1-MT 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='45 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='81 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='61 Table 4: Combining fuzzy matches with high-quality MT can improve translation with GPT-3 few-shot in-context learning, especially for low-resource and medium- resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 1-MT refers to appending fuzzy matches with the MT of the segment to be translated, while all-MT refers to additionally adding MT for each segment of the fuzzy matches along with its approved translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Human evaluation was performed for Arabic, French, and Spanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We provided the evaluators with a random sample of 500 sentences and their extracted terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' They were asked to use a 0-1 scale to determine whether each source and target term were equivalent, and whether the extracted terms were actually in the sentence pair (relevant inflex- ions are acceptable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In several cases where the evaluators marked the extracted term pair with 0, the model had made up either the source, target, or both;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' although it might be correct, it was not in the provided sentence pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In other cases, the ex- tracted term was partial, sometimes due to reach- ing the maximum length of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Nevertheless, as Table 5 illustrates, the majority of the terms in the provided sample were accurately extracted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Lang Sentences Terms Correct % EN-AR 500 2,500 2,427 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='08 EN-ES 500 2,500 2,397 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='88 EN-FR 500 2,500 2,382 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 Table 5: Human evaluation results for the terminology extrac- tion task for English-to-Arabic (EN-AR), English-to-Spanish (EN-ES), and English-to-French (EN-FR) language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 7 Terminology-Constrained MT As we saw in Section 3, adding more fuzzy matches enhances in-context learning and hence improves the translation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' However, early in the translation project, we might not have so many fuzzy matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' By incorporating domain-specific terminology, the system can produce translations that are more accurate and consistent with the terminology used in that field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this section, we investigate integrating terms in the process when there are N fuzzy matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For example, if we have only two fuzzy matches, we either ex- tract terms from these similar sentences or from a glossary, and use those that match up to 5-gram phrases in the source sentence to be translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In this work, we use the terminology extraction pro- cess elaborated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Obviously, if a pre- approved glossary is available, it can be used in- stead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We investigate three scenarios: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Few-shot translation with 2 fuzzy matches and their terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' As we do not have terms for the segment to be translated, we use terms from the 2 fuzzy matches if they are found in a set of n-grams (1-5) of the seg- ment to be translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Integrating terms into two-shot prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' using both terms and two fuzzy matches for in-context learn- (a) Zero-shot translation with max 5 glossary terms found in the source (b) Few-shot translation with both fuzzy matches and glossary terms Figure 4: Terminology-constrained MT with GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Evaluation results across EN-AR, EN-ES, EN-FR, and EN-ZH language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Integration of terms from a pre-approved glossary improves translation for both zero-shot and 2-shot scenarios, although gains from zero-shot prediction are significantly higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' ing, outperforms using fuzzy matches only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This is an example for an English-to-Spanish prompt: Terms: English: Spanish: Terms: English: Spanish: Terms: English: Spanish: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We automatically compile a glossary includ- ing all terms from the dataset, with 2+ fre- quency, and up to 5-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' If there are multi- ple targets for the same source, the term pair with the highest frequency is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Stop words and terms with empty source or tar- get sides are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The list is sorted by n-gram length, so terms with longer n-grams are prioritized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' As illustrated by Table 6, in- tegrating terms from a glossary outperforms adding terms from only two fuzzy matches, most likely due to the diversity that this op- tion offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In prompts, we experiment with adding maximum 5 terms and maximum 10 terms, which does not show a huge difference in performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' in some cases only a smaller number of terms is available in the glossary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This is an example for an English-to-Spanish prompt: Terms: English: Spanish: Terms: English: Spanish: Terms: English: Spanish: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Zero-shot translation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' without any fuzzy matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This is similar to the previous sce- nario, except that we only use terms from the glossary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In zero-shot prediction, adding terms from the glossary improves the transla- tion quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' As shown in Table 6, improve- ments are significant across all 5 language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' An English-to-Spanish prompt might look like this: Terms: = - = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' = Engligh: Spanish: 8 Conclusion In this work, we conducted several experiments to assess the performance of GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 across multi- ple translation tasks, namely adaptive MT using fuzzy matches (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Section 3), MT post-editing (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Section 5), terminology extraction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Section 6), and terminology-constrained MT (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Lang zero-shot zero-shot +terms SpBLEU 个 chrF++ 个 COMET 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='53 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='91 EN-AR 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='36 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='60 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='21 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='10 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='18 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='16 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='99 EN-ES 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='63 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='29 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='01 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='67 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='78 EN-FR 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='60 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='87 EN-ZH 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='72 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='82 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='31 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='41Lang fuzzy 2-shot fuzzy 2-shot + terms SpBLEU 个 chrF++ 个 COMET 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='17 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='57 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='36 EN-AR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='41 41.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='41 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='74 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='90 EN-FR 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='79 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='63 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='90 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='88 EN-ZH 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='12 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='60 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='18Lang System spBLEU ↑ chrF++ ↑ TER ↓ COMET ↑ EN-AR zero-shot 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='36 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='28 zero-shot + max 5 terms (glossary) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='38 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='53 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='36 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='91 fuzzy 2-shot 38.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='36 fuzzy 2-shot + max 5 terms (glossary) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='27 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='84 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='17 fuzzy 2-shot + max 10 terms (glossary) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='95 59.' metadata={'source': 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(glossary) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='99 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='18 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='21 fuzzy 2-shot 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='83 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} 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+page_content='72 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='45 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='6 zero-shot + max 10 terms (glossary) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='64 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='06 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='24 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='94 fuzzy 2-shot 46.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='41 fuzzy 2-shot + max 5 terms (glossary) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='51 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='46 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='88 fuzzy 2-shot + max 10 terms (glossary) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='31 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='25 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='39 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='57 Table 6: Terminology-constrained MT with GPT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 outperforms both zero-shot and 2-shot translation with fuzzy matches, although gains are much higher for zero-shot translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For zero-shot translation, we experimented with adding terms from a glossary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For 2-shot translation with fuzzy matches, we compared adding terms from these 2 fuzzy matches to adding terms from a glossary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' The latter revealed better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Moreover, we compared its translation quality with strong encoder-decoder MT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Generally speaking, results obtained from these experiments are very promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' While some high-resource languages such as English-to-French, English-to- Spanish and even English-to-Chinese show excel- lent results, other languages have lower support either because they are low-resource languages such as English-to-Kinyarwanda or because of issues in the GPT-3 tokenizer such as English-to- Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Nevertheless, when we used GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='5 for MT post-editing of the English-to-Arabic trans- lation obtained from OPUS, the quality signifi- cantly surpassed that obtained from both OPUS and Google Translation API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This means that dif- ferent pipelines can be adopted in production for different language pairs, based on the level of sup- port of these languages by an LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In the future, we would like to conduct simi- lar experiments with open-source LLMs such as BLOOM and GPT-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We observe that OpenAI ser- vices, including GPT-3, do not cover several coun- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Accordingly, open-source works are not only important for their quality equivalence or cost- effectiveness, but perhaps most importantly for better accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' For terminology extraction, we would like to try “phrases” instead of “terms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This would gener- ate longer strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We would like to see the effect of using such longer phrases, especially for low- resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This work mainly aims at understanding the quality and level of support that LLMs like GPT-3 can offer (out of the box) for translation across a range of diverse language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' In the future, we might consider starting with fine-tuning the model, and then conducting similar experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' This can be especially useful for low-resource languages and rare domains, and can help enhance quality and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Finally, we intend to extend human evaluation to cover not only terminology extraction, but also other aspects, especially terminology-constrained MT, to see to what extent the model adheres to the required terms, and how this affects the overall translation quality, from the perspective of profes- sional linguists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Acknowledgements This work is supported by the Science Foun- dation Ireland Centre for Research Training in Digitally-Enhanced Reality (d-real) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 18/CRT/6224, the ADAPT Centre for Digital Content Technology which is funded under the Science Foundation Ireland (SFI) Research Cen- tres Programme (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 13/RC/2106) and is co-funded under the European Regional Develop- ment Fund, and Microsoft Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' We would like to extend our sincere thanks to Julie Locquet, Senior Linguist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' Philippe Locquet, Senior Linguist and Academic Program Manager at Wordfast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' and Dr Muhammed Yaman Muhaisen, Ophthalmologist and Linguist, for conducting the evaluation of the terminology extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' References [Anastasopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content='2020] Anastasopoulos, Antonios, Alessandro Cattelan, Zi-Yi Dou, Marcello Federico, Christian Federmann, Dmitriy Genzel, Franscisco Guzm´an, Junjie Hu, Macduff Hughes, Philipp Koehn, Rosie Lazar, Will Lewis, Graham Neubig, Mengmeng Niu, Alp ¨Oktem, Eric Paquin, Grace Tang, and Sylwia Tur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FQT4oBgHgl3EQfTzY7/content/2301.13294v1.pdf'} +page_content=' 2020.' 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There are many problems with high complexity that we have to deal, which is especially true +for AI. This raises a big question: Is there a better way to deal with these highly complex problems other +than bounded by computational complexity? We believe that ideas and methods from intelligence science +can be applied to these problems and help us to exceed computational complexity. In this paper, we try to +clarify concepts, and we propose definitions such as unparticularized computing, particularized computing, +computing agents, and dynamic search. We also propose and discuss a framework, i.e., trial-and-error + +dynamic search. Number Partition Problem is a well-known NP-complete problem, and we use this +problem as an example to illustrate the ideas discussed. + +Keywords: Computational Complexity, Trial-and-Error, Dynamic Action, Intelligence, Unparticularized +Computing, Particularized Computing, Number Partition Problem + +1 +Introduction +Computing technology and theory are developing rapidly, however, the ultimate concern remains: How can we +expand our practical computing power to engineering and scientific problems, including those in artificial +intelligence? + +One focus of these problems is computational complexity. Computational complexity is a central part of +computer science, which studies how much time and how much storage space are required to compute a +problem. Usually, computational problems have scale, such as the size of the matrix, the number of bits of +integers, the environmental complexity of autonomous driving, the accuracy of visual recognition, and so on. +The computational complexity grows with the scale of the problem, and may grow relatively slowly, generally +expressed by polynomial growth, or it may grow rapidly, generally expressed by exponential growth. Problems +with exponentially increasing complexity are commonly considered to be very difficult. The most typical +example is the factorization of large integers. This is an exponentially increasing, extremely difficult +computational problem, which turn out to be the theoretical basis for cryptography. + +Problems with high computational complexity are undoubtedly very difficult, however, we cannot ignore these +problems, we have to face them. Artificial intelligence is even more inseparable from these problems of high +complexity. In almost all aspects of artificial intelligence, it is inevitable to encounter these problems. For +example, face recognition is a NP-complete problem [1]. That means that when we deal with these problems, +the requirement for computing time and storage is very high. So, we hope to adopt intelligent methods to + +1 Thanks for my wife’s consistent support + + + +reduce the requirement of time and storage. Thus, artificial intelligence is highly entangled with computation +complexity: we need to exceed computational complexity for problems in AI, but in turn, we need intelligent +methods to exceed complexity. Such entanglement will push us more and more to intelligence. This situation +could be contrasted in parallel to life that develops and improvs the intelligence while dealing with difficulties. + +But we have to first clarify the meaning. Computational complexity is a solid and rigorous theory. The +theoretical limitation set by the theory cannot be violated. However, we need to know that the limit set by the +theory is for a whole set of instances of problems, not for a single instance. Yet, we often only need to solve a +particular instance, not the whole set. We then ask: for a particular instance, is there a solution that is much +better than the generic solution for the whole class? And, how to find a good particular solution for the +particular instance? If we can do so, then, we can achieve a much better solution for a particular instance than +following the generic solution. This is what we mean when we say exceeding computational complexity. + +Exceeding computational complexity, no doubt is very difficult. In fact, whether or not it is possible to do so +is a big question. At present, there is no solid theory to support it, and certainly no definite theory to completely +deny it. Therefore, in this article, we try to make a preliminary discussion on this. We tried to clear out a +discussion framework to facilitate further exploration. In fact, what motivates us to go in this direction is: when +we were studying the number partition problem [2], we found that there are some cases that look very difficult, +but are actually easy to solve, and yet there are some cases that are indeed true hard (a very twisted and invisible +fence [3]). When we face these puzzles and think again and again, we gradually find that intelligence is looming +in it. + +In this paper, we propose the main idea: we should do “particularized computation”, not “unparticularized +computation”, for those problems with high computational complexity. And, a computing agent with +intelligence and subjectivity inside can find particularized computing for a particular instance. We propose a +framework, namely trial-and-error + dynamic action, for such a computing agent. We demonstrate this idea +with the number partition problem, which is one of the famous Karp's 25 NP-complete problems. + +2 +Limit of Computation – Computational Complexity +Any computation will require resources, be storage spaces or processor cycles, that are demanded by the +computational complexity of this computation. This can even be seen as physical barrier: without the required +resources available, it is physically impossible to execute the computation. In fact, as well known, modern +cryptography is built on such physical requirement created by computational complexity. + +However, there are different ways to deal with the limit of computation, and different ways will require different +resources. We can consider a very simple example, 4567 × 2341. If the ordinary multiplication rules are used, +16 single-digit multiplications and 12 single-digit additions are required. Such a computation is good for this +particular case, and is good for any other case . But what if our problem is 4567 × 2300? We can of course +also use the above way to do, so we need to do 16 multiplications and 12 additions. However, obviously we +can take a more compact approach, requiring only 8 multiplications and 4 additions to complete the +computation. This is because we take full advantage of the particular properties of this particular problem, the +0 in 2300, so we can use much less resources. + + + + +From this simple example, we can see that the two ways to deal with a computational task. One is to use a same +definite fixed program to all cases; the other is to use a particular method suitable for the particular case if it is +possible. Just as the example shows, the latter way requires much less resources. + +Let's look at a more realistic case. There is an integer array: Ω = {63,48,932,266,671,47,110,82,39}, we +want to divide this array into two arrays so that the sums of two arrays are equal. There is a method that can be +applied to any array. It is the exhaustive search, that is, to go through all possible partitions. The length of this +array is 9, so exhaustive search requires 2� searches, which is quite computationally expensive (at least for +manual way). However, for this particular case, we can try to solve it by a particular way. We are going to solve +it by trying and adjusting. First, we divide it into two arrays arbitrarily: Ω� =  {63,48,932,266},  Ω� = + {671,47,110,82,39}. It is easy to see that the sum of first is bigger. Then, let's make some adjustments to +make the sum of first smaller and the second bigger, like this: Ω� =  {63,48,932,110},  Ω� = + {671,47,82,39,266}. Now, the first is still bigger. However, this time, we can see it more clearly, and know +how to do the final adjustment. We get: Ω� =  {63,48,932,47,39},  Ω� =  {671,82,266,110}. The sum of the +two arrays is now equal. This way, the amount of computation is much smaller. + +That is to say, when we have a particular case, if we try to find various particular relationships in the particular +case, and utilize these relationships as much as possible, we could compute with much less resources. Therefore, +the question naturally arises, does this approach indeed save resources? This is not an easy question. We should +first make some definition carefully. + +Definition 2.1 (Unparticularized computation and particularized computation) +Suppose a computation problem, whose all instances form a set 𝑊, and we denote a computing program 𝐶 acts on the particular +instance 𝑤 ∈ 𝑊 as 𝐶(𝑤). If there is a fixed and definite computing program 𝐶, for any 𝑤 ∈ 𝑊, 𝐶(𝑤) can get the correct +result, we say that the computing program 𝐶 is a unparticularized program covering 𝑊. If for a particular instance 𝑤, there is +a computing program 𝐶�, so that 𝐶�(𝑤) can get the correct result, and can do so with as few resources as possible, we say that +𝐶� is a particularized computing program for the instance 𝑤; however, note, for u ∈ 𝑊,  𝑢 ≠ 𝑤, 𝐶�(𝑢) may not be able to +get the result (not stopping or crash), or the result may be incorrect, or the resource consumption may be much larger. + +In the terms of this definition, we can see, in the above example of number partition problem, the method of +exhaustive searching is unparticularized computing, since it works for all cases, the method of trying and +adjusting is particularized computing since it works only for the particular case, and not for other cases. + +When faced with a computation problem, the mainstream effort so far has been to try to find unparticularized +computing programs. This is reasonable, because with unparticularized computing thinking becomes simple. +For any instance, only need to "plug in" the instance into the computing program, and always obtain correct +result, no any other care is required. Simplified thinking and always correct are what people desire. For easier +problems, or with plenty of resources available, this approach is perfectly reasonable. However, such approach +is no longer appropriate when dealing with problems of high complexity. Just as the above examples show, it +is more reasonable to explore the particular properties of the particular instance, and to take full advantages of +these properties. These particular properties could be like symmetry, weaknesses, patterns, etc., and they can +be used to reduce the consumption of resources. That is to say, for problems with high complexity, we should +pursue particularized computing, rather than unparticularized computing. + + + + +OK, for a particular instance, use particularized computing, instead of unparticularized computing. Sounds +great. But where the particularized computing program comes from? For unparticularized computing, we know +how to do. It is in this way: human programmers work hard to get a program that is working for all instances. +We understand this well and are very familiar with. Whole industry is doing this for decades. However, we do +not know how to do particularized computing, it is very unfamiliar to us. Should we develop a program manually +for each particular instance? This simply is not practical. Or, should we have a finder program that can help us +to find the particularized computing program for a particular instance? Looks great. But such a finder itself will +consume resources. So, question comes: can the overall resource consumption be reduced? Thus, we need to +make some definitions about resources. + +Definition 2.2 (Resources required by unparticularized computation) +Assuming a computation problem, all instances form a set 𝑊, if 𝐶 is a unparticularized computation program covering 𝑊, for +each particular instance 𝑤 ∈ 𝑊, the resources 𝐶(𝑤)consumes is 𝑍�, then the resources required by 𝐶 is: 𝑍 = 𝑚𝑎𝑥{𝑍�| 𝑤 ∈ +𝑊}. If there are more than two unparticularized computing programs covering 𝑊, the smaller of the 2 resources required for the +2 computing programs is taken as the resources required by unparticularized computing program covering 𝑊. + +For particularized computation, we have the following proposition. + +Proposition 2.3 (Resources required by particularized computation) +Assuming a computation problem, all instances form a set 𝑊, and the resources required by unparticularized computing covering +𝑊 are 𝑍. Suppose we use a definite and fixed finder program 𝑆 to find the particularized computing program for a particular +instance, that is, for any 𝑤 ∈ 𝑊, 𝑆(𝑤) can get the particularized program 𝐶� for the instance 𝑤, then the upper bound of the +resources consumed by 𝑆(𝑤) and 𝐶�(𝑤) will be equal to 𝑍. + +Proof: 𝑆 is a definite and fixed finder program, so for any instance 𝑤 ∈ 𝑊, we first do 𝑆(𝑤) to get 𝐶�, and +then do 𝐶�(𝑤) to compute the instance. Thus, this procedure forms an unparticularized computing program +covering 𝑊, we denote it as 𝐶. Therefore, the resources required by 𝐶 are 𝑍. ■ + +This proposition tells us: it is not good to use a definite and fixed finder to find a unparticularized computing, +since by this way, no resources could be saved. According to this proposition, the resources required by +unparticularized computation is a standard standing well, which could not be easily overpass. Thus, we define +the resources required for computation as below. + +Definition 2.4 (Resources required by computation) +Assuming a computing problem, all instances form a set 𝑊, the resources required by computation for this problem are the resources +required by unparticularized computing covering 𝑊. + +That is, for an instance 𝑤 ∈ 𝑊, the computation requires resources 𝑍, which is defined by unparticularized +computing. 𝑍 is the limit dictated by computational complexity. But if we know a particularized computing +program for 𝑤 , then we could do computation with resources much less than 𝑍 . This is exceeding +computational complexity. It is great if we can do so. However, it needs us to know the particularized program +for 𝑤. If we do not know such program in advance, can we still exceed computational complexity? If so, how? +Certainly, not by a definite and fixed finder program, as Proposition 2.3 tells. Intelligence will be essential. + + + + +3 +Computing Agent and Trial-and-error + Dynamic Action +For a given particular instance, the question is how to find the particularized computing program that requires +much less resources. We cannot manually find such program, neither use a definite and fixed finder. Here are +some possibilities to have particularized program. A) We just have it. B) We have memory of a lot of such +programs and have a looking table to locate it. C) We will interact with the particular instance and then get the +particularized program from the interaction, and such process only consumes an order lower of resources. + +A) is like an oracle. This is very interesting. It shows a strong connection between particularized computing and +non-deterministic Turing machine. But we will not consider it now. B) does not work as well. It requires to +remember all instances. Normally the number of instances is very huge, to remember will need a lot of +resources. The possibility C) means the computing entity to do particularized computation has some ability. In +fact, a very strong ability: it can explore the situation, make judgment and utilize possibilities just popup. + +We would like to call such a computing entity as a computing agent, and this agent has intelligence and +subjectivity inside. Wang Pei is a researcher and advocate of AGI, according to the definition of intelligence he +advocates: intelligence is the ability to make the best adaptation under the circumstance of limited resources +[4]. Obviously, his definition of intelligence is in the same direction as that we call the ability of computing +agent as intelligence. In fact, this definition of intelligence that given by Wang Pei has a positive effect on us. +Based on these considerations, we have the following definitions. + +Definition 3.1 (Intelligence of Computing Agent) +Suppose there is a computational problem, all instances form a set 𝑊, and then suppose that the resources necessary for +unparticularized computing covering 𝑊 are 𝑍. Now there is a computational agent 𝐴 to deal with 𝑊, if for instance 𝑤 ∈ 𝑊, +𝐴 can have a particularized program 𝐶 , and 𝐶(𝑤) can be done with resources one order of magnitude lower than 𝑍 +(𝑂�𝑙𝑜𝑔(𝑍)�), then we say that 𝐴 can intelligently compute 𝑤. If 𝑉 ⊂ 𝑊 is the subset of all elements that 𝐴 can intelligently +compute, then the intelligence of 𝐴 is measured by the quantity: 𝑞(𝐴) = |𝑉|/|𝑊|. + +That is to say, a computing agent with intelligence greater than zero can break through the barriers of +computational complexity. However, a fundamental question is: Does such a computing agent really exist? To +the best of our knowledge, there is currently no theory discussing whether such a computing agent exists. +Furthermore, there is no theory that tells us how to build an intelligent computing agent. These problems are +not only major problems in computational theory but also major theoretical problems in artificial intelligence, +which require further research. Here, to make an attempt, we propose this idea: if appropriate trial-and-error +procedures and dynamic action are adopted, there is hope to form an intelligent computing agent. + +In our common sense, trial-and-error is very reasonable method, in fact, we often use it unconsciously. In the +theory of computation, Gold, Putnam, Kugel et al. insisted on using the trial-and-error method to deal with the +problem of computability [5]. Kugel called such a trial-and-error computing program a Gold-Putnam machine. +They thus developed a trial-and-error procedure for dealing with the more difficult computability problems, as +well as trying to deal with the non-computable ones. + +What exactly is trial-and-error doing? Why can it be successful? Trial-and-error is actually based on the following +facts: 1) admit that we have an unknown, but this unknown can be obtained through effort; 2) some +transformation can be used to transfer the unknown to a definite search space; 3) trial-and-error efforts is a + + + +cycle: to get feedback from trial, to seek better by feedback, and seeking is to move from one point to the next +point in the search space; 4) can reach (or get close to) the unknown in the search space. Trial-and-error is a +very effective mechanism and often an indispensable tool for solving problems. + +Back to our problem of finding particularized computation. Assuming all instances of the problem form a set +𝑊, given an instance 𝑤 ∈ 𝑊, we want to find the particularized computing program 𝐶� for 𝑤. So, here the +unknown is 𝐶�. As discussed earlier, we want to transform the unknown into some search space. There can be +many kinds of search spaces. We are here to make certain restrictions. We restrict the search space to a Boolean +vector space, that is, the search space is 𝐵�. Such restrictions are of course limited. However, if 𝑁 is large +enough, the Boolean vector space can actually cover any parameter space, and it is very common to use the +parameter space to regulate the computation (as shown by the non-deterministic Turing machine, see Cook [6] +). Therefore, it is reasonable to choose the search space as the 𝑁-dim Boolean vector space. We can change to +a different search space later if necessary. Thus, the search is to obtain the correct parameter vector 𝑝 ∈ 𝐵�, +and then the parameter vector 𝑝 will bring the particularized program 𝐶� for 𝑤 to us. Since 𝐶� depends on +the parameter vector, we can write it as 𝐶(𝑤, 𝑝). + +With the search space in hand, let's consider a trial-and-error procedure. We need to have these components +for such procedure. First, a trial-and-error program 𝑇(𝑤, 𝑝), 𝑤 ∈ 𝑊, 𝑝 ∈ 𝐵�, if the given parameter vector 𝑝 +is correct, 𝑇(𝑤, 𝑝) = 1, otherwise 𝑇(𝑤, 𝑝) = 0. This is the major feedback. But, 𝑇 also feeds back other +information 𝑡. Second, a search program 𝑆(𝑤, 𝑝, 𝑡), 𝑤 ∈ 𝑊, 𝑝 ∈ 𝐵�, 𝑆 will yield 𝑝�, which is the next point in +the search space for trial-and-error. 𝑆 also produces some other information for trial-and-error use. Third, a +computation program 𝐶(𝑤, 𝑝), 𝑤 ∈ 𝑊, 𝑝 ∈ 𝐵�, if parameter vector 𝑝 is correct, this program will be the +particularized program. With these components, the trial-and-error procedure is as follows: +1) Initially set the parameter 𝑝 = 𝑝� and start a trial-and-error cycle. +2) Trial-and-error cycle: Run trial-and-error (𝑐, 𝑡) ← 𝑇(𝑤, 𝑝), where 𝑐 is the testing result, and 𝑡 are all +other feedback information. If testing result is 1, the parameter vector is correct, exit the cycle. If +testing result is 0, continue. +3) Search 𝑆(𝑤, 𝑝, 𝑡), 𝑆 generates 𝑝�, which is the parameter vector used for the next trial-and-error. +4) The trial-and-error cycle continues until the correct parameter 𝑝 is obtained, or an error is reported. +5) If the correct parameter vector is obtained, the particularized program 𝐶(𝑤, 𝑝) is also obtained. + +Note that in the trial-and-error procedure, the most important component is 𝑆(𝑤, 𝑝, 𝑡), which will generate +next parameter vector for trial. 𝑆 could use the dumbest way, exhaustive search, to search every possible point. +However, this is not what we expected. For exhaustive search, 2� resources must be used. We expect to use +much less resources. So, we need to have a much better 𝑆, dynamic search, which has the ability to intelligently +use the feedback information from trial-and-error. In order to use the feedback information intelligently, +dynamic search needs to have its subjectivity. We discussed subjectivity and dynamic action of machine in detail +in [7]. Now we can define intelligent search. + +Definition 3.2 (Intelligent Search) +Assuming that the search space in the trial-and-error procedure is 𝑃 = 𝐵�, and the dynamic search is 𝑆(𝑤, 𝑝, 𝑡), if for a given +w, for any initial parameter 𝑝�, 𝑆 can reach the correct parameter vector 𝑝 for 𝑤 by using only 𝑂(𝑁) resources, we say that 𝑆 +can do intelligent search for 𝑤. + + + + +A computing agent with intelligent search is really intelligent. + +Proposition 3.3 (Trial-and-error + Dynamic Action) +Suppose there is a computational problem with scale 𝑁, and all instances forms a set 𝑊, and the resources required for +unparticularized computation covering 𝑊 are 𝑂(2�). Suppose that the computing agent 𝐴 consists of a program 𝐶(𝑤, 𝑝) with +parameters and trial-and-error procedure + dynamic search 𝑆. For an instance 𝑤 ∈ 𝑊, if 𝑆 can do intelligent search for 𝑤, +and the program 𝐶(𝑤, 𝑝) only needs 𝑂(𝑁) resources, then the computational agent 𝐴 has intelligence as defined in 3.1. + +It is easy to see that this proposition is true: If both 𝑆 and 𝐶(𝑤, 𝑝) only need resources of 𝑂(𝑁), so the set +𝑉 ⊂ 𝑊 specified in Definition 3.1 is not empty, thus, the computational agent 𝐴 has intelligence. So, 𝐴 can +exceed computational complexity. + +Intelligent search and computing agent are great. But, how can we get them? This is a big issue and requires a +lot of further work. In next section, we will use number partition problem as one example to shed some light +on it . + +4 +Number Partition Problem +Number partition problem is a very famous and important problem. We now use this question as an example +and apply the ideas discussed earlier in the hope that it will help us to see things better. Number partition +problem can be explained in a short sentence: given a set of natural numbers Ω, ask whether Ω can be divided +into two subsets Ω� and Ω� such that the sum of the numbers in Ω� equals the sum of the numbers in Ω�? +This problem is a typical example of P vs. NP: it is easy to verify solution but hard to find. + +Let's describe the problem in more detail. Now let the array length be 𝑁, we will consider the set Ω of length +𝑁, whose members are all natural numbers, that is, Ω ∈ 𝐼�, 𝐼 is the set of natural numbers. Then, we consider +a 𝑁-dim Boolean vector 𝑝 = (𝑝�, 𝑝�, … , 𝑝�), 𝑝� ∈ 𝐵, 𝑝 ∈ 𝐵� and for a set Ω = {𝜔�, 𝜔�, … , 𝜔�} and a +Boolean vector 𝑝, we define a quantity: +< 𝑝, Ω >= � 𝑞�ω� +� +��� +  if 𝑝�=1, 𝑞�=1; if 𝑝�=0, 𝑞�=-1 +(1) +It is easy to see that the meaning of the quantity in (1) is: partition the set into two subsets, and the quantity is +the difference between the sum of the two parts. Clearly, the parameter vector 𝑝 tells how to partition Ω into +two subsets, so, we call 𝑝 a partition vector. That is to say, this quantity < 𝑝, Ω > is actually a test whether the +partition vector 𝑝 can equally partition Ω, and this quantity also gives the feedback about how far away the +partition is from equal partition. Let's define another function: φ(Ω, 𝑝): +φ(Ω, 𝑝) = �1 𝑖𝑓 < 𝑝, Ω > = 0 +0 𝑖𝑓 < 𝑝, Ω > ≠ 0 +(2) +φ(Ω, 𝑝) is a boolean function with parameters, φ(Ω, 𝑝): 𝐼� × 𝐵� → 𝐵, that is, if the partition vector 𝑝 equally +partition Ω, the function value is 1, otherwise it is 0. So, this function is a trial-and-error function, using 𝑝 as +the parameter vector. We defined an operator ⊙ in [2], which means “trying over”. Using function φ(Ω, 𝑝) +and this operator ⊙, we can define the partition function: +𝑃𝑎𝑟�(Ω): 𝐼� → 𝐵,    𝑃𝑎𝑟�(Ω) = φ(Ω, 𝑝) ⊙ 𝑃 +(3) + + + +Here, 𝑃 is the space formed by all partition vector (here P = B�). The function means: apply all 𝑝 ∈ 𝑃 to +φ(Ω, 𝑝) to try it out. If any value in the test result is equal to 1, then the value of 𝑃𝑎𝑟�(Ω) is 1, otherwise the +value of 𝑃𝑎𝑟�(Ω) is 0. That is, the function value of 𝑃𝑎𝑟� is defined by trial-and-error. This clearly tells us that +the definition of the problem of number partition is defined by "trial-and-error + exhaustive search". So, quite +naturally, we can directly translate this definition into the computation. See pseudo code “Trial-and-Error + +Exhaustive Search”. + +Obviously, the previous trial-and-error procedure + exhaustive search is just the program expressing +𝑃𝑎𝑟�(Ω) = φ(Ω, 𝑝) ⊙ 𝑃. In fact, the program and equation are exactly the same thing. Therefore, using trial- +Pseudo Code： Trial-and-Error + Exhaustive Search +Setting: p�, Ω +Initial: N = 0, p = 𝑝� +While N < 2� +Continue to trial and error: 𝑡 ← < 𝑝, Ω > +If t = 0 then +Stop, output "Computing successful, 𝑃𝑎𝑟�(Ω) = 1, Partition vector: " 𝑝 +Else +Continue search: 𝑝� ← 𝑆, 𝑝 ← 𝑝�, 𝑁 ← 𝑁 + 1 +EndIf +EndWhile +Stop, output "Computing successful, 𝑃𝑎𝑟�(𝛺) = 0" +Pseudo Code： Trial-and-Error + Dynamic Search +Setting: p�, N���, Ω +Initial: 𝑁 = 0, 𝑝 = 𝑝� +While N < N��� +Continue to trial and error: 𝑡 ← < 𝑝, Ω > +If t = 0 then +Stop, output "Computing successful, 𝑃𝑎𝑟�(Ω) = 1, Partition vector: " 𝑝 +Else +Continue to search: (𝑝�, i) ← 𝑆(𝑡, 𝑝, Ω) +If 𝑖 = 0 then +Stop, output "Computing successful, 𝑃𝑎𝑟�(Ω) = 0" +Else 𝑖 = 1 + + +𝑝 ← 𝑝�, 𝑁 ← 𝑁 + 1 + +Else +Stop, output "Computing failed, search stops" + +EndIf +EndIf +EndWhile +Stop, output "Computing failed, out of range" + + + +and-error + search to compute 𝑃𝑎𝑟� is very natural. In the program, the search 𝑆 is an exhaustive search, i.e., +𝑆 walks through the entire 𝑃. There are many ways to implement an exhaustive search, as long as 𝑆 can traverse +the entire 𝑃. + +We would emphasize that the above computation is unparticularized computation, which is applicable to any +instance and always get correct result. It is worth noting that the resources required by this computation are +𝑂(2�). Now we have unparticularized computation for number partition problem. How can we have +particularized computation? As we discussed in Section 3, we need to transform to a search space. For number +partition problem, it is relatively easy, since the definition of partition function already contains the search +space. But we need to change the exhaustive search to dynamic search. Then, the computation program is +shown in pseudo code “Trial-and-Error + Dynamic Search”. + +Note, in trial-and-error + dynamic search, the program is different than in trial-and-error + exhaustive search +in several places. First, dynamic search does not guarantee the success of search. The program has 4 exits. Two +exists are for successful, corresponding to 𝑃𝑎𝑟�(Ω) equal to 1 or 0. The other two exist are for failures, one +failure is because the number of searches exceeds the preset value and is forced to stop, and the other failure is +because the dynamic search thinks that the search can no longer be continued. Thus, the program does not +always success. When the program successes, the value of 𝑃𝑎𝑟�(Ω) is given, otherwise the value of +𝑃𝑎𝑟�(Ω) cannot be given. It should also be noted that when the value of 𝑃𝑎𝑟�(Ω) is given as 1, the partition +vector is also given, and both value and partition vector are guaranteed to be correct. However, when the value +of 𝑃𝑎𝑟�(Ω) is 0, it may be right or wrong, because the search is not an exhaustive search, but a dynamic search +with intelligence and subjectivity, which could be wrong. In short, the computing agent does a particularized +computing for an instance (the given number array). There 3 kind results: 1) computation is done, and the +computation result is correct, 2) computation is done, but the result is wrong, 3) computation fails, and there +are 2 kind failures, one is that the search stops, and the other is out of range. + +The most important part in the program is the dynamic search 𝑆(𝑡, 𝑝, Ω). It is a computing agent with +intelligence and subjectivity. It will take a look on its inputs: 𝑡, 𝑝, Ω, where 𝑡 is the feedback information from +the testing program, and 𝑝 is partition vector currently using, and Ω is the number array that currently doing. +𝑆 will intelligently use the information to decide what is the partition vector will be used to try next time. 𝑆 will +not act in a presetting way, but it will be able to tell the current situation, explore the possibilities, and try to +give the best guess for next partition vector, and save the resources to be used. 𝑆 should be able to learn from +the situation and its mistakes as well. So, 𝑆 is definitely not a traditional search program. But the question is: +can we make such dynamic search with intelligence and subjectivity inside? + +For the number partition problem, for any given number array Ω that can be equally partitioned, there is always +a search that very quickly get the partition vector of Ω, because we can set the initial vector 𝑝� as the partition +vector. However, such searches are mundane, trivial, and not the kind of dynamic search we really want. The +dynamic search we hope is like this: starting from any initial vector 𝑝�, it will be able to reach correct results by +only using much less resources. We propose the following conjecture. + +Conjecture 4.1 (Dynamic search exists for number partition problem) +For number partition problem, there is a dynamic search 𝑆(𝑡, 𝑝, Ω) with such properties: for most integer array Ω ∈ 𝐼�, and for +any initial partition vector 𝑝�, 𝑆 will be able to reach correct results by only using 𝑂(𝑁) resources. The meaning of reaching correct + + + +results is: the trial-and-error + dynamic search will make computation done (not exit abnormally), and correctly give the value +𝑃𝑎𝑟�(Ω). + +If we can indeed demonstrate such a dynamic search, it will be a major break through in computational theory +that can be used in many engineering application areas. Not only that, finding such dynamic search will also be +a solid progress of intelligence science. However, we point out: even we find such dynamic search, the fact that +number partition problem is a NP-complete problem will not change, i.e., there are some Ω ∈ 𝐼� so that the +trial-and-error + 𝑆 will not be able to compute 𝑃𝑎𝑟�(Ω) by only using 𝑂(𝑁) resources. + +5 +Remarks +We would like to emphasize our major points again. Computational complexity sets a limit: for a computation, +there must be enough resources available for it, if there are no enough resources, the computation could not +be achieved. But this limit is for all instances of the problem. For a particular instance, it is possible to use +particularized program to achieve computation that will only requires much less resources. This is exceeding +computational complexity. However, in order to have particularized program, we need computing agent with +intelligence and subjectivity inside. + +But we have to say, it is very controversial. So far no one have made such an agent. And there is no theory to +fully support such agent yet. But we strongly believe that such computing agent does exist, and there is huge +demand for it since there are many hard problems are waiting for such agent. Thus, we try to clear mist around +this issue and establish some solid ground for further discussions. We clean up some crucial concepts such as +unparticularized and particularized computing, trial-and-error, dynamic search, etc. Number partition problem, +due to its nature, can serve as one good example for these. For this problem, we conjecture the existence of +computing agent. Note, number partition problem is one NP-complete problem. If we can find a computing +agent for it, the computing agent then can be used for many other hard problems. The direction is clear now. +We will continue research along this path and hope to reach a concrete computing agent. + +Here, we would like to mention the close relationship between particularized computing and non-deterministic +Turning machine [6]. We think this relationship is a major problem in computational theory and in artificial +intelligence. AI and exceeding computational complexity are very deeply entangled together. This is a good +thing actually. Such entanglement will strongly help the development of artificial intelligence. + +We can see the relationship between particularized computing and intelligence from another view. If there is +no difficult problem, all computations can be done by unparticularized computing, there will be no need for +intelligence. For highly complex computational problems, only a particularized computing has hope to do it. +Yet, only intelligence can make particularized computing possible. This is the reason that intelligence exists and +must exist. We can measure intelligence by how well it can create particularized computing. If we see a +computing agent can create nice particularized computing in a difficult situation, we see intelligence inside it. + +References + +[1] A. Yang, J. Wright, Y. Ma and S. Sastry, "Feature Selection in Face Recognition: A Sparse Representation +Perspective," 2007. + + + +[2] C. Xiong, "Sampling and Complexity of Partition Function," arxiv.org, 2021. +[3] S. Aaronson, "P =? NP," 2011. +[4] P. Wang, "Three fundamental misconceptions of artificial intelligence," vol. 19, no. 3, pp. 249-268, 2007. +[5] P. Kugel, "Thinking may be more than computing". +[6] S. Cook, "THE P VERSUS NP PROBLEM," 2000. +[7] C. Xiong, "Some Discussions on Subjectivity of Machine and its Function - Contributions to ICIS 2020," +in International Conference of Intelligence Science 2020, West Bengal, India, 2020. + + + + diff --git a/O9E1T4oBgHgl3EQfuAW8/content/tmp_files/load_file.txt b/O9E1T4oBgHgl3EQfuAW8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2109391f07016db4cee92edc47dc7a1b5089b6b --- /dev/null +++ b/O9E1T4oBgHgl3EQfuAW8/content/tmp_files/load_file.txt @@ -0,0 +1,380 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf,len=379 +page_content='Exceeding Computational Complexity – Trial-and-Error, Dynamic Action and Intelligence1 Chuyu Xiong Independent Researcher chuyux99@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='com ABSTRACT Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' There are many problems with high complexity that we have to deal, which is especially true for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This raises a big question: Is there a better way to deal with these highly complex problems other than bounded by computational complexity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We believe that ideas and methods from intelligence science can be applied to these problems and help us to exceed computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In this paper, we try to clarify concepts, and we propose definitions such as unparticularized computing, particularized computing, computing agents, and dynamic search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We also propose and discuss a framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=', trial-and-error + dynamic search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Number Partition Problem is a well-known NP-complete problem, and we use this problem as an example to illustrate the ideas discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Keywords: Computational Complexity, Trial-and-Error, Dynamic Action, Intelligence, Unparticularized Computing, Particularized Computing, Number Partition Problem 1 Introduction Computing technology and theory are developing rapidly, however, the ultimate concern remains: How can we expand our practical computing power to engineering and scientific problems, including those in artificial intelligence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' One focus of these problems is computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Computational complexity is a central part of computer science, which studies how much time and how much storage space are required to compute a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Usually, computational problems have scale, such as the size of the matrix, the number of bits of integers, the environmental complexity of autonomous driving, the accuracy of visual recognition, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The computational complexity grows with the scale of the problem, and may grow relatively slowly, generally expressed by polynomial growth, or it may grow rapidly, generally expressed by exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Problems with exponentially increasing complexity are commonly considered to be very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The most typical example is the factorization of large integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is an exponentially increasing, extremely difficult computational problem, which turn out to be the theoretical basis for cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Problems with high computational complexity are undoubtedly very difficult, however, we cannot ignore these problems, we have to face them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Artificial intelligence is even more inseparable from these problems of high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In almost all aspects of artificial intelligence, it is inevitable to encounter these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For example, face recognition is a NP-complete problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That means that when we deal with these problems, the requirement for computing time and storage is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, we hope to adopt intelligent methods to 1 Thanks for my wife’s consistent support reduce the requirement of time and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, artificial intelligence is highly entangled with computation complexity: we need to exceed computational complexity for problems in AI, but in turn, we need intelligent methods to exceed complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Such entanglement will push us more and more to intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This situation could be contrasted in parallel to life that develops and improvs the intelligence while dealing with difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But we have to first clarify the meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Computational complexity is a solid and rigorous theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The theoretical limitation set by the theory cannot be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, we need to know that the limit set by the theory is for a whole set of instances of problems, not for a single instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Yet, we often only need to solve a particular instance, not the whole set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We then ask: for a particular instance, is there a solution that is much better than the generic solution for the whole class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' And, how to find a good particular solution for the particular instance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If we can do so, then, we can achieve a much better solution for a particular instance than following the generic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is what we mean when we say exceeding computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Exceeding computational complexity, no doubt is very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In fact, whether or not it is possible to do so is a big question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' At present, there is no solid theory to support it, and certainly no definite theory to completely deny it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Therefore, in this article, we try to make a preliminary discussion on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We tried to clear out a discussion framework to facilitate further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In fact, what motivates us to go in this direction is: when we were studying the number partition problem [2], we found that there are some cases that look very difficult, but are actually easy to solve, and yet there are some cases that are indeed true hard (a very twisted and invisible fence [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' When we face these puzzles and think again and again, we gradually find that intelligence is looming in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In this paper, we propose the main idea: we should do “particularized computation”, not “unparticularized computation”, for those problems with high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' And, a computing agent with intelligence and subjectivity inside can find particularized computing for a particular instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We propose a framework, namely trial-and-error + dynamic action, for such a computing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=" We demonstrate this idea with the number partition problem, which is one of the famous Karp's 25 NP-complete problems." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 2 Limit of Computation – Computational Complexity Any computation will require resources, be storage spaces or processor cycles, that are demanded by the computational complexity of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This can even be seen as physical barrier: without the required resources available, it is physically impossible to execute the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In fact, as well known, modern cryptography is built on such physical requirement created by computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, there are different ways to deal with the limit of computation, and different ways will require different resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We can consider a very simple example, 4567 × 2341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If the ordinary multiplication rules are used, 16 single-digit multiplications and 12 single-digit additions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Such a computation is good for this particular case, and is good for any other case .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But what if our problem is 4567 × 2300?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We can of course also use the above way to do, so we need to do 16 multiplications and 12 additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, obviously we can take a more compact approach, requiring only 8 multiplications and 4 additions to complete the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is because we take full advantage of the particular properties of this particular problem, the 0 in 2300, so we can use much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' From this simple example, we can see that the two ways to deal with a computational task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' One is to use a same definite fixed program to all cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' the other is to use a particular method suitable for the particular case if it is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Just as the example shows, the latter way requires much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=" Let's look at a more realistic case." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' There is an integer array: Ω = {63,48,932,266,671,47,110,82,39}, we want to divide this array into two arrays so that the sums of two arrays are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' There is a method that can be applied to any array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is the exhaustive search, that is, to go through all possible partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The length of this array is 9, so exhaustive search requires 2� searches, which is quite computationally expensive (at least for manual way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, for this particular case, we can try to solve it by a particular way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We are going to solve it by trying and adjusting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' First, we divide it into two arrays arbitrarily: Ω� = {63,48,932,266}, Ω� = {671,47,110,82,39}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is easy to see that the sum of first is bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=" Then, let's make some adjustments to make the sum of first smaller and the second bigger, like this: Ω� = {63,48,932,110}, Ω� = {671,47,82,39,266}." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Now, the first is still bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, this time, we can see it more clearly, and know how to do the final adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We get: Ω� = {63,48,932,47,39}, Ω� = {671,82,266,110}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The sum of the two arrays is now equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This way, the amount of computation is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That is to say, when we have a particular case, if we try to find various particular relationships in the particular case, and utilize these relationships as much as possible, we could compute with much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Therefore, the question naturally arises, does this approach indeed save resources?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is not an easy question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We should first make some definition carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='1 (Unparticularized computation and particularized computation) Suppose a computation problem, whose all instances form a set 𝑊, and we denote a computing program 𝐶 acts on the particular instance 𝑤 ∈ 𝑊 as 𝐶(𝑤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If there is a fixed and definite computing program 𝐶, for any 𝑤 ∈ 𝑊, 𝐶(𝑤) can get the correct result, we say that the computing program 𝐶 is a unparticularized program covering 𝑊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If for a particular instance 𝑤, there is a computing program 𝐶�, so that 𝐶�(𝑤) can get the correct result, and can do so with as few resources as possible, we say that 𝐶� is a particularized computing program for the instance 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' however, note, for u ∈ 𝑊, 𝑢 ≠ 𝑤, 𝐶�(𝑢) may not be able to get the result (not stopping or crash), or the result may be incorrect, or the resource consumption may be much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In the terms of this definition, we can see, in the above example of number partition problem, the method of exhaustive searching is unparticularized computing, since it works for all cases, the method of trying and adjusting is particularized computing since it works only for the particular case, and not for other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' When faced with a computation problem, the mainstream effort so far has been to try to find unparticularized computing programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is reasonable, because with unparticularized computing thinking becomes simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For any instance, only need to "plug in" the instance into the computing program, and always obtain correct result, no any other care is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Simplified thinking and always correct are what people desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For easier problems, or with plenty of resources available, this approach is perfectly reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, such approach is no longer appropriate when dealing with problems of high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Just as the above examples show, it is more reasonable to explore the particular properties of the particular instance, and to take full advantages of these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' These particular properties could be like symmetry, weaknesses, patterns, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=', and they can be used to reduce the consumption of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That is to say, for problems with high complexity, we should pursue particularized computing, rather than unparticularized computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' OK, for a particular instance, use particularized computing, instead of unparticularized computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Sounds great.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But where the particularized computing program comes from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For unparticularized computing, we know how to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is in this way: human programmers work hard to get a program that is working for all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We understand this well and are very familiar with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Whole industry is doing this for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, we do not know how to do particularized computing, it is very unfamiliar to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Should we develop a program manually for each particular instance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This simply is not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Or, should we have a finder program that can help us to find the particularized computing program for a particular instance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Looks great.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But such a finder itself will consume resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, question comes: can the overall resource consumption be reduced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, we need to make some definitions about resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='2 (Resources required by unparticularized computation) Assuming a computation problem, all instances form a set 𝑊, if 𝐶 is a unparticularized computation program covering 𝑊, for each particular instance 𝑤 ∈ 𝑊, the resources 𝐶(𝑤)consumes is 𝑍�, then the resources required by 𝐶 is: 𝑍 = 𝑚𝑎𝑥{𝑍�| 𝑤 ∈ 𝑊}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If there are more than two unparticularized computing programs covering 𝑊, the smaller of the 2 resources required for the 2 computing programs is taken as the resources required by unparticularized computing program covering 𝑊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For particularized computation, we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='3 (Resources required by particularized computation) Assuming a computation problem, all instances form a set 𝑊, and the resources required by unparticularized computing covering 𝑊 are 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Suppose we use a definite and fixed finder program 𝑆 to find the particularized computing program for a particular instance, that is, for any 𝑤 ∈ 𝑊, 𝑆(𝑤) can get the particularized program 𝐶� for the instance 𝑤, then the upper bound of the resources consumed by 𝑆(𝑤) and 𝐶�(𝑤) will be equal to 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Proof: 𝑆 is a definite and fixed finder program, so for any instance 𝑤 ∈ 𝑊, we first do 𝑆(𝑤) to get 𝐶�, and then do 𝐶�(𝑤) to compute the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, this procedure forms an unparticularized computing program covering 𝑊, we denote it as 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Therefore, the resources required by 𝐶 are 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' ■ This proposition tells us: it is not good to use a definite and fixed finder to find a unparticularized computing, since by this way, no resources could be saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' According to this proposition, the resources required by unparticularized computation is a standard standing well, which could not be easily overpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, we define the resources required for computation as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='4 (Resources required by computation) Assuming a computing problem, all instances form a set 𝑊, the resources required by computation for this problem are the resources required by unparticularized computing covering 𝑊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That is, for an instance 𝑤 ∈ 𝑊, the computation requires resources 𝑍, which is defined by unparticularized computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑍 is the limit dictated by computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But if we know a particularized computing program for 𝑤 , then we could do computation with resources much less than 𝑍 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is exceeding computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is great if we can do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, it needs us to know the particularized program for 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If we do not know such program in advance, can we still exceed computational complexity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If so, how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Certainly, not by a definite and fixed finder program, as Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='3 tells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Intelligence will be essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 3 Computing Agent and Trial-and-error + Dynamic Action For a given particular instance, the question is how to find the particularized computing program that requires much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We cannot manually find such program, neither use a definite and fixed finder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Here are some possibilities to have particularized program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' A) We just have it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' B) We have memory of a lot of such programs and have a looking table to locate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' C) We will interact with the particular instance and then get the particularized program from the interaction, and such process only consumes an order lower of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' A) is like an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is very interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It shows a strong connection between particularized computing and non-deterministic Turing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But we will not consider it now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' B) does not work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It requires to remember all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Normally the number of instances is very huge, to remember will need a lot of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The possibility C) means the computing entity to do particularized computation has some ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In fact, a very strong ability: it can explore the situation, make judgment and utilize possibilities just popup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We would like to call such a computing entity as a computing agent, and this agent has intelligence and subjectivity inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Wang Pei is a researcher and advocate of AGI, according to the definition of intelligence he advocates: intelligence is the ability to make the best adaptation under the circumstance of limited resources [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Obviously, his definition of intelligence is in the same direction as that we call the ability of computing agent as intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In fact, this definition of intelligence that given by Wang Pei has a positive effect on us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Based on these considerations, we have the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='1 (Intelligence of Computing Agent) Suppose there is a computational problem, all instances form a set 𝑊, and then suppose that the resources necessary for unparticularized computing covering 𝑊 are 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Now there is a computational agent 𝐴 to deal with 𝑊, if for instance 𝑤 ∈ 𝑊, 𝐴 can have a particularized program 𝐶 , and 𝐶(𝑤) can be done with resources one order of magnitude lower than 𝑍 (𝑂�𝑙𝑜𝑔(𝑍)�), then we say that 𝐴 can intelligently compute 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If 𝑉 ⊂ 𝑊 is the subset of all elements that 𝐴 can intelligently compute, then the intelligence of 𝐴 is measured by the quantity: 𝑞(𝐴) = |𝑉|/|𝑊|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That is to say, a computing agent with intelligence greater than zero can break through the barriers of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, a fundamental question is: Does such a computing agent really exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' To the best of our knowledge, there is currently no theory discussing whether such a computing agent exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Furthermore, there is no theory that tells us how to build an intelligent computing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' These problems are not only major problems in computational theory but also major theoretical problems in artificial intelligence, which require further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Here, to make an attempt, we propose this idea: if appropriate trial-and-error procedures and dynamic action are adopted, there is hope to form an intelligent computing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In our common sense, trial-and-error is very reasonable method, in fact, we often use it unconsciously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In the theory of computation, Gold, Putnam, Kugel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' insisted on using the trial-and-error method to deal with the problem of computability [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Kugel called such a trial-and-error computing program a Gold-Putnam machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' They thus developed a trial-and-error procedure for dealing with the more difficult computability problems, as well as trying to deal with the non-computable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' What exactly is trial-and-error doing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Why can it be successful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Trial-and-error is actually based on the following facts: 1) admit that we have an unknown, but this unknown can be obtained through effort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 2) some transformation can be used to transfer the unknown to a definite search space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 3) trial-and-error efforts is a cycle: to get feedback from trial, to seek better by feedback, and seeking is to move from one point to the next point in the search space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 4) can reach (or get close to) the unknown in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Trial-and-error is a very effective mechanism and often an indispensable tool for solving problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Back to our problem of finding particularized computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Assuming all instances of the problem form a set 𝑊, given an instance 𝑤 ∈ 𝑊, we want to find the particularized computing program 𝐶� for 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, here the unknown is 𝐶�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' As discussed earlier, we want to transform the unknown into some search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' There can be many kinds of search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We are here to make certain restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We restrict the search space to a Boolean vector space, that is, the search space is 𝐵�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Such restrictions are of course limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, if 𝑁 is large enough, the Boolean vector space can actually cover any parameter space, and it is very common to use the parameter space to regulate the computation (as shown by the non-deterministic Turing machine, see Cook [6] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Therefore, it is reasonable to choose the search space as the 𝑁-dim Boolean vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We can change to a different search space later if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, the search is to obtain the correct parameter vector 𝑝 ∈ 𝐵�, and then the parameter vector 𝑝 will bring the particularized program 𝐶� for 𝑤 to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Since 𝐶� depends on the parameter vector, we can write it as 𝐶(𝑤, 𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=" With the search space in hand, let's consider a trial-and-error procedure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We need to have these components for such procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' First, a trial-and-error program 𝑇(𝑤, 𝑝), 𝑤 ∈ 𝑊, 𝑝 ∈ 𝐵�, if the given parameter vector 𝑝 is correct, 𝑇(𝑤, 𝑝) = 1, otherwise 𝑇(𝑤, 𝑝) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is the major feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But, 𝑇 also feeds back other information 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Second, a search program 𝑆(𝑤, 𝑝, 𝑡), 𝑤 ∈ 𝑊, 𝑝 ∈ 𝐵�, 𝑆 will yield 𝑝�, which is the next point in the search space for trial-and-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑆 also produces some other information for trial-and-error use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Third, a computation program 𝐶(𝑤, 𝑝), 𝑤 ∈ 𝑊, 𝑝 ∈ 𝐵�, if parameter vector 𝑝 is correct, this program will be the particularized program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' With these components, the trial-and-error procedure is as follows: 1) Initially set the parameter 𝑝 = 𝑝� and start a trial-and-error cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 2) Trial-and-error cycle: Run trial-and-error (𝑐, 𝑡) ← 𝑇(𝑤, 𝑝), where 𝑐 is the testing result, and 𝑡 are all other feedback information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If testing result is 1, the parameter vector is correct, exit the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If testing result is 0, continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 3) Search 𝑆(𝑤, 𝑝, 𝑡), 𝑆 generates 𝑝�, which is the parameter vector used for the next trial-and-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 4) The trial-and-error cycle continues until the correct parameter 𝑝 is obtained, or an error is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 5) If the correct parameter vector is obtained, the particularized program 𝐶(𝑤, 𝑝) is also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Note that in the trial-and-error procedure, the most important component is 𝑆(𝑤, 𝑝, 𝑡), which will generate next parameter vector for trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑆 could use the dumbest way, exhaustive search, to search every possible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, this is not what we expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For exhaustive search, 2� resources must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We expect to use much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, we need to have a much better 𝑆, dynamic search, which has the ability to intelligently use the feedback information from trial-and-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In order to use the feedback information intelligently, dynamic search needs to have its subjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We discussed subjectivity and dynamic action of machine in detail in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Now we can define intelligent search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='2 (Intelligent Search) Assuming that the search space in the trial-and-error procedure is 𝑃 = 𝐵�, and the dynamic search is 𝑆(𝑤, 𝑝, 𝑡), if for a given w, for any initial parameter 𝑝�, 𝑆 can reach the correct parameter vector 𝑝 for 𝑤 by using only 𝑂(𝑁) resources, we say that 𝑆 can do intelligent search for 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' A computing agent with intelligent search is really intelligent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='3 (Trial-and-error + Dynamic Action) Suppose there is a computational problem with scale 𝑁, and all instances forms a set 𝑊, and the resources required for unparticularized computation covering 𝑊 are 𝑂(2�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Suppose that the computing agent 𝐴 consists of a program 𝐶(𝑤, 𝑝) with parameters and trial-and-error procedure + dynamic search 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For an instance 𝑤 ∈ 𝑊, if 𝑆 can do intelligent search for 𝑤, and the program 𝐶(𝑤, 𝑝) only needs 𝑂(𝑁) resources, then the computational agent 𝐴 has intelligence as defined in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is easy to see that this proposition is true: If both 𝑆 and 𝐶(𝑤, 𝑝) only need resources of 𝑂(𝑁), so the set 𝑉 ⊂ 𝑊 specified in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='1 is not empty, thus, the computational agent 𝐴 has intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, 𝐴 can exceed computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Intelligent search and computing agent are great.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But, how can we get them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is a big issue and requires a lot of further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In next section, we will use number partition problem as one example to shed some light on it .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 4 Number Partition Problem Number partition problem is a very famous and important problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We now use this question as an example and apply the ideas discussed earlier in the hope that it will help us to see things better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Number partition problem can be explained in a short sentence: given a set of natural numbers Ω, ask whether Ω can be divided into two subsets Ω� and Ω� such that the sum of the numbers in Ω� equals the sum of the numbers in Ω�?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This problem is a typical example of P vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' NP: it is easy to verify solution but hard to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=" Let's describe the problem in more detail." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Now let the array length be 𝑁, we will consider the set Ω of length 𝑁, whose members are all natural numbers, that is, Ω ∈ 𝐼�, 𝐼 is the set of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Then, we consider a 𝑁-dim Boolean vector 𝑝 = (𝑝�, 𝑝�, … , 𝑝�), 𝑝� ∈ 𝐵, 𝑝 ∈ 𝐵� and for a set Ω = {𝜔�, 𝜔�, … , 𝜔�} and a Boolean vector 𝑝, we define a quantity: < 𝑝, Ω >= � 𝑞�ω� � ��� if 𝑝�=1, 𝑞�=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' if 𝑝�=0, 𝑞�=-1 (1) It is easy to see that the meaning of the quantity in (1) is: partition the set into two subsets, and the quantity is the difference between the sum of the two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Clearly, the parameter vector 𝑝 tells how to partition Ω into two subsets, so, we call 𝑝 a partition vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That is to say, this quantity < 𝑝, Ω > is actually a test whether the partition vector 𝑝 can equally partition Ω, and this quantity also gives the feedback about how far away the partition is from equal partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=" Let's define another function: φ(Ω, 𝑝): φ(Ω, 𝑝) = �1 𝑖𝑓 < 𝑝, Ω > = 0 0 𝑖𝑓 < 𝑝, Ω > ≠ 0 (2) φ(Ω, 𝑝) is a boolean function with parameters, φ(Ω, 𝑝): 𝐼� × 𝐵� → 𝐵, that is, if the partition vector 𝑝 equally partition Ω, the function value is 1, otherwise it is 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, this function is a trial-and-error function, using 𝑝 as the parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We defined an operator ⊙ in [2], which means “trying over”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Using function φ(Ω, 𝑝) and this operator ⊙, we can define the partition function: 𝑃𝑎𝑟�(Ω): 𝐼� → 𝐵, 𝑃𝑎𝑟�(Ω) = φ(Ω, 𝑝) ⊙ 𝑃 (3) Here, 𝑃 is the space formed by all partition vector (here P = B�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The function means: apply all 𝑝 ∈ 𝑃 to φ(Ω, 𝑝) to try it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If any value in the test result is equal to 1, then the value of 𝑃𝑎𝑟�(Ω) is 1, otherwise the value of 𝑃𝑎𝑟�(Ω) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' That is, the function value of 𝑃𝑎𝑟� is defined by trial-and-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This clearly tells us that the definition of the problem of number partition is defined by "trial-and-error + exhaustive search".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, quite naturally, we can directly translate this definition into the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' See pseudo code “Trial-and-Error + Exhaustive Search”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Obviously, the previous trial-and-error procedure + exhaustive search is just the program expressing 𝑃𝑎𝑟�(Ω) = φ(Ω, 𝑝) ⊙ 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In fact, the program and equation are exactly the same thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' using trial- Pseudo Code： Trial-and-Error + Exhaustive Search Setting: p�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Ω Initial: N = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' p = 𝑝� While N < 2� Continue to trial and error: 𝑡 ← < 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Ω > If t = 0 then Stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' output "Computing successful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑃𝑎𝑟�(Ω) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Partition vector: " 𝑝 Else Continue search: 𝑝� ← 𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑝 ← 𝑝�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑁 ← 𝑁 + 1 EndIf EndWhile Stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' output "Computing successful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑃𝑎𝑟�(𝛺) = 0" Pseudo Code： Trial-and-Error + Dynamic Search Setting: p�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' N���,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Ω Initial: 𝑁 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑝 = 𝑝� While N < N��� Continue to trial and error: 𝑡 ← < 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Ω > If t = 0 then Stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' output "Computing successful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑃𝑎𝑟�(Ω) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Partition vector: " 𝑝 Else Continue to search: (𝑝�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' i) ← 𝑆(𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Ω) If 𝑖 = 0 then Stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' output "Computing successful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑃𝑎𝑟�(Ω) = 0" Else 𝑖 = 1 𝑝 ← 𝑝�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑁 ← 𝑁 + 1 Else Stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' output "Computing failed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' search stops" EndIf EndIf EndWhile Stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' output "Computing failed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' out of range" and-error + search to compute 𝑃𝑎𝑟� is very natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In the program, the search 𝑆 is an exhaustive search, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=', 𝑆 walks through the entire 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' There are many ways to implement an exhaustive search, as long as 𝑆 can traverse the entire 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We would emphasize that the above computation is unparticularized computation, which is applicable to any instance and always get correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is worth noting that the resources required by this computation are 𝑂(2�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Now we have unparticularized computation for number partition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' How can we have particularized computation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' As we discussed in Section 3, we need to transform to a search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For number partition problem, it is relatively easy, since the definition of partition function already contains the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But we need to change the exhaustive search to dynamic search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Then, the computation program is shown in pseudo code “Trial-and-Error + Dynamic Search”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Note, in trial-and-error + dynamic search, the program is different than in trial-and-error + exhaustive search in several places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' First, dynamic search does not guarantee the success of search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The program has 4 exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Two exists are for successful, corresponding to 𝑃𝑎𝑟�(Ω) equal to 1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The other two exist are for failures, one failure is because the number of searches exceeds the preset value and is forced to stop, and the other failure is because the dynamic search thinks that the search can no longer be continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, the program does not always success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' When the program successes, the value of 𝑃𝑎𝑟�(Ω) is given, otherwise the value of 𝑃𝑎𝑟�(Ω) cannot be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It should also be noted that when the value of 𝑃𝑎𝑟�(Ω) is given as 1, the partition vector is also given, and both value and partition vector are guaranteed to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, when the value of 𝑃𝑎𝑟�(Ω) is 0, it may be right or wrong, because the search is not an exhaustive search, but a dynamic search with intelligence and subjectivity, which could be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' In short, the computing agent does a particularized computing for an instance (the given number array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' There 3 kind results: 1) computation is done, and the computation result is correct, 2) computation is done, but the result is wrong, 3) computation fails, and there are 2 kind failures, one is that the search stops, and the other is out of range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The most important part in the program is the dynamic search 𝑆(𝑡, 𝑝, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It is a computing agent with intelligence and subjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' It will take a look on its inputs: 𝑡, 𝑝, Ω, where 𝑡 is the feedback information from the testing program, and 𝑝 is partition vector currently using, and Ω is the number array that currently doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑆 will intelligently use the information to decide what is the partition vector will be used to try next time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑆 will not act in a presetting way, but it will be able to tell the current situation, explore the possibilities, and try to give the best guess for next partition vector, and save the resources to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 𝑆 should be able to learn from the situation and its mistakes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So, 𝑆 is definitely not a traditional search program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But the question is: can we make such dynamic search with intelligence and subjectivity inside?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For the number partition problem, for any given number array Ω that can be equally partitioned, there is always a search that very quickly get the partition vector of Ω, because we can set the initial vector 𝑝� as the partition vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, such searches are mundane, trivial, and not the kind of dynamic search we really want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The dynamic search we hope is like this: starting from any initial vector 𝑝�, it will be able to reach correct results by only using much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We propose the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='1 (Dynamic search exists for number partition problem) For number partition problem, there is a dynamic search 𝑆(𝑡, 𝑝, Ω) with such properties: for most integer array Ω ∈ 𝐼�, and for any initial partition vector 𝑝�, 𝑆 will be able to reach correct results by only using 𝑂(𝑁) resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The meaning of reaching correct results is: the trial-and-error + dynamic search will make computation done (not exit abnormally), and correctly give the value 𝑃𝑎𝑟�(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If we can indeed demonstrate such a dynamic search, it will be a major break through in computational theory that can be used in many engineering application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Not only that, finding such dynamic search will also be a solid progress of intelligence science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, we point out: even we find such dynamic search, the fact that number partition problem is a NP-complete problem will not change, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=', there are some Ω ∈ 𝐼� so that the trial-and-error + 𝑆 will not be able to compute 𝑃𝑎𝑟�(Ω) by only using 𝑂(𝑁) resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 5 Remarks We would like to emphasize our major points again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Computational complexity sets a limit: for a computation, there must be enough resources available for it, if there are no enough resources, the computation could not be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But this limit is for all instances of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For a particular instance, it is possible to use particularized program to achieve computation that will only requires much less resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is exceeding computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' However, in order to have particularized program, we need computing agent with intelligence and subjectivity inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But we have to say, it is very controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' So far no one have made such an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' And there is no theory to fully support such agent yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' But we strongly believe that such computing agent does exist, and there is huge demand for it since there are many hard problems are waiting for such agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Thus, we try to clear mist around this issue and establish some solid ground for further discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We clean up some crucial concepts such as unparticularized and particularized computing, trial-and-error, dynamic search, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Number partition problem, due to its nature, can serve as one good example for these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For this problem, we conjecture the existence of computing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Note, number partition problem is one NP-complete problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If we can find a computing agent for it, the computing agent then can be used for many other hard problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' The direction is clear now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We will continue research along this path and hope to reach a concrete computing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Here, we would like to mention the close relationship between particularized computing and non-deterministic Turning machine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We think this relationship is a major problem in computational theory and in artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' AI and exceeding computational complexity are very deeply entangled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is a good thing actually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Such entanglement will strongly help the development of artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We can see the relationship between particularized computing and intelligence from another view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If there is no difficult problem, all computations can be done by unparticularized computing, there will be no need for intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' For highly complex computational problems, only a particularized computing has hope to do it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Yet, only intelligence can make particularized computing possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' This is the reason that intelligence exists and must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' We can measure intelligence by how well it can create particularized computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' If we see a computing agent can create nice particularized computing in a difficult situation, we see intelligence inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Wright, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Ma and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Sastry, "Feature Selection in Face Recognition: A Sparse Representation Perspective," 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Xiong, "Sampling and Complexity of Partition Function," arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content='org, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Aaronson, "P =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' NP," 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Wang, "Three fundamental misconceptions of artificial intelligence," vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' 249-268, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Kugel, "Thinking may be more than computing".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Cook, "THE P VERSUS NP PROBLEM," 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} +page_content=' Xiong, "Some Discussions on Subjectivity of Machine and its Function - Contributions to ICIS 2020," in International Conference of Intelligence Science 2020, West Bengal, India, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfuAW8/content/2301.03384v1.pdf'} diff --git a/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf b/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9deb5ed8a92852b900480e690d8af4503bb79e67 --- /dev/null +++ b/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f4214717afbcc4a399372b1848d627f7edefa67a0f16e2981ef723cc3f784740 +size 656129 diff --git a/OtAzT4oBgHgl3EQfWfy0/vector_store/index.faiss b/OtAzT4oBgHgl3EQfWfy0/vector_store/index.faiss new file mode 100644 index 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b/OtE2T4oBgHgl3EQfrQhe/content/tmp_files/2301.04047v1.pdf.txt @@ -0,0 +1,1108 @@ +arXiv:2301.04047v1 [math.OC] 10 Jan 2023 +Local Convergence Behaviour of Generalized Gauss-Newton Multiple +Shooting, Single Shooting and Differential Dynamic Programming +Katrin Baumg¨artner, Florian Messerer, Moritz Diehl +Abstract— We revisit three classical numerical methods for +solving unconstrained optimal control problems – Multiple +Shooting (MS), Single Shooting (SS), and Differential Dynamic +Programming (DDP) – and examine their local convergence +behaviour. In particular, we show that all three methods +converge with the same linear rate if a Gauss-Newton (GN) +– or more general a Generalized Gauss-Newton (GGN) – +Hessian approximation is used, which is the case in widely +used implementations such as iLQR. +I. INTRODUCTION +Multiple Shooting, single shooting and differential dynamic +programming are three numerical methods that might be used +for solving discrete optimal control problems (OCP) that +typically arise after discretization of a continuous-time OCP: +min +x,u +N−1 +� +i=0 +li(xi, ui) + lN(xN) +(1a) +s.t. +x0 = ¯¯x0, +(1b) +xi+1 = fi(xi, ui), i = 0, . . . , N − 1, +(1c) +with states x = (x0, . . . , xN), xi ∈ Rnx, controls u = +(u0, . . . , uN−1), ui ∈ Rnu and a given initial state ¯¯x0. +The multiple shooting formulation given in (1) keeps both +the controls and the states as optimization variables. Due the +special structure of the OCP, the subproblems arising within +a Sequential Quadratic Programming (SQP) approach can be +solved efficiently via the Riccati recursion [1]. +While multiple shooting is a simultaneous approach – it +solves the simulation and optimization problem simultane- +ously – both single shooting and DDP can be considered +sequential approaches. Single shooting eliminates the states +via forward simulation and keeps only the control inputs as +optimization variables yielding an unconstrained nonlinear +program (NLP). If SSis implemented in a sparsity-exploiting +fashion, quadratic subproblems with the same sparse struc- +ture as in multiple shooting need to be solved [2]. Based on +the controls obtained from the solution of this subproblem, +an additional open-loop simulation of the nonlinear system +dynamics needs to be performed. Similarly, DDP can be +implemented by first performing a Riccati recursion based +on the very same quadratic subproblem and then simulating +This research was supported by DFG via Research Unit FOR 2401 and +project 424107692 and by the EU via ELO-X 953348. +Katrin Baumg¨artner and Florian Messerer are with the Department of Mi- +crosystems Engineering (IMTEK) and Moritz Diehl is with the Department +of Microsystems Engineering (IMTEK) and Department of Mathematics, +University Freiburg, 79110 Freiburg, Germany. +katrin.baumgaertner@imtek.uni-freiburg.de +the nonlinear system forward in time. In contrast to the open- +loop simulation in single shooting, DDP leverages the time- +varying affine feedback law that is obtained from the Riccati +recursion within the nonlinear forward simulation. +Depending on the choice of Hessian approximation that is +chosen for the quadratic subproblems, the three methods +come in different variants. Assuming convex stage and +terminal costs, we consider two common Hessian approx- +imations: exact Hessian (EH) and the Generalized Gauss- +Newton (GGN) Hessian approximation. The GGN Hessian +is a generalization of the Gauss-Newton (GN) Hessian, which +is widely used in case of quadratic stage and terminal costs, +to general convex cost functions [3], [4]. +With an exact Hessian, all three methods locally converge +with a quadratic rate. If a GGN Hessian approximation is +used, the local convergence rate – assuming that the iterates +converge – is in general linear and, as we will show in the +following, the asymptotic rate of convergence is the same +for all three methods. +A. Contribution & Outline +The contribution of this paper is to provide a unified view on +multiple shooting, SSand DDP from a numerical optimiza- +tion perspective. In particular, we show that the GGN variants +of the three methods locally converge at the same linear rate. +This rate can be exactly characterized as the smallest scalar +that satisfies two linear matrix inequalities. +After providing an overview on related work in the next +paragraph, we briefly recall the three numerical methods +in Section II highlighting their similarities and differences. +Section III analyzes the local convergence behaviour of +the three methods, which is demonstrated on an illustrative +example in Section V. +B. Related Work +The DDP algorithm using exact Hessians was originally +proposed by Mayne in 1966 [5] and further analyzed in [6]. +Proofs for quadratic convergence of DDP were first given in +1984 by [7] and [8]. In 1990, Shoemaker and Liao provided +a proof based on Bellman’s principle of optimality [9]. +Its Gauss-Newton variant, which is more commonly referred +to as iterative Linear Quadratic Regulator (iLQR), especially +within the robotics community [10], has been introduced in +[11], [12]. +The sparsity-exploiting implementation of single shooting, +which we consider here, has first been introduced in [2], [13] +using a Gauss-Newton Hessian. The Gauss-Newton variant +has also been analyzed more recently in [14]. + +In the context of direct optimal control, multiple shooting +was first suggested by Bock in 1984 [15]. Even earlier, the +multiple shooting approach has been discussed for parameter +identification problems [16] and for boundary value problems +[17]. +The quadratic convergence behaviour of the exact Hessian +variant of both multiple and single shooting directly follows +from the analysis of Newton’s method. For the Gauss- +Newton variants, local linear convergence has first been an- +alyzed in [16]. For the Generalized Gauss-Newton variants, +we refer to [4] for a detailed analysis. +For multiple and single shooting, the exact characterization +of the local contraction rate follows directly from the results +in [4], [18], where general sequential convex programming +and Generalized Gauss-Newton methods are considered. +In [19], a family of Gauss-Newton shooting methods is +introduced that combine the multiple shooting approach – +on a coarse discretization grid – with Gauss-Newton DDP +or Gauss-Newton single shooting, which is performed on +a fine discretization grid within each multiple shooting +interval. Our analysis can be easily extended to this family +of algorithms. +In [20], the local convergence behaviour of multiple shooting +and SSwith exact Hessians has been discussed in a simplified +setting, which shows different quadratic rates for the two +methods. +II. UNCONSTRAINED OPTIMAL CONTROL PROBLEM AND +NUMERICAL METHODS +In this section, we briefly recall the three numerical methods +and point out their similarities and differences. +We consider discrete optimal control problems (OCP) as +defined in (1), where we assume that both the stage costs +li, as well as the terminal cost lN are convex. If we linearize +the dynamics and approximate the objective by a quadratic +function at the current iterate (¯x, ¯u) – or (¯x, ¯u, ¯λ) for the +exact Hessian variant –, we obtain an equality constrained +Quadratic Program (QP), +min +x,u +N−1 +� +i=0 +� +q⋆ +i +r⋆ +i +�⊤� +xi +ui +� ++ 1 +2 +� +xi +ui +�⊤� +Q⋆ +i +(S⋆ +i )⊤ +S⋆ +i +R⋆ +i +�� +xi +ui +� +(2a) ++ p⊤ +NxN + 1 +2x⊤ +NPNxN +(2b) +s.t. +x0 = ¯¯x0, +(2c) +xi+1 = ai + Aixi + Biui, +i = 0, . . . , N − 1, (2d) +where the linearized dynamics are given by +Ai = ∂fi +∂xi +(¯xi, ¯ui), +Bi = ∂fi +∂ui +(¯xi, ¯ui), +(3) +ai = fi(¯xi, ¯ui) − Ai¯xi − Bi¯ui. +(4) +The cost gradients are +q⋆ +i = ∇xil(¯xi, ¯ui) − Q⋆ +i ¯xi − (S⋆ +i )⊤¯ui, +(5) +r⋆ +i = ∇uil(¯xi, ¯ui) − S⋆ +i ¯xi − R⋆ +i ¯ui, +(6) +pN = ∇xNlN(¯xN) − PN ¯xN. +(7) +The Hessian associated with the terminal stage is given by +PN = ∇2 +xilN(¯xN). +(8) +The Hessian blocks for the all other stages are given by +� +QGGN +i +(SGGN +i +)⊤ +SGGN +i +RGGN +i +� += ∇2 +(xi,ui) li(¯xi, ¯ui) +(9) +if a Generalized Gauss-Newton Hessian approximation is +used, and by +� +QEH +i +(SEH +i )⊤ +SEH +i +REH +i +� +=∇2 +(xi,ui) +� +li(¯xi, ¯ui)+¯λ⊤ +i+1fi(¯xi, ¯ui) +� +(10) +if the exact Hessian is used. Note that the dual variables ¯λi, +associated with the equality constraints in (1), are needed if +an exact Hessian is used, while they need not be computed +for the GN and GGN variant. +Multiple shooting solves an instance of the quadratic sub- +problem given in (2) in every iteration. Due to the particular +structure of this QP, it can be efficiently solved via a +backward Riccati recursion and a forward simulation based +on the linearized dynamics. +A summary of multiple shooting algorithm is give in Table I, +in the center column. The method proceeds as follows: First, +we linearize the nonlinear OCP, (3) to (10), using an exact or +GGN Hessian. Next, we perform a Riccati recursion given +as +Ki = −(Ri + B⊤ +iPi+1Bi)�1(Si + B⊤ +iPi+1Ai), +(12) +ki = −(Ri + B⊤ +iPi+1Bi)�1(ri + B⊤ +i(Pi+1ai + pi+1)), +(13) +Pi = Qi + A⊤ +iPi+1Ai + (S⊤ +i + A⊤ +iPi+1Bi)Ki, +(14) +pi = qi + A⊤ +i(Pi+1ai + pi+1) ++ K⊤ +i (ri + B⊤ +i(Pi+1ai + pi+1)), +(15) +for i = N − 1, . . . , 0. Finally, a forward simulation is +performed using the linearized system dynamics and the +linear feedback law defined by Ki, ki. The parameter α ∈ +(0, 1] is a line search parameter reducing the step size if +necessary. If multipliers are required, they are updated as +well at this final step. +Now turning to single shooting and DDP, summarized in +the left and right column of Table I, we first point out that +both DDP and single shooting require a feasible initial guess, +which is not the case for multiple shooting. Furthermore, +note that for multiple shooting, the three steps – linearization, +backward Riccati recursion, forward sweep – could be imple- +mented sequentially. If the exact Hessian is used, this does +not hold for single shooting and DDP, where the Hessian +blocks Qi, Ri, Si of stage i depend on the multiplier ¯λi+1 +computed in the previous recursive step of the very same +backward sweep. With multiple shooting, the multipliers are +part of the memory of the algorithm, which is not the case +for SSand DDP, where they need to be computed on the +fly. Thus, linearization and backward Riccati recursion are +entwined and cannot be implemented sequentially. + +TABLE I +BACKWARD AND FORWARD SWEEP OF DDP, MULTIPLE SHOOTING AND SSIN COMPARISON. +INPUT: ¯x, ¯u (feasible) +INPUT: ¯x, ¯u, ¯λ +INPUT: ¯x, ¯u (feasible) +DDP – BACKWARD SWEEP +MULTIPLE SHOOTING – BACKWARD SWEEP +SINGLE SHOOTING – BACKWARD SWEEP +¯PN, ¯pN = via eq. (8) and (7) +¯PN, ¯pN = via eq. (8) and (7) +¯PN, ¯pN = via eq. (8) and (7) +(11a) +¯λN= pN + PN ¯xN, +¯λN= pN + PN ¯xN, +(11b) +Ai, Bi, ai = via eq. (3) and (4) +Ai, Bi, ai = via eq. (3) and (4) +Ai, Bi, ai = via eq. (3) and (4) +(11c) +Qi, Ri, Si = via eq. (9) or (10) +Qi, Ri, Si = via eq. (9) or (10) +Qi, Ri, Si = via eq. (9) or (10) +(11d) +qi, ri = via eq. (5) and (6) +qi, ri = via eq. (5) and (6) +qi, ri = via eq. (5) and (6) +(11e) +Pi, pi = via eq. (14) and (15) +Pi, pi = via eq. (14) and (15) +Pi, pi = via eq. (14) and (15) +(11f) +Ki, ki = via eq. (12) and (13) +Ki, ki = via eq. (12) and (13) +Ki, ki = via eq. (12) and (13) +(11g) +¯λi= pi + Pi¯xi, +¯λi= pi + Pi¯xi, +(11h) +where i = N − 1, . . . , 0, +where i = N − 1, . . . , 0, +where i = N − 1, . . . , 0, +DDP – FORWARD SWEEP +MULTIPLE SHOOTING – FORWARD SWEEP +SINGLE SHOOTING – FORWARD SWEEP +x0 = ¯¯x0, +x0 = ¯¯x0, +x0 = ¯¯x0, +(11i) +ui = ¯ui + ki + Ki(xi − ¯xi), +ui = ¯ui + ki + Ki(xi − ¯xi), +ui = ¯ui + ki + Ki(ˆxi − ¯xi), +(11j) +xi+1 = ¯fi + Ai(xi − ¯xi) + Bi(ui − ¯ui), +ˆxi+1 = ¯fi + Ai(ˆxi − ¯xi) + Bi(ui − ¯ui),(11k) +xi+1 = f(xi, ui), +xi+1 = f(xi, ui), +(11l) +where i = 0, . . . , N − 1. +where i = 0, . . . , N − 1. +where i = 0, . . . , N − 1. +λi= ¯pi + ¯Pixi, +(11m) +where i = 0, . . . , N. +OUTPUT: x, u +OUTPUT: x, u, λ +OUTPUT: x, u +After the backward sweep, single shooting performs a linear +forward sweep to obtain the controls and a nonlinear open- +loop simulation to obtain the states. In contrast, DDP uses +the nonlinear system dynamics as well as the linear feedback +law defined by Ki, ki to perform a closed-loop forward +simulation. +III. CONVERGENCE ANALYSIS +In this section, we analyze the local convergence behaviour +of multiple shooting, single shooting and DDP. In particular, +we show that all three methods have the same linear contrac- +tion rate if a GGN Hessian approximation is used. In fact, +our analysis extends to any Hessian approximation based on +the primal variables (xi, ui). +We define the primal-dual iterate z = (x, u, λ) where λ are +the multipliers associated with the equality constraints. +Proposition 1. Let z∗ = (x∗, u∗, λ∗) be a feasible point +of (1) at which LICQ holds. The following statements are +equivalent: +(i) z∗ is KKT point of the NLP in (1). +(ii) z∗ is a fixed point of the multiple shooting iteration. +(iii) z∗ is a fixed point of the SSiteration. +(iv) z∗ is a fixed point of the DDP iteration. +Proof. We refer to, e.g., Chapter 8.8 in [21]. +All three algorithms can be defined in terms of a nonlinear +parametric root-finding problem, which we will do in the +following. In a neighbourhood of a solution, the convergence +behaviour of the iterates is governed by the spectral radius +of the Jacobian of the solution map, which is shown in the +following classical result: +Theorem 1. Let Π : Rnz → Rnz be the solution map of +the nonlinear parametric root-finding problem R(z; ¯z) = 0 +such that R(Π(¯z); ¯z) = 0 and with R is twice continuously +differentiable. +Suppose that z∗ is a fixed point of the iteration z+ = Π(z) +with +∂R +∂z1 (z∗; z∗) nonsingular. Let κ(z∗) := ρ(J(z∗)) be the +spectral radius of the matrix J(z∗) given as +J(z∗) := − +�∂R +∂z (z∗; z∗) +��1 ∂R +∂¯z (z∗; z∗). +If 0 < κ(z∗) < 1, the iterates locally converge to z∗ at +a linear rate. The asymptotic convergence rate is given by +κ(z∗). If κ(z∗) = 0, the iterates locally converge to z∗ at a +superlinear rate. If κ(z∗) > 1, the fixed point z∗ is unstable. +Proof. A Taylor expansion of the solution map Π(z) at the + +fixed point z∗ yields +zk+1 − z∗ = dΠ +d¯z (z∗)(zk − z∗) + O +� +∥zk − z∗∥2� +. +The derivative dΠ +d¯z (z∗) is given by the matrix J(z∗), which +follows from the implicit function theorem. A standard result +of linear stability analysis of nonlinear systems shows that +local convergence of the iterates zk to z∗ is determined by +the spectral radius ρ(J(z∗)), (compare e.g. [22]). +The following lemma will allow us to show that the matrix +J(z∗) in Theorem 1 is the same for all three methods. +Lemma 1. We consider two twice continuously differentiable +functions R1 : Rm × Rm → Rl, (x, y) �→ R1(x, y) and +R2 : Rm × Rm → Rl, (x, y) �→ R2(x, y). If it holds that +R1(z, z) = R2(z, z), +∂R1 +∂x (z, z) = ∂R2 +∂x (z, z), +for all z ∈ Rm, then +R1(x, y) = R2(x, y) + O +� +∥x − y∥2� +, +(16) +and in particular ∂R1 +∂y (z, z) = ∂R2 +∂y (z, z). +Proof. A first-order Taylor expansion of R1(x, y) in x +around the linearization point ¯x ∈ Rm yields +R1(x, y) =R1(¯x, y) + ∂R1 +∂x (¯x, y)(x − ¯x) + O +� +∥x − ¯x∥2� +, +R2(x, y) =R2(¯x, y) + ∂R2 +∂x (¯x, y)(x − ¯x) + O +� +∥x − ¯x∥2� +. +By subtracting these two equalities and setting y = ¯x we +directly obtain (16). Furthermore, +lim +ǫ→0 +R1(z, z+ǫd) − R1(z, z) +ǫ += lim +ǫ→0 +R2(z, z+ǫd) − R2(z, z) +ǫ ++ O(∥ǫ∥), +which implies that ∂R1 +∂y (z, z) = ∂R2 +∂y (z, z). +Equipped with the above result, we now show that multiple +shooting, single shooting and DDP locally converge at the +same linear rate if a GGN Hessian approximation is used. To +this end, we summarize the primal variables as w = (x, u) +and introduce the following notation, where we set N = 2 +w.l.o.g. to keep the notation simple, +ˆF(x, u; ¯w) = + + +¯¯x0 +− x0 +¯f0 + ¯A0(x0 − ¯x0) + ¯B0(u0 − ¯u0) − x1 +¯f1 + ¯A1(x1 − ¯x1) + ¯B1(u1 − ¯u1) − x2 + + , +F(x, u) = + + +¯¯x0 +− x0 +f(x0, u0) − x1 +f(x1, u1) − x2 + + , +denoting the linearized and nonlinear forward simulation, +and +G(x, u; ¯w) = +�¯u0 + ¯k0( ¯w) + ¯K0( ¯w)(x0 − ¯x0) − u0 +¯u1 + ¯k1( ¯w) + ¯K1( ¯w)(x1 − ¯x1) − u1 +� +, +summarizing the feedback law. The quantities ¯ki( ¯w) and +¯Ki( ¯w) are computed according to (13) and (12) respectively. +Note that they depend only on the primal iterate ¯w if a GGN +Hessian is used. +A. Multiple Shooting vs. DDP +We define the next multiple shooting iterate via w+ = +ΠGGN +MS ( ¯w) where ΠGGN +MS ( ¯w) is the solution map of the linear +root-finding problem RGGN +MS (w; ¯w) = 0 with +RGGN +MS (w; ¯w) = +� ˆF(x, u; ¯w) +G(x, u; ¯w) +� +. +(17) +Similarly, we define the next DDP iterate via w+ = ΠGGN +DDP ( ¯w) +where ΠGGN +DDP ( ¯w) is the solution map of the nonlinear root- +finding problem RGGN +DDP (w; ¯w) = 0 with +RGGN +DDP (w; ¯w) = +� +F(x, u) +G(x, u; ¯w) +� +. +(18) +Note that the only difference between (17) and (19) is the +first block where DDP uses the nonlinear dynamics while +multiple shooting uses the linearized dynamics. +Proposition 2. Consider a KKT point (w∗, λ∗) of the NLP +in (1) that satisfies LICQ and SOSC. The asymptotic linear +contraction rate κGGN +MS (w∗) of the multiple shooting iterates, +obtained via w+ = ΠGGN +MS (w), is equal to the asymptotic lin- +ear contraction rate κGGN +DDP (w∗) of the DDP iterates, obtained +via w+ = ΠGGN +DDP (w). +Proof. Theorem 1 implies that it suffices to show that the +partial derivatives of RGGN +MS (w; ¯w) and RGGN +DDP (w; ¯w) coincide +at the fixed point w∗ in order to prove that κGGN +MS (w∗) = +κGGN +DDP (w∗). We first consider the partial derivative w.r.t. ¯w. +From the definitions in (17) and (18) and together with +∂ ˆF +∂(x, u)(x∗, u∗; w∗) = +∂F +∂(x, u)(x∗, u∗; w∗), +we directly obtain +∂RGGN +MS +∂w (w∗; w∗) = +∂RGGN +DDP +∂w (w∗; w∗). To- +gether with Lemma 1, we conclude that also the partial +derivatives w.r.t. ¯w coincide at w∗. +B. Multiple Shooting vs. Single Shooting +Let y = (x, ˆx, u). We define the next SSiterate via y+ = +ˆΠGGN +SS (¯y) where ˆΠGGN +SS +is the solution map of the nonlinear +root-finding problem ˆRGGN +SS (y; ¯y) = 0 with +ˆRGGN +SS (y; ¯y) = + + +F(x, u) +ˆF(ˆx, u; ¯w) +GGGN(ˆx, u; ¯w) + +. +(19) +Similarly, we define the next multiple shooting iterate via +y+ = ˆΠGGN +MS (¯y) where ˆΠGGN +MS +is the solution map of the linear +root-finding problem ˆRGGN +MS (y; ¯y) = 0 with +ˆRGGN +MS (y; ¯y) = + + +ˆF(x, u; ¯w) +ˆF(ˆx, u; ¯w) +GGGN(ˆx, u; ¯w) + +. +(20) +Note that the definition in (20) is redundant as it includes the +same linear forward sweep twice. This is necessary only for +the comparison with single shooting: In this formulation, the +two residual maps (19) and (20) differ only in the first block +where single shooting uses a nonlinear forward simulation +while multiple shooting performs the forward simulation +based on the linearized dynamics. + +Proposition 3. Consider a KKT point (w∗, λ∗) with w∗ = +(x∗, u∗) of the NLP in (1). Let y∗ = (x∗, x∗, u∗). +The asymptotic linear contraction rate κGGN +MS (y∗) of the +multiple shooting iterates, obtained via y+ = ˆΠGGN +MS (y), is +equal to the asymptotic linear contraction rate κGGN +SS (y∗) of +the SSiterates, obtained via y+ = ΠGGN +SS (y). +Proof. We proceed as in the proof of Proposition 2 and show +that the partial derivatives of of ˆRGGN +MS (y; ¯y) and ˆRGGN +SS (y; ¯y) +coincide at the fixed point y∗. From the definitions in (20) +and (19) and together with +∂ ˆF +∂(x, u)(x∗, u∗; z∗) = +∂F +∂(x, u)(x∗, u∗; z∗), +we directly obtain +∂ ˆ +RGGN +MS +∂y (y∗; y∗) += +∂ ˆ +RGGN +SS +∂y (y∗; y∗). To- +gether with Lemma 1, we conclude that also the partial +derivatives with respect to ¯y coincide at y∗, i.e., we have +∂ ˆ +RGGN +MS +∂¯y (y∗; y∗) = ∂RGGN +SS +∂¯y (y∗; y∗). +C. Local Linear Contraction Rate +In the previous section, we have shown that multiple shoot- +ing, single shooting and DDP locally converge with the same +linear rate if a GGN Hessian is used. We now analyze +the multiple shooting iteration to further characterize this +linear rate. To this end, we consider yet another equivalent +definition of the multiple shooting iteration. We define the +next multiple shooting iterate via z+ = ˜ΠGGN +MS (¯z) where +˜ΠGGN +MS +is the solution map of the linear root-finding problem +˜RGGN +MS (z; ¯z) = 0 with +˜RGGN +MS (z; ¯z) = +� +V GGN +quad(x, u; ¯w) + ∇w ˆF(x, u; ¯w)λ +ˆF(x, u; ¯w), +� +(21) +where V GGN +quad(x, u; ¯w) is the quadratic approximation of the +objective given in (2) using a GGN Hessian, i.e., +V GGN +quad(w; ¯w) = V ( ¯w) + ∂V +∂w ( ¯w)(w − ¯w) ++ 1 +2 (w − ¯w)⊤HGGN( ¯w)(w − ¯w). +(22) +Theorem 2. Consider a KKT point z∗ = (x∗, u∗, λ∗) of the +NLP in (1). Let y∗ = (x∗, x∗, u∗). +The asymptotic linear contraction rate of the multiple shoot- +ing iterates, the SSiterates, and the DDP iterates is the same +for all three methods and given by the smallest κ that satisfies +the condition +−κ ˜ +M GGN(z∗) ⪯ ˜EGGN(z∗) ⪯ κ ˜ +M GGN(z∗), +(23) +where we have +˜ +M GGN(z∗) = Z⊤M GGN(z∗)Z, ˜EGGN(z∗) = +Z⊤EGGN(z∗)Z +with +Z +a basis of the null space of +∇wF(x∗, u∗)⊤. Here M GGN(z∗) is the Hessian approximation +and EGGN(z∗) is the deviation from the exact Hessian. +Proof. The fact that the local linear contraction rate is the +same for all three methods follows from Proposition 2 and 3. +The characterization of the rate in terms of the linear matrix +inequalities in (23) follows from Theorem 4.5 in [4] applied +to the root-finding problem in (21). +Remark 1. Note that for the OCP-structured problem at +hand, the error matrix EGGN(z∗) is given as +EGGN(z∗) = ∇2 +(x,u) +� +(λ∗)⊤F(x∗, u∗) +� +. +(24) +Considering N = 3 and reordering the optimization vari- +ables as ˜w = (x0, u0, x1, u1, x2, u2, x3), a basis Z of the +null space of ∇˜wF(x∗, u∗)⊤ is given by +Z = + + +0 +0 +0 +I +0 +0 +B∗ +0 +0 +0 +0 +I +0 +A∗ +1B∗ +0 +B∗ +1 +0 +0 +0 +I +A∗ +2A∗ +1B∗ +0 +A∗ +2B∗ +1 +B∗ +2 + + +. +Here, the Jacobians are given as A∗ +i = ∂f +∂x(x∗ +i , u∗ +i ) and B∗ +i = +∂f +∂u(x∗ +i , u∗ +i ). Moreover, the reduced error matrix ˜EGGN(z∗) = +Z⊤EGGN(z∗)Z corresponds to the error matrix of the single +shooting problem if solved in the standard dense formulation. +Corollary 1. Propositions 2 and 3 imply that +∥w+ +MS − w+ +SS∥2 = O +� +∥ ¯w − w∗∥2 +2 +� +, +(25) +if both methods start from a feasible initial guess ¯w. The +analogous result holds for multiple shooting vs. DDP and +single shooting vs. DDP. +D. Exact Hessian Variants and Quadratic Rate +If an exact Hessian is used, multiple shooting, single shooting +and DDP locally converge at a quadratic rate. In [20], the +local convergence behaviour of multiple shooting and SSwith +exact Hessians has been discussed in a simplified setting. +The authors concluded that multiple shooting formulations +lead to faster contraction rates if the nonlinearities of the +system dynamics reinforce each other, i.e. have the same +direction of curvature. SSwould lead to faster contraction +if the concatenated nonlinearities mitigate each other. They +furthermore argued that in optimal control, the nonlinearities +often reinforce each other rendering a multiple shooting +approach favorable. +IV. FURTHER REMARKS & DISCUSSION +Our main result shows that if a GGN Hessian is used all +three methods locally behave the same. In the following, +we discuss further differences and similarities, as well as +advantages and disadvantages of the three methods. +A. Initialization +Multiple shooting can start from an infeasible initial guess, +which is not possible for single shooting and DDP. The possi- +bility for infeasible initialization greatly simplifies the usage +of multiple shooting in practice as no additional routines +for finding a feasible initial guess are required. Furthermore, +infeasible initialization might improve convergence of the +method if a rough guess of how a solution trajectory might +look like is available [17]. + +B. GGN Hessian & Convex-Over-Nonlinear Objectives +We focused on a GGN Hessian approximation which comes +with several advantages: (1) The subproblems that need to be +solved in each iteration are convex and can thus be solved +reliably. (2) We require only first-order derivatives of the +dynamics. As second-order derivatives tend to be expensive +to compute, using a GGN Hessian might significantly reduce +computational cost of the method. (3) The GGN Hessian +does not require computation of the multipliers λi reducing +the complexity of the algorithm. +Moreover, the GGN approach naturally covers convex-over- +nonlinear cost where li is not convex but instead takes the +form li(xi, ui) = ψi(ri(xi, ui)) with ψi convex and ri +nonlinear. In this case, the GGN Hessian approximation is +given as +� +QGGN +i +(SGGN +i +)⊤ +SGGN +i +RGGN +i +� += ¯Ji(¯xi, ¯ui)⊤∇2φi(¯ri) ¯Ji(¯xi, ¯ui), +(26) +where ¯ri = ri(¯xi, ¯ui) and ¯Ji(¯xi, ¯ui) = +∂ri +∂(xi,ui). If the +Hessian error matrix in (24) is adapted accordingly, our local +convergence analysis is still valid in this more general case. +C. Constraints +Constraints might be incorporated into the problem formula- +tion via barrier functions or penalty functions as is done e.g. +in [23], [24], [25], [26], [27]. Note that these formulations +typically lead to convex-over-nonlinear objective functions. +D. Implementation Details +1) Forward Sweep: In the form presented here, all three +methods can be implemented in a very similar fashion if a +GGN Hessian is used. With the Riccati recursion being the +same for all three methods, only the forward sweep needs to +be adapted. One advantage of multiple shooting over SSand +DDP is the possibility to parallelize the forward sweep. +2) Line Search: If a line search strategy is used (11j) needs +to be changed to +DDP & multiple shooting +ui = ¯ui + αki + Ki(xi − ¯xi), +single shooting: +ui = ¯ui + αki + Ki(ˆxi − ¯xi), +where α ∈ (0, 1] is the step size that is successively reduced +until a sufficient decrease criterion is met. For single shooting +and DDP, the iterates are always feasible such that simply the +cost function can be considered within a sufficient decrease +approach. For multiple shooting, however, one needs to +decide on a merit function in order to combine both sufficient +decrease of the cost function and infeasibility reduction into +a scalar progress criterion. +V. NUMERICAL RESULTS +In this section, we illustrate the local convergence rate of the +MS, SS, and DDP iterates on a numerical example adopted +from [28]. +The continuous time dynamics are given as +˙x1 = x2 + u (µ + (1 − µ)x1) +˙x2 = x1 + u (µ − 4(1 − µ)x2) . +where we use µ = 0.7. We discretize the continuous +dynamics using ten steps of a Runge-Kutta integrator of +fourth order (RK4) on an integration interval of h = 0.25. +We use a quadratic cost function together with a quadratic +penalty function which penalizes control inputs that do not +satsify |ui| ≤ umax = 1, yielding the following objective: +V (x, u) = +� +i=0 +1 +2x⊤ +iQxi + 1 +2u⊤ +iRui + τβ(ui) + 1 +2x⊤ +NPxN +with Q = diag(0.5, 0.5), R = 0.8, P = diag(10, 10) and +penalty β(ui) = max(0, ui − umax)2 + min(0, ui + umax)2 +with τ = 100. Note that the objective V (x, u) comprising +the cost and penalty functions is convex and a GGN Hessian +can be used. In the following, we obtain a feasible initial +guess by simulating the nonlinear system forward in time +using the linear feedback law obtain from an LQR which +is based on the linear system obtained by linearizing at the +steady state xsteady = 0, usteady = 0. +In Fig. 1, the state trajectory after a single iteration of +multiple shooting, single shooting, and DDP is shown for +a horizon length of N = 20. All three methods start from +the same feasible initial guess that is shown in gray. While +both the SSand DDP trajectory are feasible with respect to +the dynamics constraints, the multiple shooting trajectory +exhibits gaps. The difference of the SSiterate to the multiple +shooting iterate increases significantly along the horizon. +In Fig. 2, the empirical linear contraction rate, given by +ˆκ = ∥wk+1 − wk∥ +∥wk − wk−1∥, +is shown. The exact Hessian variants require only very few +iterations to reach the convergence criterion and the empir- +ical contraction rate quickly tends to zero. The empirical +contraction rate of the three GGN variants needs a few more +iterations to converge and they converges to the same value. +Fig. 3 shows the distance to the solution of the iterate as +a function of the distance to the solution of the previous +iterate. In this log-log plot, the slope of the linear func- +tions correspond to the order of convergence. The intercepts +correspond to the rate of convergence. The GGN variants +of the three methods converge linearly at exactly the same +asymptotic rate, while the three methods using an exact +Hessian converge quadratically. +VI. CONCLUSIONS AND OUTLOOK +We provided a unified view on multiple shooting, single +shooting and DDP, three classical methods for solving un- +constrained discrete optimal control problems (OCP), from a +numerical optimization perspective. While all thee methods +can be used with different Hessian approximations, we +focused on the Generalized Gauss-Newton (GGN) Hessian, +a generalization of the widely used Gauss-Newton (GN) +Hessian to convex loss functions. We showed that the iterates +obtained with these three methods locally converge – or +diverge – to a KKT point of the OCP a the same linear +rate. + +−0.2 +0.0 +0.2 +0.4 +0.6 +x1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +x2 +GGN-MS +GGN-SS +GGN-DDP +solution +initial guess +Fig. 1. +State trajectories obtained after one iteration of GGN-MS, GGN- +SS, and GGN-DDP. All methods start from the same feasible initial guess +which is obtained by simulating the system forward in closed-loop using +the LQR feedback law associated with the steady state. The initial state is +¯¯x0 = (0.42, 0.45). +0 +5 +10 +15 +20 +iteration n +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +empirical linear contraction rate ˆκ +EH-MS +GGN-MS +EH-SS +GGN-SS +EH-DDP +GGN-DDP +Fig. 2. +Empirical contraction rate ˆκ for both the exact Hessian and GGN +Hessian variant of MS, SS and DDP. The convergence criterion is ∆w ≤ ε +with ε = 10�12. +10−8 +10−6 +10−4 +10−2 +100 +102 +∥wk − w∗∥∞ +10−8 +10−6 +10−4 +10−2 +100 +102 +∥wk+1 − w∗∥∞ +EH-MS +GGN-MS +EH-SS +GGN-SS +EH-DDP +GGN-DDP +Fig. 3. +Norm of the primal step as a function of the norm of the previous +primal step. The slope of the linear function corresponds to the order of +convergence. The intercept corresponds to the rate of convergence. +REFERENCES +[1] C. V. Rao, S. J. Wright, and J. B. Rawlings, “Application of interior- +point methods to model predictive control,” Journal of Optimization +Theory and Applications, vol. 99, pp. 723–757, 1998. +[2] A. Sideris and J. 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Diehl, Model Predictive +Control: Theory, Computation, and Design, 2nd ed. +Nob Hill, 2017. +[22] A. M. Ostrowski, Solutions of Equations in Euclidean and Banach +Spaces. +New York and London: Academic Press, 1973. +[23] A. G. Wills and W. P. Heath, “Barrier function based model predictive +control,” Automatica, vol. 40, no. 8, pp. 1415–1422, August 2004. +[24] A. Zanelli, R. Quirynen, G. Frison, and M. Diehl, “A partially +tightened real-time iteration scheme for nonlinear model predictive +control,” in Proceedings of 56th IEEE Conference on Decision and +Control, Melbourne, Australia, December 2017. + +[25] C. Feller and C. Ebenbauer, “A stabilizing iteration scheme for model +predictive control based on relaxed barrier functions,” Automatica, +vol. 80, pp. 328–339, June 2017. +[26] J. Marti-Saumell, J. Sol`a, C. Mastalli, and A. Santamaria-Navarro, +“Squash-box feasibility driven differential dynamic programming,” in +2020 IEEE/RSJ International Conference on Intelligent Robots and +Systems (IROS). +IEEE, 2020, pp. 7637–7644. +[27] K. Baumg¨artner, Y. Wang, A. Zanelli, and M. Diehl, “Fast nonlinear +model predictive control using barrier formulations and squashing with +a Generalized Gauss-Newton Hessian,” in Proceedings of the IEEE +Conference on Decision and Control (CDC), 2022. +[28] H. Chen and F. Allg¨ower, “A quasi-infinite horizon nonlinear model +predictive control scheme with guaranteed stability,” Automatica, +vol. 34, no. 10, pp. 1205–1218, 1998. + diff --git a/OtE2T4oBgHgl3EQfrQhe/content/tmp_files/load_file.txt b/OtE2T4oBgHgl3EQfrQhe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4691b41c3fcf03806b4995f1f065fed30b5809a0 --- /dev/null +++ b/OtE2T4oBgHgl3EQfrQhe/content/tmp_files/load_file.txt @@ -0,0 +1,548 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf,len=547 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='04047v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='OC] 10 Jan 2023 Local Convergence Behaviour of Generalized Gauss-Newton Multiple Shooting, Single Shooting and Differential Dynamic Programming Katrin Baumg¨artner, Florian Messerer, Moritz Diehl Abstract— We revisit three classical numerical methods for solving unconstrained optimal control problems – Multiple Shooting (MS), Single Shooting (SS), and Differential Dynamic Programming (DDP) – and examine their local convergence behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In particular, we show that all three methods converge with the same linear rate if a Gauss-Newton (GN) – or more general a Generalized Gauss-Newton (GGN) – Hessian approximation is used, which is the case in widely used implementations such as iLQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' INTRODUCTION Multiple Shooting, single shooting and differential dynamic programming are three numerical methods that might be used for solving discrete optimal control problems (OCP) that typically arise after discretization of a continuous-time OCP: min x,u N−1 � i=0 li(xi, ui) + lN(xN) (1a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' x0 = ¯¯x0, (1b) xi+1 = fi(xi, ui), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , N − 1, (1c) with states x = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , xN), xi ∈ Rnx, controls u = (u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , uN−1), ui ∈ Rnu and a given initial state ¯¯x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The multiple shooting formulation given in (1) keeps both the controls and the states as optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Due the special structure of the OCP, the subproblems arising within a Sequential Quadratic Programming (SQP) approach can be solved efficiently via the Riccati recursion [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' While multiple shooting is a simultaneous approach – it solves the simulation and optimization problem simultane- ously – both single shooting and DDP can be considered sequential approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Single shooting eliminates the states via forward simulation and keeps only the control inputs as optimization variables yielding an unconstrained nonlinear program (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If SSis implemented in a sparsity-exploiting fashion, quadratic subproblems with the same sparse struc- ture as in multiple shooting need to be solved [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Based on the controls obtained from the solution of this subproblem, an additional open-loop simulation of the nonlinear system dynamics needs to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Similarly, DDP can be implemented by first performing a Riccati recursion based on the very same quadratic subproblem and then simulating This research was supported by DFG via Research Unit FOR 2401 and project 424107692 and by the EU via ELO-X 953348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Katrin Baumg¨artner and Florian Messerer are with the Department of Mi- crosystems Engineering (IMTEK) and Moritz Diehl is with the Department of Microsystems Engineering (IMTEK) and Department of Mathematics, University Freiburg, 79110 Freiburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' katrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='baumgaertner@imtek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='uni-freiburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='de the nonlinear system forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In contrast to the open- loop simulation in single shooting, DDP leverages the time- varying affine feedback law that is obtained from the Riccati recursion within the nonlinear forward simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Depending on the choice of Hessian approximation that is chosen for the quadratic subproblems, the three methods come in different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Assuming convex stage and terminal costs, we consider two common Hessian approx- imations: exact Hessian (EH) and the Generalized Gauss- Newton (GGN) Hessian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The GGN Hessian is a generalization of the Gauss-Newton (GN) Hessian, which is widely used in case of quadratic stage and terminal costs, to general convex cost functions [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' With an exact Hessian, all three methods locally converge with a quadratic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If a GGN Hessian approximation is used, the local convergence rate – assuming that the iterates converge – is in general linear and, as we will show in the following, the asymptotic rate of convergence is the same for all three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Contribution & Outline The contribution of this paper is to provide a unified view on multiple shooting, SSand DDP from a numerical optimiza- tion perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In particular, we show that the GGN variants of the three methods locally converge at the same linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' This rate can be exactly characterized as the smallest scalar that satisfies two linear matrix inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' After providing an overview on related work in the next paragraph, we briefly recall the three numerical methods in Section II highlighting their similarities and differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Section III analyzes the local convergence behaviour of the three methods, which is demonstrated on an illustrative example in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Related Work The DDP algorithm using exact Hessians was originally proposed by Mayne in 1966 [5] and further analyzed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proofs for quadratic convergence of DDP were first given in 1984 by [7] and [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In 1990, Shoemaker and Liao provided a proof based on Bellman’s principle of optimality [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Its Gauss-Newton variant, which is more commonly referred to as iterative Linear Quadratic Regulator (iLQR), especially within the robotics community [10], has been introduced in [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The sparsity-exploiting implementation of single shooting, which we consider here, has first been introduced in [2], [13] using a Gauss-Newton Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The Gauss-Newton variant has also been analyzed more recently in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In the context of direct optimal control, multiple shooting was first suggested by Bock in 1984 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Even earlier, the multiple shooting approach has been discussed for parameter identification problems [16] and for boundary value problems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The quadratic convergence behaviour of the exact Hessian variant of both multiple and single shooting directly follows from the analysis of Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' For the Gauss- Newton variants, local linear convergence has first been an- alyzed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' For the Generalized Gauss-Newton variants, we refer to [4] for a detailed analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' For multiple and single shooting, the exact characterization of the local contraction rate follows directly from the results in [4], [18], where general sequential convex programming and Generalized Gauss-Newton methods are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In [19], a family of Gauss-Newton shooting methods is introduced that combine the multiple shooting approach – on a coarse discretization grid – with Gauss-Newton DDP or Gauss-Newton single shooting, which is performed on a fine discretization grid within each multiple shooting interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Our analysis can be easily extended to this family of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In [20], the local convergence behaviour of multiple shooting and SSwith exact Hessians has been discussed in a simplified setting, which shows different quadratic rates for the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' UNCONSTRAINED OPTIMAL CONTROL PROBLEM AND NUMERICAL METHODS In this section, we briefly recall the three numerical methods and point out their similarities and differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We consider discrete optimal control problems (OCP) as defined in (1), where we assume that both the stage costs li, as well as the terminal cost lN are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If we linearize the dynamics and approximate the objective by a quadratic function at the current iterate (¯x, ¯u) – or (¯x, ¯u, ¯λ) for the exact Hessian variant –, we obtain an equality constrained Quadratic Program (QP), min x,u N−1 � i=0 � q⋆ i r⋆ i �⊤� xi ui � + 1 2 � xi ui �⊤� Q⋆ i (S⋆ i )⊤ S⋆ i R⋆ i �� xi ui � (2a) + p⊤ NxN + 1 2x⊤ NPNxN (2b) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' x0 = ¯¯x0, (2c) xi+1 = ai + Aixi + Biui, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , N − 1, (2d) where the linearized dynamics are given by Ai = ∂fi ∂xi (¯xi, ¯ui), Bi = ∂fi ∂ui (¯xi, ¯ui), (3) ai = fi(¯xi, ¯ui) − Ai¯xi − Bi¯ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (4) The cost gradients are q⋆ i = ∇xil(¯xi, ¯ui) − Q⋆ i ¯xi − (S⋆ i )⊤¯ui, (5) r⋆ i = ∇uil(¯xi, ¯ui) − S⋆ i ¯xi − R⋆ i ¯ui, (6) pN = ∇xNlN(¯xN) − PN ¯xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (7) The Hessian associated with the terminal stage is given by PN = ∇2 xilN(¯xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (8) The Hessian blocks for the all other stages are given by � QGGN i (SGGN i )⊤ SGGN i RGGN i � = ∇2 (xi,ui) li(¯xi, ¯ui) (9) if a Generalized Gauss-Newton Hessian approximation is used, and by � QEH i (SEH i )⊤ SEH i REH i � =∇2 (xi,ui) � li(¯xi, ¯ui)+¯λ⊤ i+1fi(¯xi, ¯ui) � (10) if the exact Hessian is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Note that the dual variables ¯λi, associated with the equality constraints in (1), are needed if an exact Hessian is used, while they need not be computed for the GN and GGN variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Multiple shooting solves an instance of the quadratic sub- problem given in (2) in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Due to the particular structure of this QP, it can be efficiently solved via a backward Riccati recursion and a forward simulation based on the linearized dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A summary of multiple shooting algorithm is give in Table I, in the center column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The method proceeds as follows: First, we linearize the nonlinear OCP, (3) to (10), using an exact or GGN Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Next, we perform a Riccati recursion given as Ki = −(Ri + B⊤ iPi+1Bi)�1(Si + B⊤ iPi+1Ai), (12) ki = −(Ri + B⊤ iPi+1Bi)�1(ri + B⊤ i(Pi+1ai + pi+1)), (13) Pi = Qi + A⊤ iPi+1Ai + (S⊤ i + A⊤ iPi+1Bi)Ki, (14) pi = qi + A⊤ i(Pi+1ai + pi+1) + K⊤ i (ri + B⊤ i(Pi+1ai + pi+1)), (15) for i = N − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Finally, a forward simulation is performed using the linearized system dynamics and the linear feedback law defined by Ki, ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The parameter α ∈ (0, 1] is a line search parameter reducing the step size if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If multipliers are required, they are updated as well at this final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Now turning to single shooting and DDP, summarized in the left and right column of Table I, we first point out that both DDP and single shooting require a feasible initial guess, which is not the case for multiple shooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Furthermore, note that for multiple shooting, the three steps – linearization, backward Riccati recursion, forward sweep – could be imple- mented sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If the exact Hessian is used, this does not hold for single shooting and DDP, where the Hessian blocks Qi, Ri, Si of stage i depend on the multiplier ¯λi+1 computed in the previous recursive step of the very same backward sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' With multiple shooting, the multipliers are part of the memory of the algorithm, which is not the case for SSand DDP, where they need to be computed on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Thus, linearization and backward Riccati recursion are entwined and cannot be implemented sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' TABLE I BACKWARD AND FORWARD SWEEP OF DDP, MULTIPLE SHOOTING AND SSIN COMPARISON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' INPUT: ¯x, ¯u (feasible) INPUT: ¯x, ¯u, ¯λ INPUT: ¯x, ¯u (feasible) DDP – BACKWARD SWEEP MULTIPLE SHOOTING – BACKWARD SWEEP SINGLE SHOOTING – BACKWARD SWEEP ¯PN, ¯pN = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (8) and (7) ¯PN, ¯pN = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (8) and (7) ¯PN, ¯pN = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (8) and (7) (11a) ¯λN= pN + PN ¯xN, ¯λN= pN + PN ¯xN, (11b) Ai, Bi, ai = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (3) and (4) Ai, Bi, ai = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (3) and (4) Ai, Bi, ai = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (3) and (4) (11c) Qi, Ri, Si = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (9) or (10) Qi, Ri, Si = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (9) or (10) Qi, Ri, Si = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (9) or (10) (11d) qi, ri = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (5) and (6) qi, ri = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (5) and (6) qi, ri = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (5) and (6) (11e) Pi, pi = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (14) and (15) Pi, pi = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (14) and (15) Pi, pi = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (14) and (15) (11f) Ki, ki = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (12) and (13) Ki, ki = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (12) and (13) Ki, ki = via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (12) and (13) (11g) ¯λi= pi + Pi¯xi, ¯λi= pi + Pi¯xi, (11h) where i = N − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , 0, where i = N − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , 0, where i = N − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , 0, DDP – FORWARD SWEEP MULTIPLE SHOOTING – FORWARD SWEEP SINGLE SHOOTING – FORWARD SWEEP x0 = ¯¯x0, x0 = ¯¯x0, x0 = ¯¯x0, (11i) ui = ¯ui + ki + Ki(xi − ¯xi), ui = ¯ui + ki + Ki(xi − ¯xi), ui = ¯ui + ki + Ki(ˆxi − ¯xi), (11j) xi+1 = ¯fi + Ai(xi − ¯xi) + Bi(ui − ¯ui), ˆxi+1 = ¯fi + Ai(ˆxi − ¯xi) + Bi(ui − ¯ui),(11k) xi+1 = f(xi, ui), xi+1 = f(xi, ui), (11l) where i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' where i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' where i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' λi= ¯pi + ¯Pixi, (11m) where i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' OUTPUT: x, u OUTPUT: x, u, λ OUTPUT: x, u After the backward sweep, single shooting performs a linear forward sweep to obtain the controls and a nonlinear open- loop simulation to obtain the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In contrast, DDP uses the nonlinear system dynamics as well as the linear feedback law defined by Ki, ki to perform a closed-loop forward simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' CONVERGENCE ANALYSIS In this section, we analyze the local convergence behaviour of multiple shooting, single shooting and DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In particular, we show that all three methods have the same linear contrac- tion rate if a GGN Hessian approximation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In fact, our analysis extends to any Hessian approximation based on the primal variables (xi, ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We define the primal-dual iterate z = (x, u, λ) where λ are the multipliers associated with the equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Let z∗ = (x∗, u∗, λ∗) be a feasible point of (1) at which LICQ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The following statements are equivalent: (i) z∗ is KKT point of the NLP in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (ii) z∗ is a fixed point of the multiple shooting iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (iii) z∗ is a fixed point of the SSiteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (iv) z∗ is a fixed point of the DDP iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We refer to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=', Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='8 in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' All three algorithms can be defined in terms of a nonlinear parametric root-finding problem, which we will do in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In a neighbourhood of a solution, the convergence behaviour of the iterates is governed by the spectral radius of the Jacobian of the solution map, which is shown in the following classical result: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Let Π : Rnz → Rnz be the solution map of the nonlinear parametric root-finding problem R(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯z) = 0 such that R(Π(¯z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯z) = 0 and with R is twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Suppose that z∗ is a fixed point of the iteration z+ = Π(z) with ∂R ∂z1 (z∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' z∗) nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Let κ(z∗) := ρ(J(z∗)) be the spectral radius of the matrix J(z∗) given as J(z∗) := − �∂R ∂z (z∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' z∗) ��1 ∂R ∂¯z (z∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If 0 < κ(z∗) < 1, the iterates locally converge to z∗ at a linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The asymptotic convergence rate is given by κ(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If κ(z∗) = 0, the iterates locally converge to z∗ at a superlinear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If κ(z∗) > 1, the fixed point z∗ is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A Taylor expansion of the solution map Π(z) at the fixed point z∗ yields zk+1 − z∗ = dΠ d¯z (z∗)(zk − z∗) + O � ∥zk − z∗∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The derivative dΠ d¯z (z∗) is given by the matrix J(z∗), which follows from the implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A standard result of linear stability analysis of nonlinear systems shows that local convergence of the iterates zk to z∗ is determined by the spectral radius ρ(J(z∗)), (compare e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The following lemma will allow us to show that the matrix J(z∗) in Theorem 1 is the same for all three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We consider two twice continuously differentiable functions R1 : Rm × Rm → Rl, (x, y) �→ R1(x, y) and R2 : Rm × Rm → Rl, (x, y) �→ R2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If it holds that R1(z, z) = R2(z, z), ∂R1 ∂x (z, z) = ∂R2 ∂x (z, z), for all z ∈ Rm, then R1(x, y) = R2(x, y) + O � ∥x − y∥2� , (16) and in particular ∂R1 ∂y (z, z) = ∂R2 ∂y (z, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A first-order Taylor expansion of R1(x, y) in x around the linearization point ¯x ∈ Rm yields R1(x, y) =R1(¯x, y) + ∂R1 ∂x (¯x, y)(x − ¯x) + O � ∥x − ¯x∥2� , R2(x, y) =R2(¯x, y) + ∂R2 ∂x (¯x, y)(x − ¯x) + O � ∥x − ¯x∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' By subtracting these two equalities and setting y = ¯x we directly obtain (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Furthermore, lim ǫ→0 R1(z, z+ǫd) − R1(z, z) ǫ = lim ǫ→0 R2(z, z+ǫd) − R2(z, z) ǫ + O(∥ǫ∥), which implies that ∂R1 ∂y (z, z) = ∂R2 ∂y (z, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Equipped with the above result, we now show that multiple shooting, single shooting and DDP locally converge at the same linear rate if a GGN Hessian approximation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' To this end, we summarize the primal variables as w = (x, u) and introduce the following notation, where we set N = 2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' to keep the notation simple, ˆF(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = \uf8ee \uf8f0 ¯¯x0 − x0 ¯f0 + ¯A0(x0 − ¯x0) + ¯B0(u0 − ¯u0) − x1 ¯f1 + ¯A1(x1 − ¯x1) + ¯B1(u1 − ¯u1) − x2 \uf8f9 \uf8fb , F(x, u) = \uf8ee \uf8f0 ¯¯x0 − x0 f(x0, u0) − x1 f(x1, u1) − x2 \uf8f9 \uf8fb , denoting the linearized and nonlinear forward simulation, and G(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = �¯u0 + ¯k0( ¯w) + ¯K0( ¯w)(x0 − ¯x0) − u0 ¯u1 + ¯k1( ¯w) + ¯K1( ¯w)(x1 − ¯x1) − u1 � , summarizing the feedback law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The quantities ¯ki( ¯w) and ¯Ki( ¯w) are computed according to (13) and (12) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Note that they depend only on the primal iterate ¯w if a GGN Hessian is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Multiple Shooting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' DDP We define the next multiple shooting iterate via w+ = ΠGGN MS ( ¯w) where ΠGGN MS ( ¯w) is the solution map of the linear root-finding problem RGGN MS (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = 0 with RGGN MS (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = � ˆF(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) G(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (17) Similarly, we define the next DDP iterate via w+ = ΠGGN DDP ( ¯w) where ΠGGN DDP ( ¯w) is the solution map of the nonlinear root- finding problem RGGN DDP (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = 0 with RGGN DDP (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = � F(x, u) G(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (18) Note that the only difference between (17) and (19) is the first block where DDP uses the nonlinear dynamics while multiple shooting uses the linearized dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Consider a KKT point (w∗, λ∗) of the NLP in (1) that satisfies LICQ and SOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The asymptotic linear contraction rate κGGN MS (w∗) of the multiple shooting iterates, obtained via w+ = ΠGGN MS (w), is equal to the asymptotic lin- ear contraction rate κGGN DDP (w∗) of the DDP iterates, obtained via w+ = ΠGGN DDP (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Theorem 1 implies that it suffices to show that the partial derivatives of RGGN MS (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) and RGGN DDP (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) coincide at the fixed point w∗ in order to prove that κGGN MS (w∗) = κGGN DDP (w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We first consider the partial derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' From the definitions in (17) and (18) and together with ∂ ˆF ∂(x, u)(x∗, u∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' w∗) = ∂F ∂(x, u)(x∗, u∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' w∗), we directly obtain ∂RGGN MS ∂w (w∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' w∗) = ∂RGGN DDP ∂w (w∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' To- gether with Lemma 1, we conclude that also the partial derivatives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w coincide at w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Multiple Shooting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Single Shooting Let y = (x, ˆx, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We define the next SSiterate via y+ = ˆΠGGN SS (¯y) where ˆΠGGN SS is the solution map of the nonlinear root-finding problem ˆRGGN SS (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯y) = 0 with ˆRGGN SS (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯y) = \uf8ee \uf8f0 F(x, u) ˆF(ˆx, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) GGGN(ˆx, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) \uf8f9 \uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (19) Similarly, we define the next multiple shooting iterate via y+ = ˆΠGGN MS (¯y) where ˆΠGGN MS is the solution map of the linear root-finding problem ˆRGGN MS (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯y) = 0 with ˆRGGN MS (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯y) = \uf8ee \uf8f0 ˆF(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) ˆF(ˆx, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) GGGN(ˆx, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) \uf8f9 \uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (20) Note that the definition in (20) is redundant as it includes the same linear forward sweep twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' This is necessary only for the comparison with single shooting: In this formulation, the two residual maps (19) and (20) differ only in the first block where single shooting uses a nonlinear forward simulation while multiple shooting performs the forward simulation based on the linearized dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Consider a KKT point (w∗, λ∗) with w∗ = (x∗, u∗) of the NLP in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Let y∗ = (x∗, x∗, u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The asymptotic linear contraction rate κGGN MS (y∗) of the multiple shooting iterates, obtained via y+ = ˆΠGGN MS (y), is equal to the asymptotic linear contraction rate κGGN SS (y∗) of the SSiterates, obtained via y+ = ΠGGN SS (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We proceed as in the proof of Proposition 2 and show that the partial derivatives of of ˆRGGN MS (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯y) and ˆRGGN SS (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯y) coincide at the fixed point y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' From the definitions in (20) and (19) and together with ∂ ˆF ∂(x, u)(x∗, u∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' z∗) = ∂F ∂(x, u)(x∗, u∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' z∗), we directly obtain ∂ ˆ RGGN MS ∂y (y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' y∗) = ∂ ˆ RGGN SS ∂y (y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' To- gether with Lemma 1, we conclude that also the partial derivatives with respect to ¯y coincide at y∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=', we have ∂ ˆ RGGN MS ∂¯y (y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' y∗) = ∂RGGN SS ∂¯y (y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Local Linear Contraction Rate In the previous section, we have shown that multiple shoot- ing, single shooting and DDP locally converge with the same linear rate if a GGN Hessian is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We now analyze the multiple shooting iteration to further characterize this linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' To this end, we consider yet another equivalent definition of the multiple shooting iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We define the next multiple shooting iterate via z+ = ˜ΠGGN MS (¯z) where ˜ΠGGN MS is the solution map of the linear root-finding problem ˜RGGN MS (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯z) = 0 with ˜RGGN MS (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯z) = � V GGN quad(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) + ∇w ˆF(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w)λ ˆF(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w), � (21) where V GGN quad(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) is the quadratic approximation of the objective given in (2) using a GGN Hessian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=', V GGN quad(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' ¯w) = V ( ¯w) + ∂V ∂w ( ¯w)(w − ¯w) + 1 2 (w − ¯w)⊤HGGN( ¯w)(w − ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (22) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Consider a KKT point z∗ = (x∗, u∗, λ∗) of the NLP in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Let y∗ = (x∗, x∗, u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The asymptotic linear contraction rate of the multiple shoot- ing iterates, the SSiterates, and the DDP iterates is the same for all three methods and given by the smallest κ that satisfies the condition −κ ˜ M GGN(z∗) ⪯ ˜EGGN(z∗) ⪯ κ ˜ M GGN(z∗), (23) where we have ˜ M GGN(z∗) = Z⊤M GGN(z∗)Z, ˜EGGN(z∗) = Z⊤EGGN(z∗)Z with Z a basis of the null space of ∇wF(x∗, u∗)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Here M GGN(z∗) is the Hessian approximation and EGGN(z∗) is the deviation from the exact Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The fact that the local linear contraction rate is the same for all three methods follows from Proposition 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The characterization of the rate in terms of the linear matrix inequalities in (23) follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='5 in [4] applied to the root-finding problem in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Note that for the OCP-structured problem at hand, the error matrix EGGN(z∗) is given as EGGN(z∗) = ∇2 (x,u) � (λ∗)⊤F(x∗, u∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (24) Considering N = 3 and reordering the optimization vari- ables as ˜w = (x0, u0, x1, u1, x2, u2, x3), a basis Z of the null space of ∇˜wF(x∗, u∗)⊤ is given by Z = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 I 0 0 B∗ 0 0 0 0 I 0 A∗ 1B∗ 0 B∗ 1 0 0 0 I A∗ 2A∗ 1B∗ 0 A∗ 2B∗ 1 B∗ 2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Here, the Jacobians are given as A∗ i = ∂f ∂x(x∗ i , u∗ i ) and B∗ i = ∂f ∂u(x∗ i , u∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Moreover, the reduced error matrix ˜EGGN(z∗) = Z⊤EGGN(z∗)Z corresponds to the error matrix of the single shooting problem if solved in the standard dense formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Propositions 2 and 3 imply that ∥w+ MS − w+ SS∥2 = O � ∥ ¯w − w∗∥2 2 � , (25) if both methods start from a feasible initial guess ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The analogous result holds for multiple shooting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' DDP and single shooting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Exact Hessian Variants and Quadratic Rate If an exact Hessian is used, multiple shooting, single shooting and DDP locally converge at a quadratic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In [20], the local convergence behaviour of multiple shooting and SSwith exact Hessians has been discussed in a simplified setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The authors concluded that multiple shooting formulations lead to faster contraction rates if the nonlinearities of the system dynamics reinforce each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' have the same direction of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' SSwould lead to faster contraction if the concatenated nonlinearities mitigate each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' They furthermore argued that in optimal control, the nonlinearities often reinforce each other rendering a multiple shooting approach favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' FURTHER REMARKS & DISCUSSION Our main result shows that if a GGN Hessian is used all three methods locally behave the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In the following, we discuss further differences and similarities, as well as advantages and disadvantages of the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Initialization Multiple shooting can start from an infeasible initial guess, which is not possible for single shooting and DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The possi- bility for infeasible initialization greatly simplifies the usage of multiple shooting in practice as no additional routines for finding a feasible initial guess are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Furthermore, infeasible initialization might improve convergence of the method if a rough guess of how a solution trajectory might look like is available [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' GGN Hessian & Convex-Over-Nonlinear Objectives We focused on a GGN Hessian approximation which comes with several advantages: (1) The subproblems that need to be solved in each iteration are convex and can thus be solved reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (2) We require only first-order derivatives of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' As second-order derivatives tend to be expensive to compute, using a GGN Hessian might significantly reduce computational cost of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' (3) The GGN Hessian does not require computation of the multipliers λi reducing the complexity of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Moreover, the GGN approach naturally covers convex-over- nonlinear cost where li is not convex but instead takes the form li(xi, ui) = ψi(ri(xi, ui)) with ψi convex and ri nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In this case, the GGN Hessian approximation is given as � QGGN i (SGGN i )⊤ SGGN i RGGN i � = ¯Ji(¯xi, ¯ui)⊤∇2φi(¯ri) ¯Ji(¯xi, ¯ui), (26) where ¯ri = ri(¯xi, ¯ui) and ¯Ji(¯xi, ¯ui) = ∂ri ∂(xi,ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' If the Hessian error matrix in (24) is adapted accordingly, our local convergence analysis is still valid in this more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Constraints Constraints might be incorporated into the problem formula- tion via barrier functions or penalty functions as is done e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' in [23], [24], [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Note that these formulations typically lead to convex-over-nonlinear objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Implementation Details 1) Forward Sweep: In the form presented here, all three methods can be implemented in a very similar fashion if a GGN Hessian is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' With the Riccati recursion being the same for all three methods, only the forward sweep needs to be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' One advantage of multiple shooting over SSand DDP is the possibility to parallelize the forward sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 2) Line Search: If a line search strategy is used (11j) needs to be changed to DDP & multiple shooting ui = ¯ui + αki + Ki(xi − ¯xi), single shooting: ui = ¯ui + αki + Ki(ˆxi − ¯xi), where α ∈ (0, 1] is the step size that is successively reduced until a sufficient decrease criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' For single shooting and DDP, the iterates are always feasible such that simply the cost function can be considered within a sufficient decrease approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' For multiple shooting, however, one needs to decide on a merit function in order to combine both sufficient decrease of the cost function and infeasibility reduction into a scalar progress criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we illustrate the local convergence rate of the MS, SS, and DDP iterates on a numerical example adopted from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The continuous time dynamics are given as ˙x1 = x2 + u (µ + (1 − µ)x1) ˙x2 = x1 + u (µ − 4(1 − µ)x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' where we use µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We discretize the continuous dynamics using ten steps of a Runge-Kutta integrator of fourth order (RK4) on an integration interval of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We use a quadratic cost function together with a quadratic penalty function which penalizes control inputs that do not satsify |ui| ≤ umax = 1, yielding the following objective: V (x, u) = � i=0 1 2x⊤ iQxi + 1 2u⊤ iRui + τβ(ui) + 1 2x⊤ NPxN with Q = diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='5), R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='8, P = diag(10, 10) and penalty β(ui) = max(0, ui − umax)2 + min(0, ui + umax)2 with τ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Note that the objective V (x, u) comprising the cost and penalty functions is convex and a GGN Hessian can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In the following, we obtain a feasible initial guess by simulating the nonlinear system forward in time using the linear feedback law obtain from an LQR which is based on the linear system obtained by linearizing at the steady state xsteady = 0, usteady = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 1, the state trajectory after a single iteration of multiple shooting, single shooting, and DDP is shown for a horizon length of N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' All three methods start from the same feasible initial guess that is shown in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' While both the SSand DDP trajectory are feasible with respect to the dynamics constraints, the multiple shooting trajectory exhibits gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The difference of the SSiterate to the multiple shooting iterate increases significantly along the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 2, the empirical linear contraction rate, given by ˆκ = ∥wk+1 − wk∥ ∥wk − wk−1∥, is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The exact Hessian variants require only very few iterations to reach the convergence criterion and the empir- ical contraction rate quickly tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The empirical contraction rate of the three GGN variants needs a few more iterations to converge and they converges to the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 3 shows the distance to the solution of the iterate as a function of the distance to the solution of the previous iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' In this log-log plot, the slope of the linear func- tions correspond to the order of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The intercepts correspond to the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The GGN variants of the three methods converge linearly at exactly the same asymptotic rate, while the three methods using an exact Hessian converge quadratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK We provided a unified view on multiple shooting, single shooting and DDP, three classical methods for solving un- constrained discrete optimal control problems (OCP), from a numerical optimization perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' While all thee methods can be used with different Hessian approximations, we focused on the Generalized Gauss-Newton (GGN) Hessian, a generalization of the widely used Gauss-Newton (GN) Hessian to convex loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' We showed that the iterates obtained with these three methods locally converge – or diverge – to a KKT point of the OCP a the same linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='6 x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='6 x2 GGN-MS GGN-SS GGN-DDP solution initial guess Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' State trajectories obtained after one iteration of GGN-MS, GGN- SS, and GGN-DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' All methods start from the same feasible initial guess which is obtained by simulating the system forward in closed-loop using the LQR feedback law associated with the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The initial state is ¯¯x0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='42, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 0 5 10 15 20 iteration n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content='2 empirical linear contraction rate ˆκ EH-MS GGN-MS EH-SS GGN-SS EH-DDP GGN-DDP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Empirical contraction rate ˆκ for both the exact Hessian and GGN Hessian variant of MS, SS and DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The convergence criterion is ∆w ≤ ε with ε = 10�12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 10−8 10−6 10−4 10−2 100 102 ∥wk − w∗∥∞ 10−8 10−6 10−4 10−2 100 102 ∥wk+1 − w∗∥∞ EH-MS GGN-MS EH-SS GGN-SS EH-DDP GGN-DDP Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Norm of the primal step as a function of the norm of the previous primal step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The slope of the linear function corresponds to the order of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' The intercept corresponds to the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' Rao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} +page_content=' 1205–1218, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE2T4oBgHgl3EQfrQhe/content/2301.04047v1.pdf'} diff --git a/PtFIT4oBgHgl3EQffCtm/content/tmp_files/2301.11277v1.pdf.txt b/PtFIT4oBgHgl3EQffCtm/content/tmp_files/2301.11277v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..693ea8eee13af1a435e41bc646e1f7210ea57789 --- /dev/null +++ b/PtFIT4oBgHgl3EQffCtm/content/tmp_files/2301.11277v1.pdf.txt @@ -0,0 +1,1938 @@ +arXiv:2301.11277v1 [cond-mat.other] 25 Jan 2023 +Room-temperature spin glass behavior in zinc ferrite epitaxial thin films +Julia Lumetzberger,1 Verena Ney,1 Anna Zhakarova,2 Nieli Daffe,2 Daniel Primetzhofer,3 and Andreas Ney1, ∗ +1Johannes Kepler University Linz, Institute for Semiconductor and +Solid State Physics, Altenberger Strasse 69, 4040 Linz, Austria +2Swiss Light Source (SLS), Paul Scherrer Institut, 5232 Villigen PSI, Switzerland +3Department of Physics and Astronomy, +˙Angstr¨om Laboratory, +Uppsala University, Box 516, SE-751 20 Uppsala, Sweden +(Dated: January 27, 2023) +Zinc ferrite (ZnFe2O4) epitaxial thin films were grown by reactive magnetron sputtering on +MgAl2O4 and Al2O3 substrates varying a range of preparation parameters. The resulting structural +and magnetic properties were investigated using a range of experimental techniques confirming epi- +taxial growth of ZnFe2O4 with the nominal stoichiometric composition and long range magnetic +order at and above room temperature. The main preparation parameter influencing the tempera- +ture Tf of the bifurcation between M(T ) curves under field cooled and zero-field cooled conditions +was found to be the growth rate of the films, while growth temperature or the Ar:O2 ratio did not +systematically influence Tf. Furthermore Tf was found to be systematically higher for MgAl2O4 as +substrate and Tf extends to above room temperature. While in some samples Tf seems to be more +likely correlated with superparamagentism, the highest Tf occurs in ZnFe2O4 epitaxial films where +experimental signatures of magnetic glassiness can be found. Element-selective X-ray magnetic cir- +cular dichroism measurements aim at associating the magnetic glassiness with the occurrence of a +different valence state and lattice site incorporation of Fe pointing to a complex interplay of various +competing magnetic interactions in ZnFe2O4. +I. +INTRODUCTION +Zinc ferrite (ZnFe2O4) belongs to the crystallographic +group of normal spinels of the form AB2O4 where in the +ideal case the A-cation (Zn2+) exclusively occupies the +tetrahedral (Td) lattice sites as Zn2+ +T d while the B-cation +(Fe3+) is found on the octahedral (Oh) sites as Fe3+ +Oh. +The magnetic properties of zinc ferrite have been un- +der investigation for quite some time revealing a rather +complex situation. Early studies of the bulk material re- +port antiferromagnetic (AFM) order with a very low N´eel +temperature of 9 K [1, 2]. Later-on it was demonstrated +by magnetic neutron scattering experiments on ZnFe2O4 +single crystals that even in perfect crystals geometrical +frustration leads to an unusual magnetic behavior [3]. In +particular, it was pointed out, that the Fe3+ +Oh sublattice +can be regarded to be similar to various pyrochlores or +Laves phases which are known for their intrinsic geomet- +rical frustration [3]. The situation becomes even more +complex when defects such as inversion are considered +which are expected to occur, in particular, in thin films +of ZnFe2O4. +A partial inversion in ZnFe2O4 has the +stoichiometric formula of [Zn1−δFeδ]T d[ZnδFe2−δ]OhO4, +where δ denotes the degree of inversion. In the ideal case +of δ = 0, i. e., no inversion, there is only the weak AFM +superexchange interaction between Fe3+ +Oh which is usually +denoted as JBB [4] plus the geometrical frustration men- +tioned before [3]. +JBB can be held responsible for the +AFM order at low temperatures in bulk single crystals. +For a finite degree of inversion there is an additional, +∗Electronic address: andreas.ney@jku.at +much stronger AFM superexchange interaction JAB be- +tween Fe3+ on Td (also called A-site) and Oh (or B-site) +sites, i. e., Fe3+ +T d and Fe3+ +Oh [4], which leads to ferrimag- +netism for incomplete inversion [5]. The additional JAA +exchange between the Fe3+ +T d is the weakest [6]. However, +if the additional Fe3+ +T d is not compensated by Zn2+ +Oh, i. e., +if there is some degree of deviation from the ideal stoi- +chiometry of ZnFe2O4, some finite amount of Fe2+ +Oh has to +form because of charge neutrality. This results in an ad- +ditional double exchange (DE) over the Oh (or B-) sites +JDE +BB between Fe3+ +Oh and Fe2+ +Oh, which results in spin cant- +ing in magnetite [4] or in non-stoichiometric ZnFe2O4 [5]. +In many cases there are reports on some finite degree of +inversion in ZnFe2O4 and an upper limit of δ = 0.6 has +been found by the analysis of the magnetic moment of +the Fe [7], X-ray absorption spectroscopy (XAS) [8] or +via Rietveld refinement of X-ray diffraction (XRD) data +[9]. Therefore, the magnetic order in ZnFe2O4 can be +expected to be highly complex, especially for thin films, +where the presence of various kinds of defects such inver- +sion and/or off-stoichiometry can be expected. +The growth of zinc ferrite in thin film form is moti- +vated by a range of possible applications such as gas sen- +sors, photo catalytic disinfection or other photo-catalytic +applications, see [9, 10] and Refs. +therein. +Besides +that, zinc ferrite thin films can also be considered as an +interesting semiconducting material in spintronics with +tunable magnetic properties [4, 6, 11–13]. +A variety +of reports of different types of magnetic order in zinc +ferrite can be found throughout the literature ranging +from ferro(i)magnetic [4, 6, 13–17], to superparamagnetic +[8, 11, 18, 20, 21] and to spin glass behavior [7, 9, 19, 22]. +Note, that in some cases superparamagnetism with inter- +particle interactions is associated with a so-called cluster- + +2 +glass behavior [7, 8, 19]. +However, only a few studies +report on characteristic experimental signatures of mag- +netic glassiness [7, 9, 19, 22], while others report only +temperature-dependent magnetization [M(T )] measure- +ments under different field-cooling conditions which how- +ever could also be associated with superparamagnetism, +e. g. +[8]. +Finally, the control of defects and thus the +magnetic properties was reported to be experimentally +achievable by varying different preparation parameters, +i. e., oxygen partial pressure [6, 12, 17, 18, 21], stoichio- +metric composition [4], post-growth thermal treatment +[19, 20] or deposition rate [7]. +Similar to the reported types of magnetism in zinc fer- +rite, also the techniques for sample preparation span a +wide range from pulsed laser deposition (PLD) [4, 6, 7, +11–13, 16, 17, 20], over reactive magnetron sputtering +(RMS) [8, 14, 15, 18, 19, 21] for thin film growth, to ball +milling [9] and solid state reaction [22] for bulklike sam- +ples. Likewise, a range of different substrates has been +used for thin film growth amongst them are MgAl2O4 +[12], c-plane Al2O3 [7, 11], a-plane Al2O3 [16], SrTiO3 +[13, 17, 20], MgO [4, 6, 11], Si(001) and Si(111) [14, 21] +and glass substrates [8, 15, 18, 19]. +It is remarkable, +that spin glass behavior has mainly been reported for +bulk ZnFe2O4 [3] or bulklike nanopowders [9, 22] while +for thin film samples mostly a cluster glass is inferred +[7, 8, 19]. Among the reports of cluster glass behavior +only one is based on thin film growth by PLD on single +crystalline substrates [7], while the others rely on sput- +tered polycrystalline ZnFe2O4 samples [8, 19] making a +larger amount of defects expectable, e. g., due to an in- +trinsically large number of grain boundaries. Finally, also +the temperature range, where magnetic glassiness is ob- +served ranges from below 20 K for the single crystalline +ZnFe2O4 in [3, 22], over around 100 K for the annealed +ZnFe2O4 nanopowder in [9] up to 300 K for the PLD +grown ZnFe2O4 epitaxial films at deposition rates above +3 nm/s which drops down to below 100 K for rates below +2 nm/s [7]. +Here, we report on epitaxial thin film growth of +ZnFe2O4 by RMS on two different substrates namely +MgAl2O4 and c-plane Al2O3. Various preparation pa- +rameters have been varied in order to control the for- +mation of defects in a systematic way for epitaxial thin +film samples. In agreement with [7] the most relevant +preparation parameter is found to be the deposition rate. +For high deposition rates ZnFe2O4 films exhibit spin- +glass like behavior up to rather high temperatures on +MgAl2O4 substrates which is shifted to lower tempera- +tures on Al2O3 substrates. In contrast, the stoichiom- +etry on ZnFe2O4 is maintained throughout the sample +series and also the oxygen partial pressure were found to +play a minor role in the resulting magnetic properties. +The spin-glass behavior is associated with a significant +amount of inversion up to δ ∼ 0.3, corroborating earlier +reports [8, 9]. In addition, a significant magnetic polar- +ization of the Zn cation is found at room temperature +by means of element selective magnetometry indicating +that the microscopic origin of the magnetic properties of +ZnFe2O4 is even more complex. +II. +EXPERIMENTAL DETAILS +Zinc ferrite was fabricated using reactive magnetron +sputtering (RMS) from an oxide target having the +nominal composition of ZnFe2O4. +The epitaxial thin +films were grown on doubleside polished single crys- +talline +spinel [MgAl2O4(001)] and +c-plane sapphire +[Al2O3(0001)] substrates in an ultrahigh vacuum (UHV) +chamber with a base pressure of 4 × 10−8 mbar and a +working pressure of 4 × 10−3 mbar. +To determine the +ideal growth parameters, the deposition temperature was +varied from room temperature (RT) to 550 °C, the Ar:O2 +ratio from 10 : 0 to 10 : 0.5, and the sputtering power from +20 W to 100 W, which corresponds to a growth rate from +0.36 to 3.69 nm/min. The nominal thickness is kept at +40 nm and is controlled via a quartz crystal microbalance +which is at room temperature so that the actual thickness +of most of the films is by 10-20% lower because of the el- +evated temperature of the substrate during growth. The +structural properties of the films were investigated by X- +ray diffraction (XRD) measurements with a Pananalyti- +cal X’Pert MRD recording ω − 2θ scans and symmetric +as well as asymmetric reciprocal space maps (RSM). The +chemical composition was determined by ion beam anal- +ysis, i.e. Rutherford backscattering spectrometry (RBS) +using a 2 MeV He+ primary beam at the Tandem Labora- +tory at Uppsala University. To disentangle the element +specific contributions, the spectra were analyzed using +the SIMNRA software [23]. Details of the experimental +setup are described elsewhere [24]. Furthermore, Elec- +tron Recoil Detection (ERDA) with a primary ion beam +of 36 MeV iodine ions was employed to rule out contam- +inations with light elements like H or C. +The magnetic properties were measured by integral +superconducting quantum interference device (SQUID) +magnetometry using a Quantum Design MPMS-XL5 sys- +tem applying the magnetic field in the film plane. The +M(H) curves were recorded in range of ±5 T at 300 K +and 2 K and M(T ) curves have been recorded from 2 K +up to 395 K at 10 mT while warming after a cool down +in 5 T (FH), under nominally zero-field cooled conditions +(ZFC) as well while cooling down in 10 mT (field cooled, +FC). Additionally, waiting time experiments were per- +formed analogous to the ones in [25] by cooling down +the sample in zero field and introducing a waiting time +twait at various waiting temperatures Twait which is typ- +ically 10 000s. Then a M(T ) curve identical to a ZFC +curve without waiting time was subsequently recorded. +Subtracting these two curves represents a typical waiting +time experiment for spin glasses where a so-called ZFC +memory—or hole-burning—effect can be seen by a dip +in the difference of the magnetization with and without +waiting time around Twait [25]. A second memory ex- +periment already used before for ZnFe2O4 in [9] was per- + +3 +formed in addition, in which a M(T ) curve is recorded +under FC conditions with 10 mT and a waiting time twait +at nominally zero field is inserted at several Twait before +the FC curve is resumed at 10 mT. Then a subsequent +M(T ) is recorded in 10 mT while warming. The typical +signature of a spin glass in these so-called FC memory +experiments is a relaxation of the magnetization at Twait +in the FC curve and the presence of an inflection point +around Twait in the subsequent M(T ) curve while warm- +ing [9]. Note, however, that also superparamagnets ex- +hibit similar signatures in the FC memory experiments +[26]. All M(H) and M(T ) data were corrected for the +diamagnetic background of the substrate which was de- +termined from the M(H) curves a high magnetic field at +300 K [27]. The M(H) curves at 2 K for samples grown +on MgAl2O4 had to be corrected for an additional param- +agnetic contribution which was determined form a M(H) +measurement of bare MgAl2O4 from the same batch of +samples. In general, zero field conditions are referred to +nominally 0.0 mT after the superconducting magnet had +been reset (magnet reset option of the MPMS) and any +applied magnetic field is afterwards limited to 10 mT; +this assures a residual pinned magnetic field of typi- +cally 0.1 mT or less [28]. +A cooling and heating rate +of 1 K/min is used for all M(T ) measurements in both +FC and ZFC memory experiments. Note, that the typ- +ical frequency-dependent ac-susceptibility measurements +like in [7, 19, 22] were not available for the used SQUID. +The element specific magnetic properties have been +investigated +by +x-ray +absorption +near +edge +spec- +troscopy (XANES) and x-ray magnetic circular dichro- +ism (XMCD) measurements which were performed at the +Xtreme beamline at the Swiss Light Source (SLS) [29]. +The XMCD spectra were recorded at the Fe L3/2- and Zn +L3/2-edges at 300 K under 20° grazing incidence in total +electron yield. For the Fe-edges the magnetic field was +set to 5 T and only the circular polarization has been +switched to obtain the XMCD. For the Zn-edges the di- +rection of the magnetic field has been reversed as well +to minimize artifacts. The XMCD spectra at the Fe L3 +edge are compared to simulations carried out by mul- +tiplet ligand field theory using the CTM4XAS package +[30]. These simulations have been used before to deter- +mine the site occupancy and formal oxidation state of +Fe and Ni in nickel ferrites [31] and Zn/Al doped nickel +ferrites [32]. For the present work the simulation param- +eters for Fe2+ +Oh, Fe3+ +Oh, and Fe3+ +T d are identical to those in +[32] and details on the simulations can be found there. +III. +EXPERIMENTAL RESULTS +The structural properties of the ZnFe2O4 thin films +were analyzed by symmetric ω − 2θ scans using XRD. +Figure 1(a) shows a comparison of the diffractograms of +zinc ferrite grown on MgAl2O4 and Al2O3 where the lat- +ter is shifted upward for clarity. The samples were grown +with a nominal thickness of 40 nm at a substrate tem- +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +35 +40 +45 +800 +1000 +1200 +1400 +1600 +0 +50 +100 +150 +200 + - 2 + (�) +In te n s ity (a r b . u n its ) + + 80W on MgAl +2 +O +4 + 80W on Al +2 +O +3 +ZnFe +2 +O +4 +(311) +Al +2 +O +3 +(006) +ZnFe +2 +O +4 +(004) +MgAl +2 +O +4 +(004) +(a) +(b) +in te n s ity (c o u n ts ) +energy (keV) + 80W on MgAl +2 +O +4 + 80W on Al +2 +O +3 + simulation + Fe + Zn +2 MeV He ++ +FIG. 1: (a) Structural characterization by X-ray diffraction: +comparison of the symmetric ω − 2θ scans of ZnFe2O4 films +grown at 80 W on MgAl2O4 (black) and Al2O3 (red). (b) +Chemical composition determined by means of RBS spectra +recorded with a 2 MeV He+ primary ion beam of ZnFe2O4 +films grown at 80 W on MgAl2O4 (black) and Al2O3 (red). +perature of TS = 450 °C with an Ar:O2 ratio of 10 : 0.5 +and a sputtering power of 80 W. The XRD scan of the +samples grown on MgAl2O4 exhibits a (004) reflex at +41.79±0.09° with a full width at half maximum (FWHM) +of 0.6 °, corresponding to a perpendicular lattice param- +eter a⊥ = (8.64 ± 0.02) �A. The sample grown on Al2O3 +exhibits the (311) reflex at 35.61° corresponding to a per- +pendicular lattice parameter of a⊥ = (7.28±0.02)�A. For +this sample weak Laue oscillations can be seen indicat- +ing a smoother growth compared to the sample grown +on MgAl2O4. +In addition, an asymmetric RSM along +the (¯1¯15) plane has been recorded for a 100 nm sam- +ple grown on MgAl2O4 with a sputtering power of 60 W +(not shown). It reveals an in-plane lattice parameter of +a∥ = (8.32±0.05)�A which provides evidence that the film +is relaxed, since the film peak does not align with the sub- +strate peak. The reflection from the MgAl2O4 substrate +corresponds to a lattice parameter of asub = 8.08 �A, +which implies a lattice mismatch of ∼ 4.3 % with re- + +4 +spect to bulk ZnFe2O4 (a0 = 8.441 �A) (JCPDS card No. +82-1049). +A comparison between various films grown +on MgAl2O4 and Al2O3 indicates no significant differ- +ence in crystalline quality with similar FWHM despite +the change in texture from (004) on MgAl2O4 to (311) +on Al2O3. No other reflexes can be found underlining +highly textured growth of ZnFe2O4 for both substrates +and the samples are devoid of other crystalline phases. +Furthermore, the chemical composition of ZnFe2O4 on +both substrates is determined using RBS. In Fig. 1(b) +the RBS data of the 80 W sample grown on MgAl2O4 +(black squares) is shown in comparison to the sample +grown on Al2O3 (red circles). Both ZnFe2O4 films have +no deviation from the nominal stoichiometry within the +uncertainties of the measurement technique. Finally, the +samples are investigated using ERDA to check for con- +taminations of light elements like H or C but neither +element could be detected (not shown). We can there- +fore conclude that our ZnFe2O4 samples have the nom- +inal stoichiometric composition, grow epitaxially on ei- +ther substrate and are devoid of a significant amount of +secondary phases or contaminants within the detection +limits of XRD and ERDA. +In a first step, the magnetic properties are investigated +using standard M(H) curves which are shown in Fig. 2 +recorded at (a) 300 K and (b) 2 K for ZnFe2O4 grown +at 60 W on MgAl2O4 (black squares) and Al2O3 (red +circles). ZnFe2O4 grown on Al2O3 has a higher magne- +tization of MS = 130 ± 15 kA/m compared to growth +on MgAl2O4 where MS = 110 ± 15 kA/m at 300 K. +For both samples MS increases to above 200 kA/m at +2 K. For the M(H) curves recorded at 2 K for ZnFe2O4 +grown on MgAl2O4 the full squares denote the data when +only the diamagnetic contribution has been subtracted. +Note, that in a previous publication on ZnFe2O4 grown +on MgAl2O4 this apparently paramagnetic behavior has +been attributed to cationic disorder of the Fe3+ in [12]. +However, if a bare MgAl2O4 substrate is measured, one +also measures a net-paramagnetic behavior after subtrac- +tion of the diamagnetism so that one has to attribute this +paramagnetic contribution to the MgAl2O4 substrate it- +self. The open squares are the data where also the mea- +sured paramagnetic background of the bare MgAl2O4 has +been subtracted and no obvious paramagnetic contribu- +tion of the ZnFe2O4 is visible any more. The insets in +Fig. 2 enlarge the low-field behavior of the M(H) curves. +While they are virtually anhysteretic at 300 K a clear hys- +teresis with a coercive field of Hc = 80 ± 10 mT is found +for ZnFe2O4 films on either substrate. This behavior at +2 K is consistent with most of the ZnFe2O4 films grown +on a range of different substrates reported throughout the +literature reporting ferro(i)magnetism or superparamag- +netism both in terms of magnetization as well as coercive +field at low temperatures and clearly rules out the pure +antiferromagnetic behavior of bulk ZnFe2O4. +Figure 3 shows the M(T ) behavior recorded at 10 mT +under FC conditions (full symbols) as well as after ZFC +conditions (open symbols) for the ZnFe2O4 grown at +-4 +-2 +0 +2 +4 +-400 +-200 +0 +200 +400 +-150 +-100 +-50 +0 +50 +100 +150 +-4 +-2 +0 +2 +4 +-50 +0 +50 +-0.04 +0.00 +0.04 +-0.2 +0.0 +0.2 +-200 +0 +200 + + +M (k A /m ) +0 +H (T) + MgAl +2 +O +4 + corrected + Al +2 +O +3 +(b) +T = 2 K + + + +0 +H (T) +M (k A /m ) +60 W ZnFe +2 +O +4 +/ + MgAl +2 +O +4 + Al +2 +O +3 +(a) +T = 300 K + + + + + + + + +FIG. 2: SQUID measurements of M(H) curves shown for the +60 W ZnFe2O4 grown on MgAl2O4 (black squares) and Al2O3 +(red circles) at (a) 300 K and (b) at 2 K. (a) shows an M(H) +curve at RT (b). At 2 K the paramagnetic contribution of the +MgAl2O4 substrate has been subtracted (open squares). The +insets enlarge the measurements at low fields. +60 W on MgAl2O4 (a) and Al2O3 (b), i. e., the identi- +cal pair of samples as in Fig. 2. Both samples exhibit +a clear bifurcation between the FC and ZFC curves in- +dicating a blocking or spin-freezing peak at a temper- +ature of Tf = 290 K for ZnFe2O4/MgAl2O4 (a) and at +Tf = 190 K for ZnFe2O4/Al2O3 (b). +Both 60 W sam- +ples together with the two 80 W samples shown in Fig. +1 are part of a sample series where only the sputtering +power and thus the deposition rate has been changed +while all other growth parameters have been kept con- +stant. +In terms of magnetization as well as coercivity +all samples from these series show comparable magnetic +behavior. The only systematic dependency on the sput- +tering power is an increase of the measured Tf with in- +creasing sputtering power. +The insets in Fig. 3 show +the measured Tf as a function of the sputtering power +for ZnFe2O4/MgAl2O4 (a) as well as for ZnFe2O4/Al2O3 +(b). Irrespective of the comparable increase with sputter- +ing power the overall values of Tf are systematically lower +for the Al2O3 substrate by about 100 K. Note that the +obtained Tf for the samples grown on either substrate, +are well above the values usually reported for zinc ferrite + +5 +20 +40 +60 +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +20 +40 +60 +80 +20 +40 +60 +80 +170 +180 +190 +200 +µ +0 +H = 10 mT +(a) +T (K) +M (k A /m ) +60 W ZnFe +2 +O +4 +/MgAl +2 +O +4 + FC + ZFC +T +f µ +0 +H = 10 mT +20 +40 +60 +80 +100 +260 +280 +300 +320 +T +f + (K ) +sputter power (W) +M gAl +2 +O +4 +(b) +T (K) +M (k A /m ) +60 W ZnFe +2 +O +4 +/Al +2 +O +3 + FC + ZFC +T +f +Al +2 +O +3 +T +f + (K ) +sputter power (W) +FIG. 3: M(T ) curves recorded at 10 mT under field cooled +(FC, full symbols) and after zero-field cooled (ZFC, open sym- +bols) conditions shown for the 60 W ZnFe2O4 grown on (a) +MgAl2O4 (black squares) and (b) Al2O3 (red circles) sub- +strates. The insets show the dependence of Tf on the sput- +tering power for both substrates, respectively. +[9, 17, 20, 22]. Only in few cases the Tf is found at such +elevated temperatures [7, 19] and a controllable shift of +Tf is only reported in [7] so far; however, the drop in Tf +with decreasing deposition rate in [7] is by a factor of two +more pronounced compared with the present case. Note, +that the power series was grown by varying the sputter +power nonmonotonously so that a dependence of Tf on +the growth sequence—and thus target degradation—can +be ruled out. +In a second step, the dependence of Tf on other growth +parameters shall be briefly summarized. +Figure 4(a) +compiles the sample series as a function of the growth +temperature Tgrowth from RT up to 550 °C for ZnFe2O4 +grown on MgAl2O4 at a sputtering power of 60 W and +an Ar:O2 ratio of 10 : 0.5. The XRD shows no signifi- +cant changes for Tgrowth ≥ 300 °C while the sample at +RT appears to be virtually amorphous. The inset shows +MS at 300 K and Tf as determined from SQUID mea- +surements analogous to Figs. 2 and 3. Tf is found to +be about constant around 250 K with a slight tendency +to decrease for higher Tgrowth; the only exception is the +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +41 +42 +43 +44 +45 +41 +42 +43 +44 +45 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +0.0 +0.2 +0.4 +100 +150 +200 +250 +300 +0 +200 +400 +50 +100 +150 +200 +250 +300 + - 2 + (�) + +in te n s ity (a r b . u n its ) +T +grow th + 550�C + 500�C + 450�C + 400�C + 300�C + 200�C + RT +MgAl +2 +O +4 +(004) +ZnFe +2 +O +4 +(004) +(a) +(b) +MgAl +2 +O +4 +(004) +ZnFe +2 +O +4 +(004) +in te n s ity (a r b . u n its ) + - 2 + (�) +Ar:O +2 + (sccm) + 10:0.5 + 10:0.35 + 10:0.25 + 10:0.15 + 10:0 + +O +2 + flow (sccm) +M +s + (k A /m ) / T +f + (K ) +M +s + (k A /m ) / T +f + (K ) +T +grow th + (�C) +FIG. 4: (a) Structural and resulting magnetic properties as +a function of the growth temperature Tgrowth for ZnFe2O4 +grown on MgAl2O4. The dependence of the identical param- +eters as a function of the Ar:O2 ratio is shown in (b). Ms +(black squares) and Tf (red circles) shown in the insets were +extracted from SQUID measurements. +amorphous sample at RT where Tf is clearly reduced. In +contrast, MS steadily increases with increasing Tgrowth +which can be taken as an indication for an increasing +amount of inversion, i. e., of Fe3+ +T d in analogy with [7, 9]. +However; for the present sample series this increasing MS +with Tgrowth has no obvious influence on the observed +Tf. This trend is opposite to the annealing series in [9], +where the amount of inversion and thus resulting MS +is decreasing with increasing annealing temperatures for +nanopowdered ZnFe2O4. Note, that increasing Tgrowth +in epitaxial growth typically leads to a decrease in ac- +tual thickness compared to the nominal one which would +lead to a decrease in MS which was calculated from the +nominal thickness. On the other hand, this increase in +MS can also be associated with an increase in the order +temperature which is in all cases above 400 K and thus +beyond the accessible temperature range of the SQUID +magnetometer and thus unknown. +A second ZnFe2O4 +sample series was grown on +MgAl2O4 at a sputtering power of 60 W and a fixed + +6 +Tgrowth of 450 °C while varying the Ar:O2 ratio which +is compiled in Fig. 4(b). The XRD of all samples does +not show any significant changes with increasing oxygen +content. +In the inset the resulting MS at 300 K and +Tf are shown. While MS is independent on the Ar:O2 +ratio within error bars, Tf does not show a conclusive +trend, mostly because of a rather high Tf for the sam- +ple at Ar:O2 ratio of 10 : 0.15. Disregarding this, a faint +increase within errorbars may be inferred but there is +no pronounced dependence of Tf on the Ar:O2 ratio, es- +pecially if this is compared with the dependence on the +growth rate shown in Fig. 3(a). This finding is rather in- +teresting, since in [7] the growth rate has been associated +with a deficiency in oxygen leading to an increase in Tf. +However, all samples were found to be highly resistive +above the GΩ-range (not shown). +Therefore, a signif- +icant amount of oxygen vacancies can be ruled out for +the entire series because the two samples grown without +and with maximum oxygen partial pressure have virtu- +ally identical physical properties where the high oxygen +partial pressure in RMS should safely rule out any oxygen +deficiency. In turn, the dependence of Tf on the growth +rate, which is consistently found in [7] and Fig. 3(a), can- +not depend on the existence of oxygen vacancies for RMS +grown ZnFe2O4. To summarize this part, the two sample +series shown in Fig. 4 underline, that the relevant prepa- +ration parameter to control Tf is the sputtering power +and thus growth rate, while Tgrowth and the Ar:O2 ratio +play a minor role in the resulting magnetic properties, in +particular, Tf is not directly controllable via oxygen va- +cancies. Therefore, in the following only samples grown +at Ar:O2 ratio of 10 : 0.5 and Tgrowth = 450 °C, as those +in Figs. 1 to 3, will be discussed further. +In a next step, the actual type of magnetic order shall +be determined because the reports in the literature range +from ferro(i)magnetism, over superparamagnetism, to a +cluster glass or spin-glass behavior. +As pointed out +above, MS for the present set of samples is found to +be consistent with most of the reports found through- +out the literature, while the bifurcation at Tf is found +at rather elevated temperatures. Figure 5(a) shows the +M(T ) behavior recorded under FC conditions (full sym- +bols) as well as after ZFC conditions (open symbols) for +the ZnFe2O4 grown at 80 W on MgAl2O4 for an external +field of 5 mT (black squares) and 10 mT (red circles). As +expected Tf increases with decreasing external magnetic +field which is consistent with both spin-freezing as well as +superparamagnetism. Reducing the external field further +to fields of 0.2 to 0.5 mT), where most of the spin glass +experiments are typically carried out [25], Tf gets very +close to the maximum attainable temperature of 400 K +of the SQUID magnetometer (not shown). Therefore, all +subsequent experiments are only carried out at fields of +5 mT or 10 mT to keep the maximum achievable temper- +ature well above Tf. Note, that unfortunately the SQUID +does not allow to go above the magnetic order tempera- +ture which is in all cases above 400 K. +To get a first estimate on the existence of magnetic +100 +200 +300 +400 +10 +20 +30 +40 +50 +0 +20 +40 +60 +80 +0 +100 +200 +300 +400 +T +w ait +T +w ait +T +w ait +T +w ait +(b) + +M ( +e m u ) +T (K) + FC (10 mT) IS + FH (10 mT) +t +wait + = 10 +4 + s (0 mT) +80 W ZnFe +2 +O +4 +/MgAl +2 +O +4 +T +f +T (K) + +M ( +e m u ) + FC (5 mT) + ZFC (5 mT) + FC (10 mT) + ZFC (10 mT) +(a) +80 W ZnFe +2 +O +4 +/MgAl +2 +O +4 + + + + +FIG. 5: (a) M(T ) curves of the 80 W ZnFe2O4 grown on +MgAl2O4 recorded under field cooled (FC, full symbols) and +after zero-field cooled (ZFC, open symbols) conditions at +5 mT (black squares) and 10 mT (red circles). +(b) M(T ) +curves recorded while cooling at 10 mT (FC) with intermit- +tent stops (IS, open squares) with various Twait with twait of +10,000 s marked by arrows. The M(T ) curve while warming +is subsequently recorded at 10 mT (red line). +glassiness, the FC memory sequence used in [9] was car- +ried out. Figure 5(b) shows the FC M(T ) curve recorded +at 10 mT with intermittent stops (IS, open symbols) at +various Twait which are marked with arrows. Here the +field was reduced to 0 mT for a waiting time twait of +10,000 s. +Then the field was set to 10 mT again and +the cooling down is resumed. +Subsequently M(T ) is +measured at 10 mT while heating at the same rate as +during FC without any IS (FH, full line). +In the FC +curve clear steps can be seen for most Twait which are +most pronounced just below the maximum of the ZFC +curve in Fig. 5(a), i. e., also below Tf, while they are vir- +tually absent above Tf. +Note, that in Fig. 5 we show +the magnetic data in emu to demonstrate the absolute +size of the steps in comparison to the detection limit of +the SQUID of 2 − 4 · 10−7 emu [27, 28]. +These steps +demonstrate magnetic relaxation during twait. More im- +portant, the subsequent FH curve shows clear inflection +points around Twait, and the inset enlarges the two most + +7 +prominent ones. Therefore, there is a first experimen- +tal evidence for magnetic glassiness in epitaxial ZnFe2O4 +analogous to ZnFe2O4 nanopowders in [9]; however, this +glassiness extents to rather high temperatures which are +only comparable to those reported in [7]. A caveat is still, +that such a behavior can also be observed and modeled +in superparamagnetic samples as discussed in detail in +[26] where only subtleties in these type of FC memory +sequences allow to distinguish a superparamagnet from +a superspin-glass. Therefore ZFC memory experiments +are needed in addition. +Figure 6 provides additional experimental evidence for +magnetic glassiness of the same sample by ZFC memory +experiments adopted after [25]. Here the sample is cooled +down under ZFC conditions once without any waiting +time and once cooling is stopped at Twait = 270 K for +varying waiting times twait from 500 s to 50,000 s. Sub- +sequently, an M(T ) curve is measured at 5 mT while +warming (FH). In Fig. 6(a) the difference ∆M between +the FH without and with Twait is plotted for all twait. +Note, that ∆M is provided in emu and the visible scat- +ter in the difference data is around 1 − 2 · 10−8 emu, +which demonstrates the high reproducibility of the data +recorded with the SQUID magnetometer. It should be +stressed that this is only possible if the magnet is reset +before the measurement to eliminate any trapped flux. In +all subsequent measurements one has to avoid magnetic +fields larger than 10 mT so that the nominal and actual +field are identical for all measurements within 0.1 mT be- +tween which ∆M is taken. For twait of 500 s and 1,000 s +an increase of ∆M below Tf is visible with a maximum +around the maximum of the ZFC M(T ) curve, i. e., ∆M +follows the shape of the ZFC curve. However, the max- +imum of the ZFC curve for the given experimental con- +ditions is around 300 K while the maximum of the ∆M- +curve is around 225 K, i. e., shifted to lower temperatures +and does not go back to zero. This low temperature in- +crease of ∆M is difficult to be explained in a straight- +forward manner, because the nominally ZFC conditions +only correspond to less than 0.1 mT [28]. Therefore the +difference between the ZFC with an without twait of 500 s +at 270 K, i. .e., below Tf implies that the system is allowed +to spend additional 500 s close to the freezing tempera- +ture in a tiny, but finite field. If one considers a super- +paramagnetic ensemble close to its blocking temperature +this implies more time for thermally activated switching +in a tiny field which imprints a tiny additional magnetiza- +tion because the residual field induces a slight imbalance +in the probability of switching parallel and antiparallel to +it. A superparamagnetic ensemble would further imply +relatively fast characteristic time-scales for the switching +attempts. This would be in accordance that ∆M with +twait of 500 s and 1.000 s are virtually identical, because +all the switching events are already done, while without +twait the system is ramped through the blocking temper- +ature with a rate of 60 s/K, so much less switching events +can take place around the blocking temperature, where +the tiny residual field is sufficient to aid the thermally +100 +150 +200 +250 +300 +350 +-0.1 +0.0 +0.1 +0.2 +-0.2 +0.0 +0.2 +0.4 +100 +150 +200 +250 +300 +350 +(b) +t +wait + = 10,000 s + +M ( +e m u ) +T (K) + 320 K + 300 K + 280 K + 260 K + 240 K + 220 K + 200 K + 160 K +T +w ait + = +80 W +ZnFe +2 +O +4 +/MgAl +2 +O +4 +t +w ait + = + +T (K) +FH (5 mT) +af ter ZFC in 0 mT + +M ( +e m u ) + 500 s + 1,000 s + 5,000 s + 10,000 s + 50,000 s +T +wait + = 270 K +(a) +FIG. 6: Characteristic hole-burning experiment for the 80 W +ZnFe2O4 sample grown on MgAl2O4. (a) shows the depen- +dence on the waiting time twait for a fixed waiting tempera- +ture Twait of 270 K while (b) shows the dependence on Twait +for a fixed twait of 10,000 s. +activated switching events. The remaining low tempera- +ture increase is thus the frozen-in result of more switching +events close to Tf resulting in a waiting-time imprinted +additional magnetization. This is further corroborated +by the fact, that the low-temperature increase is found +to decrease with decreasing Twait, i. e., a waiting further +below Tf and thus in a region with potentially slower +dynamics, see Fig. 6(b). We thus infer that this low tem- +perature increase of ∆M is most likely to be indicative +of a superparamagnetic-like behavior with relatively fast +dynamics rather than classical rejuvenation effects in su- +perferromagnets as discussed in [25]. +Beyond this low-temperature increase of ∆M seen for +all twait in Fig. 6(a), there is a minimum evolving with +increasing twait becoming clearly visible at 5,000 s and +being most pronounced at 50,000 s. This is a typical char- +acteristic of a (super)spin glass as discussed in [25, 26]; +however, the minimum is shifted to lower temperatures +compared to Twait by about 20 K. This shift is also seen, +irrespective of the actual Twait, see also Fig 7(a) further +below. Figure 6(b) shows ∆M curves for a fixed twait of + +8 +100 +150 +200 +250 +300 +350 +-0.2 +-0.1 +0.0 +0.1 +-0.6 +-0.4 +-0.2 +0.0 +100 +150 +200 +250 +300 +350 +(b) +t +wait + = 10,000 s + +M ( +e m u ) +T (K) + 300 K + 280 K + 260 K + 240 K + 220 K + 200 K + 160 K +T +w ait + = +80 W +ZnFe +2 +O +4 +/MgAl +2 +O +4 +t +w ait + = + +T (K) +FH (5 mT) +af ter ZFC in 0 mT + +M ( +e m u ) + 1,000 s + 5,000 s + 10,000 s + 50,000 s +T +wait + = 270 K +(a) +FIG. 7: +Identical set of data as in Fig. 6 for the 80 W +ZnFe2O4 sample grown on MgAl2O4 for (a) varying twait for +Twait = 270 K (magenta dahed line) and (b) varying Twait for +twait =10,000 s; however, ∆M is taken differently (see text). +10,000 s for various Twait. For Twait above Tf no minimum +is visible and only the low temperature increase can be +seen. In contrast, a clear minimum is observable which +is strongest for Twait of 280 K, i. e., close to Tf. For lower +Twait is becomes less pronounced and the minimum shifts +to lower temperatures, which are however always below +the respective Twait, e. g., the minimum in ∆M for Twait +of 160 K is at 150 K (orange pentagons). Therefore, the +80 W ZnFe2O4 sample grown on MgAl2O4 shows all the +experimental characteristics of magnetic glassiness. On +the one hand a FC memory effect characteristic of su- +perparamagnets and (super)spin glasses [26] which have +been reported for ZnFe2O4 before [9], see Fig. 5(b). On +the other hand, a twait-dependent minimum is observed, +which is known as hole-burning experiment [25] and is +absent in superparamagnets but is seen in (super)spin- +glasses [26]. +On the other hand, the low-temperature +increase of ∆M for short twait or Twait above Tf resemble +more of superparamagnetic-like behavior. However, we +will show in the following that superparamagnetic-like +behavior and magnetic glassiness coexist. +Figure 7 provides an alternative way of presenting the +identical results as in Fig. 6. In this case ∆M is taken +as the difference between all data with respect to (a) +twait =500 s and (b) Twait = 320 K. In other words, +here ∆M(T ) should only contain the magnetic glassiness +since the superparamagnetic behavior—which is reflected +by the low temperature increase of ∆M seen for short +twait in Fig. 6(a), or twait well above Tf in Fig. 6(b)— +is subtracted and thus only the slow, glassy dynamics +can be seen. Figure 7(a) reveals that ∆M(T ) of twait +of 500 s and 1,000 s are virtually identical, since only a +zero line is visible. In other words, the fast dynamics of +the superparamagnet are over while the slow dynamics +of the glassiness have not yet set in, both referring to +the experimental accuracy. Having thus subtracted the +fast dynamics the evolving dip in ∆M(T ) with twait of +5,000 s and higher nicely represents the remaining mag- +netic glassiness with its characteristic slow dynamics and +ZFC memory effect. It is furthermore visible that the +minimum in ∆M(T ) does not align with Twait which is +indicated by the dashed magenta line in Fig. 7(a). Twait +merely appears to align with the inflection point of the +high-temperature side of ∆M(T ) which is also seen in +the Twait-dependence of ∆M(T ) in Fig. 7(b). Also here, +the low-temperature increase of ∆M seen in Fig. 6(b) +is fully removed by taking ∆M always with respect to +Twait = 320 K. Note, that in Fig. 7(b) Twait = 300 K +is not nicely visible but the dip in ∆M(T ) is very weak +and clearly less pronounced compared to the others and +in fact may only reflect the limits of reproducibility of +these types of SQUID experiments; one should keep in +mind that two ZFC M(T ) curves like in Fig. 5(a) are sub- +tracted from each other, i. e., the signal size and thus the +relative accuracy of each data point varies (slightly) over +the entire T -range which can easily affect difference sig- +nals of the order of 1·10−7 emu. Nevertheless, Figs. 6 and +7 nicely demonstrate that in ZnFe2O4 superparamagnetic +and glassy behavior coexist and can be separated from +each other. This is quite remarkable, since an epitaxial +film of ZnFe2O4 is structurally quite distinct from a su- +perparamagnetic ensemble like horse-spleen ferritin or a +superspin-glass like a dense ensemble like Fe3N nanopar- +ticles which were both investigated in [26]. Yet, ZnFe2O4 +epitaxial thin films exhibit both types of magnetic or- +der at the same time. Therefore, the observed magnetic +glassiness appears to be better described in terms of a +cluster glass like in [7], i. e., a superparamagnetic-like en- +semble with (frustrated) intercluster interactions. How- +ever, these interactions have to be inhomogeneous and +disordered throughout the sample and in contrast to the +nanopowder in [9] they have no obvious structural origin. +One has to therefore conclude that they stem from local +variations of the cation distribution, i. e., from chemical +or A/B disorder and thus they crucially depend on a fi- +nite amount of inversion. This in turn also explains why +highly crystalline bulk ZnFe2O4 samples in [1–3] exhibit +quite distinct magnetic properties. +Since we have seen that Tf is a function of the growth +power during the sputtering process, the two power se- + +9 +0.0 +0.5 +1.0 +1.5 +2.0 +100 +200 +300 +400 +100 +200 +300 +-0.2 +0.0 +50 +100 +150 +200 +250 +0.0 +0.5 +1.0 +100 +200 +0.0 +0.5 +t +wait + = 10,000 s +T (K) +M (k A /m ) +sputter power + 100 W + 80 W + 60 W + 40 W + 20 W +MgAl +2 +O +4 +(a) + + +t +wait + = 10,000 s +(b) +M (k A /m ) +T (K) +sputter power + 80 W + 60 W + 40 W + 20 W +Al +2 +O +3 + + + + +FIG. 8: +Comparison of the hole-burning experiments for +ZnFe2O4 grown on MgAl2O4 (a) and Al2O3 (b) as a func- +tion of sputter power (details see text). The insets show the +high sputter power samples only. +ries of ZnFe2O4 samples grown on MgAl2O4 and Al2O3 +shall be directly compared. +For that we have chosen +to perform the hole-burning ZFC waiting experiments +of Fig. 6(b) on the identical relative temperature scale +for each sample. In other words, the highest and lowest +temperature of the M(T ) curves as well as Twait have +been chosen to be a the same relative temperature with +respect to Tf to assure that the samples spent compa- +rable time-spans in regions with comparable magnetiza- +tion dynamics. +Note, that in addition the full experi- +ment for all Twait of Fig. 6(b) on an absolute temper- +ature scale have also been performed (not shown), but +the direct comparison in essence reveals the identical re- +sult. Figure 8 shows the ∆M curves for the power-series +of ZnFe2O4 grown on MgAl2O4 (a) and Al2O3 (b) for +twait of 10,000 s; the insets enlarge the samples grown at +high sputtering powers. Irrespective of the substrate the +samples grown at sputtering powers of 20 W and 40 W +do only show the low-temperature increase of ∆M, i. e., +mostly superparamagnetic-like behavior; for ZnFe2O4 on +Al2O3 a faint and broad minimum is visible which how- +ever does not show a clear shift with Twait or a pro- +nounced dependence with twait. Therefore, we consider +0 +1 +700 +705 +710 +715 +720 +725 +730 +-0.2 +0.0 +0.2 +0.4 +700 +705 +710 +715 +720 +725 +730 +0 +1 +-0.2 +0.0 +0.2 +0.4 +Fe L +3/2 +-edges +T = 300 K +20� grazing +(a) +photon energy (eV) +n o r m . X A N E S +; +; + 80 W + 20 W +X M C D +ZnFe +2 +O +4 +/MgAl +2 +O +4 +36% Fe +2+ +Oh +/32% Fe +3+ +Oh +/33% Fe +3+ +Td +32% Fe +2+ +Oh +/38% Fe +3+ +Oh +/30% Fe +3+ +Td +Fe L +3/2 +-edges +T = 300 K +20� grazing +; +; +(b) +n o r m . X A N E S +photon energy (eV) + 80 W + 20 W +X M C D +34% Fe +2+ +Oh +/48% Fe +3+ +Oh +/18% Fe +3+ +Td +28% Fe +2+ +Oh +/28% Fe +3+ +Oh +/44% Fe +3+ +Td +ZnFe +2 +O +4 +/Al +2 +O +3 +FIG. 9: Normalized XANES and XMCD spectra recorded at +the Fe L3/2-edges under grazing incidence at 300 K for the +80 W and 20 W ZnFe2O4 samples grown on (a) MgAl2O4 and +(b) Al2O3, respectively. The XMCD spectra have also been +simulated to determine the relative amount of the individual +Fe species (see text). +this part as inconclusive, i. e., not as clear experimental +evidence for glassiness. In contrast, the samples grown +at 60 W and higher all show a hole-burning behavior in +the ZFC memory experiments which is pronounced for +ZnFe2O4 on MgAl2O4, see inset of Fig. 8(a), but rather +weak for ZnFe2O4 on Al2O3, for which only a faint min- +imum can be seen, see inset of Fig. 8(b). Therefore, in +ZnFe2O4 on Al2O3 only superparamagnetic-like can be +inferred and signatures of magnetic glassiness are faint +and limited to high sputtering powers. This goes hand- +in-hand with a more pronounced maximum in the ZFC +curves, see Fig. 3(b) and an increased magnetization, see +Fig. 2(a). +In contrast, ZnFe2O4 on MgAl2O4 exhibits +a clear transition from superparamagnetic-like behavior +at low sputtering powers with clear signs of magnetic +glassiness existing at high sputtering powers, i. e., growth +rates. +To ultimately clarify what causes the discrep- +ancy in the magnetic properties for ZnFe2O4 grown on +MgAl2O4 and Al2O3 as well as at low and high sputtering +powers, the 20 W and the 80 W samples were subjected +to an element-selective magnetic characterization using +XMCD. + +10 +Figure 9 shows the measured XANES and XMCD spec- +tra at the Fe L3/2-edges for ZnFe2O4 grown on MgAl2O4 +(a) and Al2O3 (b) for the samples grown at 20 W and +80 W, respectively. The XMCD at the Fe L3-edge has +been also simulated by respective multiplet ligand field +theory using the CTM4XAS code using the identical +parameters as in [32]. +In brief, the negative peaks in +the XMCD spectrum are stemming from the octahedral +contributions FeOh, where Fe2+ +Oh is mostly seen at lower +(706.6 eV) and Fe3+ +Oh at higher (708.5 eV) photon ener- +gies; the positive peak at 707.8 eV can be assigned to +Fe3+ +T d. The experimental XMCD can be reproduced by +adjusting the relative concentrations of Fe3+ +Oh, Fe2+ +Oh, and +Fe3+ +T d to match the experimental XMCD; the results of +this are given in Fig. 9. It can be seen in Fig. 9(a), that +there are no pronounced differences between the exper- +imental XMCD spectra of the Fe L3/2-edge XMCD for +ZnFe2O4/MgAl2O4 grown at either 20 W or 80 W as well +as for the respective results of the simulation. About one +third of the Fe is located on tetrahedral sites, i. e., the de- +gree of inversion δ is around 0.3 for both sputtering pow- +ers. Also a significant amount of Fe2+ +Oh is found, which +would suggest a strong contribution from a JDE +BB double +exchange interaction which appears to be slightly larger +for the 80 W sample which exhibits the magnetic glassi- +ness in comparison to the 20 W sample, which only shows +the superparamagnetic-like low temperature increase of +∆M. The ZnFe2O4/Al2O3 samples in Fig. 9(b) exhibit a +different behavior. Here the 80 W sample has a strongly +reduced contribution of Fe3+ +T d compared to the FeOh com- +pared to the 20 W sample which has the highest relative +content of Fe3+ +T d. On the other hand, the actual positive +peak in the XMCD is of identical size in both samples. It +therefore appears, the XMCD intensity for the FeOh is +reduced while the amount of Fe3+ +T d remains constant. This +may appear as contradiction at first sight, since the rel- +ative contents may suggest different degrees of inversion +for the two samples. However, one should keep in mind +that the magnetic superexchange interaction on the octa- +hedral sites JBB is weakly antiferromagnetic while double +exchange leads to spin canting [4, 5]. Since the magnetic +order is observed up to above room temperature for all +samples in this work, the JAB superexchange mechanism +has to play a significant role, which is consistent with a fi- +nite degree of inversion of the order of 0.3. In that light, +the presence of a finite amount of inversion giving rise +to Fe3+ +T d is a prerequisite for magnetic order at elevated +temperatures but does not play a decisive role for the +presence of magnetic glassiness, since Fe3+ +T d is found in all +four samples while glassiness is only found in the 80 W +samples, in particular in those grown on MgAl2O4. In +addition, the presence of Fe2+ +Oh in all samples further sug- +gests the presence of an additional JDE +BB double exchange +mechanism associated with spin canting. Here the rela- +tive amount of Fe2+ +Oh increases only slightly from the 20 W +sample on Al2O3 over 20 W on MgAl2O4, 80 W on Al2O3 +to 80 W on MgAl2O4, i. e., it follows the trend of in- +creasing glassiness of the samples. However, the changes +1010 +1020 +1030 +1040 +1050 +1060 +0 +1 +-2 +0 +2 +4 +6 +0 +1 +1010 +1020 +1030 +1040 +1050 +1060 +-2 +0 +2 +4 +6 +n o r m . X A N E S +photon energy (eV) +20W ZnFe +2 +O +4 +/Al +2 +O +3 +80W ZnFe +2 +O +4 +/Al +2 +O +3 +X M C D (% ) +20W ZnFe +2 +O +4 +/MgAl +2 +O +4 +Zn L +3/2 +-edges +T = 300 K +20� grazing +80W ZnFe +2 +O +4 +/MgAl +2 +O +4 +(b) +photon energy (eV) +n o r m . X A N E S +(a) +Zn L +3/2 +-edges +T = 300 K +20� grazing +X M C D (% ) +FIG. 10: Normalized XANES and XMCD spectra recorded +at the Zn L3/2-edges under grazing incidence at 300 K for the +80 W and 20 W ZnFe2O4 samples grown on (a) MgAl2O4 and +(b) Al2O3, respectively. +are rather small and the significance of determining such +small changes with multiplet ligand field simulations is +limited. +Obviously, there is no straightforward mech- +anism for the occurrence of magnetic glassiness which +can be derived form the XMCD spectra at the Fe L3/2- +edges. A finite degree of inversion has to play a role but +mostly for the high order temperatures observed. The +existence of glassiness appears to be linked to a delicate +balance of the various competing exchange interactions +as well as the local cationic configuration which has to +be inhomogeneous throughout the sample as discussed +above. Obviously the growth rate influences mostly the +latter where the highest growth rates favor glassiness, +most likely via increased local cationic disorder, in par- +ticlular in ZnFe2O4 on MgAl2O4 substrates. +Figure 10 shows the measured XANES and XMCD +spectra at the Zn L3/2-edges of the identical set of sam- +ples as in Fig. 9. The XANES for ZnFe2O4 on MgAl2O4 +in Fig. 10(a) is rather similar to the one of ZnFe2O4 on +Al2O3 in (b). All four samples exhibit a finite XMCD +with comparable spectral shape; all XMCD spectra were +derived by reversing both, helicity of the light as well +as the magnetic field and it was verified that the XMCD +spectrum nicely reverses with reversing external field (not + +11 +shown). The size of the Zn L3/2-edge XMCD follows the +amount of Fe3+ +T d as seen in Fig. 9, i. e., the Zn XMCD +is largest for the 20 W sample on Al2O3, which has the +highest relative Fe3+ +T d content and it is lowest for the 20 W +sample on Al2O3 which has the lowest relative Fe3+ +T d con- +tent. It is thus reasonable to assume that the magnetic +polarization of Zn in ZnFe2O4 is mostly associated with +Zn2+ +Oh. In turn, this implies that a weakly polarized cation +substitutes for a strongly polarized one thus reducing the +effective exchange. +This is consistent with the exper- +imental observation that the 80 W ZnFe2O4/MgAl2O4 +has the highest Tf and the lowest magnetic polarization +of Zn while the highest Zn polarization in the 20 W +ZnFe2O4/Al2O3 sample is associated with the lowest Tf. +To verify this hypothesis, more sophisticated theoretical +calculations beyond the multiplet ligand field codes is re- +quired where the individual spectroscopic signatures in +the Zn L3/2-edge XANES and XMCD can be associated +with the actual Zn species which however goes beyond +the scope of this paper. Nevertheless, it is already evident +that a too high degree of inversion as seen by strong mag- +netic polarization of the Zn together with a high relative +content of Fe3+ +T d is unfavorable for both, high Tf as well as +magnetic glassiness and high growth rates appear to be +an experimental means to control/limit excessive inver- +sion but at the same time assure sufficient local cationic +disorder to induce magnetic glassiness. +IV. +DISCUSSION AND CONCLUSION +ZnFe2O4 epitaxial thin films have been grown on +MgAl2O4 and Al2O3 substrates with varying preparation +conditions. All samples were investigated with respect to +their basic structural and magnetic properties and long +range magnetic order was found above room tempera- +ture for all samples. The stoichiometric composition of +the samples was verified using RBS. A clear bifurcation +between M(T ) curves under FC and ZFC conditions is +found at Tf, which is systematically higher for ZnFe2O4 +on MgAl2O4 by about 100 K. Tf is found to systemati- +cally increase with increasing the sputtering power and +thus growth rate in agreement with [7]. The Ar:O2 ratio +was not found to influence neither Tf nor Ms in a system- +atic manner; increasing Tgrowth increases only Ms while +Tf exhibits no systematic changes. +An in-depth study of the magnetic properties us- +ing FC as well as ZFC memory experiments reveals +magnetic glassiness for samples grown at high sput- +ter powers. +The glassiness is more pronounced for +ZnFe2O4/MgAl2O4 compared to ZnFe2O4/Al2O3, where +the signatures of magnetic glassiness beyond those in +FC memory experiments are generally weak. At lower +growth rates the signatures of glassiness are absent and a +low-temperature increase of ∆M is observed which points +towards superparamagnetic-like behavior and the signa- +tures in FC memory experiments are weak and the ZFC +memory experiments shows no hole burning effect in ac- +cordance with the expectations for superparamagnetic +samples [26]. In contrast, at high growth rates, in par- +ticular for ZnFe2O4/MgAl2O4 ZFC memory experiments +show an additional hole burning effect which is charac- +teristic for spin glasses [25] and superspin glasses [26]. +Since the glassiness coexists with superparamagnetic-like +signatures, in particular, the low temperature increase of +∆M, the structural properties of the ZnFe2O4 epitaxial +films are quite different from the nanoparticle ensembles +in [26] the observed magnetic properties are described +best as cluster glass in analogy to comparable observa- +tions for epitaxial ZnFe2O4 in [7]. +An in-depth characterization based on XANES and +XMCD reveals that a finite magnetic polarization at the +Zn L3/2 edges exists in all ZnFe2O4 samples which adds +more complexity to the magnetic interactions beyond the +usually discussed Fe-based exchange. At the Fe L3/2 the +XMCD is used to extract the relative concentrations of +Fe3+ +T d, Fe3+ +Oh, and Fe2+ +Oh by means of multiplet ligand field +simulations as done before [31, 32]. The abundance of +Fe3+ +T d correlates well with the size of the magnetic polar- +ization of Zn and thus both can serve as a measure for the +degree of inversion. For highest inversion Tf is found to be +lowest and signatures of magnetic glassiness are absent. +In contrast, the sample with the strongest signatures of +magnetic glassiness, the 80 W ZnFe2O4/MgAl2O4, ex- +hibits no significant changes in the Fe L3/2-edge XMCD +compared to the superparamagnetic-like 20 W sample. +The most prominent tendency appears to be the rel- +ative amount of Fe2+ +Oh which can be associated with a +double-exchange mechanism which was held responsible +for spin canting [4, 5]. Since the differences in XMCD be- +tween the respective samples are small, this correlation +between glassiness and Fe2+ +Oh cannot be taken as signifi- +cant but merely a starting point for more elaborate the- +oretical work to understand the details of the obtained +XMCD spectra beyond the multiplet ligand field simula- +tions. Most likely the cluster glass behavior in epitaxial +ZnFe2O4 cannot be assigned to the actual structure of +the materials like in common superparamagnets or dense +nanoparticle ensembles [26] but due to local variations +of the stoichiometry, leading to an inhomogeneous local +cation distribution throughout the sample. +As a con- +sequence, the local magnetic moments in ZnFe2O4 are +disordered due to partial inversion and partially canted +due to the presence of Fe2+ +Oh, which leads to characteristic +signatures of a cluster glass at rather high temperatures +which is mostly controllable by the growth rate as re- +ported for ZnFe2O4 before [7]. +Acknowledgments +J. L. gratefully acknowledges funding by FWF project +ORD-49 at the initial stage of this work. A.Z. acknowl- +edges the financial support by the Swiss National Science +Foundation (SNSF) under Project No. 200021-169467. +The X-ray absorption measurements were performed on + +12 +the EPFL/PSI X-Treme beamline at the Swiss Light +Source, Paul Scherrer Institut, Villigen, Switzerland. In +addition, support by VR-RFI (Contracts No. 2017-00646 +9 and No. 2019-00191) and the Swedish Foundation for +Strategic Research (SSF, Contract No. RIF14-0053) sup- +porting accelerator operation at Uppsala University is +gratefully acknowledged. The research leading to this re- +sult has been supported by the RADIATE project under +the Grant Agreement No. 824096 from the EU Research +and Innovation programme HORIZON 2020. +[1] J. M. Hastings and L. M. Corliss, Rev. Mod. Phys. 25, +114 (1953). +[2] J. M. Hastings and L. M. Corliss, Phys. Rev. 102, 1460 +(1956). +[3] K. Kamazawa, Y. Tsunoda, H. Kadowaki, and K. Kohn +Phys. Rev. B 68, 024412 (2003) +[4] D. Venkateshvaran, +M. Althammer, +A. 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B 102, 054402 (2020). + diff --git a/PtFIT4oBgHgl3EQffCtm/content/tmp_files/load_file.txt b/PtFIT4oBgHgl3EQffCtm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b68a87bde9262426aef668e122895fbc06ef811f --- /dev/null +++ b/PtFIT4oBgHgl3EQffCtm/content/tmp_files/load_file.txt @@ -0,0 +1,827 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf,len=826 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='11277v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='other] 25 Jan 2023 Room-temperature spin glass behavior in zinc ferrite epitaxial thin films Julia Lumetzberger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 Verena Ney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 Anna Zhakarova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 Nieli Daffe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 Daniel Primetzhofer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='3 and Andreas Ney1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' ∗ 1Johannes Kepler University Linz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Institute for Semiconductor and Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Altenberger Strasse 69,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 4040 Linz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Austria 2Swiss Light Source (SLS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Paul Scherrer Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 5232 Villigen PSI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Switzerland 3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' ˙Angstr¨om Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Uppsala University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Box 516,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' SE-751 20 Uppsala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Sweden (Dated: January 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2023) Zinc ferrite (ZnFe2O4) epitaxial thin films were grown by reactive magnetron sputtering on MgAl2O4 and Al2O3 substrates varying a range of preparation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The resulting structural and magnetic properties were investigated using a range of experimental techniques confirming epi- taxial growth of ZnFe2O4 with the nominal stoichiometric composition and long range magnetic order at and above room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The main preparation parameter influencing the tempera- ture Tf of the bifurcation between M(T ) curves under field cooled and zero-field cooled conditions was found to be the growth rate of the films, while growth temperature or the Ar:O2 ratio did not systematically influence Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Furthermore Tf was found to be systematically higher for MgAl2O4 as substrate and Tf extends to above room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' While in some samples Tf seems to be more likely correlated with superparamagentism, the highest Tf occurs in ZnFe2O4 epitaxial films where experimental signatures of magnetic glassiness can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Element-selective X-ray magnetic cir- cular dichroism measurements aim at associating the magnetic glassiness with the occurrence of a different valence state and lattice site incorporation of Fe pointing to a complex interplay of various competing magnetic interactions in ZnFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' INTRODUCTION Zinc ferrite (ZnFe2O4) belongs to the crystallographic group of normal spinels of the form AB2O4 where in the ideal case the A-cation (Zn2+) exclusively occupies the tetrahedral (Td) lattice sites as Zn2+ T d while the B-cation (Fe3+) is found on the octahedral (Oh) sites as Fe3+ Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The magnetic properties of zinc ferrite have been un- der investigation for quite some time revealing a rather complex situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Early studies of the bulk material re- port antiferromagnetic (AFM) order with a very low N´eel temperature of 9 K [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Later-on it was demonstrated by magnetic neutron scattering experiments on ZnFe2O4 single crystals that even in perfect crystals geometrical frustration leads to an unusual magnetic behavior [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In particular, it was pointed out, that the Fe3+ Oh sublattice can be regarded to be similar to various pyrochlores or Laves phases which are known for their intrinsic geomet- rical frustration [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The situation becomes even more complex when defects such as inversion are considered which are expected to occur, in particular, in thin films of ZnFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A partial inversion in ZnFe2O4 has the stoichiometric formula of [Zn1−δFeδ]T d[ZnδFe2−δ]OhO4, where δ denotes the degree of inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In the ideal case of δ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', no inversion, there is only the weak AFM superexchange interaction between Fe3+ Oh which is usually denoted as JBB [4] plus the geometrical frustration men- tioned before [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' JBB can be held responsible for the AFM order at low temperatures in bulk single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For a finite degree of inversion there is an additional, ∗Electronic address: andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='ney@jku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='at much stronger AFM superexchange interaction JAB be- tween Fe3+ on Td (also called A-site) and Oh (or B-site) sites, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', Fe3+ T d and Fe3+ Oh [4], which leads to ferrimag- netism for incomplete inversion [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The additional JAA exchange between the Fe3+ T d is the weakest [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, if the additional Fe3+ T d is not compensated by Zn2+ Oh, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', if there is some degree of deviation from the ideal stoi- chiometry of ZnFe2O4, some finite amount of Fe2+ Oh has to form because of charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This results in an ad- ditional double exchange (DE) over the Oh (or B-) sites JDE BB between Fe3+ Oh and Fe2+ Oh, which results in spin cant- ing in magnetite [4] or in non-stoichiometric ZnFe2O4 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In many cases there are reports on some finite degree of inversion in ZnFe2O4 and an upper limit of δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='6 has been found by the analysis of the magnetic moment of the Fe [7], X-ray absorption spectroscopy (XAS) [8] or via Rietveld refinement of X-ray diffraction (XRD) data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, the magnetic order in ZnFe2O4 can be expected to be highly complex, especially for thin films, where the presence of various kinds of defects such inver- sion and/or off-stoichiometry can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The growth of zinc ferrite in thin film form is moti- vated by a range of possible applications such as gas sen- sors, photo catalytic disinfection or other photo-catalytic applications, see [9, 10] and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Besides that, zinc ferrite thin films can also be considered as an interesting semiconducting material in spintronics with tunable magnetic properties [4, 6, 11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A variety of reports of different types of magnetic order in zinc ferrite can be found throughout the literature ranging from ferro(i)magnetic [4, 6, 13–17], to superparamagnetic [8, 11, 18, 20, 21] and to spin glass behavior [7, 9, 19, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that in some cases superparamagnetism with inter- particle interactions is associated with a so-called cluster- 2 glass behavior [7, 8, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, only a few studies report on characteristic experimental signatures of mag- netic glassiness [7, 9, 19, 22], while others report only temperature-dependent magnetization [M(T )] measure- ments under different field-cooling conditions which how- ever could also be associated with superparamagnetism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Finally, the control of defects and thus the magnetic properties was reported to be experimentally achievable by varying different preparation parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', oxygen partial pressure [6, 12, 17, 18, 21], stoichio- metric composition [4], post-growth thermal treatment [19, 20] or deposition rate [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Similar to the reported types of magnetism in zinc fer- rite, also the techniques for sample preparation span a wide range from pulsed laser deposition (PLD) [4, 6, 7, 11–13, 16, 17, 20], over reactive magnetron sputtering (RMS) [8, 14, 15, 18, 19, 21] for thin film growth, to ball milling [9] and solid state reaction [22] for bulklike sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Likewise, a range of different substrates has been used for thin film growth amongst them are MgAl2O4 [12], c-plane Al2O3 [7, 11], a-plane Al2O3 [16], SrTiO3 [13, 17, 20], MgO [4, 6, 11], Si(001) and Si(111) [14, 21] and glass substrates [8, 15, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It is remarkable, that spin glass behavior has mainly been reported for bulk ZnFe2O4 [3] or bulklike nanopowders [9, 22] while for thin film samples mostly a cluster glass is inferred [7, 8, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Among the reports of cluster glass behavior only one is based on thin film growth by PLD on single crystalline substrates [7], while the others rely on sput- tered polycrystalline ZnFe2O4 samples [8, 19] making a larger amount of defects expectable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', due to an in- trinsically large number of grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Finally, also the temperature range, where magnetic glassiness is ob- served ranges from below 20 K for the single crystalline ZnFe2O4 in [3, 22], over around 100 K for the annealed ZnFe2O4 nanopowder in [9] up to 300 K for the PLD grown ZnFe2O4 epitaxial films at deposition rates above 3 nm/s which drops down to below 100 K for rates below 2 nm/s [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Here, we report on epitaxial thin film growth of ZnFe2O4 by RMS on two different substrates namely MgAl2O4 and c-plane Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Various preparation pa- rameters have been varied in order to control the for- mation of defects in a systematic way for epitaxial thin film samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In agreement with [7] the most relevant preparation parameter is found to be the deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For high deposition rates ZnFe2O4 films exhibit spin- glass like behavior up to rather high temperatures on MgAl2O4 substrates which is shifted to lower tempera- tures on Al2O3 substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, the stoichiom- etry on ZnFe2O4 is maintained throughout the sample series and also the oxygen partial pressure were found to play a minor role in the resulting magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The spin-glass behavior is associated with a significant amount of inversion up to δ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='3, corroborating earlier reports [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In addition, a significant magnetic polar- ization of the Zn cation is found at room temperature by means of element selective magnetometry indicating that the microscopic origin of the magnetic properties of ZnFe2O4 is even more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' EXPERIMENTAL DETAILS Zinc ferrite was fabricated using reactive magnetron sputtering (RMS) from an oxide target having the nominal composition of ZnFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The epitaxial thin films were grown on doubleside polished single crys- talline spinel [MgAl2O4(001)] and c-plane sapphire [Al2O3(0001)] substrates in an ultrahigh vacuum (UHV) chamber with a base pressure of 4 × 10−8 mbar and a working pressure of 4 × 10−3 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' To determine the ideal growth parameters, the deposition temperature was varied from room temperature (RT) to 550 °C, the Ar:O2 ratio from 10 : 0 to 10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5, and the sputtering power from 20 W to 100 W, which corresponds to a growth rate from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='36 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='69 nm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The nominal thickness is kept at 40 nm and is controlled via a quartz crystal microbalance which is at room temperature so that the actual thickness of most of the films is by 10-20% lower because of the el- evated temperature of the substrate during growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The structural properties of the films were investigated by X- ray diffraction (XRD) measurements with a Pananalyti- cal X’Pert MRD recording ω − 2θ scans and symmetric as well as asymmetric reciprocal space maps (RSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The chemical composition was determined by ion beam anal- ysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Rutherford backscattering spectrometry (RBS) using a 2 MeV He+ primary beam at the Tandem Labora- tory at Uppsala University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' To disentangle the element specific contributions, the spectra were analyzed using the SIMNRA software [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Details of the experimental setup are described elsewhere [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Furthermore, Elec- tron Recoil Detection (ERDA) with a primary ion beam of 36 MeV iodine ions was employed to rule out contam- inations with light elements like H or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The magnetic properties were measured by integral superconducting quantum interference device (SQUID) magnetometry using a Quantum Design MPMS-XL5 sys- tem applying the magnetic field in the film plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The M(H) curves were recorded in range of ±5 T at 300 K and 2 K and M(T ) curves have been recorded from 2 K up to 395 K at 10 mT while warming after a cool down in 5 T (FH), under nominally zero-field cooled conditions (ZFC) as well while cooling down in 10 mT (field cooled, FC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Additionally, waiting time experiments were per- formed analogous to the ones in [25] by cooling down the sample in zero field and introducing a waiting time twait at various waiting temperatures Twait which is typ- ically 10 000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Then a M(T ) curve identical to a ZFC curve without waiting time was subsequently recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Subtracting these two curves represents a typical waiting time experiment for spin glasses where a so-called ZFC memory—or hole-burning—effect can be seen by a dip in the difference of the magnetization with and without waiting time around Twait [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A second memory ex- periment already used before for ZnFe2O4 in [9] was per- 3 formed in addition, in which a M(T ) curve is recorded under FC conditions with 10 mT and a waiting time twait at nominally zero field is inserted at several Twait before the FC curve is resumed at 10 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Then a subsequent M(T ) is recorded in 10 mT while warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The typical signature of a spin glass in these so-called FC memory experiments is a relaxation of the magnetization at Twait in the FC curve and the presence of an inflection point around Twait in the subsequent M(T ) curve while warm- ing [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, however, that also superparamagnets ex- hibit similar signatures in the FC memory experiments [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' All M(H) and M(T ) data were corrected for the diamagnetic background of the substrate which was de- termined from the M(H) curves a high magnetic field at 300 K [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The M(H) curves at 2 K for samples grown on MgAl2O4 had to be corrected for an additional param- agnetic contribution which was determined form a M(H) measurement of bare MgAl2O4 from the same batch of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In general, zero field conditions are referred to nominally 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 mT after the superconducting magnet had been reset (magnet reset option of the MPMS) and any applied magnetic field is afterwards limited to 10 mT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' this assures a residual pinned magnetic field of typi- cally 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 mT or less [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A cooling and heating rate of 1 K/min is used for all M(T ) measurements in both FC and ZFC memory experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that the typ- ical frequency-dependent ac-susceptibility measurements like in [7, 19, 22] were not available for the used SQUID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The element specific magnetic properties have been investigated by x-ray absorption near edge spec- troscopy (XANES) and x-ray magnetic circular dichro- ism (XMCD) measurements which were performed at the Xtreme beamline at the Swiss Light Source (SLS) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XMCD spectra were recorded at the Fe L3/2- and Zn L3/2-edges at 300 K under 20° grazing incidence in total electron yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For the Fe-edges the magnetic field was set to 5 T and only the circular polarization has been switched to obtain the XMCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For the Zn-edges the di- rection of the magnetic field has been reversed as well to minimize artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XMCD spectra at the Fe L3 edge are compared to simulations carried out by mul- tiplet ligand field theory using the CTM4XAS package [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' These simulations have been used before to deter- mine the site occupancy and formal oxidation state of Fe and Ni in nickel ferrites [31] and Zn/Al doped nickel ferrites [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For the present work the simulation param- eters for Fe2+ Oh, Fe3+ Oh, and Fe3+ T d are identical to those in [32] and details on the simulations can be found there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' EXPERIMENTAL RESULTS The structural properties of the ZnFe2O4 thin films were analyzed by symmetric ω − 2θ scans using XRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 1(a) shows a comparison of the diffractograms of zinc ferrite grown on MgAl2O4 and Al2O3 where the lat- ter is shifted upward for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The samples were grown with a nominal thickness of 40 nm at a substrate tem- 10 0 10 1 10 2 10 3 10 4 10 5 10 6 35 40 45 800 1000 1200 1400 1600 0 50 100 150 200 2 (�) In te n s ity (a r b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' u n its ) 80W on MgAl 2 O 4 80W on Al 2 O 3 ZnFe 2 O 4 (311) Al 2 O 3 (006) ZnFe 2 O 4 (004) MgAl 2 O 4 (004) (a) (b) in te n s ity (c o u n ts ) energy (keV) 80W on MgAl 2 O 4 80W on Al 2 O 3 simulation Fe Zn 2 MeV He + FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 1: (a) Structural characterization by X-ray diffraction: comparison of the symmetric ω − 2θ scans of ZnFe2O4 films grown at 80 W on MgAl2O4 (black) and Al2O3 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' (b) Chemical composition determined by means of RBS spectra recorded with a 2 MeV He+ primary ion beam of ZnFe2O4 films grown at 80 W on MgAl2O4 (black) and Al2O3 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' perature of TS = 450 °C with an Ar:O2 ratio of 10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 and a sputtering power of 80 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XRD scan of the samples grown on MgAl2O4 exhibits a (004) reflex at 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='09° with a full width at half maximum (FWHM) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='6 °, corresponding to a perpendicular lattice param- eter a⊥ = (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='02) �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The sample grown on Al2O3 exhibits the (311) reflex at 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='61° corresponding to a per- pendicular lattice parameter of a⊥ = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='02)�A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For this sample weak Laue oscillations can be seen indicat- ing a smoother growth compared to the sample grown on MgAl2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In addition, an asymmetric RSM along the (¯1¯15) plane has been recorded for a 100 nm sam- ple grown on MgAl2O4 with a sputtering power of 60 W (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It reveals an in-plane lattice parameter of a∥ = (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='05)�A which provides evidence that the film is relaxed, since the film peak does not align with the sub- strate peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The reflection from the MgAl2O4 substrate corresponds to a lattice parameter of asub = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='08 �A, which implies a lattice mismatch of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='3 % with re- 4 spect to bulk ZnFe2O4 (a0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='441 �A) (JCPDS card No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 82-1049).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A comparison between various films grown on MgAl2O4 and Al2O3 indicates no significant differ- ence in crystalline quality with similar FWHM despite the change in texture from (004) on MgAl2O4 to (311) on Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' No other reflexes can be found underlining highly textured growth of ZnFe2O4 for both substrates and the samples are devoid of other crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Furthermore, the chemical composition of ZnFe2O4 on both substrates is determined using RBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 1(b) the RBS data of the 80 W sample grown on MgAl2O4 (black squares) is shown in comparison to the sample grown on Al2O3 (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Both ZnFe2O4 films have no deviation from the nominal stoichiometry within the uncertainties of the measurement technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Finally, the samples are investigated using ERDA to check for con- taminations of light elements like H or C but neither element could be detected (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' We can there- fore conclude that our ZnFe2O4 samples have the nom- inal stoichiometric composition, grow epitaxially on ei- ther substrate and are devoid of a significant amount of secondary phases or contaminants within the detection limits of XRD and ERDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In a first step, the magnetic properties are investigated using standard M(H) curves which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2 recorded at (a) 300 K and (b) 2 K for ZnFe2O4 grown at 60 W on MgAl2O4 (black squares) and Al2O3 (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' ZnFe2O4 grown on Al2O3 has a higher magne- tization of MS = 130 ± 15 kA/m compared to growth on MgAl2O4 where MS = 110 ± 15 kA/m at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For both samples MS increases to above 200 kA/m at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For the M(H) curves recorded at 2 K for ZnFe2O4 grown on MgAl2O4 the full squares denote the data when only the diamagnetic contribution has been subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that in a previous publication on ZnFe2O4 grown on MgAl2O4 this apparently paramagnetic behavior has been attributed to cationic disorder of the Fe3+ in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, if a bare MgAl2O4 substrate is measured, one also measures a net-paramagnetic behavior after subtrac- tion of the diamagnetism so that one has to attribute this paramagnetic contribution to the MgAl2O4 substrate it- self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The open squares are the data where also the mea- sured paramagnetic background of the bare MgAl2O4 has been subtracted and no obvious paramagnetic contribu- tion of the ZnFe2O4 is visible any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2 enlarge the low-field behavior of the M(H) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' While they are virtually anhysteretic at 300 K a clear hys- teresis with a coercive field of Hc = 80 ± 10 mT is found for ZnFe2O4 films on either substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This behavior at 2 K is consistent with most of the ZnFe2O4 films grown on a range of different substrates reported throughout the literature reporting ferro(i)magnetism or superparamag- netism both in terms of magnetization as well as coercive field at low temperatures and clearly rules out the pure antiferromagnetic behavior of bulk ZnFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 3 shows the M(T ) behavior recorded at 10 mT under FC conditions (full symbols) as well as after ZFC conditions (open symbols) for the ZnFe2O4 grown at 4 2 0 2 4 400 200 0 200 400 150 100 50 0 50 100 150 4 2 0 2 4 50 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 200 0 200 M (k A /m ) 0 H (T) MgAl 2 O 4 corrected Al 2 O 3 (b) T = 2 K 0 H (T) M (k A /m ) 60 W ZnFe 2 O 4 / MgAl 2 O 4 Al 2 O 3 (a) T = 300 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2: SQUID measurements of M(H) curves shown for the 60 W ZnFe2O4 grown on MgAl2O4 (black squares) and Al2O3 (red circles) at (a) 300 K and (b) at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' (a) shows an M(H) curve at RT (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' At 2 K the paramagnetic contribution of the MgAl2O4 substrate has been subtracted (open squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The insets enlarge the measurements at low fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 60 W on MgAl2O4 (a) and Al2O3 (b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', the identi- cal pair of samples as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Both samples exhibit a clear bifurcation between the FC and ZFC curves in- dicating a blocking or spin-freezing peak at a temper- ature of Tf = 290 K for ZnFe2O4/MgAl2O4 (a) and at Tf = 190 K for ZnFe2O4/Al2O3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Both 60 W sam- ples together with the two 80 W samples shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 1 are part of a sample series where only the sputtering power and thus the deposition rate has been changed while all other growth parameters have been kept con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In terms of magnetization as well as coercivity all samples from these series show comparable magnetic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The only systematic dependency on the sput- tering power is an increase of the measured Tf with in- creasing sputtering power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 3 show the measured Tf as a function of the sputtering power for ZnFe2O4/MgAl2O4 (a) as well as for ZnFe2O4/Al2O3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Irrespective of the comparable increase with sputter- ing power the overall values of Tf are systematically lower for the Al2O3 substrate by about 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note that the obtained Tf for the samples grown on either substrate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' are well above the values usually reported for zinc ferrite 5 20 40 60 0 100 200 300 400 0 100 200 300 400 20 40 60 80 20 40 60 80 170 180 190 200 µ 0 H = 10 mT (a) T (K) M (k A /m ) 60 W ZnFe 2 O 4 /MgAl 2 O 4 FC ZFC T f µ 0 H = 10 mT 20 40 60 80 100 260 280 300 320 T f (K ) sputter power (W) M gAl 2 O 4 (b) T (K) M (k A /m ) 60 W ZnFe 2 O 4 /Al 2 O 3 FC ZFC T f Al 2 O 3 T f (K ) sputter power (W) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 3: M(T ) curves recorded at 10 mT under field cooled (FC, full symbols) and after zero-field cooled (ZFC, open sym- bols) conditions shown for the 60 W ZnFe2O4 grown on (a) MgAl2O4 (black squares) and (b) Al2O3 (red circles) sub- strates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The insets show the dependence of Tf on the sput- tering power for both substrates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' [9, 17, 20, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Only in few cases the Tf is found at such elevated temperatures [7, 19] and a controllable shift of Tf is only reported in [7] so far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' however, the drop in Tf with decreasing deposition rate in [7] is by a factor of two more pronounced compared with the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that the power series was grown by varying the sputter power nonmonotonously so that a dependence of Tf on the growth sequence—and thus target degradation—can be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In a second step, the dependence of Tf on other growth parameters shall be briefly summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 4(a) compiles the sample series as a function of the growth temperature Tgrowth from RT up to 550 °C for ZnFe2O4 grown on MgAl2O4 at a sputtering power of 60 W and an Ar:O2 ratio of 10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XRD shows no signifi- cant changes for Tgrowth ≥ 300 °C while the sample at RT appears to be virtually amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The inset shows MS at 300 K and Tf as determined from SQUID mea- surements analogous to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Tf is found to be about constant around 250 K with a slight tendency to decrease for higher Tgrowth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' the only exception is the 10 1 10 2 10 3 10 4 10 5 10 6 10 7 41 42 43 44 45 41 42 43 44 45 10 1 10 2 10 3 10 4 10 5 10 6 10 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='4 100 150 200 250 300 0 200 400 50 100 150 200 250 300 2 (�) in te n s ity (a r b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' u n its ) T grow th 550�C 500�C 450�C 400�C 300�C 200�C RT MgAl 2 O 4 (004) ZnFe 2 O 4 (004) (a) (b) MgAl 2 O 4 (004) ZnFe 2 O 4 (004) in te n s ity (a r b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' u n its ) 2 (�) Ar:O 2 (sccm) 10:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 10:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='35 10:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='25 10:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='15 10:0 O 2 flow (sccm) M s (k A /m ) / T f (K ) M s (k A /m ) / T f (K ) T grow th (�C) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 4: (a) Structural and resulting magnetic properties as a function of the growth temperature Tgrowth for ZnFe2O4 grown on MgAl2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The dependence of the identical param- eters as a function of the Ar:O2 ratio is shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Ms (black squares) and Tf (red circles) shown in the insets were extracted from SQUID measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' amorphous sample at RT where Tf is clearly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, MS steadily increases with increasing Tgrowth which can be taken as an indication for an increasing amount of inversion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', of Fe3+ T d in analogy with [7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' for the present sample series this increasing MS with Tgrowth has no obvious influence on the observed Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This trend is opposite to the annealing series in [9], where the amount of inversion and thus resulting MS is decreasing with increasing annealing temperatures for nanopowdered ZnFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that increasing Tgrowth in epitaxial growth typically leads to a decrease in ac- tual thickness compared to the nominal one which would lead to a decrease in MS which was calculated from the nominal thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' On the other hand, this increase in MS can also be associated with an increase in the order temperature which is in all cases above 400 K and thus beyond the accessible temperature range of the SQUID magnetometer and thus unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A second ZnFe2O4 sample series was grown on MgAl2O4 at a sputtering power of 60 W and a fixed 6 Tgrowth of 450 °C while varying the Ar:O2 ratio which is compiled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XRD of all samples does not show any significant changes with increasing oxygen content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In the inset the resulting MS at 300 K and Tf are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' While MS is independent on the Ar:O2 ratio within error bars, Tf does not show a conclusive trend, mostly because of a rather high Tf for the sam- ple at Ar:O2 ratio of 10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Disregarding this, a faint increase within errorbars may be inferred but there is no pronounced dependence of Tf on the Ar:O2 ratio, es- pecially if this is compared with the dependence on the growth rate shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This finding is rather in- teresting, since in [7] the growth rate has been associated with a deficiency in oxygen leading to an increase in Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, all samples were found to be highly resistive above the GΩ-range (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, a signif- icant amount of oxygen vacancies can be ruled out for the entire series because the two samples grown without and with maximum oxygen partial pressure have virtu- ally identical physical properties where the high oxygen partial pressure in RMS should safely rule out any oxygen deficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In turn, the dependence of Tf on the growth rate, which is consistently found in [7] and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 3(a), can- not depend on the existence of oxygen vacancies for RMS grown ZnFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' To summarize this part, the two sample series shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 4 underline, that the relevant prepa- ration parameter to control Tf is the sputtering power and thus growth rate, while Tgrowth and the Ar:O2 ratio play a minor role in the resulting magnetic properties, in particular, Tf is not directly controllable via oxygen va- cancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, in the following only samples grown at Ar:O2 ratio of 10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 and Tgrowth = 450 °C, as those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 1 to 3, will be discussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In a next step, the actual type of magnetic order shall be determined because the reports in the literature range from ferro(i)magnetism, over superparamagnetism, to a cluster glass or spin-glass behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' As pointed out above, MS for the present set of samples is found to be consistent with most of the reports found through- out the literature, while the bifurcation at Tf is found at rather elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 5(a) shows the M(T ) behavior recorded under FC conditions (full sym- bols) as well as after ZFC conditions (open symbols) for the ZnFe2O4 grown at 80 W on MgAl2O4 for an external field of 5 mT (black squares) and 10 mT (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' As expected Tf increases with decreasing external magnetic field which is consistent with both spin-freezing as well as superparamagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Reducing the external field further to fields of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 mT), where most of the spin glass experiments are typically carried out [25], Tf gets very close to the maximum attainable temperature of 400 K of the SQUID magnetometer (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, all subsequent experiments are only carried out at fields of 5 mT or 10 mT to keep the maximum achievable temper- ature well above Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that unfortunately the SQUID does not allow to go above the magnetic order tempera- ture which is in all cases above 400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' To get a first estimate on the existence of magnetic 100 200 300 400 10 20 30 40 50 0 20 40 60 80 0 100 200 300 400 T w ait T w ait T w ait T w ait (b) M ( e m u ) T (K) FC (10 mT) IS FH (10 mT) t wait = 10 4 s (0 mT) 80 W ZnFe 2 O 4 /MgAl 2 O 4 T f T (K) M ( e m u ) FC (5 mT) ZFC (5 mT) FC (10 mT) ZFC (10 mT) (a) 80 W ZnFe 2 O 4 /MgAl 2 O 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 5: (a) M(T ) curves of the 80 W ZnFe2O4 grown on MgAl2O4 recorded under field cooled (FC, full symbols) and after zero-field cooled (ZFC, open symbols) conditions at 5 mT (black squares) and 10 mT (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' (b) M(T ) curves recorded while cooling at 10 mT (FC) with intermit- tent stops (IS, open squares) with various Twait with twait of 10,000 s marked by arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The M(T ) curve while warming is subsequently recorded at 10 mT (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' glassiness, the FC memory sequence used in [9] was car- ried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 5(b) shows the FC M(T ) curve recorded at 10 mT with intermittent stops (IS, open symbols) at various Twait which are marked with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Here the field was reduced to 0 mT for a waiting time twait of 10,000 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Then the field was set to 10 mT again and the cooling down is resumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Subsequently M(T ) is measured at 10 mT while heating at the same rate as during FC without any IS (FH, full line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In the FC curve clear steps can be seen for most Twait which are most pronounced just below the maximum of the ZFC curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 5(a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', also below Tf, while they are vir- tually absent above Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 5 we show the magnetic data in emu to demonstrate the absolute size of the steps in comparison to the detection limit of the SQUID of 2 − 4 · 10−7 emu [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' These steps demonstrate magnetic relaxation during twait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' More im- portant, the subsequent FH curve shows clear inflection points around Twait, and the inset enlarges the two most 7 prominent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, there is a first experimen- tal evidence for magnetic glassiness in epitaxial ZnFe2O4 analogous to ZnFe2O4 nanopowders in [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' however, this glassiness extents to rather high temperatures which are only comparable to those reported in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A caveat is still, that such a behavior can also be observed and modeled in superparamagnetic samples as discussed in detail in [26] where only subtleties in these type of FC memory sequences allow to distinguish a superparamagnet from a superspin-glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore ZFC memory experiments are needed in addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 6 provides additional experimental evidence for magnetic glassiness of the same sample by ZFC memory experiments adopted after [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Here the sample is cooled down under ZFC conditions once without any waiting time and once cooling is stopped at Twait = 270 K for varying waiting times twait from 500 s to 50,000 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Sub- sequently, an M(T ) curve is measured at 5 mT while warming (FH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(a) the difference ∆M between the FH without and with Twait is plotted for all twait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that ∆M is provided in emu and the visible scat- ter in the difference data is around 1 − 2 · 10−8 emu, which demonstrates the high reproducibility of the data recorded with the SQUID magnetometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It should be stressed that this is only possible if the magnet is reset before the measurement to eliminate any trapped flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In all subsequent measurements one has to avoid magnetic fields larger than 10 mT so that the nominal and actual field are identical for all measurements within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 mT be- tween which ∆M is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For twait of 500 s and 1,000 s an increase of ∆M below Tf is visible with a maximum around the maximum of the ZFC M(T ) curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', ∆M follows the shape of the ZFC curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, the max- imum of the ZFC curve for the given experimental con- ditions is around 300 K while the maximum of the ∆M- curve is around 225 K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', shifted to lower temperatures and does not go back to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This low temperature in- crease of ∆M is difficult to be explained in a straight- forward manner, because the nominally ZFC conditions only correspond to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 mT [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore the difference between the ZFC with an without twait of 500 s at 270 K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', below Tf implies that the system is allowed to spend additional 500 s close to the freezing tempera- ture in a tiny, but finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' If one considers a super- paramagnetic ensemble close to its blocking temperature this implies more time for thermally activated switching in a tiny field which imprints a tiny additional magnetiza- tion because the residual field induces a slight imbalance in the probability of switching parallel and antiparallel to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A superparamagnetic ensemble would further imply relatively fast characteristic time-scales for the switching attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This would be in accordance that ∆M with twait of 500 s and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='000 s are virtually identical, because all the switching events are already done, while without twait the system is ramped through the blocking temper- ature with a rate of 60 s/K, so much less switching events can take place around the blocking temperature, where the tiny residual field is sufficient to aid the thermally 100 150 200 250 300 350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='4 100 150 200 250 300 350 (b) t wait = 10,000 s M ( e m u ) T (K) 320 K 300 K 280 K 260 K 240 K 220 K 200 K 160 K T w ait = 80 W ZnFe 2 O 4 /MgAl 2 O 4 t w ait = T (K) FH (5 mT) af ter ZFC in 0 mT M ( e m u ) 500 s 1,000 s 5,000 s 10,000 s 50,000 s T wait = 270 K (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6: Characteristic hole-burning experiment for the 80 W ZnFe2O4 sample grown on MgAl2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' (a) shows the depen- dence on the waiting time twait for a fixed waiting tempera- ture Twait of 270 K while (b) shows the dependence on Twait for a fixed twait of 10,000 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' activated switching events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The remaining low tempera- ture increase is thus the frozen-in result of more switching events close to Tf resulting in a waiting-time imprinted additional magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This is further corroborated by the fact, that the low-temperature increase is found to decrease with decreasing Twait, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', a waiting further below Tf and thus in a region with potentially slower dynamics, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' We thus infer that this low tem- perature increase of ∆M is most likely to be indicative of a superparamagnetic-like behavior with relatively fast dynamics rather than classical rejuvenation effects in su- perferromagnets as discussed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Beyond this low-temperature increase of ∆M seen for all twait in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(a), there is a minimum evolving with increasing twait becoming clearly visible at 5,000 s and being most pronounced at 50,000 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This is a typical char- acteristic of a (super)spin glass as discussed in [25, 26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' however, the minimum is shifted to lower temperatures compared to Twait by about 20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This shift is also seen, irrespective of the actual Twait, see also Fig 7(a) further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 6(b) shows ∆M curves for a fixed twait of 8 100 150 200 250 300 350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 100 150 200 250 300 350 (b) t wait = 10,000 s M ( e m u ) T (K) 300 K 280 K 260 K 240 K 220 K 200 K 160 K T w ait = 80 W ZnFe 2 O 4 /MgAl 2 O 4 t w ait = T (K) FH (5 mT) af ter ZFC in 0 mT M ( e m u ) 1,000 s 5,000 s 10,000 s 50,000 s T wait = 270 K (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 7: Identical set of data as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6 for the 80 W ZnFe2O4 sample grown on MgAl2O4 for (a) varying twait for Twait = 270 K (magenta dahed line) and (b) varying Twait for twait =10,000 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' however, ∆M is taken differently (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 10,000 s for various Twait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For Twait above Tf no minimum is visible and only the low temperature increase can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, a clear minimum is observable which is strongest for Twait of 280 K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', close to Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For lower Twait is becomes less pronounced and the minimum shifts to lower temperatures, which are however always below the respective Twait, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', the minimum in ∆M for Twait of 160 K is at 150 K (orange pentagons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, the 80 W ZnFe2O4 sample grown on MgAl2O4 shows all the experimental characteristics of magnetic glassiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' On the one hand a FC memory effect characteristic of su- perparamagnets and (super)spin glasses [26] which have been reported for ZnFe2O4 before [9], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' On the other hand, a twait-dependent minimum is observed, which is known as hole-burning experiment [25] and is absent in superparamagnets but is seen in (super)spin- glasses [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' On the other hand, the low-temperature increase of ∆M for short twait or Twait above Tf resemble more of superparamagnetic-like behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, we will show in the following that superparamagnetic-like behavior and magnetic glassiness coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 7 provides an alternative way of presenting the identical results as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In this case ∆M is taken as the difference between all data with respect to (a) twait =500 s and (b) Twait = 320 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In other words, here ∆M(T ) should only contain the magnetic glassiness since the superparamagnetic behavior—which is reflected by the low temperature increase of ∆M seen for short twait in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(a), or twait well above Tf in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(b)— is subtracted and thus only the slow, glassy dynamics can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 7(a) reveals that ∆M(T ) of twait of 500 s and 1,000 s are virtually identical, since only a zero line is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In other words, the fast dynamics of the superparamagnet are over while the slow dynamics of the glassiness have not yet set in, both referring to the experimental accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Having thus subtracted the fast dynamics the evolving dip in ∆M(T ) with twait of 5,000 s and higher nicely represents the remaining mag- netic glassiness with its characteristic slow dynamics and ZFC memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It is furthermore visible that the minimum in ∆M(T ) does not align with Twait which is indicated by the dashed magenta line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Twait merely appears to align with the inflection point of the high-temperature side of ∆M(T ) which is also seen in the Twait-dependence of ∆M(T ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Also here, the low-temperature increase of ∆M seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(b) is fully removed by taking ∆M always with respect to Twait = 320 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 7(b) Twait = 300 K is not nicely visible but the dip in ∆M(T ) is very weak and clearly less pronounced compared to the others and in fact may only reflect the limits of reproducibility of these types of SQUID experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' one should keep in mind that two ZFC M(T ) curves like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 5(a) are sub- tracted from each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', the signal size and thus the relative accuracy of each data point varies (slightly) over the entire T -range which can easily affect difference sig- nals of the order of 1·10−7 emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Nevertheless, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6 and 7 nicely demonstrate that in ZnFe2O4 superparamagnetic and glassy behavior coexist and can be separated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This is quite remarkable, since an epitaxial film of ZnFe2O4 is structurally quite distinct from a su- perparamagnetic ensemble like horse-spleen ferritin or a superspin-glass like a dense ensemble like Fe3N nanopar- ticles which were both investigated in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Yet, ZnFe2O4 epitaxial thin films exhibit both types of magnetic or- der at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, the observed magnetic glassiness appears to be better described in terms of a cluster glass like in [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', a superparamagnetic-like en- semble with (frustrated) intercluster interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' How- ever, these interactions have to be inhomogeneous and disordered throughout the sample and in contrast to the nanopowder in [9] they have no obvious structural origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' One has to therefore conclude that they stem from local variations of the cation distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', from chemical or A/B disorder and thus they crucially depend on a fi- nite amount of inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This in turn also explains why highly crystalline bulk ZnFe2O4 samples in [1–3] exhibit quite distinct magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Since we have seen that Tf is a function of the growth power during the sputtering process, the two power se- 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 100 200 300 400 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 t wait = 10,000 s T (K) M (k A /m ) sputter power 100 W 80 W 60 W 40 W 20 W MgAl 2 O 4 (a) t wait = 10,000 s (b) M (k A /m ) T (K) sputter power 80 W 60 W 40 W 20 W Al 2 O 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 8: Comparison of the hole-burning experiments for ZnFe2O4 grown on MgAl2O4 (a) and Al2O3 (b) as a func- tion of sputter power (details see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The insets show the high sputter power samples only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' ries of ZnFe2O4 samples grown on MgAl2O4 and Al2O3 shall be directly compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For that we have chosen to perform the hole-burning ZFC waiting experiments of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(b) on the identical relative temperature scale for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In other words, the highest and lowest temperature of the M(T ) curves as well as Twait have been chosen to be a the same relative temperature with respect to Tf to assure that the samples spent compa- rable time-spans in regions with comparable magnetiza- tion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Note, that in addition the full experi- ment for all Twait of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 6(b) on an absolute temper- ature scale have also been performed (not shown), but the direct comparison in essence reveals the identical re- sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 8 shows the ∆M curves for the power-series of ZnFe2O4 grown on MgAl2O4 (a) and Al2O3 (b) for twait of 10,000 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' the insets enlarge the samples grown at high sputtering powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Irrespective of the substrate the samples grown at sputtering powers of 20 W and 40 W do only show the low-temperature increase of ∆M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', mostly superparamagnetic-like behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' for ZnFe2O4 on Al2O3 a faint and broad minimum is visible which how- ever does not show a clear shift with Twait or a pro- nounced dependence with twait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, we consider 0 1 700 705 710 715 720 725 730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='4 700 705 710 715 720 725 730 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='4 Fe L 3/2 edges T = 300 K 20� grazing (a) photon energy (eV) n o r m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' X A N E S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 80 W 20 W X M C D ZnFe 2 O 4 /MgAl 2 O 4 36% Fe 2+ Oh /32% Fe 3+ Oh /33% Fe 3+ Td 32% Fe 2+ Oh /38% Fe 3+ Oh /30% Fe 3+ Td Fe L 3/2 edges T = 300 K 20� grazing ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' (b) n o r m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' X A N E S photon energy (eV) 80 W 20 W X M C D 34% Fe 2+ Oh /48% Fe 3+ Oh /18% Fe 3+ Td 28% Fe 2+ Oh /28% Fe 3+ Oh /44% Fe 3+ Td ZnFe 2 O 4 /Al 2 O 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 9: Normalized XANES and XMCD spectra recorded at the Fe L3/2-edges under grazing incidence at 300 K for the 80 W and 20 W ZnFe2O4 samples grown on (a) MgAl2O4 and (b) Al2O3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XMCD spectra have also been simulated to determine the relative amount of the individual Fe species (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' this part as inconclusive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', not as clear experimental evidence for glassiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, the samples grown at 60 W and higher all show a hole-burning behavior in the ZFC memory experiments which is pronounced for ZnFe2O4 on MgAl2O4, see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 8(a), but rather weak for ZnFe2O4 on Al2O3, for which only a faint min- imum can be seen, see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Therefore, in ZnFe2O4 on Al2O3 only superparamagnetic-like can be inferred and signatures of magnetic glassiness are faint and limited to high sputtering powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This goes hand- in-hand with a more pronounced maximum in the ZFC curves, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 3(b) and an increased magnetization, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, ZnFe2O4 on MgAl2O4 exhibits a clear transition from superparamagnetic-like behavior at low sputtering powers with clear signs of magnetic glassiness existing at high sputtering powers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' To ultimately clarify what causes the discrep- ancy in the magnetic properties for ZnFe2O4 grown on MgAl2O4 and Al2O3 as well as at low and high sputtering powers, the 20 W and the 80 W samples were subjected to an element-selective magnetic characterization using XMCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 10 Figure 9 shows the measured XANES and XMCD spec- tra at the Fe L3/2-edges for ZnFe2O4 grown on MgAl2O4 (a) and Al2O3 (b) for the samples grown at 20 W and 80 W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XMCD at the Fe L3-edge has been also simulated by respective multiplet ligand field theory using the CTM4XAS code using the identical parameters as in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In brief, the negative peaks in the XMCD spectrum are stemming from the octahedral contributions FeOh, where Fe2+ Oh is mostly seen at lower (706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='6 eV) and Fe3+ Oh at higher (708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='5 eV) photon ener- gies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' the positive peak at 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='8 eV can be assigned to Fe3+ T d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The experimental XMCD can be reproduced by adjusting the relative concentrations of Fe3+ Oh, Fe2+ Oh, and Fe3+ T d to match the experimental XMCD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' the results of this are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 9(a), that there are no pronounced differences between the exper- imental XMCD spectra of the Fe L3/2-edge XMCD for ZnFe2O4/MgAl2O4 grown at either 20 W or 80 W as well as for the respective results of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' About one third of the Fe is located on tetrahedral sites, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', the de- gree of inversion δ is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='3 for both sputtering pow- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Also a significant amount of Fe2+ Oh is found, which would suggest a strong contribution from a JDE BB double exchange interaction which appears to be slightly larger for the 80 W sample which exhibits the magnetic glassi- ness in comparison to the 20 W sample, which only shows the superparamagnetic-like low temperature increase of ∆M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The ZnFe2O4/Al2O3 samples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 9(b) exhibit a different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Here the 80 W sample has a strongly reduced contribution of Fe3+ T d compared to the FeOh com- pared to the 20 W sample which has the highest relative content of Fe3+ T d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' On the other hand, the actual positive peak in the XMCD is of identical size in both samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It therefore appears, the XMCD intensity for the FeOh is reduced while the amount of Fe3+ T d remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This may appear as contradiction at first sight, since the rel- ative contents may suggest different degrees of inversion for the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, one should keep in mind that the magnetic superexchange interaction on the octa- hedral sites JBB is weakly antiferromagnetic while double exchange leads to spin canting [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Since the magnetic order is observed up to above room temperature for all samples in this work, the JAB superexchange mechanism has to play a significant role, which is consistent with a fi- nite degree of inversion of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In that light, the presence of a finite amount of inversion giving rise to Fe3+ T d is a prerequisite for magnetic order at elevated temperatures but does not play a decisive role for the presence of magnetic glassiness, since Fe3+ T d is found in all four samples while glassiness is only found in the 80 W samples, in particular in those grown on MgAl2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In addition, the presence of Fe2+ Oh in all samples further sug- gests the presence of an additional JDE BB double exchange mechanism associated with spin canting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Here the rela- tive amount of Fe2+ Oh increases only slightly from the 20 W sample on Al2O3 over 20 W on MgAl2O4, 80 W on Al2O3 to 80 W on MgAl2O4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', it follows the trend of in- creasing glassiness of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' However, the changes 1010 1020 1030 1040 1050 1060 0 1 2 0 2 4 6 0 1 1010 1020 1030 1040 1050 1060 2 0 2 4 6 n o r m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' X A N E S photon energy (eV) 20W ZnFe 2 O 4 /Al 2 O 3 80W ZnFe 2 O 4 /Al 2 O 3 X M C D (% ) 20W ZnFe 2 O 4 /MgAl 2 O 4 Zn L 3/2 edges T = 300 K 20� grazing 80W ZnFe 2 O 4 /MgAl 2 O 4 (b) photon energy (eV) n o r m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' X A N E S (a) Zn L 3/2 edges T = 300 K 20� grazing X M C D (% ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 10: Normalized XANES and XMCD spectra recorded at the Zn L3/2-edges under grazing incidence at 300 K for the 80 W and 20 W ZnFe2O4 samples grown on (a) MgAl2O4 and (b) Al2O3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' are rather small and the significance of determining such small changes with multiplet ligand field simulations is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Obviously, there is no straightforward mech- anism for the occurrence of magnetic glassiness which can be derived form the XMCD spectra at the Fe L3/2- edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A finite degree of inversion has to play a role but mostly for the high order temperatures observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The existence of glassiness appears to be linked to a delicate balance of the various competing exchange interactions as well as the local cationic configuration which has to be inhomogeneous throughout the sample as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Obviously the growth rate influences mostly the latter where the highest growth rates favor glassiness, most likely via increased local cationic disorder, in par- ticlular in ZnFe2O4 on MgAl2O4 substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Figure 10 shows the measured XANES and XMCD spectra at the Zn L3/2-edges of the identical set of sam- ples as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The XANES for ZnFe2O4 on MgAl2O4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 10(a) is rather similar to the one of ZnFe2O4 on Al2O3 in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' All four samples exhibit a finite XMCD with comparable spectral shape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' all XMCD spectra were derived by reversing both, helicity of the light as well as the magnetic field and it was verified that the XMCD spectrum nicely reverses with reversing external field (not 11 shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The size of the Zn L3/2-edge XMCD follows the amount of Fe3+ T d as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 9, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=', the Zn XMCD is largest for the 20 W sample on Al2O3, which has the highest relative Fe3+ T d content and it is lowest for the 20 W sample on Al2O3 which has the lowest relative Fe3+ T d con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' It is thus reasonable to assume that the magnetic polarization of Zn in ZnFe2O4 is mostly associated with Zn2+ Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In turn, this implies that a weakly polarized cation substitutes for a strongly polarized one thus reducing the effective exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' This is consistent with the exper- imental observation that the 80 W ZnFe2O4/MgAl2O4 has the highest Tf and the lowest magnetic polarization of Zn while the highest Zn polarization in the 20 W ZnFe2O4/Al2O3 sample is associated with the lowest Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' To verify this hypothesis, more sophisticated theoretical calculations beyond the multiplet ligand field codes is re- quired where the individual spectroscopic signatures in the Zn L3/2-edge XANES and XMCD can be associated with the actual Zn species which however goes beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Nevertheless, it is already evident that a too high degree of inversion as seen by strong mag- netic polarization of the Zn together with a high relative content of Fe3+ T d is unfavorable for both, high Tf as well as magnetic glassiness and high growth rates appear to be an experimental means to control/limit excessive inver- sion but at the same time assure sufficient local cationic disorder to induce magnetic glassiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION ZnFe2O4 epitaxial thin films have been grown on MgAl2O4 and Al2O3 substrates with varying preparation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' All samples were investigated with respect to their basic structural and magnetic properties and long range magnetic order was found above room tempera- ture for all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The stoichiometric composition of the samples was verified using RBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A clear bifurcation between M(T ) curves under FC and ZFC conditions is found at Tf, which is systematically higher for ZnFe2O4 on MgAl2O4 by about 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Tf is found to systemati- cally increase with increasing the sputtering power and thus growth rate in agreement with [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The Ar:O2 ratio was not found to influence neither Tf nor Ms in a system- atic manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' increasing Tgrowth increases only Ms while Tf exhibits no systematic changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' An in-depth study of the magnetic properties us- ing FC as well as ZFC memory experiments reveals magnetic glassiness for samples grown at high sput- ter powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The glassiness is more pronounced for ZnFe2O4/MgAl2O4 compared to ZnFe2O4/Al2O3, where the signatures of magnetic glassiness beyond those in FC memory experiments are generally weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' At lower growth rates the signatures of glassiness are absent and a low-temperature increase of ∆M is observed which points towards superparamagnetic-like behavior and the signa- tures in FC memory experiments are weak and the ZFC memory experiments shows no hole burning effect in ac- cordance with the expectations for superparamagnetic samples [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, at high growth rates, in par- ticular for ZnFe2O4/MgAl2O4 ZFC memory experiments show an additional hole burning effect which is charac- teristic for spin glasses [25] and superspin glasses [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Since the glassiness coexists with superparamagnetic-like signatures, in particular, the low temperature increase of ∆M, the structural properties of the ZnFe2O4 epitaxial films are quite different from the nanoparticle ensembles in [26] the observed magnetic properties are described best as cluster glass in analogy to comparable observa- tions for epitaxial ZnFe2O4 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' An in-depth characterization based on XANES and XMCD reveals that a finite magnetic polarization at the Zn L3/2 edges exists in all ZnFe2O4 samples which adds more complexity to the magnetic interactions beyond the usually discussed Fe-based exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' At the Fe L3/2 the XMCD is used to extract the relative concentrations of Fe3+ T d, Fe3+ Oh, and Fe2+ Oh by means of multiplet ligand field simulations as done before [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The abundance of Fe3+ T d correlates well with the size of the magnetic polar- ization of Zn and thus both can serve as a measure for the degree of inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' For highest inversion Tf is found to be lowest and signatures of magnetic glassiness are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In contrast, the sample with the strongest signatures of magnetic glassiness, the 80 W ZnFe2O4/MgAl2O4, ex- hibits no significant changes in the Fe L3/2-edge XMCD compared to the superparamagnetic-like 20 W sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The most prominent tendency appears to be the rel- ative amount of Fe2+ Oh which can be associated with a double-exchange mechanism which was held responsible for spin canting [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Since the differences in XMCD be- tween the respective samples are small, this correlation between glassiness and Fe2+ Oh cannot be taken as signifi- cant but merely a starting point for more elaborate the- oretical work to understand the details of the obtained XMCD spectra beyond the multiplet ligand field simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Most likely the cluster glass behavior in epitaxial ZnFe2O4 cannot be assigned to the actual structure of the materials like in common superparamagnets or dense nanoparticle ensembles [26] but due to local variations of the stoichiometry, leading to an inhomogeneous local cation distribution throughout the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' As a con- sequence, the local magnetic moments in ZnFe2O4 are disordered due to partial inversion and partially canted due to the presence of Fe2+ Oh, which leads to characteristic signatures of a cluster glass at rather high temperatures which is mostly controllable by the growth rate as re- ported for ZnFe2O4 before [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Acknowledgments J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' gratefully acknowledges funding by FWF project ORD-49 at the initial stage of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' acknowl- edges the financial support by the Swiss National Science Foundation (SNSF) under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 200021-169467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The X-ray absorption measurements were performed on 12 the EPFL/PSI X-Treme beamline at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' In addition, support by VR-RFI (Contracts No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2017-00646 9 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 2019-00191) and the Swedish Foundation for Strategic Research (SSF, Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' RIF14-0053) sup- porting accelerator operation at Uppsala University is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' The research leading to this re- sult has been supported by the RADIATE project under the Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' 824096 from the EU Research and Innovation programme HORIZON 2020.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' B 79, 134405 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Zviagin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Grundmann, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Schmidt-Grund, physica status solidi (b) 257, 1900630 (2020).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Heluani, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' B 84, 064404 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Yamamoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Tanaka, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Lorenz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Ziese, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Esquinazi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Grundmann, Thin Solid Films 527, 273 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFIT4oBgHgl3EQffCtm/content/2301.11277v1.pdf'} +page_content=' Guo, C.' metadata={'source': 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N. Appleton,1 P. Guillard,2, 3 Bjorn Emonts,4 Francois Boulanger,5 Aditya Togi,6 William T. Reach,7 +Kathleen Alatalo,8 M. Cluver,9, 10 T. Diaz Santos,11 P-A Duc,12 S.Gallagher,13 P. Ogle,8 E. O’Sullivan,14 +K. Voggel,12 and C. Xu15, 16 +1Caltech/IPAC, MC 314-6, 1200 E. California Blvd., Pasadena, CA 91125, USA. apple@ipac.caltech.edu +2Sorbonne Universit´e, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98bis bd Arago, 75014 Paris, France +3Institut Universitaire de France, Minist`ere de l’Enseignement Sup´erieur et de la Recherche, 1 rue Descartes, 75231 Paris Cedex 05, +France +4National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903 +5UPMC Universite Paris 06, ´Ecole Normale Sup´erieure, 75005 Paris, France +6Texas State University, 601 University Dr, San Marcos, TX 78666, USA +7Universities Space Research Association, NASA Ames Research Center MS 232-11, Mountain View, CA 94035, USA +8STScI, 3700 San Martin Drive, Baltimore, MD 21218 +9Centre for Astrophysics and Supercomputing, Swinburne University of Technology, John Street, Hawthorn, 3122, Australia +10Department of Physics and Astronomy, University of the Western Cape, Robert Sobukwe Road, Bellville, 7535, South Africa +11Institute of Astrophysics, Foundation for Research and Technology-Hellas (FORTH), Heraklion, 70013, Greece, and School of Sciences, +European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus. +12Universit´e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg (ObAS), UMR 7550, 67000 Strasbourg, France +13Institute for Earth and Space Exploration, Western University, 1151 Richmond St., London, ON N6A 3K7, Canada +14Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA +15Chinese Academy of Sciences South America Center for Astronomy, National Astronomical Observatories, CAS, Beijing 100101, +People’s Republic of China +16National Astronomical Observatories, Chinese Academy of Sciences (NAOC), 20A Datun Road, Chaoyang District, Beijing 100101, +People’s Republic of China +Abstract +We combine JWST and HST imaging with ALMA CO(2-1) spectroscopy to study the highly tur- +bulent multi-phase intergalactic medium (IGM) in Stephan’s Quintet on 25-150 pc scales. Previous +Spitzer observations revealed luminous H2 line cooling across a 45 kpc-long filament, created by a gi- +ant shock-wave, following the collision with an intruder galaxy NGC 7318b. We demonstrate that the +MIRI/F1000W/F770W filters are dominated by 0-0 S(3) H2 and a combination of PAH and 0-0 S(5) H2 +emission. They reveal the dissipation of kinetic energy as massive clouds experience collisions, interac- +tions and likely destruction/re-cycling within different phases of the IGM. In one kpc-scaled structure, +warm H2 formed a triangular-shaped head and tail of compressed and stripped gas behind a narrow +shell of cold H2. In another region, two cold molecular clumps with very different velocities are con- +nected by an arrow-shaped stream of warm, probably shocked, H2 suggesting a cloud-cloud collision is +occurring. In both regions, a high warm-to-cold molecular gas fraction indicates that the cold clouds +are being disrupted and converted into warm gas. We also map gas associated with an apparently +forming dwarf galaxy. We suggest that the primary mechanism for exciting strong mid-IR H2 lines +throughout Stephan’s Quintet is through a fog of warm gas created by the shattering of denser cold +molecular clouds and mixing/recycling in the post-shocked gas. Without spectroscopy, JWST cannot +provide a complete picture of the kinematics and excitation of the shocked warm gas, but it reveals +the rich variety of ways that different gas phases interact with one another in Stephan’s Quintet. +1. INTRODUCTION +Since the discovery of a filament of radio continuum +in the intergalactic medium (IGM) of the Stephan’s +Quintet (Allen & Hartsuiker 1972), this compact galaxy +group has been studied at many wavelengths to try to +better understand that remarkable nature of its multi- +phase intergalactic medium. As suspected in the early +studies (Moles et al. 1997; Sulentic et al. 2001; Xu et al. +1999), the overall picture that has emerged is that one +of the galaxies, NGC 7318b, is colliding with the diffuse +arXiv:2301.02928v1 [astro-ph.GA] 7 Jan 2023 + +2 +Appleton et al. +intergalactic medium of the main group at a very high +velocity creating a giant (45 kpc) filament of shocked gas +seen from the X-rays (Trinchieri et al. 2005; O’Sullivan +et al. 2009), in the UV (Xu et al. 2005) and in ionized gas +emissions (Xu et al. 2003; Iglesias-P´aramo et al. 2012; +Konstantopoulos et al. 2014; Duarte Puertas et al. 2019, +2021). Studies of the stellar populations along the giant +shocked structure also suggested that star clusters are +beginning to form there (Gallagher et al. 2001; Fedotov +et al. 2011), although at a low rate, perhaps because +some of the forming clusters lie inside bubbles of highly +excited ionized gas (Konstantopoulos et al. 2014). +The physical conditions within the gas along the main +shock front are far from simple and require more de- +tailed study. Mid-IR spectroscopy of the main filament +showed that the entire filament is radiating strongly in +pure rotational lines of molecular hydrogen (Appleton +et al. 2006; Cluver et al. 2010; See Figure 1). An expla- +nation of the excitation of such strong, L(H2) > 1041 erg +s−1, molecular hydrogen emission from a region experi- +encing fast shocks from the intruder’s high velocity was +presented by Guillard et al. (2009). The model assumes +that the main intruder shock propagates into a clumpy +pre-shock medium, heating low-density regions to X-ray +temperatures, but causing mildly over-dense regions to +collapse to form H2 on grains on timescales shorter than +the dynamical timescale of the collision. These molec- +ular clouds experience low-velocity molecular shocks as +they continue to dissipate mechanical energy from their +surroundings. The interaction of these warm H2 clouds +with their surroundings is a key area that we will in- +vestigate with the James Web Space Telescope (JWST) +and Atacama Large Millimeter Array (ALMA) in this +paper. +A more detailed analysis of the H2 excitation proper- +ties of the warm H2 in Stephan’s Quintet was discussed +in Appleton et al. (2017), where it was shown that in +order to heat so much H2 (109M⊙) at low temperatures +(150-400 K), a mix of low-velocity magnetic C-shocks +(∼5-10 km s−1) and faster (15-25 km s−1) J-shocks were +needed. +Very broad spectral linewidths have been observed in +the Stephan’s Quintet filament. Guillard et al. (2012) +used the IRAM 30m telescope to detect CO emission +from several regions in the main filament and the bridge, +and revealed line profiles of several hundred km s−1 in +at least three separate kinematic components. In ad- +dition to gas detected at the systemic velocity of the +intruder NGC 7318b (5774 km s−1), and that of the re- +maining group members (6600 km s−1), broad lines of +CO emission were also detected at intermediate veloci- +ties, suggesting that molecular gas has formed out of dif- +fuse shocked gas. Recent UV spectroscopic observations +with the Hubble Space Telescope’s (HST) Cosmic Ori- +gins Spectrograph (COS) targeted five regions along the +main filament and in the bridge, and detected extremely +broad Lyα emission (FWHM exceeding 1500 km s−1) +over small sampled regions of ∼1 kpc scale (Guillard +et al. 2022). The lines were even broader in some cases +than those seen in the [CII] line (Appleton et al. 2013) +using Herschel, suggesting some resonant scattering of +UV photons. The detection of broad-line, powerful Lyα +emission, broad [CII] neutral gas, and broad CO emis- +sion strongly supports the picture of a highly turbulent +medium containing many gas phases resulting from a +turbulent cascade of energy from the large to the small +scales. +Although the Early Release Observations (ERO) +made by JWST of Stephan’s Quintet cover the entire +inner group (including NGC 7319, NGC 7318a/b, NGC +7317 and the foreground galaxy NGC 7320), this pa- +per will concentrate on the main shocked filament and +bridge that were observed by Spitzer in spectral mapping +mode (Cluver et al. 2010; Appleton et al. 2017) using +the InfraRed Spectrograph (IRS; Houck et al. 2004). As +shown in Figure 1, this included the main north-south +H2 filament between NGC 7318b/a and NGC 7319, as +well as the H2 bridge connecting NGC 7319 to the main +filament. It is known that much of the gas, including HI, +ionized gas and molecular gas lies outside the main body +of the galaxies. Although the previous IRS observations +of the mid-IR lines covered much of this gas, it did not +cover all. The Long-low IRS coverage included all of the +area shown in Figure 1, but IRS Short-low coverage was +restricted to only the the main H2 filament and part of +the H2 bridge (see § 3.1). +Data from the JWST and ALMA allow us to directly +compare, for the first time at comparable resolution, the +distribution of warm 0-0S(3) H2 (as we will demonstrate +via the F1000W MIRI Band) to cooler molecular hydro- +gen mapped through the low-J CO(2-1) line, as well as +ionized gas emission (Hα with WFC3 HST, and Paα +with NIRCam F200W). Very hot gas, which forms an +important IGM component in Stephan’s Quintet is de- +tected in X-rays throughout Stephan’s Quintet, and es- +pecially in the main shock. Its distribution cannot be +compared in detail to our current observation, because +even the deepest Chandra images (O’Sullivan et al. 2009) +do not detect enough photons at arcsecond scales to al- +low meaningful comparison. +These observations will help us gain an understanding +of how turbulent energy, driven mainly by the intruder +galaxy, is dissipated from large driving scales to smaller +dissipation scales, and how this affects the cooling of + +The Multi-phase IGM in Stephan’s Quintet +3 +the gas through different gas phases and temperature +regimes. +In particular, how can we explain so much +energy flowing out through low-velocity shocks needed +to produce the high radiant line luminosity from warm +molecular hydrogen discovered by Spitzer. Although we +do not expect to probe down to the smallest dissipation +scales, we will show that we already see differences in +the distribution of different thermal phases in the IGM +at ∼150 pc scales probed by ALMA and JWST. +The current paper will not try to address all aspects +of the extended emission detected by JWST, but will +primarily concentrate on two main objectives. Firstly, +we compare the JWST MIRI images with the Spitzer +IRS spectra and spectral images, to identify the domi- +nant features present in the three MIRI filters (F770W, +F1000W and F1500W). Secondly, we compare the dis- +tribution of warm molecular gas, cold molecular gas and +ionized gas in three contrasting regions within the main +IGM between NGC 7319 and NGC 7318b/a. The three +regions we have chosen to emphasize in this paper were +observed with ALMA in CO(2-1). The primary beams +(fields-of-view) of the three ALMA pointings and their +associated field numbers are also identified on Figure 1.1. +We will the regions in more detail in §4. Although we +touch on the sparsity of recent star formation in two out +of three of the studied fields, we leave a full discussion +of the star formation properties and cluster formation +to a later paper. +In § 2 we will describe observations made with JWST, +HST, the Atacama Large Millimeter Array (ALMA) +and the Combined Array for Research in Millimeter- +wave Astronomy (CARMA). Given that we do not yet +have spectroscopy of the IGM in Stephan’s Quintet, we +present, in § 3 a discussion of what each of the MIRI +and NIRCam images is likely to contain by comparing +the JWST images with Spitzer spectral maps of espe- +cially warm H2 and PAH bands. In § 4 we present the +results of our zoomed-in study of three major emission +complexes in the intergalactic medium observed at high +resolution by ALMA in the CO (2-1) line. This allows +us to synthesize from JWST, ALMA, and HST, the best +possible information we have about the nature of the +warm and cold molecular gas, the ionized gas and the +formation of star clusters, all at subarcsec resolution. +§ 4 also includes a discussion of the relative fraction of +warm and cold molecular gas obtained for the regions +using both the JWST and ALMA data. In § 5 we dis- +cuss the results in the context of our understanding of +1 Other fields were originally proposed in the ALMA proposal but +only these three regions were actually observed +turbulence and shocks which we believe largely domi- +nate the gas dynamics of this multi-phase IGM. In § 6 +we present our conclusions. +We assume in this paper a distance to Stephan’s Quin- +tet of 94 Mpc (for H0 = 70 km s−1Mpc−1 and an +assumed group heliocentric systemic velocity of 6600 +km s−1) for consistency with previous work (e. g. Apple- +ton et al. 2017; Guillard et al. 2022). At this distance 1 +arcsec corresponds to a linear scale of 456 pc. The over- +all scale of the giant north-south intergalactic filament +in Stephan’s Quintet is 47 kpc. +2. OBSERVATIONS +2.1. JWST images +The first description of the NIRCam and MIRI +ERO observations, and the choice of Stephan’s Quin- +tet as a target (PID 2732) is presented in Pon- +toppidan et al. (2022). +The NIRCam images were +taken in 6 filters (F090W/F277W, F150W/F356W, +F200W/F444W), with an integration time of approxi- +mately 20 min for each band. The FULLBOX 5-points +“5TIGHT” dither pattern was used, resulting in a large +rectangle mosaic of 3 pairs of dithered tiles, covering +6.3 × 7.3 arcmin2. +The MIRI image covers a much +smaller field of view than the NIRCam mosaic, i.e. the +central galaxies, NGC7318a/b, NGC7319 (a Seyfert 2 +galaxy), and NGC7320 (foreground galaxy), using 4 tiles +in three MIRI bands, F770W, F1000W, and F1500W. +The integration times were about 22 min per filter. The +large cycling 8-points dither pattern was used to maxi- +mize the spatial coverage of a single tile. +We worked with the level 2b images retrieved from +MAST2. +2.2. ALMA Observations +Observations +of +CO(2-1) +in +the +three +fields +of +Stephan’s Quintet were made with the ALMA 12m ar- +ray in a mosaic observing mode on 21 July 2015 (ID: +2015.1.00241; PI: P. Guillard). +The total integration +time for the mosaic was 35 minutes. The spectral setup +consisted of four spectral windows of 1.875 GHz with 3.9 +MHz channels. One of the spectral windows was centred +on the redshifted CO(2-1) (νrest = 230.538 GHz), while +the remaining three spectral windows covered line-free +continuum emission. The standard ALMA calibration +plan included bandpass, phase, and flux calibration. +2 The data were obtained from the Mikulski Archive for Space +Telescopes at the Space Telescope Science Institute, which is op- +erated by the Association of Universities for Research in Astron- +omy, Inc., under NASA contract NAS 5-03127 for JWST. + +4 +Appleton et al. +Figure 1. +Stephan’s Quintet: A grey-scale representation of the NIRCam F150W ERO image of Stephan’s Quintet with +overlays of the Spitzer IRS image of the 0-0 S(1) H2 line (blue contours) and the half-power primary beam size of the ALMA +CO (2-1) observations (red circles) described in § 2.2. Contours of H2 emission are in units of 0.3, 0.53, 0.75, 0.98, 1.2, 1.43, +1.65, 1.88 and 2.1 MJy sr−1 from Cluver et al. (2010). +The data were calibrated with the scriptForPI.py cal- +ibration script that was included with the archival data, +using the ALMA calibration pipeline version r32491 that +is included with the Common Astronomy Software Ap- +plications (CASA) version 4.3.1 (CASA Team et al. +2022). After calibration, the continuum emission was +subtracted in the (u,v)-domain by fitting a straight line +to the line-free channels. +The line-data where subse- +quently imaged using the CLEAN algorithm to produce +data cubes with 20 km s−1 channels and a resolution of +0.36′′ × 0.21′′ (PA ∼ 10◦). The root-mean-square (rms) +noise in these data cubes is 0.5 mJy beam−1 chan−1. +A primary beam correction was applied to produce ac- +curate flux measurements. The half-power beam width +of the primary beam is 22.8 arcsec. Total intensity maps +of the CO(2-1) emission where made by integrating the +signal across the channels where line emission was de- +tected. +2.3. CARMA CO Observations +Observations of Stephan’s Quintet were made with +CARMA on 1 Aug 2010 (program c0593-PI. K. Alatalo) +using 15 antennae in the CO (1-0) line. The original data +was processed with MIRIAD (Sault et al. 1995). Data +reduction and imaging were done in identical fashion +to Alatalo et al. (2013). The original data was cleaned +(H¨ogbom 1974), which resulted in a restored synthe- +sized beam of 4.1 x 3.3 arcsec2. Velocities from extracted +spectra have been converted to heliocentric velocities as- +suming a shift of -11 km s−1 from lsrk to heliocentric. +Velocities are all quoted using the optical definition of +velocity. + +SQ-A +NGC 7319 + +H2 Bridge +Main H2 +Filament +Field 5 +Field 6 +ALMA CO (2-1) +Primary Beams 0-0 S(1) +H2 +NGC7318b/a +50arcsecsThe Multi-phase IGM in Stephan’s Quintet +5 +2.4. HST Archival Images +We de-archived calibrated images from the HST +MAST archive for filters F665W and F336W with the +WFC3/UV instrument. For F665N, the image encom- +passes the Hα line, and includes data based on five +dithered observations, each taken with integration times +of 5200s each in July and August 2009, and was obtained +as part of a WFC3 Early Release Observations cam- +paign (SM4/ERO). For F336W, the observations were +obtained as part of proposal id 12301 (P. I. = S. Gal- +lagher) on 2011-10-27, with a total integration time of +14474 s. The resolution of the WFC3/UV observations +is comparable with that achieved by NIRCam (0.05-0.06 +arcsec). For F665W and F336W, the expected FWHM +of a point source is 0.07 arcsec, and 0.075 arcsec respec- +tively, corresponding to ∼30 pc at D = 98 Mpc. +2.5. WCS Alignment of the JWST Images +The observations were obtained from the JWST and +HST MAST archive, and most of the images required +small adjustments to the WCS coordinates for proper +alignment. We initially used GAIA DR3 stars to align +one image (the HST WFC3 F665N) to the DR3 sys- +tem using the STScI software package WCSTweak. The +NIRCam images were also similarly aligned. However, +for MIRI, the same method did not produce good re- +sults, probably because of the smaller number of stars +used for alignment. +We used a refinement scheme to +solve this problem as we need good alignment between +all the images. This was especially important for one of +the target fields that required continuum subtraction of +the F1000W and F770W images with F1500W. First +we created small subimages of the F1000W, F770W +and F1500W images around the Field 6 area using +the NRAO software package CASA. The subimage con- +tained at least 3 GAIA DR3 stars close to the target of +interest. These subimages were then aligned more care- +fully by making small pixel-level shifts in each image +to bring the subimages into close alignment with each +other and the positions of the GAIA stars. We applied +the same method to Field 5 and 4 (see § 4). The fi- +nal results produced local MIRI images which aligned +to better than 1 MIRI pixel (0.3 arcsec) over the small +scale of the extracted regions. +The results were vali- +dated where the CO ALMA images aligned with clearly +related features. +Following Papadopoulos et al. (2008), the uncer- +tainty in the absolute astrometry of the ALMA data +is δθbas = (δB · ∆k)/B ≈ (δφbas/2π)⟨Θbeam⟩. The phase +error δφbas ∼ (2π/λ)(δB · δk), with |δB| ∼ 0.1mm the +assumed typical error in the baseline length, |δk| = 4.26◦ +the distance to the phase calibrator J2216+3518, +λ ∼ 1.33 mm the wavelength of the redshifted CO(2- +1) emission, and Θbeam ∼ 0.36′′ the major axis of the +synthesized beam. This results in δθbas ∼ 0.02′′, which +means that the uncertainty in the absolute astrometry +of the ALMA data is small compared to that of the HST +and JWST data3. +3. INTERPRETING THE EMISSION OBSERVED +THROUGH THE MIRI AND NIRCAM FILTERS +3.1. Comparison with Spitzer IRS spectral mapping in +H2 and PAH Bands +Under conditions found in the disks of normal galax- +ies, strong 7.7 µm and 8.6 µm Polycyclic Aromatic +Hydrocarbon (PAH) features are expected to domi- +nate the MIRI F770W filter for low-redshift galaxies. +In Stephan’s Quintet (Appleton et al. 2006; Guillard +et al. 2009; Cluver et al. 2010; Appleton et al. 2017), +much of the mid-IR spectrum of the IGM is dominated +by strong pure rotational H2 lines, and emission from +[SiII]λ34.8µm. +Unlike normal star formation regions, +the majority of the main IGM filament in Stephan’s +Quintet exhibits very weak 7.7µm PAH complex emis- +sion and weak nebular lines like [SIII]λ18.7,33.5µm (Clu- +ver et al. 2010). This is the result of low star formation +rates in the main filament (measured to be between 0.05 +and 0.08 M⊙ yr−1; Cluver et al. 2010), and emission +lines more typical of fast shocks than HII regions (e. +g. Konstantopoulos et al. 2014). Examples of these H2 +dominated spectra have been presented in Figures 8, 13 +and 14 of Cluver et al. (2010), and Figure 15 and 16 of +Appleton et al. (2017). The Spitzer IRS spectra showed +that only a small minority of places along the main fil- +ament (including parts of the SQ-A region) are spectra +found with line flux ratios more typical of star forming +regions. +In Figure 2a-c, we compare the filter transmission +band-passes for the MIRI imaging bands (F770W, +F1000W and F1500W) directly with the Spitzer IRS +spectra of three zoomed-in regions near the center of +each of the ALMA CO (2-1) primary beam pointings +(Fields 4, 5 and 6) shown in Figure 1. The regions are +typical of the diversity of mid-IR spectral properties en- +countered in the IGM of SQ in general, and are the +focus of this paper. The MIRI filter F770W is likely to +contain emission primarily from the 7.7µm PAH com- +plex in regions where star formation dominates, but in +other regions, especially those with strong H2 emission, +the 0-0 S(5) can also potentially contribute, as in Fig- +3 See also: +https://help.almascience.org/kb/articles/what-is-the- +absolute-astrometric-accuracy-of-alma + +6 +Appleton et al. +Figure 2. Spitzer IRS spectra of three 5.5 x 5.5 arcsec2 regions centered on the three fields studied at high resolution in +CO (2-1) emission with ALMA and at 7.7 and 10 µm with JWST MIRI imaging (see Figure 5). The MIRI broad band filter +transmissions are superimposed to emphasize that the F1000W MIRI filter detects almost exclusively emission from the S(3) +line and that the 7.7 µm band captures a combination of the weak 7.7mum PAH complex and emission from S(5) H2. The +15 µm MIRI bandpass detects mainly faint dust continuum and some (weak) [NeIII] emission. ALMA Field 5 was not covered +by the Short-Lo IRS module. +ure 3a. The F1000W filter, on the other hand, almost +exclusively contains line emission from the S(3) mid-IR +pure rotational H2 line (See also Figures 15e and h of +Appleton et al. 2017). Based on the presented spectra, +we expect the MIRI F1500W filter to be dominated by +faint dust continuum from the general IGM, as well as +from warm dust from embedded star clusters. +A clean separation between the mainly PAH domi- +nated emission in the F770W MIRI filter and the 0- +0S(3)H2-dominated F1000W band can be seen in many +regions along the main IGM filament and in the bridge +region in Figure 3. Here we overlay the contour maps +obtained by isolating a PAH band4 from the IRS spec- +tral map from Spitzer in Figure 3a (white contours) with +a two-color image of the JWST MIRI F1000W (green) +and the MIRI F770W (orange). In Figure 3b we over- +lay the 0-0 S(3) H2 contours (red) over the same im- +age. The overlays show that the white contours of PAH +emission from Spitzer follow quite closely the brighter +emission in the F770W dominated regions, whereas the +red contours of H2 follow closely the green emission +originating mainly from the F1000W filter. This figure +demonstrates how the 10µm MIRI filter, as we expected +from the individual IRS specta, detects preferentially +H2 emission, while the F770W filter detects those re- +gions with dominated by PAH emission. As shown in +Cluver et al. (2010) the PAH emission is relatively faint +in the main shock, but this figure emphasise those re- +gions with stronger PAH emission. This near-isolation +of the H2 emission in many regions of the IGM is a re- +sult of the relatively narrow width of the F1000W filter, +4 We used the IRS 11.3µm PAH rather than the 7.7µm PAH, be- +cause the signal to noise ratio was much better in this PAH fea- +ture in the Spitzer data. +and the lack of other lines or bands that can seriously +contaminate it for the redshift of Stephan’s Quintet. +3.2. Dust continuum +Observations using the F1500W MIRI filter are ex- +pected to detect mainly warm dust emission, similar to +that mapped by Spitzer at 24µm with MIPS (see Guil- +lard et al. 2010). +However, in the SQ-A region (Fig- +ure 2b), the dust continuum is also contaminated by +[NeIII] emission. As noted above the very strong 0-0S(1) +line is, fortuitously, redshifted out of the F1500W band. +Before discussing results for each of the zoomed-in re- +gions in the next section, we mention the removal of +faint dust emission from F1000W and F770W at the +center of Field 6 (near the center of the main IGM fila- +ment in SQ). In this field, faint dust continuum in the +F1500W filter was seen in the vicinity of the structures +we are interested in (see § 4). +We therefore removed +the faint dust continuum from both the F1000W and +F770W images, using the Spitzer IRS spectrum of the +region as a guide. First we created small sub-images of +the field in all three bands centered on the main struc- +ture of interest and corrected them for small offsets in +the WCS as discussed in §2. After removing a median +sky background from each subimage, we then subtracted +the F1500W image from both the F770W and F1000W +images, scaling the continuum by a factor of 0.9 to ac- +count from a slight rise in continuum seen between 7.7, +10 and 15 µm in the IRS spectrum of the region. This +allowed us to better define the distribution the H2 and +PAH features. The effect was quite small (the dust emis- +sion was quite weak compared with the emission seen in +the 7.7 and 10µm bands) on the resulting fluxes of the +F1000W and F770W band. +Without spectroscopy of +the region on the same spatial scale as the observed fea- +tures, this method is necessarily imperfect, and where +we estimate the fluxes for the features observed in Field + +7 +3.5 +Mid-Filament +Northern SF Region +0-0 S(1) +Bridge nr NGC 7319 +[SI] +[Sill] +3.5 +H2 +0 sa ALMA Field 6 +F9 +ALMA Field 4 +ALMA Field 5 +3.0 +H2 + MIRI FILTERS +2.5 +(MJy s +5 - +F770W +[Sil +MIRI FILTERS +2.5 - +F1000W +F1500W +0-0 S(1) +2.0 +F770W +F1000W +F1500W +4 - +2.0 - +F1500W +Density ( +0-0 S(3) +0-0 S(2) +1.5 +H2 +0-0 S(0) +0-0 S(0) +0-0 S(0) +[Nel] +H2 +1.0 - +H2 +D +PAH +1.0 - +[Nell] +0.5 +0.0 +0.0 +10 +15 +20 +25 +30 +35 +40 5 +10 +15 +20 +25 +30 +35 +40 +15 +20 +25 +30 +35 +5 +Observed Wavelength (um) +Observed Wavelength (um) +Observed Wavelength (μum)The Multi-phase IGM in Stephan’s Quintet +7 +6 in the next section, we include larger uncertainties to +take this into account. For the sub-region near the cen- +ter of Field 5, no obvious dust structure was seen in +the F1500W band near the structures of interest, and so +we did not attempt to continuum subtract the H2 and +PAH-dominated emission from the F1000W and F770W +images. For Field 4 — centered in the SQ-A star forming +region — our attempts at subtracting a scaled version of +the dust dust continuum led to a severe over-subtraction +of the emission in the F770W and F1000W bands for +most reasonable dust scale factors. This is likely due to +a combination of factors. Firstly at that position (see +Figure 2b), we have shown that the 15µm band shows +emission from not only dust continuum, but also rela- +tively strong [NeIII], as well as broad PAH-band emis- +sion from the 17µm “plateau” complex (see Smith et al. +2007). These additional contaminants make it difficult +to isolate the warm dust component in the 15µm image. +Therefore, for this star formation dominated field, we +conclude that the F1000W and F770W MIRI filters are +hopelessly contaminated by dust emission in way that +cannot easily be corrected without high resolution spec- +troscopic data from the MIRI MRS. This significantly +limits our discussion of the molecular hydrogen proper- +ties in Field 4. For that region we limit our discussion +of the warm H2 properties determined from the Spitzer +IRS observations. +3.3. Near-IR images +We also present images in this paper obtained through +the NIRCam F200W filter. +F150W provides a near- +IR continuum band relatively clear of emission lines at +the redshift of Stephan’s Quintet (0.022 < z < 0.025), +with the well-known near-infrared [Fe II] lines falling +just blueward or redward of the filter transmission. +The bandpass of the F200W filter is sensitive to emis- +sion from the hydrogen recombination line Paα, and +starlight. Diffuse emission from this line becomes obvi- +ous when we compare some of those images with those +obtained in the F150W NIRCam filter, and by compar- +ison with the HST Hα images. +JWST/MIRI, which is able to detect warm H2 +through the pure rotational transitions, can significantly +improve our knowledge of how energy is dissipated in +the IGM down to the 0.3 arcsec scales of giant molecular +cloud complexes (∼140 pc). NIRCam, with its ability to +probe down to scales of 0.05-0.06 arcsec (23-27 pc at 1.5 +and 2µm respectively) can probe ionized gas and stellar +associations and clusters at scales at which sporadic star +formation along the main filament is observed. +4. RESULTS +We study in detail three regions which lie near the +centers of the ALMA CO (2-1) observing fields. They +are highlighted in Figure 4 as boxes (dotted lines) super- +imposed on one of the publicly released ERO images of +the inner part of the group. Field 6, which is dominated +by 10µm H2 emission (green color in this figure), was +targeted by HST COS (Guillard et al. 2022) and shows +extremely broad Lyα emission, broad [CII]158µm emis- +sion and contains some of the the warmest H2 observed +by Spitzer (e. g. Cluver et al. 2010). It is representa- +tive of a shock-heated region in the main N/S filament. +Field 5 is another H2-dominated region which lies out- +side the main north/south H2 filament, but observations +at other wavelengths (e. g. single disk CO observations +by Guillard et al. (2012), and Herschel [CII] observations +of Appleton et al. 2013) show that this gas is also highly +turbulent. It forms part of the bridge of H2 emission +seen by Spitzer. +Field 4, by contrast, lies in a well- +studied extragalactic star forming region, SQ-A (or the +Northern Star Forming region) and has been studied ex- +tensively at optical wavelengths (Gallagher et al. 2001; +Xu et al. 2003). Its star forming nature can be inferred +from its bluish-white appearance in the false color map +of Figure 4 due to a combination of weaker H2, strong +PAH 7.7µm emission and dust continuum (See Cluver +et al. 2010; Appleton et al. 2017). +4.1. Analysis of the three regions +4.1.1. Field 6: Within the main shock-dominated +North/South filament +In Field 6 (Figure 5a) we show the integrated CO (2-1) +emission from cold molecular gas superimposed on the +warm H2 emission. The warm H2 gas (attributed to the +0-0 S(3) line) is distributed in an elongated structure +extending over 4-5 arcsecs (∼ 2 kpc at D = 94 Mpc) +with a triangular-shaped hot-spot to the south-west, and +a series of fainter clumps extending to the east. +The +overall distribution of warm H2 has the appearance of +head-tail structure. The CO emission (white contours) +displays a clumpy narrow shell-like structure associated +with the warm H2 head, and a scattering of CO clumps +along the tail, extending over the same overall extent, +but not correlating in detail with the hotspots in the +warm H2. +Based on the published IRS spectra, Figure 5b is inter- +preted as a mix of H2 (0-0S(5)) and weak PAH emission. +The dominant triangular-shaped head structure seen in +the 10µm image is also strongly represented here. +A +second major clump of 7.7 µm emission is seen to the +NW, which is weak at 10µm. It is plausible that this +feature is PAH dominated, whereas the emission from +the compact head (which is seen also at 10µm) may be + +8 +Appleton et al. +Figure 3. Stephan’s Quintet: A false-color representation of the MIRI F770W and F1000W JWST images with contours of +rest-frame (a) 11.3µm PAH and (b) 9.66µm emission derived from spectral cubes obtained from the Spitzer IRS Short-low +mapping of Stephan’s Quintet. The white and red solid lines show the extent of the IRS spectral mapping (See Cluver et al. +2010). The figure demonstrates how the S(3) line isolated in the Spitzer cube follows closely the 10µm MIRI emission (green), +and the PAH features follow the MIRI 7.7µm image which contains contributions from the 7.7µm PAH and H2 emission (See +also Appleton et al. 2017, and Figure 2 of this paper). +dominated by warmer than normal H2 which would emit +strongly at in the 0-0 S(5) line. Without mid-IR spec- +troscopy, this cannot be verified. +This distribution of ionized gas can be inferred from +the narrow-band HST F665N filter images (dominated +by Hα emission) in Figure 5c) and the F200W NIR- +Cam image (dominated by Paα emission and near-IR +starlight) in Figure 5d. +Both images show significant extended (presumed ion- +ized gas) emission associated with the head of the warm +H2 including a protrusion, or “spike” labeled in Fig- +ure 5c). The feature is also seen in CO emission. Ex- +tended(ionized) gas also follows more closely the distri- +bution of the warm H2 in the tail, than it does the CO +emission. This is well demonstrated in the composite +image Figure 5f, which shows an RGB representation +of the 10µm (warm H2 , CO emission (cold H2) and +F200W (ionized gas and stars). +Gemini optical spec- +troscopy (Konstantopoulos et al. 2014) which covered +this region at a spatial resolution of 1 arcsec strongly +suggested that the ionized gas is shock-excited, which +would explain why the ionized gas follows the warm H2 +emission in the head and the clumpy tail, and is less +correlated with the CO emission. +The spatial resolution of the F200W image is far supe- +rior to the HST image, and it is clear that there are sev- +eral compact sources on scales of < 0.05-0.6 arcsec (23- +27 pc) embedded in the ionized gas component. These +sources are likely stellar associations, or unresolved su- +per star clusters (Gallagher et al. 2001). They are gen- +erally rather sparsely distributed and poorly correlated +with the warm or cool molecular hydrogen. +One exception is a group of compact sources and ex- +tended F200W emission associated with the southern tip +of the NW clump of CO emission identified in Figure 5c. +These sources may be evidence of recent star formation +associated with the head-tail structure. Evidence of this +comes from the bright u-band sources associated with +the same region shown in Figure 5e. +Although a full +analysis of the NIRCamcolors of the point sources in +Stephan’s Quintet will be discussed in a separate pa- +per, we know these blue regions have the optical colors +of young clusters with ages estimated to be between 3- +5 Myrs based on previous HST UBVmVI photometry +(Fedotov 2014; see Table A4 for more details). The ex- +istence of young star formation near the NW CO clump +may also explain why that region exhibits PAH emission, +since PAHs are believed to be excited by UV radiation +from young stellar associations. +The kinematics of the diffuse ionized gas centered +on the head from optical spectroscopy shows extremely +broad line widths (>800-1200 km s−1) in several key +emission lines (e. +g. +[OIII]5007, [OI]6300, and Hα) +(Konstantopoulos et al. 2014; Guillard et al. 2022). The +broad line-widths have been attributed to a combination +of turbulence and large-scale motions associated with +gas along the line-of-sight. Even broader wings in the +Lyα profile were also seen at this position. We will ar- + +b +a +NGC 7318b +NGC 731&a +NGC 7319 +MIRI F1000W +MIRI F1000W +Spitzer iRS- 9.66um/(0 -0S(3) +MIRI F770W +Spitzer IRS PAH (11.3um) +MIRI F770WThe Multi-phase IGM in Stephan’s Quintet +9 +Figure 4. Stephan’s Quintet: A false-color representation of the MIRI F770W, F1000W and F1500W JWST image of the inner +Stephan’s Quintet, showing the three regions that are highlighted in the discussion that are detected near the primary beam +centers in the ALMA CO (2-1) emission line observations. Field 6 is representative of the center of the main filament where +previous observations with Spitzer show that the molecular hydrogen has the highest temperature. Field 5 lies in the bridge +region between NGC 7319 and the main shock identified in 1. Finally, in Field 4, a bright region in the SQ-A star forming +region is highlighted (see text). +gue in § 5 that the warm H2 emission detected by JWST +at this position may be responsible for UV scattering of +Lyα emission within the turbulent regions, and if so, +should exhibit broad line-widths. +While we cannot yet measure the kinematics of the +warm gas, the CO observations provide kinematic infor- +mation about the cold molecular phase. Figure 6a and +b shows the integrated CO surface density (moment 0) +map and the mean velocity field (moment 1) map of the +CO emission respectively. The systemic velocity of the +CO emission falls between the velocity of the intruder +galaxy NGC 7318b (V(int) = 5770 km s−1) and that +of the velocity barycenter of the main group (V(group) += 6600 km s−1). This is consistent with gas which has +been decelerated in the collision of the intruder with the +group-wide gas. +However, the linewidth of the entire +region in CO is less that 120 km s−1, as shown in the +integrated ALMA CO (2-1) spectrum of the entire region +and the CARMA CO (1-0) spectrum shown in Figure 6c. +This width is significantly smaller than that seen in the +ionized gas, implying that the cold molecular gas does +not take part in the turbulent motions seen in the ion- +ized gas phase. More extensive kinematic information +for the CO emission is presented in the Appendix-A, +where we show individual spectra extracted from many +regions (Figure A1), along with tabulated properties of +the CO emission on the scale of the ALMA beam (Ta- +ble A1). The tip of the head structure has the highest +radial velocity, and the clumps show a trend to lower ve- +locities as one moves east into the tail, reaching the low- +est velocity in one of the most easterly clumps. We will +argue later that this suggests material is being stripped +from the head into the tail through ramp-pressure strip- +ping with a hot medium. +The velocity dispersion5 of the individual clumps +around the structure show a mix of unusually broad line +widths (up to 100 km s−1 FWHM) and narrow lines +(< 40 km s−1). +In Table A1, we show that the re- +gion near the “spike” feature seen in the ionized gas, +is unusually broad (FWHM = 102 km s−1). Away from +5 Hereby measured as the FWHM of the emission line profile, or +2.35 × the sigma of these mainly Gaussian lines. + +MIRI Filters +SQ-A +F770W +F1000WF1500W +Field 4 +NGC7318a +NGC 7319 +NGC7318b +Field 6 +Field 510 +Appleton et al. +Figure 5. A zoom-in on the center of the ALMA Field 6 (see Figure 1) comparing (white) contours of CO (2-1) emission to (a) +continuum-subtracted pure warm 0-0S(3) H2, (b) continuum-subtracted warm 0-0 S(5) H2 plus 7.7µm PAH complex emission, +(c) Hα emission from F665N WFC3 HST, (d) mix of IR (red compact) stellar sources and Paα emission in the NIRCam f200w +filter, (e) blue star clusters from F336W WFC3/UV HST (Gallagher et al. 2001), (f) a composite RGB color representation +of the F1000W (mainly warm H2), ALMA CO (2-1) emission (cool H2), and the F200W (Paα plus star clusters) images. The +MIRI identifications of the contributing lines/PAH bands are clear from the Spitzer IRS spectra of a region 5.5 x 5.5 arcsec2 in +Figure 2a, which spatially covers this entire region. All images were carefully aligned using GAIA DR3 stars close to the region +shown, leading to relative uncertainties in astrometry of ∼ 0.15 arcsec. The F770W and F1000W JWST images were smoothed +to the same resolution as the F1500W image before subtracting the carefully-aligned dust continuum. The white scale bar is 3 +arcsec in length. A more detailed description of the CO emission and its kinematics is given in Figure A1. +the head, several regions in the tail of the structure +show line widths ranging from 60-90 km s−1(FWHM) +(see Figure A1) to values lower than 40 km s−1. Given +the masses of the individual molecular clumps of ∼few× +106M⊙ (Table A1), it is unlikely that the clumps with +high velocity dispersion are gravitational bound6. The +broader lines are consistent with turbulent motions on +the sub-arcsec scale in the colder gas component, while +others have much narrower lines. +6 For example, for a clouds of diameter 100 pc (just less than the +resolution of ALMA) and given a typical gas surface density of +Σgas = 170 M⊙pc−2), the velocity dispersion for a cloud in grav- +itational (virial) equilibrium would be σ2 = (3/5)πGRcloudΣgas. +To be in equilibrium σ = 8.3 km s−1, or a FWHM ∼ 20 km s−1. + +Triangular +head +F1000W.JWST +F77OW.JWST +lump +Spike +F665NHS +f200w.IWS1 +Blue = warm H2 Green = Cool H2 +Red = ionized gas +F336W HST +CompositeThe Multi-phase IGM in Stephan’s Quintet +11 +Figure 6. ALMA Field 6 region (labeled in Figure 4): (a) CO (2-1) integrated emission, (b) the intensity-weight mean velocity +map of the same region, and (c) the ALMA CO (2-1) and CARMA CO (1-0) spectrum of the region 5 x 5 arcsec centered on +same region. Effective synthesized beam-shapes (0.36 x 0.21 arcsec2 FWHM) are shown graphically in the left-hand corner of +each figure. Arrows indicate the radial velocity of the intruder V(int) and the barycentric velocity of the main group (V(gr). +One such region occurs in the CO emission clump +closest to the bright blue star clusters described ear- +lier. Here the CO emission is essentially unresolved (40 +km s−1or less) perhaps suggesting turbulent motions are +calmer there. Table A1 presents information about the +line fluxes and estimated H2 masses for all the regions +measured in detail with ALMA. +4.1.2. Field 5: A shocked structure in the bridge region +Field 5 lies outside the main molecular filament in +Stephan’s Quintet, and is closer to large face-on galaxy +NGC 7319. It forms part of an apparent bridge of H2 +emission discussed by Cluver et al. (2010). The JWST +images (e. g. Figure 4) show that it is the most easterly +of series of irregular shock-dominated (appearing green +in that image) clouds that run approximately East/West +across the field and may not be physically connected. +In contrast to Field 6 which contained a single coher- +ent CO structure, Field 5 contains three separate bright +structures visible in the CO maps (labeled in Figure 7a). +These consist of a clumpy CO ring about 1 arcsec across +(450 pc), a compact elongated core 1.5 arcsec to the SE +of the ring, and a broken linear filament of gas running +north-south 0.5 arcsec further east of the compact core. +The direction of the elongation of the core points to- +wards the center of the ring. Both Figure 7(a) and (b) +show that the compact CO core lies near the brightest +emission at the center of the arrow-shaped 10µm (warm +H2) structure. As the arrow-shaped emission narrows +down towards the NW, it connects to the northern part +of the CO emission from the ring. Figure 7(c) shows dif- +fuse Hα emission from the compact CO core but little +emission from the ring. The higher spatial resolution of +the NIRCam F200W image shows that the correspond- +ing Paα in the core is also elongated Figure 7(d). Faint +F200W emission is also seen in the brightest CO ring +clump. The U-band image, Figure 7e, shows no obvi- +ous emission near any of the CO molecular structures. +Although not shown here, the F150W NIRCam image +shows point-like sources in the northern part of the CO +ring, suggesting that the F200W image is revealing a +mix of HII regions and compact sources within the ring. +The composite image, showing the three gas phases in +one figure emphasizes the warm H2 connection between +the ring and the core (Figure 7f). +A string of clumpy sources at 2µm is clearly seen ap- +proximately 1 arcsec to the east and north of the ring +(F200W image). +It is not clear if these sources have +any real connection to the CO structures, as some could +be background galaxies. One of the clumps has a faint +U-band association (see Appendix-B and Table A3) sug- +gesting recent star formation. +Near-IR spectroscopy +may help to determine if they are are physically associ- +ated with Stephan’s Quintet. +The kinematics of CO gas in the compact core is strik- +ingly different from that of the ring and the linear struc- +ture. Figure 8a and b, shows the integrated and radial +velocity map of the CO (2-1) emission, and spectra of +the three different CO emitting structures are presented +in Figure 8c. The velocities within the partial ring in- +crease clockwise around the ring from 6631 km s−1 in +the north, reaching the highest value in the south and +east quadrant of 6688 km s−1. In contrast, the compact +core (which lies near the peak in the warm H2 emission) +has a much lower (negative 250 km s−1) radial velocity +than the ring (∼ 6400 km s−1). The core also shows a +velocity shear along its length of about 30 km s−1over +0.5 arcsec (200 pc), as shown in the Appendix-A. Finally +the third linear structure to the east of the compact core, +shares velocities similar velocity to that of the ring, with +velocities ranging from 6620 in the south to 6680 in the +north. + +Velocity (Helio) km/s +0.02 +0.06 +0.1 +0.14 +0.18 +6018 +6038 +6058 +6078 +6098 +MS) +ALMA_CO(2-1) +CO +(2-1) +b +CARMA_CO(1-0) +33:58:24 +0.025 +V(int) +V(gr) +tion +0.020 +33:58:23 +clinati +0.015 +0.010 +33:58:22 +0.005 +Flux +0.000 +00 +33:58:21 +J20( +1-0.005 +5600 +5800 +6000 +6200 +6400 +6600 +22:36:00.1 +35:59.9 +59.8 +59.7 +22:36:00.1 +35:59.9 +59.8 +59.7 +Velocity (Helio) km/s +J2000 Right Ascension (HMS) +J2000 Right Ascension (HMS)12 +Appleton et al. +Figure 7. A zoom-in on the center of the ALMA Field 5 (see Figure 1) comparing (white) contours of CO (2-1) emission +to (a) pure warm 0-0S(3) H2, (b) warm 0-0 S(5) H2 plus 7.7µm PAH complex emission, (c) Hα emission from F665N WFC3 +HST, (d) mix of IR (red compact) stellar sources and Paα emission in the NIRCam f200w filter, (e) blue star clusters from +F336W WFC3/UV HST, (f) a composite RGB color representation of the F1000W (mainly warm H2), ALMA CO (2-1) emission +(cool H2), and the F200W (Paα plus star clusters) images. The MIRI identifications of the contributing lines/PAH bands are +likely from the Spitzer IRS spectra of a region 5.5 x 5.5 arcsec2 in Figure 2, which in this case only includes the longer IRS LL +wavelengths. All images were carefully aligned using GAIA DR3 stars close to the region shown, leading to relative uncertainties +in astrometry of ∼ 0.15 arcsec. The F770W and F1000W JWST images were smoothed to the same resolution as the F1500W +image before subtracting the dust continuum. The white scale bar is 3 arcsec in length. +An in-depth view of the kinematics of Field 5 in given +in the Appendix-A, including individual spectra, Figure +A2, and a table of CO properties, Table A2. The spec- +tra show that individual CO clumps in all of the three +CO structures exhibit broad CO line-widths at the scale +of the ALMA beam ( ∼150 pc). The compact core it- +self has a velocity dispersion of 145 km s−1(some of this +may be due to the gradient seen along the core), and +all components of the ring have line-widths which lie in +the range 60-120 km s−1(FWHM). The linear structure +shows narrower line profiles in the range 48-87 km s−1. +Appendix-A also shows that although the majority of +the gas in the ring has much higher radial velocities than +the core structure, one small segment of the ring (at the +point nearest the core) has a velocity similar to the core. +This is supported by the CARMA spectrum, Figure 8d, +which shows a blueward component associated with the +average spectrum of the ring. This strengthens the idea +that the two CO structures are somehow related despite +their discrepant radial velocities. We will discuss in the +next section the possibility that the warm bridge con- +necting the CO ring and core may be splashed warm + +a +ring +linear +structure +compact +F1000W JWST +F770WJWST +core +d +F665N HST +F200W JWST +Blue = Warm H2 +Green = Cool H2 +Red = ionized gas +*+ +F336W HST +CompositeThe Multi-phase IGM in Stephan’s Quintet +13 +Figure 8. ALMA Field 5 region: (a) the CO (2-1) integrated emission at a resolution of 0.36 x 0.21 arcsec2, and (b) the +intensity-weight mean velocity maps of the same region, (c) the ALMA CO (2-1) spectra of the three different extracted regions +shown as a red polygons in (a). Note the very different systemic velocity of the compact structure B. In (d) we overplot the +CARMA CO (1-0) over a 5 x 5 arcsec2 aperture (blue line), compared with the velocity of the ring C. The effective ALMA +beam shapes (FWHM ellipses) are shown graphically in the bottom left hand corner of the images. The black vertical line on +the spectra show the barycentric systemic velocity of the V(group). +Figure 9. Extended CO (2-1) emission for ALMA Field 5 over a larger area than Figure 7. The CO emission (white contours) +have been smoothed to a resolution 1 x 1 arcsec2. (a) warm 0-0 S(3) H2 , (b) warm 0-0 S(5) plus 7.7µm complex PAH emission, +and (b) a smoothed version of the extended velocity field, again showing the unusually low velocity for the compact core +compared with and the other emission regions to the south. The long filament connecting the brighter CO structures is seen in +the warm H2 and faint H2+PAH emission. the filaments velocity is shown schematically because it is weak, but has the same +velocity as the majority of the other structures except the compact core. The white scale bar is 3 arcsec in length. +molecular gas caused by the collision of a dense molec- +ular cloud moving at high (blueward) velocity with re- +spect to more diffuse gas associated with material in the +group. +Some of the CO emission in Field 5 may be associated +with a larger filament and other CO clumps, as seen +in a larger-scale (10-15 arcsec, 4.5-7 kpc) image, Fig- +ure 9a, where we present the ALMA CO emission map +smoothed (to 1 x 1 arcsec2) to bring out fainter fea- +tures. This reveals a very faint filament of CO gas and +several fainter southern CO concentrations. Both the fil- +ament and the southern CO structures have faint warm +H2 counterparts. Filaments like this one, in this case ex- +tending over 6-10 arcseconds in scale (3kpc-5kpc), seem +to be common in the MIRI maps of SQ (e. +g. +Fig- +ure 4). Figure 9b shows again the velocity field of the +ring, compact core and linear structure, this time in re- +lation to the larger-scale structure in the region. This +further emphasises how the compact core has such a dif- +ferent velocity from all the other structure in the region. +The velocity of the filament is poorly determined in CO +because it is so faint, but seems to have a velocity simi- +lar to that of the average velocity of the group. Mid-IR +spectroscopy would be needed to provide more informa- + +(Jy/beam-km/s) +(km/s) Heliocentric +0.05 +0.2 +0.25 +12.5 +0 +0.1 +0.15 +0.3 +6280 +6383 +6488 +6590 +6700 +(DMS) +Flux Density (mjy) +ALMA Reg A +C +0.0 +ALMA Reg B +27.5 +27.5 +ALMA Reg C +7.5 +6160 km/s +VGROUP +a +b +J2000 Declination + 5.0 +J2000 Declination +2.5 +26.5 +26.5 +0.0 +-2.5 +6200 +6400 +6600 +6800 +7000 +20 +25.5 +CARMA CO(1-0) +p +25.5 + 100 K)d +Transition +(µJy) +(arcsec2) +(µm +) +(×10−18Wm−2 ) +(×1032 W) +(×106 M⊙) +[1] +[2] +[3] +[4] +[5] +[6] +[7] +[8] +[9] +[10] +F5-R1 +F1000W +0-0S(3) +44.4 +2.8 +2.0 +2.64 +2.8 +[4.2]b +5.6 +F5-R2 +F1000W +0-0S(3) +16.5 +0.6 +2.0 +0.99 +1.0 +[4.2]b +2.1 +F5-R3 +F1000W +0-0S(3) +6.9 +0.9 +2.0 +0.41 +0.4 +[4.2]b +0.9 +F5-R4 +F1000W +0-0S(3) +19.7 +2.7 +2.0 +1.18 +1.3 +[4.2]b +2.5 +F5-R5 +F1000W +0-0S(3) +13.8 +2.6 +2.0 +0.83 +0.9 +[4.2]b +1.8 +F5-IRSarea +F1000W +0-0S(3) +86.5 +30.3 +2.0 +5.18 +5.5 +4.22 +11.0 +F5-IRS +LLe +0-0S(1) +- +30.3 +- +4.01e +4.2 +- +- +F5-IRS +LLe +0.0S(0) +- +30.3 +- +0.50e +0.5 +- +- +F6-All +F1000W +0-0S(3) +58.4 +11.0 +2.0 +4.30 +4.5 +[4.5]b +14.4 +F6-R1 +F1000W +0-0S(3) +19.6 +1.6 +2.0 +1.18 +1.2 +[4.5]b +4.0 +F6-R2 +F1000W +0-0S(3) +6.0 +1.3 +2.0 +0.36 +0.4 +[4.5]b +1.2 +F6-R3 +F1000W +0-0S(3) +5.36 +0.7 +2.0 +0.32 +0.3 +[4.5]b +1.1 +F6-R4 +F1000W +0-0S(3) +7.33 +1.0 +2.0 +0.44 +0.5 +[4.5]b +1.5 +F6-R5 +F1000W +0-0S(3) +7.73 +1.3 +2.0 +0.46 +0.5 +[4.5]b +1.5 +F6-IRSbox +F1000W +0-0S(3) +76.0 +30.3 +2.0 +4.56 +4.8 +4.5 +15.3 +F4-IRSonlyf +SL-LL +0-0S(3) +- +30.3 +- +2.1 +2.2 +5.0 +14.8 +aThe regions for F5 (Field 5) and F6 (Field 6) are shown graphically in Figure 12. +b The flux density is measured from the surface brightness in the JWST MIRI image over the apertures shown in Figure 12, assuming a pixel size +of 0.11 x 0.11 arcsec2. +c Power law index n, assuming the warm H2 column density follows a power law with temperature of the form δN ∝ T −nδT, following the work +of Togi & Smith (2016), and as discussed for Stephan’s Quintet by Appleton et al. (2017). +dEstimated mass in warm H2 for T> 100 K assuming a value for the powerlaw index, n, for the resolved regions that is the same as that measured +by the IRS over a larger 5.5 x 5.5 arcsec2 region. These warm H2 mass estimates for T> 100 K employ the method of Togi & Smith (2016). For +Field 6, n was derived from the full mapping with both LL and SL IRS modules described in (Appleton et al. 2017). For Field 5, was measured +using Spitzer IRS Long-low data for 0-0S(0) and 0-0S(1), combined with the flux measured for 0-0S(3) from the MIRI data. +eLine fluxes measure directly from extracted IRS spectrum from Long-Low (LL) IRS spectrograph shown in Figure 2b. +fUnambiguous H2 emission was not easily extracted from the F1000W data for this region (see text). This entry is taken from the IRS data from +Appleton et al. (2017) for comparison with the other regions. Note the large value for the power law index (n) indicative of cooler H2 temperatures +near the northern tip of the SQ filament in the SQ-A region. +Daddi et al. 2015). We assumed a value of 0.8 to be +similar to that measured in the highly turbulent Taffy +Bridge (Zhu et al. 2007). Column 6 provides an estimate +of the H2 mass for an assumed value of the X-factor, +XCO. For the purposes on the discussion, we present +the H2 masses in Column 6, and the total gas mass in +Column 7 assuming XCOis the average Galactic value. +This value is likely to be significantly overestimated in +regions of strong shocks and turbulence. The footnotes +to the table provide more details about the definitions +and assumptions used in the calculations. +Bearing in mind the uncertainties in both the warm +and cold molecular masses, we can estimate the ratio +of M(H2)warm/ M(H2)cold by comparing the warm gas +masses estimated by JWST with those from ALMA. +For the F5R1 regions (the arrow-head structure in +Field 5) the ratio of the warm to cold H2 masses is +5.6/(25.6+17.7) = 0.13. +This assumes that both the + +The Multi-phase IGM in Stephan’s Quintet +19 +Table 2. Integrated properties of CO emission in Field 5 and 6 over same areas as Table 1 +Region +Vhelio +FWHM +Σ(Sv∆V ) +Σ(Sv∆V )a +M(H2)a +Mgasb +CO(2-1) +CO(1-0) +×106 +×106 +(km s−1) +(km s−1) +(Jy-km s−1) +(Jy-km s−1) +(M⊙) +(M⊙) +[1] +[2] +[3] +[4] +[5] +[6] +[7] +F5R1A +6401 (±6) +101 (±15) +0.9 (±0.1) +0.28 (±0.05) +18.8 (±3.0) +25.6 (±4.1) +F5R1B +6646 (±4) +73 (±10) +0.6 (±0.1) +0.19 (±0.03) +13.0 (±2.0) +17.7 (±2.7) +F5R2 +6388 (±4) +119 (±10) +0.8 (±0.1) +0.25 (±0.02) +16.7 (±1.5) +22.7 (±2.0) +F5R3 +6656 (±2) +105 (± 5) +1.5 (±0.1) +0.47 (±0.02) +31.5 (±1.5) +42.8 (±2.0) +F5R4 +6618 (±4) +84 (± 8) +1.1 (±0.1) +0.33 (±0.04) +22.4 (±2.4) +30.4 (±3.3) +F5R5 +6537 (±5) +67 (±12) +0.6 (±0.1) +0.20 (±0.04) +13.5 (±2.7) +18.3 (±3.6) +F6-ALL +6054 (±2) +91 (±4) +4.7 (±0.2) +1.5 (±0.1) +99.1 (±4.4) +134.8 (±6.0) +F6R1 +6073 (±1) +47 (±2) +1.1 (±0.1) +0.3 (±0.0) +23.0 (±1.1) +31.3 (±1.4) +F6R2 +6040 (±2) +93 (±4) +1.2 (±0.1) +0.4 (±0.0) +26.1 (±1.3) +35.4 (±1.7) +F6R3 +6051 (±2) +41 (±5) +0.2 (±0.0) +0.1 (±0.0) +4.3 (±0.6) +5.9 (±0.8) +F6R4 +6009 (±2) +37 (±4) +0.3 (±0.0) +0.1 (±0.0) +5.9 (±0.7) +8.1 (±1.0) +F6R5 +6103 (±4) +48 (±10) +0.2 (±0.1) +0.1 (±0.0) +4.9 (±1.1) +6.7 (±1.5) +aWe assume a conversion between a line flux at CO (1-0) to that of CO (2-1) of +SCO(1−0)/SCO(2−1) = (ν1−0/ν2−1)2(r21)−1, where ν1−0 and ν2−1 are the rest frequencies of +the transitions, and r21 is assumed to be 0.8, similar to that measured in the Taffy galaxy +bridge (Zhu et al. 2007), which shares many similarities with the gas in Stephan’s Quintet +(see Appleton et al. 2022). The H2 masses presented here are for an XCO value = XCO,20 = +N(H2)/ICO = 2 × 1020 cm−2 (K−km s−1)−1, the standard value assumed for our Galaxy. +In the text we discuss how this may not be applicable in such a turbulent region. MH2 = +7.72×103D2Σ(Sv∆V)(1+z)−1, where D = 94 Mpc, and Σ(Sv∆V ) is the CO(1−0) line integral +with Sv in Jy and the velocity V of the gas in km s−1. We assume z = 0.02. +b Total gas mass Mgas = MH2 × 1.36, includes a 36% correction for Helium (Bolatto et al. +2013). +CO linear filament and the CO core both contribute to +the total mass. Its not clear that the linear structure +is actually detected in warm CO, so the ratio could be +much higher if that object was excluded (0.22). How- +ever, in reality, since XCO is likely to be much smaller +than Galactic (say 1/5 XCO,20), then the fraction of +warm gas with T > 100 K is likely to be very large +(almost equal to the cold H2). These values are signifi- +cantly greater than that seen in normal galaxies. F5R3, +which includes the partial ring of CO emission has a +warm/cold H2 ratio of 0.02, which is much more like a +normal galaxy. +Regions F5R4 and R5 show warm to +cold ratios somewhat in between the two extremes (0.08 +and 0.1). +These regions lie at the opposite end of a +warm filament (only weakly detected in CO) that con- +nects them to the other Field 5 structures. The warm +triangular hotspot in Field 6 shows an average ratio of +warm to cold H2 = 0.12. Again, if we assume XCO is +1/5 of the Galactic value, this would imply a signifi- +cant warm component. F6R2, has a much lower ratio +of warm to cold gas (0.03), a number which is closer +to that of normal star forming galaxies. It is interest- +ing that this is the region which emits strongly in the +F770W filter, which is consistent with this containing a +PDR. The other clumps in the tail have unusually large +values of warm to cold CO (in the range 0.18-0.22), re- +flecting the relative faintness of the CO in the clumps +in the tail of Field 6. The overall conclusion is that ex- +cept for the regions that show obvious star formation +(PAH emission and compact star clusters embedded in +the emission) the warm to cold H2 ratios are unusually +large for both Field 5 and Field 6. We will discuss this +further in the discussion in the context of shocks and +turbulence. +5. DISCUSSION +SQ is a fantastic example to study the dynamical in- +teraction between gas phases in a group environment. + +20 +Appleton et al. +The big leap forward in spatial resolution offered by the +JWST images compared to Spitzer sheds light on the +morphological structures of the multi-phase gas, and the +physical processes at the origin of the formation and ex- +citation of the molecular gas in the IGM. We discuss +what we are learning about those two aspects in the +following sections. +5.1. Morphology of the molecular gas emission and +large-scale kinematics +In order to explain the formation of large masses of +molecular gas and high luminosity in the pure rotational +ground-state transitions of H2, Guillard et al. (2009) +presented a model in which a large-scale driving shock +wave (travelling at 600-800 km s−1) from the intruder +passes into a multi-phase medium of gas in a filament +of tidal debris caused by a previous tidal interactions +between group members (see Figure 13 for a schematic +view). +We summarize here that scenario, which was +based on observations with limited spatial coverage and +resolution with Spitzer, and then discuss what we are +learning from the new JWST plus ALMA data presented +here about the link between the large-scale dynamics +in the group, and the formation and excitation of the +molecular gas in the IGM. +The large-scale shock wave driven by the intruder, +NGC7318b, passes over the tidal, multiphase filament, +heating low density gas (nH < 2 × 10−2 cm−3) to X-ray +emitting temperatures (see O’Sullivan et al. 2009), and +destroying any dust grains present there (illustrated in +Fig. 13 as the light orange medium). Regions in the pre- +shock region with over-densities greater than 10 (typi- +cally H I +gas with nH > +0.1 cm−3) compared to the +volume-filling gas are heated to lower ( < 106 K) tem- +peratures, can cool and form molecules on dust grains as +they survive the slower shocks (sketched as the dark blue +clouds in Fig. 13). In the next section, we discuss that +the dynamical interaction between the cold and hot gas +can lead to the same physical conditions down stream +from the shock, as thermally unstable gas is produced +within the turbulent mixing layers (Gronke & Oh 2022), +leading to a cycle of molecular clouds breaking up and +reforming in the flow. This is sketched as the turbu- +lent tail in the zoom-in panel on the right-hand side of +Fig. 13. +An efficient transfer of the bulk kinetic energy of the +galaxy collision to turbulent energy within the molecular +gas, associated with a continuous cold gas destruction- +reformation cycle, is required to make H2 a dominant +coolant of the postshock gas (Guillard et al. 2009). In +the next section we discuss how the spatial decoupling +between cold and warm gas at GMC scales, revealed by +the comparison between ALMA and JWST images, is +shedding light on the physical processes that drive this +energy transfer mediated by thermal instability and a +dynamical interaction between gas phases. +5.2. Cycling and redistribution of matter across gas +phases: shattering and growth of cold clouds in +the IGM +The ALMA and JWST data presented in this paper +reveals complex and turbulent structures, with large ve- +locity dispersions (40 to 100 km s−1) at GMC scales, +and a spatial decoupling of cold and warm molecular +gas. +Although a quantitative interpretation of these +data is difficult and beyond the scope of this paper, we +qualitatively discuss a scenario that relate the cold and +warm H2 distributions to theoretical work of cold clouds +shattering in their dynamical interaction with the hot +volume-filling hot plasma and subsequent gas cooling +(Gronke & Oh 2020). +When the shock wave hits the multiphase IGM fila- +ment, the clouds are compressed on a crushing timescale +tcrush = Lc/Vc, where Vc is the shock velocity in the +cloud and Lc the size of the preshock cloud. +The +shocked clouds experience dynamical instabilities and +mixing at the interface with the hot gas flow, as well +as thermal instabilities induced by gas cooling, which +provide an additional fragmentation mechanism occur- +ring over the gas cooling timescale, tcool, as the shock +moves into the cloud. The gas stripped from the cloud +may cool and condense as a result of thermal instabil- +ity. This occurs if the gas cooling time tcool is smaller +than tcrush. There is a wide range of parameter space +(typically nH≈ 0.1 − 100 cm−3, Lc = 1 − 100 pc) where +tcrush > tcool, i.e. molecular gas can form before the +shock has crossed the cloud and thus before dynamical +instabilities fully develop (see Fig. 4 in Guillard et al. +2009). +Because of thermal instability, the post-shock clouds +shatter in many clumplets, which then are seeds for fur- +ther cooling-induced condensation, but also mixing (Far- +ber & Gronke 2022a). In this parameter space where +the cooling time is shorter than the cloud destruction +timescale, this shattering process causes the gas to cy- +cle through the ISM phases with the reformation of cold +clumpy gas. Cloud growth depends on density, as well as +on local heating processes (Jennings et al. 2022). Given +the large masses of diffuse molecular gas observed in the +IGM where atomic H I gas is undetected, qualitatively, +the post-shock physical conditions must be in the den- +sity and irradiation regime where the cloud evolution +is dominated by shattering and perhaps cooling-driven +coagulation. + +The Multi-phase IGM in Stephan’s Quintet +21 +Figure 13. Sketch of the scenario leading to the formation of multiphase gas in the post-shock region. The relative motion +between the intruder, NGC 7318b, and the tidally-stripped filament drives a large-scale shock (40 kpc-long, symbolised with the +red line) in the low-density gas (the light blue filament). The pre-shock medium being multiphase, shock velocities are slower +in the denser (atomic) regions (darker blue clouds), allowing molecular gas (dark blue clumps) to form in less than the crushing +timescale of the H I clouds (see Sect. 5.1 and Guillard et al. 2009). We argue in Sect. 5.2 and 5.3 that the small-scale (a few +100 pc) H2 and CO structures observed JWST and ALMA may be the result of the fragmentation of the clouds and subsequent +reformation of a “foggy” molecular gas (symbolised as tiny green droplets). At small scales, as sketched in the zoom-in panel on +the right, the dynamical interaction between the flow of hot gas (red arrow) and the resulting cloudlets drives a cycling across +ISM phases, leading to the formation of extended and turbulent H2 tails. +Many of the structures detected with JWST along +the main north-south molecular filament are shaped like +cometary tails, with Field 6 being a prime example (see +Fig. 12). They could be signatures of a misty H2 out- +flow driven by the dynamical interaction between cold +and hot ISM phases, e.g. ram pressure stripping of CO +clouds as the hot X-ray emitting gas flows past them. +This is illustrated in the zoom-in panel of Fig; 13. The +high abundance of warm H2 relative to the CO derived +cold gas mass discussed in § 4.1 for both Field 6 and +Field 5 lends support to this idea, despite their differ- +ent spatial positions in the SQ group. We will return +to these differences later. The pressure-driven flow ve- +locity is larger than the turbulent gas velocity because +the droplets are tiny. At the scale of an H2 droplet, the +difference between the turbulent velocity dispersion and +the flow velocity is even larger. Farber & Gronke (2022a) +suggest that in typical CGM or galaxy cluster environ- +ments, cloud shattering during the cooling process could +be responsible for in-situ formation of a fog of warm +(1000 K) gas embedded in hot (107−8 K) plasma, with +molecular shattering length scales of the order of 0.1– +100 pc. This would imply that fragments formed in such +manner are very small, with a low volume filling frac- +tion. A detailed investigation of the physical conditions, +which could be achieved with JWST H2 spectroscopy, +would allow us to determine whether the cloudlets can +coagulate, or whether the molecular streams we observe +remain misty. +5.3. Multi-scale kinematical structures and survival of +the cold gas +The very complex, multi-scale kinematical and mor- +phological structures observed at high resolution with +JWST and ALMA points towards a highly clumpy +(misty) structure of the molecular gas in the IGM of +SQ. This could explain why the Lyα line emission is +so strong and the line profile so broad (Guillard et al. +2022). Indeed, the cloudlets must have a small volume- +filling fraction but their surface filling factor must be +close to unity so Lyα photons can escape through mul- + +pre-shock +Main fast shock +Post-shock turbulent medium +tidal filament +Hot X-ray gas +Molecular clumps +Hot gas flow +Cloud shattering and formation of a +multiphase misty turbulent flow +Diffuse gas +Hl clouds22 +Appleton et al. +tiple scatterings off the clump surfaces without being +absorbed in the inter-clump, dust-free plasma. +The ALMA and JWST observations also shed light +on a puzzle raised by the H2 observations by Spitzer +(Appleton et al. 2006), as well as single-dish CO ob- +servations, which showed extreme velocity dispersion +(≈ 1000 km s−1) of the molecular gas on kpc-scales. +Those large scale velocity gradients, also observed in +other gas phases, could partly originate from coherent +bulk flows (Guillard et al. 2022). The H2 molecules can +only survive at shock velocities ≲ 20 km s−1, which is +much smaller than those velocities at kpc scales, but +also smaller than the linewidths of the CO clouds. The +ALMA observations presented in this paper indeed show +that the CO clumps have velocity dispersions in the +range of 40–100 km s−1. Those velocities are also higher +than the shock velocities needed to account for the exci- +tation of the pure rotational lines of H2 of the order of +5-20 km s−1 (Guillard et al. 2009; Lesaffre et al. 2013; +Appleton et al. 2017). +The tiny H2 droplets are unlikely to survive over the +length of the tails, which supports the idea that they are +continuously destroyed and reformed out of the more +diffuse gas. +This could explain why some of the CO +clumps (in the tail of field 6 for instance) have high ve- +locity dispersions (see Fig. A1), as they are reformed out +of pressure-driven outflow from the parent cloud. This +will need to be further investigated with future H2 spec- +troscopy with JWST to study the velocity gradients of +the warm H2 gas at the scales of the ALMA observa- +tions. Spectroscopy would also allow us to see whether +this fog of small warm clouds are being accelerated in +the bulk flow of the hot X-ray emitting gas, and would +likely differ from the motion of the massive CO clouds. +The temperature of the warm droplets may also increase +along the tail as the foggy clouds progressively mix and +become heated in the faster flow. This might be evident +in the excitation diagram of H2 when a full complement +of H2 rotational lines can be measured. Those obser- +vations will motivate a direct comparison with future +numerical simulations. +The comparison between the warm and cold molecu- +lar gas spatial distributions in the IGM of SQ show that +the powerful mid-IR emission originally discovered with +Spitzer spectroscopy does not primarily have a one to +one correlation with the cold CO(1-0) emitting gas. This +supports the view of the model put forward in Guillard +et al. (2009), which involves a turbulent energy cascade +and dissipation of mechanical energy within the warm +H2 phase, which are both driven by the dynamical in- +teraction between the hot and cold gas phases. +The +complex morphology and kinematics of the warm and +cold H2 structures shows that this dynamical interac- +tion occurs over a wide range of scales. This is expected +from an energy cascade over many orders of magnitudes +in spatial scales, from tens of kpc at the injection scale +of mechanical energy from the galaxy collision to sub-pc +dissipation scale through line emission (Guillard et al. +2009). +5.4. Does the proposed molecular cloud outflow model +apply across the whole SQ system? +As we have stressed earlier, the high luminosity in +the warm molecular hydrogen lines over a large part +of Stephan’s Quintet system requires a common mecha- +nism to convert a fraction of the kinetic energy resulting +from the large-scale shock wave into rotational molecular +hydrogen via low-velocity shocks (Guillard et al. 2009; +Appleton et al. 2017). We have seen that, of the three re- +gions we studied in detail here, the one that most closely +fits the proposed cloudy outflow model is the head-tail +structure in Field 6, where we seem to be witnessing a +possible break-up of a large cold molecular cloud into +warmer H2 in both the head and tail. +Although modeling of the actual structure of a cold +molecular cloud in a hot wind will require further sim- +ulation, its likely that cometary structures should be +common along the main shock wave, and the diver- +sity of structures should be similar to those seen in the +cloud-survival sequences of models by Farber & Gronke +(2022b). Those model sequences, where clouds are en- +trained in a fast wind, show that in some cases the cooler +head and tail structures can survive for long periods, +whereas in other cases, depending on the properties of +the clouds, they can dissolve fairly quickly. Indeed, one +of the models has a morphology remarkably similar to +the shell-like CO and warm H2 structure in the head of +Field 6, including the spike feature at the head of the +clump. Many of the shapes of the bright H2 emitting +clumps seen along the main shock front in Stephan’s +Quintet (Figure 4) show suggestions of head/tail struc- +tures, and have filamentary structure very reminiscent +of the modeled shapes seen when the head/tail struc- +tures survive. Others have shapes similar to the models +where the clouds have lost their heads, and are in the +process of dissolving. Therefore it is plausible that our +picture of the foggy outflowing cloud picture in appli- +cable to a large number of cloud structures seen in the +main shock. +In addition, Figure 4 shows head-tail structures far +to the south and west of the main SQ-A region. One +of those tails extends over 30-40 arcsecs (10-20 kpc!). +These structures may represent the interaction of dense +molecular clumps with the outer hot halo of the intruder + +The Multi-phase IGM in Stephan’s Quintet +23 +far downstream of the main shock. More observations of +the cold and warm molecular gas, and their kinematics +will be needed to test this idea. +In the case of our study of the Field 4 region in SQ-A, +it is less clear that the cloud outflow model is applica- +ble, although we note that a faint irregular tail is seen +extending to the east of the main concentration of gas +and stars. +Xu et al. (1999) has previously suggested +that SQ-A is too young to have formed as a tidal dwarf +galaxy, but is more likely a starburst region created as +part of the main shocked filament. +The H2 gas tem- +perature measured by Spitzer in SQ-A is much lower +than other regions in the main filament (Appleton et al. +2017), and the gas may have cooled to the point that +stars can form in large quantities there. Although un- +derstanding how those stars form is beyond the scope +of this paper, we have shown in Guillard et al. (2009), +that the large (> 50 pc), dense (nH > 100 cm−3) pre- +shock clouds are expected to be gravitationally unstable +in the hot gas at this pressure (see their Fig. 4). There- +fore, their evolution will be different than the more dif- +fuse gas, and should trigger star formation. This may +explain star formation in SQ-A, and other star forming +regions scattered in the IGM. Those large pre-existing +clouds have crushing times longer than the SQ collision +age (believed to be a few tens of millions of years). How- +ever, such clouds must be rare, otherwise we would see +major bursts of star formation all over the main fila- +ment, which is not observed. +The case of Field 5 (in the bridge between NGC 7319 +and the main filament) is interesting. As we have shown +in § 4, the morphology of the warm and cold gas is not +so obviously a head-tail structure, although there is a +warm gas filament running through the region. +This +region, may owe its overall morphology and kinematics +to another process–for example the collision of a dense +molecular concentration (the compact core) with a dif- +fuse region near NGC 7319, driving a ring into the tar- +get material. The elongation of the compact CO emit- +ting core along the direction which points towards the +center of the CO emitting ring, and the existence of +a clump of CO in the ring at the same velocity as the +core may support a dynamical connection. Future warm +H2 spectroscopy will determine whether the warm gas +connecting the two CO structures with disparate radial +velocities are related, or merely chance alignments of +two different cloud systems along the line of sight. Such +events, if they occur, are likely to be relatively rare, and +unlikely to provide a generic mechanism for the dissipa- +tion of collisional kinetic energy into widespread warm +H2 emission. +Finally we come to the complex network of filaments +around NGC 7319, which, for the most part, appear pink +in the false color representation of Figure 4. This implies +that the 7.7µm and 15µm emission is much stronger here +than in the regions which are warm H2 dominated. +The irregular radial structure of the filaments might +suggest a nuclear origin connected with the AGN in +NGC 7319. +The JWST MRS data in the central re- +gion of the AGN indeed reveal a spatially-resolved ion- +ized and molecular outflow, with kinematical evidence of +an interaction between the AGN jet and the host ISM +(Pereira-Santaella et al. 2022). Perhaps previous jet ac- +tivity from the AGN has somehow formed these strange +filaments. +6. CONCLUSIONS +We have made a detailed comparison, at sub-arcsec +resolution, between previous HST, recent ERO JWST +NIRCam and MIRI imaging and ALMA CO (2-1) imag- +ing spectroscopy of three targeted regions approximately +3 kpc in size in the large-scale (40-50 kpc) intergalactic +gas in Stephan’s Quintet. Based on a comparison with +a full spectral map by Spitzer and observations made by +JWST, we demonstrated that much of the gas along the +main molecular filament detected in the F1000W/MIRI +filter is dominated by strong emission from the 0-0 S(3) +line originating in warm molecular hydrogen. This al- +lowed us to reveal differences in spatial distribution of +the warm H2 gas phase, measured by JWST, and the +colder CO-emitting gas, at comparable subarcsec reso- +lution in three regions targeted by ALMA. Our study +has come to the following conclusions: +• The three zoomed-in regions we studied show a +great variety of structure in the warm H2 when +compared with the cold molecular gas on scales +from 150 pc to 3 kpc. In many cases diffuse ion- +ized gas (observed by HST and F200W/NIRCam) +follows the warm H2 distribution in the two struc- +tures believed to be shock-dominated (ALMA +Field 5 and 6), whereas the cold molecular gas was +more clumpy, and not always strongly correlated +with the warm H2. Both structures show an over- +abundance of warm H2 over the colder gas with +reasonable assumption about the conversion from +CO line intensity to total H2 mass. In the star for- +mation dominated region (at the center of ALMA +Field 4), the 10µm and 7.7µ bands are well corre- +lated, as expected in regions dominated by PDRs. +• In the (ALMA Field 6) region observed near the +center of the main shocked filament, the warm H2 +emission resembles a head-tail structure. Strong + +24 +Appleton et al. +CO emission forms a partial shell of cold H2 which +is embedded in a bright extended hotspot of warm +H2 (F1000W emission), containing approximately +4 × 106 +M⊙ of warm gas if we assume the ex- +citation of the gas is similar to that inferred on +a larger scale by Spitzer. This triangular-shaped +head also shows extended ionized gas. The tail is +composed of clumps of warm and cold H2 emission +scattered in two streams eastwards from the head. +Near the apex of the triangular warm H2 head and +in some regions along the tail, CO linewidths of 60 +< FWHM (km s−1) < 100 km s−1 +are observed +suggesting quite turbulent (probably gravitational +unbound) gas. We suggest that we are witness- +ing the shattering of a dense cold molecular gas +cloud in a fast hot wind, leading a fog of warm +H2 (see below). +We believe that this process is +a commonly occurring phenomenon in Stephan’s +Quintet, judging by the existence of other head- +tail strcutures along the main shock and elsewhere +in the group. +• In another shock-dominated warm H2 +region +(ALMA Field 5) which is not part of the main +molecular filament, we notice a different mor- +phology. +Warm molecular gas appears to con- +nect together a compact elongated CO emitting +clump which points towards the center of a ring +of CO emission separated by 1.5 arcsec (∼ 700 +pc). The two CO structures have very different +radial velocities, with the compact core having +strongly blueshifted emission with respect to the +ring. Both the ring and the core show broad (∼ +100 km s−1FWHM) CO line-widths, suggesting +that they are highly turbulent. +One possibility +is that a dense molecular cloud (perhaps from the +outer parts of the intruder galaxy) has impacted +a low-density gas filament in the outer parts on +NGC 7319, entraining gas and dissipating kinetic +energy which heats the bridge between them. Such +events are probably rare. Alternatively, the con- +nection between the two CO clouds by the warm +gas may be entirely coincidental. +• The distribution of warm, cold and ionized gas +in the SQ-A star forming region (ALMA Field 4) +is more typical of the gas distribution of a small +dwarf galaxy. The warm and cold molecular gas +are contained within a roughly elongated stellar +distribution (as observed by NIRCam). The lack +of clear rotation in the CO-emitting gas may argue +against this object being created in a tidal stream. +Instead, we suggest it may be more consistent with +the collapse of a rare pre-existing dense molecular +structure which has become gravitationally unsta- +ble after it was compressed by the passage of the +large-scale shock, leading to a starburst. +We present a conceptual picture for the transfer +of kinetic energy from the intruder galaxy’s large +scale shock into a pre-existing multi-phase tidal +filament via the formation and subsequent molec- +ular cloud shattering of cold molecular clouds in +a post-shock layer. After the main shock passes +over a multi-phase tidal filament, both X-ray emit- +ting hot gas, and dense molecular clouds can form +together if the molecular formation time on dust +grains is shorter than the cloud-crushing time of +the clouds. +However, as the molecular clouds +grow, the velocity of the denser forming clouds +relative to the diffuse hot gas causes them to ex- +perience ram-pressure stripping from the hot com- +ponent. Models show that in this situation, the +cold molecular clouds can shatter into a fog of +tiny molecular clouds, heated by an exchange in +energy with the hot medium. The fog clouds are +expected to have low internal velocity dispersion +which we associate with the clouds that radiate +mid-IR H2 emission (low shock velocities in the +molecular gas). The warm and cold gas associated +with ALMA Field 6 is likely one example of many +such cold clouds being turned into a warm H2 fog +all along the main SQ large-scale shock front. Re- +gions outside the main molecular shock front also +seem to show a head-tail morphology (for exam- +ple one extremely long, 20kpc tail to the SW of +the SQ-A star forming region). These may be ex- +amples of a similar process, where the flow of hot +gas may originate in the relative motion of the +intruder galaxy’s hot halo relative to possible pre- +existing molecular clouds in the system. Future +mid-IR spectroscopy will allow us to test this pic- +ture, as the kinematics of the warm H2 emitting +fog is likely to be very different from that of the +cold clouds, as they rapidly accelerate up to the +flow velocity of the hot gas, before eventually be- +ing destroyed. + +The Multi-phase IGM in Stephan’s Quintet +25 +We acknowledge the remarkable dedication of the sci- +entists and engineers who made the James Web Space +Telescope possible, and especially the Early Release +Observation (ERO) science team upon which some of +our paper is based. +This work is based, in part, on +observations made with the NASA/ESA/CSA James +Webb Space Telescope. The data were obtained from +the Mikulski Archive for Space Telescopes at the Space +Telescope Science Institute, which is operated by the +Association of Universities for Research in Astronomy, +Inc., under NASA contract NAS 5-03127 for JWST. +These observations are associated with program # +2732. +This paper makes use ALMA data from pro- +gram ID: 2015.1.00241 (P.I. P. Guillard). ALMA is a +partnership of ESO (representing its member states), +NSF (USA) and NINS (Japan), together with NRC +(Canada), MOST and ASIAA (Taiwan), and KASI (Re- +public of Korea), in cooperation with the Republic of +Chile. +The Joint ALMA Observatory is operated by +ESO, AUI/NRAO and NAOJ. The National Radio As- +tronomy Observatory is a facility of the National Sci- +ence Foundation operated under cooperative agreement +by Associated Universities, Inc. +E.O’S. acknowledges +support from NASA through Chandra Award Number +GO8-19112A. +APPENDIX +A. ALMA AND CARMA CO KINEMATICS OF THE ALMA FIELD REGIONS 4,5 AND 6 +In this Appendix we present Table A1, Table A2 and Table A3, which provide observed and derived properties of +individually extracted spectra for Field6, Field 5 and Field4 respectively based on the CO (2-1) ALMA data. The +regions (e. +g. +F6reg1) refer to the defined region extracted from each of the ALMA fields shown in Figure A1, +Figure A2, and Figure A3 respectively. Those figures show a grey-scale representation of the surface density map of +the CO integrated emission with elliptical (blue) beam-size extraction areas marked. The spectra and fitted Gaussian +profiles are also shown. At the bottom of Figure A1 and Figure A2 we present the integrated emission profiles. For +Figure A3, the integrated spectrum is F4p1. +In each table we provide the following information. Column 1 is the region name, Column 2 is the heliocentric +velocity of the emission, with uncertainty, Column 3 is the FWHM (and uncertainty) of the spectrum in km s−1, and +Columns 4 and 5 provide the CO line integral in Jy-km s−1 for the observed CO (2-1) emission profile and the derived +equivalent CO (1-0) line integral respectively. The assumption for the conversion to a CO (1-0) line integral is given +in the footnotes. Column 6 give the total (cold) H2 mass in each extracted region assuming a Galactic value for the +conversion from CO line intensity to molecular column density (see footnotes). Column 7 gives the total gas mass +assuming 36% Helium content. +In the shock dominated regions Field 5 and 6, the typical mass within the ALMA beam within the brighter regions +is a few ×106M⊙, whereas the brightest regions in Field 4 (SQ-A) are a factor of 8 to ten times higher, indicating +significantly higher hydrogen columns in SQ-A. This difference may be even higher if the value for Xco is larger in the +shock-dominated regions. + +26 +Appleton et al. +Table A1. Properties of CO emission for beam-sized extractions in Field 6 (see Figure A1) +Region +Vhelio +FWHM +Σ(Sv∆V ) +Σ(Sv∆V )a +M(H2)a +Mgasb +CO(2-1) +CO(1-0) +×106 +×106 +(km s−1) +(km s−1) +(mJy-km s−1) +(mJy-km s−1) +(M⊙) +(M⊙) +F6reg1 +6052 (±1) +36 (±3) +92 (± 8) +29 (±2) +1.9 (±0.2) +2.6 (±0.2) +F6reg2 +6067 (±2) +27 (±5) +38 (± 7) +12 (±2) +0.8 (±0.1) +1.1 (±0.2) +F6reg3 +6069 (±2) +40 (±4) +98 (±10) +31 (±3) +2.0 (±0.2) +2.8 (±0.3) +F6reg4 +6071 (±1) +46 (±3) +113 (± 9) +35 (±3) +2.4 (±0.2) +3.2 (±0.3) +F6reg5 +6065 (±2) +56 (±6) +88 (±10) +28 (±3) +1.8 (±0.2) +2.5 (±0.3) +F6reg6 +6082 (±2) +50 (±4) +113 (±10) +35 (±3) +2.4 (±0.2) +3.2 (±0.3) +F6reg7 +6086 (±6) +102 (±13) +85 (±12) +27 (±4) +1.8 (±0.2) +2.4 (±0.3) +F6reg8 +6068 (±2) +34 (±4) +78 (±9) +24 (±3) +1.6 (±0.2) +2.2 (±0.3) +F6reg9 +6072 (±2) +42 (±4) +75 (±8) +23 (±3) +1.6 (±0.2) +2.1 (±0.2) +F6reg10 +6068 (±2) +37 (±4) +68 (±8) +21 (±3) +1.4 (±0.2) +1.9 (±0.2) +F6reg11 +6058 (±1) +43 (±3) +99 (±8) +31 (±3) +2.1 (±0.2) +2.8 (±0.2) +F6reg12 +6061 (±3) +77 (±7) +96 (±10) +30 (±3) +2.0 (±0.2) +2.7 (±0.3) +F6reg13 +6051 (±3) +66 (±6) +107 (±11) +34 (±4) +2.2 (±0.2) +3.1 (±0.3) +F6reg14 +6053 (±1) +41 (±3) +113 (±9) +35 (±3) +2.4 (±0.2) +3.2 (±0.3) +F6reg15 +6021 (±3) +91 (±7) +156 (±14) +49 (±4) +3.3 (±0.3) +4.4 (±0.4) +F6reg16 +6013 (±2) +79 (±6) +144 (±11) +45 (±4) +3.0 (±0.2) +4.1 (±0.3) +F6reg17 +6014 (±3) +52 (±7) +64 (±10) +20 (±3) +1.3 (±0.2) +1.8 (±0.3) +F6reg18 +6007 (±1) +20 (±3) +42 (±6) +13 (±2) +0.9 (±0.1) +1.2 (±0.2) +F6reg19 +6007 (±1) +30 (±3) +72 (±8) +23 (±3) +1.5 (±0.2) +2.1 (±0.2) +F6reg20 +6012 (±3) +56 (±6) +88 (±11) +27 (±3) +1.8 (±0.2) +2.5 (±0.3) +Integratedc +– +– +2380 (±230) +743 (±70) +49.69 (±5.0) +67.6 (±7.0) +aWe assume a conversion between a line flux at CO (1-0) to that of CO (2-1) of SCO(1−0) /SCO(2−1) += (ν1−0/ν2−1)2 (r21)−1, where ν1−0 and ν2−1, where ν1−0 and ν2−1 are the rest frequencies of the +transitions, and r21 is assumed to be 0.8, similar to that measured in the Taffy galaxy bridge (Zhu +et al. 2007), which shares many similarities with the gas in Stephan’s Quintet (see Appleton et al. +2022). The H2 masses presented here are for an XCO value = XCO,20 = N(H2)/ICO = 2 × 1020 cm−2 +(K−km s−1)−1, the standard value assumed for our Galaxy. In the text we discuss how this may +not be applicable in such a turbulent region. MH2 = 7.72 × 103D2Σ(Sv ∆V )(1+z)−1, where D = +94 Mpc, and Σ(Sv∆V ) is the CO(1−0) line integral with Sv in Jy and the velocity V of the gas in +km s−1. We assume z = 0.02. +b Total gas mass Mgas includes a 36% correction for Helium (Bolatto et al. 2013). +c Integral properties of the whole structure are based on the summed extracted spectrum obtained +from the cyan polygons shown in the inset to Figure A1 and the last spectrum in the same figure. +B. PHOTOMETRY AND AGE DETERMINATION +FROM THE FEDOTOV (2014) CATALOG +Photometry of the Stephan’s Quintet system was per- +formed by Fedotov (2014) using UBVmVI photometry +based on observations made using HST and the F336W +(U), F438W (B), F547W (Vm), F606W (V) and F814W +(I) with WFC3/UV. The magnitudes where converted +to the Vega system assuming zeropoints of 23.484 (U), +24.974 (B), 24.748 (Vm), 25.987 (V) and 24.680 (I). Cor- +rections for foreground extinction assumed A336 = 0.353 +mag, A438 = 0.288 mag, A606 = 0.197 mag, and A814 = +0.122 mag. Super Star Clusters(SSCs) were extracted +from the images using DAOfind. +At the distance of +Stephan’s Quintet, the WFC3 resolution means that ob- +jects as large as 12 pc would be classified as SSCs, but +may include more than one cluster. SSCs were selected +with a color cut of B-V < 1.5 or V-I < 1.0 in order + +The Multi-phase IGM in Stephan’s Quintet +27 +Table A2. Properties of CO emission for beam-sized extractions in Field 5 (see Figure A2) +Region +Vhelio +FWHM +Σ(Sv∆V ) +Σ(Sv∆V )a +M(H2)a +Mgasb +CO(2-1) +CO(1-0) +×106 +×106 +(km s−1) +(km s−1) +(mJy-km s−1) +(mJy-km s−1) +(M⊙) +(M⊙) +F5reg1 +6416 (±10) +145 (±23) +0.10 (±0.02) +0.030 (±0.005) +2.0 (±0.3) +2.7 (±0.5) +F5reg2 +6399 (±4) +117 (±10) +0.15 (±0.01) +0.045 (±0.004) +3.0 (±0.3) +4.1 (±0.4) +F5reg3 +6388 (±5) +108 (±12) +0.14 (±0.02) +0.045 (±0.005) +3.0 (±0.4) +4.1 (±0.5) +F5reg4 +6412 (±6) +75 (±14) +0.06 (±0.01) +0.018 (±0.004) +1.2 (±0.2) +1.7 (±0.3) +F5reg5 +6631 (±4) +102 (±10) +0.15 (±0.02) +0.047 (±0.005) +3.2 (±0.3) +4.3 (±0.4) +F5reg6 +6651 (±4) +101 (±8) +0.14 (±0.01) +0.045 (±0.004) +3.0 (±0.3) +4.1 (±0.4) +F5reg7 +6640 (±3) +107 (±8) +0.17 (±0.01) +0.052 (±0.004) +3.5 (±0.3) +4.7 (±0.4) +F5reg8 +6666 (±2) +60 (±6) +0.10 (±0.01) +0.030 (±0.003) +2.0 (±0.2) +2.7 (±0.3) +F5reg9 +6679 (±2) +61 (±6) +0.09 (±0.01) +0.029 (±0.003) +1.9 (±0.2) +2.6 (±0.3) +F5reg10 +6688 (±12) +177 (±29) +0.12 (±0.04) +0.037 (±0.014) +2.5 (±0.8) +3.4 (±1.2) +F5reg11d +– +– +– +– +– +– +F5reg12d +– +– +– +– +– +– +F5reg13 +6628 (±6) +79 (±15) +0.07 (±0.01) +0.021 (±0.004) +1.4 (±0.3) +1.9 (±0.4) +F5reg14Ae +6419 (±9) +87 (±21) +0.052 (±0.02) +0.016 (±0.008) +1.1 (±0.6) +1.5 (±0.8) +F5reg14Be +6671 (±5) +59 (±12) +0.05 (±0.01) +0.015 (±0.003) +1.0 (±0.2) +1.4 (±0.3) +F5reg15 +6671 (±5) +59 (±12) +0.05 (±0.01) +0.015 (±0.003) +1.0 (±0.2) +1.4 (±0.3) +F5reg16 +6680 (±3) +48 (±6) +0.07 (±0.01) +0.021 (±0.003) +1.4 (±0.2) +1.9 (±0.3) +Integrated +– +– +2300 (±230) +718 (±70) +48.1 (±4) +6.54 (±7) +aWe assume a conversion between a line flux at CO (1-0) to that of CO (2-1) of SCO(1−0) /SCO(2−1) = +(ν1−0/ν2−1)2 (r21)−1, where ν1−0 and ν2−1 are the rest frequencies of the transitions, and r21 is as- +sumed to be 0.8, similar to that measured in the Taffy galaxy bridge (Zhu et al. 2007), which shares +many similarities with the gas in Stephan’s Quintet (see Appleton et al. 2022). The H2 masses +presented here are for an XCO value = XCO,20 = N(H2)/ICO = 2 × 1020 cm−2 (K−km s−1)−1, +the standard value assumed for our Galaxy. In the text we discuss how this may not be applicable +in such a turbulent region. +MH2 = 7.72 × 103D2Σ(Sv ∆V )(1+z)−1, where D = 94 Mpc, and +Σ(Sv∆V ) is the CO(1−0) line integral with Sv in Jy and the velocity V of the gas in km s−1. We +assume z = 0.02. +b Total gas mass includes a 36% correction for Helium (Bolatto et al. 2013). +c Integral properties of the whole structure are based on the summed extracted spectrum obtained +from the cyan polygons shown in the inset to Figure A2 and the last spectrum in the same figure. +dlow signal to noise region +eTwo features present. 14A = faint broad emission, 14B = brighter higher velocity emission. +to minimize detection of foreground objects. In addi- +tion, to be considered a SSC, the object had to have a +photometric error of < 0.3 mag, fullfill a sharpness cri- +terion, and exhibit a goodness-of-fit criterion based on +the shape of the object compared with the point-spread- +function. +Table B4 presents B, Vm, V, I photometry for objects +we identify in the U-band images which correspond to +the youngest objects associated with the three studied +fields. For Field 4 (SQ-A) most of the objects with U- +band detections were of moderate age (F4A-F) ranging +from 72-360 Myrs. NIRCam detects a large number of +potential star clusters in the bidy of this likely forming +dwarf galaxy for which we defer discussion to another +paper. In Field 5, only one object (F5D) is clearly de- +tected in U-band with an age of 8 Myr. This SSC is +one of the objects that lies in the string of potentially +background galaxies to the East and North of the CO +ring. In Field 6, F6A and B are the bright young stellar +objects (3-5 Myr) with young ages seen associated with + +28 +Appleton et al. +Table A3. Properties of CO emission for beam-sized extractions in Field 4 (see Figure A3) +Region +Vhelio +FWHM +Σ(Sv∆V ) +Σ(Sv∆V )a +M(H2)a +Mgasb +CO(2-1) +CO(1-0) +×106 +×106 +(km s−1) +(km s−1) +(mJy-km s−1) +(mJy-km s−1) +(M⊙) +(M⊙) +F4reg2 +6708 (±1) +40 (±3) +0.53 (±0.04) +0.166 (±0.013) +11.1 (±0.9) +15.1 (±1.2) +F4reg3 +6731 (±1) +37 (±2) +0.59 (±0.04) +0.183 (±0.012) +12.3 (±0.8) +16.7 (±1.1) +F4reg4 +6706 (±1) +43 (±3) +0.19 (±0.02) +0.058 (±0.005) +3.9 (±0.3) +5.3 (±0.5) +F4reg5 +6702 (±1) +35 (±3) +0.15 (±0.02) +0.048 (±0.005) +3.2 (±0.3) +4.4 (±0.5) +F4reg6 +6731 (±1) +35 (±3) +0.17 (±0.02) +0.054 (±0.006) +3.6 (±0.4) +5.0 (±0.5) +F4reg7 +6736 (±1) +35 (±2) +0.21 (±0.02) +0.066 (±0.005) +4.4 (±0.3) +6.0 (±0.4) +F4reg1(int)c +6719 (±1) +47 (±1) +1.4 (±0.04) +0.422 (±0.012) +28.2 (±0.8) +38.4 (±1.1) +aWe assume a conversion between a line flux at CO (1-0) to that of CO (2-1) of SCO(1−0) /SCO(2−1) += (ν1−0/ν2−1)2 (r21)−1, where ν1−0 and ν2−1 are the ratios of the rest frequencies of the transitions, +and r21 is assumed to be 0.8, similar to that measured in the Taffy galaxy bridge (Zhu et al. 2007), +which shares many similarities with the gas in Stephan’s Quintet (see Appleton et al. 2022). The H2 +masses presented here are for an XCO value = XCO,20 = N(H2)/ICO = 2 × 1020 cm−2 (K−km s−1)−1, +the standard value assumed for our Galaxy. In the text we discuss how this may not be applicable in +such a turbulent region. MH2 = 7.72 × 103D2Σ(Sv ∆V )(1+z)−1, where D = 94 Mpc, and Σ(Sv∆V ) +is the CO(1−0) line integral with Sv in Jy and the velocity V of the gas in km s−1. We assume z = +0.02. +b Total gas mass Mgas = MH2 × 1.36, includes a 36% correction for Helium (Bolatto et al. 2013). +c Integral properties of the whole structure are based on the summed extracted spectrum obtained +from position 1 (see Figure A3.) +the PAH region in the NW CO clump structure seen +in Figure5c. +The other clumps (F6C, F6D and F6E) +as quite close to the triangular head of the warm H2 +emission, and also exhibit young ages. The RA and Dec +positions are from the original catalog, and were found +to be shifted slightly relative to the GAIA coordinate +system used in the paper. +The method used by (Fedotov 2014) to estimate the +intrinsic Av and age of the clusters was by compari- +son of the SED with population synthesis models. The +use of the U-band and the V574 filter, helps to break +the age-reddening degeneracy, as well as avoiding sig- +nificant contamination with Hα emission in the F606W +filter. The age and extinction of each cluster was found +by least-squares fitting the SEDs to predictions from the +2009 updates to the Bruzual & Charlot (2003) models +with an assumed Chabrier (2003) initial mass function. +The best fits values for E(B-V), Av and age forthe clus- +ters are given in the Table. +REFERENCES +Alatalo, K., Davis, T. A., Bureau, M., et al. 2013, MNRAS, +432, 1796, doi: 10.1093/mnras/sts299 +Allen, R. J., & Hartsuiker, J. W. 1972, Nature, 239, 324, +doi: 10.1038/239324a0 +Appleton, P. 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J. +2008, ApJ, 684, 845, doi: 10.1086/590233 +Pereira-Santaella, M., ´Alvarez-M´arquez, J., Garc´ıa-Bernete, +I., et al. 2022, A&A, 665, L11, +doi: 10.1051/0004-6361/202244725 + +30 +Appleton et al. +Figure A1. Field 6 extracted spectra for CO (2-1) spectra covering, the CO beam-size extraction regions (blue ellipses on the +inset) labeled in in inset at the center of the figure. The inset shows a grey-scale representation of the CO integrated emission. +The cyan polygons around the emission show the extracted areas which were summed for an integrated spectrum over the whole +region. Gaussian fits to the CO line profiles are shown superimposed on the spectra with a red line. Region 21 shows faint +emission and was not fit. The red and blue vertical lines indicate the average velocity of the intruder galaxy NGC 7318b, and +the average group velocity of the main SQ members respectively. The integrated spectrum obtained over the cyan polygons is +also presented as the last spectrum. The integrated properties of all the spectra are given in Table A1. + +fit +fit +fit +F6reg1. +F6reg2. +F6reg3. +F6reg4. +VINT +VGROUP +VINT +VGROUP +VGROUP +VINT +VGROUP +VINT +0 +0 +0 +M +0 +AMI +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +fit +fit +fit +5 +fit +F6reg5. +F6reg6. +F6reg7. +F6reg8. +VINT +VGROUP +VINT +VGROUP +ViNT +VGROUP +VINT +VGROUP +MA +WM +WAA +0 +VIAN +0 +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) + (iun Aasua xni +fit +fit +fit +F6reg9. +F6reg10 +F6reg11 +F6reg12 +VINT +VGROUP +VINT +VGROUP +VINT +VGROUP +VINT +VGROUP +MAAA. +ANAA/Y +0 +AA +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +fit +Field 6. +(iu) aisua xni +fit +F6reg13 +F6reg14 +VINT +VGROUP +VINT +VGROUP +0 +MMAW +Velocity (Vhelio) +Velocity (Vhelio) +fit +fit +F6reg15 +F6reg16 +VINT +VGROUP +VINT +VGROUP +0 +M/W +Velocity (Vhelio) +Velocity (Vhelio) +fit +fit +fit +fit +F6reg17 +F6reg18 +F6reg19 +F6reg20 +VINT +VGROUI +VINT +VGROUP +VGROUP +VINT +VGROUP +VINT +0 +0 +aw +0 +0 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +Velocity (Vhelio) +5 +Integral +F6reg21 +20 +VINT +VGROUP +VGBOUP +0 +c +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +Velocity (Vhelio) +Velocity (Vhelio)The Multi-phase IGM in Stephan’s Quintet +31 +Figure A2. Field 5 extracted spectra for CO (2-1) spectra covering the CO beam-size extraction regions (green ellipses) labeled +on the inset at the center of the figure. The inset shows a grey-scale representation of the CO integrated emission, with contours +of the same emission superimposed in red. Gaussian fits to the CO line profiles are shown overlayed on each of the spectra with +a red line. Regions 11 and 12 are too faint to fit. The red and blue vertical lines indicate the average velocity of the intruder +galaxy NGC 7318b, and the average group velocity of the main SQ members respectively. The integrated spectrum obtained +over the cyan polygons is also presented as the last spectrum. The integrated properties of all the spectra are given in Table A2. +Note that Regions 1, 2, 3, and 4 have significantly different lower velocities than the rest of the emission. Region 14 also shows +a double profile (labeled A and B), with faint emission from the low velocity structure as well as the higher velocity structure +(13, 14, 15 and 16). + +fit +Reg 1 +fit +Reg 2 +fit +Reg 3 +fit +Reg 4 +Reg 5 +fit +6 +9 +9 +6 +VINT +VGROUP +VINT +VGROUP +VINT +VINT +VGROUP +VINT +VGROUP +0 +0 +MM +AMM +W +A /LA. +M.N +VV +0 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +Velocity km/s (opt) +Velocity km/s (opt) +Velocity km/s (opt) +Velocity km/s (opt) +Velocity km/s (opt) +fit +fit +Reg 7 +6 +Reg 6 +6 +arcsec +VINT +VGROUP +VINT +VGROUP +1 +1 +AMANM +C +0 +5500 +6000 +6500 +7000 +5500 +6000 +7000 +Velocity km/s (opt) +Velocity km/s (opt) +fit +9 +Reg 8 +Reg 9 +6 +ViNT +VGROUP +VINT +VGROUP +Field 5 +1 +0 +0 +A.M +M MA +5500 +6000 +6500 +7000 +Velocity km/s (opt) +5500 +6000 +6500 +7000 +Velocity km/s (opt) +fit +fit +9 +Reg 10 +6 +Reg 11 +6 +Reg12 +6 +Reg13 +Reg14 +9 +VINT +VGROUP +VINT +VGROUP +2 +VINT +VGROUP +VINT +VGROUP +VINT +VGROUP +1 +1 +1 +1 +0 +YA +0 +W +AA +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +5500 +6000 +6500 +Velocity km/s (opt) +Velocity km/s (opt) +Velocity km/s (opt) +7000 +Velocity km/s (opt) +Velocity km/s (opt) +fit +fit +Reg 15 +Reg 16 +Integrated +VINT +VGROUP +VINT +VGROUP +1 +1 +LANA +AK +M +5500 +6000 +6500 +7000 +5500 +6000 +6500 +7000 +6000 +6200 +6400 +6600 +6800 +7000 +Velocity km/s (opt) +Velocity km/s (opt) +Velocity km/s (Helio)32 +Appleton et al. +Figure A3. Field 4 extracted spectra for CO (2-1) spectra covering the CO extraction regions (green ellipses) labeled on the +inset at the center of the figure. The inset shows a grey-scale representation of the CO integrated emission, with contours of +the same emission superimposed in red. Gaussian fits to the CO line profiles are shown overlayed on each of the spectra with +a red line. The red and black vertical lines indicate the average velocity of the intruder galaxy NGC 7318b, and the average +group velocity of the main SQ members respectively. + +30 +30 +fit +Pos 3 +fit +Pos 2 +25 +25 +15 +VINT +10 +VGROUP +VINT +VGROUP +5 +5 +IM +11 +6000 +6500 +7000 +7500 +6000 +6500 +7000 +7500 +Velocity km/s (opt) +Velocitykm/s (opt) +12 +12 +fit +VGROUP +fit +VINT +2 arcsec +VINT +VGROUP +10 +10 +Pos 4 +Pos 6 +Flux Density (mly) +8 +2 +3 +Flux Density (my) +8 +6 +6 +4 +4 +2 +LAMAMI +-2 +Field 4 +-2 +6000 +6500 +7000 +7500 +6000 +6500 +7000 +7500 +Velocity km/s (opt) +Velocitykm/s (opt) +12 +30 +12 +fit +Integrated +fit +fit +VINT +VGROUP +VINT +VGROUP +10 +25 +10 +Pos 1 +Flux Density (mjy) +Pos 5 +Flux Density (mjy) +Pos 7 +8 +8 +6 +6 +15 +4 +4 +10 +VINT +VGROUA +2 +2 +5 +0 +AWM +2 +6000 +6500 +7000 +7500 +6000 +6500 +7000 +7500 +6000 +6500 +7000 +7500 +Velocity km/s (opt) +Velocity km/s (opt) +Velocitykm/s (opt)The Multi-phase IGM in Stephan’s Quintet +33 +Pontoppidan, K., Blome, C., Braun, H., et al. 2022, arXiv +e-prints, arXiv:2207.13067. +https://arxiv.org/abs/2207.13067 +Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in +Astronomical Society of the Pacific Conference Series, +Vol. 77, Astronomical Data Analysis Software and +Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. +Hayes, 433. https://arxiv.org/abs/astro-ph/0612759 +Smith, J. D. T., Draine, B. T., Dale, D. A., et al. 2007, +ApJ, 656, 770, doi: 10.1086/510549 +Sulentic, J. W., Rosado, M., Dultzin-Hacyan, D., et al. +2001, AJ, 122, 2993, doi: 10.1086/324455 +Togi, A., & Smith, J. D. T. 2016, ApJ, 830, 18, +doi: 10.3847/0004-637X/830/1/18 +Trinchieri, G., Sulentic, J., Pietsch, W., & Breitschwerdt, D. +2005, A&A, 444, 697, doi: 10.1051/0004-6361:20052910 +Xu, C., Sulentic, J. W., & Tuffs, R. 1999, ApJ, 512, 178, +doi: 10.1086/306771 +Xu, C. K., Lu, N., Condon, J. J., Dopita, M., & Tuffs, R. J. +2003, ApJ, 595, 665, doi: 10.1086/377445 +Xu, C. K., Iglesias-P´aramo, J., Burgarella, D., et al. 2005, +ApJL, 619, L95, doi: 10.1086/425130 +Zhu, M., Gao, Y., Seaquist, E. R., & Dunne, L. 2007, AJ, +134, 118, doi: 10.1086/517996 + diff --git a/RtE1T4oBgHgl3EQfHgPM/content/tmp_files/load_file.txt b/RtE1T4oBgHgl3EQfHgPM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df582ee120a1814f93f25bd86be92a52fa88ae05 --- /dev/null +++ b/RtE1T4oBgHgl3EQfHgPM/content/tmp_files/load_file.txt @@ -0,0 +1,2467 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf,len=2466 +page_content='Draft version January 10, 2023 Typeset using LATEX twocolumn style in AASTeX631 Multi-phase gas interactions on subarcsec scales in the shocked IGM of Stephan’s Quintet with JWST and ALMA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Appleton,1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Guillard,2, 3 Bjorn Emonts,4 Francois Boulanger,5 Aditya Togi,6 William T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Reach,7 Kathleen Alatalo,8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Cluver,9, 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Diaz Santos,11 P-A Duc,12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='Gallagher,13 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Ogle,8 E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' CAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Beijing 100101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' People’s Republic of China 16National Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Chinese Academy of Sciences (NAOC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 20A Datun Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Chaoyang District,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Beijing 100101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' People’s Republic of China Abstract We combine JWST and HST imaging with ALMA CO(2-1) spectroscopy to study the highly tur- bulent multi-phase intergalactic medium (IGM) in Stephan’s Quintet on 25-150 pc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Previous Spitzer observations revealed luminous H2 line cooling across a 45 kpc-long filament, created by a gi- ant shock-wave, following the collision with an intruder galaxy NGC 7318b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We demonstrate that the MIRI/F1000W/F770W filters are dominated by 0-0 S(3) H2 and a combination of PAH and 0-0 S(5) H2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' They reveal the dissipation of kinetic energy as massive clouds experience collisions, interac- tions and likely destruction/re-cycling within different phases of the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In one kpc-scaled structure, warm H2 formed a triangular-shaped head and tail of compressed and stripped gas behind a narrow shell of cold H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In another region, two cold molecular clumps with very different velocities are con- nected by an arrow-shaped stream of warm, probably shocked, H2 suggesting a cloud-cloud collision is occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In both regions, a high warm-to-cold molecular gas fraction indicates that the cold clouds are being disrupted and converted into warm gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We also map gas associated with an apparently forming dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We suggest that the primary mechanism for exciting strong mid-IR H2 lines throughout Stephan’s Quintet is through a fog of warm gas created by the shattering of denser cold molecular clouds and mixing/recycling in the post-shocked gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Without spectroscopy, JWST cannot provide a complete picture of the kinematics and excitation of the shocked warm gas, but it reveals the rich variety of ways that different gas phases interact with one another in Stephan’s Quintet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' INTRODUCTION Since the discovery of a filament of radio continuum in the intergalactic medium (IGM) of the Stephan’s Quintet (Allen & Hartsuiker 1972), this compact galaxy group has been studied at many wavelengths to try to better understand that remarkable nature of its multi- phase intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' As suspected in the early studies (Moles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Sulentic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 1999), the overall picture that has emerged is that one of the galaxies, NGC 7318b, is colliding with the diffuse arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='02928v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='GA] 7 Jan 2023 2 Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' intergalactic medium of the main group at a very high velocity creating a giant (45 kpc) filament of shocked gas seen from the X-rays (Trinchieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' O’Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2009), in the UV (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2005) and in ionized gas emissions (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Iglesias-P´aramo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Konstantopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Duarte Puertas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Studies of the stellar populations along the giant shocked structure also suggested that star clusters are beginning to form there (Gallagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Fedotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2011), although at a low rate, perhaps because some of the forming clusters lie inside bubbles of highly excited ionized gas (Konstantopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The physical conditions within the gas along the main shock front are far from simple and require more de- tailed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Mid-IR spectroscopy of the main filament showed that the entire filament is radiating strongly in pure rotational lines of molecular hydrogen (Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' See Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' An expla- nation of the excitation of such strong, L(H2) > 1041 erg s−1, molecular hydrogen emission from a region experi- encing fast shocks from the intruder’s high velocity was presented by Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The model assumes that the main intruder shock propagates into a clumpy pre-shock medium, heating low-density regions to X-ray temperatures, but causing mildly over-dense regions to collapse to form H2 on grains on timescales shorter than the dynamical timescale of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These molec- ular clouds experience low-velocity molecular shocks as they continue to dissipate mechanical energy from their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The interaction of these warm H2 clouds with their surroundings is a key area that we will in- vestigate with the James Web Space Telescope (JWST) and Atacama Large Millimeter Array (ALMA) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A more detailed analysis of the H2 excitation proper- ties of the warm H2 in Stephan’s Quintet was discussed in Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2017), where it was shown that in order to heat so much H2 (109M⊙) at low temperatures (150-400 K), a mix of low-velocity magnetic C-shocks (∼5-10 km s−1) and faster (15-25 km s−1) J-shocks were needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Very broad spectral linewidths have been observed in the Stephan’s Quintet filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2012) used the IRAM 30m telescope to detect CO emission from several regions in the main filament and the bridge, and revealed line profiles of several hundred km s−1 in at least three separate kinematic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In ad- dition to gas detected at the systemic velocity of the intruder NGC 7318b (5774 km s−1), and that of the re- maining group members (6600 km s−1), broad lines of CO emission were also detected at intermediate veloci- ties, suggesting that molecular gas has formed out of dif- fuse shocked gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Recent UV spectroscopic observations with the Hubble Space Telescope’s (HST) Cosmic Ori- gins Spectrograph (COS) targeted five regions along the main filament and in the bridge, and detected extremely broad Lyα emission (FWHM exceeding 1500 km s−1) over small sampled regions of ∼1 kpc scale (Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The lines were even broader in some cases than those seen in the [CII] line (Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2013) using Herschel, suggesting some resonant scattering of UV photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The detection of broad-line, powerful Lyα emission, broad [CII] neutral gas, and broad CO emis- sion strongly supports the picture of a highly turbulent medium containing many gas phases resulting from a turbulent cascade of energy from the large to the small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Although the Early Release Observations (ERO) made by JWST of Stephan’s Quintet cover the entire inner group (including NGC 7319, NGC 7318a/b, NGC 7317 and the foreground galaxy NGC 7320), this pa- per will concentrate on the main shocked filament and bridge that were observed by Spitzer in spectral mapping mode (Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2017) using the InfraRed Spectrograph (IRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Houck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' As shown in Figure 1, this included the main north-south H2 filament between NGC 7318b/a and NGC 7319, as well as the H2 bridge connecting NGC 7319 to the main filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' It is known that much of the gas, including HI, ionized gas and molecular gas lies outside the main body of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Although the previous IRS observations of the mid-IR lines covered much of this gas, it did not cover all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The Long-low IRS coverage included all of the area shown in Figure 1, but IRS Short-low coverage was restricted to only the the main H2 filament and part of the H2 bridge (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Data from the JWST and ALMA allow us to directly compare, for the first time at comparable resolution, the distribution of warm 0-0S(3) H2 (as we will demonstrate via the F1000W MIRI Band) to cooler molecular hydro- gen mapped through the low-J CO(2-1) line, as well as ionized gas emission (Hα with WFC3 HST, and Paα with NIRCam F200W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Very hot gas, which forms an important IGM component in Stephan’s Quintet is de- tected in X-rays throughout Stephan’s Quintet, and es- pecially in the main shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Its distribution cannot be compared in detail to our current observation, because even the deepest Chandra images (O’Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2009) do not detect enough photons at arcsecond scales to al- low meaningful comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These observations will help us gain an understanding of how turbulent energy, driven mainly by the intruder galaxy, is dissipated from large driving scales to smaller dissipation scales, and how this affects the cooling of The Multi-phase IGM in Stephan’s Quintet 3 the gas through different gas phases and temperature regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In particular, how can we explain so much energy flowing out through low-velocity shocks needed to produce the high radiant line luminosity from warm molecular hydrogen discovered by Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Although we do not expect to probe down to the smallest dissipation scales, we will show that we already see differences in the distribution of different thermal phases in the IGM at ∼150 pc scales probed by ALMA and JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The current paper will not try to address all aspects of the extended emission detected by JWST, but will primarily concentrate on two main objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Firstly, we compare the JWST MIRI images with the Spitzer IRS spectra and spectral images, to identify the domi- nant features present in the three MIRI filters (F770W, F1000W and F1500W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Secondly, we compare the dis- tribution of warm molecular gas, cold molecular gas and ionized gas in three contrasting regions within the main IGM between NGC 7319 and NGC 7318b/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The three regions we have chosen to emphasize in this paper were observed with ALMA in CO(2-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The primary beams (fields-of-view) of the three ALMA pointings and their associated field numbers are also identified on Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We will the regions in more detail in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Although we touch on the sparsity of recent star formation in two out of three of the studied fields, we leave a full discussion of the star formation properties and cluster formation to a later paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In § 2 we will describe observations made with JWST, HST, the Atacama Large Millimeter Array (ALMA) and the Combined Array for Research in Millimeter- wave Astronomy (CARMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Given that we do not yet have spectroscopy of the IGM in Stephan’s Quintet, we present, in § 3 a discussion of what each of the MIRI and NIRCam images is likely to contain by comparing the JWST images with Spitzer spectral maps of espe- cially warm H2 and PAH bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In § 4 we present the results of our zoomed-in study of three major emission complexes in the intergalactic medium observed at high resolution by ALMA in the CO (2-1) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This allows us to synthesize from JWST, ALMA, and HST, the best possible information we have about the nature of the warm and cold molecular gas, the ionized gas and the formation of star clusters, all at subarcsec resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' § 4 also includes a discussion of the relative fraction of warm and cold molecular gas obtained for the regions using both the JWST and ALMA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In § 5 we dis- cuss the results in the context of our understanding of 1 Other fields were originally proposed in the ALMA proposal but only these three regions were actually observed turbulence and shocks which we believe largely domi- nate the gas dynamics of this multi-phase IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In § 6 we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We assume in this paper a distance to Stephan’s Quin- tet of 94 Mpc (for H0 = 70 km s−1Mpc−1 and an assumed group heliocentric systemic velocity of 6600 km s−1) for consistency with previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Apple- ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' At this distance 1 arcsec corresponds to a linear scale of 456 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The over- all scale of the giant north-south intergalactic filament in Stephan’s Quintet is 47 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' OBSERVATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' JWST images The first description of the NIRCam and MIRI ERO observations, and the choice of Stephan’s Quin- tet as a target (PID 2732) is presented in Pon- toppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The NIRCam images were taken in 6 filters (F090W/F277W, F150W/F356W, F200W/F444W), with an integration time of approxi- mately 20 min for each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The FULLBOX 5-points “5TIGHT” dither pattern was used, resulting in a large rectangle mosaic of 3 pairs of dithered tiles, covering 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The MIRI image covers a much smaller field of view than the NIRCam mosaic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' the central galaxies, NGC7318a/b, NGC7319 (a Seyfert 2 galaxy), and NGC7320 (foreground galaxy), using 4 tiles in three MIRI bands, F770W, F1000W, and F1500W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The integration times were about 22 min per filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The large cycling 8-points dither pattern was used to maxi- mize the spatial coverage of a single tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We worked with the level 2b images retrieved from MAST2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' ALMA Observations Observations of CO(2-1) in the three fields of Stephan’s Quintet were made with the ALMA 12m ar- ray in a mosaic observing mode on 21 July 2015 (ID: 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='00241;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' PI: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Guillard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The total integration time for the mosaic was 35 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The spectral setup consisted of four spectral windows of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='875 GHz with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='9 MHz channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' One of the spectral windows was centred on the redshifted CO(2-1) (νrest = 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='538 GHz), while the remaining three spectral windows covered line-free continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The standard ALMA calibration plan included bandpass, phase, and flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2 The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is op- erated by the Association of Universities for Research in Astron- omy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 4 Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Stephan’s Quintet: A grey-scale representation of the NIRCam F150W ERO image of Stephan’s Quintet with overlays of the Spitzer IRS image of the 0-0 S(1) H2 line (blue contours) and the half-power primary beam size of the ALMA CO (2-1) observations (red circles) described in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Contours of H2 emission are in units of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='53, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='98, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='43, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='65, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='88 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 MJy sr−1 from Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The data were calibrated with the scriptForPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='py cal- ibration script that was included with the archival data, using the ALMA calibration pipeline version r32491 that is included with the Common Astronomy Software Ap- plications (CASA) version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 (CASA Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' After calibration, the continuum emission was subtracted in the (u,v)-domain by fitting a straight line to the line-free channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The line-data where subse- quently imaged using the CLEAN algorithm to produce data cubes with 20 km s−1 channels and a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='36′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='21′′ (PA ∼ 10◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The root-mean-square (rms) noise in these data cubes is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 mJy beam−1 chan−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A primary beam correction was applied to produce ac- curate flux measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The half-power beam width of the primary beam is 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='8 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Total intensity maps of the CO(2-1) emission where made by integrating the signal across the channels where line emission was de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' CARMA CO Observations Observations of Stephan’s Quintet were made with CARMA on 1 Aug 2010 (program c0593-PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Alatalo) using 15 antennae in the CO (1-0) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The original data was processed with MIRIAD (Sault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Data reduction and imaging were done in identical fashion to Alatalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The original data was cleaned (H¨ogbom 1974), which resulted in a restored synthe- sized beam of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 arcsec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Velocities from extracted spectra have been converted to heliocentric velocities as- suming a shift of -11 km s−1 from lsrk to heliocentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Velocities are all quoted using the optical definition of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' SQ-A NGC 7319 H2 Bridge Main H2 Filament Field 5 Field 6 ALMA CO (2-1) Primary Beams 0-0 S(1) H2 NGC7318b/a 50arcsecsThe Multi-phase IGM in Stephan’s Quintet 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' HST Archival Images We de-archived calibrated images from the HST MAST archive for filters F665W and F336W with the WFC3/UV instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' For F665N, the image encom- passes the Hα line, and includes data based on five dithered observations, each taken with integration times of 5200s each in July and August 2009, and was obtained as part of a WFC3 Early Release Observations cam- paign (SM4/ERO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' For F336W, the observations were obtained as part of proposal id 12301 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Gal- lagher) on 2011-10-27, with a total integration time of 14474 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The resolution of the WFC3/UV observations is comparable with that achieved by NIRCam (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='06 arcsec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' For F665W and F336W, the expected FWHM of a point source is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='07 arcsec, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='075 arcsec respec- tively, corresponding to ∼30 pc at D = 98 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' WCS Alignment of the JWST Images The observations were obtained from the JWST and HST MAST archive, and most of the images required small adjustments to the WCS coordinates for proper alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We initially used GAIA DR3 stars to align one image (the HST WFC3 F665N) to the DR3 sys- tem using the STScI software package WCSTweak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The NIRCam images were also similarly aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' However, for MIRI, the same method did not produce good re- sults, probably because of the smaller number of stars used for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We used a refinement scheme to solve this problem as we need good alignment between all the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This was especially important for one of the target fields that required continuum subtraction of the F1000W and F770W images with F1500W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' First we created small subimages of the F1000W, F770W and F1500W images around the Field 6 area using the NRAO software package CASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The subimage con- tained at least 3 GAIA DR3 stars close to the target of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These subimages were then aligned more care- fully by making small pixel-level shifts in each image to bring the subimages into close alignment with each other and the positions of the GAIA stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We applied the same method to Field 5 and 4 (see § 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The fi- nal results produced local MIRI images which aligned to better than 1 MIRI pixel (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 arcsec) over the small scale of the extracted regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The results were vali- dated where the CO ALMA images aligned with clearly related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Following Papadopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2008), the uncer- tainty in the absolute astrometry of the ALMA data is δθbas = (δB · ∆k)/B ≈ (δφbas/2π)⟨Θbeam⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The phase error δφbas ∼ (2π/λ)(δB · δk), with |δB| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1mm the assumed typical error in the baseline length, |δk| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='26◦ the distance to the phase calibrator J2216+3518, λ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='33 mm the wavelength of the redshifted CO(2- 1) emission, and Θbeam ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='36′′ the major axis of the synthesized beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This results in δθbas ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='02′′, which means that the uncertainty in the absolute astrometry of the ALMA data is small compared to that of the HST and JWST data3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' INTERPRETING THE EMISSION OBSERVED THROUGH THE MIRI AND NIRCAM FILTERS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Comparison with Spitzer IRS spectral mapping in H2 and PAH Bands Under conditions found in the disks of normal galax- ies, strong 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 µm and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='6 µm Polycyclic Aromatic Hydrocarbon (PAH) features are expected to domi- nate the MIRI F770W filter for low-redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In Stephan’s Quintet (Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2017), much of the mid-IR spectrum of the IGM is dominated by strong pure rotational H2 lines, and emission from [SiII]λ34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='8µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Unlike normal star formation regions, the majority of the main IGM filament in Stephan’s Quintet exhibits very weak 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm PAH complex emis- sion and weak nebular lines like [SIII]λ18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5µm (Clu- ver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This is the result of low star formation rates in the main filament (measured to be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='08 M⊙ yr−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010), and emission lines more typical of fast shocks than HII regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Konstantopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Examples of these H2 dominated spectra have been presented in Figures 8, 13 and 14 of Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2010), and Figure 15 and 16 of Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The Spitzer IRS spectra showed that only a small minority of places along the main fil- ament (including parts of the SQ-A region) are spectra found with line flux ratios more typical of star forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In Figure 2a-c, we compare the filter transmission band-passes for the MIRI imaging bands (F770W, F1000W and F1500W) directly with the Spitzer IRS spectra of three zoomed-in regions near the center of each of the ALMA CO (2-1) primary beam pointings (Fields 4, 5 and 6) shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The regions are typical of the diversity of mid-IR spectral properties en- countered in the IGM of SQ in general, and are the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The MIRI filter F770W is likely to contain emission primarily from the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm PAH com- plex in regions where star formation dominates, but in other regions, especially those with strong H2 emission, the 0-0 S(5) can also potentially contribute, as in Fig- 3 See also: https://help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='almascience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='org/kb/articles/what-is-the- absolute-astrometric-accuracy-of-alma 6 Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Spitzer IRS spectra of three 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 arcsec2 regions centered on the three fields studied at high resolution in CO (2-1) emission with ALMA and at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 and 10 µm with JWST MIRI imaging (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The MIRI broad band filter transmissions are superimposed to emphasize that the F1000W MIRI filter detects almost exclusively emission from the S(3) line and that the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 µm band captures a combination of the weak 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7mum PAH complex and emission from S(5) H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The 15 µm MIRI bandpass detects mainly faint dust continuum and some (weak) [NeIII] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' ALMA Field 5 was not covered by the Short-Lo IRS module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' ure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The F1000W filter, on the other hand, almost exclusively contains line emission from the S(3) mid-IR pure rotational H2 line (See also Figures 15e and h of Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Based on the presented spectra, we expect the MIRI F1500W filter to be dominated by faint dust continuum from the general IGM, as well as from warm dust from embedded star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A clean separation between the mainly PAH domi- nated emission in the F770W MIRI filter and the 0- 0S(3)H2-dominated F1000W band can be seen in many regions along the main IGM filament and in the bridge region in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Here we overlay the contour maps obtained by isolating a PAH band4 from the IRS spec- tral map from Spitzer in Figure 3a (white contours) with a two-color image of the JWST MIRI F1000W (green) and the MIRI F770W (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In Figure 3b we over- lay the 0-0 S(3) H2 contours (red) over the same im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The overlays show that the white contours of PAH emission from Spitzer follow quite closely the brighter emission in the F770W dominated regions, whereas the red contours of H2 follow closely the green emission originating mainly from the F1000W filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This figure demonstrates how the 10µm MIRI filter, as we expected from the individual IRS specta, detects preferentially H2 emission, while the F770W filter detects those re- gions with dominated by PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' As shown in Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2010) the PAH emission is relatively faint in the main shock, but this figure emphasise those re- gions with stronger PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This near-isolation of the H2 emission in many regions of the IGM is a re- sult of the relatively narrow width of the F1000W filter, 4 We used the IRS 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3µm PAH rather than the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm PAH, be- cause the signal to noise ratio was much better in this PAH fea- ture in the Spitzer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' and the lack of other lines or bands that can seriously contaminate it for the redshift of Stephan’s Quintet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Dust continuum Observations using the F1500W MIRI filter are ex- pected to detect mainly warm dust emission, similar to that mapped by Spitzer at 24µm with MIPS (see Guil- lard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' However, in the SQ-A region (Fig- ure 2b), the dust continuum is also contaminated by [NeIII] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' As noted above the very strong 0-0S(1) line is, fortuitously, redshifted out of the F1500W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Before discussing results for each of the zoomed-in re- gions in the next section, we mention the removal of faint dust emission from F1000W and F770W at the center of Field 6 (near the center of the main IGM fila- ment in SQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In this field, faint dust continuum in the F1500W filter was seen in the vicinity of the structures we are interested in (see § 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We therefore removed the faint dust continuum from both the F1000W and F770W images, using the Spitzer IRS spectrum of the region as a guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' First we created small sub-images of the field in all three bands centered on the main struc- ture of interest and corrected them for small offsets in the WCS as discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' After removing a median sky background from each subimage, we then subtracted the F1500W image from both the F770W and F1000W images, scaling the continuum by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='9 to ac- count from a slight rise in continuum seen between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7, 10 and 15 µm in the IRS spectrum of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This allowed us to better define the distribution the H2 and PAH features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The effect was quite small (the dust emis- sion was quite weak compared with the emission seen in the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 and 10µm bands) on the resulting fluxes of the F1000W and F770W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Without spectroscopy of the region on the same spatial scale as the observed fea- tures, this method is necessarily imperfect, and where we estimate the fluxes for the features observed in Field 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 Mid-Filament Northern SF Region 0-0 S(1) Bridge nr NGC 7319 [SI] [Sill] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 H2 0 sa ALMA Field 6 F9 ALMA Field 4 ALMA Field 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 H2 MIRI FILTERS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 (MJy s 5 - F770W [Sil MIRI FILTERS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 - F1000W F1500W 0-0 S(1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 F770W F1000W F1500W 4 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 - F1500W Density ( 0-0 S(3) 0-0 S(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 H2 0-0 S(0) 0-0 S(0) 0-0 S(0) [Nel] H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 - H2 D PAH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 - [Nell] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 15 20 25 30 35 5 Observed Wavelength (um) Observed Wavelength (um) Observed Wavelength (μum)The Multi-phase IGM in Stephan’s Quintet 7 6 in the next section, we include larger uncertainties to take this into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' For the sub-region near the cen- ter of Field 5, no obvious dust structure was seen in the F1500W band near the structures of interest, and so we did not attempt to continuum subtract the H2 and PAH-dominated emission from the F1000W and F770W images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' For Field 4 — centered in the SQ-A star forming region — our attempts at subtracting a scaled version of the dust dust continuum led to a severe over-subtraction of the emission in the F770W and F1000W bands for most reasonable dust scale factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This is likely due to a combination of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Firstly at that position (see Figure 2b), we have shown that the 15µm band shows emission from not only dust continuum, but also rela- tively strong [NeIII], as well as broad PAH-band emis- sion from the 17µm “plateau” complex (see Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These additional contaminants make it difficult to isolate the warm dust component in the 15µm image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Therefore, for this star formation dominated field, we conclude that the F1000W and F770W MIRI filters are hopelessly contaminated by dust emission in way that cannot easily be corrected without high resolution spec- troscopic data from the MIRI MRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This significantly limits our discussion of the molecular hydrogen proper- ties in Field 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' For that region we limit our discussion of the warm H2 properties determined from the Spitzer IRS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Near-IR images We also present images in this paper obtained through the NIRCam F200W filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' F150W provides a near- IR continuum band relatively clear of emission lines at the redshift of Stephan’s Quintet (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='022 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='025), with the well-known near-infrared [Fe II] lines falling just blueward or redward of the filter transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The bandpass of the F200W filter is sensitive to emis- sion from the hydrogen recombination line Paα, and starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Diffuse emission from this line becomes obvi- ous when we compare some of those images with those obtained in the F150W NIRCam filter, and by compar- ison with the HST Hα images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' JWST/MIRI, which is able to detect warm H2 through the pure rotational transitions, can significantly improve our knowledge of how energy is dissipated in the IGM down to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 arcsec scales of giant molecular cloud complexes (∼140 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' NIRCam, with its ability to probe down to scales of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='06 arcsec (23-27 pc at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 and 2µm respectively) can probe ionized gas and stellar associations and clusters at scales at which sporadic star formation along the main filament is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' RESULTS We study in detail three regions which lie near the centers of the ALMA CO (2-1) observing fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' They are highlighted in Figure 4 as boxes (dotted lines) super- imposed on one of the publicly released ERO images of the inner part of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 6, which is dominated by 10µm H2 emission (green color in this figure), was targeted by HST COS (Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2022) and shows extremely broad Lyα emission, broad [CII]158µm emis- sion and contains some of the the warmest H2 observed by Spitzer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' It is representa- tive of a shock-heated region in the main N/S filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 5 is another H2-dominated region which lies out- side the main north/south H2 filament, but observations at other wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' single disk CO observations by Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2012), and Herschel [CII] observations of Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2013) show that this gas is also highly turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' It forms part of the bridge of H2 emission seen by Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 4, by contrast, lies in a well- studied extragalactic star forming region, SQ-A (or the Northern Star Forming region) and has been studied ex- tensively at optical wavelengths (Gallagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Its star forming nature can be inferred from its bluish-white appearance in the false color map of Figure 4 due to a combination of weaker H2, strong PAH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm emission and dust continuum (See Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Analysis of the three regions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 6: Within the main shock-dominated North/South filament In Field 6 (Figure 5a) we show the integrated CO (2-1) emission from cold molecular gas superimposed on the warm H2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The warm H2 gas (attributed to the 0-0 S(3) line) is distributed in an elongated structure extending over 4-5 arcsecs (∼ 2 kpc at D = 94 Mpc) with a triangular-shaped hot-spot to the south-west, and a series of fainter clumps extending to the east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The overall distribution of warm H2 has the appearance of head-tail structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The CO emission (white contours) displays a clumpy narrow shell-like structure associated with the warm H2 head, and a scattering of CO clumps along the tail, extending over the same overall extent, but not correlating in detail with the hotspots in the warm H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Based on the published IRS spectra, Figure 5b is inter- preted as a mix of H2 (0-0S(5)) and weak PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The dominant triangular-shaped head structure seen in the 10µm image is also strongly represented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A second major clump of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 µm emission is seen to the NW, which is weak at 10µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' It is plausible that this feature is PAH dominated, whereas the emission from the compact head (which is seen also at 10µm) may be 8 Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Stephan’s Quintet: A false-color representation of the MIRI F770W and F1000W JWST images with contours of rest-frame (a) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3µm PAH and (b) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='66µm emission derived from spectral cubes obtained from the Spitzer IRS Short-low mapping of Stephan’s Quintet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The white and red solid lines show the extent of the IRS spectral mapping (See Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The figure demonstrates how the S(3) line isolated in the Spitzer cube follows closely the 10µm MIRI emission (green), and the PAH features follow the MIRI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm image which contains contributions from the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm PAH and H2 emission (See also Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2017, and Figure 2 of this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' dominated by warmer than normal H2 which would emit strongly at in the 0-0 S(5) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Without mid-IR spec- troscopy, this cannot be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This distribution of ionized gas can be inferred from the narrow-band HST F665N filter images (dominated by Hα emission) in Figure 5c) and the F200W NIR- Cam image (dominated by Paα emission and near-IR starlight) in Figure 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Both images show significant extended (presumed ion- ized gas) emission associated with the head of the warm H2 including a protrusion, or “spike” labeled in Fig- ure 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The feature is also seen in CO emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Ex- tended(ionized) gas also follows more closely the distri- bution of the warm H2 in the tail, than it does the CO emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This is well demonstrated in the composite image Figure 5f, which shows an RGB representation of the 10µm (warm H2 , CO emission (cold H2) and F200W (ionized gas and stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Gemini optical spec- troscopy (Konstantopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2014) which covered this region at a spatial resolution of 1 arcsec strongly suggested that the ionized gas is shock-excited, which would explain why the ionized gas follows the warm H2 emission in the head and the clumpy tail, and is less correlated with the CO emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The spatial resolution of the F200W image is far supe- rior to the HST image, and it is clear that there are sev- eral compact sources on scales of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='6 arcsec (23- 27 pc) embedded in the ionized gas component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These sources are likely stellar associations, or unresolved su- per star clusters (Gallagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' They are gen- erally rather sparsely distributed and poorly correlated with the warm or cool molecular hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' One exception is a group of compact sources and ex- tended F200W emission associated with the southern tip of the NW clump of CO emission identified in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These sources may be evidence of recent star formation associated with the head-tail structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Evidence of this comes from the bright u-band sources associated with the same region shown in Figure 5e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Although a full analysis of the NIRCamcolors of the point sources in Stephan’s Quintet will be discussed in a separate pa- per, we know these blue regions have the optical colors of young clusters with ages estimated to be between 3- 5 Myrs based on previous HST UBVmVI photometry (Fedotov 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' see Table A4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The ex- istence of young star formation near the NW CO clump may also explain why that region exhibits PAH emission, since PAHs are believed to be excited by UV radiation from young stellar associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The kinematics of the diffuse ionized gas centered on the head from optical spectroscopy shows extremely broad line widths (>800-1200 km s−1) in several key emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' [OIII]5007, [OI]6300, and Hα) (Konstantopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The broad line-widths have been attributed to a combination of turbulence and large-scale motions associated with gas along the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Even broader wings in the Lyα profile were also seen at this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We will ar- b a NGC 7318b NGC 731&a NGC 7319 MIRI F1000W MIRI F1000W Spitzer iRS- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='66um/(0 -0S(3) MIRI F770W Spitzer IRS PAH (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3um) MIRI F770WThe Multi-phase IGM in Stephan’s Quintet 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Stephan’s Quintet: A false-color representation of the MIRI F770W, F1000W and F1500W JWST image of the inner Stephan’s Quintet, showing the three regions that are highlighted in the discussion that are detected near the primary beam centers in the ALMA CO (2-1) emission line observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 6 is representative of the center of the main filament where previous observations with Spitzer show that the molecular hydrogen has the highest temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 5 lies in the bridge region between NGC 7319 and the main shock identified in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Finally, in Field 4, a bright region in the SQ-A star forming region is highlighted (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' gue in § 5 that the warm H2 emission detected by JWST at this position may be responsible for UV scattering of Lyα emission within the turbulent regions, and if so, should exhibit broad line-widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' While we cannot yet measure the kinematics of the warm gas, the CO observations provide kinematic infor- mation about the cold molecular phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 6a and b shows the integrated CO surface density (moment 0) map and the mean velocity field (moment 1) map of the CO emission respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The systemic velocity of the CO emission falls between the velocity of the intruder galaxy NGC 7318b (V(int) = 5770 km s−1) and that of the velocity barycenter of the main group (V(group) = 6600 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This is consistent with gas which has been decelerated in the collision of the intruder with the group-wide gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' However, the linewidth of the entire region in CO is less that 120 km s−1, as shown in the integrated ALMA CO (2-1) spectrum of the entire region and the CARMA CO (1-0) spectrum shown in Figure 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This width is significantly smaller than that seen in the ionized gas, implying that the cold molecular gas does not take part in the turbulent motions seen in the ion- ized gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' More extensive kinematic information for the CO emission is presented in the Appendix-A, where we show individual spectra extracted from many regions (Figure A1), along with tabulated properties of the CO emission on the scale of the ALMA beam (Ta- ble A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The tip of the head structure has the highest radial velocity, and the clumps show a trend to lower ve- locities as one moves east into the tail, reaching the low- est velocity in one of the most easterly clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We will argue later that this suggests material is being stripped from the head into the tail through ramp-pressure strip- ping with a hot medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The velocity dispersion5 of the individual clumps around the structure show a mix of unusually broad line widths (up to 100 km s−1 FWHM) and narrow lines (< 40 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In Table A1, we show that the re- gion near the “spike” feature seen in the ionized gas, is unusually broad (FWHM = 102 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Away from 5 Hereby measured as the FWHM of the emission line profile, or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='35 × the sigma of these mainly Gaussian lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' MIRI Filters SQ-A F770W F1000WF1500W Field 4 NGC7318a NGC 7319 NGC7318b Field 6 Field 510 Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A zoom-in on the center of the ALMA Field 6 (see Figure 1) comparing (white) contours of CO (2-1) emission to (a) continuum-subtracted pure warm 0-0S(3) H2, (b) continuum-subtracted warm 0-0 S(5) H2 plus 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm PAH complex emission, (c) Hα emission from F665N WFC3 HST, (d) mix of IR (red compact) stellar sources and Paα emission in the NIRCam f200w filter, (e) blue star clusters from F336W WFC3/UV HST (Gallagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 2001), (f) a composite RGB color representation of the F1000W (mainly warm H2), ALMA CO (2-1) emission (cool H2), and the F200W (Paα plus star clusters) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The MIRI identifications of the contributing lines/PAH bands are clear from the Spitzer IRS spectra of a region 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 arcsec2 in Figure 2a, which spatially covers this entire region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' All images were carefully aligned using GAIA DR3 stars close to the region shown, leading to relative uncertainties in astrometry of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='15 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The F770W and F1000W JWST images were smoothed to the same resolution as the F1500W image before subtracting the carefully-aligned dust continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The white scale bar is 3 arcsec in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A more detailed description of the CO emission and its kinematics is given in Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' the head, several regions in the tail of the structure show line widths ranging from 60-90 km s−1(FWHM) (see Figure A1) to values lower than 40 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Given the masses of the individual molecular clumps of ∼few× 106M⊙ (Table A1), it is unlikely that the clumps with high velocity dispersion are gravitational bound6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The broader lines are consistent with turbulent motions on the sub-arcsec scale in the colder gas component, while others have much narrower lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 6 For example, for a clouds of diameter 100 pc (just less than the resolution of ALMA) and given a typical gas surface density of Σgas = 170 M⊙pc−2), the velocity dispersion for a cloud in grav- itational (virial) equilibrium would be σ2 = (3/5)πGRcloudΣgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' To be in equilibrium σ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 km s−1, or a FWHM ∼ 20 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Triangular head F1000W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='JWST F77OW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='JWST lump Spike F665NHS f200w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='IWS1 Blue = warm H2 Green = Cool H2 Red = ionized gas F336W HST CompositeThe Multi-phase IGM in Stephan’s Quintet 11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' ALMA Field 6 region (labeled in Figure 4): (a) CO (2-1) integrated emission, (b) the intensity-weight mean velocity map of the same region, and (c) the ALMA CO (2-1) and CARMA CO (1-0) spectrum of the region 5 x 5 arcsec centered on same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Effective synthesized beam-shapes (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='36 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='21 arcsec2 FWHM) are shown graphically in the left-hand corner of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Arrows indicate the radial velocity of the intruder V(int) and the barycentric velocity of the main group (V(gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' One such region occurs in the CO emission clump closest to the bright blue star clusters described ear- lier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Here the CO emission is essentially unresolved (40 km s−1or less) perhaps suggesting turbulent motions are calmer there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Table A1 presents information about the line fluxes and estimated H2 masses for all the regions measured in detail with ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Field 5: A shocked structure in the bridge region Field 5 lies outside the main molecular filament in Stephan’s Quintet, and is closer to large face-on galaxy NGC 7319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' It forms part of an apparent bridge of H2 emission discussed by Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The JWST images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 4) show that it is the most easterly of series of irregular shock-dominated (appearing green in that image) clouds that run approximately East/West across the field and may not be physically connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In contrast to Field 6 which contained a single coher- ent CO structure, Field 5 contains three separate bright structures visible in the CO maps (labeled in Figure 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' These consist of a clumpy CO ring about 1 arcsec across (450 pc), a compact elongated core 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 arcsec to the SE of the ring, and a broken linear filament of gas running north-south 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 arcsec further east of the compact core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The direction of the elongation of the core points to- wards the center of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Both Figure 7(a) and (b) show that the compact CO core lies near the brightest emission at the center of the arrow-shaped 10µm (warm H2) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' As the arrow-shaped emission narrows down towards the NW, it connects to the northern part of the CO emission from the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 7(c) shows dif- fuse Hα emission from the compact CO core but little emission from the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The higher spatial resolution of the NIRCam F200W image shows that the correspond- ing Paα in the core is also elongated Figure 7(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Faint F200W emission is also seen in the brightest CO ring clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The U-band image, Figure 7e, shows no obvi- ous emission near any of the CO molecular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Although not shown here, the F150W NIRCam image shows point-like sources in the northern part of the CO ring, suggesting that the F200W image is revealing a mix of HII regions and compact sources within the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The composite image, showing the three gas phases in one figure emphasizes the warm H2 connection between the ring and the core (Figure 7f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A string of clumpy sources at 2µm is clearly seen ap- proximately 1 arcsec to the east and north of the ring (F200W image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' It is not clear if these sources have any real connection to the CO structures, as some could be background galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' One of the clumps has a faint U-band association (see Appendix-B and Table A3) sug- gesting recent star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Near-IR spectroscopy may help to determine if they are are physically associ- ated with Stephan’s Quintet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The kinematics of CO gas in the compact core is strik- ingly different from that of the ring and the linear struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 8a and b, shows the integrated and radial velocity map of the CO (2-1) emission, and spectra of the three different CO emitting structures are presented in Figure 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The velocities within the partial ring in- crease clockwise around the ring from 6631 km s−1 in the north, reaching the highest value in the south and east quadrant of 6688 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In contrast, the compact core (which lies near the peak in the warm H2 emission) has a much lower (negative 250 km s−1) radial velocity than the ring (∼ 6400 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The core also shows a velocity shear along its length of about 30 km s−1over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 arcsec (200 pc), as shown in the Appendix-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Finally the third linear structure to the east of the compact core, shares velocities similar velocity to that of the ring, with velocities ranging from 6620 in the south to 6680 in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Velocity (Helio) km/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='18 6018 6038 6058 6078 6098 MS) ALMA_CO(2-1) CO (2-1) b CARMA_CO(1-0) 33:58:24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='025 V(int) V(gr) tion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='020 33:58:23 clinati 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='010 33:58:22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='005 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='000 00 33:58:21 J20( 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='005 5600 5800 6000 6200 6400 6600 22:36:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 35:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 22:36:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 35:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7 Velocity (Helio) km/s J2000 Right Ascension (HMS) J2000 Right Ascension (HMS)12 Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' A zoom-in on the center of the ALMA Field 5 (see Figure 1) comparing (white) contours of CO (2-1) emission to (a) pure warm 0-0S(3) H2, (b) warm 0-0 S(5) H2 plus 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm PAH complex emission, (c) Hα emission from F665N WFC3 HST, (d) mix of IR (red compact) stellar sources and Paα emission in the NIRCam f200w filter, (e) blue star clusters from F336W WFC3/UV HST, (f) a composite RGB color representation of the F1000W (mainly warm H2), ALMA CO (2-1) emission (cool H2), and the F200W (Paα plus star clusters) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The MIRI identifications of the contributing lines/PAH bands are likely from the Spitzer IRS spectra of a region 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 arcsec2 in Figure 2, which in this case only includes the longer IRS LL wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' All images were carefully aligned using GAIA DR3 stars close to the region shown, leading to relative uncertainties in astrometry of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='15 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The F770W and F1000W JWST images were smoothed to the same resolution as the F1500W image before subtracting the dust continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The white scale bar is 3 arcsec in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' An in-depth view of the kinematics of Field 5 in given in the Appendix-A, including individual spectra, Figure A2, and a table of CO properties, Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The spec- tra show that individual CO clumps in all of the three CO structures exhibit broad CO line-widths at the scale of the ALMA beam ( ∼150 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The compact core it- self has a velocity dispersion of 145 km s−1(some of this may be due to the gradient seen along the core), and all components of the ring have line-widths which lie in the range 60-120 km s−1(FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The linear structure shows narrower line profiles in the range 48-87 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Appendix-A also shows that although the majority of the gas in the ring has much higher radial velocities than the core structure, one small segment of the ring (at the point nearest the core) has a velocity similar to the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This is supported by the CARMA spectrum, Figure 8d, which shows a blueward component associated with the average spectrum of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This strengthens the idea that the two CO structures are somehow related despite their discrepant radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' We will discuss in the next section the possibility that the warm bridge con- necting the CO ring and core may be splashed warm a ring linear structure compact F1000W JWST F770WJWST core d F665N HST F200W JWST Blue = Warm H2 Green = Cool H2 Red = ionized gas + F336W HST CompositeThe Multi-phase IGM in Stephan’s Quintet 13 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' ALMA Field 5 region: (a) the CO (2-1) integrated emission at a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='36 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='21 arcsec2, and (b) the intensity-weight mean velocity maps of the same region, (c) the ALMA CO (2-1) spectra of the three different extracted regions shown as a red polygons in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Note the very different systemic velocity of the compact structure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' In (d) we overplot the CARMA CO (1-0) over a 5 x 5 arcsec2 aperture (blue line), compared with the velocity of the ring C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The effective ALMA beam shapes (FWHM ellipses) are shown graphically in the bottom left hand corner of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The black vertical line on the spectra show the barycentric systemic velocity of the V(group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Extended CO (2-1) emission for ALMA Field 5 over a larger area than Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The CO emission (white contours) have been smoothed to a resolution 1 x 1 arcsec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' (a) warm 0-0 S(3) H2 , (b) warm 0-0 S(5) plus 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='7µm complex PAH emission, and (b) a smoothed version of the extended velocity field, again showing the unusually low velocity for the compact core compared with and the other emission regions to the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The long filament connecting the brighter CO structures is seen in the warm H2 and faint H2+PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' the filaments velocity is shown schematically because it is weak, but has the same velocity as the majority of the other structures except the compact core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The white scale bar is 3 arcsec in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' molecular gas caused by the collision of a dense molec- ular cloud moving at high (blueward) velocity with re- spect to more diffuse gas associated with material in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Some of the CO emission in Field 5 may be associated with a larger filament and other CO clumps, as seen in a larger-scale (10-15 arcsec, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5-7 kpc) image, Fig- ure 9a, where we present the ALMA CO emission map smoothed (to 1 x 1 arcsec2) to bring out fainter fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This reveals a very faint filament of CO gas and several fainter southern CO concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Both the fil- ament and the southern CO structures have faint warm H2 counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Filaments like this one, in this case ex- tending over 6-10 arcseconds in scale (3kpc-5kpc), seem to be common in the MIRI maps of SQ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Fig- ure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Figure 9b shows again the velocity field of the ring, compact core and linear structure, this time in re- lation to the larger-scale structure in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' This further emphasises how the compact core has such a dif- ferent velocity from all the other structure in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' The velocity of the filament is poorly determined in CO because it is so faint, but seems to have a velocity simi- lar to that of the average velocity of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content=' Mid-IR spectroscopy would be needed to provide more informa- (Jy/beam-km/s) (km/s) Heliocentric 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='25 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='3 6280 6383 6488 6590 6700 (DMS) Flux Density (mjy) ALMA Reg A C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 ALMA Reg B 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 ALMA Reg C 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 6160 km/s VGROUP a b J2000 Declination 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 J2000 Declination 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 6200 6400 6600 6800 7000 20 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 CARMA CO(1-0) p 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE1T4oBgHgl3EQfHgPM/content/2301.02928v1.pdf'} +page_content='5 Ω/2 +we use UIFBL(r,θ) = UIFBL(r,Ω−θ). +Figure 4 is the plot of Eq. (12) for Ω = 3π/2 assuming SiO2 as the dielectric with a relative +permittivity of 3.9. The metal is located in the upper left corner region where x > 0nm and y < +0nm. Far removed from the corner, the traditional IFBL energy −e2/(16πεr) is recovered but +8 + +Ω +θ +UIFBL/(−e2/(16πεr)) +a +π +π/2 +1 +b +π +θ +csc(θ) +c +2π +π +2/π ≈ 0.63 +d 3π/2 +π (or π/2) +6−2/ +√ +3 ≈ 0.85 +e 4π/3 +π +8 +√ +3/9+1/π −3/4 ≈ 1.11 +f +2π +3π/2 +1/π +1/2 ≈ 0.82 +g 3π/2 Ω/2 = 3π/4 2+2 +√ +2−16 +√ +3/9 ≈ 0.74 +h 4π/3 Ω/2 = 2π/3 +4/ +√ +3−3/2 ≈ 0.81 +i 2π/3 Ω/2 = π/3 +4 +√ +3/9+2/π ≈ 1.41 +j π/2 +Ω/2 = π/4 +2 +√ +2−1 ≈ 1.83 +k π/3 +Ω/2 = π/6 +5−4/ +√ +3 ≈ 2.69 +TABLE I: Image-force barrier lowering for various angles between the metal surface, Ω, and for +various angles between the metal and the 2D material, θ. +FIG. 5: Illustrations depicting 10 different configurations considered in Table I using the same +color scheme as Fig. 1. The metal surface is separated by an angle Ω and the 2D semiconductor +has an angle θ with the upper-most metal surface. Illustrations are rotated such that the +semiconductor is horizontal. +at the corner, the IFBL effect is reduced. Visual inspection shows that the barrier can easily be +lowered by more than 0.1 eV due to the IFBL effect, which could improve contact resistance by +orders of magnitude. +Finally, we evaluate some special cases of angles between the metal and the 2D semiconductor +in Table I. Fig. 5 illustrates all the geometries listed in the table, except for b, where the θ is +variable. Taking Ω = π and θ = π/2, the usual IFBL expressionU(a) +IFBL = −e2/(16πr) is recovered. +9 + +(g) +(h) +() +(G) +(k)(a) +(c) +(d) +(e) +(f)Taking Ω = 3π/2 and θ = π, as would be the case in Fig. 1b, the image potential only reduces by +15%. Choosing the 2D material to be in the same plane as one of the metal plates, i.e θ = π as +is the case in Fig. 5c-e, IFBL will become stronger than the Fig.5a case once Ω < 1.384π. And +for Ω = 4π/3 or Fig. 5e, i.e. when there is an angle π/3 between the metal and the 2D material, +the experienced IFBL is roughly 11% stronger. The configuration in Fig. 5k, which has the 2D +material in between two plates with an angle π/3, yields an improvement of the IFBL by a factor +2.69, which is the highest of all cases we consider. +In summary, we determined the IFBL energy emerging from a metal with surfaces separated +by an angle Ω. We showed how the IFBL energy can be determined directly and using the method +of images provided a cone-manifold space is used. To model contact resistance to 2D materials +accurately, IFBL should be incorporated and we provided a numerical implementation of Eq. (12) +using Gauss-Laguerre quadrature. We considered 10 configurations with various angles between +the metal surfaces and the 2D material where the IFBL can be evaluated analytically. We found +that the IFBL was weakened by a factor 6 − 2 +√ +3 ≈ 0.85 for a top metal contact with its surface +perpendicular to the 2D material and found that IFBL is strengthened by 11% for a metal contact- +ing the semiconductor with an angle π/3 between them. Further, smaller angles between the metal +and semiconductor can more than double IFBL. Fabricating contacts to 2D materials with a small +angle between the metal and the 2D material should significantly improve contact resistance. In +all cases, the strength of the IFBL is also scaled with the permittivity of the material surround- +ing the 2D material. 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Zucker, “Table of the zeros and weight factors of the first fifteen laguerre +polynomials,” Bulletin of the American Mathematical Society, vol. 55, no. 10, pp. 1004–1012, +1949. +13 + diff --git a/TdE0T4oBgHgl3EQfUwDn/vector_store/index.pkl b/TdE0T4oBgHgl3EQfUwDn/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..cb7539570ca4912d71ef1db9ead1cb14acda9245 --- /dev/null +++ b/TdE0T4oBgHgl3EQfUwDn/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0a7191fa4745711d54ddbbad1b31da780f3901407ed04c0c2de0c3c17af284ad +size 219724 diff --git a/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf b/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..96f3323579c6e177e7d762cb941c2dfdfa9bcf25 --- /dev/null +++ b/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0200094ffdc1a7ae11711060ac6b71c8a22c1664b104478d302c8712d5292a18 +size 606178 diff --git a/UNA0T4oBgHgl3EQfEf9Z/vector_store/index.faiss b/UNA0T4oBgHgl3EQfEf9Z/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b0e9647ec6357f02e1ea234264deb6ae170155eb --- /dev/null +++ b/UNA0T4oBgHgl3EQfEf9Z/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:12d028f8dee0bc18f47c0a52707c399fd603fc38cbd9f5c647a6f01ff50d3bab +size 1572909 diff --git a/UNA0T4oBgHgl3EQfEf9Z/vector_store/index.pkl b/UNA0T4oBgHgl3EQfEf9Z/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..180e4dde6c7e0768b436a5e8df3e6b7603ece081 --- /dev/null +++ b/UNA0T4oBgHgl3EQfEf9Z/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c0c45dd5df3940a54f082a97a8961577b7c993ced2a37ba474c4971d7edca709 +size 71541 diff --git a/UdAyT4oBgHgl3EQfuvkh/vector_store/index.pkl b/UdAyT4oBgHgl3EQfuvkh/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d89478fe75433e67bf511f8e00ca843bd891072e --- /dev/null +++ b/UdAyT4oBgHgl3EQfuvkh/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:65cdc16b962051a52ed791e6969e59c0d4202ab87579acf6245b91ec528dbdd2 +size 153226 diff --git a/UdE1T4oBgHgl3EQfIgMu/content/2301.02939v1.pdf b/UdE1T4oBgHgl3EQfIgMu/content/2301.02939v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0a7debd980d1b2253e912309cda84e8cd273afde --- /dev/null +++ b/UdE1T4oBgHgl3EQfIgMu/content/2301.02939v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a4d1fd7272964aa521743f4ed3c88da7283e535e65abdf214b08c682b73f3789 +size 802406 diff --git a/UdE1T4oBgHgl3EQfIgMu/vector_store/index.faiss b/UdE1T4oBgHgl3EQfIgMu/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..81f3c0aa2fc1941641e5a3c26eea76de86e72a28 --- /dev/null +++ b/UdE1T4oBgHgl3EQfIgMu/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7be8ed922dc40f371c26ca6c4a26f96bc6076fafe7360e462c4f5cb167691339 +size 3932205 diff --git a/UdE1T4oBgHgl3EQfIgMu/vector_store/index.pkl b/UdE1T4oBgHgl3EQfIgMu/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1e9c7381a781267f020da1d54e9dd3b63e407da5 --- /dev/null +++ b/UdE1T4oBgHgl3EQfIgMu/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1ec27ac93cc7cbcd913caa6ae3611b6e9dd901abe2803fee2e137350d9547243 +size 132157 diff --git a/UdFIT4oBgHgl3EQfgCsL/content/tmp_files/2301.11281v1.pdf.txt b/UdFIT4oBgHgl3EQfgCsL/content/tmp_files/2301.11281v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7542daee39aa2477bf6672f2158535cb0805502b --- /dev/null +++ b/UdFIT4oBgHgl3EQfgCsL/content/tmp_files/2301.11281v1.pdf.txt @@ -0,0 +1,1305 @@ +Quantifying Hidden Symmetry in the Tetragonal +CH3NH3PbI3 Perovskite +Kuntal Talit and David A. Strubbe∗ +Department of Physics, University of California, Merced, 5200 N. Lake Rd., Merced, CA +95343 +E-mail: dstrubbe@ucmerced.edu +Phone: +1-209-228-4481 +Abstract +The assignment of an exact space group to the tetragonal CH3NH3PbI3 perovskite +structure is experimentally challenging and controversial in the literature. Average ori- +entation of the methylammonium ion that gives symmetry to the experimental mea- +surement is not captured in a static density functional theory calculation, although +the quasi-I4cm and quasi-I4/mcm structures are commonly used in calculations. In +this work we have developed a methodology to quantify the hidden symmetry of these +structures using group theory, to enable use of symmetries in understanding spec- +troscopy and other properties. We study the approximate symmetry of vibrational +modes, including analysis of degenerate representations, as well as the dielectric, elas- +tic, electro-optic, Born effective charge, and Raman tensors and the dynamical matrix. +Comparing to each subgroup of the full tetragonal D4h, our results show that the +quasi-I4cm is best described by the expected corresponding point group C4v, whereas +the quasi-I4/mcm (despite corresponding to point group D4h) is best described by the +lower symmetry C2v. Our methodology can be useful generally for analysis of other +soft hybrid materials or any approximately symmetric material. +1 +arXiv:2301.11281v1 [cond-mat.mtrl-sci] 26 Jan 2023 + +Hybrid organometallic perovskites are one of the most researched material for solar cell +application in last decade. There are more than sixteen thousand research documents pub- +lished between 2009 to 2019 regarding perovskite solar cell.1 A huge amount of research has +been done towards low-cost fabrication, increasing the photo-conversion efficiency, making +active layer materials etc. But in-depth understanding about some fundamental aspects is +still missing. The exact symmetry of room-temperature tetragonal methylamonium lead +iodide (MAPI) is one of them. There is still debate about the space group symmetry of the +tetragonal MAPI. Some reports suggest that the structure is ferroelectric or polar having +quasi-I4cm space group symmetry2,3 while others have found tetragonal MAPI structures +that are antiferroelectric or antipolar in nature, having quasi-I4/mcm space group symme- +try4–6(Fig. 1) There are also reports that identified the space group as I4/m.7 There are +experiments that reports the structure to have space groups I422 and P42212 which are +subgroups of I4/mcm.8 +It is important to know the symmetry of the tetragonal structure because symmetry is an +essential tool to understand the spectroscopy and other properties of hybrid perovskites.9 An +important example relates to the electronic properties: the two different structures I4cm and +I4/mcm have different electronic properties. I4/mcm is a centrosymmetric structure with +inversion symmetry and theoretically it should not produce Rashba splitting in the band- +structure while I4cm is a non-centrosymmetric structure without the inversion symmetry +and it produces significant Rashba splitting.10 A DFT study concluded that any calculated +significant Rashba splitting in case of the I4/mcm structure is incorrect and may be due +to incorrect structural relaxation.10 Another DFT study determined that the energy differ- +ence between a quasi-I4cm and a quasi-I4/mcm structure is very small (0.1 eV) and they +can coexist in a single crystal with domains of altering tilting directions which can further +dynamically interchange into each other at room temperature crossing some energy barrier +that caused due to the specific interaction between the MA+ ion and the inorganic cage.11 +A source of difficulty in determining the symmetry experimentally is that when the mate- +2 + +rial goes through a phase transition from cubic to tetragonal structure due to temperature +changes, it loses some symmetry elements which can gives rise to twinning along the lost +symmetry element.12 In experiment, we mainly get the overall symmetry of the whole crys- +tal, but sometimes the unit cell might have different symmetry due to such twinning within +the crystal. +In case of the experimental structure, each MA+ ion is statistically distributed with a +fractional occupation of 25% for each of 4 orientation.8 This arrangement provides symmetry. +To do any theoretical calculation we must take a snapshot of multiple possible orientations +of the MA+ ion and at that moment we lose all the symmetry. Quarti et al. have done a +detailed study of the tetragonal structure and found that a set of polar structures (I4cm) are +more stable than the apolar (I4/mcm) ones.11 The most stable structure reported in their +study is a polar structure with I4cm space group symmetry and this structure is used by +other works13,14 (though seems to be described as I4/mcm). +Figure 1: Tetragonal MAPI with different space group symmetries: (a) I4cm (C4v) structure, +(b) experimental structure (D4) with partial occupancies, and (c) I4/mcm (D4h), having the +full symmetry of the tetragonal structure. +In case of the low-temperature orthorhombic structure, 4 MA+ ions in the unit cell are +static which gives it a perfect D2h symmetry. At high temperatures, the random spinning +of the MA+ ion within the cage makes the structure pseudo-cubic, and is not even close +to any symmetry, complicating theoretical analyses.15 For the room-temperature tetragonal +3 + +(a) I4cm +(b) I422 +(c) I4/mcmstructure, the average over space and time of this random spinning makes this structure +quasi-I4cm or quasi-I4/mcm. So, the tetragonal MAPI does not have any exact symmetry, +but is considered to have approximate symmetry. Previous literature however has not quan- +titatively assessed the symmetry of these structures, to describe rigorously how close they +are to I4cm, I4/mcm, or any other space group. In this work, We want to quantify how well +symmetries such as I4cm or I4/mcm describe the structures, and find the highest degree of +approximate symmetry that can be used to describe properties of this tetragonal structure. +To identify the hidden symmetry in the structure we have checked the symmetry from dif- +ferent aspects: (a) atomic coordinates, (b) vibrational modes, (c) elastic tensor (or stiffness +matrix), (d) dielectric tensor, (e) electro-optic tensor, (f) dynamical matrix, (g) Born effec- +tive charge tensors, and (h) atomic Raman tensors. The elastic, dielectric, and electro-optic +tensors provide global mechanical and electronic properties, whereas the coordinates and dy- +namical matrix provide atom-resolved structural properties, and the Born effective charges +and atomic Raman tensors provide atom-resolved mixed structural/electronic properties. We +use these assessments of symmetry in structural, electronic, and vibrational aspects in com- +bination to identify the most appropriate symmetry subgroup description of the tetragonal +structure. +For any crystal structure that has exact symmetry vibrational modes can be classified +according to irreducible representations, but this cannot be done when the structure does +not have any symmetry. In this work, we have developed a method to calculate the approx- +imate irreducible representation of the vibrational modes for an approximately symmetric +structure. We use the approximate symmetry of the crystal structure and its character table +as our input and use group theory to calculate approximate characters in the character table +and thereby calculate the irreducible representations of the vibrational modes. We have +calculated the contributions of irreducible representations for each phonon mode of tetrag- +onal MAPI which can be helpful for spectroscopic studies. As a test of our methodology, +we have also calculated the same for perfectly symmetric orthorhombic MAPI and TiO2 +4 + +and it gives correct irreducible representations for both the systems compared with Quan- +tum ESPRESSO results. Our methodology can be useful to calculate hidden symmetry and +approximate mode irreducible representations for any approximately symmetric structure. +We have studied two different tetragonal structures, quasi-I4cm13 and quasi-I4/mcm,16 +using density functional theory. +Computational details, similar to our previous work on +strain effects in cubic MAPI,15 are given in the supporting information. To check the initial +symmetry of the structures we have used FINDSYM.17,18 The result is given in Table S1. +As we already know that the theoretical structure does not have any symmetry due to the +different orientations of the MA+ ions within the structure, we have removed all the MA+ +ions from the I4cm structure and checked the symmetry of the Pb-I cage only. With some +tolerance with respect to the lattice and the atomic positions, we found that the Pb-I cage +still holds the D4h point group symmetry. One thing to notice here is that the Pb-I cage +and the whole structure both have symmetry Cs which is a subgroup of D4h, even with low +tolerance values. We will come back to this point while explaining phonon mode symmetries. +For the quasi-I4/mcm structure, even with low tolerance values, the predicted symmetry by +FINDSYM is C2v which is orthorhombic symmetry and lower in symmetry than D4h. This +gives an indication that the tetragonal structure may be better described using some lower +symmetry subgroups of D4h. +Next, we consider three tensors which provide global (not atom-resolved) properties of +the system, the elastic, dielectric, and electro-optic tensors. Examining the full stiffness +tensor Cij, 6 × 6 in Voigt notation, shows symmetry in mechanical response. Our calculated +tensors for quasi-I4cm and the quasi-I4/mcm structures are shown in Fig. S1. For tetragonal +(I) crystal system19 we should have nonzero elements C11 = C22, C33, C44 = C55, C66, C12, +and C13 = C23. The stiffness tensor for quasi-I4cm structure closely follows the tetragonal +(I) symmetry, except there are small off-diagonal values. For the stiffness tensor of the quasi- +I4/mcm structure, all the diagonal values are different and C13 is not same as C23. This is +not even close to tetragonal (I) symmetry, but more like orthorhombic symmetry. Applying +5 + +symmetry rotations that belongs to D4h point group to the stiffness matrix it is possible to +quantify how each symmetry is obeyed by the stiffness matrix of both the structures. +We have calculated the static electronic (ϵ∞) and electronic+ionic contribution (ϵ0) of +the dielectric tensor for our tetragonal MAPI structures (Fig. +S2). +Both show similar +symmetry properties. For a perfectly symmetric tetragonal structure we should have only +the diagonal values with (ϵ11 = ϵ22). Calculated off-diagonal values are also an indication +that the structure is not properly symmetric. Although the off-diagonal elements are close to +zero for I4/mcm structure, I4cm obeys the tetragonal symmetry better than I4/mcm. The +dielectric tensor for the I4cm structure also is consistent with S4, D2d, C4, C4v, D4 and D4h +point group symmetries. For I4/mcm, the dielectric tensor is consistent with C2v, D2, and +D2h point group symmetries. +The static non-linear electro-optic tensor χ(2) is a sensitive probe of symmetry, partic- +ularly centrosymmetry,10 since all tensor elements would vanish in the presence of exact +inversion symmetry. The calculated values are in Rydberg atomic units. For the quasi- +I4/mcm structure all the values are close to zero except for χ(2) +zzz, ≈ 30.19 a.u.. More values +are nonzero for I4cm, a clear indication that it is non-centrosymmetric. We have further +checked all the symmetries that χ(2) for a non-centrosymmetric structure should obey.20 We +see that χ(2) +zzz = 22.73 a.u. is the largest element, and χ(2) +xxx ≈ χ(2) +yyy ≈ 3.125 a.u. are other +large elements, which are supposed to be zero for all tetragonal symmetries. These nonzero +values are consistent with Cs, C4 and C4v, and D4h point groups. +Our tetragonal structures in this work do not have any exact symmetry and hence no ir- +reducible representations for their vibrational modes. We can still calculate the approximate +mode irreducible representations using help of group theory. The well known formula for +decomposition of reducible representation into its corresponding irreducible representations +is given in Eq. 1.21 The number of times the irreducible representation Γj appears in the +reducible representation is given by aj, where h is the order of the point group, Ck denotes +a class in the point group, Nk is the number of elements in Ck and χ(Γj)(Ck) represents the +6 + +Figure 2: Deviation from symmetry for each operation of D4h for (a) Born effective charges, +∆Z, (b) atomic Raman tensors, ∆R, and (c) dynamical matrices, ∆D, of I4cm and I4/mcm +structures. +character of the irreducible representation Γj for a symmetry operation in class Ck. +aj = 1 +h +� +k +Nk[χ(Γj)(Ck)]∗χ(Ck) +(1) +The second orthogonality rule for the columns of the character table is given in Eq. 2. +� +j +[χ(Γj)(Ck)]∗χ(Γj)(Ck′) = h +Nk +δkk′ +(2) +For k = E (identity operation), Nk = 1. So we can rewrite Eq. 2 as +� +j +[χ(Γj)(E)]∗χ(Γj)(Ck′) = hδEk′ +(3) +χ(Γj)(E) = 1 for A or B (non degenerate) irreducible representation, χ(Γj)(E) = 2 for E +(doubly degenerate) and χ(Γj)(E) = 3 for T (triply degenerate) irreducible representations. +7 + +14cm +I4/mcm +(a) +3.2 +3.2 +2.329 +2.329 +2.1392.139 +2.1652.166 +2.166 +2.168 + 2.0992.099 2.0992.099 2.161 +2.161 +1.9511.951 +1.558 +1.56 +1.3821.382 +1.382 +1.382 +0.025 +0.0 /p +(2) v(1) C4(1) C4(2) v(2) +C(2) a(1)C2C2(1) S4(1) S4(2) C2(2) C(1)n +v(1)(1) C4(1) C4(2)α(2)C2 +v(2)n +()() (z)s ()s (z) (z) +(b) +Symmetry +Symmetry +1.6 +1.6 +1.1971.197 +1.1231.123 +1.156 +1.156 +1.151 +1.151 +1.0051.0061.0061.006 +0.982 0.982 0.982 0.982 +0.827 +0.817 +0.817 +0.8170.817 +0.797 +0.797 +0.7 +0.7 +0m +0.008 +0.005 +d(2) αv(2) C4(1) C4(2) αv(1)n +(T)P0(z) (z)s (L)"s () () +hC(2) a(2) C4(1) C4(2) oa(1) C(1) S4(2) S4(1) C9(2) C2o(2) +C(2) +gv(1) +E +i +E +C(1) +(c) +Symmetry +Symmetry +1.6 +1.6 +1.2071.207 +1.207 +1.2091.209 +1.2071.208 +1.208 +1.195 +1.1951.205 +1.205 +1.205 +1.205 +1.214 +1.214 +1.0071.007 +0.9570.957 +0.713 +0.713 +0.679 0.68 +0.713 +0.713 +0.68 +0.68 +0.002 +00 +oa(2) ov(2) C4(1) C4(2) v(1) C2od(1) C(1) S4(1) S4(2) C(2) OhC2(1) C(2) +v(1) α(1) C4(1) C4(2) α(2)C2v(2) +(z)() ()s (z)"s () () +F +F +Symmetry +SymmetryMultiplying Eq. (1) with χ(Γj)(E) and summing over j we get +� +j +χ(Γj)(E)aj = 1 +h +� +k +� +j +χ(Γj)(E)[χ(Γj)(Ck)]∗χ(Ck) += 1 +h +� +k +hδEk′χ(Ck) += χ(E) +(4) +It is interesting to note that when we sum over the contributions (aj) of all irreducible +representations for any mode, it turns out exactly 1 for non-degenerate, 2 for doubly degen- +erate, and 3 for triply degenerate modes. To make the sum 1 for all the modes we have to +divide χ(E) by 2 for for doubly degenerate, and by 3 for triply degenerate modes.We have +used Eq. 1 to calculate aj and then use Eq. 4 to find out the proportion of irreducible +representations for each mode. +Using the above methodology, we have calculated the approximate irreducible representa- +tions for I4cm and I4/mcm structures. The main process is explained in the form of a simple +flow chart (Fig. 3). We started with the I4cm structure. We have relaxed the structure +using as mentioned in the computational method section. Density functional perturbation +theory (DFPT) is used to calculate the phonon modes at q = 0. The acoustic sum rule +(ASR) is applied using the dynmat.x code as implemented in Quantum ESPRESSO. We +have taken the position coordinates of the relaxed structure and its calculated phonon mode +vectors as input. The closest symmetry of the structure we considered is I4/mcm (or D4h +point group), because this is the highest symmetry in tetragonal structure and if we calcu- +late this once, we can always get results for I4cm as it is a subgroup of I4/mcm. From the +character table of D4h, we get all the symmetry operations (16 in our case) and the target +irreducible representations.22 From the space group we find all the fractional translations +that are involved. We have constructed all the 3 × 3 rotational matrices (Mαβ) and the +fractional translation vector (⃗t) to apply on the original atomic positions (⃗r) of the crystal +unit cell as r′ +α = �3 +β=1 Mαβ(rβ + tβ) where α and β denote the x, y, and z directions. To +8 + +Figure 3: Approximate phonon mode symmetry calculation flow chart. +make the calculations simple, we started with the Pb-I cage only. The orientation of the +MA+ ions in the structure breaks the symmetry, but the Pb-I cage still holds the D4h point +group symmetry within certain tolerance values (Table S1). After removing the MA+ ion +from the structure, we have applied all the symmetry operations on the structure to find +how they swap atoms. When we apply a rotation to the crystal structure, if for example, +a carbon atom (C1) takes the place of another carbon atom (C2), we say C1 and C2 are +swapped atoms of each other with respect to that rotation. Vibration modes should obey +certain symmetry operations based on the symmetry of the crystal structure. We apply +the symmetry transformation to the vibrational mode Cartesian vectors and calculate the +projection of the transformed mode vector to the original one for each atom and the value +9 + +Coordinates of crystal structure, +phonon mode eigenvectors +Guess a point group +Find symmetry operations (“class" in the character table), +fractional translations (from space group), and +prepare the transformation matrices. +Apply transformation matrices and fractional translation +to all atomic coordinates in the unit cell +r = B=1 Mαβ(rβ +ts) +Map the transformed atom index +for each atom with the original one +For each atom and symmetry operation calculate the projection +of the rotated/transformed phonon mode vector with +its original one as follows +X(Ch) = i,i,a,b UiaMapKi,Uip +Check +x is an +character table +Yes +integer for all +for +symmetry operation. +irreducible +representation +ON1 +Using group theory calculate the contribution of each +irreducible representation to this reducible representation.will give us the character value χ corresponding to that symmetry class for that mode. The +equation for calculating the projection is +χ(Ck) = +� +i,i′,α,β +UiαMαβ(Ck)Ki,i′(Ck)Uiβ +(5) +where i and i′ denote the atom index of the original and transformed atoms respectively, +Ki,i′(Ck) denotes the matrix that transforms i to i′, Uiα denotes the mode vector for atom i +in direction α, and Ck denotes the symmetry class for which χ is been calculated. +To calculate the character we need the mode eigenvector for that particular mode. Once +we remove the MA+ ions we need to re-normalize the mode vectors ( ⃗Ui) for Pb and I as +⃗Vi = ⃗Ui/ +��N +i=1| ⃗Ui|2 where i is the atom index and N is the total number of atoms after +removing all the MA+ ions. We calculated the value of χ for all symmetry classes and for +each mode of the tetragonal MAPI. As our structure is not exactly symmetric, we did not +expect to get integer values for χ for all the symmetry classes, in fact our calculated values +are in fractions. So, we need to find a different way rather than checking character table for +a direct match as we have already mentioned in the flowchart(Fig.3). +For each phonon mode we have calculated χ(Ck) for all symmetry class Ck belonging +to the point group D4h and prepared a table which we call calculated character of modes +because it is like a character table but with character values in fractions rather than in +integers as we normally see in a character table. +Each row of this calculated character +of mode table is treated as a reducible representation and we decompose them into the +irreducible representations using group theory (Eq. 1). +We noticed that the sum of contributions of all the irreducible representations become +1 for all the modes. We gave a theoretical explanation why this occurs using group theory +(Eq. 4). If we just sum up the contributions (aj) for all irreducible representations we end +up getting sum as 2’s and 3’s for doubly degenerate or triply degenerate modes which is a +problem because in that case the contributions of irreducible representations for each mode +10 + +do not sum up to one, which makes it hard to compare between all the modes. We have +studied it further by decomposing the degenerate modes into a possible combination of two +symmetrized non-degenerate modes by looking at how the basis functions (x,y) transform +with different symmetry operations and repeated the same calculations for calculating aj +and this time it gave the sum as 1 but our symmetrized combination is just one of the many +possible permutations of how (x,y) basis can transform under the symmetry operations. It +become even harder when the character value become imaginary in some cases, for example, +in the character table for C3 point group the degenerate irreducible representation is a +symmetrized combination of 1, ei2π/3, and e−i2π/3. +This issue is known as the doubling +problem.23 On the other hand, we see that in our formulation of Eq. 4 we just have to divide +the sum by 2 for the doubly degenerate mode as χ(Γj)(CE) = 2 and this makes the sum of all +the irreducible representations for each mode (including the degenerate ones) as 1. We use +this treatment, in which case we do not need to split the degenerate mode into an arbitrary +basis. +As a test case of our method, we checked orthorhombic MAPI, whose modes are all +non-degenerate, and TiO2, which has some doubly degenerate modes. Our method is able +to calculate the mode irreducible representations for these two exactly symmetric structures +(Fig. S5), comparable to the Quantum ESPRESSO ph.x output of the mode irreducible rep- +resentations. We are able to calculate the irreducible representations exactly even without +considering the hydrogen atoms in orthorhombic MAPI. This result suggests it is reason- +able to try to calculate mode irreducible representations of the tetragonal structure without +considering the H atoms. +Now applying the method to tetragonal MAPI: the contribution of irreducible represen- +tations for each mode of Pb-I cage is shown in Fig. 4(a). Because the mid and high frequency +modes do not have much Pb-I vibrations it is not enough to get irreducible representations +for all the modes of tetragonal MAPI just using only Pb-I cage. It also indicates that the high +frequency modes are purely molecular modes. We need to consider the molecular vibrations +11 + +456789 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +119 +120 +121 +122 +123 +124 +125 +126 +127 +128 +129 +130 +131 +132 +133 +134 +135 +136 +137 +138 +139 +140 +141 +142 +143 +144 +mode numbers +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +irrep contributions +A1g +A2g +B1g +B2g +Eg +A1u +A2u +B1u +B2u +Eu +(a) I4cm (Pb-I cage only) +123456789 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +119 +120 +121 +122 +123 +124 +125 +126 +127 +128 +129 +130 +131 +132 +133 +134 +135 +136 +137 +138 +139 +140 +141 +142 +143 +144 +mode numbers +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +irrep contributions +A1g +A2g +B1g +B2g +Eg +A1u +A2u +B1u +B2u +Eu +(b) I4cm complete sructure +123456789 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +119 +120 +121 +122 +123 +124 +125 +126 +127 +128 +129 +130 +131 +132 +133 +134 +135 +136 +137 +138 +139 +140 +141 +142 +143 +144 +mode numbers +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +irrep contributions +A1g +A2g +B1g +B2g +Eg +A1u +A2u +B1u +B2u +Eu +(c) I4/mcm complete structure +Figure 4: Contributions of different irreducible representations for each mode in (a) Pb-I +cage only structure of I4cm symmetry, (b) full I4cm structure, and (c) full I4/mcm structure +calculated considering the highest symmetry D4h of the tetragonal structure. +if we want to calculate the irreducible representations correctly for mid and high frequency +modes. We decided to keep the C and N of the MA+ ion with the Pb-I cage and not con- +sider the H atoms which are randomly oriented anyway and hard to track after the rotational +symmetry operation on the structure as they are more in number and close to each other in +space. Our reason of not considering H is also supported by the idea that, for orthorhombic +MAPI we are able to calculate exact irreducible representations for each mode even without +considering the H atoms in the structure and we have also checked that the contribution +12 + +of the H atoms in each mode eigenvectors for both orthorhombic and tetragonal structure +looks similar and the H mainly affects the high-frequency modes (Fig. S3). We followed +the same process as we mentioned earlier for Pb-I cage and calculated the contributions +of the irreducible representations for phonon modes of both I4cm and I4/mcm tetragonal +structures. The result is given in Fig. 4(b,c). We can see that for low-frequency modes, +both Pb-I cage-only calculation and the entire structure (except H) give similar result, for +mid-frequency region the molecular modes change the irreducible representations that are +coming from Pb-I only. It can be also seen that some modes obey the symmetry better than +the others. +To assess the degree to which each vibrational mode obeys symmetry, we construct a +quantity T (Ck) that is equal to the number of modes N = 144 for a perfectly obeyed +symmetry operation Ck. In the absence of degeneracy, all characters are ±1, and so the sum +of the squared characters χi of modes for any class Ck should be equal to the total number of +modes, T (Ck) = �N +i=1 χ2 +i (Ck) = N. However, degenerate modes should be treated together, +as their individual characters are arbitrary and only the sum of their characters is meaningful. +For doubly degenerate modes (no triple degeneracies occur in D4h), this sum can be −2, 0, +or 2. In this case, rather than χ2 +i + χ2 +i+1 in the sum, we use 2(|χi + χi+1| − 1)2, which gives +a contribution of 2 for the two modes together for anhy of these 3 possible ideal values, and +preserves the idea of a total sum of N. How do we identify degenerate modes in the presence +of approximate symmetry? Almost all the modes have some contributions from the doubly +degenerate Eg and Eu representations of D4h. We consider a mode degenerate if the sum +of the contributions from Eg and Eu is greater than 80%, and there is a pair of consecutive +modes close in frequency. The results for T (Ck) are given in Fig. S4. We can see that some +of the symmetry operations such as C4, C′ +2, S4, σv, and σd have values close to 144, while +others are as low as half this. We see that T (Ck) is the same for each member of a class, as +expected. +To consider whether the vibrational modes of the ostensibly I4/mcm structure are best +13 + +described by I4/mcm symmetry or some other subgroup, we have assessed which symmetry +operations are obeyed by the modes. For example, we can see that σd is obeyed better than +rest of the operations. So subgroup Cs clearly applies well. To check more rigorously, we +have calculated the contribution of irreducible representations of vibrational modes based +on each subgroup, and ranked each subgroup based on a value (RG) as given in Eq. 6. +RG = +Nmodes +� +ν +Nirreps +� +i +µ2 +ν,i +(6) +Here µν,i is the contribution of the irreducible representation i for mode ν, and the squaring +is analogous to the inverse participation ratio. The sum should be equal to or less than +the total number of modes, which is 144 in our case but will be less as our structure is not +properly symmetric. This is because for a perfect irreducible representation of a mode, the +maximum value of µν,i can be 1. +The plot for the rank of each subgroup is given in Fig. 5. Based on the symmetry of the +stiffness tensor (C), dielectric tensor tensor (ϵ), electro-optic tensor (χ(2)), and calculated +rank of the subgroup we can see that C4v is the best symmetry point group for I4cm and +C2v for I4/mcm structure. +Given the limitations of the method above for analyzing symmetry of vibrational modes, +we also investigated the symmetry directly of atom-resolved tensors of the system, in partic- +ular the Born effective charge tensors (Zαij), atomic Raman tensors (Rijkα) and dynamical +matrix (Ds +iαjβ). Here α, β are the atom indices (including only Pb, I, C, and N atoms) and +i, j represent the Cartesian x, y, and z directions. These tensors are the source of IR and +Raman spectroscopy, and in the case of Z and R, provide a mixed structural/electronic prop- +erty. Each tensor was calculated using density functional perturbation theory in Quantum +ESPRESSO. We can quantify deviations from symmetry by transforming the tensor under a +symmetry operation and calculating the deviation from the original. If the structure obeys +the symmetry perfectly, then deviations from symmetry ∆Z (Eq. 7), ∆R (Eq. 8), and ∆D +14 + +Figure 5: Ranking of subgroups of D4h in declining order of satisfaction of symmetry by the +vibrational modes of the two MAPI structures. The subgroups are also annotated with a +tensor (ϵ, C, and χ(2)) if the tensor’s symmetry properties are consistent with that subgroup. +(Eq. 9) should be zero for each symmetry operation. +∆Z = +� +� +� +�� +αij +�����Zαij − +� +α′i′j′ +Kii′Kjj′Mαα′Zα′i′j′ +����� +2 +(7) +∆R = +� +� +� +�� +ijkα +�����Rijkα − +� +i′j′k′α′ +Kii′Kjj′Kkk′Mαα′Rα′i′j′k′ +����� +2 +(8) +∆D = +� +� +� +�� +ijαβ +�����Ds +iαjβ − +� +i′j′α′β′ +Kii′Mαα′Kjj′Mββ′Ds +α′i′j′β′ +����� +2 +(9) +Calculated deviation from symmetry for both I4cm and I4/mcm structures are shown in Fig. +2, giving quite similar values in the two cases. σd for I4cm and σv for I4/mcm are obeyed +almost perfectly. Here we can see that the symmetries that belong to the same class (e.g. +15 + +(a) +160 +symmetry +142.87 +140 +I4cm structure +120 +108.5 +107.85 +107.06 +107.05 +101.16 +101.11 +100.48 +100 +Contribution to +85.72 +84.05 +80- +79.65 +79.51 +67.79 +67.13 +66.38 +66.38 +60 +61.83 +44.13 +40 +20 +0 +Cs(od2) +C2 +Cs(od) +D2d +C4 C4v/(/4cm)D4 +Cs(oh) +Cs(ov2) Cs(ov) +D2 +C2h +C4h +D4h +C2v +Dzh +3 +3 +3 +3 +Sub-group +3 +3 +c +c +c +c +(b) +(2) +(2) +X +160 +symmetry +143.31 +140 +I4/mcm structure +120 +109.68 +109.27 +109.13 +108.33 +108.33 +107.5 +107.34 +107.33 +100 +94.85 +93.85 +87.89 +87.73 +80 +77.19 +76.81 +76.72 +76.03 +76.03 +Contribution +60 +40 +20 +0 +Cs(oy) +C2 +Cs(ov2) +C2v +C4 C4v/(i4cm) S4 +D4 +D2d +Ci +Cs(oh) Cs(d2) Cs(oa) +C2h +D2 +D2h +C4h +D4h +3 +Sub-group +3 +3 +c +c +c +.(2)C4(1) and C4(2)) give the same value for the deviation of symmetry, which is because those +operations are related by the highly satisfied symmetries σd for I4cm and σv for I4/mcm. +To assess the significance of the other deviations, we can compare to the Frobenius norm of +each tensor (Eqs. 10, 11, 12). +||Z|| = +�� +αij +|Zαij|2 +(10) +||R|| = +�� +ijkα +|Rijkα|2 +(11) +||D|| = +�� +ijαβ +��Ds +iαjβ +��2 +(12) +We find that ||Z|| = 21.83 a.u.−1 for I4cm and 21.91 a.u.−1 for I4/mcm, ||R|| = 0.9956 a.u.−1 +for I4cm and 1.024 a.u.−1 for I4/mcm, and ||D|| = 5.607 Ryd2 for I4cm and 5.603 Ryd2 for +I4/mcm. Deviations are small compared to the Frobenius norm for Z and D, indicating all +symmetry operations are approximately valid, but on the same order as the Frobenius norm +for R, indicating poor satisfaction of symmetries. As a result, we can expect approximate +symmetries in tetragonal MAPI to be useful for analysis of IR spectroscopy but not very +useful for analysis of Raman spectroscopy. The close satisfaction of σ symmetries indicates +that Cs is an appropriate point group in both cases, but the similar values of the deviation +for other operations does not help to distinguish further what higher symmetry may be ap- +propriate, and so our picture of the approximate symmetry remains based on the vibrational +modes and global tensors. +We have analyzed hidden symmetry in theoretical structures of tetragonal MAPI, via a +group theory analysis of vibrational modes and by rotation of response tensors, to quan- +tify approximate symmetries that can be used to understand its spectroscopy and other +properties. Theoretical calculations have proposed predominant structures referred to as +quasi-I4cm or quasi-I4/mcm, but neither possesses any exact symmetry in its atomic co- +ordinates. Nevertheless, by looking at symmetry in various perspectives and considering +16 + +subgroups of the full tetragonal symmetry (D4h point group), we find that the quasi-I4cm +structure can indeed be best described by approximate I4cm (C4v point group) symme- +try, whereas the quasi-I4/mcm structure is best described not by I4/mcm but by the lower +symmetry of the C2v subgroup. Our methodology allows us to quantify the approximate +symmetry of different vibrational modes, by analysis into irreducible representations, and +to quantify the degree to which each symmetry operation is satisfied by the modes. We +also assessed the symmetry of global response tensors (dielectric, elastic, and electro-optic) +and atom-resolved response tensors (Born effective charge, Raman, and dynamical matrix) +to develop a combined picture of the usable symmetries in this material. We exclude the +H atoms from the analysis due to rapid cation rotations except at very low temperature. +Our methodology can be useful to rigorously quantify approximate symmetry in a material, +for example doped structures, polycrystalline materials or even amorphous materials and be +helpful to understand spectroscopy. Such an approach is particularly important for novel +soft semiconductors such as low-dimensional hybrid perovskites24 and other organic metal +halide hybrid materials,25 which typically feature symmetry-breaking cation rotations ex- +cept at the lowest temperatures, and yet still have enough symmetry for group theory to be +useful in analysis. Our methodology can be used for a variety of properties such analysis of +strain effects on Raman spectra,15 in which we previously established approximate isotropic +symmetry in amorphous Si,26 or for excited-state forces and exciton-phonon couplings. +Acknowledgement +This material is based upon work supported by the Air Force Office of Scientific Research +under award number FA9550-19-1-0236. This work used computational resources from the +Multi-Environment Computer for Exploration and Discovery (MERCED) cluster at UC +Merced, funded by National Science Foundation Grant No. ACI-1429783, and the National +Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office +17 + +of Science User Facility operated under Contract No. DE-AC02-05CH11231. +References +(1) Shikoh, A. S.; Polyakov, A. A Quantitative Analysis of the Research Trends in Per- +ovskite Solar Cells in 2009–2019. Phys. Status Solidi A 2020, 217, 2000441. +(2) Stoumpos, C. C.; Malliakas, C. 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Stress +effects on the Raman spectrum of an amorphous material: Theory and experiment on +a-Si:H. Phys. Rev. B 2015, 92, 241202(R). +20 + +Graphical TOC Entry +21 + +Superimposed original Pb-l cage with its rotated (C2z) form +107.54 cm-1 +109.86 cm-1 +briginal +rotated +position +position +u +Z +x(C2z) = + 0), if Cm(e1, · · · , em) > 0 for any orthonormal +frame {ei}. When m = 2 (resp. m = 3), the m-intermediate curvature is also called +bi-Ricci (resp. tri-Ricci) curvature. +We note that Cm(e1, · · · , em) only depends on the subspace which {e1, · · · , em} spans. +To see this, we orthogonally split TpM = V ⊕ V ⊥ where V = span{e1, · · · , em}. The +metric tensor also splits as g = g1 ⊕ g2. We have Cm(e1, · · · , em) = ⟨Rijkl , 1 +2(g1)il(g1)jk + +(g1)il(g2)jk⟩. More properties of intermediate curvature can be found in [2]. +A model space for intermediate curvatures is the product of flat torus and round sphere +M = T m × Sn−m: M has positive Cm+1 but only non-negative Cm. It is natural to ask +1The meaning of the phrase “intermediate curvature” may differ in other topics of study. +1 + +a topological obstruction problem: whether T m × Sn−m admit metrics with positive Cm. +This question extends two well-known results. For the case m = 1 (i.e. Ric > 0), Bonnet- +Myers’ theorem or Bochner’s formula implies negative answer. The case m = n − 1 (i.e. +M = T n with positive scalar curvature) is Geroch’s conjecture, and the answer is also +negative as shown by Schoen-Yau [13] [14] and Gromov-Lawson [8]. The intermediate +cases are recently proved by Brendle-Hirsch-Johne [2] up to dimension 7. +Theorem 1.2 ([2]). For n ≤ 7, there is no metric on T m × Mn−m with Cm > 0, where +Mn−m is any (n − m)-dimensional closed manifold. +Chu-Kwong-Lee [4] later proved a corresponding rigidity statement. +Theorem 1.3 ([4]). Let n ≤ 5. If a metric g on T m × Mn−m satisfies Cm ≥ 0, then g +splits isometrically as a product of a flat T m with a metric on Mn−m with nonnegative +Ricci curvature. +This paper is focused on the dimension constraints that appear in the above theorems. +It turns out that counterexamples exist in higher dimensions. The following is one main +theorem of this paper. +Theorem 1.4. Assume m ≥ 2, n ≥ m+2. Let F = F(x1, · · · , xm) be a positive function +on the torus T m. Consider the following metric on Sn−m × T m: +g = ǫ2F 2h + F −2 n−m +m−1 +m +� +i=1 +dx2 +i , +(1.1) +where h is the standard metric on Sn−m. Then we have +Cm +� ∂x1 +|∂x1|, · · · , ∂xm +|∂xm| +� += (n − m) +�(n − m)(m − 2) +2(m − 1) +− 1 +� +F 2 n−m +m−1 −2 +m +� +i=1 +�∂F +∂xi +�2. +If n(m − 2) ≥ m2 − 2 and ǫ is sufficiently small (depending on the C2 norm of log F), +then g satisfies Cm ≥ 0 everywhere. If further n(m − 2) > m2 − 2, then Cm > 0 at all +non-critical points of F. +The relation n(m − 2) ≥ m2 − 2 obtained here is consistent with the algebraic in- +equalities obtained in [2, Lemma 3.14] and [4, Lemma 2.3]. From Theorem 1.4, we obtain +metrics on T 4 × S4 and T 3 × S5 with almost positive intermediate curvature. To resolve +the issue occurred at the critical points of F, we can further perturb the metric so that Cm +is positive everywhere. This confirms the optimality of the condition n ≤ 7 in Theorem +1.2. +Corollary 1.5. Assume all the conditions in Theorem 1.4, and n(m − 2) > m2 − 2. If +f is a Morse function on T m, then there exists a small perturbation of g that satisfies +Cm > 0 strictly. +For Theorem 1.3, we obtain non-trivial metrics on T 3 × S4 and T 4 × S3 with non- +negative intermediate curvatures, giving counterexamples when n ≥ 7. The case n = 6 is +not included in Theorem 1.3. However, the proof in [4] can be improved and thus extend +to n = 6. +Theorem 1.6. The statement of Theorem 1.3 holds for n = 6 as well. +2 + +Given the formula (1.1), it is an elementary work to verify all the conclusions. There- +fore, finding (1.1) is the most important aspect of this result. To better explain the idea, +let us briefly summarize the proof of Theorem 1.2 in [2]. If there exists a metric as stated +in the theorem, we find a chain of hypersurfaces M ⊃ Σ1 ⊃ · · · ⊃ Σm, with Σ1 being a +stable minimal hypersurface in M, and each Σi being a minimizer of a certain weighted +area functional on Σi−1. Combining all the stability inequalities of Σi, one obtains an +inequality of the form +0 ≤ +� +Σm +� +− Cm − V1 − V2 − · · · +� +, +(1.2) +where V1, V2, · · · are algebraic expressions involving the second fundamental forms of Σi. +We would obtain a contradiction if Vi ≥ 0 for all i. However, this is true only when a +certain dimensional inequality is satisfied. +The observation here is that: the validity of the algebraic inequalities Vi ≥ 0 suggests +the existence of counterexamples. If Vi has the chance to be negative, we try to reverse +engineer a metric that captures the negative part. This leaves some room in (1.2) that +allows positive Cm. Then we try to achieve equality case for all the other inequalities +involved. Therefore, we are forced to have Cm ≡ −(V1 + V2 + · · · ) > 0. This process +finally yields (1.1). A more detailed explanation is contained in subsection 2.1. +The next problem concerns diameter bounds under the presence of uniformly positive +intermediate curvatures. By Bonnet-Myers’ theorem, manifolds with uniformly positive +Ricci curvature has bounded diameter. When generalizing to bi-Ricci curvature, one needs +to pass to a stable minimal surface. +Theorem 1.7 (Shen-Ye [15]). Let M be a (complete) manifold with dimension n ≤ 5 and +bi-Ricci curvature C2 ≥ λ > 0. Then any two-sided stable minimal hypersurface in M has +diameter ≤ C(n)λ−1/2. +We note that Theorem 1.7 can be interpreted intrinsically in terms of Σ (without +mentioning the ambient manifold M). +Definition 1.8. On a Riemannian manifold M, we define the minimum Ricci curvature +function +R0(p) := +min +e∈TpM,|e|=1RicM(e, e), +p ∈ M. +Equivalently, R0 is the minimal eigenvalue of Ricci curvature at each point. +Lemma 1.9. Suppose M satisfies bi-Ricci curvature lower bound C2 ≥ λ, and Σ ⊂ M is +a stable minimal surface. Then +λ1(−∆Σ + R0) ≥ λ, +(1.3) +which means +� +Σ |∇ϕ|2 + R0ϕ2 ≥ λ +� +Σ ϕ2 for all ϕ ∈ C∞(Σ). +We may understand Σ as having “Ric ≥ λ” in a weaker sense. We wish to obtain a +Bonnet-Myers’ theorem from (1.3). The second main result of this paper is that such result +holds only for dimension ≤ 4. We can generalize condition 1.3 by adding a coefficient in +front of R0, and this helps us see clearly of the optimal form of constraint. +3 + +Theorem 1.10. Let β > 0, n ≤ 7. Suppose an n-dimensional complete manifold Σ +satisfies +λ1(−∆Σ + βR0) ≥ λ > 0, +then we have diam(Σ) ≤ C(n, β)λ−1/2 if + + + + + +β ≥ 1 +2 +(n = 3), +β > 1 +4(n − 1) +(n ̸= 3). +(1.4) +There exist non-compact counterexamples when (1.4) is not satisfied. +The condition n ≤ 7 is due to the smoothness issue of area-minimizing currents. +Theorem 1.7 is a consequence of the β = 1 case, hence the condition n ≤ 5 there is +necessary. A proof of Theorem 1.10 (without counterexamples) was given in Shen-Ye [16] +using weighted minimal geodesics with weight function u1/β. Condition (1.3) is equivalent +to the notion of conformal Ricci curvature lower bounds introduced in [16]. +Here we give a new proof of Theorem 1.10 based on the µ-bubble technique. One bene- +fit of this proof is that counterexamples can be easily reverse-engineered. An introduction +to µ-bubbles, including references, is included in subsection 3.1. +We observe that, the usage of µ-bubbles imposes a more strict dimension constraint +than minimal surfaces. For example, we can show that Sn−1 × S1 (n ≤ 7) does not admit +metrics with λ1(−∆+ βR0) > 0 when β ≥ 1 − +1 +n−1 (see Remark 3.3). The proof only uses +(generalized) minimal surfaces, so the constraint for β is weakened compared to (1.4). +The gap from 1 − +1 +n−1 to 1 +4(n − 1) indicates essential difference between Sn−1 × S1 and +Sn−1 × R in regard of condition (1.3). +Finally, we relate intermediate curvature to the notion of macroscopic dimension. A +manifold M is said to have macroscopic dimension ≤ k, or to have finite Urysohn k- +width, if there exists a simplicial complex X with dimension ≤ k, and a continuous map +f : M → X, such that each fiber f −1(p) has uniformly bounded diameter. +Clearly +manifolds with Ric ≥ λ > 0 have finite Urysohn 0-width. Gromov conjectured that an +n-manifold with uniformly positive scalar curvature has finite Urysohn (n−2)-width. The +dimension 3 case has been answered affirmatively [8] [9] [11]. In higher dimensions this +problem is still open. +Interpolating between Ricci and scalar curvature, we would expect that manifolds with +uniformly positive m-intermediate curvature has finite Urysohn (m − 1)-width. Apply- +ing Chodosh-Li’s slice-and-dice argument [3], we show affirmatively the case m = 2, in +dimension up to 5. +Theorem 1.11. Let M be a complete simply-connected manifold with dimension n ≤ 5 +and bi-Ricci curvature C2 ≥ λ > 0. Then M has finite Urysohn 1-width. +In dimension higher than 5, there exists counterexamples in the same spirit as in +Theorem 1.10. +Theorem 1.12. When n > 5, there exists a complete metric on Sn−2×R2 with uniformly +positive bi-Ricci curvature and infinite Urysohn 1-width. +This paper is organized as follows. In section 2 we discuss the case of non-negative +Cm. This section contains a detailed explanation of the reverse engineering process, and +4 + +the proof of Theorem 1.4. In section 3 we discuss the case of uniformly positive Cm. This +section contains an introduction to µ-bubbles, and the proof of Lemma 1.9, Theorem +1.10 - 1.12. In Section 4, we prove the rigidity in dimension 6 (Theorem 1.6). +Notations. We use M to denote a manifold, and Σ denote a hypersurface. We use +N to denote the unit normal vector of Σ, and A, H denotes the second fundamental form +and mean curvature. The sign convention is A(X, Y ) = ⟨∇XN , Y ⟩, H = trΣ A. These +notations apply to all sections except Section 4, where we adopt the same notation as in +the references. +Acknowledgements. The author would like to thank Sven Hirsch for inspiring con- +versations and comments on previous drafts. He also thanks Jianchun Chu for discussions +on the work [4]. +2 +The Case of Nonnegative Intermediate Curvature +2.1 +Reverse engineering of counterexamples +In this section, we use the example T 3×S5 to demonstrate the reverse engineering process +mentioned in the introduction part. To better show the idea, we re-write the argument +in [2] using more convenient notations. In what follows, we use subscript to denote the +dimension in which an object lives. For example, u7 denotes a function on M7, N7 (resp. +A6 and H6) denotes the unit normal vector (resp. second fundamental form and mean +curvature) of M6 ⊂ M7, and ∇5 denote the gradient on M5. +Assume that M8 = T 3 × S5 has positive tri-Ricci curvature. Ignoring the possible +singularity, suppose that we can find a smooth area minimizer M7 ⊂ M8 in the homology +class of T 2 × S5. By the stability inequality for minimal surfaces, there exists u7 > 0 on +M7 such that +∆7u7 ≤ +� +− |A7|2 − Ric8(N8, N8) +� +u7. +(2.1) +Then, let M6 ⊂ M7 be a minimizer of the functional M �→ +� +M u7 in the homology class +of S1 × S5. The stability inequality implies that there exists u6 > 0 on M6 such that +∇6 · +� +u7∇6u6 +� +≤ +� +− |A6|2u7 − Ric7(N7, N7) + ∆7u7 − ∆6u7 +� +u6. +(2.2) +Finally, let M5 ⊂ M6 be a minimizer of M �→ +� +M u6u7 in the homology class of S5. The +stability inequality gives +0 ≤ +� +M5 u6u7|∇5ϕ|2 + +� +M5 +� +− |A5|2u6u7 − Ric6(N6, N6)u6u7 + ∆6(u6u7) − ∆5(u6u7) +� +ϕ2 +(2.3) +for any ϕ. We let ϕ = (u6u7)−1, and then combine (2.3) with (2.2) (2.1). Note that +∆6(u6u7) = ∇6 · (u7∇6u6) + u6∆6u7 + ∇5u6 · ∇5u7 + ∂u6 +∂N6 +· ∂u7 +∂N6 +, +and the Gauss equations +Ric7(N7, N7) = Ric8(N7, N7) − R8(N7, N8, N8, N7) − A2 +7(N7, N7), +5 + +Ric6(N6, N6) = Ric8(N6, N6) − R8(N6, N7, N7, N6) − R8(N6, N8, N8, N6) ++ H6A6(N6, N6) − A2 +6(N6, N6) − A2 +7(N6, N6) +− A7(N6, N6)A7(N7, N7) + A7(N6, N7)2. +After rearranging the terms, we obtain +0 ≤ − +� +M5 C3(N6, N7, N8)ϕ +− +� +M5 +� +(u6u7)−3|∇5(u6u7)|2 − (∇5u6 · ∇5u7)(u6u7)−2� +− +� +M5 +� +|A5|2 − u−1 +6 +∂u6 +∂N6 +· u−1 +7 +∂u7 +∂N6 +� +ϕ +− +� +M5 +� +|A6|2 + H6A6(N6, N6) − A2 +6(N6, N6) +� +ϕ +− +� +M5 +� +|A7|2 − A2 +7(N6, N6) − A7(N6, N6)A7(N7, N7) + A7(N6, N7)2� +ϕ +=: − +� +M5 C3(N6, N7, N8)ϕ − (I + V3 + V2 + V1). +(2.4) +This inequality is equivalent to Lemma 3.4, 3.10 in [2], and there the terms V1, V2, V3 are +shown to be non-negative in lower dimensions. However, it is not true that V1, V2, V3 ≥ 0 +in the current dimension. For example, for the term V3 we have +V3 ≥ H2 +5 +5 − ∂ log u6 +∂N6 +· ∂ log u7 +∂N6 += 1 +5 +�∂ log u6 +∂N6 ++ ∂ log u7 +∂N6 +�2 +− ∂ log u6 +∂N6 +· ∂ log u7 +∂N6 +? +≥ 0, +which is clearly not true by letting u6 = u7 and A5 be a multiple of g|M5. Similarly, V2, V1 +can be negative. For example, by letting A6, A7 have diagonal form with +A6(e1, e1) = · · · = A6(e5, e5) = x, +A6(N6, N6) = −5 +2x, +(2.5) +and +A7(e1, e1) = · · · = A7(e5, e5) = y, +A7(N6, N6) = A7(N7, N7) = −5 +2y, +(2.6) +we obtain V2 < 0, V1 < 0. Among the many available choices, (2.5) (2.6) are chosen so +that the resulting metric has the simplest form. +Next, we collect and combine together the information obtained. Consider a metric +on T 3 × S5 of the following form: +g = f 2 +1h + f 2 +2dx2 + f 2 +3dy2 + f 2 +4 dz2, +where f1, f2, f3, f4 are functions in x, y, z, and h is the standard metric on S5. +The first condition we impose on g is that, any coordinate slice +M5 = {x = x0, y = y0, z = z0} ⊂ M6 = {y = y0, z = z0} ⊂ M7 = {z = z0} +is a slicing of weighted minimal surfaces introduced above. This gives many hints on the +choice of fi. First, the mean curvature of M7 is identically zero along the flow ∂/∂z = f4N8 +by our condition, hence +∆7f4 ≡ −(|A2 +7| + Ric8(N8, N8))f4 +6 + +by the variation formula of mean curvature. Compared with (2.1), we have a natural choice +f4 = u7. We also know that (2.1) should be equality. Second, the area +� +f 5 +1f2f3 dx dy +must be constant in z. Hence we let f 5 +1f2f3 be pointwise constant. +Followed by the same analysis on M6, we obtain f2 = u6, and f 5 +1f2u7 is constant +(which implies f3 = f4 = u7), and that (2.2) is an equality. By spherical symmetry, (2.3) +is equality for constant ϕ. Therefore, all the inequalities (2.1) (2.1) (2.3) (2.4) achieve +equality, this forces +C3(N6, N7, N8) = −(V1 + V2 + V3). +Next, we interpret conditions (2.5) (2.6) in terms of f1, f2, f3, f4. Note that +A6(ei, ei) = f −1 +3 +∂ log f1 +∂y +, A6(N6, N6) = f −1 +3 +∂ log f2 +∂y +(2.5) +−−−→ f2 = f −5/2 +1 +, +and +A7(ei, ei) = f −1 +4 +∂ log f1 +∂z +, A7(N7, N7) = f −1 +4 +∂ log f3 +∂z +(2.6) +−−−→ f3 = f −5/2 +1 +. +Under these choices we have V1 < 0, V2 < 0, V3 < 0. We finally obtain +f1 = F(x, y, z), f2 = f3 = f4 = F(x, y, z)−5/2, +which is exactly (1.1). One could repeat this argument on the general cases T m × Sn−m. +A much faster way is to directly consider g = F 2h + F −s � dx2 +i . The choice s = 2 n−m +m−1 is +determined by cancelling all the second derivatives of F in Cm(∂x1, · · · , ∂xm). +2.2 +Counterexamples in dimension ≥ 7 +Notations. For convenience, for the rest of this section we let k = n − m, so that the +underlying manifold is T m × Sk. Greek indices α, β, · · · are assumed to be in the Sk +direction, while Latin indices i, j, · · · are in the T m directions. The indices p, q, · · · can +be in either direction. Denote by {eα}k +α=1 an orthonormal frame for g tangent to Sk, +and denote ei = +∂xi +|∂xi|. Denote the partial derivatives Fi = +∂F +∂xi, and dF 2 = � +i F 2 +i . Let +s = +2k +m−1. Therefore, the metric under consideration is +g = ǫ2F 2hαβ + F −s � +i +dx2 +i . +(2.7) +Proof of Theorem 1.4. +We first compute Cm( ∂x1 +|∂x1|, · · · , ∂xm +|∂xm|). The Christoffel symbols of g are: +Γi +αβ = −ǫ2F s+1Fihαβ, +Γβ +αi = F −1Fiδβ +α, +Γα +ij = Γj +iα = 0, +Γi +ij = −s +2F −1Fj, +(when i ̸= j) Γj +ii = s +2F −1Fj. +7 + +From this we compute the curvature components (Einstein summation is not used): +gjjRi +ijj = s +2F s−1(Fii + Fjj) − s +2F s−2(F 2 +i + F 2 +j ) − s2 +4 F s−2 � +k̸=i,j +F 2 +k +(i ̸= j), +gααRi +iαα = −F s−1Fii + s +2F s−2(−F 2 +i + +� +j̸=i +F 2 +j ), +gααRj +iαα = −F s−1Fij − sF s−2FiFj +(i ̸= j), +Rα +ijk = Ri +αβγ = Rβ +ijα = 0, +Rη +αβγ = +� +ǫ−2F −2 − F s−2 � +i +F 2 +i +�� +δη +αgβγ − δη +βgαγ +� +. +Moreover, Rl +ijk is always a polynomial combination of ∂2F +F +and ∂F +F . Finally we compute +Cm +� ∂x1 +|∂x1|, · · · , ∂xm +|∂xm| +� += +� +i 0, +∀p ∈ Z. +(2.11) +To see this, we recall the general formula for variation of metrics: for a family of metrics +gt with ˙g = +dg +dt , we have ˙Γr +pq := +d +dtΓr +pq = +1 +2grs(∇ip˙gqs + ∇q ˙gps − ∇s ˙gpq), and +d +dtRs +pqr = +∇p ˙Γs +qr − ∇q ˙Γs +pr. Applying ˙g = ψg, we obtain +d +dtRp +pqq = −1 +2(∇2 +ppψ + ∇2 +qqψ) (p ̸= q). Note +that ∇2 +iiψ < 0 and ∇2 +ααψ = 0 at p. Also ˙g = 0 at p, hence +d +dt +� +Cm(p)( ∂x1 +|∂x1|g +, · · · , ∂xm +|∂xm|g +) +���� +t=0 = +� +i 0. +This implies +d +dtCm ≥ a − b(|x| + |z|) in the local coordinate chart defined above, for +some a, b > 0 depending on F, ψ. Therefore, there exists a small neighborhood Vp ∋ p +and a sufficiently small tp (depending on F, ψ) for which +Cm(gt) ≥ C(|x|2 + |z|2) + 1 +2at − 2bt(|x| + |z|), +∀t < tp. +Hence for small t we have Cm(gt) > 0 in Vp. For each p ∈ Z we obtain such a small +neighborhood Vp. In the open cover � +p Vp ⊃ Z, we choose a finite subcover {Vpi} and +obtain a uniform t0 ≪ 1 such that Cm(gt) > 0 in � Vpi when t ≤ t0. Since Cm(g0) > 0 on +E \ � Vpi, by compactness we have Cm(gt) > 0 on E \ � Vpi when t ≤ t1, for a sufficiently +small t1. This shows Cm(gmin(t0,t1)) > 0 on the whole E. +□ +3 +The Case of Uniformly Positive Intermediate Cur- +vature +3.1 +A brief introduction to µ-bubbles +Let M be a Riemannian manifold. A µ-bubble is a hypersurface Σ ⊂ M with prescribed +mean curvature H = h, where h is a function on M to be chosen. In variational perspec- +tive, we consider the generalized perimeter functional +E0(Ω) := |∂Ω| − +� +Ω +h. +(3.1) +10 + +When Ω is a critical point of E0, its boundary Σ = ∂Ω is a µ-bubble. The second variation +formula of E0 at a critical point is computed to be +δ2E0(Ω)(ϕ) = +� +Σ +� +|∇Σϕ|2 − +� +|A|2 + Ric(N, N) + ∂h +∂N +� +ϕ2� +. +(3.2) +A µ-bubble is called stable if the second variation (3.2) is always non-negative. To +demonstrate how stable µ-bubbles interact with uniformly positive curvatures, here we +prove that a stable µ-bubble in a manifold with C2 ≥ λ > 0 satisfies a Bonnet-Myers’ +theorem, for a properly chosen function h. +Let e be a measurable unit vector field on Σ such that RicΣ(e, e) = R0 everywhere. +The stability inequality implies +0 ≤ +� +Σ +� +|∇Σϕ|2 + R0ϕ2 − +� +RicΣ(e, e) + |A|2 + RicM(N, N) + hN +� +ϕ2� += +� +Σ +� +|∇Σϕ|2 + R0ϕ2 − C2(e, N)ϕ2 + |∇Mh|ϕ2 − +� +HA(e, e) − A2(e, e) + |A|2� +ϕ2� +. +When dim(M) = n ≤ 5 ⇔ dim(Σ) ≤ 4, we claim that +HA(e, e) − A2(e, e) + |A|2 ≥ C(n)H2 +(C(n) > 0). +Proof of the claim. Let {e1, · · · , en−2, e} form an orthonormal frame of Σ. Then +HA(e, e) − A2(e, e) + |A|2 ≥ A(e, e)2 + A(e, e) +n−2 +� +i=1 +A(ei, ei) + +n−2 +� +i=1 +A(ei, ei)2. +Using Young’s inequality: +A(e, e)A(ei, ei) ≥ −4 +5A(ei, ei)2 − 5 +16A(e, e)2. +Hence +HA(e, e) − A2(e, e) + |A|2 ≥ min +�1 +5, 1 − 5 +16(n − 2) +�� +A(e, e)2 + +� +i +A(ei, ei)2� +≥ C(n)H2. +(Note: in dimension 6 we can only obtain HA(e, e) − A2(e, e) + |A|2 ≥ 0.) +□ +Therefore, we have +0 ≤ +� +Σ +� +|∇Σϕ|2 + R0ϕ2 − λ +2ϕ2 − +� +C(n)h2 + λ +2 − |∇Mh| +� +ϕ2� +, +∀ϕ. +(3.3) +Lemma 3.1. Suppose dim(M) ≤ 5, and M satisfies C2 ≥ λ > 0. If the prescribed mean +curvature function h satisfies +|∇Mh| ≤ C(n)h2 + λ +2, +(3.4) +then any connected stable µ-bubble satisfies λ1(−∆Σ + R0) ≥ +1 +2λ, hence has bounded +diameter by Theorem 1.10. +□ +11 + +One meaningful choice for h satisfying (3.4) is the following: given a domain S ⊂ M +with smooth boundary, we set +h(x) = +� +λ/2C(n) cot +�� +C(n)λ/8 d(x) +� +, +(3.5) +where d ∈ C∞ is a smoothing of d(−, ∂S) such that |∇d| ≤ 2 and 1 +2d(x, ∂S) ≤ d(x) ≤ +2d(x, ∂S). We note that h is infinite at ∂S and away from S by distance π +� +32/C(n)λ = +C′(n)λ−1/2. The infinity here plays the role of barrier, so any µ-bubble must lie within +distance C′(n)λ−1/2 from S. This observation and Lemma 3.1 are essential for geometric +applications of µ-bubbles. +However, the energy (3.1) needs to be renormalized in order to be well-defined. Also, +in subsequent sections we will need the weighted version of µ-bubbles. +Definition 3.2 (weighted µ-bubble). Let Ω+, Ω− be two disjoint domains in Σ. Suppose +u > 0 is a smooth function. Let h be a smooth prescribed mean curvature function on +Σ \ (Ω− ∪ Ω+), with h|∂Ω+ = ∞, h|∂Ω− = −∞. Let Ω0 be a fixed domain that contains Ω+ +and is disjoint from Ω−. Consider the following functional acting on all open sets Ω with +Ω∆Ω0 ⊂⊂ Σ \ (Ω− ∪ Ω+): +Eu(Ω) := +� +∂Ω +u dl − +� +M +(χΩ − χΩ0)hu dA. +(3.6) +The boundary of a critical point of Eu is called a µ-bubble with weight u. +It is shown in [3] [17] that a global minimizer of (3.6) always exists. Since h is infinite +on ∂Ω±, a µ-bubble always lie between Ω+ and Ω−. By geometric measure theory, a +µ-bubble is always a C2,α hypersurface. The first variation of (3.6) gives +H = h − u−1uN, +(3.7) +and the second variation of (3.6) at a critical point gives +δ2Eu(ϕ) = +� +Σ +� +u|∇Σϕ|2 − +� +|A| + Ric(N, N) + hN +� +uϕ2 ++ (∆u − ∆Σu)ϕ2 − huNϕ2� +. +(3.8) +Proof of (3.8). The first variation of (3.6) is computed to be +δEu(ϕ) = +� +Σ +� +Hu + uN − hu +� +ϕ +Since Σ is a critical point of Eu, the second variation contains the following terms: +δ2Eu(ϕ) = +� +Σ +�dH +dt u + HuNϕ + duN +dt − hNuϕ − huNϕ +� +ϕ +(3.9) +Finally, it is an elementary fact that +dH +dt = −∆ϕ − +� +|A|2 + Ric(N, N) +� +ϕ, +duN +dt += ∇2u(N, N)ϕ − ⟨∇Σu , ∇Σϕ⟩. +Combining these with (3.9) and using integration by part, we obtain (3.8). +□ +12 + +Notes. µ-bubbles were first considered by Gromov [5] [7] to prove metric inequalities +on scalar curvature. Recently, this technique has found many applications in problems +involving scalar curvature. +J. Zhu [17] utilized µ-bubbles to obtain several geometric +inequalities. +Lesourd-Unger-Yau [10] applied the µ-bubble technique to prove a posi- +tive mass theorem with arbitrary ends. Chodosh-Li [3] and Gromov [6] recently proved +that aspherical 4- and 5-manifolds do not admit metrics with positive scalar curvature. +Definition 3.2 closely follows the ones in [3] [17]. +3.2 +Results on diameter bounds +Proof of Lemma 1.9. Let Σ ⊂ M as in the statement of lemma. The stability inequality +gives +0 ≤ +� +Σ +� +|∇ϕ|2 − +� +|A|2 + RicM(N, N) +� +ϕ2� +, +∀ϕ ∈ C∞(Σ), +(3.10) +where we use N to denote the unit normal vector. +We make a measurable choice of +unit vector field e on Σ, such that RicΣ(e, e) = R0 at every point. By Gauss equation +RicΣ(e, e) = RicM(e, e) − RM(N, e, e, N) − A2(e, e), from (3.10) we obtain +0 ≤ +� +Σ +� +|∇ϕ|2 + R0ϕ2 − +� +RicΣ(e, e) + |A|2 + RicM(N, N) +� +ϕ2� += +� +Σ +� +|∇ϕ|2 + R0ϕ2 − C2,M(e, N)ϕ2 − +� +|A|2 − A2(e, e) +� +ϕ2� +≤ +� +Σ +� +|∇ϕ|2 + R0ϕ2 − λϕ2� +, +∀ϕ ∈ C∞(Σ). +□ +Proof of Theorem 1.10. +Part 1: proof of the diameter bound. +Given the eigenvalue condition in the statement of the theorem, we let u satisfy +∆u ≤ βR0u − λu, +Consider the weight function v = u1/β, which satisfies the inequality +∆v ≤ R0v − λβ−1v + (1 − β)v−1|∇v|2. +Let h be a prescribed mean curvature function to be determined. Let S ⊂ Σ be a +stable µ-bubble for the energy functional (3.6) with weight v. Then S satisfies the mean +curvature condition (we denote RicNN = RicΣ(N, N) and vN = ∂v +∂N for short) +H = h − v−1vN, +(3.11) +as well as the stability inequality +0 ≤ +� +S +� +v|∇Sϕ|2 − (|A|2 + RicΣ(N, N) + hN)vϕ2 + (∆Σv − ∆Sv)ϕ2 − hvNϕ2� +, +∀ϕ. +13 + +Choosing ϕ = v−1 in the stability inequality, we obtain +0 ≤ +� +S +� +− v−3|∇Qv|2 − +�(h − v−1vN)2 +n − 1 ++ R0 − |∇Σh| +� +v−1 ++ R0v−1 − λβ−1v−1 + (1 − β)v−3� +|∇Qv|2 + v2 +N +� +− hv−2vN +� += +� +S +� +− βv−3|∇Qv|2 − +� +1 +n − 1h2 + λβ−1 − |∇Σh| +� +v−1 ++ (1 − β − +1 +n − 1)v−3v2 +N − n − 3 +n − 1hv−2vN +� +(3.12) +We need β ≥ 1 − +1 +n−1, so that the v−3v2 +N term is non-positive. For n = 3 this is the only +constraint for β. When n ̸= 3, we apply Young’s inequality to the last term: +− n − 3 +n − 1hv−2vN ≤ ( +1 +n − 1 + β − 1)v−3v2 +N + 1 +4 +�n − 3 +n − 1)2 +h2v−1 +1 +n−1 + β − 1. +(3.13) +This introduce a positive coefficient in the h2v−1 term. We need the coefficient of h2v−1 +to be strictly negative, hence +1 +4 +�n − 3 +n − 1)2 +1 +1 +n−1 + β − 1 − +1 +n − 1 < 0 ⇔ β > 1 − +1 +n − 1 + (n − 3)2 +4(n − 1) = n − 1 +4 +. +Thus we obtain condition (1.4). One reaches a contradiction in (3.12) if h satisfies +|∇Σh| < +� +1 +n − 1 − 1 +4 +�n − 3 +n − 1)2 +1 +1 +n−1 + β − 1 +� +h2 + λβ−1 =: C2h2 + λβ−1, +C2 > 0. +(3.14) +Let p ∈ M, and Ω+ be a small geodesic ball near p. Set +h(x) = +� +λβ−1/C2 cot +�� +C2λβ−1/4 d(x) +� +, +where d(x) is a smoothing of d(x, ∂B) such that |∇d| ≤ 2, 1 +2d(x, ∂B) ≤ d(x) ≤ 2d(x, ∂B). +Then h satisfies (3.14). +If diam(Σ) > 4π/ +� +C2λβ−1, then Ω− = {q ∈ Σ : d(q, p) > +2π/ +� +C2λβ−1} is non-empty for some choice of p. +Then the µ-bubble problem (3.6) +is well-defined, and we obtain a contradiction from the above argument. +This shows +diam(Σ) ≤ 4π/ +� +C2λβ−1. +Remark 3.3. We can show that Sn−1×S1 does not admit metrics with λ1(−∆+βR0) > 0, +when β ≥ 1 − +1 +n−1. Otherwise, we find an minimizer of Σ �→ +� +Σ v in the homology class +of Sn−1, and obtain contradiction from (3.12) with h = 0. +Part 2: construction of counterexamples. +Suppose β does not satisfy (1.4). We first assume n = dim(Σ) ̸= 3. Consider the data +g = dr2 + ǫ2f(r)2g (r ∈ R), +v = v(r), +h = h(r), +where g is the standard metric on Sn−1. It is not hard to compute +Ric(∂r, ∂r) = −(n − 1)f ′′ +f , +Ric(e, e) = (n − 2)ǫ−2f −2 − (n − 2)f ′2 + ff ′′ +f 2 +, +(3.15) +14 + +where e is any unit vector tangent to Sn−1. Based on similar analysis as in subsection +2.1, we let f, v, h solve the following equations: first, the equality in (3.14), i.e. +h′ = −C2h2 − λβ−1. +(3.16) +We assume here that β > 1− +1 +n−1, and the case β ≤ 1− +1 +n−1 is treated later. Thus we have +C2 ≤ 0, and the solution to (3.16) has the form of hyperbolic tangent functions. Second, +we let v satisfy the equality case in (3.13), namely +v′ = −1 +2 +n − 3 +n − 1 +hv +1 +n−1 + β − 1 =: −C3hv +(C3 ̸= 0). +(3.17) +Finally, we let every constant r slice to be a µ-bubble, namely +(n − 1)f ′ +f = h − v′ +v . +(3.18) +It is elementary to verify that the solution to (3.16) (3.17) (3.18) satisfies +v′′ + (n − 1)f ′ +f v′ + (n − 1)f ′′ +f v + λβ−1v − (1 − β)v−1(v′)2 = 0, +which implies that u = vβ solves the desired equation: +∆u = β Ric(∂r, ∂r)u − λu. +(3.19) +To complete the construction, we need to show that Ric(∂r, ∂r) is the minimal Ricci +curvature at each point. +When n = 2 this is clear. +Now we assume n ≥ 4. +When +C2 = 0 ⇔ β = 1 +4(n − 1), the exact solution to the above system is + + + + + + + + + + + +h(r) = −λβ−1r, +u(r) = exp +�1 +2C3λr2� +, +f(r) = exp +� +− 1 + C3 +2(n − 1)λβ−1r2� +. +(3.20) +In light of equation (3.15), we want f to be exponentially decaying at r → ∞, so that +ǫ−2f −2 dominates the other terms. In this way we obtain Ric(e, e) ≫ Ric(∂r, ∂r) for small +ǫ, and the desired result follows. Observe that β > 1 − +1 +n−1 ⇒ C3 > 0, hence f indeed +decays. For C2 < 0 ⇔ β ∈ (1 − +1 +n−1, 1 +4(n − 1)), the exact solution is + + + + + + + + + + + + + + + + + +h(r) = +� +λβ−1 +−C2 +coth +�� +−C2λβ−1r +� +, +u(r) = cosh +�� +−C2λβ−1r +�−C3β/C2 +, +f(r) = cosh +�� +−C2λβ−1r +�(1+C3)/C2(n−1) +. +(3.21) +Again, we have C3 > 0 so f decays at infinity. +To resolve the case β ∈ (0, 1 − +1 +n−1], we note the following fact: when β > β′ > 0, we +have λ1(−∆ + βR0) ≥ λ ⇒ λ1(−∆ + β′R0) ≥ λβ′/β. Indeed, +∆u ≤ βR0u − λu ⇒ ∆uβ′/β ≤ β′R0uβ′/β − λβ′ +β uβ′/β, +∀u > 0. +(3.22) +15 + +Therefore, formula (3.20) with β = 1 +4(n − 1) is also a counterexample for β ≤ 1 − +1 +n−1. +This settles the case n ̸= 3. +When n = 3 and β < +1 +2, equations (3.13) (3.14) are not used. Looking at (3.12) +directly, we need to solve (3.18) along with the following equation: +(1 +2 − β)v−3v2 +N = (1 +2h2 + λβ−1 − |h′|)v−1. +(3.23) +A possible choice is + + + + + + + + + + + + + +h(r) = −λβ−1r, +u(r) = exp +� +λr2 +2√1 − 2β +� +, +f(r) = exp +� +− 1 +4( +1 +√1 − 2β + 1)λβ−1r2� +, +so u, f satisfies (3.19). There are many choices for (3.23). For this choice f decays expo- +nentially at infinity, so the minimum of Ricci curvature is attained at Ric(∂r, ∂r). +□ +3.3 +Results on Macroscopic Dimension +Proof of Theorem 1.11. +The proof is an application of Chodosh and Li’s slice-and-dice argument [3]. +Let Ω0 ⊂ M be a small geodesic ball. Consider the minimizer of the µ-bubble problem +(3.6) with unit weight u = 1 and mean curvature function h given by (3.5) with S = +Ω0. We obtain a stable µ-bubble Ω1 ⊃ Ω0. Replacing Ω1 by its connected component +containing Ω0, we may assume Ω1 is connected. Then consider the µ-bubble problem now +with the choice of domain S = N(Ω1, 1), the distance 1 collar neighborhood of Ω1. We +obtain another stable µ-bubble Ω2 ⊃ Ω1. Repeating this process, we obtain an exhausting +chain of domains Ω0 ⊂ Ω1 ⊂ Ω2 ⊂ · · ·, with each Ωi connected. Denote Σi = ∂Ωi. +Let Σij be a conneted component of Σi. By simply-connectedness of M, Σij is sepa- +rating, i.e. M \ Σij has two connected components. One component of M \ Σij contains +Ωi while the other does not. Let M \(� +i Σi) = � +ij Uij be a decomposition into connected +components, with Uij ⊂ Ωi+1 \ Ωi. Hence ∂Uij contains exactly one connected component +Σij of Σi. The connection pattern of Uij thus forms a tree. +By construction we have Uij ⊂ N(Σ, 1 + C(n)λ−1/2). By the separating property of +Σij, we actually have Uij ⊂ N(Σij, 1 + C(n)λ−1/2). By Lemma 3.1 we have diam(Σij) ≤ +C(n)λ−1/2. Hence diam(Uij) is uniformly bounded. We thicken the set Uij by taking +union with a small distance neighborhood of Q: let �Uij = Uij ∪ N(Σij, ǫij). Thus {�Uij} +form an open cover. When ǫij ≪ 1 and ǫij ≪ mink d(Σij, Σik), there is no overlap among +three sets. Let X be the nerve of {�Uij}, thus X is a 1-dimensional complex. +Finally, let {fij} be a partition of unity subordinate to {�Uij}. By linear interpolation +we obtain a continuous map Φ : M → X. For each point p, the preimage Φ−1(p) is +contained in a single open set in {�Uij}, hence has uniformly bounded diameter. +□ +Proof of Theorem 1.12. +16 + +Choose β = 1, n ≥ 4 in the counterexamples constructed above. For n = 4 we choose +(3.20), and for n ≥ 5 we choose (3.21). Note that u(r) ≥ 1 under these choices. Consider +M = R2 × Sn−1 with the metric +g = u(r)2dt2 + dr2 + ǫ2f(r)2g, +where g denotes the round spherical metric. Under our choice g has infinite Urysohn 1- +width: for any point p ∈ Sn−1, the map (R2, gEuc) → R2 × {p} ⊂ (M, g) is expanding. If +g has finite Urysohn 1-width, then so does R2. But R2 is known to have infinite Urysohn +1-width. +It remains to confirm that g has uniformly positive bi-Ricci curvature. We compute +the curvature components of g: +sec(u−1∂t, ∂r) = −u′′ +u , +sec(∂r, u−1∂t) = −u′′ +u , +sec(e, e′) = ǫ−2f −2 − (f ′)2 +f 2 , +sec(∂r, e) = −f ′′ +f , +sec(u−1∂t, e) = −f ′u′ +fu . +(3.24) +where e ⊥ e′ are any two unit vectors tangent to Sn−1. All the cross curvature terms +Rpqrs (q ̸= r) vanish. As expected from the construction, we have +C2(∂r, u−1∂t) = −u′′ +u − (n − 1)f ′′ +f − (n − 1)u′f ′ +uf ≡ λ. +To control C2(X1, X2) where X1, X2 are not tangent to ∂r, ∂t, we repeat the argument in +(2.10) and above. The estimates of curvature components are replaced by (3.24). Note +that ǫ−2f −2 has at least exponential growth at r → ∞, while all the other terms in (3.24) +have at most polynomial growth. Following the argument from (2.9) to (2.10), one shows +that C2 ≥ λ when ǫ is sufficiently small. +□ +4 +Rigidity in Dimension 6 +Proof of Theorem 1.6. +In the proof of Theorem 1.3 in [4], all the statements only need n(m − 2) < m2 − 2, +except Proposition 3.1 where one further needs n(m − 1) < m(m + 1). +The former +inequality only requires n ≤ 6, while the latter requires n ≤ 5. The inequality n(m−1) < +m(m + 1) comes from the usage of µ-bubbles in the proof of equation (3.8) in [4] (see +(3.11) and below). As we already noticed in previous sections, µ-bubbles imposes more +strict constraints on the dimension. +Here we prove equation (3.8) without using µ-bubbles, thus extend the result to n = 6. +The remaining parts of the proof in [4] are left unchanged. The idea here is inspired by +the scalar curvature rigidity theorem of Bray-Brendle-Neves [1] (see also [12, Section 2.4] +and references therein). +For consistency, in this section we adopt the same notations as in [4]. We briefly +summarize the setups for equation (3.8) in [4]. Let M = T m × Mn−m be a manifold with +Cm ≥ 0. We inductively construct a chain of hypersurfaces M = Σ0 ⊃ Σ1 ⊃ · · · ⊃ Σm +and positive weight functions ρi ∈ C∞(Σ0) that satisfies the conditions in [4, Definition +17 + +2.1]. More precisely, the starting weight is ρ0 = 1 and Σ1 ⊂ M is a minimal hypersurface, +and each Σk minimizes the weighted area functional +Hn−k +ρk−1(Σ) = +� +Σ +ρk−1 +in a certain homology class in Σk−1. Therefore, the generalized mean curvature +Hρk−1(Σk) := HΣk + ⟨∇Σk−1 log ρk−1 , νk⟩ +is zero on each Σk, where νk is the normal vector of Σk ⊂ Σk−1. By analyzing the stability +inequalities, a list of rigidity statements are obtained on Σk, see [4, Proposition 2.1]. Next, +one constructs a foliation of hypersurfaces {Σm,t ⊂ Σm−1}0≤t≤ǫ starting from Σm,0 = Σm, +such that the generalized mean curvature Hρm−1(Σm,t) is constant on each Σm,t. Let ϕtν +be the variational vector field, and denote by Hρ(t) the generalized mean curvature. We +have ϕ0 = 1, hence ϕt > 0 for sufficiently small t. Equation (3.8) states that +Hρ(t) ≤ 0 +(4.1) +for all t, thus the weighted volume of Σm,t is non-increasing. +By the minimality of +Σm,0 = Σm, all the slices Σm,t are minimizing, thus the rigidity results [4, Proposition +2.1] propagate along the foliation. From this we finally obtain the splitting result. +The new proof of (4.1) is by directly computing dHρ(t)/dt. By definition we have +dHρ(t) +dt += −∆Σmϕt − +� +|hΣm|2 + RicΣm−1(νm, νm) +� +ϕt ++ ϕt∇2 +Σm−1 log ρm−1(νm, νm) − ⟨∇Σmϕt , ∇Σm log ρm−1⟩ +(4.2) +where we write Σm = Σm,t for short. Multiplying both sides by ϕ−1 +t +and integrating over +Σm,t, we obtain +dHρ(t) +dt +� +Σm,t +ϕ−1 +t += +� +Σm,t +� +− ϕ−1 +t ∆Σmϕt − |hΣm|2 − RicΣm−1(νm, νm) ++ ∆Σm−1 log ρm−1 − ∆Σm log ρm−1 − HΣm⟨∇Σm−1 log ρm−1 , νm⟩ +− ⟨∇Σm log ϕt , ∇Σm log ρm−1⟩ +� +Let �ρm = ϕtρm−1, from integration by part we obtain +� +Σm,t +� +− ϕ−1 +t ∆Σmϕt − ⟨∇Σm log ϕt , ∇Σm log ρm−1⟩ +� += − +� +Σm,t +⟨∇Σm log ϕt , ∇Σm log �ρm⟩ +Therefore, +dHρ(t) +dt +� +Σm,t +ϕ−1 +t += +� +Σm,t +� +− |hΣm|2 − RicΣm−1(νm, νm) − ⟨∇Σm log ϕt , ∇Σm log �ρm⟩ ++ ∆Σm−1 log ρm−1 − HΣm,t⟨∇Σm−1 log ρm−1 , νm⟩ +� +We want to show that the right hand side is non-positive. This is shown by following the +main argument in [2]. However, since the bottom slice Σm,t has nonzero generalized mean +18 + +curvature, we have to re-estimate some of the terms. Applying the slicing identities [2, +Lemma 3.4] for 1 ≤ k ≤ m − 1 (which does not involve Σm,t), we obtain +dHρ(t) +dt +� +Σm,t +ϕ−1 +t +≤ +� +Σm,t +� +− |hΣm|2 − RicΣm−1(νm, νm) − ⟨∇Σm log ϕt , ∇Σm log �ρm⟩ +− HΣm,t⟨∇Σm−1 log ρm−1 , νm⟩ + +m−1 +� +k=2 +H2 +Σk − +m−1 +� +k=1 +λk +− +m−1 +� +k=1 +� +|hΣk|2 + RicΣk−1(νk, νk) + ⟨∇Σk log ρk , ∇Σk log wk⟩ +�� += +� +Σm,t +� +− (Λ + G + �E + R) − |hΣm|2 − ⟨∇Σm log ϕt , ∇Σm log �ρm⟩ +− HΣm,t⟨∇Σm−1 log ρm−1 , νm⟩ +� +where the terms Λ, G, R are the same as in [2, Lemma 3.4], and �E := �m−1 +k=1 |hΣk|2 − +�m−1 +k=2 H2 +Σk differs from E by removing the Σm terms. Lemma 3.7 in [2] used the condition +that Σm has zero generalized mean curvature. Here the inequality is adjusted as +G ≥ +m−1 +� +k=2 +�1 +2 + +1 +2(k − 1) +� +H2 +Σk + +m +2(m − 1) +� +|∇Σm log ρm−1|2 + ⟨∇Σm−1 log ρm−1 , νm⟩2� +. +The identity for R [2, Lemma 3.8], the grouping identity [2, Lemma 3.10], as well as the +curvature inequalities for top and intermediate slices [2, Lemma 3.11, 3.12], are unaffected. +What remains are the terms on the bottom slice Σm, which are collected as follows: +� +Σm,t +� +− |hΣm|2 − ⟨∇Σm log ϕt , ∇Σm log �ρm⟩ − HΣm⟨∇Σm−1 log ρm−1 , νm⟩ +− +m +2(m − 1) +� +|∇Σm log ρm−1|2 + ⟨∇Σm−1 log ρm−1 , νm⟩2�� +≤ +� +Σm,t +� +− H2 +Σm +n − m − HΣm⟨∇Σm−1 log ρm−1 , νm⟩ − +m +2(m − 1)⟨∇Σm−1 log ρm−1 , νm⟩2 +− +m +2(m − 1)|∇Σm log ρm−1|2 − ⟨∇Σm log ϕt , ∇Σm(log ρm−1 + log ϕt)⟩ +� +. +For this to be non-positive, we need +2m +(n − m)(m − 1) ≥ 1 ⇔ n(m − 1) ≤ m2 + m, +(4.3) +which is satisfied for n = 6, m ≤ n − 1. Hence dHρ(t)/dt ≤ 0. +□ +References +[1] H. Bray, S. Brendle, A. Neves, Rigidity of area-minimizing two spheres in three- +manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830. +[2] S. +Brendle, +S. +Hirsch, +F. +Johne, +A generalization of Geroch’s conjecture, +https://arxiv.org/abs/2207.08617 (2022). +19 + +[3] O. Chodosh, C. Li, Generalized soap bubbles and the topology of manifolds with pos- +itive scalar curvature, https://arxiv.org/abs/2008.11888 (2020). +[4] J. Chu, K. Kwong, M. Lee, Rigidity on non-negative intermediate curvature, +https://arxiv.org/abs/2208.12240 (2022). +[5] M. Gromov, Metric Inequalities with Scalar Curvature, Geom. Funct. Anal. 28 (2018), +no.3, 645–726. +[6] M. Gromov, No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds, +https://arxiv.org/abs/2009.05332 (2020). +[7] M. +Gromov, +Four +lectures +on +scalar +curvature, +https://arxiv.org/abs/1908.10612 (2020). +[8] M. Gromov, H. B. Lawson, Jr., Positive scalar curvature and the Dirac operator on +complete Riemannian manifolds, Inst. Hautes ´Etudes Sci. Publ. Math. No. 58 (1983), +83–196. +[9] M. Katz, The First Diameter of 3-Manifolds of Positive Scalar Curvature, Proc. +Amer. Math. Soc. 104 (1988), no. 2, 591–595. +[10] M. Lesourd, R. Unger, S.-T. Yau, The Positive Mass Theorem with Arbitrary Ends, +https://arxiv.org/abs/2103.02744 (2021). +[11] Y. Liokumovich, D. Maximo, Waist inequality for 3-manifolds with positive scalar +curvature, https://arxiv.org/abs/2012.12478 (2021). +[12] D. Lee, Geometric Relativity, Graduate Studies in Mathematics, 201. American +Mathematical Society, Providence, RI, 2019. xii+361 pp. +[13] R. Schoen, S.-T. Yau, On the structure of manifolds with positive scalar curvature, +Manuscripta Math. 28 (1979), no. 1-3, 159–183. +[14] R. Schoen, S.-T. Yau, Positive Scalar Curvature and Minimal Hypersurface Singu- +larities, https://arxiv.org/abs/1704.05490 (2017). +[15] Y. Shen, R. Ye, On stable minimal surfaces in manifolds of positive bi-Ricci curva- +tures, Duke Math J. 85 (1996), no.1, 109-116. +[16] Y. Shen, R. Ye, On the geometry and topology of manifolds of positive bi-Ricci cur- +vature, https://arxiv.org/abs/dg-ga/9708014 (1997). +[17] J. Zhu, Width estimate and doubly warped product, Trans. Amer. Math. Soc. 374 +(2021), 1497-1511. +Kai Xu, +Department of Mathematics, Duke University, Durham, NC 27708, USA, +Email address: kx35@math.duke.edu. +20 + diff --git a/YtE0T4oBgHgl3EQf3wK0/content/tmp_files/load_file.txt b/YtE0T4oBgHgl3EQf3wK0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..658edc81b926ff0eaaca85c7dfb8dd11066c4b95 --- /dev/null +++ b/YtE0T4oBgHgl3EQf3wK0/content/tmp_files/load_file.txt @@ -0,0 +1,738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf,len=737 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='02730v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='DG] 6 Jan 2023 Dimension Constraints in Some Problems Involving Intermediate Curvature Kai Xu Abstract In [2] Brendle-Hirsch-Johne proved that T m × Sn−m does not admit metrics with positive m-intermediate curvature when n ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Chu-Kwong-Lee showed in [4] a corresponding rigidity statement when n ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In this paper, we show optimality of the dimension constraints by giving concrete counterexamples in n ≥ 7 and extending the rigidity result to n = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension ≤ 5 and bi- Ricci curvature ≥ 1 have finite Urysohn 1-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Counterexamples are constructed in dimension ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 1 Introduction The intermediate curvature 1 of a Riemannian manifold is a curvature quantity that lies between Ricci and scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For {ei}n i=1 an orthonormal frame at a point on a manifold M, we organize the sectional curvatures sec(ei, ej) in an n × n matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Thus Ricci curvature is the sum of entries in the first row, while scalar curvature is twice the sum of entries in the upper triangular part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We define the m-intermediate curvature as the sum of the first m rows of the upper triangular part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let M be an n-manifold, and 1 ≤ m ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Suppose {e1, e2, · · · , en} is an orthonormal frame at p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The m-intermediate curvature of M, denoted by Cm, is defined by Cm(e1, · · · , em) := m � p=1 n � q=p+1 R(ep, eq, eq, ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We say that M has m-intermediate curvature lower bound λ (abbreviated as Cm ≥ λ), if Cm(e1, · · · , em) ≥ λ for any orthonormal frame {ei} at any p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We say that M has positive m-intermediate curvature ( Cm > 0), if Cm(e1, · · · , em) > 0 for any orthonormal frame {ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' When m = 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' m = 3), the m-intermediate curvature is also called bi-Ricci (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' tri-Ricci) curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We note that Cm(e1, · · · , em) only depends on the subspace which {e1, · · · , em} spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' To see this, we orthogonally split TpM = V ⊕ V ⊥ where V = span{e1, · · · , em}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The metric tensor also splits as g = g1 ⊕ g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We have Cm(e1, · · · , em) = ⟨Rijkl , 1 2(g1)il(g1)jk + (g1)il(g2)jk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' More properties of intermediate curvature can be found in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' A model space for intermediate curvatures is the product of flat torus and round sphere M = T m × Sn−m: M has positive Cm+1 but only non-negative Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' It is natural to ask 1The meaning of the phrase “intermediate curvature” may differ in other topics of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 1 a topological obstruction problem: whether T m × Sn−m admit metrics with positive Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This question extends two well-known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For the case m = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Ric > 0), Bonnet- Myers’ theorem or Bochner’s formula implies negative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The case m = n − 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' M = T n with positive scalar curvature) is Geroch’s conjecture, and the answer is also negative as shown by Schoen-Yau [13] [14] and Gromov-Lawson [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The intermediate cases are recently proved by Brendle-Hirsch-Johne [2] up to dimension 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2 ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For n ≤ 7, there is no metric on T m × Mn−m with Cm > 0, where Mn−m is any (n − m)-dimensional closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Chu-Kwong-Lee [4] later proved a corresponding rigidity statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let n ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' If a metric g on T m × Mn−m satisfies Cm ≥ 0, then g splits isometrically as a product of a flat T m with a metric on Mn−m with nonnegative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This paper is focused on the dimension constraints that appear in the above theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' It turns out that counterexamples exist in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The following is one main theorem of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Assume m ≥ 2, n ≥ m+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let F = F(x1, · · · , xm) be a positive function on the torus T m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Consider the following metric on Sn−m × T m: g = ǫ2F 2h + F −2 n−m m−1 m � i=1 dx2 i , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1) where h is the standard metric on Sn−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Then we have Cm � ∂x1 |∂x1|, · · · , ∂xm |∂xm| � = (n − m) �(n − m)(m − 2) 2(m − 1) − 1 � F 2 n−m m−1 −2 m � i=1 �∂F ∂xi �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' If n(m − 2) ≥ m2 − 2 and ǫ is sufficiently small (depending on the C2 norm of log F), then g satisfies Cm ≥ 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' If further n(m − 2) > m2 − 2, then Cm > 0 at all non-critical points of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The relation n(m − 2) ≥ m2 − 2 obtained here is consistent with the algebraic in- equalities obtained in [2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='14] and [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4, we obtain metrics on T 4 × S4 and T 3 × S5 with almost positive intermediate curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' To resolve the issue occurred at the critical points of F, we can further perturb the metric so that Cm is positive everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This confirms the optimality of the condition n ≤ 7 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Assume all the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4, and n(m − 2) > m2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' If f is a Morse function on T m, then there exists a small perturbation of g that satisfies Cm > 0 strictly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3, we obtain non-trivial metrics on T 3 × S4 and T 4 × S3 with non- negative intermediate curvatures, giving counterexamples when n ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The case n = 6 is not included in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' However, the proof in [4] can be improved and thus extend to n = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3 holds for n = 6 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 2 Given the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1), it is an elementary work to verify all the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' There- fore, finding (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1) is the most important aspect of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' To better explain the idea, let us briefly summarize the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2 in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' If there exists a metric as stated in the theorem, we find a chain of hypersurfaces M ⊃ Σ1 ⊃ · · · ⊃ Σm, with Σ1 being a stable minimal hypersurface in M, and each Σi being a minimizer of a certain weighted area functional on Σi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Combining all the stability inequalities of Σi, one obtains an inequality of the form 0 ≤ � Σm � − Cm − V1 − V2 − · · · � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2) where V1, V2, · · · are algebraic expressions involving the second fundamental forms of Σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We would obtain a contradiction if Vi ≥ 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' However, this is true only when a certain dimensional inequality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The observation here is that: the validity of the algebraic inequalities Vi ≥ 0 suggests the existence of counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' If Vi has the chance to be negative, we try to reverse engineer a metric that captures the negative part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This leaves some room in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2) that allows positive Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Then we try to achieve equality case for all the other inequalities involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Therefore, we are forced to have Cm ≡ −(V1 + V2 + · · · ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This process finally yields (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' A more detailed explanation is contained in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The next problem concerns diameter bounds under the presence of uniformly positive intermediate curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' By Bonnet-Myers’ theorem, manifolds with uniformly positive Ricci curvature has bounded diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' When generalizing to bi-Ricci curvature, one needs to pass to a stable minimal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='7 (Shen-Ye [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let M be a (complete) manifold with dimension n ≤ 5 and bi-Ricci curvature C2 ≥ λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Then any two-sided stable minimal hypersurface in M has diameter ≤ C(n)λ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='7 can be interpreted intrinsically in terms of Σ (without mentioning the ambient manifold M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' On a Riemannian manifold M, we define the minimum Ricci curvature function R0(p) := min e∈TpM,|e|=1RicM(e, e), p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Equivalently, R0 is the minimal eigenvalue of Ricci curvature at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Suppose M satisfies bi-Ricci curvature lower bound C2 ≥ λ, and Σ ⊂ M is a stable minimal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Then λ1(−∆Σ + R0) ≥ λ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3) which means � Σ |∇ϕ|2 + R0ϕ2 ≥ λ � Σ ϕ2 for all ϕ ∈ C∞(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We may understand Σ as having “Ric ≥ λ” in a weaker sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We wish to obtain a Bonnet-Myers’ theorem from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The second main result of this paper is that such result holds only for dimension ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We can generalize condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3 by adding a coefficient in front of R0, and this helps us see clearly of the optimal form of constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let β > 0, n ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Suppose an n-dimensional complete manifold Σ satisfies λ1(−∆Σ + βR0) ≥ λ > 0, then we have diam(Σ) ≤ C(n, β)λ−1/2 if \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 β ≥ 1 2 (n = 3), β > 1 4(n − 1) (n ̸= 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4) There exist non-compact counterexamples when (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The condition n ≤ 7 is due to the smoothness issue of area-minimizing currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='7 is a consequence of the β = 1 case, hence the condition n ≤ 5 there is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' A proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='10 (without counterexamples) was given in Shen-Ye [16] using weighted minimal geodesics with weight function u1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3) is equivalent to the notion of conformal Ricci curvature lower bounds introduced in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Here we give a new proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='10 based on the µ-bubble technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' One bene- fit of this proof is that counterexamples can be easily reverse-engineered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' An introduction to µ-bubbles, including references, is included in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We observe that, the usage of µ-bubbles imposes a more strict dimension constraint than minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For example, we can show that Sn−1 × S1 (n ≤ 7) does not admit metrics with λ1(−∆+ βR0) > 0 when β ≥ 1 − 1 n−1 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The proof only uses (generalized) minimal surfaces, so the constraint for β is weakened compared to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The gap from 1 − 1 n−1 to 1 4(n − 1) indicates essential difference between Sn−1 × S1 and Sn−1 × R in regard of condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Finally, we relate intermediate curvature to the notion of macroscopic dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' A manifold M is said to have macroscopic dimension ≤ k, or to have finite Urysohn k- width, if there exists a simplicial complex X with dimension ≤ k, and a continuous map f : M → X, such that each fiber f −1(p) has uniformly bounded diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Clearly manifolds with Ric ≥ λ > 0 have finite Urysohn 0-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Gromov conjectured that an n-manifold with uniformly positive scalar curvature has finite Urysohn (n−2)-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The dimension 3 case has been answered affirmatively [8] [9] [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In higher dimensions this problem is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Interpolating between Ricci and scalar curvature, we would expect that manifolds with uniformly positive m-intermediate curvature has finite Urysohn (m − 1)-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Apply- ing Chodosh-Li’s slice-and-dice argument [3], we show affirmatively the case m = 2, in dimension up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let M be a complete simply-connected manifold with dimension n ≤ 5 and bi-Ricci curvature C2 ≥ λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Then M has finite Urysohn 1-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In dimension higher than 5, there exists counterexamples in the same spirit as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' When n > 5, there exists a complete metric on Sn−2×R2 with uniformly positive bi-Ricci curvature and infinite Urysohn 1-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In section 2 we discuss the case of non-negative Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This section contains a detailed explanation of the reverse engineering process, and 4 the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In section 3 we discuss the case of uniformly positive Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This section contains an introduction to µ-bubbles, and the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='10 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In Section 4, we prove the rigidity in dimension 6 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We use M to denote a manifold, and Σ denote a hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We use N to denote the unit normal vector of Σ, and A, H denotes the second fundamental form and mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The sign convention is A(X, Y ) = ⟨∇XN , Y ⟩, H = trΣ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' These notations apply to all sections except Section 4, where we adopt the same notation as in the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The author would like to thank Sven Hirsch for inspiring con- versations and comments on previous drafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' He also thanks Jianchun Chu for discussions on the work [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 2 The Case of Nonnegative Intermediate Curvature 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1 Reverse engineering of counterexamples In this section, we use the example T 3×S5 to demonstrate the reverse engineering process mentioned in the introduction part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' To better show the idea, we re-write the argument in [2] using more convenient notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' In what follows, we use subscript to denote the dimension in which an object lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For example, u7 denotes a function on M7, N7 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' A6 and H6) denotes the unit normal vector (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' second fundamental form and mean curvature) of M6 ⊂ M7, and ∇5 denote the gradient on M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Assume that M8 = T 3 × S5 has positive tri-Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Ignoring the possible singularity, suppose that we can find a smooth area minimizer M7 ⊂ M8 in the homology class of T 2 × S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' By the stability inequality for minimal surfaces, there exists u7 > 0 on M7 such that ∆7u7 ≤ � − |A7|2 − Ric8(N8, N8) � u7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1) Then, let M6 ⊂ M7 be a minimizer of the functional M �→ � M u7 in the homology class of S1 × S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The stability inequality implies that there exists u6 > 0 on M6 such that ∇6 · � u7∇6u6 � ≤ � − |A6|2u7 − Ric7(N7, N7) + ∆7u7 − ∆6u7 � u6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2) Finally, let M5 ⊂ M6 be a minimizer of M �→ � M u6u7 in the homology class of S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The stability inequality gives 0 ≤ � M5 u6u7|∇5ϕ|2 + � M5 � − |A5|2u6u7 − Ric6(N6, N6)u6u7 + ∆6(u6u7) − ∆5(u6u7) � ϕ2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3) for any ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We let ϕ = (u6u7)−1, and then combine (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Note that ∆6(u6u7) = ∇6 · (u7∇6u6) + u6∆6u7 + ∇5u6 · ∇5u7 + ∂u6 ∂N6 ∂u7 ∂N6 , and the Gauss equations Ric7(N7, N7) = Ric8(N7, N7) − R8(N7, N8, N8, N7) − A2 7(N7, N7), 5 Ric6(N6, N6) = Ric8(N6, N6) − R8(N6, N7, N7, N6) − R8(N6, N8, N8, N6) + H6A6(N6, N6) − A2 6(N6, N6) − A2 7(N6, N6) − A7(N6, N6)A7(N7, N7) + A7(N6, N7)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' After rearranging the terms, we obtain 0 ≤ − � M5 C3(N6, N7, N8)ϕ − � M5 � (u6u7)−3|∇5(u6u7)|2 − (∇5u6 · ∇5u7)(u6u7)−2� − � M5 � |A5|2 − u−1 6 ∂u6 ∂N6 u−1 7 ∂u7 ∂N6 � ϕ − � M5 � |A6|2 + H6A6(N6, N6) − A2 6(N6, N6) � ϕ − � M5 � |A7|2 − A2 7(N6, N6) − A7(N6, N6)A7(N7, N7) + A7(N6, N7)2� ϕ =: − � M5 C3(N6, N7, N8)ϕ − (I + V3 + V2 + V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4) This inequality is equivalent to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='10 in [2], and there the terms V1, V2, V3 are shown to be non-negative in lower dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' However, it is not true that V1, V2, V3 ≥ 0 in the current dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For example, for the term V3 we have V3 ≥ H2 5 5 − ∂ log u6 ∂N6 ∂ log u7 ∂N6 = 1 5 �∂ log u6 ∂N6 + ∂ log u7 ∂N6 �2 − ∂ log u6 ∂N6 ∂ log u7 ∂N6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' ≥ 0, which is clearly not true by letting u6 = u7 and A5 be a multiple of g|M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Similarly, V2, V1 can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For example, by letting A6, A7 have diagonal form with A6(e1, e1) = · · · = A6(e5, e5) = x, A6(N6, N6) = −5 2x, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='5) and A7(e1, e1) = · · · = A7(e5, e5) = y, A7(N6, N6) = A7(N7, N7) = −5 2y, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='6) we obtain V2 < 0, V1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Among the many available choices, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='5) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='6) are chosen so that the resulting metric has the simplest form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Next, we collect and combine together the information obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Consider a metric on T 3 × S5 of the following form: g = f 2 1h + f 2 2dx2 + f 2 3dy2 + f 2 4 dz2, where f1, f2, f3, f4 are functions in x, y, z, and h is the standard metric on S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The first condition we impose on g is that, any coordinate slice M5 = {x = x0, y = y0, z = z0} ⊂ M6 = {y = y0, z = z0} ⊂ M7 = {z = z0} is a slicing of weighted minimal surfaces introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' This gives many hints on the choice of fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' First, the mean curvature of M7 is identically zero along the flow ∂/∂z = f4N8 by our condition, hence ∆7f4 ≡ −(|A2 7| + Ric8(N8, N8))f4 6 by the variation formula of mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Compared with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1), we have a natural choice f4 = u7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We also know that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1) should be equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Second, the area � f 5 1f2f3 dx dy must be constant in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Hence we let f 5 1f2f3 be pointwise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Followed by the same analysis on M6, we obtain f2 = u6, and f 5 1f2u7 is constant (which implies f3 = f4 = u7), and that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' By spherical symmetry, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3) is equality for constant ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Therefore, all the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='3) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4) achieve equality, this forces C3(N6, N7, N8) = −(V1 + V2 + V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Next, we interpret conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='5) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='6) in terms of f1, f2, f3, f4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Note that A6(ei, ei) = f −1 3 ∂ log f1 ∂y , A6(N6, N6) = f −1 3 ∂ log f2 ∂y (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='5) −−−→ f2 = f −5/2 1 , and A7(ei, ei) = f −1 4 ∂ log f1 ∂z , A7(N7, N7) = f −1 4 ∂ log f3 ∂z (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='6) −−−→ f3 = f −5/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Under these choices we have V1 < 0, V2 < 0, V3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We finally obtain f1 = F(x, y, z), f2 = f3 = f4 = F(x, y, z)−5/2, which is exactly (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' One could repeat this argument on the general cases T m × Sn−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' A much faster way is to directly consider g = F 2h + F −s � dx2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The choice s = 2 n−m m−1 is determined by cancelling all the second derivatives of F in Cm(∂x1, · · · , ∂xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='2 Counterexamples in dimension ≥ 7 Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' For convenience, for the rest of this section we let k = n − m, so that the underlying manifold is T m × Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Greek indices α, β, · · · are assumed to be in the Sk direction, while Latin indices i, j, · · · are in the T m directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The indices p, q, · · · can be in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Denote by {eα}k α=1 an orthonormal frame for g tangent to Sk, and denote ei = ∂xi |∂xi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Denote the partial derivatives Fi = ∂F ∂xi, and dF 2 = � i F 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Let s = 2k m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Therefore, the metric under consideration is g = ǫ2F 2hαβ + F −s � i dx2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='7) Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' We first compute Cm( ∂x1 |∂x1|, · · · , ∂xm |∂xm|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' The Christoffel symbols of g are: Γi αβ = −ǫ2F s+1Fihαβ, Γβ αi = F −1Fiδβ α, Γα ij = Γj iα = 0, Γi ij = −s 2F −1Fj, (when i ̸= j) Γj ii = s 2F −1Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' 7 From this we compute the curvature components (Einstein summation is not used): gjjRi ijj = s 2F s−1(Fii + Fjj) − s 2F s−2(F 2 i + F 2 j ) − s2 4 F s−2 � k̸=i,j F 2 k (i ̸= j), gααRi iαα = −F s−1Fii + s 2F s−2(−F 2 i + � j̸=i F 2 j ), gααRj iαα = −F s−1Fij − sF s−2FiFj (i ̸= j), Rα ijk = Ri αβγ = Rβ ijα = 0, Rη αβγ = � ǫ−2F −2 − F s−2 � i F 2 i �� δη αgβγ − δη βgαγ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Moreover, Rl ijk is always a polynomial combination of ∂2F F and ∂F F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE0T4oBgHgl3EQf3wK0/content/2301.02730v1.pdf'} +page_content=' Finally we compute Cm � ∂x1 |∂x1|, · · · , ∂xm |∂xm| � = � i 0. Then since the P n +0 (cosh ρ) term appears in a squared term, all contributions +with n > 0 are cancelled and all that is left is the n = 0 contribution +I(0, ρ) = +π2 +2 sinh ρΓ(2ǫ)[P 0 +0 (coth ρ)]2 = +π2 +2 sinh ρ +1 +2ǫ. +(3.15) +Thus the infinite tower of poles with τ = i(n − K) collapses into just a single n = 0 pole contribution at τ = 0 when +K = 0. To confirm that I(0, ρ) is singular we write it out explicitly using the exact form given in (1.3), viz. the ǫ → 0 +limit +I(0, ρ) = +π +2 sinh ρ +� ∞ +0 +dτ +τ 2 → +π +2 sinh ρ +� ∞ +0 +dτ +τ 2 + ǫ2 = +π +2 sinh ρ +π +2ǫ = +π2 +4 sinh ρ +1 +ǫ , +(3.16) +to thus diverge at τ = 0. To compare directly with (3.15) we evaluate the integral in (3.16) as a contour integral. +While (1.3) is the K = 0 limit of (2.3), since Γ(iτ − K)Γ(−iτ − K) contains poles in both the upper- and lower-half +planes we must take one of the τ = 0 poles in I(0, ρ) to lie in the upper-half plane and the other in the lower-half +plane. Thus closing above gives +I(0, ρ) = +π +4 sinh ρ +� ∞ +−∞ +dτ +(τ 2 + ǫ2) = +π +4 sinh ρ +� ∞ +−∞ +dτ +(τ − iǫ)(τ + iǫ) = +2iπ2 +4 sinh ρ +1 +2iǫ = +π2 +4 sinh ρ +1 +ǫ , +(3.17) +which we recognize as (3.15). Thus even for the normalization integral an infinite number of poles collapses into a +single one at the K = 0 exceptional point. +Acknowledgments +One of us (PDM) wishes to acknowledge helpful discussions with Dr. U. G¨unther. +[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York (1972). +[2] I. S. Gradshteyn and I. M. Ryzhik,Tables of Integrals, Series, and Products, Academic Press, New York (1980). +[3] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin (1966). +[4] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). +[5] C. M. Bender, S. Boettcher and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999). +[6] C. M.Bender, D. C. Brody and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002). +[7] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007). +[8] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008). +[9] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou and D. N. +Christodoulides, Phys. Rev. Lett. 103, 093902 (2009). +[10] B. Peng, S. K. ¨Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender and L. Yang, +Nature Physics 10, 394 (2014). +[11] S. A. Cummer, J. Christensen and A. Al`u, Nature Reviews Material 1, 16001 (2016). +[12] P. D. Mannheim, J. Phys. A: Math. Theor. 51, 315302 (2018). +[13] R. +El-Ganainy, +K. +G. +Makris, +M. +Khajavikhan, +Z. +H. +Musslimani, +S. +Rotter +and +D. +N. +Christodoulides, +Nature Phys. 14, 11 (2018). +[14] C. M. Bender, PT Symmetry in Quantum And Classical Physics, World Scientific Press, Singapore (2018). +[15] E. M. Graefe, U. G¨unther, H. J. Korsch and A. E. Niederle, J. Phys. A: Math. Theor. 41, 255206 (2008). +[16] C. M. Bender and P. D. Mannheim, Phys. Rev. D 78, 025022 (2008). +[17] C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. 100, 110402 (2008). +[18] P. D. Mannheim, Prog. Part. Nucl. Phys. 56, 340 (2006). +[19] P. D. Mannheim, Prog. Part. Nucl. Phys. 94, 125 (2017). +[20] P. D. Mannheim, Phys. Rev. D 102, 123535 (2020). +[21] A. Amarasinghe, T. Liu, D. A. Norman and P. D. Mannheim, Phys. Rev. D 103, 104022 (2021). +[22] If both −ν −1/2 and K are nonnegative integers the P −ν−1/2 +K +become ordinary Legendre polynomials P m +ℓ , and they vanish +if m > ℓ. +[23] H. S. Cohl, T. H. Dang and T. M. Dunster, SIGMA 14, 136 (2018). + diff --git a/Z9E2T4oBgHgl3EQfvgiL/content/tmp_files/load_file.txt b/Z9E2T4oBgHgl3EQfvgiL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd2f6bacbdf5ab67d0d1ca993764c0835cef62d1 --- /dev/null +++ b/Z9E2T4oBgHgl3EQfvgiL/content/tmp_files/load_file.txt @@ -0,0 +1,398 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf,len=397 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='04092v1 [math-ph] 8 Jan 2023 Exceptional points for associated Legendre functions of the second kind Tianye Liu, Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Norman and Philip D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Mannheim Department of Physics, University of Connecticut, Storrs, CT 06269, USA tianye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='liu@uconn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='edu,daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='norman@uconn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='edu, philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='mannheim@uconn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='edu (Dated: January 8 2023) We consider the complex ν plane structure of the associated Legendre function of the second kind Q−1/2−K ν (cosh ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' We find that for any noninteger value for K Q−1/2−K ν (cosh ρ) has an infinite number of poles in the complex ν plane, but for any negative integer K there are no poles at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For K = 0 or any positive integer K there is only a finite number of poles, with there only being one single pole (at ν = 0) when K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' This pattern is characteristic of the exceptional points that appear in a wide variety of physical contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, unusually for theories with exceptional points, Q−1/2−K ν (cosh ρ) has an infinite number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Other than in the PT -symmetry Jordan-block case, exceptional points usually occur at complex values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' While not being Jordan- block exceptional points themselves, the exceptional points associated with the Q−1/2−K ν (cosh ρ) nonetheless occur at real values of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' INTRODUCTION The associated Legendre functions are solutions to the second-order differential equation � d2 dρ2 + cosh ρ sinh ρ d dρ − ν(ν + 1) − µ2 sinh2 ρ � F(ν, µ, coshρ) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='1) and as such they generalize the standard Legendre functions to noninteger ν and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Being second-order differential equations they have two classes of solutions, the first kind being the P µ ν (z), the second kind being the Qµ ν(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' In general these solutions can be expressed in terms of hypergeometric functions [1, 2]: P µ ν (cosh ρ) = 1 Γ(1 − µ) �cosh ρ + 1 cosh ρ − 1 �µ/2 F(−ν, ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' 1 − µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (1 − cosh ρ)/2), Qµ ν(cosh ρ) = eiµππ1/2Γ(ν + µ + 1) 2ν+1Γ(ν + 3/2) (sinh ρ)µ(cosh ρ)−ν−µ−1F(ν/2 + µ/2 + 1, ν/2 + µ/2 + 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' ν + 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' 1/ cosh2 ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) However, for some specific values of µ and ν they can be expressed in terms of elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus for µ = −1/2 and arbitrary ν we have [1] P −1/2 ν (cosh ρ) = � 1 2π sinh ρ �1/2 (e(ν+1/2)ρ − e−(ν+1/2)ρ) ν + 1/2 , Q−1/2 ν (cosh ρ) = −i � π 2 sinh ρ �1/2 e−(ν+1/2)ρ ν + 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3) The function P −1/2 ν (cosh ρ) is well behaved at ρ = 0, while the function Q−1/2 ν (cosh ρ) is singular at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Similarly, the function P −1/2 ν (cosh ρ) is well behaved at ν = −1/2, while the function Q−1/2 ν (cosh ρ) is singular at ν = −1/2, having a single pole there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' An analogous though qualitatively different behavior is met for the general µ = −1/2 − K with arbitrary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Specifically, while P −1/2−K ν (cosh ρ) remains singularity free, it is the purpose of this paper to show that for noninteger K Q−1/2−K ν (cosh ρ) develops not just one or even two but actually an infinite number of poles in the complex ν plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Moreover, we will also show that for any negative integer K there are no poles at all, while for zero or positive integer K there is only a finite number of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus for the Q−1/2−K ν (cosh ρ) functions the points with integer K form an infinite family of exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Exceptional points are singular points in parameter space where the behavior of a system undergoes a qualitative change [3], and such singular points had typically only been known to occur at complex values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' 2 However, with the emergence of the PT symmetry program pioneered by Bender and collaborators both theoretically and experimentally [4–14], exceptional points have gained prominence, with the PT setting providing not just a very informative framework for illustrating some of the general features associated with exceptional points, in the PT case the singularities often occur when parameters are real, to thus make them experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The PT program itself is based on the recognition that the physical consistency of a quantum mechanical theory does not require that its Hamiltonian be Hermitian, despite the fact that with Hermiticity one has both reality of energy eigenvalues and conservation of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, while Hermiticity implies the reality of eigenvalues, there is no converse theorem that says that eigenvalues cannot all be real if the Hamiltonian is not Hermitian, with Hermiticity only being sufficient for reality but not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Similarly, while using the Dirac inner product (the overlap of a ket with its Hermitian conjugate bra) yields probability conservation in the Hermitian Hamiltonian case, there is no theorem that says that one cannot obtain probability conservation with a non-Hermitian Hamiltonian and a different inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' With the development of the PT program it was realized that the necessary condition for the reality of eigenvalues is that the Hamiltonian have an antilinear symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' While any antilinear symmetry would suffice for this purpose, because it was for PT symmetry (parity P is linear and time reversal T is antilinear) that this was first identified, theories of this type are generically referred to as PT theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Moreover, in the antilinear symmetry case an inner product that is built out of the overlap of a ket with its antilinear conjugate bra is time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' While antilinear symmetry thus reproduces the key features of Hermitian quantum mechanics it actually goes further as it can also achieve things that cannot be obtained in the Hermitian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' In particular if a Hamiltonian H has an antilinear symmetry and obeys AH = HA where A is an antilinear operator (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' an operator that not only acts on operators and states but also complex conjugates complex numbers), then its eigenstates obey both H|ψ⟩ = E|ψ⟩ and AH|ψ⟩ = AE|ψ⟩ = HA|ψ⟩ = E∗A|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus, as first noted by Wigner in his study of time reversal invariance, for every eigenvalue E with eigenvector |ψ⟩ there is an eigenvalue E∗ with complex conjugate eigenvector A|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' One can thus have real eigenvalues (E = E∗) or eigenvalues that come in complex conjugate pairs (E ̸= E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' This latter option has spurred a large amount of activity especially in the field of experimental optics, as realized through physical systems with (balanced) gain and loss (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' the review of [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' In many PT studies one deals with Hamiltonians in which parameters can be varied continuously, and one can go from domains in which E and E∗ form a complex conjugate pair to domains in which energies are real (the case with E = E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' At the transition point something quite unusual happens, namely not only do the energies become equal (ER + iEI → ER, ER − iEI → ER) the complex conjugate eigenvectors become equal too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus at the transition point there is a loss of eigenvectors, and the Hamiltonian becomes of nondiagonalizable Jordan-block form with its set of eigenvectors being incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (The emergence of the full set of eigenvectors as we go away from the transition point is referred to as an unfolding, and is studied for the case of a finite number of eigenvectors in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=') Away from the transition point and on either side of it the eigenvectors do form complete bases, and the discontinuity at the transition point is in the loss of Hamiltonian eigenvectors, with the lost eigenvectors becoming nonstationary solutions to the Schr¨odinger equation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The point in parameter space at which the Hamiltonian becomes Jordan block is thus an exceptional point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' It is outside the Hermitian Hamiltonian framework as Hermitian Hamiltonians always have complete sets of eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' And not only do exceptional points go beyond Hermitian quantum mechanics, systems with exceptional points have now been observed experimentally [13], to thus make this option for quantum theory fully viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' On the theoretical side these remarks are of relevance to the construction of a quantum-mechanically viable theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' In the standard second-order Einstein gravity theory radiative corrections generate fourth-order gravity terms, and it had been thought that such theories have states of negative Dirac inner product and a consequent loss of unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, it turns out that these theories are not Hermitian theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Rather, they are PT theories, with the PT theory inner product being time independent, positive definite, and fully unitary [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' If one now switches off the second-order term, what remains is a pure fourth-order theory of gravity, such as the conformal gravity theory of interest to us in this paper, with a Hamiltonian that turns out to be Jordan block [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus the limit in which we switch off the second-order term is a singular limit, with the gravity theory then being at an exceptional point, while still remaining unitary [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For conformal gravity the action is of the form [18, 19] IW = −αg � d4x (−g)1/2CλµνκCλµνκ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) where Cλµνκ = Rλµνκ − 1 2 (gλνRµκ − gλκRµν − gµνRλκ + gµκRλν) + 1 6Rα α (gλνgµκ − gλκgµν) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='5) is the conformal Weyl tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The gravitational coupling constant αg is dimensionless, with the theory thus not only being unitary, it is also power-counting renormalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Through its exceptional point structure conformal gravity is thus offered as a fully consistent theory of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' 3 When the conformal gravity theory is applied to cosmology the three-curvature of the associated Robertson-Walker background geometry is found to be negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=', topologically open [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Moreover, fluctuations around this background are found to obey none other than the associated Legendre function equation given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='1) with ρ being the conformal time [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (Technically one finds that the relevant ν are of the conical function form ν = −1/2 + iτ with a real τ that lies in the range 0 ≤ τ ≤ ∞, so one can consider analyticity in ν or in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=') As we now show, the conformal gravity fluctuation equations have an exceptional point structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus conformal gravity actually has two exceptional point structures associated with it, one for the background and the other for the fluctuations around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' While it was conformal gravity that led the current authors to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='1), the reader can regard our paper purely as a study of modes propagating in a space of constant negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' What makes the Q−1/2−K ν (cosh ρ) so interesting is that being at an exceptional point typically involves the loss of a finite number of dynamical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, for the Q−1/2−K ν (cosh ρ) with integer K we lose an infinite number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Moreover, with every integer value of K itself being an exceptional point, in total we even have an infinite number of exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' And not only that, even though the exceptional points associated with the Q−1/2−K ν (cosh ρ) are not themselves of the Jordan-block type, they are nonetheless real, even though non-Jordan-block exceptional points typically are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Associated Legendre functions of the second kind thus provide an interesting theoretical laboratory for exploring some of the general features of exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' COMPLEX ν PLANE STRUCTURE OF ASSOCIATED LEGENDRE FUNCTIONS FUNCTIONS There are various ways to explore the analytic structure of the Q−1/2−K ν (cosh ρ) in the complex ν plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The most direct is to note that since the location of the ν plane poles does not depend on the value of cosh ρ, we can study Q−1/2−K ν (cosh ρ) at large cosh ρ since then it can be expressed in terms of elementary functions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' [2] Q−1/2−K ν (cosh ρ) → −iπ1/2e−iπKΓ(ν + 1/2 − K) 2ν+1Γ(ν + 3/2) (cosh ρ)−ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='1) Q−1/2−K ν (cosh ρ) thus possesses poles at ν = K − 1/2, K − 3/2, K − 5/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' and zeroes at ν = −3/2, −5/2, −7/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. Since the positions of the zeroes are independent of K, for any noninteger K Q−1/2−K ν (cosh ρ) has an infinite number of complex ν plane poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, when K is an integer some of the poles can be cancelled, with all of them being cancelled when K is a negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' When K = 0 one pole remains (at ν = −1/2, just as shown in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3)), when K = 1 two poles remain (at ν = −1/2, −3/2), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The full pattern is K = −2 no poles, −2 < K < −1 infinite set of poles, K = −1 no poles, −1 < K < 0 infinite set of poles, K = 0 one pole, 0 < K < 1 infinite set of poles, K = 1 two poles, 1 < K < 2 infinite set of poles, K = 2 three poles, 2 < K < 3 infinite set of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) We thus obtain an infinite set of exceptional points, one for each integer value of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' An alternate way to explore the complex ν plane structure of the Q−1/2−K ν (cosh ρ) is to first relate the Q−1/2−K ν (cosh ρ) to the P −ν−1/2 K (coth ρ) through the so-called Whipple relation [1, 2] Q−1/2−K ν (cosh ρ) = −ie−iKπ � π 2 sinh ρ �1/2 Γ(ν + 1/2 − K)P −ν−1/2 K (coth ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3) And then on showing that the P −ν−1/2 K (coth ρ) have no complex ν plane poles, we can then identify the poles of Q−1/2−K ν (coth ρ) as the poles of Γ(ν + 1/2 − K), unless they are cancelled by any relevant zeroes that the P −ν−1/2 K (coth ρ) might and in fact do have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' To explore the complex ν plane structure of the P −ν−1/2 K (coth ρ) we use their representation in terms of hypergeo- metric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For the hypergeometric representation we have to convert coth ρ to cosh ρ in order to use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' To this end we set coth ρ = cosh α, so that sinh ρ = 1/ sinh α and cosh ρ = coth α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Since (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) would hold for cosh α, for coth ρ we obtain P −ν−1/2 K (coth ρ) = 1 Γ(ν + 3/2)e−(ν+1/2)ρF(−K, K + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' ν + 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (1 − coth ρ)/2) 4 = 1 Γ(ν + 3/2)e−(ν+1/2)ρ � 1 − K(K + 1)(1 − coth ρ) 2(ν + 3/2) − K(−K + 1)(K + 1)(K + 2)(1 − coth ρ)2 8(ν + 3/2)(ν + 5/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. � = e−(ν+1/2)ρ � 1 Γ(ν + 3/2) − K(K + 1)(1 − coth ρ) 2Γ(ν + 5/2) − K(−K + 1)(K + 1)(K + 2)(1 − coth ρ)2 8Γ(ν + 7/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) Thus no individual term in P −ν−1/2 K (coth ρ) has any poles in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, P −ν−1/2 K (coth ρ) does have zeroes due to the presence of the 1/Γ(ν + 3/2) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' We thus obtain exactly the same Γ(ν + 1/2 − K)/Γ(ν + 3/2) patten for the Q−1/2−K ν (cosh ρ) as obtained from the large cosh ρ expansion, just as we should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' While the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) only has a finite number of terms in it if K is an integer, it is still possible that with noninteger K the then infinite sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) might diverge even though no individual term in it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, this is not the case since through the Γ(ν +1/2−K)/Γ(ν +3/2) factor that appears in the large cosh ρ limit all the poles have already been accounted for, and thus no further poles can be generated from the hypergeometric expansion approach that have not already been obtained from the selfsame Γ(ν + 1/2 − K)/Γ(ν + 3/2) factor that it equally possesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' From analysis of the last line in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) we can understand the pattern we have found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The 1/Γ(ν+3/2) term generates zeros at ν = −3/2, −5/2, −7/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. The 1/Γ(ν + 5/2) term generates zeros at ν = −5/2, −7/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. The 1/Γ(ν + 7/2) term generates zeros at ν = −7/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. If K = 0 all terms in the last line in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) other than the 1/Γ(ν + 3/2) term are cancelled, the zeroes begin at ν = −3/2, and with Γ(ν + 1/2 − K) → Γ(ν + 1/2) there is just one pole (at ν = −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' If K = 1 all terms other than the 1/Γ(ν + 3/2) and 1/Γ(ν + 5/2) are cancelled, so now the zeroes begin at ν = −5/2 (1/Γ(ν + 5/2) does not vanish at ν = −3/2), and with Γ(ν + 1/2 − K) → Γ(ν − 1/2) there are now two poles (at ν = −1/2, ν = −3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' This pattern repeats for K = 3 and so on, with each increase in K by one leading to one more pole [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' If K = −1 only the 1/Γ(ν + 3/2) term is not cancelled and zeros at ν = −3/2, −5/2, −7/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' At the same time Γ(ν + 1/2 − K) → Γ(ν + 3/2) and so all poles are cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' This then repeats for K = −2, K = −3 and so on with all poles being cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' If K is not an integer then none of the poles in Γ(ν + 1/2 − K) can be cancelled and there is an infinite number of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' To understand how we are able to go from an infinite number of poles if K is not an integer to a finite number if K is an integer we consider K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' As long as K is not zero but is very close to zero, every term in the last line in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) is nonzero and there is no cancellation of any pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' When we let K go to zero every term in this last line is cancelled except the very first one, and all but one of the poles are cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' It is in this way that the integer K become exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' CONTOUR INTEGRALS INVOLVING Q−1/2−K −1/2+iτ(cosh ρ) On setting ν = −1/2 + iτ we can rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='1) in the form � d2 dρ2 + cosh ρ sinh ρ d dρ + 1 4 − (−1/2 − K)2 sinh2 ρ � Q−1/2−K −1/2+iτ(cosh ρ) = −τ2Q−1/2−K −1/2+iτ(cosh ρ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='1) and treat it as an eigenvalue equation for eigenfunction Q−1/2−K −1/2+iτ(cosh ρ) and eigenvalue −τ2, where 0 ≤ τ ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3) we obtain Q−1/2−K −1/2+iτ(cosh ρ) = −ie−iKπ � π 2 sinh ρ �1/2 Γ(iτ − K)P −iτ K (coth ρ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) and then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) it follows that Q−1/2−K −1/2 (cosh ρ) = −ie−iKπ � π 2 sinh ρ �1/2 Γ(−K)P 0 K(coth ρ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3) with Q−1/2−K −1/2 (cosh ρ) becoming singular at K = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. For the Q−1/2−K −1/2+iτ(cosh ρ) modes the normalization is given by I(K, ρ) = � ∞ 0 dτQ−1/2−K −1/2+iτ(cosh ρ)[Q−1/2−K −1/2+iτ(cosh ρ)]∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) As we now show, an analysis of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) that will dovetail with the singularity in Q−1/2−K −1/2 (cosh ρ) at K = 0 will provide us with further insight into the nature of exceptional point at K = 0, where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) has a pole at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' 5 To this end use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='2) enables us to rewrite the integrand in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='4) as � Q−1/2−K −1/2+iτ(cosh ρ) � � Q−1/2−K −1/2+iτ(cosh ρ) �∗ = π 2 sinh ρΓ(iτ − K)Γ(−iτ − K)P −iτ K (coth ρ)P iτ K (coth ρ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='5) so that I(K, ρ) = π 2 sinh ρ � ∞ 0 dτΓ(iτ − K)Γ(−iτ − K)P −iτ K (coth ρ)P iτ K (coth ρ) = π 4 sinh ρ � ∞ −∞ dτΓ(iτ − K)Γ(−iτ − K)P −iτ K (coth ρ)P iτ K (coth ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='6) For large τ the integrand behaves as [23] � Q−1/2−K −1/2+iτ(cosh χ) � � Q−1/2−K −1/2+iτ(cosh χ) �∗ → π 2 sinh χτ −2−2K, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='7) and thus the integral will exist if −1 − 2K < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='e −1/2 < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For any general K that satisfies this condition, we can integrate I(K, ρ) as an integral on the real τ axis, though it can only be done numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' However, the integral can also be done as a contour integral, and this will enable us to monitor how the infinite number of poles collapse into a single one when K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' With the asymptotic bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='7) holding for |τ|, it holds on the circle at infinity in the complex τ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The circle contribution to a contour integration will thus be suppressed if −1 < K, a constraint that holds for our previous −1/2 < K condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For −1/2 < K we can close the contour in either the upper- or lower-half complex τ planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For definitiveness we shall close above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' For −1/2 < K < 0 the poles in Γ(iτ − K) are in the upper-half plane while those in Γ(−iτ − K) are in the lower-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' The poles in Γ(iτ − K) are on the imaginary τ axis, being of the form τ = i(n − K), where n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. Closing the contour in the upper-half plane then gives I(K, ρ) = π2i 2 sinh ρ n=∞ � n=0 (−1)n in!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Γ(n − 2K)P K−n K (coth ρ)P n−K K (coth ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='8) In order to set K = 0 we need to treat the ensuing pole at iτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Since K had been taken to be negative, K approaches K = 0 from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' We thus set K = −ǫ where ǫ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus at K = 0 we set I(0, ρ) = π2 2 sinh ρ n=∞ � n=0 (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Γ(n + 2ǫ)P −n 0 (coth ρ)P n 0 (coth ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='9) To evaluate this expression we need to determine P −n 0 (coth ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Recalling the general relation [1, 2] P −µ ν (z) = Γ(ν − µ + 1) Γ(ν + µ + 1) � P µ ν (z) − 2 π e−iµπ sin(µπ)Qµ ν(z) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='10) we obtain P −n 0 (z) = Γ(−n + 1) Γ(n + 1) � P n 0 (z) − 2 π e−inπ sin(nπ)Qn 0 (z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='11) We can eliminate the sin(nπ)Qn 0(z) term since Qn 0(z) is not singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Specifically, we can determine Qn 0(z) using the relations Q0(z) = 1 2 ln �z + 1 z − 1 � , Qn 0(z) = (z2 − 1)n/2 dn dzn Q0(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='12) and thus confirm that Qn ν(z) does not develop a complex ν plane singularity as ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Consequently, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='9) respectively reduce to P −n 0 (z) = Γ(−n + 1) Γ(n + 1) P n 0 (z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='13) I(0, ρ) = π2 2 sinh ρ n=∞ � n=0 (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Γ(n + 2ǫ)Γ(−n + 1) Γ(n + 1) [P n 0 (coth ρ)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='14) 6 Now Γ(−n + 1) has single poles at n = 1, n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='. However the P n 0 (cosh ρ) with integer n are ordinary Legendre polynomials, and they are zero if n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Then since the P n 0 (cosh ρ) term appears in a squared term, all contributions with n > 0 are cancelled and all that is left is the n = 0 contribution I(0, ρ) = π2 2 sinh ρΓ(2ǫ)[P 0 0 (coth ρ)]2 = π2 2 sinh ρ 1 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='15) Thus the infinite tower of poles with τ = i(n − K) collapses into just a single n = 0 pole contribution at τ = 0 when K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' To confirm that I(0, ρ) is singular we write it out explicitly using the exact form given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' the ǫ → 0 limit I(0, ρ) = π 2 sinh ρ � ∞ 0 dτ τ 2 → π 2 sinh ρ � ∞ 0 dτ τ 2 + ǫ2 = π 2 sinh ρ π 2ǫ = π2 4 sinh ρ 1 ǫ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='16) to thus diverge at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' To compare directly with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='15) we evaluate the integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='16) as a contour integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' While (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3) is the K = 0 limit of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='3), since Γ(iτ − K)Γ(−iτ − K) contains poles in both the upper- and lower-half planes we must take one of the τ = 0 poles in I(0, ρ) to lie in the upper-half plane and the other in the lower-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus closing above gives I(0, ρ) = π 4 sinh ρ � ∞ −∞ dτ (τ 2 + ǫ2) = π 4 sinh ρ � ∞ −∞ dτ (τ − iǫ)(τ + iǫ) = 2iπ2 4 sinh ρ 1 2iǫ = π2 4 sinh ρ 1 ǫ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='17) which we recognize as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Thus even for the normalization integral an infinite number of poles collapses into a single one at the K = 0 exceptional point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Acknowledgments One of us (PDM) wishes to acknowledge helpful discussions with Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' G¨unther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Abramowitz and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' Stegun, Handbook of Mathematical Functions, Dover Publications, New York (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfvgiL/content/2301.04092v1.pdf'} +page_content=' [2] I.' metadata={'source': 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We tested it on complex multidimen- +sional arrays taken, among other, from real-world +applications, including unconstrained binary opti- +mization and optimal control problems, for which +the possible number of elements is up to 2100 ele- +ments. In numerical experiments, both on analytic +model functions and on complex problems, our +algorithm outperform existing popular discrete +optimization methods (Particle Swarm Optimiza- +tion, Covariance Matrix Adaptation, Differential +Evolution and others). Moreover, we take the +same set of hyperparameters of our algorithm for +all numerical applications. +1. Introduction +The multidimensional optimization problem is one of the +most common in machine learning. It includes the important +case of discrete optimization (Parker & Rardin, 2014): +min +x f(x), +s.t. x ∈ {0, 1, . . . , N}d, +(1) +which occurs when searching for the minimum or maximum +element in an implicitly given multidimensional tensor1, +*Equal contribution 1 Skolkovo Institute of Science and Tech- +nology, Moscow, Russia +2 AIRI, Moscow, Russia . Correspon- +dence to: Anastasia Batsheva . +1A tensor is just a multidimensional array with a number of +dimensions d (d ≥ 1). A two-dimensional tensor (d = 2) is +a matrix, and when d = 1 it is a vector. For scalars we use +including when considering the discretization of functions +from a continuous argument. Multidimensional discrete +optimization problems are still computationally difficult in +the case of complex target functions and large dimensions, +and efficient direct gradient-free optimization procedures +are highly needed. +The development of methods for low-rank tensor approxima- +tions has made it possible to introduce fundamentally new +approaches for the approximation, storage and operation +with multidimensional tensors (Cichocki et al., 2016; 2017). +One of the common methods for low-rank approximation +is the tensor train (TT) decomposition (Oseledets, 2011), +which allows to bypass the curse of dimensionality. Many +useful algorithms (e. g., element-wise addition, multiplica- +tion, solution of linear systems, convolution, integration, +etc.) have effective implementations for tensors given in +the TT-representation. The complexity of these algorithms +turns out to be polynomial in dimension and mode size if +the TT-ranks are bounded. It makes TT-decomposition ex- +tremely popular in a wide range of applications, including +computational mathematics and machine learning. +In the last few years, several new discrete optimization algo- +rithms based on the TT-format have been proposed (Sozykin +et al., 2022; Selvanayagam et al., 2022; Shetty et al., 2022; +Chertkov et al., 2022a). However, the development of new +tensor train-based approaches for optimization is possible +based on recently proposed methods for working with proba- +bility distributions and sampling in the TT-format (Novikov +normal font (a, b, c, . . .), we denote vectors with bold letters +(a, b, c, . . .), we use upper case letters (A, B, C, . . .) for matri- +ces, and calligraphic upper case letters (A, B, C, . . .) for tensors +with d > 2. The (n1, n2, . . . , nd)th entry of a d-dimensional +tensor A ∈ RN1×N2×...×Nd is denoted by A[n1, n2, . . . , nd], +where nk = 1, 2, . . . , Nk (k = 1, 2, . . . , d), and Nk is a size +of the k-th mode. The mode-k slice of such tensor is denoted +by A[n1, . . . , nk−1, :, nk+1, . . . , nd], and it is a vector of the +length Nk. +arXiv:2301.12162v1 [math.NA] 28 Jan 2023 + +PROTES: Probabilistic Optimization with Tensor Sampling +et al., 2021; Dolgov et al., 2020). Within the framework of +this approach, it is possible to specify a multidimensional +discrete probability distribution in the TT-format, followed +by efficient sampling from it and updating its parameters to +approximate the minimum in a better way. +To summarize, our main contributions are the following: +• We develop a new method PROTES for optimization +(finding the minimum or maximum value) of multi- +dimensional data arrays and discretized multivariate +functions based on a sampling method from a proba- +bility distribution in the TT-format; +• We apply2 our approach for several multidimensional +QUBO and optimal control problems to demonstrate +its performance and compare it with various popular +discrete optimization algorithms. We used the same +set of hyperparameters of our algorithm for all numeri- +cal experiments, and obtained the best result in all 20 +considered problems. +2. Method +Our task is to minimize the given multivariate discrete func- +tion f. Denote an argument of the function f by a vector +x = {x1, x2, . . . , xd}. First, we make a monotonic trans- +formation F[f](x) of the function f that returns a function +that has the maximum where the given function f has a +minimum. A reasonable choice is the transformation using +the Fermi-Dirac function: +F[f](x) = +1 +exp +� +(f(x) − fmin − E)/T +� ++ 1, +(2) +where fmin is the minimum of f, T > 0 is a parameter +and E > 0 is some small threshold. Then the problem is +reduced to finding the maximum of the function F[f]. To do +this, we first find a maximum of the following expectation +max +θ +EξθF(ξθ), +(3) +where a family of random variables ξθ has a parameterised +distribution function pθ(x). +We do it with the help of +2The program code with numerical examples is available in the +repository https://github.com/anabatsh/TT_pro. +the gradient ascent method and the REINFORCE trick al- +gorithm (Williams, 1992) to approximate the gradient of +EξθF(ξθ). According to this algorithm, we can estimate +this gradient by the following Monte-Carlo-like expression +∇θEξθF(ξθ) ≈ 1 +N +N +� +i=1 +F(xi)∇θ log pθ(xi). +(4) +The values {xi}N +1 in the last expression are independent +realization of the random variable ξθ. +When we find the optimal values of θ then we expect the op- +timal distribution pθ to have a peak at the point of maximum +function F. Thus we can obtain the argument of its max- +imum by sampling from this distribution. For very small +values of T, only a few terms contribute to the sum (4), +namely those xi for which f(xi) − fmin < E is hold. For +these values of x, F is close to 1, while for the other sam- +ples its value is 0. Thus, we can discard all other samples +and keep a few samples with the best value of the optimized +function. This approach is the essence of our method. +Loss function. So, we came to the following loss function +in the form +�L = − +k +� +j=1 +log pθ(x(j)), +(5) +where instead of the parameter E we use a fixed number k +of the best sample, i. e. such samples in which the value of +the target function f is the smallest. +Parametrization of the density using TT-format. The +crucial step is what distribution to choose as pθ in low- +parametric way. The distribution has to have two properties: +we should be able to sample from the distribution in a fast +way, and we should be able to compute the log-likelihood. +We propose to use TT-format to represent the density: +pθ(n) = 1 +Z P[n], Z = +� +n +P[n]. +(6) +Thus, parameters θ are just flatten elements of all cores +of this TT-representation P. Note that the algorithm for +sampling from the density given in the TT-format allows +sampling in the case of the initially non-normalized tensor +as well (Dolgov et al., 2019). So, the tensor values may not +be normalized. + +PROTES: Probabilistic Optimization with Tensor Sampling +Figure 1: Schematic representation of the TT-decomposition. The top picture demonstrates the calculation of the specific +tensor element (n1, . . . , nd) from its TT-representation, and the bottom picture presents the related tensor network diagram. +Algorithm 1 Multidimensional tensor optimizer PROTES. +Data: the function fY(n), that computes the value of the tar- +get tensor Y ∈ RN1×N2×...×Nd at the multi-index n = +[n1, n2, . . . , nd]⊤; the maximum number of requests M; +the number of generated samples per iteration K; the num- +ber of selected candidates per iteration k; the number of +SGD steps s; the GD learning rate h; the TT-rank of the +probability tensor R. +Result: the d-dimensional multi-index n(opt), which relates to +the approximated tensor minimum (or maximum) value. +// Generate a random rank-R tensor: +P = tt random([N1 × N2 × . . . × Nd], R) ∈ RN1×N2×...×Nd +for m = 1 to M/K do +// Generate K samples from P: +n1, n2, . . . nK = tt sample(P, K) +// Compute the objective function values in samples: +y1 = f(n1), y2 = f(n2), . . . yK = f(nK), +// Update the current optimal multi-index from samples: +n(opt) = indopt([y1, y2, . . . , yK], [n1, n2, . . . , nK]) +// Select top-k (minimum or maximum)) samples: +i1, i2, . . . , ik = argopt([y1, y2, . . . , yK], k) +// Perform s steps of gradient ascend with loss function (5): +P ← gd(P, �L, q) +return n(opt). +Details of the proposed algorithm. The steps of the pro- +posed algorithm are as follows. First (if the problem is +unconstrained, if constraints are imposed, see below) we set +the random elements of the TT cores uniformly from the +interval [0, 1]. +Then we sample K values from the tensor P and calculate +the value of the target function f on them. From these +samples, we take a set {x(j)}k +j=1 of size k < K of those on +which the target function is smallest. Values k and K are the +hyperparameters of the algorithm, as well as the tensor ranks +for the probability. If any conditions are imposed in the task, +then we immediately throw samples that do not satisfy this +condition. From the remaining samples, we construct the +loss function (5), and make several gradient ascend steps for +the elements of the tensor cores. Then, these iterations with +sampling continue a given number of times. We summarize +this steps in Algorithm 1, where we present the details of +the PROTES implementation. +After a sufficient number of iterations, we expect the ten- +sor P to represent an almost delta-function with a pro- +nounced peak in the value of the minimum of the target +function. Therefore, this value will be sampled, since the +probability of sampling other values will be small. +Constraints. A very nice property of the proposed approach +is that it can be adapted to handle constraints, as we will +show in numerical examples. Suppose, we have some ad- +missible set of indices. One option is just to remove such +samples from the top-k values. However, in some cases +the probability of sampling indices that are admissible is +very low (for example, in the condition that a binary string +should have at least 3 consecutive 1), so this approach will +not work. Instead, if the constraint permits we use the al- +gorithm for the for constructive construction of tensors in +TT format by a known analytic function defining the con- +straints, described in (Ryzhakov & Oseledets, 2022). Once +the indicator tensor (1 if the index is admissible and 0 if it +is not) is built in the TT-format, we can just initialize the +starting distribution by it, and it will be guaranteed that the +samples always belong to the admissible set. + +Na +nd +R3 +Rd-1 +N3 +R2 +R2 +Rd-2 +n3 +Rd- +R1 +R1 +N1 +N2 +N3 +Na-1 +Nd +n2 +ni +N2 +nd +ni +n3 +n2 +nd-1 +N1 +VE RNixN2x...xNa +G1 +G2 +G3 +Gd-1 +Gd +N3 +： +. +N2 +R1 +R2 +R3 +Rd-2 +Rd-1 +G1 +G2 +G3 +Gd-1 +Gd +Nd +.·. +N1 +Ni +N2 +N3 +Na-1 +NdPROTES: Probabilistic Optimization with Tensor Sampling +What did we get. In the end, we get an algorithm that tries +to solve the optimization problem without gradients, can +work with constraints and can be used to optimize functions +both with discrete and continuous variables. Continuous +variables are treated through discretization on a grid, re- +ducing the task to the discrete optimization. The method +belongs to the class of evolutionary algorithms, but it is very +easy to derive, easy to implement and as we will see is quite +competitive on different examples from different domains. +The main advantage of PROTES is the usage of tensor-train +decomposition to represent the proposal density, which is +a very expressive way to represent densities that can be +even non-local. For example, mixture of Gaussians can be +represented with very small rank. Now we will discuss the +properties of the tensor-train decomposition in more details. +2.1. Tensor Train Decomposition +Let us dwell on the concepts and the properties of the Tensor +Train format. A d-way tensor Y ∈ RN1×N2×...×Nd is said +to be in the TT-format (Oseledets, 2011), if its elements are +represented by the following formula +Y[n1, n2, . . . , nd] = +R1 +� +r1=1 +R2 +� +r2=1 +· · · +Rd−1 +� +rd−1=1 +G1[1, n1, r1] G2[r1, n2, r2] . . . +Gd−1[rd−2, nd−1, rd−1] Gd[rd−1, nd, 1], +(7) +where (n1, n2, . . . , nd) is a multi-index (ni = 1, 2, . . . , Ni +for i = 1, 2, . . . , d), integers R0, R1, . . . , Rd (with con- +vention R0 = Rd = 1) are named TT-ranks, and three- +dimensional tensors Gi ∈ RRi−1×Ni×Ri (i = 1, 2, . . . , d) +are named TT-cores. The TT-decomposition (7) allows to +represent a tensor or a discretized multivariable function +in a compact and descriptive low-parameter form, which is +linear in dimension d (see illustration on Figure 1), i. e., it +has less than d · maxi=1,...,d(NiR2 +i ) parameters. +Many useful algorithms (e. g., element-wise addition, mul- +tiplication, solution of linear systems, convolution, inte- +gration, etc.) have effective implementations for tensors +given in the TT representation. The complexity of these +algorithms turns out to be polynomial in dimension and +mode size if the TT-ranks are bounded. +It makes TT- +decomposition extremely popular in a wide range of appli- +cations, including computational mathematics and machine +learning. A detailed description of the TT-format and related +algorithms are given in works (Oseledets, 2011; Cichocki +et al., 2016). +3. Related Work and Baselines +Below we give a brief analysis of universal approaches to +discrete optimization and then discuss the methods based +on low-rank tensor approximations, which have become +popular in the last few years. +3.1. Discrete Optimization and Gradient Free Methods +In recent years machine learning algorithms achieved im- +pressive results in various applications. The important part +of these algorithms is an optimization procedure. In many +situations, the problem-specific target function is not differ- +entiable, too complex, or its gradients are not helpful due to +the non-convex nature of the problem (Kolda et al., 2003; +Alarie et al., 2021), and standard well-known gradient-based +methods cannot be applied directly. The examples include +hyper-parameter selection during the training of neural mod- +els, training neural networks with discrete weights or with +non-differentiable loss functions, policy optimization in rein- +forcement learning, and many other optimization problems +in complex scenarios. In all these contexts, efficient direct +gradient-free optimization procedures are highly needed. +In the case of high dimensional optimization, evolutionary +strategies (ES) (Doerr & Neumann, 2021) are one of the +most advanced methods of black-box optimization. This +approach aims to optimize the parameters of the search +distribution, typically a multidimensional Gaussian, to max- +imize the objective function. Finite difference schemes are +commonly used to approximate gradients of the parame- +ters of the search distribution. Numerous works proposed +techniques to improve the convergence of ES (Nesterov & +Spokoiny, 2017), for example, second-order natural gra- +dient (Wierstra et al., 2014), or the history of recent up- +dates (Covariance Matrix Adaptation Evolution Strategy; +CMA-ES) (Hansen, 2006), or even surrogate gradients (Ma- +heswaranathan et al., 2019) may be used to generate updates. + +PROTES: Probabilistic Optimization with Tensor Sampling +There is a large variety of other heuristic methods for find- +ing the global extremum. In particular, we note such pop- +ular approaches as NoisyBandit (Scarlett et al., 2017), Par- +ticle Swarm Optimization (PSO) (Kennedy & Eberhart, +1995), Simultaneous Perturbation Stochastic Approxima- +tion (SPSA) (Maryak & Chin, 2001), Differential Evolution +(DE) (Storn & Price, 1997) and scrambled-Hammersley +(scr-Hammersley) (Hammersley, 1960). +3.2. Tensor Based Optimization Methods +Recently, TT-decomposition has been actively used for the +optimization of multidimensional arrays and multivariable +functions. An iterative method TTOpt based on the maxi- +mum volume approach is proposed in the work (Sozykin +et al., 2022). TTOpt is based on the theorem of sufficient +proximity of the maximum modulo element of the maxi- +mum volume submatrix (i.e. the submatrix having the maxi- +mum modulus of the determinant) to the maximum modulo +element of the tensor. Based on this observation, tensor +elements are sampled by analogy with the TT-cross method +from specially selected successive unfoldings of the tensor, +and then the search for the optimum is carried out among +the elements of the obtained maximum volume submatrices. +To be able to find the minimum element, dynamic mapping +of the tensor elements is carried out, which converts the +minimum values into maximum ones. The authors applied +this approach to the problem of optimizing the weights of +neural networks in the framework of reinforcement learn- +ing problems in (Sozykin et al., 2022) and to the QUBO +problem in (Nikitin et al., 2022). A similar optimization +approach was also considered in (Selvanayagam et al., 2022) +with practical applications of the method for optimizing the +housings of electronic devices and in (Shetty et al., 2022) +for optimizing the movement in space of robotic arms. +A new algorithm Optima-TT based on the probabilis- +tic sampling from the TT-tensor was proposed in recent +work (Chertkov et al., 2022a). This approach makes it possi- +ble to obtain a very accurate approximation for the optimum +of the given TT-tensor within the framework of successive +tensor multiplication of TT-cores with an intelligent selec- +tion of potential candidates for the optimum. However, we +note that this method is intended for directly optimizing +the TT-tensors, which means that its success strongly de- +pends on the quality of the approximation of the original +multidimensional data array. For this purpose, one of the ap- +proximation methods in the TT-format (TT-SVD, TT-ALS, +TT-cross, etc.) should be used. We note, that the dependence +of the quality of the optimization result on the accuracy of +the tensor approximation was not studied for this method. +3.3. List of Baselines +Taking into account the analysis of discrete optimization +methods, as baselines we consider two tensor based opti- +mization methods: TTOpt3 (BS1) and Optima-TT4 (BS2), +and five popular gradient free optimization algorithms from +the nevergrad framework (Bennet et al., 2021)5: OnePlu- +sOne (BS3), PSO (BS4), NoisyBandit (BS5), SPSA (BS6) +and Portfolio approach (BS7), which is based on the combi- +nation of CMA-ES, DE and scr-Hammersley methods. +4. Numerical Experiments +To evaluate the effectiveness of the proposed approach +PROTES, we carried out a series of 20 numerical exper- +iments for various formulations of model problems. The +results are presented in Table 1. For our method and the 7 +baselines described in the previous section, we report for +each model problem the resulting approximation to the min- +imum value, and in the last row of the table, for clarity, we +present the number of wins for each of the methods. +For all the considered problems, we used the default set +of parameters for baselines, and for our method we fixed +the following parameter values: the number of generated +samples per iteration is K = 50; the number of selected +candidates from K samples per iteration is k = 5; the +number of GD steps is s = 100; the GD learning rate is +h = 10−4; the TT-rank of the probability tensor is R = 5. +For all methods, the limit on the number of requests to the +objective function was fixed at the value M = 104. +3We used implementation of the method from https:// +github.com/AndreiChertkov/ttopt. +4We used implementation from https://github.com/ +AndreiChertkov/teneva. The TT-tensor for optimization +was generated by TT-cross method. +5See +https://github.com/facebookresearch/ +nevergrad + +PROTES: Probabilistic Optimization with Tensor Sampling +Table 1: Minimization result for all selected benchmarks (P-01 – P-20). We report the values obtained by the proposed +method PROTES and by all considered baselines (BS1 – BS7). For each benchmark, the best result is highlighted in bold. +The last row presents the number of best results for each baseline. +OUR +BS-1 +BS-2 +BS-3 +BS-4 +BS-5 +BS-6 +BS-7 +ANALYTIC +FUNCTIONS +P-01 +9.1E+00 +9.1E+00 +9.1E+00 +9.1E+00 +9.1E+00 +2.0E+01 +9.1E+00 +9.1E+00 +P-02 +1.7E+00 +1.7E+00 +1.7E+00 +2.7E+00 +1.7E+00 +5.4E+00 +2.4E+00 +1.9E+00 +P-03 +-9.8E-01 +-9.8E-01 +-9.8E-01 +-9.8E-01 +-9.8E-01 +-6.8E-01 +-9.8E-01 +-9.8E-01 +P-04 +4.5E+00 +4.5E+00 +4.5E+00 +4.5E+00 +4.5E+00 +7.4E+01 +4.5E+00 +4.5E+00 +P-05 +-5.4E+00 +-5.4E+00 +-5.4E+00 +-3.9E+00 +-5.0E+00 +-1.9E+00 +-1.6E+00 +-5.3E+00 +P-06 +1.6E-01 +1.6E-01 +1.6E-01 +1.6E-01 +1.6E-01 +1.9E-01 +4.4E-01 +1.6E-01 +P-07 +2.3E+07 +2.3E+07 +2.3E+07 +4.8E+07 +2.9E+07 +9.6E+09 +1.4E+11 +2.3E+07 +P-08 +2.7E+01 +2.7E+01 +2.7E+01 +6.7E+01 +2.7E+01 +8.3E+01 +1.0E+02 +2.7E+01 +P-09 +1.2E+00 +1.2E+00 +1.2E+00 +1.4E+00 +1.2E+00 +2.5E+00 +1.7E+00 +1.2E+00 +P-10 +4.4E+02 +4.4E+02 +4.4E+02 +8.3E+02 +7.6E+02 +1.7E+03 +2.8E+03 +4.4E+02 +QUBO +P-11 +-3.6E+02 +-3.5E+02 +-3.4E+02 +-3.2E+02 +-3.4E+02 +-3.2E+02 +0.0E+00 +-3.6E+02 +P-12 +-5.9E+03 +-5.9E+03 +-5.9E+03 +-5.2E+03 +-5.7E+03 +-5.4E+03 +-5.9E+03 +-5.9E+03 +P-13 +-3.8E+00 +-3.7E+00 +-3.4E+00 +-2.8E+00 +1.1E+01 +7.4E+02 +-1.2E+00 +-3.8E+00 +P-14 +-3.1E+03 +-2.9E+03 +-2.2E+03 +-2.5E+03 +-2.9E+03 +-2.6E+03 +-2.9E+03 +-3.0E+03 +CONTROL +P-15 +6.8E-03 +8.4E-03 +5.5E-01 +1.4E-02 +9.9E-03 +1.7E-02 +1.7E-01 +7.9E-03 +P-16 +1.4E-02 +3.0E-02 +2.3E-01 +4.3E-02 +1.7E-02 +4.9E-02 +2.5E-01 +1.5E-02 +P-17 +3.0E-02 +3.4E-01 +2.1E+00 +5.0E-02 +3.2E-02 +1.1E-01 +1.4E+00 +3.6E-02 +CONTROL ++CONSTR. +P-18 +1.3E-02 +1.5E-02 +FAIL +4.8E-02 +9.1E-02 +FAIL +2.5E-01 +5.6E-02 +P-19 +1.7E-02 +1.6E+00 +FAIL +FAIL +FAIL +FAIL +FAIL +FAIL +P-20 +4.7E-02 +FAIL +FAIL +FAIL +FAIL +FAIL +FAIL +FAIL +WINS +20 +11 +11 +4 +7 +0 +4 +11 +As can be seen from Table 1, method PROTES, in contrast +to alternative approaches, gives a consistently top result for +all model problems. Further, in separate subsections, we +describe in more detail the considered model problems and +discuss the corresponding numerical results. +4.1. Multivariable Analytic Functions +First, we consider the optimization task for various ten- +sors arising from the discretization of multivariable analytic +functions. We consider 10 popular benchmarks: Ackley (P- +01), Alpine (P-02), Exponential (P-03), Grienwank (P-04), +Michalewicz (P-05), Piston6 (P-06), Qing (P-07), Rastri- +gin (P-08), Schaffer (P-09) and Schwefel (P-10). These +functions have a complex landscape and are often used in +problems of evaluating the effectiveness of optimization +algorithms (Dieterich & Hartke, 2012; Jamil & Yang, 2013). +This set of functions is also well suited for the study of +tensor based methods, since the functions have a varying +6This function corresponds to the practical problem of mod- +eling the time that takes a piston to complete one cycle within a +cylinder; the description of related parameters and their ranges can +be found in (Zankin et al., 2018; Chertkov et al., 2022b). +low-rank structure (Chertkov et al., 2022b; Str¨ossner et al., +2022). We consider the 7-dimensional case (since this is the +dimension of the Piston function) and discretization on a +uniform grid with 16 nodes. +As follows from the results presented in Table 1 (bench- +marks P-1, P-2, ..., P-10), our method, like the other two +tensor approaches (BS-2 and BS-3), gave the most accurate +solution for all model problems. The most sophisticated +approach from the nevergrad package (BS-7) turned out to +be the next in accuracy (the method did not converge only +in two cases out of ten). +4.2. Quadratic Unconstrained Binary Optimization +Quadratic +Unconstrained +Binary +Optimization +(QUBO) (Glover et al., 2022) is a widely known +NP-hard problem which unifies a rich variety of combina- +torial optimization problems from finance and economics +applications to machine learning and quantum computing. +QUBO formulation in a very natural manner using penalty +functions, yielding exact model representations in contrast +to the approximate representations produced by customary + +PROTES: Probabilistic Optimization with Tensor Sampling +uses of penalty functions. The standard QUBO problem +can be formulated as follows: +F(x) = xT Qx → min +x , +s.t. x ∈ {0, 1}d, +(8) +where x is a vector of binary decision variables of the length +d and Q ∈ Rd×d is a square matrix of constants. In all +experiments we select dimension d = 50. +We consider the following QUBO problems from qubo- +gen7 package: Max-Cut Problem (P-11; search for partition +for undirected graph into two sets such that the number of +edges between the two sets is a large as possible), Mini- +mum Vertex Cover Problem (P-12; search for a cover with +a minimum number of vertices in the subset of the graph +vertices such that each edge in the graph is incident) and +Quadratic Knapsack Problem (P-13; search for a subset of +maximum profit that satisfies the budget limitations from a +set of potential projects with specified interactions between +pairs of projects). +We also considered one more benchmark (P-14) from the +work (Dong et al., 2021) (problem k3; d = 50), where angle- +modulated bat algorithm (AMBA) algorithm was proposed +for high-dimensional binary optimization problems with +engineering application to antenna topology optimization. +This is the ordinary binary knapsack problem with fixed +(randomly generated) weights wi ∈ [5, 20], profits pi ∈ +[50, 100] (i = 1, 2, . . . , d) and the maximum capacity C = +1000. +The proposed method PROTES for all four considered prob- +lems ( P-11, P-12, P-13, P-14) gives the best result, as can be +seen from Table 1. The baseline BS-7 again turned out to be +the next in accuracy. We note that the exact solution for P-14 +is provided in (Dong et al., 2021) (−3103). Several methods, +including the one proposed by the authors, were compared +in (Dong et al., 2021) for this problem: BPSO (−2854), +BBA (−2976), AMBA (−2956), A-AMBA (−2961), P- +AMBA (−2989). Thus, only the PROTES for this problem +converges to the global optimum. To demonstrate the con- +vergence behavior of different methods depending on the +number of performed requests to the objective function, we +7See https://github.com/tamuhey/qubogen +Figure 2: Convergence of optimization methods depending +on the number of requests to the objective function. +present the corresponding graph8 on Figure 2. +4.3. Optimal Control +Suppose we have a state variable x ∈ R controlled by a +binary variable i called control (i. e. it’s just a switch with +modes ”off” = 0 and ”on” = 1) over some discrete interval of +time [0, T]. The state at time t + 1 is based on the previous +state x(t) and control i(t) as follows x(t + 1) : ˙x(t) = +f(x(t), i(t)), where the function f is called an equation +function. The Optimal Control problem is to find such a +sequence of controls i∗ = [i∗(0), i∗(1), . . . , i∗(T)] over +our time interval [0, T] that minimizes a special function F. +This sequence is called an optimal solution and the function +F is called an objective function. Any other sequence +i = [i(0), i(1), . . . , i(T)] we will call just a solution. +Formulating the problem mathematically, we need to find +such a solution +F(x, i) → min +x,i , +s.t. +� +� +� +� +� +x(0) = x0, +˙x(t) = f(x(t), i(t)), +i(t) ∈ {0, 1} +(9) +here x = [x(0), x(1), . . . , x(T)] is a state variable path. We +8For clarity of demonstration, the values of the objective func- +tion on the graph are inverted. + +Our +3000 +BS-1 +ttt +BS-2 +BS-3 +2800 +BS-4 +BS-5 +2600 +BS-6 +BS-7 +2400 +Exact +2200 +2000 +10° +10 +10 +3 +10 +4 +10 +Number of requestsPROTES: Probabilistic Optimization with Tensor Sampling +assume that function f is nonlinear. In this case, finding an +optimal solution raises a lot of difficulties. +Note, that we can easily reformulate the above-described +problem as a tensor minimum finding. Since we consider +only binary vectors of the solution i ∈ {0, 1}T , we have +the limited number of the possible variants equal to 2T +and, hence, only the limited number of objective values +F(x, i) = F(x(i), i) = ˆF(i) ∈ R2T is possible. It means +that we can represent the set of all solutions in the form of +tensor: +F[i] = F[i0, i1, . . . , iT −1] = ˆF(i) +And the optimal solution i∗ is nothing more than the multi- +index of the minimum of this tensor. +In numerical applications we consider the nonlinear equa- +tion function f(x, i) = x3 − i. Function F we set as +F(x, i) = 1 +2 +T +� +t=0 +(x(t) − xref)2 +The initial and the reference state are fixed at values x0 = +0.8, xref = 0.7. +For T, we considered several values, such as 25, 50 and 100 +(benchmarks P-15, P-16 and P-17 respectively). As follows +from the results presented in Table 1, our method, gave the +most accurate solution in all three cases. +4.4. Optimal Control with Constraints +As mentioned earlier, some conditions or constraints may be +imposed on the solution. There are two types of constraints: +the first is the constraint that only the solution i must satisfy +(therefore we will call it control constraint). It is usually +set in the form P(i) = True\False. The second type +of constraints are those of the form h(x(t), i(t)) ∀t, called +path constraints. For example, we can require that x(t) +does not exceed some given value M at each step. Thus, +the second type of constraints differs from the first in that +there is a dependence not only on the control, but also on +the state variable. +We only consider constraints of the first type and define P +as follows: the control variable can take value 1 no less +than 3 times during the whole time interval. Formally, this +can be written as follows: +P = +� +i ∈ [0, 1]N +���� +i(t) ≥ i(t − 1) − i(t − 2) +i(t) ≥ i(t − 1) − i(t − 3) +� +However, not all remaining solutions (i.e. satisfying condi- +tion P) fit, if the differential equation is f(x, i) = x3 − i, +since for some values of x(t), i(t) reachable during the so- +lution process, the next state x(t + 1) may not be found. +In this case, in numerical experiments, the target function +returns a very large number as a stub. To account for this +condition in our algorithm, we construct the initial distribu- +tion in the form of an indicator tensor as described in the +implementation details section of the algorithm. The details +of this construction are in the Appendix. +The numerical results for T = 25 (P-18), T = 50 (P-19) +and T = 100 (P-20) are presented in Table 1. Only our +method successfully found the local optimum in all three +cases. +5. Conclusions +We presented an optimization algorithm based on sampling +from the probability density defined in the Tensor Train +format. For all numerical experiments presented in the paper, +we used the same set of hyperparameters, so our algorithm is +rather universal. In order to take into account the constraints, +as in the problem of optimal control with constraints, we +only considered them in the form of a specially selected +initial approximation (special form of an indicator tensor in +TT format); further on, the algorithm did not consider the +constraints explicitly. This approach allows us to extend the +capabilities of the algorithm by using the properties of the +Tensor Train representation. Numerical algorithms show +that we outperform many popular optimization methods. +The main direction in our future work is scaling of the +method to large dimensions. For d ≥ 1000 we have encoun- +tered numerous technical difficulties, which can be allevi- +ated by other tensor formats (such as Hierachical Tucker, +which can be parallelized over d) and more efficient im- +plementations of the optimization method (now we used +standard autodifferentiation without special tensor optimiza- +tion methods such as Riemannian optimization), + +PROTES: Probabilistic Optimization with Tensor Sampling +Acknowledgements +The +work +was +supported +by +the +Analytical +cen- +ter under the RF Government (subsidy agreement +000000D730321P5Q0002, Grant No. +70-2021-00145 +02.11.2021). +References +Alarie, S., Audet, C., Gheribi, A. E., Kokkolaras, M., and Le +Digabel, S. Two decades of blackbox optimization appli- +cations. EURO Journal on Computational Optimization, +9:100011, 2021. ISSN 2192-4406. +Bennet, P., Doerr, C., Moreau, A., Rapin, J., Teytaud, F., +and Teytaud, O. Nevergrad: Black-box optimization +platform. 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For all cores except the last one, we use derivative functions in the form +f k +0 (x) = +� +1, +x = 0 or x = l +None, +else, +(10) +f k +1 (x) = min(l, x + 1), +(11) +and for the last core +f d +0 (x) = +� +1, +x = 0 or x = l +0, +else, +(12) +f d +1 (x) = +� +1, +x ≥ l − 1 +0, +else. +(13) +A tensor in the TT-format built on such derivative functions is 0 if there are less than l ones among its arguments in a row, +and 1 in all other cases. + diff --git a/_dFLT4oBgHgl3EQfwC-M/content/tmp_files/load_file.txt b/_dFLT4oBgHgl3EQfwC-M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c7d28460854909a850f7362b36ee3b8a7ac19f2 --- /dev/null +++ b/_dFLT4oBgHgl3EQfwC-M/content/tmp_files/load_file.txt @@ -0,0 +1,805 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf,len=804 +page_content='PROTES: Probabilistic Optimization with Tensor Sampling Anastasia Batsheva * 1 Andrei Chertkov * 1 Gleb Ryzhakov * 1 Ivan Oseledets 1 2 Abstract We develop new method PROTES for optimiza- tion of the multidimensional arrays and dis- cretized multivariable functions, which is based on a probabilistic sampling from a probability den- sity function given in the low-parametric tensor train format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' We tested it on complex multidimen- sional arrays taken, among other, from real-world applications, including unconstrained binary opti- mization and optimal control problems, for which the possible number of elements is up to 2100 ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' In numerical experiments, both on analytic model functions and on complex problems, our algorithm outperform existing popular discrete optimization methods (Particle Swarm Optimiza- tion, Covariance Matrix Adaptation, Differential Evolution and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' Moreover, we take the same set of hyperparameters of our algorithm for all numerical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' Introduction The multidimensional optimization problem is one of the most common in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' It includes the important case of discrete optimization (Parker & Rardin, 2014): min x f(x), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' x ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' , N}d, (1) which occurs when searching for the minimum or maximum element in an implicitly given multidimensional tensor1, Equal contribution 1 Skolkovo Institute of Science and Tech- nology, Moscow, Russia 2 AIRI, Moscow, Russia .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFLT4oBgHgl3EQfwC-M/content/2301.12162v1.pdf'} +page_content=' Correspon- dence to: Anastasia Batsheva