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+Application Of ADNN For Background Subtraction In
+Smart Surveillance System
+University Of Alberta
+Department of Computing Science
+Piyush Batra, Gagan Raj Singh, Neeraj Goyal
+December 7, 2022
+Brief Abstract
+Object movement identification is one of the most researched problems in the
+field of computer vision. In this task, we try to classify a pixel as foreground or
+background. Even though numerous traditional machine learning and deep learning
+methods already exist for this problem, the two major issues with most of them
+are the need for large amounts of ground truth data and their inferior performance
+on unseen videos. Since every pixel of every frame has to be labeled, acquiring
+large amounts of data for these techniques gets rather expensive. Recently, Zhao
+et al. [1] proposed one of a kind Arithmetic Distribution Neural Network (ADNN)
+for universal background subtraction which utilizes probability information from
+the histogram of temporal pixels and achieves promising results. Building onto
+this work, we developed an intelligent video surveillance system that uses ADNN
+architecture for motion detection, trims the video with parts only containing motion,
+and performs anomaly detection on the trimmed video.
+1
+Literature review
+Motion detection aims to find regions related to moving objects, and background subtraction is a
+widely used technique for this task. Herein, every pixel of each video frame is compared against
+their historical counterparts or a background model, depending on the technique, and then
+classified into foreground or background. Pixels that differ significantly from the reference are
+classified as moving objects or foreground, and static pixels are referred to as background. This
+section will discuss some of the previously proposed methods related to background subtraction,
+video surveillance, and anomaly detection.
+Many background modeling techniques based on mathematical theories, like the temporal
+average [2], temporal median [3], or the histogram over time [4], have been proposed for motion
+detection by a stationary camera. But these are not robust to challenges in surveillance videos
+such as dynamic background, object shadow, camera jitter, weather conditions (either snow or
+rain), and variations in illumination. To overcome these problems, several motion-detection
+methods, like Temporal Differencing [5], Three-frame Difference [6], Gaussian mixture model [7],
+1
+
+DSTEI [9], etc, have been presented over the past years.
+Temporal Differencing, proposed by Cheung et al. [5], is used for detecting temporal changes
+in intensity in video frames. However, its main drawback is that the detected objects are
+incomplete and poorly presented.
+In the Gaussian mixture model proposed by Stauffer and Grimson [7], the temporal histogram
+of each pixel is modeled using a mixture of K Gaussian distributions to precisely model a dynamic
+background. This method produced a real-time tracker which can deal with lighting changes,
+repetitive motions from clutter, and long-term scene changes. Later, Chan et al. [8] proposed a
+Generalized Stauffer–Grimson (GSG) algorithm for background subtraction in dynamic scenes.
+In this method, the statistics required for online learning of dynamic texture are derived from
+generalizing the GMM proposed by Stauffer and Grimson [7].
+The Difference-based Spatio-Temporal Entropy Image (DSTEI) by Jing et al. [9] is an entropy-
+based method for human motion detection. A Spatio-temporal histogram is generated by
+accumulated pixels obtained by the difference between consecutive images. This histogram is
+then normalized to calculate the degree of randomness and magnitude of entropy to denote
+the significance of motion. In this method, noises are assumed to follow Gaussian distribution.
+However, these assumptions, such as heavy shadows or sudden illumination changes, will be
+violated in some cases.
+Soumyadip Sengupta et al. [10] proposed a background matting technique that generated
+high-quality foreground and alpha mattes in natural settings. In this method, a deep learning
+framework is developed and trained on synthetic-composite data and then adapted to actual data
+using an adversarial network. Even though providing an additional photo of the background
+requires a small amount of foresight, it is far less tedious than creating a trimap for traditional
+matting methods.
+In 2017, Dan Yang et al. [11] proposed a multi-feature background approach for complex
+video scenes that measures the stability of features and then selects different dominant features
+to model the background from the pixel and time-sequence domains. This Stability of Adaptive
+Features approach showed promising results on both complex and baseline scenes.
+For applications of background subtraction in real time, Z. Kuang et al. [12] proposed a
+combination of the Horn-Schunck optical-flow estimation technique [13] and autoencoder neural
+networks that solve the problem of motion blur in real-time background subtraction during
+video conferencing. This method uses an optical-flow-based model to extract motion features
+between every two frames and then combine these features with the appearance feature from
+the original frame. An encoder-decoder network in combination with CNN is then used to learn
+and predict a mask output for the human head and shoulders for background subtraction.
+Similarly, for real-time background subtraction, DK Yadav et al. [14] proposed a Pixel
+Intensity Based (PIBBS) system that first models the background, then extracts moving objects
+with a threshold and updates the background using a feedback-based background updation
+scheme. To improve the detection quality, this system also uses morphological operators as the
+last step.
+Bruno Sauvalle et al. [15] proposed using an autoencoder to model the background of a
+video as a low-dimensional manifold. The output of this autoencoder is then compared with
+the original image to compute the segmentation masks. In this method, the autoencoder is also
+trained to predict the background noise, which allows it to compute a pixel-dependent threshold
+for each frame to perform the foreground segmentation. Without using temporal or motion
+information, this method could perform at par with state-of-the-art solutions on CDnet 2014 [16]
+2
+
+and LASIESTA [17] datasets.
+To overcome the problem of camera jitter and sudden changes in illumination, Ye Tao et al.
+[18] proposed a generative architecture for unsupervised deep background modeling, which
+learns the parameters automatically and uses intensity and optical flow features between a
+reference and a target frame. This system generates a background with a probabilistic heat map
+of the color values for a given input frame. This method could also be applied to unseen videos
+without re-training. When tested, this method shows promising results over state-of-the-art
+[19][20][21] methods on the SBMnet dataset[22].
+Guanfang Dong et al. [23] proposed a novel denoising neural network model called Feature-
+guided Denoising Convolutional Neural Network (FDCNN) to denoise the images produced
+by portable devices. This technique employed a hierarchical denoising framework driven
+by a feature masking layer. The feature extraction algorithm used in this method is based
+on Explainable Artificial Intelligence (XAI) for medical images. Similarly, Yingnan Ma et al.
+[24] proposed an Edge-guided Denoising Convolutional Neural Network which can preserve
+important edge information in ultrasound images when removing noise. This method increases
+the recognition of various organs in ultrasound images.
+Jhony H. Giraldo et al. [25] proposed a new algorithm called Graph Background Subtraction
+(GraphBGS). It is composed of instance segmentation, background initialization, graph construc-
+tion, and graph sampling. Unlike Deep Learning methods for background subtraction which
+require vast amounts of data, this method is a semi-supervised algorithm inspired by the theory
+of recovery of graph signals.
+To generate descriptions of human actions and their interactions, Zijian Kuang et al. [26]
+proposed a technique that utilizes an Actor Relation Graph (ARG) based model with novel
+improvements for group activity recognition. This method also used MobileNet as the backbone
+to extract features from each video frame.
+To accurately perform background subtraction in a freely moving camera, Zhao et al. [27]
+developed a novel method called “the integration of foreground and background cues.” The
+underlying motivation in this technique is to utilize the exclusiveness between these cues to
+compensate for their corresponding defects. The foreground is segmented by combining super-
+pixels with proximity under multiple levels.
+As video resolution and, subsequently, the video size is increasing daily, Ruixing et al. [28]
+proposed a method to compute the optimal image resolution adaptively. This is achieved by
+exploiting the correlation between an image’s gray-value distribution and resolution. This
+approach was proposed to increase the performance of multi-object online tracking and learning.
+A novel tracklet reliability assessment metric was also introduced in this paper to eliminate the
+incorrect samples and can recover occluded targets.
+As a unique application of neural networks in multimedia, C. Sun et al. [29] proposed a 2-step
+product re-identification (Re-ID) method which involves image feature extraction and a feature
+search and retrieval engine. To extract the features of the input image, a novel AlphaAlexNet, an
+extended version of the AlexNet, is being used. Vearch, a visual search system, is used as the
+image search similarity engine. The new model - AlphaAlexNet, demonstrated improved object
+detection accuracy of Vearch.
+To classify two distributions without using just histograms and incorporating a deep learning
+network to learn and classify distributions automatically, Chunqiu Zhao and Anup Basu [30]
+proposed a novel vessel segmentation method based on distribution learning using a spatial
+distribution descriptor (RPoSP) under multiple scales. Here, statistical distributions are indirectly
+forced as an input to a CNN for distribution learning. The proposed approach showed promising
+3
+
+results when compared to existing state-of-the-art methods[31][32] on the DRIVE[33] dataset.
+Yongxin Ge et al. [34] proposed the Deep Variation Transformation Network (DVTN) model,
+which uses pixel variations to detect the background. This model assigns the probability to
+each pixel, and then by using thresholding, it computes whether it’s background or foreground.
+This model compares the pixel variation instead of distributions. Previously used models
+in background detection usually fail when they encounter similar observations, causing false
+detections. The DVTN analyzes the pixel variations in a new space, where the above observations
+are classified easily. This model outperforms the traditional background detection models by
+showing astonishing results on the CDnet2014 dataset.
+However, all of the methods mentioned above require either a large amount of ground truth
+data or result in inadequate performance on unseen videos. Zhao and Basu [35] proposed a
+Deep Pixel Distribution Learning (DPDL) technique to overcome these issues. Unlike typical
+approaches, which compare new frames to a formulated background model, this technique
+focuses on comparing pixels’ current and historical frames. This method uses a novel pixel-
+based feature called the Random Permutation of Temporal Pixels (RPoTP) to represent the
+distribution of past observations for a particular pixel. Subsequently, a CNN is used to learn
+whether the current pixel is foreground or background. Adding on to this method, Zhao et al.
+[36] later proposed a new Dynamic Deep Pixel Distribution Learning (D-DPDL) technique. In
+this method, the RPoTP feature is dynamically permuted in this method for every training epoch.
+To compensate for the random noise generated in this process, a Bayesian Refinement model is
+used and improve the accuracy.
+Zhao et al. [1] also proposed an Arithmetic Distribution Neural Network architecture demon-
+strating even better performance than the D-DPDL method. The input in the ADNN method is
+histograms of subtractions between current pixels and their historical counterparts. The sum
+and product arithmetic distribution layers proposed here demonstrate a better ability to clas-
+sify distributions than the convolutional layers in D-DPDL. Moreover, the number of learning
+parameters used in ADNN architecture (0.1 Million) is significantly less than that used in the
+D-DPDL method (7 Million).
+Coming onto detecting anomalies in videos, Virender Singh et al. [37] proposed an approach
+to detect variation from the norm in real-world CCTV recordings. This method uses two deep
+learning models (CNN and RNN) to learn a general anomaly detection model with a poorly
+labeled dataset. The training dataset has been doubled by flipping the videos horizontally, thus
+increasing the testing accuracy. The overall accuracy of the model is 97.23
+Y Fan et al. [38] proposed a technique that first converts the video clips of an ongoing event
+into Dynamic Images, which can simultaneously capture the appearance and temporal evolution
+of the occurrence. The approach uses dynamic images of two categories of video clips and
+involves training a detector based on deep-learning techniques.
+Yu Tian et al. [39] proposed a weakly-supervised anomaly detection algorithm, Robust
+Temporal Feature Magnitude learning (RTFM), aiming to identify snippets containing abnormal
+events. This method trains a feature magnitude learning function to effectively recognize the
+positive instances, substantially enhancing the robustness of this method to the negative instances
+from abnormal videos. RTFM achieves significantly improved subtle anomaly discriminability
+and sample efficiency.
+The Weakly Supervised Video Anomaly Detection(WSVAD) [40]-[42] method for anomaly
+detection suffers from the wrong identification of normal and abnormal instances during the
+training process. Kapil Deshpande et al. [43] proposed better-quality transformer-based features
+named Videoswin Features, followed by an attention layer to capture long and short-range
+4
+
+dependencies in the temporal domain. This method extracts better-quality features from available
+videos resulting in better performance.
+2
+Method
+In this work, we implemented an Arithmetic Distribution Neural Network [1] to develop a video
+surveillance system for identifying object movement in a static video. In this ADNN model,
+the arithmetic operations are utilized to introduce the arithmetic distribution layers, including
+the product and sum distribution layers. Outputs from these layers are combined and passed
+through a classifier for accurate classification. We chose this architecture because it requires
+training only one network, with limited training data, and it works well with unseen test videos.
+Figure 1: The flow diagram of our proposed approach
+Upon successful object movement detection using background subtraction, we further ana-
+lyzed the results obtained from ADNN to filter out their anomalous activities.
+2.1
+Motion detection - Arithmetic Distribution Neural Network
+In this work, we used ADNN proposed by Zhao et al. [1] to detect motion in the input surveil-
+lance video. This paper proposed arithmetic distribution layers, which are a new type of network
+layer that is designed to improve distribution analysis in classification tasks. These layers, which
+include product and sum distribution layers, are an alternative to convolution layers. During
+the forward pass of the proposed arithmetic distribution layers, the input distributions are
+processed using the distributions in the learning kernels to generate the output distributions.
+In the backpropagation process, the gradient of the distributions in the learning kernels with
+respect to the network output is calculated to update the learning kernels. These operations are
+based on histograms and arithmetic distribution operations rather than the matrix arithmetic
+operations used in traditional convolution layers.
+To improve the accuracy of the foreground mask generated, an improved Bayesian refinement
+model is used. This model takes into account the correlations between pixels by using a mixture
+of Gaussian approximation functions rather than just Euclidean distance, as in the original
+Bayesian refinement model. The Bayesian refinement model is used to iteratively refine the
+foreground mask, with the output of the arithmetic distribution neural network serving as the
+initial binary mask for the iteration process.
+5
+
+Sum Distribution
+Layer
+Convolution
+Classifier
+Anomalydetection
+ramework
+Basedonpixe
+classification
+A Set of trimmed videos
+Input Video
+with activity
+A Set of anomaly
+detected videos
+Product
+Distribution LayerFigure 2: Arithmetic distribution neural network for background subtraction
+After obtaining the refined foreground masks from the ADNN architecture, we utilize a
+python script to generate a trimmed video from a set of input frames by using a threshold value
+on the frames generated by the Bayesian refinement model. The threshold value determines the
+minimum number of white pixels (foreground pixels) that must be present in a frame in order
+for it to be included in the trimmed video. For this work, we are using a threshold value of 5% to
+generate the trimmed videos.
+Figure 3: Generation of trimmed video from input frames passed through the ADNN (arithmetic
+distribution neural network)
+2.2
+Anomaly Detection
+Following the works of Waqas Sultani et al. [44], we have put into use their novel Multiple
+Instance Learning framework for the second part of our system. Once we obtain the trimmed
+video from the previous step, we use that as the input in this step. In this, a training set of
+positive (containing an abnormality someplace) and negative (having no anomaly) videos are
+used to train the anomaly detection model. Then each video is divided into a sequence of
+non-overlapping temporal segments.
+6
+
+Convolution,
+Full Connection
+Relu
+Convolution
+Product
+Distribution Layer
+Foreground
+Size:B × 3 × 201 × 2
+Background
+Size:B × 3 × 201 × 1
+Size:B × 2×1 × 1
+Size:B × 10 × 201 × 1
+Sum Distribution
+Layer
+Size:B × 512 × 1 × 1
+Datadimensions:BatchSize×Channels×Width×Height
+Size:B × 3 × 201 × 2Frames from ADNN Output
+CombinedTrimmed Videc
+Motion Video1
+Motion Video 2
+Framesfrom inputvideo
+Input VideoFigure 4: Anomaly detection flow diagram
+Each video in the training set can be represented as a bag, and each video segment represents
+an instance in the bag. After extracting C3D features from video segments using a pre-trained
+3D convNet, a fully connected neural network is trained using the novel ranking loss function; it
+computes the ranking loss between the top-rated occurrences in the positive bag and the negative
+bag.
+In conclusion, the proposed method for detecting anomalies in surveillance videos consists
+of two main steps. First, the ADNN architecture is used to detect motion in the input video
+and generate a refined foreground mask. This mask is then used to create a trimmed video,
+which is used as input for the second step of the system. In this step, we used a pre-trained
+multiple instance learning model trained on a set of positive and negative videos and used to
+classify each temporal segment in the test video as normal or anomalous. The predicted scores
+for each segment are then combined to generate a prediction (anomaly graph) for the entire
+video. By combining these two approaches, the system is able to effectively detect abnormalities
+in surveillance videos, even when they only occur for a short period of time or are only present
+in a small number of segments.
+3
+Results
+In this section, we will discuss our experimental results for two different videos. Table 1 compares
+the full video and trimmed video for two different videos, labeled Video 1 and Video 2. For
+Video 1, the full video had a duration of 06:37 minutes, a size of 90.5 MB, and contained 11937
+frames. The anomaly detection process for this video took 789 seconds. The trimmed video
+for Video 2 had a duration of 04:09 minutes, a size of 68.5 MB, and contained 7470 frames. The
+anomaly detection process for this video took 540 seconds, which is lower than the time taken
+for the full video.
+For Video 2, the full video had a duration of 04:59 minutes, a size of 40.6 MB, and contained
+8990 frames. The anomaly detection process for this video took 610 seconds. On the other hand,
+the trimmed video for Video 2 had a duration of 1:04 minutes, a size of 10.2 MB, and contained
+1950 frames. The anomaly detection process for this video took 137 seconds, which is also lower
+than the time taken for the full video.
+7
+
+Positive bag
+Instance scores in positive bag
+Anomaly video
+Bag instance (video segment)
+4096
+MIL Ranking Loss with sparsity
+ and smoothness constraints
+609
+512
+N
+C3D feature extraction
+32
+for each video segment
+0.1
+...
+32 temporal segments
+32 temporal segments
+(anomaly score)
+...
+pre-trained 3D ConvNet
+Normal video
+Negative bag
+Instance scores in negative bagThe graphs obtained after anomaly detection are shown below. These are the relative anomaly
+scores of each video segment (32 in this case). We can see that the anomalous regions in the
+trimmed video are more focused, and there are comparatively fewer inactive regions. Moreover,
+the overall structure of the graphs is similar for both the original and trimmed videos, indicating
+that trimming down the video does not affect the anomaly identification and the relative scores
+of different segments.
+Figure 5: The graphs indicate anomaly scores of the video2 (left) and its trimmed version (right)
+Overall, the results in Table 1 show that the trimmed videos had shorter durations and
+smaller sizes compared to the full videos. Additionally, the anomaly detection process for the
+trimmed videos took much less time than the full videos in both examples. This suggests that
+using trimmed videos leads to a more efficient anomaly detection process.
+4
+Discussion
+The results presented above demonstrate the effectiveness of our ADNN-based video surveillance
+system in identifying object movement and filtering out anomalous activities. As shown, the
+trimmed videos had shorter durations, smaller sizes and required less time for anomaly detection
+compared to the full videos in both examples. This suggests that the ADNN model and the use
+of trimmed videos lead to a more efficient and effective video surveillance system. Additionally,
+the ADNN model we employed has the advantage of requiring only limited training data and
+8
+
+Duration
+Size
+Frames
+Anomaly
+(mm: ss)
+(MB)
+Detection
+(cpu - sec)
+Full Video
+06:37
+90.5
+11937
+789
+Video 1
+Trimmed Video (Combined)
+04:09
+68.5
+7470
+540
+Full Video
+04:59
+40.6
+8990
+610
+Vide02
+Trimmed Video (Combined)
+1:04
+10.2
+1950
+137
+Table I: Comparison of results for trimmed and full-length videos0.40 
+0.35 
+0.8
+0.30 
+0.25 
+0.6 
+0.20 
+0.4 
+0.15
+0.10 
+0.2
+0.05 
+0.00 
+0.0 
+5
+10
+15
+20
+25
+30
+n
+10
+15
+20
+25
+30being able to perform well with unseen test videos. This makes it a suitable choice for practical
+implementation in real-world scenarios.
+In conclusion, our ADNN-based video surveillance system has demonstrated its ability to
+accurately detect object movement and filter out anomalous activities, making it a promising
+solution for video surveillance applications.
+5
+Future Work
+In the future, we plan to work on making the ADNN model more efficient at inferring foreground
+masks, as it currently takes a significant amount of time to process videos. This will be a major
+challenge, but we believe it is necessary in order to make the system more practical and useful in
+real-world scenarios.
+Additionally, we will work on generating a better test dataset to further evaluate the adapt-
+ability of this system. This will help us to better understand the limitations and potential
+improvements of the system. Overall, the goal would be to improve the efficiency of the ADNN
+model in order to make it a useful tool for video surveillance and anomaly detection applications.
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+

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+Training a Deep Q-Learning Agent Inside a
+Generic Constraint Programming Solver
+Tom Marty1,2, Tristan François2, Pierre Tessier2, Louis Gautier2,
+Louis-Martin Rousseau1, and Quentin Cappart1
+1 Polytechnique Montréal, Montreal, Canada
+{tom.marty,louis-martin.rousseau,quentin.cappart}@polymtl.ca
+2 Ecole Polytechnique, Palaiseau, France
+{tristan.francois,pierre.tessier,louis.gautier}@polytechnique.edu
+Abstract. Constraint programming is known for being an efficient ap-
+proach to solving combinatorial problems. Important design choices in
+a solver are the branching heuristics, designed to lead the search to the
+best solutions in a minimum amount of time. However, developing these
+heuristics is a time-consuming process that requires problem-specific ex-
+pertise. This observation has motivated many efforts to use machine
+learning to learn efficient heuristics without expert intervention automat-
+ically. To our knowledge, it is still an open research question. Although
+several generic variable-selection heuristics are available in the literature,
+the options for a generic value-selection heuristic are more scarce. In this
+paper, we propose to tackle this issue by introducing a generic learning
+procedure that can be used to obtain a value-selection heuristic inside
+a constraint programming solver. This has been achieved thanks to the
+combination of a deep Q-learning algorithm, a tailored reward signal, and
+a heterogeneous graph neural network architecture. Experiments on graph
+coloring, maximum independent set, and maximum cut problems show
+that our framework is able to find better solutions close to optimality
+without requiring a large number of backtracks while being generic.
+Keywords: Constraint Programming · Reinforcement Learning · Graph
+Representation Learning
+1
+Introduction
+Combinatorial optimization has countless industrial applications, such as schedul-
+ing, routing, or finance. Unfortunately, most of these problems are NP-hard and,
+thereby, challenging to solve efficiently in practice. It is why finding good solu-
+tions have motivated intense research efforts for many years. Traditional methods
+for tackling them are somehow based on a search procedure: A clever enumer-
+ation of the solution space is performed to find a feasible and possibly optimal
+solution. Among these methods, constraint programming (CP) is an exact algo-
+rithm. It constitutes a popular approach as they offer the possibility to find the
+optimal solution or good feasible approximations by stopping the search early.
+arXiv:2301.01913v1  [cs.AI]  5 Jan 2023
+
+2
+T. Marty et al.
+An additional asset is its declarative paradigm in modeling, which makes the
+technology easier for the end user to grasp. This aspect has been greatly facili-
+tated by introducing solver-agnostic modeling languages, such as MiniZinc [30].
+Aligned with this goal, the propagation engine inside a CP solver is mostly hid-
+den from the end user. However, ensuring a generic search procedure is trickier
+as non-trivial heuristics must be designed to make the solving process efficient
+for an arbitrary problem. That being said, generic variable-selection heuristics
+(i.e., selecting the next variable to branch on) have been successfully designed.
+Notable examples are impact-based search [32] or activity-based search [26]. On
+the other hand, there is no similar generic heuristic for the value-selection (i.e.,
+selecting the next value to branch one). As a concrete example, the current
+version of MiniZinc3 does not propose generic value-selection heuristics, except
+in(out)domain. In practice, this heuristic is often designed thanks to problem-
+specific expert knowledge, which is often out of reach for end-users that do not
+have a solid background in artificial intelligence.
+In another context, machine learning (ML) has been recently considered
+for automating the design of branching heuristics, both in constraint program-
+ming [11], mixed-integer programming [14,20], or SAT solving [35]. Specifically,
+reinforcement learning (RL) [38] or imitation learning [19] approaches, often
+combined with deep learning [23], have gained special attention. Although this
+idea seems appealing, this is not an easy task to achieve in practice as several
+technical considerations must be taken into account in order to ensure both the
+efficiency and the genericity of the approach. In constraint programming, we
+identified three questions to resolve when learning a generic branching heuristic
+inside a solver. They are as follows:
+1. How to train the machine learning model? An intuitive way is to leverage an
+RL agent that would explore the tree search by making branching decisions
+and rewarding it based on the quality of the solution found on a terminal
+node. For getting a certificate of optimality, this would typically be done with
+a depth-first search traversal of the tree. However, as pointed out by several
+authors [33,36], the backtracking operations inside a solver raise difficulties
+when formalizing the task as a Markov decision process and may require
+redefining it. Besides, this training scheme intensifies the credit assignment
+problem [27], which is ubiquitous in reinforcement learning.
+2. How to evaluate the quality of a value selection? A core component of an
+RL environment is the reward function, which gives a score to each decision
+performed. The end goal for the agent is to perform a sequence of decisions
+leading to the best-accumulated sum of rewards. In our case, an intuitive
+solution would be to reward the agent according to the quality of the solution
+found. However, this information is only available at terminal nodes, and
+only a zero reward is provided in branching nodes. This is related to sparse
+reward problematic, which is known to complicate the training process.
+3. How to learn from a CP model? This question relates to the type of archi-
+tecture that can be used to obtain a value-selection heuristic from a search
+3 https://www.minizinc.org/doc-2.5.5/en/lib-stdlib.html
+
+Training a DQN Agent Inside a Generic CP Solver
+3
+node (i.e., a partially solved CP model). A promising direction has been
+proposed by Gasse et al. [14] for binary mixed-integer programs. They intro-
+duced a bipartite graph linking variables and constraints (i.e., the two types
+of nodes) when a variable is involved in a given constraint. The subsequent
+architecture is a heterogeneous graph neural network. However, this encod-
+ing is not directly applicable in constraint programming, as a CP model
+generally involves non-binary variables and combinatorial constraints. This
+has been partially addressed by Chalumeau et al. [12], who introduced a
+tripartite graph where variables, values, and constraints are specific types
+of nodes. Their approach, however, suffers from a lack of genericity as their
+method requires retraining when the number of variables changes.
+To our knowledge, answering such questions is still an open challenge in the
+research community. This paper proposes to progress in this direction. It intro-
+duces a generic learning procedure that can be used to obtain a value-selection
+heuristic from a constraint programming model given as input. The approach
+has been designed to be generic in that it can be used in whatever the CP model
+is. In practice, a specific way to extract features from a constraint should be
+designed for any available constraint, but this has to be done only once per
+constraint type. In this proof of concept, we limit our experiments to three com-
+binatorial optimization problems, namely graph coloring, maximum independent
+set, and maximum cut. Specifically, we propose three main contributions, each
+dedicated to addressing one of the aforementioned difficulties. They are as fol-
+lows: (1) a learning procedure, based on restarts, for training a reinforcement
+learning agent directly inside a CP solver, (2) a reward function able to assign
+non-zero intermediate rewards based on the propagation that has been carried
+out on the node, and (3) a neural architecture based on a tripartite graph and a
+heterogeneous graph neural network. Experimental results show that combining
+these three ideas enables the search to find good solutions without requiring
+many backtracks. As shown by limiting the search tree’s size with a budget.
+The paper is structured as follows. The next section presents other ap-
+proaches related to our contribution. Then, Section 3 introduces succinctly tech-
+nical background on reinforcement learning and graph neural networks. The core
+contribution is then presented in Section 4. Finally, Section 5 provides experi-
+mental results and closes with a discussion of the results and of some limitations.
+2
+Related Work
+Bengio et al. [5] identified three ways to leverage machine learning for com-
+binatorial optimization. First, end-to-end learning aims to solve the problem
+only with a trained ML model. This has been, for instance, considered for the
+traveling salesman problem [4,21]. However, such an approach does not guar-
+antee the validity nor the optimality of the solution returned. Second, learning
+to configure is dedicated to providing insights to a solver before its execution.
+This can be, for instance, the decision to linearize the problem in the context
+of quadratic programs [7] or to learn when a decomposition is appropriate [22].
+
+4
+T. Marty et al.
+This approach is also referred to as parameter tuning [18]. We refer to the initial
+survey for extended information about these two families of approaches. Third,
+learning within a search procedure uses machine learning within the solver. Our
+contribution belongs to this last category of methods. Although the idea of com-
+bining learning and searching for solving combinatorial optimization problems
+was already discussed in the nineties [31], it has re-emerged only recently with
+the rise of deep learning. Most combinatorial optimization solvers are based
+on branch-and-bound and backtracking. In this context, ML is often used with
+branching rules to follow. Imitation learning [19] has been for instance used to
+replicate the expensive strong branching strategy for mixed-integer programming
+solvers [14,20]. One limitation of imitation learning is that the performances are
+bounded by the relevance of the imitated strategy, which remains heuristic and
+perfectible [37]. This opens the door for RL approaches that have the guarantee
+to find the best branching strategy [25] eventually. A branching strategy can be
+split into two challenging decisions, variable selection, and value selection. Both
+of them have been addressed by reinforcement learning approaches.
+Concerning the learning for selecting the next variable to branch on, Song
+et al. [36] propose to combine a double deep Q-network algorithm [40] with a
+graph neural network for carrying out this task. The approach is trained to
+minimize the expected number of nodes to reach a leaf node using the first-fail
+principle. Although this is a good proxy for pruning a maximum of infeasible
+solutions for a constraint satisfaction problem, it does not extend naturally to
+optimization variants, for which one should consider a trade-off between the
+quality of the solution found and the number of nodes required to reach that
+solution. For the value-selection heuristic, Cappart et al. [11] proposed to train
+a model with reinforcement learning outside the CP solver and to integrate
+the agent, once trained, subsequently in the solver. This has been achieved by
+reaping the benefits of a dynamic programming formulation of a combinatorial
+problem. An important limitation of this work is that no information related to
+the CP solver, such as the propagation achieved on a node, can be used to drive
+the decision. Chalumeau et al. [12] proposed to carry out the learning inside the
+solver. The model is trained to find the optimal solution and to prove it with
+the least number of search nodes possible. However, this goal is disconnected
+from finding the best solution as quickly as possible and is practically hard
+to achieve, even with a good heuristic. A more realistic goal is to find a good
+solution quickly without closing the search. This is how the contribution of this
+paper is positioned.
+We would like to point out that learning how to branch is not the only way to
+leverage ML inside a combinatorial optimization solver. Related works have also
+been proposed on learning tight optimization bounds [8,10] or for accelerating
+column generation approaches [29]. A recurrent design choice is an architecture
+based on graph neural networks. We refer to the following survey for more in-
+formation about combinatorial optimization with graph neural networks [9].
+
+Training a DQN Agent Inside a Generic CP Solver
+5
+3
+Technical Background
+3.1
+Reinforcement Learning
+Let ⟨S, A, T, R⟩ be a 4-tuple representing a Markov decision process where S is
+the set of states in the environment, A is the set of actions that the agent can
+do, T : S × A → S is a transition function leading the agent from one state
+to another, given the action taken, and R : S × A → R is a reward function
+of taking action from a specific state. The sequence [s1, . . . , sT ] from the initial
+state (s1) of an agent towards a terminal state (sT ) is referred to as an episode.
+The returned reward within a partial episode [st, . . . , sT ] can be formalized as
+follows: Gt = �T
+i=t R(si, ai). We intentionally omitted the discounting factor
+as we do not want to discount the late rewards in our application. The agent
+is governed by a policy π : S → A, which indicates the action that must be
+taken on a given state. The agent’s goal is to find the policy that will lead
+it to maximize the accumulated reward until a terminal state is reached. The
+core idea of reinforcement learning is to determine this policy by letting the
+agent interact with the environment and by increasing the probability of taking
+action if it leads to high amounts of subsequent rewards. There are a plethora
+of reinforcement learning algorithms dedicated to this task, such as trust region
+policy optimization [34] or soft actor-critic [15]. We refer to SpinningUp website
+for explanations of the main algorithms [1].
+This section presents the core principles of deep Q-learning [28], which is
+the algorithm used in this paper. The idea of this algorithm is to compute
+an action-value function Qπ(st, at) = Gt. Intuitively, this function gives the
+accumulated reward that the agent will obtain when performing the action a
+at state s while subsequently following a policy π. The output of this func-
+tion for a specific action is referred to as a Q-value. Provided that the action-
+value function can be computed exactly, the optimal policy π⋆ turns to be sim-
+ply the selection of the action having the highest Q-value on a specific state:
+π∗ = argmaxπQπ(s, a), ∀(s, a) ∈ (S, A). Although the exact computation of Q-
+values can theoretically be performed, a specific value must be computed for each
+pair of states and actions, which is not tractable for realistic situations. It is why
+a tremendous amount of work has been carried out to approximate accurately
+and efficiently Q-values. Among them, deep Q-learning aims to provide a neural
+estimator ˆQ(s, a, θ) ≈ Q(s, a), where θ is a tensor of parameters that must be
+learned during a training phase. This algorithm is commonly enriched with other
+mechanisms dedicated to speed-up or stabilizing the training process, such as
+the double deep Q-network variant [40]. Concerning the neural architecture, we
+opted for a graph neural network, which is explained in the next section.
+3.2
+Graph Neural Network
+Intuitively, the goal of a graph neural network (GNN) is to embed information
+contained in a graph (e.g., the structure of the graph, spatial properties, features
+of the nodes, etc.) into a d-dimensional tensor for each node u ∈ V of the graph.
+
+6
+T. Marty et al.
+To do so, information on a node is iteratively refined by aggregating information
+from neighboring nodes. Each iteration of aggregation is referred to as a layer
+of the GNN and involves parameters that must be learned. Let hk
+u ∈ Rd×1 be
+the tensor representation of node a u at layer k of the GNN, hk+1
+u
+∈ Rl×1 be
+the tensor representation of this node at the next layer (l being the dimension
+of a node at the layer k + 1), and θ1 ∈ Rl×d and θ2 ∈ Rl×d be two matrices of
+parameters, respectively. Each GNN layer carries out the following update:
+hk+1
+u
+= g
+�
+θ1hk
+u ⋆ (
+�
+v∈N(u)
+θ2hk
+v)
+�
+∀u ∈ V
+(1)
+Three operations are involved in this update: (1) � is an aggregation operator
+that is dedicated to aggregating information of neighbors (e.g., mean-pooling or
+sum-pooling), (2) ⋆ is a merging which enables to combine of the information
+of a node with the ones from the neighbors (e.g., a concatenation), and (3) g is
+an element-wise non-linear activation function, such as the ones commonly used
+in fully connected neural networks (e.g., ReLU). Without loss of generality, the
+bias term is not included in the equation. A concrete implementation of a GNN
+turns out to define these three functions adequately. The training is carried out
+in a fully connected neural network through back-propagation and an optimizer
+based on gradient descent.
+4
+Learning a Value-Selection Heuristic Inside a Solver
+This section presents how a value-selection heuristic can be learned with re-
+inforcement learning in a CP solver from a model given as input. This is the
+core contribution of this paper. Three mechanisms are introduced: (1) a train-
+ing procedure based on restarts, (2) a reward function leveraging propagation of
+domains, and (3) a heterogeneous graph neural network architecture. They are
+described individually in the next subsections. They have been implemented in
+recently introduced SeaPearl.jl solver [12]. The main specificity of this solver
+is to natively integrate support for learning inside the search procedure. This
+greatly facilitates the prototyping of new search algorithms based on learning.
+4.1
+Restart-Based Training
+Generally speaking, the performance of a reinforcement learning agent is tightly
+correlated with the definition on an episode. This corresponds to the agent’s
+interactions with the CP solver’s search procedure and is related to the goal
+desired for the agent. Two options are discussed in this section, (1) an episode
+based on depth-first search, which has been introduced by Chalumeau et al. [12],
+and (2) an episode based on restarts is one contribution of the paper.
+Building branching heuristics for solving exact combinatorial optimization
+problems often concurrently targets two objectives: finding quickly good solu-
+tions and proving the optimality of a solution. The approach of Chalumeau et
+
+Training a DQN Agent Inside a Generic CP Solver
+7
+al. [12] relies heavily on the second objective and aims to minimize the number
+of visited search nodes before proving optimality (e.g. closing the search). To do
+so, they defined a training episode as a complete solving process carried out by
+the depth-first search of a solver and penalized through the reward function the
+generation of each node. This is illustrated in the left picture of Fig. 1. However,
+this approach suffers from an important difficulty. An episode only terminates
+when the search is completed, which is often intractable for realistic problems as
+it requires exploring an exponentially large search tree. This is especially prob-
+lematic during the training phase, where the heuristic is still mediocre. This has
+also been pointed out by Scavuzzo et al. [33] for mixed-integer programming.
+Fig. 1: The two training procedures (left: depth-first search, right: restart-based)
+Unlike this approach, we propose to train the model to find good solutions
+quickly. To do so, we followed the approach proposed by Cappart et al. [11]:
+an episode is defined as a diving heuristic. No backtrack is allowed; the episode
+stops when a complete solution is found or when a failure is generated. Once
+the episode is terminated, a restart from the root node is performed, and a
+new episode is generated, whereas the name of restart-based episode. This is
+illustrated in the right picture of Fig. 1. One limitation of Cappart et al. [11] is
+that episodes are executed outside the CP solver during the training and are then
+unable to use information based on the propagation for the branching. Inspired
+by Song et al. [36] for variable-selection heuristics, we addressed this limitation
+by executing each episode inside the solver during the training. Formally, this
+requires defining the dynamics of the environment as a Markov Decision Process
+(i.e., a tuple ⟨S, A, T, R⟩, see Section 3.1).
+Set of states Let P = ⟨X, D(X), C, O⟩ be the expression of a combinatorial
+optimization problem (COP), defined by its variables (X), the domains (D),
+its constraints (C), and an objective function (O). Each state st ∈ S is
+defined as the pair st = (Pt, xt), where Pt is a partially solved COP (i.e.,
+some variables may have been assigned), and xt ∈ X is a variable selected for
+branching, at step t of the episode. The initial state s1 ∈ S corresponds to
+the situation after the execution of the fix-point. A terminal node is reached
+either if all the variables are assigned (∀x ∈ X : |Dt(x)| = 1), or if a failure
+
+Decision branching
+Back-tracking
+Reward Signal
+End of the episode8
+T. Marty et al.
+is detected (∃x ∈ X : |Dt(x)| = 0). The variable selected for branching is
+obtained through a standard heuristic such as first-fail.
+Set of actions Given a state st = (Pt, xt), an action at corresponds to the
+selection of a value v ∈ D(xt) for branching at step t. Finding the most
+promising value to branch on is the problem addressed in this paper.
+Transition function Given a state st = (Pt, xt) and an action at = v, the
+transition function executes three successive operations. First, it assigns the
+value v to the variable x (i.e., D(xt+1) = v). Second, it executes the fix-
+point on Pt in order to prune the domains (i.e., Pt+1 = fixPoint(Pt)). Third,
+it selects the next variable to branch on (i.e., xt+1 = nextVariable(Pt)). This
+results in a new state st+1 = (Pt+1, xt+1). Integrating the propagation inside
+the transition is the main difference from the work of Cappart et al. [11].
+Reward function The function is defined separately in Section 4.2.
+Concerning the training, we opted for a double deep Q-learning algorithm,
+known to perform well for discrete action space, but other RL algorithms could
+also be used. Finally, we compared both training procedures for the maximum
+independent set problem with instances with 50 nodes using performance pro-
+files [13]. The ratio is computed using the optimal solution as a reference. As a
+non-learned baseline, we added the performances of an agent performing only
+random decisions. Training is carried out on randomly generated Barabási-Albert
+graphs [2]. Evaluation is performed on 20 other graphs following the same dis-
+tribution. A detailed explanation of the experimental protocol is proposed in
+Section 5. Figure 2 shows performance profiles. As expected, we observe that
+our new agent (single dive learning) can obtain better solutions quickly, with a
+comparable ability to prove optimality compared to Chalumeau et al. [12].
+(a) Value of the solution obtained.
+(b) Node visited until optimality.
+Fig. 2: Comparison of both training methods on max. independent set (50 nodes).
+4.2
+Propagation-Based Reward
+The goal of the reward is to lead the agent to good solutions to the combinato-
+rial problem. Based on our training procedure, an intuitive function is to reward
+
+1.0
+20 instances
+0.8
+0.6
+0.4
+Single Dive Learning
+0.2
+DFS-based Learning
+Random
+0.0
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+Within this factor of the best score1.0
+20 instances
+0.8
+0.6
+Proportion of the
+0.4
+Single Dive Learning
+0.2
+DFS-based Learning
+Random
+0.0
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+Within this factor of the smallest number of node visitedTraining a DQN Agent Inside a Generic CP Solver
+9
+the agent proportionally to the quality of the solution found at the end of an
+episode. In case of an infeasible solution is found, a penalty can be given. The
+main drawback of this rewarding scheme is that this information is only available
+at terminal nodes, and only a zero reward is provided in branching nodes. This is
+related to the sparse reward problem, which is known to complicate the training
+process [39]. To address this difficulty, we propose a new rewarding scheme based
+on the domain reduction of the objective variable (i.e., the variable that must be
+minimized or maximized). This happens either thanks to the branching assign-
+ment or the application of the fix-point. There are two main components: (1)
+an intermediate reward (rmid) collected at branching nodes, and (2) a terminal
+reward (rend) collected only at the end of an episode.
+Fig. 3: Intermediate reward when four values are pruned from the domain.
+Assuming a minimization problem, the intermediate reward follows two prin-
+ciples: each domain reduction of the largest values of the domain is rewarded,
+and each domain reduction of the lowest values of the domain is penalized. The
+rationale is to lead the agent to a situation where the minimum cost can be even-
+tually obtained while removing costly solutions. It is formalized in Equations (2)
+to (4), where rmid
+t
+is the reward obtained at step t, and is illustrated in Fig 3. As
+shown in Equation 5, the terminal reward is set to -1 if the leaf node corresponds
+to an infeasible solution and 0 if it is feasible. Finally, the total reward (racc)
+accumulated during an episode of T steps is the sum of all intermediate rewards
+with the final term, as proposed in Equation (6).
+rub
+t
+= #
+�
+v ∈ Dt(xobj)
+��� v /∈ Dt+1(xobj) ∧ v > max
+�
+Dt(xobj)
+��
+(2)
+rlb
+t = #
+�
+v ∈ Dt(xobj)
+��� v /∈ Dt+1(xobj) ∧ v < min
+�
+Dt(xobj)
+��
+(3)
+rmid
+t
+= rub
+t − rlb
+t
+��D1(xobj)
+��
+(4)
+rend
+t
+= −1 if unfeasible solution found (0 otherwise)
+(5)
+racc =
+T −1
+�
+t=1
+rmid
+t
++ rend
+T
+(6)
+An experimental analysis of this new reward scheme (propagation-based re-
+ward) is carried out for the maximum cut, graph coloring, and maximum inde-
+pendent set problems. As a baseline, we consider a reward (score reward) that
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+D+
+1
+2
+3
+4
+5
+6
+7
+↓
+3-1
+2
+D++1
+2
+3
+4
+10
+1010
+T. Marty et al.
+only gives a value at terminal nodes (rend
+T ) without an intermediate reward. Be-
+sides, we present the values of the optimal solution and the solutions obtained
+by a random value-selection heuristic. Fig. 4 shows the evolution of the objective
+value (y-axis, averaged on 20 instances of the validation step) with the training
+time (number of episodes in the x-axis). Instances are Barabási-Albert randomly
+generated graphs with 50 nodes. Except for the reward scheme, the other parts
+of the architecture are unchanged. We observe that the propagation-based reward
+provides a more stable training (Fig. 4a) and can converge to a better model or,
+at least, to an equally good model as the sparse reward (Figs. 4b and 4c).
+(a) Graph coloring.
+(b) Maximum cut.
+(c) Max. independent set.
+Fig. 4: Training curve for the two rewarding schemes.
+4.3
+Heterogeneous Graph Neural Network Architecture
+An important part of our framework is the neural network architecture that we
+designed to perform a prediction of the next value to branch on. A high-level
+representation is proposed in Fig. 5. Four steps are carried out: (1) a CP model
+encoder, (2) a graph neural network encoder, (3) a neural network decoder, and
+(4) an action-selection policy. They are detailed in the next subsections.
+Fig. 5: High-level overview of the neural architecture designed.
+
+50
+Propagation-based reward
+Score reward
+40
+Optimal Score
+Random
+30
+20
+10
+0
+2
+4
+6
+8
+10
+Training episode
+(x1000)0
+Propagation-based reward
+-20
+Score reward
+Optimal Score
+-40
+Random
+-60
+-80
+-100
+-120
+0
+2
+4
+6
+8
+10
+Training episode (x1000)-2
+-3
+Propagation-based reward
+Score reward
+-4
+Optimal Score
+-5
+Random
+-6
+-7
+-8
+-9
+-10
+0
+2
+4
+6
+10
+Training episode (x1000)GNN Encoder (2)
+NN Decoder (3)
+Solver state and
+Predicted
+selected variable
+Q-Table
+St = (Pt,&t)
+*=3
+Extract
+*=2
+value
+GNN layers
+CP Encoder
+features
+*=1
+Q(St, X1 = 3)
+X1
+*=3
+Q(St, Xi = 2)
+X1
+X1
+*=1
+Q(St, Xi = 1)
+Extract
+variable
+X1
+C1
+C2
+Action-Selection
+features
+4'
+PolicyTraining a DQN Agent Inside a Generic CP Solver
+11
+Step 1: CP Model Encoder This module’s core idea is to learn for any CP
+model given as input, unlike Cappart et al. [11], who require a specific encod-
+ing for each combinatorial problem. This has been achieved for mixed-integer
+programs thanks to a bipartite graph representation [14] and by Chalumeau et
+al. [12] for CP models thanks to a tripartite graph. This last work does not lever-
+age any feature related to the variables, values, or constraints. We built upon
+this last approach by adding such features. Specifically, let P = ⟨X, D(X), C, O⟩
+be the combinatorial problem we want to encode. The idea consists in building
+a simple undirected graph G(V1, V2, V3, f1, f2, f3, E1, E2) encoding all the infor-
+mation of Pt from a state st = (Pt, xt). In this representation, V1, V2, and V3 are
+three types of vertices, f1, f2, and f3 are three vectors of features, and E1 with
+E2 are two distinct sets of edges. This yields a graph with three types of nodes
+decorated with features. The first part of the encoding we propose is as follows:
+(1) each variable, constraint, and value corresponds to a specific type of node
+(V1 = X, V2 = C, and V3 = D), (2) each time a variable x ∈ V1 is involved in
+a constraint c ∈ V2, an edge (x, c) ∈ E1 is added between both nodes, (3) each
+time a value v ∈ V3 is in the domain of a variable x ∈ V1, an edge (v, x) ∈ E2 is
+added between both nodes. This gives a tripartite graph representation of a CP
+model generically. This is illustrated in Fig. 6. The second part of the encoding
+is to add features to each node. Intuitively, the features will provide meaningful
+information and thus improve the quality of the model. The features we consid-
+ered are proposed below. We note that we can easily extend this encoding by
+integrating new features.
+1. Features attached to variables (f1): the current domain size, the initial do-
+main size, a binary indication if the variable is already assigned, and a binary
+indication if the variable corresponds to the objective.
+2. Features attached to constraints (f2): the constraint type (one-hot encoding),
+and a binary indication if the constraint propagation has reduced domains.
+3. Features attached to values (f3): its numerical value.
+Fig. 6: Representation computed by the CP encoder on a simple example.
+Step 2: Graph Neural Network Encoder Once the CP model has been
+encoded as a graph, the next step is to embed this representation as a latent
+
+Solver State
+Tripartite Heterogeneous Graph
+Vi
+V2
+XiE1,2.X2E1,21,X3E[1,2,3]
+CP Encoder
+fi(X1)
+f2(C))
+.C1 = X1 ≤ X2
+fi(X2)
+.C2 = X2 ≤X3
+f2(Ca)
+.C3 = AllDifferent(Xi, X2, X3)
+fi(Xs)
+f2(C2)
+E
+Ei12
+T. Marty et al.
+vector of features for each node of the graph (see Section 3.2). We propose
+to carry out this operation with a graph neural network. Unlike the standard
+prediction scheme presented in Equation (1), our graph has three types of nodes.
+For this reason, we opted for a heterogeneous architecture. Concretely, a specific
+convolution is carried out for each node type. The architecture is detailed in
+Equations (7) to (9), where � is the sum-pooling or mean-pooling aggregation,
+operator (.∥.) is a concatenation of vectors, Nx(n) is the set of neighbouring
+nodes of n from V1 (variable), Nc(n) is the set of neighbouring nodes of n from V2
+(constraint), Nv(n) is the set of neighbouring nodes of n from V3 (value), θk
+1,...,10
+are weight matrices at layer k, and g is the leakyReLU activation function [24].
+Another difference with the canonical GNN equation is the integration of skip
+connections (h0
+x, h0
+c, and h0
+c) allowing to keep at each layer information from
+the input features. This technique is ubiquitous in deep convolutional networks
+such as in ResNet [17]. Finally, the initial embedding are initialized as follows:
+h0
+x = θ11f1, h0
+c = θ12f2, and h0
+v = θ13f3, where θ11,...,13 are new weight matrices.
+hk+1
+x
+= g
+�
+θk
+1h0
+x
+�� θk
+2hk
+x
+�� (
+�
+c∈Nc(x)
+θk
+3hk
+c)
+�� (
+�
+v∈Nv(x)
+θk
+4hk
+v)
+�
+∀x ∈ V1
+(7)
+hk+1
+c
+= g
+�
+θk
+5h0
+c
+�� θk
+6hk
+c
+�� (
+�
+x∈Nx(c)
+θk
+7hk
+x)
+�
+∀c ∈ V2
+(8)
+hk+1
+v
+= g
+�
+θk
+8h0
+v
+�� θk
+9hk
+v
+�� (
+�
+x∈Nx(v)
+θk
+10hk
+x)
+�
+∀v ∈ V3
+(9)
+Step 3: Neural Network Decoder At this step, a d-dimensional tensor is
+obtained for each node of the graph. Let x ∈ V1 be the node representing the
+current variable selected for branching, and Vx ⊆ V3 the subset of nodes repre-
+senting the values available for x (i.e., the values that are in the domain of the
+variable). The goal of the decoder is to predict a Q-value (see Section 3.1) for each
+v ∈ Vx. The computation is formalized in Equation (10), where hK
+x and hK
+v are
+the node embedding of variable x and value v, respectively, after K iterations of
+the GNN architecture. The functions ϕx : Rd → Rl, ϕv : Rd → Rl, ϕq : R2l → R
+are fully-connected neural networks. Such a Q-value must computed for each
+value v ∈ Vx. It is internally done thanks to matrix operations, allowing a more
+efficient computation.
+ˆQ(hK
+x , hK
+v ) = ϕq
+�
+ϕx(hK
+x )
+�� ϕv(hK
+v )
+�
+∀v ∈ Vx
+(10)
+Step 4: Action-Selection Policy Once all the Q-values have been computed
+for the current variable, the branching policy π on variable x consists simply
+by taking the highest Q-value, according to the standard Q-learning algorithm
+shown in Equation (11). Once trained, this value should represent the branching
+choice leading to the best decision according to the reward of Equation (6).
+π(v|x) = argmaxv∈Vx ˆQ(hK
+x , hK
+v )
+(11)
+
+Training a DQN Agent Inside a Generic CP Solver
+13
+Assembling all the pieces together, this architecture gives a generic approach to
+obtain a data-driven value-selection heuristic inside a CP solver. Concerning the
+search strategy, we propose to embed our predictions inside an iterative limited
+discrepancy search (ILDS) [16]. This strategy is commonly used when we are
+confident on the quality of the heuristic. The core idea is to restrict the number
+of branching choices deviating from the heuristic (i.e., a discrepancy). By doing
+so, the search will explore a subset of solutions expected to be good while giving
+a chance to reconsider the value-heuristic selection which is nevertheless prone
+to errors. This mechanism is enriched with a procedure that iteratively increases
+the number of discrepancies allowed once a level has been explored.
+5
+Experiments
+The goal of the experiments is to evaluate the quality of the learned value-
+selection heuristic and the efficiency of the approach when solving combinatorial
+optimization problems. Three problems are considered: graph coloring (COL),
+maximum independent set (MIS), and maximum cut (MAXCUT).
+5.1
+Experimental Protocol
+Three configurations are proposed for each problem: small (20 to 30 nodes),
+medium (40 to 50 nodes) and large (80 to 100 nodes) instances, except for
+MAXCUT which was already challenging for the medium size. Training is carried
+out on randomly generated Barabási-Albert graph [2] with a density factor vary-
+ing between 4 and 15 according to the size of the instances. A specific model is
+trained for each configuration using randomly generated instances. Evaluation is
+then performed on 20 other graphs following the same distributions. The models
+are trained on an Nvidia Tesla V100 32Go GPU until convergences. It took up
+to 72 hours of training time for the most difficult cases (graph coloring with
+80 nodes) and less than 1 hour for the simplest cases (graph coloring with 20
+nodes). Each operation of the CP solver during training and evaluation is carried
+out on a CPU Intel Xeon Silver 4116 at 2.10GHz. The approach has been imple-
+mented in Julia and is integrated into the solver Seapearl. The implementation
+is available on GitHub with MIT open-source licence4.
+Our approach (ILDS-Learned) is compared with the optimal solution (OPT)
+which is obtained by an exact solver, with a standard depth-first search strategy
+based on a random value selection (DFS-Random), and with the application of
+the learned heuristic without any backtrack (Dive-Learned). A standard first-
+fail variable-selection heuristic is used for all the methods. Finally, a maximum
+budget in terms of the number of nodes visited is enforced. The idea is to show
+that we can obtain solutions close to optimality with few backtracks.
+4 https://github.com/corail-research/SeaPearl.jl
+
+14
+T. Marty et al.
+5.2
+Quantitative Results
+Table 1 summarizes the main results of our approach. A first observation is that
+the learned value-selection, even without backtrack (Dive-Learned) can find solu-
+tions close to optimality. For instance, a single dive for MAXCUT with 50 nodes
+yields a solution with an optimality gap of 17% in less than 1 second, whereas
+DFS-Random required 22 seconds and roughly 51,000 nodes explored to find a
+solution with the same gap. Within the same budget, ILDS-Learned improves
+the solution but with a longer execution time. This increased computation time
+is mainly due to the fact that calling the graph neural network architecture
+(Section 4.3) at each tree search node is more expensive than calling a random
+heuristic. Experimental results are also proposed in Fig. 7 using performance
+profiles [13] for the hardest instances of each problem (80 for graph coloring, 100
+for max independent set, and 50 for maximum cut. The ratio is computed using
+the optimal solution as a reference. Within the same maximal number of nodes
+visited (1000), we observe that ILDS-Learned dominate DFS-Random. Besides,
+when restricting ten times the budget for ILDS-Learned, we still perform better
+than the competitor.
+Table 1: Results for the three problems. For each configuration, the average
+result (rounded) on the 20 test instances is reported, and a specific node budget
+is enforced for DFS-Random and ILDS-Learned. Gap indicates the optimality gap,
+Node gives the number of nodes explored before finding the best solution within
+the budget, and Time gives the time, in seconds, before finding this solution.
+DFS-Random
+Dive-Learned
+ILDS-Learned
+(with budget)
+(no backtrack)
+(with budget)
+Size
+OPT Gap
+Node Time Gap
+Time Gap
+Node Time Budget
+COL
+20
+4.95 2,33
+85
+< 1 0.00
+< 1 0.00
+21
+< 1
+102
+40
+7.90 0.00
+1,559
+< 1 0.00
+< 1 0.00
+41
+< 1
+104
+80
+12.00 0.00
+6,698
+11 0.02
+2 0.00
+85
+2
+104
+MIS
+30
+10.20 0.00
+291
+< 1 0.05
+< 1 0.00
+41
+< 1
+104
+50
+14.90 0.00
+8,011
+< 1 0.19
+< 1 0.00
+2,749
+2
+105
+100
+21.75 0.09 51,174
+7 0.19
+< 1 0.01 28,483
+170
+105
+MAXCUT
+20
+46.45 0.03
+5,071
+< 1 0.19
+< 1 0.03
+3,059
+2
+104
+50 135.19 0.17 51,222
+22 0.17
+< 1 0.10 35,977
+172
+105
+5.3
+Discussions
+Previous experiments showed the capacity of our approach to obtain a value-
+selection heuristic in a CP solver, thanks to historical instances of the same
+distribution. Unlike many related works based on imitation learning [14,20], the
+training is not supervised and thus does not require labels from an expert (e.g.,
+
+Training a DQN Agent Inside a Generic CP Solver
+15
+(a) Graph coloring.
+(b) Maximum cut.
+(c) Max. independent set.
+Fig. 7: Best solutions found on largest instances for the three problems.
+an expensive heuristic or an exact solving). One major difficulty encountered
+by our approach is the increased computation time due to the inference of the
+graph neural network at each node of the tree search. A first solution would
+be to reduce the complexity of the model by compressing its knowledge, e.g.,
+using network pruning tools [41]. Another idea is to call the model only in a
+few nodes, in a similar fashion as Cappart et al. [8] did for decision-diagram-
+based branch-and-bound [6]. The last idea to tackle this scaling issue would be
+to restrict the learning only to small instances and transfer the model to solve
+larger instances. The architecture has been designed to do so, and experiments
+on this aspect are part of future work. A second difficulty is the size of the CP
+encoding as a tripartite graph, which involves a specific node for each variable,
+value, and constraint. This grows proportionally with the problem size and slows
+down the training phase. As a concrete example, graph coloring instances with
+80 nodes require 72 hours of training time. An interesting research question is
+how to build such a generic encoding more compactly.
+6
+Conclusion
+The efficiency of constraint programming solvers is partially due to the branch-
+ing heuristics used to guide the search. Unlike the variable selection, there is
+no available generic and efficient heuristic for the value selection. In practice,
+value-selection heuristics are often designed thanks to problem-specific expert
+knowledge, often out of reach for non-practitioners. In this paper, we proposed
+a learning-based approach for obtaining such a heuristic thanks to historical
+data, characterized by problem instances following the same distribution of the
+one that must be solved. This has been achieved thanks to the combination of
+a restart-based training procedure, a non-sparse reward signal, and a hetero-
+geneous graph neural network architecture. Experiments on three combinato-
+rial optimization problems show that the framework can find better solutions
+close to optimality without requiring many backtracks. Several limitations (e.g.,
+tractability for larger instances) have been identified, and addressing them is
+part of future work. We also plan to consider other combinatorial problems,
+such as the ones proposed in XCSP3 competitions [3].
+
+20 instances
+1.0
+0.8
+0.6
+Proportion of the 2
+0.4
+RL Agent - ILDS - 100
+0.2
+RL Agent - ILDS - 1000
+Random - DFS - 1oo0
+0.0
+1
+2
+3
+4
+5
+6
+Within this factor of the best score1.0
+0.8
+0.6
+0.4
+RL Agent - ILDS - 1o0
+0.2
+RL Agent - ILDS - 1000
+Random - DFS - 1oo0
+0.0
+1.0
+1.1
+1.2
+1.3
+1.4
+Within this factor of the best score1.0
+0.8
+0.6
+0.4
+RL Agent - ILDS - 1o0
+0.2
+RL Agent - ILDS - 1000
+Random - DFS - 1oo0
+0.0
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+Within this factor of the best score16
+T. Marty et al.
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+

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+Springer Nature 2021 LATEX template
+Auditing citation polarization during the
+COVID-19 pandemic
+Taekho You1, Jinseo Park2, June Young Lee2 and Jinhyuk
+Yun3*
+1Institute for Social Data Science, Pohang University of Science
+and Technology, Cheongam-ro 77, Pohang, 37673,
+Gyeongsangbukdo, Republic of Korea.
+2Center for Global R&D Data Analysis, Korea Institute of
+Science and Technology Information, Hoegi-ro 66,
+Dongdaemun-gu, 02456 Seoul, Republic of Korea.
+3*School of AI Convergence, Soongsil University, Sangdo-ro 369,
+Dongjak-gu, 06978 Seoul, Republic of Korea.
+*Corresponding author(s). E-mail(s): [email protected];
+Abstract
+The recent pandemic stimulated scientists to publish a significant amount
+of research that created a surge of citations of COVID-19-related papers
+in a short time, leading to an abrupt inflation of the journal impact fac-
+tor (IF). By auditing the complete set of COVID-19-related publications
+in the Web of Science, we reveal here that COVID-19-related research
+worsened the polarization of academic journals: the IF before the pan-
+demic was proportional to the increment of IF, which had the effect of
+increasing inequality while retaining the journal rankings. We also found
+that the most highly cited studies related to COVID-19 were published
+in prestigious journals at the onset of the epidemic, independent of their
+innate importance or quality. Through the present quantitative investi-
+gation, our findings caution against the belief that quantitative metrics,
+particularly IF, can indicate the significance of individual papers. Rather,
+such metrics reflect the social attention given to a particular study.
+1
+arXiv:2301.01926v1  [cs.DL]  5 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Auditing citation polarization during the COVID-19 pandemic
+1 Introduction
+The recent pandemic has boosted COVID-19-related research, which has led
+to a growing number of researchers publishing COVID-19-related papers [1].
+During the pandemic, as of 2021 more than 4% of published research papers
+focused on COVID-19 [1]. The availability of COVID-19-related research has
+supported the public to overcome the current pandemic.
+The expansion of this new research field has had a substantial impact
+on the scholarly publishing ecosystem. COVID-19-related papers received a
+large number of citations in a short period, causing a dramatic shift in
+citation counts. Specifically, some journals benefited from publishing COVID-
+19-related research because it significantly increased their mean citation rate.
+As an illustrative example, the Lancet more than doubled its impact factor
+(IF) from 79.323 to 202.731, according to the 2021 Journal Citation Reports
+(JCR) released in June 2022. It has been contended that COVID-19-related
+papers have inflated the citation-based metrics; indeed, some journals have
+increased their IF by more than tenfold [2].
+Consequently, the long-lasting IF controversy has reemerged. Due to the
+heavy-tailed nature of citation, which is sometimes referred to as the rich-get-
+richer effect, many critics argue that IFs do not accurately reflect the impact
+of scientific items because they rely solely upon mean citation counts [3, 4].
+In response, alternative metrics have been proposed [5, 6]. Moreover, although
+the IF metric was designed to measure the performance of journals rather
+than single papers [7], it is nevertheless frequently misunderstood to reflect
+the quality of an individual paper [8]. The spreading of these misunderstand-
+ings has increased unintended dynamics in the conduction and evaluation of
+research [9, 10], even leading to cases of malpractice [11].
+Resolving this IF controversy from COVID-19-related papers necessitates
+a deep comprehension of citation dynamics in academia, such as the extent to
+which COVID-19 publications affect journal IFs and who benefits more from
+publishing COVID-19-related papers. The Matthew effect [12], also known
+as the rich-get-richer effect, gives valuable insight into the accumulation of
+rewards in academia [13, 14, 15, 16, 17]. Previous studies demonstrated that a
+little variation in early stages leads to a substantial difference in the productiv-
+ity and citations of authors and journals in later stages [18, 14, 15]. Moreover,
+a paper is more likely to receive citations when published in a prestigious jour-
+nal that has a high IF [19]. Citation inequality results from the widening gaps
+in return from such small, initial differences [20, 21, 17]. We believe that the
+emergence of the COVID-19 research field presents an excellent opportunity
+to comprehend scholarly dynamics in response to external societal influence.
+In this study, we quantitatively exhibit the impact of COVID-19-related
+papers on the citation ecosystem to aid in resolving the long-lasting debates on
+the IF metric. For this purpose, we investigate the changes in IF by the publi-
+cation of COVID-19-related papers considering prior journal IF. We find that
+COVID-19-related papers received more citations than other papers, and we
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+3
+100
+101
+102
+103
+104
+Number of citations received
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Cumulative density function
+COVID-19-related papers (2020)
+Non-COVID-19-related papers (2020)
+COVID-19-related papers (2021)
+Non-COVID-19-related papers (2021)
+2019
+2020
+2021
+Year
+0.0
+0.2
+0.4
+0.6
+0.8
+Citations between COVID-19-related papers
+83.1 %
+90.9 %
+83.9 %
+5.8 %
+45.3 %
+48.3 %
+A
+B
+Citations received
+References citing
+Fig.
+1 Difference
+in
+citation
+distribution
+between
+COVID-19-related
+and
+non-COVID-19-related papers. A Citation distribution of COVID-19-related and non-
+COVID-19-related papers that contribute to the annual IF calculation. For example, the
+distribution of citations in 2021 includes citations received for papers published in 2019 and
+2020 from the papers published in 2021. A distribution of the yearly citation pattern is
+displayed in Fig. S2. B Citation origin of COVID-19-related papers. We display both the
+percentage of citations received from other COVID-19-related papers and references citing
+other COVID-19 publications.
+also show that although most of the citations originated from other COVID-
+19-related papers, the degree of benefit to the journals differs by the prestige of
+the journals reflected in their prior IFs. We reveal that the number of COVID-
+19-related papers and the extent of the increase in journal IF are nearly
+uncorrelated, while the IFs of prestigious journals with high IFs increased more
+than those of low-IF journals. Lastly, we find that the majority of highly cited
+COVID-19 publications were published during the earliest stages of the pan-
+demic, selected by prestige journals. Taken together, the results demonstrate
+that the benefits of publishing COVID-19-related research were granted mainly
+to the prestige journals, which may aggravate citation inequality in academia.
+2 Results
+2.1 Citation homophily between COVID-19-related
+papers
+During the pandemic, COVID-19-related papers have increased their share in
+academia. In 2019, only 350 papers (0.013%) were related to COVID-19, many
+of which were mainly focused on other coronaviruses, based on our search
+query (see Methods for step-by-step details on gathering COVID-19-related
+papers). As the virus spread, their share increased to 89,112 (2.004%) and
+162,256 (4.194%) in 2020 and 2021, respectively. Moreover, COVID-19-related
+research occupied a major fraction of all citations across academia. Such papers
+published in 2020 received 2,654,613 citations until the end of 2021, which is
+
+Springer Nature 2021 LATEX template
+4
+Auditing citation polarization during the COVID-19 pandemic
+13.8% among the total 19,203,421 citations in 2020 and 2021. But not only
+gaining a high share, COVID-19-related publications also received immediate
+citations: those published in 2021 received 787,009 citations out of the total
+6,457,473 citations in 2021 (12.2%). This same trend even extended down to
+the monthly citation level, as displayed in Fig. S1. After publication, 31.8% of
+the citations of COVID-19-related papers arose within 6 months, while 22.2%
+did so for non-COVID-19 papers. Compared to the statistics indicating that
+COVID-19-related papers produced just 4.1% and 6.9% of references within
+the same time period (2020 and 2021, respectively), this proportion of received
+citations is high.
+The increased attention given to COVID-19 research resulted in a citation
+distribution with a longer tail than other research. The two-year citation dis-
+tribution shows that COVID-19-related papers received more citations than
+non-COVID-19-related papers in a given year (see Fig. 1A for the merged
+distribution along with Fig. S2 displaying separated distributions). When we
+assume that the citation distribution follows a simple power law (y ∼ xk),
+the COVID-19-related papers show k ≃ 1.9 and k ≃ 2.7 for 2020 and 2021,
+respectively (see Methods for the detailed computation), while non-COVID-19-
+related papers have an exponent of 3.2 and 3.3 for 2020 and 2021, respectively.
+The lower exponents indicate that the proportion of COVID-19-related papers
+with extremely high citation counts is greater than that of non-COVID-19-
+related papers. Consequently, COVID-19-related papers also received more
+citations on average. COVID-19-related research received an average of 22.6
+(2020) and 21.8 (2021) citations, while non-COVID-19-related papers received
+4.9 (2020) and 5.2 (2021) citations. This result is consistent with a previous
+observation using SCOPUS [1].
+We also find a homophily of citations, namely that the high citation counts
+of COVID-19-related papers are attributable to other COVID-19-related
+papers. We observed a high rate of citation exchange between publications
+related to COVID-19, in which more than 40% of the references in these
+papers cite other COVID-19-related papers, excluding 2019 (Fig. 1B). The
+homophily is much stronger when we consider the received citations, where
+90.9% of citations to COVID-19-related papers published in 2020 were from
+other COVID-19-related papers. This finding indicates that the rising amount
+of COVID-19-related research in 2020 and 2021 resulted in a number of such
+papers receiving a substantial number of citations.
+2.2 Contribution of COVID-19-related research to IF
+inflation
+Several highly cited COVID-19-related studies may bolster the publishing jour-
+nals’ IF. To quantify this, we calculate the IF in terms of the existence and
+number of COVID-19-related papers (see Methods for IF calculation). We mea-
+sure two different types of IFs and compare them to estimate the advantage of
+publishing COVID-19-related papers: IF excluding COVID-19-related papers
+and IF including them. We observe that only 763 journals (16%) among those
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+5
+10
+2
+10
+1
+100
+101
+102
+Impact factor (IF)
+10
+5
+10
+3
+10
+1
+101
+Surplus IF by COVID-19-related papers
+A
+B
+100
+101
+102
+103
+Number of COVID-19-related papers
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Surplus IF by a single COVID-19-related paper
+Fig. 2 Surplus impact factor (IF) by COVID-19-related publications. A Journal
+impact factor increase by publishing COVID-19-related papers, where the simple superlinear
+growth y ∼ x1.7 can characterize the growth pattern (dotted line). Here, we applied a simple
+linear regression method to the logarithm of the values of interest to estimate the power-law
+scaling relationship between the IF and its surplus by COVID-19-related papers, assuming
+a simple power-law scaling of y = Cxk. B Increase in IF per COVID-19-related paper in
+proportion to the number of COVID-19-related papers published in journals. The increase is
+calculated by dividing the absolute difference in IF between papers including and excluding
+COVID-19 by the number of COVID-19-related papers in the journals. In both A and B,
+the red dots represent the average value of surplus IF and the error bars show the standard
+deviation in log-scale.
+publishing one or more COVID-19-related papers in 2019 and 2020 dropped in
+IF in 2021, while the other 4,004 journals (84%) enhanced their IFs through
+the publication of COVID-19-related papers in the same period. For the for-
+mer, even though the journals decreased in IF by publishing COVID-19-related
+papers, the amount of decrease was limited. Only one of these 763 journals
+(CA-A CANCER JOURNAL FOR CLINICIANS) dropped in IF by more
+than 1 (Fig. S3).
+Individual scientists have a greater tendency to cite widespread, popular
+journals than less popular journals due to psychological, sociological, and eco-
+nomic factors, leading to the rich-get-richer phenomenon of citation [22]. For
+the COVID-19-related papers, we find that the surplus IF is proportional to
+the prior IF (Fig. 2). High correlation exists between IF and its surplus (Pear-
+son r = 0.670, Fig. 2A), and their relationship is even superlinear (y = x1.7).
+This pattern is also verified when we consider the relative advantage of IFs by
+dividing the surplus IF by the prior journal IF, which also shows a positive
+correlation (Fig. S4).
+However, publishing numerous papers on COVID-19 did not necessar-
+ily increase the journal IF. Instead, as more COVID-19-related papers were
+published, the gain in IF per COVID-19-related paper decreased (Fig. 2B).
+Journals that published only one COVID-19-related paper in 2019 and 2020
+increased their IF by 0.12 on average, whereas journals that published over
+
+Springer Nature 2021 LATEX template
+6
+Auditing citation polarization during the COVID-19 pandemic
+500 papers in the same period increased their IF by only 0.0009. For example,
+the Lancet, the journal with the highest IF in JCR 2021, doubled its IF (from
+93.04 to 189.25) while publishing only 46 COVID-19-related papers (9.4% of
+all citable items) in 2019 and 2020. To take one more extreme example, one
+journal that published only one COVID-19-related paper in 2019 and 2020
+increased its IF by 37, whereas the journal that published the largest num-
+ber of COVID-19-related papers (730 papers) improved its IF by only 1.02. In
+other words, while a single well-chosen paper published during the pandemic
+could have potentially resulted in a significant increase in IF, publishing a large
+number of COVID-19-related papers did not provide many benefits. Although
+allocating more shares to COVID-19-related papers correlates positively with
+the rise in the IFs of journals, the correlation is slight (Pearson r = 0.110; see
+Fig. S5).
+To confirm that COVID-19 research has legitimately increased journal
+IFs, we examine the correlation between IFs across years taking into account
+the existence of COVID-19-related papers. When excluding COVID-19-related
+papers, the correlation between two consecutive years (Pearson r = 0.957
+between 2019 and 2020 and Pearson r = 0.925 between 2020 and 2021) is
+significantly high. The correlation between the IF in 2021 excluding COVID-
+19-related papers and the IF in 2020 with COVID-19-related papers is r =
+0.926. Thus, the overall trend of journal IF without papers related to COVID-
+19 did not change significantly, and this high correlation suggests the existence
+of a linear relationship between the IFs for two consecutive years. Incorporat-
+ing COVID-19-related papers, however, reduces the correlation between the
+IFs for 2020 and 2021 (Pearson r = 0.850). Note that we also observe a lower
+correlation between the IF for 2021 with COVID-19-related papers and the
+IF for 2020 without COVID-19-related papers (Pearson r = 0.849), indicating
+non-linear relationships between the IFs of the two consecutive years. Given
+that journals with a higher prior IF received a greater increase in IF from
+COVID-19-related papers (Fig. 2A), the publication of COVID-19 research
+may contribute to the polarization of journal IFs.
+2.3 The Matthew effect of IF polarization during the
+pandemic
+In the preceding sections, we demonstrated that the publication of COVID-19-
+related research had a positive correlation with journal IFs, while the amount
+of increment had a strong correlation with the prior IFs of the journals (Fig. 2).
+One may wonder how much the overall journal landscape, i.e., the journal
+rankings, has changed due to the surplus IFs, or conversely, the magnitude
+of the change in IF based on the journal ranking [23]. To demonstrate the
+influence of COVID-19-related publications on the landscape of JCR rankings,
+we compare the ratio of surplus IF in 2021 considering the journal rank in their
+research categories. On average, the publication of COVID-19-related papers
+increased the journal IF by 15.2% (dotted line in Fig. 3). The IFs of the top
+10% prestige journals increased by 39.4%, while the IFs of the bottom 10%
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+7
+100
+101
+Increase ratio
+Top 10%
+10-20%
+20-30%
+30-40%
+40-50%
+50-60%
+60-70%
+70-80%
+80-90%
+90-100%
+Journal Ranking
+Fig. 3 Relative ratio of surplus IF from publishing COVID-19-related papers by
+the 2021 journal rankings for JCR categories. The ratio was calculated by subtracting
+the IF for 2021 excluding COVID-19-related papers from the IF for 2021 including COVID-
+19-related papers and then dividing this value by the prior IF. The dotted line indicates
+the average surplus IF from publishing COVID-19-related research (15.2%). Here, the boxes
+represent the quartiles of the dataset except for points determined to be outliers.
+journals increased by only 9.6% on average. Most journals increased their IF
+by less than the average increase (15.2%) except for the top 20%. On average,
+higher-ranked journals gained more citations, and this trend is robust across
+all categories (Table S1).
+The majority of journals with a significant increase in IF due to COVID-
+19-related publications were already high-IF journals. Among all journals, 132
+increased their IF by greater than twofold. 55.3% of these 132 journals are
+in the top 10% of at least one of their research categories. Only five journals
+fall within the bottom 10%. In terms of research category, 102 of the 132
+journals (77.3%) are classified into Clinical Medicine since the majority of
+COVID-19-related papers (70.9%) were published in this category.
+This proportion of highly cited articles in prestigious journals has the
+potential to exacerbate citation polarization. Indeed, we found that more
+COVID-19-related articles were published in prestigious journals with a high
+
+Springer Nature 2021 LATEX template
+8
+Auditing citation polarization during the COVID-19 pandemic
+IF than in other journals (Fig. 4A). While the top 10% ranked journals pub-
+lished 26.3% (51,976) of all COVID-19-related papers from 2019 to 2021, the
+bottom 90% to 100% ranked journals published only 3.5% (6,977) in the same
+period. We also observe that the share of COVID-19-related papers decreases
+as the journal ranking falls (Fig. 4A). Moreover, the proportion of highly cited
+papers exacerbates the disparity. Eighty-four percent (86) of the 102 papers
+with over 1000 citations were published by the top 10% journals, while jour-
+nals ranked 10 to 20 % published only eight of these studies. No papers with
+over 1000 citations were published in the bottom 50% journals.
+Citation is a stochastic multiplicative process, whereby papers with a higher
+citation count are more likely to receive additional citations. Even with simi-
+lar content between papers, those published in a more prominent location are
+more likely to be cited [19]. In addition, earlier works may receive more cita-
+tions because citations are cumulative. Indeed, we find that the majority of
+highly cited COVID-19-related papers were published during the early stage
+of the pandemic (early 2020), as shown in Fig. 4B. The number of COVID-19-
+related publications gradually increased as the pandemic progressed (see the
+blue line in Fig. 4B). Despite this, papers with higher numbers of citations were
+generally published earlier. In conjunction with the finding that highly cited
+studies were likely to be published in prestigious journals, we may conclude
+that COVID-19-related studies were originally introduced in high IF journals,
+and that lower IF journals then cited the previous publications from high IF
+journals.
+This growing pattern of citations could worsen the polarization of aca-
+demic journals. Publication of COVID-19-related research gave a significantly
+greater benefit to journals with higher IFs than to those with lower IFs. Con-
+sequently, the relative position (rank) of the majority of journals shows only
+minor changes, although the overall IF of all journals tended to increase dur-
+ing the pandemic. Only 39.0% of journals that published COVID-19-related
+papers moved to a higher rank by including COVID-19-related research in one
+of their subject categories; among them, only 51.8% changed their IF quantile
+to a higher one (Fig. S6). The other 61.0% of journals maintained or decreased
+their ranking. In addition, significant increases or decreases in the ranks of
+journals were rarely observed (Fig. S6). The majority (90%) of the top 10%
+ranked journals maintained their position regardless of COVID-19 research,
+while the other 10% fell into the 10% to 20% group.
+In summary, i) the IF of journals increased overall by publishing COVID-
+19-related research, ii) journals with higher IFs received greater benefits by
+publishing COVID-19-related research, and iii) the relative ranks of jour-
+nals did not change significantly from publishing COVID-19-related research.
+These findings lead to an interesting question: Did the publication of COVID-
+19-related research actually increase the polarization of journals? To answer
+this, we applied the Gini coefficient [24], a well-known measure of income
+inequality, to the distribution of journal IFs. In our investigation, the Gini
+coefficient measures the distribution of citations across journals within a JCR
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+9
+All
+>10
+>100
+>1000
+Number of citations received
+0.0
+0.2
+0.4
+0.6
+0.8
+Fraction of COVID-19-related papers
+A
+B
+C
+Top 10%
+10-20%
+20-30%
+30-40%
+40-50%
+50-60%
+60-70%
+70-80%
+80-90%
+90-100%
+2020
+2021
+2022
+Year
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+Probability density function
+all
+>10
+>100
+>1000
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Gini coefficient excluding COVID-19-related papers
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Gini coefficient including COVID-19-related papers
+Number of COVID-19
+-related papers
+0
+800
+1600
+2400
+3200
+Gini coefficient changes
+Increased
+Decreased
+Fig. 4 Distribution of COVID-19-related papers and their disparities. A Distribu-
+tion of COVID-19-related research by journal ranking. As the number of citations increases,
+the likelihood of papers belonging to high-IF journals increases. 26.4% of all papers were
+published in the top 10% ranked journals, whereas 85.3% of papers with more than 1000 cita-
+tions were published in the top 10% ranked journals. B Distribution of COVID-19-related
+papers by publication date. From the beginning of the pandemic to mid-2020, the number of
+papers increased. Approximately the same number of papers were published between then
+and the end of 2021. Most of the highly cited papers (> 1000 citations) were published in
+early 2020. C Plot of the Gini coefficient of the IF distribution by JCR category. Each dot
+represents a JCR category. The Gini coefficient is computed using the IF distribution of
+journals in a particular category including and excluding COVID-19-related papers. Blue
+(orange) dots indicate an increase (decrease) in the Gini coefficient by publishing COVID-
+19-related research. The size of the dots is proportional to the number of COVID-19-related
+studies published in the category.
+category, ranging from 0 for the lowest heterogeneity (when all journals receive
+the same average number of citations) to 1 for the highest heterogeneity
+(when only a single journal receives all citations). The trend illustrated by the
+difference in the Gini coefficient as a function of the number of COVID-19-
+related papers (see Figs. 4 and S7) implies that the disparity in the number
+of citations between journals increases as the number of COVID-19-related
+papers published increases. In conclusion, based on the present snapshot of
+the Web of Science (WOS) dataset, we found that the general pattern of het-
+erogeneity, or polarization, among journals rises as the number of published
+COVID-19-related papers increases.
+3 Discussion
+From the outset of the global COVID-19 pandemic, many scholars pursued
+the topic and published a massive number of studies in an unprecedentedly
+short period. We discovered a trend that, as a result of the intensive publica-
+tion, COVID-19-related papers acquired more citations than papers in other
+domains, which reflects its considerable attention in academia. We uncovered
+two significant consequences that may have led to a more severe polarization
+of journals in terms of citations. First, 84% of journals that published COVID-
+19-related papers in 2019 and 2020 increased their impact factors. Second,
+prestigious journals were more likely to publish highly cited COVID-19-related
+papers than other journals (Fig. 3).
+
+Springer Nature 2021 LATEX template
+10
+Auditing citation polarization during the COVID-19 pandemic
+Nonetheless, we demonstrated that publishing a large number of COVID-
+19-related papers did not immediately boost a journal’s IF. Increasing numbers
+of COVID-19-related papers published in a journal tended to diminish the
+citation impact of a single COVID-19-related article. In addition, we found
+that prestigious journals with a high prior IF gained more benefit (increased
+IF) from publishing COVID-19-related research, and also that the publications
+receiving the highest number of citations were predominantly published in
+prestige journals during the early stages of the pandemic. Given that not all
+COVID-19-related publications increased their journal’s IF, one may assume
+that prestige journals simply have accepted and published more significant
+research. However, considering that some papers published in prestige journals
+were ultimately retracted [25, 26], the high number of citations given to these
+journals is not only based on the significance of the works but also based in
+part on the visibility of these journals, which can worsen the polarization of
+academic publishing (Fig. 4).
+As we could not explicitly assess the quality of each paper due to the scale
+of the dataset, it is unclear which of the two aforementioned characteristics
+(quality or visibility) has a larger impact on the current disparity in benefit
+from publishing COVID-19-related research. We believe that a more in-depth
+investigation of the relationship between research quality (or significance) and
+citations may be necessary to increase the impact of our findings. Also, a more
+detailed understanding of such correlation should form the basis of explaining
+complex citation dynamics, yet we leave this for future research.
+Despite its limitations, this study can provide important insights into
+citation dynamics and its effects on global events. Because of the rich-get-
+richer nature of citations, papers published in prestigious journals tend to
+receive more citations. As the relative ranking of the journals did not change
+significantly despite the increase in the overall IFs of journals publishing
+COVID-19-related research, fluctuations in IF may not well reflect the actual
+impact of academic publications. This effect predominantly benefited well-
+established journals, while other journals did not experience benefits to the
+same extent (Fig. 4). Our research indicates that IFs are vulnerable to exter-
+nal events. The majority of the recent IF changes are attributable to citations
+of COVID-19-related publications; consequently, after the pandemic is over,
+the majority of the journals may revert to their pre-pandemic IF levels. It is
+challenging to evaluate academic journals or other participants (researchers,
+institutions, etc.) using basic statistics because doing so reflects only a portion
+of actual scientific achievements. Therefore, the simplified metrics employed
+by some governments [27] should be accompanied by a comprehensive and
+qualitative analysis of journals and individual papers for assessment.
+The use of quantitative indicators such as the IF metric has been under
+debate. The San Francisco Declaration on Research Assessment (DORA),
+which serves as the starting point for these discussions, explicitly states that
+the use of journal-based measures (such as IFs) should be avoided to act as a
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+11
+proxy for the quality of individual research publications, to evaluate the con-
+tributions of an individual scientist, or to make hiring, promotion, or funding
+choices. In practice, however, many funders and institutions employ journal-
+based measures or the number of citations as markers for evaluation rather
+than assessing the quality of individual papers. The polarization of citations
+observed in this study demonstrates the inherent hazard of such indicators.
+The IF metric is not a stable index against external shocks; it might fluctuate
+temporarily and then revert following external factors. Along with the other
+well-known limitations of IF, such as skewed citation distributions within jour-
+nals [28, 29], the vulnerability of the IF metric as we found here indicates
+that it is increasingly inappropriate to consider journal IF as a proxy for an
+individual paper’s quality.
+During the current pandemic, the rapid release of COVID-19-related works
+resulted in less-qualified academic outputs to the public, leading to the retrac-
+tion of many publications [30, 31]. Unfortunately, this issue happened not only
+in journals with a reputation for a weak review process or low publishing dif-
+ficulty but also in prestigious journals that are widely respected. Worse still,
+these retracted works earned a substantial number of citations and extensive
+media attention [32]. The general public may assume that papers published in
+academic journals are trustworthy and may likewise trust secondary sources
+such as scientific news reporting the results of academic findings. In the current
+context of appraising science and technology, there is a chance that content
+published in journals with strong indicators will be considered more reputable.
+Scientists must inform the public that citation measures and journals are not
+equivalent to the quality of individual research publications. In other words,
+the number of citations should not be the defining characteristic of quality
+research. The contemporary ecosystem of research and technology is seemingly
+supported by scientists’ mutual trust and goodwill, and the public may view
+the scientific community’s findings with a similar level of confidence. Combined
+with the stability issue of the IF metric identified in this study, shouldn’t the
+current practice of over-reliance on citation indices be discontinued so as not
+to break this chain of trust? For this reason, we believe that responsible action
+based on actual societal influence is essential for all members of academia, as
+opposed to merely producing popular research to boost citation impact and
+one’s professional reputation.
+Methods
+Data
+We used publications and citation data from the XML dump of the Web of
+Science Core Collection, which is dated back to 2017 and updated until the
+26th week of 2022. The data includes complete copies of Science Citation
+Index Expanded (SCIE), Social Sciences Citation Index (SSCI), and Arts &
+Humanities Citation Index (AHCI), along with the Emerging Sources Citation
+
+Springer Nature 2021 LATEX template
+12
+Auditing citation polarization during the COVID-19 pandemic
+Index (ESCI). The data comprises 16,957,120 articles, 82,317 journals, and
+116,086,223 references retrieved from papers published between 2017 and 2022.
+COVID-19-related publications
+COVID-19-related papers were retrieved from the Web of Science database
+(WOS,
+urlhttps://www.webofscience.com/)
+using
+the
+following
+search
+queries
+provided
+by
+Dimensions
+(https://www.dimensions.ai/covid19/):
+"2019-nCoV" OR "COVID-19" OR \SARS-CoV-2" OR "HCoV-2019" OR
+"hcov" OR "NCOVID-19" OR "severe acute respiratory syndrome
+coronavirus 2" OR "severe acute respiratory syndrome corona
+virus 2" OR \coronavirus disease 2019" OR (("coronavirus" OR
+"corona virus") AND (Wuhan OR China OR novel)). We limited the publi-
+cations from 2019. A total of 251, 718 COVID-19-related papers were collected
+on 4 July 2022. We consider all other papers in the WOS that were not
+retrieved from the above searching process as non-COVID-19-related papers.
+Estimation of the power-law exponent
+In this study, the power-law exponents of the citation distribution in Fig. 1A
+were estimated using the Python package named powerlaw [33]. Although
+all the citation distributions in Fig. 1A seem to be heavy-tailed distribu-
+tions, which are commonly referred to as the power law, we verified that
+the distributions sincerely follow the power law via comparison with alterna-
+tive distributions (e.g., log-normal or exponential). In the comparison with
+the exponential distribution, all distributions were found to be more likely
+to be power-law distributions rather than exponential (p < 0.001). However,
+comparison with the log-normal distribution was unclear. Only non-COVID-
+19-related papers published in 2021 better fit the power-law distribution in a
+statistically significant manner, while the other three were inconclusive (p var-
+ied 0.48–0.60). In this study, we estimated the power-law exponent with the
+assumption of a simple power law (y ∼ xk) regardless of the best fit distribu-
+tion, as we were more interested in comparing the thickness of the tails than
+in determining the exact exponents.
+Reproduction of the journal impact factors
+Although we extracted the total number of publications in the WOS with
+a complete copy of the WOS provided by Clarivate, minor differences can
+be presented mainly because the WOS does not report detailed methods to
+filter the dataset, e.g., dump dates and the coverage of citable items. Thus,
+to reproduce and estimate the journal impact factors (IFs), we followed the
+method used for the JCR impact factor [2] but with an in-house XML copy of
+the Web of Science, as follows:
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+13
+IF = citations received by items published in the past 2 years
+number of citable items published in the past 2 years .
+(1)
+We limited the citable items to those belonging to the journals indexed in
+SCI-Expanded, SSCI, and A&HCI. We also considered as citable items only
+articles, review papers, and proceedings papers in terms of publication type;
+however, publication types were not considered when computing the number
+of citations.
+Note that, as of 2020, Clarivate Inc. now considers early access publications
+as regular publications and includes them in the calculation of IF. For instance,
+if an article is published as early access in 2020 and officially published in
+2021, then the article is counted as a citable item published in 2020, taking
+into account the references as the citations occurred in 2020. The article is not
+considered in 2021.
+With this procedure, we successfully reproduced IF scores highly correlated
+with the IFs provided by Clarivate JCR (Pearson r = 0.99; see Fig. S8). In
+this study, we refer to the value computed from Eq. 1 as IF instead of the
+impact factor provided by JCR unless otherwise specified. When computing
+the IFs excluding COVID-19-related papers, we counted out the COVID-19-
+related citable items and their references from the denominator and numerator
+in Eq. 1, respectively.
+Acknowledgement
+This research was supported by the MSIT (Ministry of Science and ICT),
+Republic of Korea, under the Innovative Human Resource Development
+for Local Intellectualization support program (IITP-2022-RS-2022-00156360)
+supervised by the IITP (Institute for Information & Communications Tech-
+nology Planning & Evaluation). This work was also supported by the National
+Research Foundation of Korea (NRF) funded by the Korean government (grant
+No. NRF-2022R1C1C2004277 (T.Y.) and 2022R1A2C1091324 (J.Y.)). The
+Korea Institute of Science and Technology Information (KISTI) also supported
+this research with grant No. K-23-L03-C01 (J.Y.L., J.P.) and by providing
+KREONET, a high-speed Internet connection. The funders had no role in the
+study design, data collection and analysis, decision to publish, or preparation
+of the manuscript.
+Ethics declarations
+Competing interests
+The authors declare no competing interests.
+
+Springer Nature 2021 LATEX template
+14
+Auditing citation polarization during the COVID-19 pandemic
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+Aziz Bahey, A. & Al-Thani, H. Publications and retracted articles of
+covid-19 pharmacotherapy-related research: A systematic review. Science
+Progress 104(2):00368504211016,936 (2021).
+[32] Khan, H., Gupta, P., Zimba, O. & Gupta, L. Bibliometric and altmet-
+ric analysis of retracted articles on covid-19. Journal of Korean Medical
+Science 37(6) (2022).
+[33] Alstott, J., Bullmore, E. & Plenz., D. powerlaw: a python package for
+analysis of heavy-tailed distributions. PloS One 9(1):e85,777 (2014).
+
+Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+1
+Supplementary Information for
+Auditing citation polarization during the COVID-19
+pandemic
+Taekho You, Jinseo Park, June Young Lee, Jinhyuk Yun∗
+∗Corresponding author. Email: [email protected]
+This PDF file includes:
+Supplementary table S1
+Figures. S1 to S8
+
+Springer Nature 2021 LATEX template
+2
+Auditing citation polarization during the COVID-19 pandemic
+Figure S1 Probability density function of the citation time difference between
+the publication and the citation month of the papers. For the plot, the COVID-19-
+related and non-COVID-19-related papers published in 2020 and 2021 were used.
+Figure
+S2 Citation
+distribution
+of
+COVID-19-related
+and
+non-COVID-19-
+related papers. The citation distribution that can contribute to the annual IF calculation
+(left) is the same distribution as in Fig. 1A. The separated citation distributions in one-
+year (middle) and two-year (right) time gaps show the same pattern. Both plots show that
+COVID-19-related papers have a heavier-tailed distribution.
+
+Non-COviD-19-relatedpapers
+COVID-19-relatedpapers
+10'0
+Probability density function
+0.06
+0.05
+0.04
+0.03
+乙O'0
+0.01
+0.00
+0
+5
+10
+15
+20
+Citationsafterpublication (month)100
+100
+100
+10-1
+10-1
+10-1,
+Function
+10-2,
+10-2
+Density
+10-3
+10-3,
+10-3,
+Cumulative
+10-4
+10-4
+10-4 
+Non-COVID-19 papers (2017)
+10-5,
+Non-COVID-19 papers (2017+2018)
+10-5 
+Non-COVID-19 papers (2018)
+10-5
+cOVID-19 papers (2018+2019)
+COVID-19 papers (2019)
+Non-COVID-19 papers (2017)
+Non-COVID-19 papers (2018+2019)
+Non-COVID-19 papers (2019)
+Non-COVID-19 papers (2018)
+10-6
+COVID-19 papers (2019+2020)
+COVID-19 papers (2020)
+COVID-19 papers (2019)
+10-6
+Non-COVID-19 papers (2019+2020)
+Non-COVID-19 papers (2020)
+Non-COVID-19 papers (2019)
+100
+101
+102
+103
+104
+100
+101
+102
+103
+104
+100
+101
+102
+103
+104
+Number of Citations Received
+Number of Citations Received
+Number of Citations ReceivedSpringer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+3
+Table S1 Relative ratio of surplus impact factor (IF) from publishing COVID-19-related papers by journal category classified by JCR. The bold
+numbers represent the highest ratio of surplus IF in each category.
+Journal Category
+Top 10%
+10–20%
+20–30%
+30–40%
+40–50%
+50–60%
+60–70%
+70–80%
+80–90%
+90–100%
+Agricultural Science
+1.664
+1.098
+1.111
+1.040
+1.002
+1.029
+1.078
+1.051
+1.039
+1.062
+Arts & Humanities, Interdisciplinary
+1.209
+1.399
+1.146
+1.046
+0.999
+1.086
+1.165
+0.995
+0.990
+0.982
+Biology & Biochemistry
+1.286
+1.103
+1.101
+1.090
+1.064
+1.067
+1.115
+1.062
+1.066
+1.081
+Chemistry
+1.104
+1.043
+1.040
+1.042
+1.038
+1.055
+1.063
+1.026
+1.046
+1.109
+Clinical Medicine
+1.508
+1.251
+1.189
+1.162
+1.152
+1.127
+1.142
+1.106
+1.102
+1.111
+Computer Science
+1.112
+1.118
+1.125
+1.079
+1.061
+1.038
+1.090
+1.041
+1.071
+1.053
+Economics & Business
+1.423
+1.161
+1.180
+1.115
+1.105
+1.088
+1.103
+1.075
+1.078
+1.024
+Engineering
+1.044
+1.109
+1.041
+1.048
+1.029
+1.053
+1.027
+1.026
+1.037
+1.146
+Environment/Ecology
+1.388
+1.209
+1.201
+1.135
+1.129
+1.112
+1.126
+1.106
+1.074
+1.121
+Geosciences
+1.024
+1.012
+1.037
+1.018
+1.018
+1.002
+1.021
+1.033
+1.041
+1.001
+History & Archaeology
+1.162
+1.165
+1.121
+1.072
+1.114
+0.994
+1.069
+1.003
+0.988
+0.969
+Literature & Language
+1.106
+1.175
+1.111
+1.205
+1.026
+1.136
+1.110
+1.050
+1.019
+1.045
+Material Science
+1.017
+1.018
+1.010
+1.024
+1.009
+1.010
+1.008
+1.011
+1.024
+1.054
+Mathematics
+1.053
+1.046
+1.139
+1.039
+1.052
+1.062
+1.061
+1.016
+1.080
+1.013
+Multidisciplinary
+1.102
+1.086
+1.081
+1.081
+1.056
+1.066
+1.057
+1.048
+1.061
+1.030
+Philosophy & Religion
+1.232
+1.219
+1.221
+1.155
+1.067
+1.141
+1.131
+1.010
+1.066
+1.046
+Physics
+1.058
+1.057
+1.039
+1.025
+1.018
+1.030
+1.030
+1.028
+1.014
+1.073
+Plant & Animal Science
+1.102
+1.046
+1.055
+1.020
+1.068
+1.063
+1.096
+1.022
+1.039
+1.027
+Psychiatry/Psychology
+1.666
+1.341
+1.183
+1.153
+1.121
+1.227
+1.137
+1.176
+1.137
+1.136
+Social Sciences, General
+1.389
+1.218
+1.182
+1.147
+1.130
+1.073
+1.087
+1.070
+1.088
+1.033
+Visual & Performing Arts
+1.088
+1.095
+1.069
+1.109
+1.049
+1.138
+1.038
+1.041
+1.025
+1.115
+
+Springer Nature 2021 LATEX template
+4
+Auditing citation polarization during the COVID-19 pandemic
+Figure S3 Probability density function of journals with IFs that increased or
+decreased by publishing COVID-19-related papers. The IFs of 763 journals (16%)
+decreased while the IFs of 4004 journals (84%) increased. A The pdf for the absolute increase
+in IF. B The pdf for the relative increase in IF divided by the IF excluding COVID-19-related
+papers.
+Figure S4 Relative IF increase by publishing COVID-19-related papers relative
+to the journal’s IF. The extent of IF increase is divided by the IF excluding COVID-19-
+related papers. The red squares and error bars respectively show the average value and the
+standard deviation of the increased IF in log-scale.
+
+Increased
+103
+Increased
+Decreased
+Decreased
+102
+102
+Probability density function
+101
+Probability density function
+101
+100
+100
+10
+10-1
+10-2
+10-2,
+10-3
+10-4
+10-3
+10-5
+10-3
+10-1
+100
+101
+102
+10-5
+10-3
+10-1
+100
+101
+Increase (decrease) of IF
+Relative increase fdecreasej of IF101
+papers
+100
+[9-related 
+by COVID-1!
+10-2
+10-2
+10-1
+100
+101
+102
+Impact factor (IF)Springer Nature 2021 LATEX template
+Auditing citation polarization during the COVID-19 pandemic
+5
+Figure S5 Fraction of COVID-19-related papers published in journals by the
+journal IFs. The red squares and error bars respectively show the average value and the
+standard deviation of the fraction of COVID-19-related papers in log-scale. The Pearson
+correlation is 0.110.
+Figure S6 Change in journal ranking by publishing COVID-19-related papers.
+The values show the change rate between ranking groups by publishing COVID-19-related
+papers. Both Pearson and Spearman rank correlations of the IF ranks are 0.99. A Rank
+change of all journals that published COVID-19-related papers in their category. B Rank
+change of the journals that published COVID-19-related papers, where less than 10% of the
+journals published COVID-19-related papers in their category.
+
+A
+B
+ 1.0
+papers
+Top 10%
+0.90
+0.11
+EO0
+0.02
+0.01
+0.00
+0.00
+0.00
+0.01
+0.00
+Top 10%
+1.00
+0.08
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+paper
+0.8 
+10-20%
+0.10
+0.75
+0.13
+0.03
+0.02
+0.01
+0.01
+0.00
+0.00
+0.00
+10-20%
+0.00
+0.92
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+ COVID-19
+ COVID-19
+ 0.8 
+20-30%
+0.00
+0.14
+0.68
+0.15
+0.06
+0.02
+0.01
+1O'0
+0.01
+0.00
+20-30%
+0.00
+0.00
+1.00
+0.12
+0.00
+0.08
+0.00
+0.00
+0.00
+0.00
+30-40%
+0.00
+0.01
+0.02
+0.01
+0.00
+0.00
+ 0.6 
+0.14
+0.64
+0.14
+0.06
+30-40%
+0.00
+0.00
+0.00
+0.88
+0.17
+0.00
+0.00
+0.00
+0.00
+0.00
+including
+including
+ 0.6 
+40-50%
+0.00
+0.00
+0.00
+0.15
+0.61
+0.13
+0.05
+0.02
+0.01
+0.00
+40-50%
+0.00
+0.00
+0.00
+0.00
+0.83
+0.00
+0.09
+0.00
+0.00
+0.00
+I ranking by IF i
+50-60%
+0.00
+0.00
+0.00
+0.01
+0.16
+0.63
+0.12
+0.05
+0.01
+0.00
+0.4
+50-60%
+0.00
+0.00
+0.00
+0.00
+0.00
+0.92
+0.18
+0.00
+0.00
+0.00
+F
+by
+ 0.4
+60-70%
+0.00
+0.00
+0.00
+0.00
+0.01
+0.15
+0.64
+0.14
+0.05 
+0.01
+60-70%
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.73
+0.09
+0.00
+0.00
+I ranking !
+70-80%
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.13
+0.65
+0.12
+0.02
+70-80%
+0.00
+0.00
+0.00
+0.00 
+0.00
+0.00
+0.00
+0.91
+0.09
+0.00
+ 0.2 
+ 0.2 
+80-90%
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.11
+0.11
+euuno
+0.71
+80-90%
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+0.82
+0.00
+0.00
+0.00
+0.00
+0.00 
+0.00
+0.00
+0.00
+0.00
+0.07
+0.86
+90-100%
+0.00
+0.00
+0.00
+0.00
+0.00 
+0.00 
+0.00
+0.00
+0.09
+1.00
+0.0
+ 0.0
+oo
+ol
+ola
+oo
+ola
+10
+lo
+olo
+olo
+ofo
+30°
+20
+TOP
+70
+-100°
+ToF
+30
+JournalrankingbyIF
+excluding
+COVID-19
+papers
+Journal ranking
+by IF
+excluding 
+COVID-19
+papers100
+Fraction of COVID-19-related papers
+10-4
+10-2
+10-1
+100
+101
+102
+Impact factorSpringer Nature 2021 LATEX template
+6
+Auditing citation polarization during the COVID-19 pandemic
+Figure S7 Change in the Gini coefficients of the JCR categories by the number
+of published COVID-19-related papers. A positive correlation between the number
+of COVID-19-related papers and the changes in the Gini coefficients of the categories is
+observed (Pearson r = 0.545).
+Figure S8 Correlation between the IFs provided by JCR and those calculated
+in this paper. The Pearson correlation is high for both years 2020 and 2021.
+
+work
+0.20
+19-related
+COVID-1
+0.15
+Ag
+0.10
+increased
+coefficient 
+0.05
+00'0
+Gini
+100
+101
+102
+103
+Number of COVID-19-related papers2020
+2021
+Pearson:0.997
+Pearson:0.998
+500
+250
+400
+200
+lated
+150
+Icul
+lcul
+cal
+ 200
+F
+100
+100
+50
+0
+0
+100
+200
+300
+400
+500
+0
+50
+100
+150
+200
+250
+300
+IF provided by JCR
+IF provided by JCR

5tAzT4oBgHgl3EQf9_5e/content/tmp_files/2301.01927v1.pdf.txt

@@ -0,0 +1,1066 @@
+Towards simultaneous coherent radiation in the
+visible and microwave bands with doped
+molecular crystals
+Hao Wu,†,‡,§ Tong Li,†,‡,§ Zhang-Qi Yin,† Jiyang Ma,∗,†,‡ Xu-Ri Yao,†,‡ Bo
+Zhang,†,‡ Mark Oxborrow,¶ and Qing Zhao†,‡
+†Center for Quantum Technology Research and Key Laboratory of Advanced Optoelectronic
+Quantum Architecture and Measurements (MOE), School of Physics, Beijing Institute of
+Technology, Beijing 100081, China
+‡Beijing Academy of Quantum Information Sciences, Beijing 100193, China
+¶Department of Materials, Imperial College London, South Kensington, SW7 2AZ London,
+United Kingdom
+§These authors contributed equally: Hao Wu, Tong Li
+E-mail: [email protected]
+Abstract
+Coherent sources exploiting the stimulated emission of non-equilibrium quantum
+systems, i.e. gain media, have proven indispensable for advancing fundamental research
+and engineering. The operating electromagnetic bands of such coherent sources have
+been continuously enriched for increasing demands. Nevertheless, for a single bench-
+top coherent source, simultaneous generation of radiation in multiple bands, especially
+when the bands are widely separated, present formidable challenges with a single gain
+medium. Here, we propose a mechanism of simultaneously realizing the stimulated
+1
+arXiv:2301.01927v1  [physics.optics]  5 Jan 2023
+
+emission of radiation in the visible and microwave bands, i.e. lasing and masing ac-
+tions, at ambient conditions by utilizing photoexcited singlet and triplet states of the
+pentacene molecules that are doped in p-terphenyl. The possibility is validated by the
+observed amplified spontaneous emission (ASE) at 645 nm with a narrow linewidth
+around 1 nm from the pentacene-doped p-terphenyl crystal used for masing at 1.45
+GHz and consolidated by a 20-fold-lower threshold of ASE compared to the reported
+masing threshold. The overall threshold of the pentacene-based multiband coherent
+source can be optimized by appropriate alignment of the pump-light polarization with
+the pentacene’s transition dipole moment. Our work not only shows a great promise on
+immediate realization of multiband coherent sources but also establishes an intriguing
+solid-state platform for fundamental research of quantum optics in multiple frequency
+domains.
+Introduction
+Coherent sources that capable of generating coherent electromagnetic radiation have served
+as crucial ingredients enabling numerous breakthroughs in the fields of physical, environmen-
+tal, and biological sciences. A well-established approach to achieve coherent electromagnetic
+radiation is to exploit the stimulated emission process1 induced by interactions between elec-
+tromagnetic fields and matters. Stemming from the discovery of the stimulated emission,
+optical lasers spanning from the ultraviolet to the near-infrared region, together with their
+forerunners and microwave analogs, masers, have become indispensable coherent sources for
+applications in telecommunication,2 metrology,3,4 sensing,5–7 machining8 and quantum infor-
+mation.9,10 In addition, the recent development of terahertz11 and X-ray12 lasers has offered
+alluring promise of shedding light on astronomical observations, spectroscopy and structural
+biology. The rich variety of applications require distinct coherent sources operational across
+the extremely wide electromagnetic spectrum from microwave to X-rays. Even though the
+underlying mechanism of those coherent sources is the same (i.e. the stimulated emission),
+2
+
+their practical realizations are completely different in terms of the components, structures
+and scale, rendering simultaneous generation of coherent radiation in widely separated spec-
+trum bands with a single source hitherto unattainable on the bench top.
+Gain media, constituted by the quantum systems with non-degenerate energy levels, are
+the core determining radiation wavelengths of the stimulated emission, therefore, vital for
+realizing multiband coherent sources. It is not scarce for quantum systems to possess multiple
+transitions across different spectrum bands by their intrinsic properties (e.g. large quantum
+numbers13) and/or external manipulations with electric/magnetic fields.14 Ruby (chromium
+ions-doped aluminum oxide) is a representative gain medium which can be employed for
+both solid-state masers15 and lasers,16 i.e. capable of generating coherent radiation in the
+microwave and visible regions, while the vastly different experimental setups, especially the
+cryogenic and magnetic-field requirements for ruby masers, have resulted in no demonstration
+of simultaneous maser and laser actions of ruby to date. More recently, the negatively charged
+nitrogen-vacancy defects (NV−) in diamond have been demonstrated to be promising solid-
+state maser17 and laser18 gain media at ambient conditions. However, the preparations and
+material properties of the NV− diamonds employed for the maser and laser applications
+are rather different. The diamonds were synthesized via the routes of high pressure high
+temperature (HPHT) and chemical vapor deposition (CVD) for the NV− laser and maser,
+respectively, which gives rise to the different concentrations of the doped nitrogen atoms and
+NV−. The concentration of the gain media, i.e. NV−, required for the laser action is 1.4
+p.p.m. which is about four folds higher than that (0.36 p.p.m.) used for the NV− maser.
+With respect to the performance of the NV− diamond based coherent sources, the relatively
+weak output (∼1 pW) of the NV− maser17 and the broad spectral linewidth (20 nm) of the
+NV− laser18 still need to be substantially improved for practical considerations. In addition
+to the solid-state systems, due to the rich energy structures, the stimulated emission of
+radiation ranging from the microwave to infrared (IR) domain have been observed in vapor
+of Rydberg atoms19,20 but their ability of simultaneous generation of coherent radiation in
+3
+
+multiple wavelength domains is still not evident. Moreover, the limited number of atoms
+has also restricted the power of such coherent sources. The output power of the Rydberg
+masers has been reported to be up to 10 pW21 and the pulse energy of a Rydberg IR laser
+was approximately 1 µJ.20
+Compared to the gain media mentioned above, doped molecular crystals combines the key
+advantages of the inorganic solid-state systems (e.g. robustness and ease of integration) with
+rich opportunities of tuning the energy levels of the quantum systems, like Rydberg atoms,
+through bottom-up engineering of the guests and host matrices.22,23 While sustaining the
+satisfying optical, magnetic and electronic properties, the doping concentrations of molecular
+crystals are tunable up to tens of thousands p.p.m.24 implying the great potential of realizing
+powerful coherent sources with such gain media. Among numerous doped molecular crystals,
+pentacene-doped p-terphenyl (Pc:Ptp) has attracted extensive attention especially in the
+last decade.
+Pc:Ptp is a low-cost and easy-fabricated single organic crystal, which can
+be produced in bulk with a high quality.25 Due to the doping structure, the functional
+dopants, i.e. pentacene molecules, are well protected by the matrix of p-terphenyl, which
+not only possess great steadiness and durability but also emerge appealing photophysical
+properties (which are lacking in neat pentacene) well suited for multidisciplinary applications,
+such as photovoltaics,24 dynamic nuclear polarization (DNP)26 and quantum information
+processing.10,27,28 In particular, Pc:Ptp is the first, and so far the only doped molecular crystal
+capable of masing at room temperature in Earth’s field29 and the superior maser output at
+a level of milliwatt (i.e. 109 times higher than that of NV− masers) has yet to be surpassed
+by other room-temperature maser gain media. The maser application as well as the recent
+applications mentioned above mainly exploits the photoexcited triplet states of pentacene in
+p-terphenyl. It is worth noting that the singlet states of pentacene are also intriguing due to
+their photoluminescent properties, owing to which the stimulated emission of radiation in the
+visible region manifesting as amplified spontaneous emission (ASE) have been observed30,31
+implying the potential of the pentacene molecules to be suitable gain media for lasing as
+4
+
+well. However, since the ASE phenomena were observed with pentacene doped in trans-1,4-
+distyrylbenzene (trans-DSB)30 and 1,4-bis(2-cyano styryl)benzene (2-CSB),31 respectively,
+instead of p-terphenyl, and the dynamics of pentacene’s triplet spins can be modulated by
+the host effects,32 the suitability of these two host matrices for the maser action of pentacene
+remains elusive.
+In this work, we investigate the optical emission properties of the Pc:Ptp crystals em-
+ployed in the previous maser studies6,27,33 and first demonstrate the ASE process in Pc:Ptp at
+645 nm, where molecular crystals rarely reached, under the experimental conditions identical
+to those required for the maser action. Combining spectral analysis with nanosecond time-
+resolved characterizations, the ASE properties of Pc:Ptp are systematically studied at room
+temperature. The obtained ASE spectrum shows an extremely narrow linewidth around 1
+nm that is the narrowest among the reported molecular crystals revealing ASE behaviors.
+Since ASE is a prerequisite for lasing, our results provide solid evidence that Pc:Ptp can
+serve as a solid-state multiband gain medium for emitting coherent microwave and visible
+light simultaneously at ambient conditions.
+Results
+Structural and optical properties of Pc:Ptp
+Throughout the study, pink Pc:Ptp single crystals, shown in Fig. 1a, with a doping concen-
+tration of 1000 p.p.m. was used, which is the same as that employed for room-temperature
+masers.6,27,33 The relatively high doping concentration allowing sufficient pentacene molecules
+for microwave and optical gains arises from the similar molecular packing coefficients of pen-
+tacene and p-terphenyl which favors the formation of solid solutions with these two organic
+substances.34,35 The molecular packing coefficient k can be expressed as k = (z × V0)/V ,
+where z is the number of molecules in the unit cell, V0 and V are the volumes of the molecule
+and unit cell, respectively. The k values of pentacene and p-terphenyl have been determined
+5
+
+b
+c
+400 μm
+ab plane
+a
+d
+e
+S0
+S1
+510 nm
+550 nm
+590 nm
+475 nm
+ 
+λ    = 645 nm
+T2
+T1
+Tx
+Ty
+Tz
+ Intersystem crossing
+Non-
+radiative 
+transition
+Radiative 
+transition
+Maser transition
+  
+fxz = 1.45 GHz
+Internal 
+conversion
+Site 1
+Site 2
+b
+a
+c
+x
+z
+y
+500
+600
+700
+800
+0.0
+0.5
+1.0
+Normalized intensity (a.u.)
+Wavelength (nm)
+ Fluorescence
+ Fitting
+27.6 
+ 0.3 nm
+400
+500
+600
+700
+0.0
+0.5
+1.0
+Normalized absorbance (a.u.)
+Wavelength (nm)
+ Absorbance
+Laser transition
+0-1
+}
+475 nm
+510 nm
+550 nm
+590 nm
+645 nm
+599 nm
+701 nm
+±
+1 mm
+Figure 1: Material characterizations of Pc:Ptp and mechanism of simultaneous
+coherent radiation. a Optical microscopic image of a Pc:Ptp crystal under white-light
+illumination. The cleavage plane, i.e. ab plane, of the crystal is labelled. b Crystal struc-
+ture of the host matrix, p-terphenyl (blue) with substitutionally doped pentacene molecules
+(pink). The two inequivalent doping sites as well as the molecular axes of pentacene are
+labelled. c UV/vis absorption and d Fluorescence spectra of Pc:Ptp with all characteristic
+peaks labelled. The linewidth, i.e. FWHM of the strongest fluorescent peak is determined
+by a Lorentzian fitting. Inset: optical microscopic image of a fluorescent Pc:Ptp with the
+excitation light filtered. e Proposed mechanism of simultaneous lasing and masing actions
+in Pc:Ptp exploiting the transitions of stimulated emission highlighted in the pentacene’s
+singlet (pink) and triplet (orange) manifolds.
+6
+
+to be 0.743 and 0.751 that brings them adjacent to the closest packing condition where k=
+0.74.34 As shown in Fig. 1b, at room temperature, pentacene molecules are substitutionally
+doped in the monoclinic unit cell of p-terphenyl with two inequivalent doping sites.36 The
+structural properties of Pc:Ptp crystals are dominated by the host matrix, i.e. p-terphenyl.
+Therefore, Pc:Ptp also possesses an intrinsic laminar structure with a cleavage (001) (i.e.
+ab) plane25 where the molecules stand (see Fig. 1b). The ‘head-to-head’ packing against
+the ab plane results in the weaker intermolecular interactions compared to the π − π inter-
+actions along the molecular xy plane where x and y denote the long and short axes of the
+molecules, respectively. The cleavage property facilitates the fabrication of Pc:Ptp crystals
+with distinguishable crystal planes. As exhibited in Fig. 1a, the large crystal facet can be
+straightforwardly determined to be the (001) plane due to the obvious delamination shown
+on the edge.
+The optical properties of Pc:Ptp were investigated by measuring its absorption and flu-
+orescence spectra as illustrated in Fig. 1c and 1d. It can be found that the doped crystal
+reveals explicit absorption at wavelengths of 475, 510, 550 and 590 nm, which correspond to
+the characteristic transitions between pentacene’s excited and ground singlet states25,37 as
+depicted in Fig. 1e. According to the absorption spectrum, the highest absorbance locates
+at 590 nm was referred to determine the optimal wavelength for optically pumping Pc:Ptp in
+the following measurements. In terms of the photoluminescent property of Pc:Ptp, Fig. 1d
+indicates that the crystal can generate intense fluorescence at wavelengths of 599 and 645
+nm, corresponding to the 0-0 and 0-1 transitions,30 respectively, as well as a weak emission
+band central at 701 nm under illumination of a green light-emitting diode (LED). The small
+peak near 550 nm is attributed to the emission of LED which was not completed filtered
+during the fluorescence measurements. The full width at half maximum (FWHM) of the
+strongest fluorescence peak at 645 nm is measured to be 27.6±0.3 nm.
+7
+
+Mechanism of the simultaneous lasing and masing actions in Pc:Ptp
+Combining the optical properties obtained above with the reported properties of Pc:Ptp
+masers,29,38 we propose the mechanism of realizing simultaneous lasing and masing actions
+by exploiting both the photoexcited singlet and triplet states of Pc:Ptp crystals. As schemat-
+ically demonstrated in Fig. 1e, the pentacene molecules in the ground singlet state can be
+efficiently promoted to the excited singlet state with an optical pumping at 590 nm, by which
+the population inversion is achieved in the singlet manifold for the stimulated emission of ra-
+diation in the visible region, e.g. 599 or 645 nm where the strong fluorescence was observed.
+In the meantime, due to the spin-orbit coupling, pentacene molecules can also transfer to
+the excited triplet state (T2) via the intersystem crossing with a yield of 62.5% at room
+temperature39 and rapidly decay to the lowest triplet state (T1) via the internal conversion.
+Since the triplet state is metastable, the pentacene molecules will eventually decay back to
+the ground singlet state by either the radiative or non-radiative T1 →S0 transition.40,41 In
+Earth’s field, T1 is non-degenerate and constituted by three sublevels Tx, Ty and Tz due to
+the dipolar interactions of pentacene’s triplet electron spins. The resonance frequencies of
+the triplet sublevels governed by the zero-field-splitting (ZFS) parameters have been deter-
+mined by electron paramagnetic resonance (EPR) measurements42,43 to be 1.45 GHz, 1.344
+GHz and 106.5 MHz for Tx ↔Tz, Ty ↔Tz and Tx ↔Ty transitions, respectively. An alluring
+property of the pentacene’s lowest triplet state is that, upon its generation, the populations
+of the triplet electrons follow a non-Boltzmann distribution in the three sublevels with a ratio
+of Px : Py : Pz= 0.76:0.16:0.0844 at room temperature. The strong population inversion and
+relatively slow spin-lattice relaxation43 between the Tx and Tz sublevels can be exploited for
+realizing the stimulated emission of radiation in the microwave region, i.e. masing. There-
+fore, the optical pumping of Pc:Ptp can simultaneously introduce population inversions in
+both singlet and triplet manifolds fulfilling the prerequisites of the lasing and masing actions
+at ambient conditions.
+8
+
+Spectral analysis of the ASE of Pc:Ptp
+As the masing action has been successfully demonstrated with the Pc:Ptp crystal,27,33 we
+verify the mechanism of multiband coherent radiation proposed above by investigating the
+feasibility of the stimulated emission in the pentacene’s singlet states under the experimental
+conditions similar to that for the maser studies. We measured the emission spectra of Pc:Ptp
+under the optical pumping of a nanosecond pulsed laser which has proven to be sufficiently
+powerful for achieving the threshold of Pc:Ptp masers.38 The pump laser was focused onto
+the cleavage plane of the crystal with a beam diameter of about 5 mm (see Supplementary
+Fig. 2 for the detailed experimental setup). At different pump intensities, the emission
+spectra shown in Fig. 2a were recorded by collecting the emitted light from the edge of the
+sample (see inset in Fig. 2a). It can be seen that at the peak wavelength (i.e. 645 nm) of
+the Pc:Ptp’s emission spectra, the intensity gradually increases while the spectral linewidth
+equal to the FWHM gets narrower with the increment of pump energies implying the ASE
+process occurs. The incomplete emission peaks at the wavelength of 600 nm is due to the
+long pass filter with a cut-on wavelength of 600 nm used to filter out the pump light at 590
+nm. In contrast, the filter used in the fluorescence measurements has a cut-on wavelength
+of 550 nm leading to a complete peak at 600 nm, as shown in Fig. 2a.
+To characterize the ASE process of Pc:Ptp, the intensity as well as the FWHM of the
+strongest emission peak at 645 nm was plotted as a function of the pump intensity as shown
+in Fig. 2b. There are two distinct areas in Fig. 2b, which were respectively fitted by linear
+equations, resulting in a kink behavior of laser-like thresholds. The difference between ASE
+and lasing processes is that in general, there is an optical cavity in the composition of a
+laser, while ASE is the stimulated emission that occurs without a cavity.74 We define the
+kink intensity as the ASE threshold of Pc:Ptp, which is 1.47 mJ cm−2. From the slopes of
+the two fitted lines, it is evident that below the threshold, with the increment of the pump
+intensity, the emission intensity increases slightly while the FWHM narrows rapidly, whereas
+the situation changes oppositely above the threshold. This is because once the pump intensity
+9
+
+a
+b
+d
+c
+e
+Optical pumping
+Emission
+1.2
+Pump intensity 
+0.6
+1.8
+2.4
+0.0
+0.5
+1.0
+ Emission intensity
+ FWHM
+ Fitting
+mJ cm-2
+Normalized intensity (a.u.)
+0
+4
+8
+12
+ FWHM (nm)
+(
+)
+640
+642
+644
+646
+648
+650
+0.0
+4.0
+8.0
+Emission intensity (a.u.)
+Wavelength (nm)
+ Experiment
+ Fitting
+1.33 nm
+400
+500
+600
+700
+2
+4
+6
+8
+10
+67
+31
+72
+69
+49
+71
+70
+69
+68
+66
+65
+49
+65
+64
+63
+62
+61
+60
+59
+58
+57
+49
+4950
+54
+53
+52
+51
+56
+56
+55
+48
+47
+46
+FWHM (nm)
+Wavelength (nm)
+Pc:Ptp
+45
+550
+600
+650
+700
+750
+0.0
+0.5
+1.0
+1.5
+Emission intensity (a.u.)
+Wavelength (nm)
+ Fluorescence spectrum
+ 0.58 mJ cm-2
+ 0.80 mJ cm-2
+ 1.09 mJ cm-2
+ 1.43 mJ cm-2
+(
+400
+500
+600
+700
+800
+900
+0.0
+2.0
+4.0
+6.0
+Emission intensity (a.u.)
+Wavelength (nm)
+ 2.20 mJ cm-2
+Figure 2: Spectral analysis of Pc:Ptp’s ASE process. a Effect of pump intensity in
+the emission spectrum of Pc:Ptp. Inset: schematic of the experimental setup. b Dependence
+of the intensity (pink dots) and FWHM (orange triangles) of the emission peak at 645 nm
+on pump intensity. The kink behavior of the dependence reveals the ASE process whose
+threshold is determined by the point of intersection between the two linear fitting lines
+(black dotted).
+c Emission spectrum of Pc:Ptp measured above the ASE threshold.
+d
+Zoomed-in view of the 645-nm ASE peak (highlighted by a pink dotted box in c) whose
+FWHM is determined by a Lorentzian fitting (black dotted curve). e Comparison of the
+wavelength and FWHM of Pc:Ptp’s narrowest ASE peak with those of the reported organic
+single crystals31,45–72 reviewed in ref.73 Details of the reported organic single crystals are
+summarized in Supplementary Information. The regime of ASE wavelength rarely reached
+by organic single crystals is highlighted with a black dotted box.
+10
+
+exceeds the threshold, the stimulated radiation near the wavelength of 645 nm is substantially
+enhanced, resulting in a rapid increase in the emission intensity in the vicinity of 645 nm
+manifesting a narrowing of the entire emission spectrum while the reduction of FWHM is
+limited by the optical inhomogeneous broadening of pentacene molecules. In Fig. 2c and
+2d, the narrowest emission spectrum arising from Pc:Ptp’s ASE process was obtained with
+a pump intensity of 2.2 mJ cm−2. Compared with the normal fluorescence spectrum of the
+Pc:Ptp crystal in Fig. 2a, the ASE spectra in Fig. 2c and 2d show a substantial narrowing
+of the linewidth from 25 to 1.33 nm, which clearly reveals the feasibility of Pc:Ptp for
+achieving the stimulated emission of radiation in the visible region. Most importantly, the
+ASE threshold determined here, i.e. 1.47 mJ cm−2, is much lower than the maser threshold38
+of Pc:Ptp, 26.3 mJ cm−2 measured with the similar optical pump source, which indicates the
+simultaneous lasing and masing actions can be realized once the maser threshold is fulfilled.
+In addition, we have also compared the ASE performance of Pc:Ptp with various organic
+single crystals in terms of the ASE wavelength and linewidth.73 In Fig. 2e, we summarized
+the reported organic single crystals with evident ASE behaviors of which the measured
+emission linewidths are below 10 nm. The types of the referred crystals can be found in
+the Supplementary Information. As demonstrated in Fig. 2e, the ASE wavelengths of these
+organic single crystals are distributed in various visible bands, especially in the wavelength
+range of 400-600 nm, but leave the regime between 600 and 700 nm (i.e. red-color regime)
+rarely reached. Thus, the ASE wavelength of Pc: Ptp at 645 nm is a good supplement to fill
+in this almost blank regime. Moreover, among the listed crystals, Pc:Ptp has the narrowest
+ASE linewidth of 1.33 nm, as per our knowledge, which is even comparable to some organic
+crystal lasers.75–79 The outstanding monochromaticity reveals the potential of Pc:Ptp to be
+a novel organic solid-state laser gain media.
+11
+
+a
+c
+b
+d
+Exp.
+Sim.
+τem= 6.7 ns
+T1
+|1>
+|3>
+|2>
+|4>
+|5>
+P
+W32
+W23
+A32
+k43
+k35
+k21
+k51
+τem= 3.4 ns
+S1
+S0
+2
+4
+6
+8
+3
+4
+5
+6
+7
+Pump intensity mJ cm-2
+Emission lifetime (ns)
+4
+8
+12
+16
+20
+ W32 ( × 107 s-1)
+τem, cal= 6.7 ns
+τem, cal= 3.4 ns
+τem, cal = A32+W32+k35
+1
+0.0
+0.5
+1.0
+Normalized intensity (a.u.)
+ 1.13 mJ cm-2
+ 2.20 mJ cm-2
+ 3.16 mJ cm-2
+ 4.09 mJ cm-2
+ 5.12 mJ cm-2
+ 6.14 mJ cm-2
+ 7.16 mJ cm-2
+Pump enhancement
+0
+10
+20
+30
+40
+0.0
+0.5
+1.0
+ 1.13 mJ cm-2
+ 2.20 mJ cm-2
+ 3.16 mJ cm-2
+ 4.09 mJ cm-2
+ 5.12 mJ cm-2
+ 6.14 mJ cm-2
+ 7.16 mJ cm-2
+Pump enhancement
+Normalized intensity (a.u.)
+Time (ns)
+(k52)
+(
+)
+Figure 3: Kinetic analysis of Pc:Ptp’s ASE process. a Pump-intensity-dependent
+emission decays of Pc:Ptp measured at 645 nm. The emission lifetimes are obtained with an
+exponential fitting. b Five-level kinetic model accounting for the pump-intensity-dependent
+emission decays of Pc:Ptp. c Simulation results of the pump-intensity-dependent emission
+decay of Pc:Ptp on the basis of the five-level model in b. d Simulated rates of stimulated
+emission W32 (pink) as a function of pump intensity. The pump-intensity-dependent emission
+lifetimes (orange) are calculated according to the equation embedded.
+12
+
+Kinetic analysis of the ASE of Pc:Ptp
+Emission lifetimes are important parameters that can be employed to interpret the kinetic
+processes involved in the electronic states upon photoexcitation. We therefore further an-
+alyze the kinetic behaviors of the ASE process of Pc:Ptp based on the emission lifetime
+measurements. The associated experimental setup can be found in the Supplementary In-
+formation Fig. 3. As the fluorescence lifetime of Pc:Ptp has been estimated to be around
+9 ns39 at room temperature, a photodetector with a time resolution of 1 ns resolution was
+employed in our setup for capturing the kinetic process accurately. Fig. 3a shows the emis-
+sion decays obtained with different pump intensities. By exponential fittings of the decay
+curves, we found the emission lifetime was decreased from 6.7 to 3.4 ns (as indicated by
+the black arrow in Fig. 3a) with enhanced optical pumping. The emission lifetime of 6.7 ns
+obtained with the relative weak pumping is close to the reported value of 9 ns.39 The faster
+decays observed with the stronger optical pumping implies a pump-intensity-dependent ki-
+netic process that is included in the emission process. This behavior is consistent with the
+characteristic of an ASE process that the higher pump intensity will lead to enhanced stim-
+ulated emission induced by the increased photons generated from spontaneous emission. To
+fully characterize the observed pump-intensity-dependent emission decays, we constructed a
+five-level kinetic model comprising both singlet and triplet states of the pentacene molecules
+as demonstrated in Fig. 3b. The origins of the photoexcited singlet and triplet states are
+similar to that demonstrated in Fig. 1c. To reduce the complexity of the kinetic model,
+the numbers of the vibrational levels included in the ground (S0) and first excited singlet
+states (S1) were decreased to two, as illustrated by |1⟩ and |2⟩ of S0, and |3⟩ and |4⟩ of S1
+in Fig. 3b. In addition, due to the extremely fast internal conversion between T2 and T1 in
+a time scale of femtosecond to picosecond,80 the model was further simplified by assuming
+the direct intersystem crossing from S1 to T1, i.e. from the lowest vibrational level of S1,
+|3⟩ to |5⟩ shown in Fig. 3b. Thus, the kinetic processes involved in the five-level model are
+the optical pumping (|1⟩ → |4⟩), the relaxation between the vibrational levels in the singlet
+13
+
+manifold (|4⟩ → |3⟩ and |2⟩ → |1⟩), the spontaneous emission (|3⟩ → |2⟩), the simulated
+emission (|3⟩ → |2⟩) and absorption (|2⟩ → |3⟩) and the intersystem crossing (|3⟩ → |5⟩ and
+|5⟩ → |1⟩ (|2⟩)).
+Based on the kinetic processes, we derived a set of coupled rate equations to simulate
+the observed emission decays as a function of the pump intensity (see Supplementary In-
+formation). We found the simulated decay curves shown in Fig. 3c can well reproduce the
+measured emission decays as well as the dependence of the emission lifetimes with the pump
+intensity by a set of stimulated transition rates, W32 and W23 (see Fig. 3b and 3d). The
+stimulated emission rate, W32, obtained from the simulation shows an almost linear increase
+from 4×107 to 1.8×108 s−1 (exceeding the spontaneous emission rate,39 A32 = 4.2×107 s−1)
+with the enhanced pump intensity that reveals the transition of the dominant kinetic process
+in the emission decay from the spontaneous emission to the stimulated emission, i.e. ASE
+occurs. The emission lifetimes τem,cal in Fig. 3d were calculated with τem,cal =
+1
+A32+W32+k35
+where k35 = 6.9 × 107 s−1 is the rate of the intersystem crossing.39
+Optimization of the ASE efficiency
+It is known that the efficiency of the transition of molecules in the ground singlet state to the
+excited singlet state can be maximized by aligning the polarization of pump light with the
+molecules’ transition dipole moments.81 For the pentacene molecules doped in p-terphenyl,
+the pentacene’s short axis (i.e. the y axis in Fig. 1b), almost parallel to the ab cleavage
+plane of the crystal,25 coincides with the transition dipole moment of the lowest spin allowed
+transition of pentacene.82 Therefore, we further attempted to optimize the ASE efficiency
+by enhancing the singlet transition probability of the pentacene’s molecules which would
+benefit the realization of a low-threshold Pc:Ptp laser in the future.
+Fig. 4a schematically illustrates the experimental setup (see Methods and Supplementary
+Information for more details) where a horizontally polarized laser beam was focused by a
+combination of a reversely placed beam expander and a convex lens and propagated perpen-
+14
+
+0
+100
+200
+300
+1.2
+1.4
+1.6
+1.8
+ASE threshold (mJ cm   )
+-2
+Angle (°)
+1.2
+1.8
+2.4
+3.0
+  Experiment
+  Fitting
+  95% Confidence interval
+Emission intensity (a.u.)
+a
+b
+c
+Beam expander
+(reversely placed)
+Convex lens
+Rotational stage
+Pump laser
+Pc:Ptp
+Horizontal polarization
+33.7°
+33.7°
+θ
+Light polarization
+Transition dipole moment
+PLmax
+PLmin
+b  axis
++
+-
+ab
+1
+D
+ab
+2
+D
+ab plane
+d
+0°
+Figure 4: Dependence of ASE performance on the alignment of crystal orien-
+tation with pump-light polarization. a Schematic of the experimental setup. b The
+emission intensity (upper subplot) and ASE threshold (lower subplot) of Pc:Ptp’s ASE peak
+at 645 nm as a function of the rotated angle of the crystal. The measured data sets (squares
+and triangles) are fitted by sinusoidal functions (solid curves) with a period of 180◦ and 95%
+confidence interval bands (shadowed areas). Error bars denote the standard errors of the
+data. c Schematic diagram illustrating the orientations of pentacene’s short molecular axes
+(purple sticks), i.e., the transition dipole moments (dashed lines) projected into the Pc:Ptp
+crystal’s ab plane. The angle between pentacene’s short axes and light polarization (solid
+lines with arrows) is defined by θ. The maximum and minimum emission intensities obtained
+respectively with θ = 33.7◦ and −56.3◦ are indicated in the left bottom corner.
+15
+
+dicular with respect to the cleavage plane (i.e. ab plane) of the Pc:Ptp crystal fixed on a
+rotational sample disk. Under the excitation of a fixed pump intensity exceeding the ASE
+threshold, the strong emission intensity resulting from the ASE process shown in Fig. 4b was
+measured when the crystal was rotated within the plane where the cleavage facet locates.
+It can be seen that the emission intensity shows an angular dependence, and the periodic
+behavior can be fitted by a sinusoidal function with a period of 180◦, i.e. the maximum
+and minimum emission intensities occur with a 90◦ interval. The orthogonal correlation
+can be explained by the convolution effect of the alignments between the light polarization
+and the transition dipole moments of the pentacene molecules doped in two inequivalent
+sites. As shown in Fig. 4c, the projections of the pentacene’s transition dipole moments
+into the ab plane, Dab
+i
+(i = 1 and 2) are parallel to the short molecular axes of the two
+groups of pentacene molecules, and thus, symmetrical about the crystal b-axis according to
+the room-temperature crystal structure of p-terphenyl.83 The angles of the two transition
+dipole moments with respect to the b-axis are both 33.7◦. By assuming the transition dipole
+moments of the two groups of pentacene molecules only vary in terms of the orientation,
+the obtained emission intensity is proportional to |Dab
+1 · E|2 + |Dab
+2 · E|2 where E is the
+electric field vector of the laser light in the ab plane.84 By denoting the angle between the
+light polarization and one of the transition dipole moments to be θ (−90◦ < θ ≤ 90◦),
+the emission intensity is found to be proportional to 1 + cos[( 2θ
+180π) − 67.4
+180 π] cos ( 67.4
+180 π) which
+implies a modulation of the emission intensity with a period of 180◦ and matches with our
+measurements. It can also be found that the maximum and minimum emission intensities
+should be obtained when θ = 33.7◦ and −56.3◦ which correspond to the scenarios where the
+light polarization is parallel and perpendicular to the b axis, respectively.
+Moreover, the ASE threshold was measured as a function of the rotation angle which
+reveals a similar periodic trend and orthogonal relationship but with a ∼ 90◦ offset of the
+extreme points with respect to those in Fig. 4d. This offset is due to that the highest singlet
+transition probability indicated by the strongest emission intensity in Fig. 4d facilitates the
+16
+
+buildup of the population inversion in the singlet manifold and thus reduces the threshold
+of achieving the ASE process, and vice versa.
+Therefore, by adjusting the angle of the light polarization with respect to the pentacene’s
+transition dipole moment, the highest singlet transition probability can be achieved, offering
+the advantages of a two-fold enhancement of the emission intensity and a reduction of the
+ASE threshold by around 30% (see Fig. 4b and 4d) compared to those measured at an
+orthogonal position. This strategy will be beneficial for lowering not only the lasing threshold
+of Pc:Ptp, but also its masing threshold, because the facilitated transition to the excited
+singlet state can also lead to the more efficient generation of the pentacene’s triplet states
+via the intersystem crossing.
+Discussion
+In summary, our study reveals the unexplored potential of Pc:Ptp crystals as room-temperature
+laser gain media which has been overlooked in previous fluorescent and magnetic-resonance
+spectroscopic studies44,81,85 on Pc:Ptp’s photoexcited spin states. Even without an optical
+cavity, the stimulated emission observed at 645 nm with a narrow linewidth around 1 nm
+shows a great promise of Pc:Ptp lasers to fill the wavelength gap of the existing organic
+solid-state lasers. Most importantly, since Pc:Ptp masers have been realized with the identi-
+cal crystals and optical-pumping conditions,38 our findings prove the feasibility of achieving
+simultaneous lasing and masing actions with Pc:Ptp crystals at room temperature.
+The next step will be to fabricate a multiband coherent device by incorporating a Pc:Ptp
+crystal with a hybrid cavity architecture supporting both resonances at 645 nm and 1.45 GHz.
+Considering the volumes of the three-dimensional (3D) dielectric microwave cavities29,86 and
+the Pc:Ptp crystals employed in the Pc:Ptp masers, a Fabry-P´erot optical cavity could
+be a compatible choice for promoting the lasing action while not perturbing the microwave
+electromagnetic modes in the 3D dielectric cavities. The pumping threshold of the multiband
+17
+
+coherent radiations can be minimized by appropriate alignments of the pentacene’s short
+molecular axis with the polarization of the optical pumping, as well as the magnetic field
+of the electromagnetic mode in the microwave cavity.86 We envision that the correlation
+and manipulation of the optical and microwave photons simultaneously generated by the
+proposed multiband coherent source are worth being investigated for fundamental tests of
+quantum optics, the possibility of phase locking for development of self-referenced frequency
+combs and optimization of the solid-state quantum sensors exploiting the nonlinear behaviors
+of the stimulated emission in either the microwave6 or visible5 band.
+Methods
+Sample preparation
+A Pc:Ptp single crystal with a doping concentration of 1000 p.p.m. was grown with the
+Bridgman method as reported in ref.29 The as-grown Pc:Ptp crystal was cut to obtain a
+cleavage facet which was successively polished by abrasive papers, 0.1-µm cerium oxide
+powder and 0.05-µm aluminum oxide powder. The surface parallel to the finished facet was
+polished by repeating the above procedures.
+Optical characterizations
+The UV/vis absorption spectrum of Pc:Ptp was collected using a UV-visible-near infrared
+spectrophotometer (Lambda 1050+, PerkinElmer). The fluorescence spectrum of Pc:Ptp
+was collected using a home-built setup whose block diagram is shown in Supplementary
+Information Fig.1. A green LED source was used to illuminate the sample, and the fluo-
+rescence spectrum was collected by an optical spectrum analyzer (Maya 2000 Pro, Ocean
+Optics, resolution 1 nm). The optical microscopic images were taken by a Complementary
+Metal-Oxide-Semiconductor Transistor (CMOS) camera (AP-MV-UH1080, Apico).
+18
+
+ASE measurements
+The ASE properties of Pc:Ptp were determined via a home-built setup illustrated in Supple-
+mentary Information Fig.2. An optical parametric oscillator (OPO) (BBOPO-Vis, Deyang
+Tech.
+Inc., pulse duration 7 ns) pumped by an Nd:YAG Q-switched laser (Nimma-900,
+Beamtech, repetition rate 10 Hz) with horizontal polarized output at 590 nm was used for
+the ASE measurements. The OPO output beam was focused on the sample surface by a
+reversely placed beam expander (2x) and a convex lens with a focal length of 20 cm. The
+beam diameter was 5 mm, which completely covered the sample surface. A 50/50 beam split-
+ter was used to divert the pump light for measuring the pump energy with a energy meter
+(BGS6321, Beijing Institute of Optoelectronic Technology). A long-pass filter with cut-on
+wavelength of 600 nm was used to eliminate the pump light. The ASE signals were collected
+by an optical fiber connected to a high-resolution spectrometer (SpectraPro HRS-750, Pro
+EM 512B, Teledyne Princeton Instruments).
+Emission lifetime measurements
+The experimental setup was similar to that used for the ASE measurements except the
+spectrometer was replaced by a photodetector (DET10A2, Thorlabs, resolution 1 ns). The
+time-domain emission signals under several different pump energies were collected by an
+oscilloscope (WAVERUNNER 6KA, LeCroy).
+Orientation-dependent emission measurements
+The setup was the same as the ASE measurements. The Pc:Ptp crystal was fixed on the
+center of a rotational disk (HRSP40-L, Heng Yang Optics, 0.1◦ resolution) so that the in-
+cident light can propagate perpendicular to the cleavage plane. The crystal was rotated
+with an interval of 30◦ between each measurement. The emission spectra of Pc:Ptp were
+measured at different rotation angles under the same pump intensity of 1.53 mJ cm−2. The
+19
+
+ASE thresholds were measured by varying the pump intensity at different rotation angles
+with an interval of 30◦.
+Acknowledgement
+The authors sincerely thank Shamil Mirkhanov for stimulating discussions and Tan Wang and
+FORTEC Technology (HK) Co.Ltd. for providing us with the monochromator SpectraPro
+HRS-750. H.W. acknowledges financial support from the National Science Foundation of
+China (NSFC) (12204040) and the China Postdoctoral Science Foundation (YJ20210035 and
+2021M700439). Jiyang Ma acknowledge financial support from China National Postdoctral
+Program for Innovative Talents (BX20200057).
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+

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@@ -0,0 +1,1492 @@
+1 
+ 
+An Analysis of Honeypots and their Impact as a 
+Cyber Deception Tactic 
+Daniel Zielinski, Hisham A. Kholidy 
+State University of New York (SUNY) Polytechnic Institute, 
+College of Engineering, Network and Computer Security Department, Utica, NY USA 
+[email protected], [email protected] 
+ 
+Abstract— This paper explores deploying a cyber 
+honeypot system to learn how cyber defenders can use a 
+honeypot system as a deception mechanism to gather 
+intelligence. Defenders can gather intelligence about an 
+attacker such as the autonomous system that the 
+attacker’s IP is allocated from, the way the attacker is 
+trying to penetrate the system, what different types of 
+attacks are being used, the commands the attacker is 
+running once they are inside the honeypot, and what 
+malware the attacker is downloading to the deployed 
+system. We demonstrate an experiment to implement a 
+honeypot system that can lure in attackers and gather all 
+the information mentioned above. The data collected is 
+then thoroughly analyzed and explained to understand 
+all this information. This experiment can be recreated 
+and makes use of many open-source tools to successfully 
+create a honeypot system. 
+Keywords—honeypots, deception mechanism, autonomous 
+system, deceptive honeypot system, open-source tools 
+I. 
+INTRODUCTION 
+Cyber honeypots can be very useful tools when trying to 
+lure in attackers and gain a defensive advantage by learning 
+how attackers are operating and what information they are 
+seeking. Honeypots are very common systems that many 
+enterprise organizations use to improve their security 
+program and gain insight into the common attacks they face. 
+When using a honeypot, defenders can gain various 
+amounts of information about the attackers targeting them. 
+Defenders can then use this information to better protect 
+their “real” environments. Defenders can gain insight into 
+the most popular types of attacks that are being used to 
+break into their honeypot systems and then make sure that 
+their actual systems are properly protected against these 
+attacks. Defenders can also learn what data the attackers are 
+trying to reach, or what exactly the motive of the attackers 
+is. In addition, defenders will be able to learn the different 
+types of malware that attackers are installing onto the 
+honeypot systems. 
+Modern honeypots are not able to look or be configured 
+the same way that they have in the past. This is 
+because over time attackers started learning different ways 
+to detect if what they were attacking is a honeypot. Once an 
+attacker figures out that what they are attacking is a 
+honeypot, they will no longer use their time or resources on 
+the honeypot. This will then make it where the defender will 
+not be able to gain the information they are seeking about 
+the attacker, and the attacker will not use up a great number 
+of resources on the honeypot. Therefore, modern honeypot 
+systems must be deceptive. They need to look, run, and 
+operate like the “real” system. An effective way of doing so 
+is by looking at your actual system and then placing data 
+that looks the same but is fake into your honeypot. This will 
+make attackers spend a lot of time trying to exploit your 
+honeypot and make them spend time searching for artifacts 
+such as valid credentials. This will also help dry up a lot of 
+the attackers’ resources and make them waste a lot of 
+money. By the time the attackers realize that they have been 
+caught in a honeypot trap, they may become extremely 
+frustrated and decide to use their remaining resources on a 
+different target to prevent them from falling into another 
+honeytrap that you or your organization may have set up. 
+In the next sections, We will be explaining the details of a 
+honeypot and the architecture of a modern deceptive 
+honeypot system. We will also go into the different types of 
+information a defender can gain from using a honeypot and 
+how this information can be leveraged. We then conduct an 
+experiment in which we use a Virtual Private Server to 
+deploy an open-source honeypot system that utilizes many 
+different honeypot containers to create an effective system. 
+This system also utilizes many other features such as an 
+IPS/IDS and a data visualization dashboard to make it easier 
+to analyze data. 
+II. 
+BACKGROUND 
+In this section, we will discuss the network architecture 
+of where to place a honeypot, different types of honeypots, 
+types of data honeypots can collect, and techniques to 
+achieve deception within the honeypot itself. 
+A. Honeypot Network Architecture 
+When deciding where to place a honeypot on a network 
+you must be very careful. Placing a honeypot inside an 
+internal network can be a major security design flaw. This is 
+because it would allow an attacker to easily gain access to 
+your internal network by exploiting the honeypot. An 
+
+2 
+ 
+attacker would then be able to move laterally throughout the 
+network to find real systems and servers. In essence, this 
+would make it much easier for the attacker because you 
+would be “inviting” them into your internal local area 
+network (LAN). Instead, a honeypot should be placed inside 
+a network’s Demilitarized Zone (DMZ). 
+ 
+Figure 1: Architecture of a network with a honeypot deployed [1]. 
+ 
+A DMZ is an isolated part of the network that is connected 
+to the internet and is typically where public-facing services 
+are placed. In this example, a web server, DNS server, and 
+FTP server are all placed inside the DMZ along with the 
+honeypot [1]. The DMZ is separated on both sides with a 
+firewall. The firewall between the DMZ and Internet is 
+typically a weaker firewall that allows traffic to outside 
+users to do tasks like visiting a website hosted or interacting 
+with available files. This firewall will not allow in all traffic 
+but will have the necessary ports open to allow 
+functionality. The firewall between the DMZ and the 
+production network is a much more hardened firewall. This 
+firewall will not let outside, or unauthorized users bypass it 
+easily. In an enterprise setting, most organizations will have 
+an Intrusion Detection System (IDS), or Intrusion 
+Prevention System (IPS) configured to alert on attacks being 
+made against this firewall [2]. This is to keep cyber analysts 
+alert of any attacks that might be going on against their 
+network. The production network may have sensitive data 
+flowing through it and may have systems that store 
+important data, so it is important to keep the honeypot 
+isolated from it. 
+B. Honeypot Characteristics 
+There are different types of modern honeypots that one 
+can deploy into their environments. However, all the 
+honeypots do have some things in common such as being 
+low-maintenance, low cost, and easy to deploy [3]. The 
+reason honeypots need to need to be low maintenance is 
+that, from an organizational perspective, engineers cannot 
+be using up large amounts of their time manually making 
+changes and fixes to the honeypot. They have many other 
+duties to take care of and should be able to look at and 
+analyze data collected from the honeypot easily. This is also 
+connected to why a honeypot is easy to deploy. You should 
+take your time deploying a honeypot to ensure that 
+everything is working properly, but honeypots are easy to 
+deploy because more time and focus should be spent on 
+securing the real environment. If anything goes wrong with 
+your honeypot you should be able to just reboot it and most 
+of the time it will fix itself. Furthermore, a honeypot is often 
+relatively low cost compared to the number of resources an 
+individual or organization may have. Now, one can make a 
+massive honeypot that is extremely expensive, but it would 
+not make any sense from a financial perspective or provide 
+a great amount of additional benefit. Most individuals and 
+organizations deploy honeypots that are not very expensive 
+since they can provide great benefits at a low cost. This also 
+then allows for a great number of financial resources to be 
+spent on other things like employees, software, and other 
+systems. 
+C. Types of Honeypots Based on their Design 
+Many different types of Honeypots can be deployed with 
+various complexities. 
+ 
+Figure 2: Types of Honeypots [Based on Design] [4]. 
+ 
+A pure honeypot is meant to be a full-scale replica of a 
+production environment that contains fake data that is meant 
+to pose as real data. A high-interactive honeypot is like a 
+pure honeypot because it runs many different real-like 
+services, but it does not hold as much data and is not a 
+replica of a full-scale production environment. A mild- 
+interaction honeypot is different than a high-interaction 
+honeypot because it just emulates aspects of the application 
+layer but does not have its own OS. This can be used to stall 
+or confuse attackers who are trying to attack your systems 
+[5]. Finally, a low-interaction honeypot is very lightweight 
+and can be used to match a small number of services and 
+applications that are in use. This type of honeypot can be 
+used to keep track of UDP, TCP, and ICMP ports and 
+
+TYPESOFHONEYPOT
+[Based on the design]
+Low-interaction Honeypots
+ Medium-interaction Honeypots
+High-interaction Honeypots
+Pure HoneypotsDMZ
+Webserver
+DNSserver
+Internet
+Firewall
+Firewall
+Production
+Honeypot
+network
+FTPserver3 
+ 
+services. In which we make can make use of things like fake 
+databases, data, and files as bait to trap attackers to 
+understand the attacks that happen in real-time [4]. 
+D. Types of Honeypots Based on their Technologies 
+There are different types of honeypots based on the 
+deception technologies that they utilize. 
+ 
+ 
+Figure 3: Types of Honeypots [Based on their deception technology] [4]. 
+ 
+A malware honeypot is a type of honeypot that is meant 
+to identify and trap malware inside a network or system. A 
+sophisticated malware honeypot will recognize the 
+malware’s signature and alert based on its Common 
+Vulnerabilities and Exposures (CVE) ID. It will also keep a 
+count of the CVEs to help recognize the most used malware 
+that is being installed onto a system. A database honeypot is 
+exactly what it sounds like – a honeypot that appears to be a 
+vulnerable database. Often it will be something like an SQL 
+database that will allow things like injection attacks to occur 
+to attract attackers looking to gain sensitive information that 
+can be stored in a database such as credit card numbers. A 
+spider honeypot is installed to trap web crawlers that target 
+web applications with the intent of stealing data. An email 
+honeypot is a fake email server that utilizes hoax email 
+addresses and emails to attract attackers to interact with it. 
+Any suspicious email received by attackers can be scanned, 
+and their email addresses can be then blacklisted on the real 
+email servers. A spam honeypot is like an email honeypot 
+but is meant to attract spammers to exploit vulnerable email 
+elements and give details about their activities. Finally, a 
+honeynet is a honeypot system that can contain various 
+types of honeypots mentioned. It will often aggregate data 
+from all the honeypots into a central location to make 
+alerting and monitoring easier [4]. 
+E. Data that Honeypots Collect 
+Honeypots can collect a large amount of data that can be 
+very useful in establishing a more secure security program 
+as well as serve to be very useful for research purposes. It is 
+often useful to have a service configured that aggregates this 
+data into a dashboard that makes it easier to look at and 
+understand. 
+One of the most basic artifacts that a honeypot can 
+collect is the attacker’s IP address. This is a useful artifact 
+because from the IP address we can see things like the 
+general location of the attacker to help identify if potentially 
+many attacks are coming from a specific area. Additionally, 
+we can see if this attacker is launching many attacks on our 
+systems, and we can see how long they have been attacking 
+our systems [6]. We can also block this IP address from 
+accessing our real network. Finally, from the IP address, we 
+can find the attacker’s Autonomous System Number (ASN) 
+to see what Autonomous System the attacker’s IP is 
+allocated from. 
+Another artifact we can gather about the attacker is what 
+Operating System (OS) they are using on their host. We 
+may not be able to figure out the exact version of what OS 
+they are using, but we will be able to figure out a general 
+version of the OS they are using. For instance, we can see if 
+the attacker is running Windows 7 or if the attacker is 
+running a Linux version from 2.2.x-3.x. 
+Many attackers utilize automation tools to try to break 
+into a server. 
+ 
+Figure 4: A sample Dictionary Attack trying common passwords on an 
+SSH server [7]. 
+They often run dictionary attacks, a type of attack that 
+tries to log in to a server using a list of commonly used or 
+default usernames and passwords, to crack into your system. 
+We can use a honeypot to see what passwords and 
+usernames the attackers are trying to use to log in and even 
+take it a step further by making a count of each username 
+and password attempted to find the most popular credentials 
+attackers are trying to use. 
+Furthermore, when an attacker gains access to a 
+honeypot system we can see what commands they are 
+running once they break in. This can help us figure out what 
+their motives may be and what they are looking for. We can 
+also see what they are downloading onto the system. They 
+
+ACCoUNTCHECK:[ssh)Host:192.168.18.132(1of1,complete)User:owmedb(1of1,complete
+Password:123456(1of3546complete
+ICcoUNTCHECK:[ssh)Host:192.168.18.132(1of1,complete)User:ownedb(1of1,ecomplete
+Password:12345(2of3546complete)
+AccoUNTCHECK:[ssh)Host:192.168.18.132(1of1,0complete)User:Ounedb(1of1,0complete
+Password:password(3of3546complete)
+ICcoUNTCHECK:[ssh]Host:192.168.18.132(1of1,0complete)User:ownedb(1of1,0complete
+Password:password1（4of3546complete）
+ACCouNTCHECK:ssh)Host:192.168.18.132(1of1,0complete)User:ownedb(1of1,0complete
+Pas5word:123456789（5of3546complete
+ACCoUNTCHECK:[ssh)Host:192.168.18.132(1of1,complete)User:ownedb(1of1,complete
+as5word:12345678(6of3546complete
+CCoUNTCHECK:[ssh)Host:192.168.18.132(1of1,ecomplete)User:ownedb(1of1,ecomplete
+assword:1234567890(7of3546complete)
+ACCoUNTCHECK:[ssh)Host:192.168.18.132(1of1,complete)User:ownedb(1of1,ecomplete
+assword:abc123(8of3546.complete
+ACcoUNTCHECK:[ssh)Host:192.168.18.132(1of1,0complete)User:ownedb(1of1,@complete
+Password:computer(9of3546complete)
+ICoUNTCHECK:[ssh)Host:192.168.18.132(1of1,0complete)User:ownedb(1of1,0complete
+Password:Th3Basics(10of3546complete
+ACCOUNTFOUND:[ssh]Host:192.168.18.132User:OwnedbPassword:Th3B@sics[SUCCESS]
+rootebt:~#TypesofHoneypot
+Based on theirdeception technology
+MalwareHoneypots
+Database Honeypots
+Spam Honeypots
+Email Honeypots
+SpiderHoneypots
+Honeynet Honeypots4 
+ 
+can download worms or viruses onto the system, or maybe 
+even agents that will try to clean up log files to keep them 
+undetected so they can persist more. They even might install 
+things like rootkits to help with this persistence. In essence, 
+any time the attacker attempts to interact with the honeypot 
+system, you can access and track that information. 
+F. Honeypot Deception Techniques 
+Many different techniques can be utilized to achieve 
+deception within a honeypot system to trick attackers. To 
+start, a commonly used technique is to not allow open 
+access to a server. If attackers can connect to your server on 
+the first try, it is often a sign that you are inviting them in, 
+and it may cause them to be very suspicious. To solve this, 
+make it that after many failed attempts they will finally be 
+able to gain access and login into your system. This will 
+make them think that their brute-forcing or dictionary attack 
+worked, but it still took a while for it to be a success. This 
+will cause them to have less suspicion. 
+Another more complex technique is to utilize honey 
+credentials. 
+ 
+Figure 5: Honey Credentials being stored in memory [8]. 
+ 
+Honey credentials help catch malicious actors by 
+injecting fake credentials into a system’s memory. When an 
+attacker gains access to your network and finds the honey 
+credentials, they will attempt to use them. Since these 
+credentials don’t exist, any attempt to use them can trigger 
+an alert and notify you immediately. In a targeted attack, the 
+attacker will be able to dump recovered honey credentials 
+from the system’s memory through privilege escalation or a 
+system flaw. The attacker will then attempt to perform 
+lateral movement into the fake objects, resulting in their 
+exposure and making it easy to trace them [9]. 
+Finally, the most important deception technique is to 
+make the honeypot blend in and look realistic. Earlier we 
+mentioned the different types of honeypots based on their 
+deception technology. It is important to be able to look at 
+these honeypots from the attacker’s perspective. For 
+instance, an SSH server must look like an actual SSH server 
+that you or your organization would utilize. If it looks fake 
+no one will take the bait and they will be able to easily detect 
+what they are attacking is a honeypot. You should be 
+configuring and storing data on your honeypot that you 
+would store on your actual systems but just make sure it is 
+fake information [10]. You can install real services you 
+would normally use onto your honeypot, but just make sure 
+they do not contain sensitive information. By having your 
+honeypot blend in, it will keep your attacker occupied and 
+cause them to reveal more information about themselves and 
+what their motives are. 
+III. T-POT HONEYPOT SYSTEM FRAMEWORK 
+In this section, we detail the honeypot system we will be 
+deploying to gather data and intelligence about attackers. 
+A. System Fundamentals 
+T-Pot is an open-source all-in-one honeypot platform 
+that runs on Debian Linux. The honeypot daemons as well 
+as other support components are dockered. This allows T- 
+Pot to run multiple honeypot daemons and tools on the same 
+network interface while maintaining a small footprint and 
+while constraining each honeypot within its own 
+environment [11]. Documentation to the platform can be 
+found at https://github.com/telekom-security/tpotce. T-Pot 
+uses docker images for 25 different honeypots to create a 
+massive honeypot system. All the honeypots each represent 
+different systems. For instance, cowrie is a medium-to-high 
+interaction SSH and Telnet honeypot designed to log brute 
+force attacks and the shell interaction performed by the 
+attacker [12]. In addition, another example is the honeytrap 
+honeypot is a network security tool written to observe 
+attacks against TCP or UDP services [13]. You can read 
+about all 25 different honeypots deployed in the system on 
+the documentation page for T-Pot. For the most part, 
+understanding each honeypot will not be very important 
+because we will mainly focus on the aggregated data 
+collected for the whole system. 
+B. Tools Utilized by T-Pot 
+T-Pot uses a variety of different tools to aggregate, alert, 
+monitor, and analyze data collected. 
+ 
+Figure 6: Description of tools used by T-Pot [11]. 
+
+mitrikate20alptax6foeto)
+msy.
+[0000003]
+Primary
+Username
+adninistrator
+Donain
+NTLM
+microsoft
+-
+76d202631
+SHAI
+bc059168c2d8d1200c
+tspkg:
+KO
+wdigest
+Username
+adninistrator
+Donain
+Passuord
+microsoft.con
+(nuil)
+kerberos :
+Csername
+adninistrator
+Donain
+microsoft.con
+Password
+ssp:
+superpass
+credman :
+AuthenticationIdt
+Q91755333（00000090：05781345)
+UOTSSOC
+NewCredentials fron 0
+Jser Name
+HindouaBMarkB
+nark
+Domsin
+SID
+S-1-5-21-1469176257-2007698836-3967804884-1001
+mSY.
+(00800003]
+Primary
+Username
+root
+Donain
+:
+NTLM
+211394dc229g20
+200018a69ac04
+SHA1
+Sa71afbecd987668b917e4425c82ac29a0e39b93
+tspkg:
+KO
+wdsgest
+Username
+root
+Donain
+Iinux,org
+Password
+:
+(ltnu)
+livessp
+kerberos :
+Username
+root
+Donain
+linux.org
+Password
+notreallythepassword
+ssp:
+credman :
+2
+HACyberchefawebappforencryption,encoding,compressionanddataanalysis
+ELK stack to beautifulyvisualizeallthe events captured byT-Pot
+ElasticsearchHeadawebfrontendforbrowsingandinteractingwithanElasticSearchcluster
+Fattapysharkbasedscriptforextractingnetworkmetadataandfingerprintsfrompcapfilesandlivenetwork
+traffic.
+Spiderfoot a open source intelligenceautomation tool
+·Suricata a Network Security Monitoring engine5 
+ 
+All the tools mentioned in Figure 6 can be further 
+analyzed by examining the information found on the T-Pot 
+documentation page. For our experiment, we will focus on 
+the ELK stack and Suricata tools. ELK stack has a tool 
+named Kibana, which we will utilize to visualize all the data 
+collected by the different honeypots. We can use Kibana to 
+analyze the data for each individual honeypot as well as 
+look at an aggregated view of information collected from all 
+the honeypots deployed in one dashboard [14]. Suricata is a 
+network security monitoring engine that we can use to 
+monitor and alert us about suspicious activity occurring 
+within the system [15]. The information from Suricata will 
+be viewable and aggregated into the Kibana Dashboard. As 
+mentioned previously and displayed in Figure 6 there are 
+many other tools pre-built into T-Pot, but we will not be 
+using them as they are not necessary to understand and 
+analyze the data collected. 
+C. Deploying the T-Pot Honeypot System 
+The system requirements for deploying the system are as 
+follows: 8 GB Ram, 128 GB SSD, Network via DHCP, and 
+a non-proxied internet connection. You can download the 
+Pre-built ISO Image (~50 MB), or you can create your own 
+ISO Image that allows you to customize the system to fit 
+your needs better. Since we are doing this for research 
+purposes to see what information we can learn about 
+attackers and not using the system in an enterprise 
+environment, xutilized the Pre-built ISO Image. 
+To deploy the honeypot system, I used a Virtual Private 
+Server (VPS). I uploaded the ISO file to the VPS provider’s 
+website and then began installation. The reason I used a 
+VPS is to avoid large amounts of unwanted traffic coming 
+towards my personal network. I also did not want to reveal 
+my actual public IP address to attackers. 
+Once the ISO image finished installing in my VM the 
+honeypot system was fully functional and ready to lure in 
+attackers. To reach the T-Pot landing page to gain access to 
+all the tools deployed in the system I had to go into a web 
+browser and enter “https://<my.ip>:64297”. I then logged in 
+with the credentials created during installation. 
+ 
+Figure 7: T-Pot Landing Page. 
+ 
+Now that the honeypot is fully deployed attackers will 
+begin to attack the honeypot. We can view data collected 
+from the attacks in the Kibana dashboard. I will wait 
+approximately 3 weeks before analyzing the data just to let a 
+good amount of data accumulate. In the next section, we 
+will test some of the functionality on the honeypot by 
+running some attacks to make sure the honeypot is 
+configured properly before waiting the 3 weeks to analyze 
+the data collected. 
+IV. TESTING HONEYPOT SYSTEM 
+FUNCTIONALITY 
+In this section, we will test some of the functionalities of 
+the honeypot by running a Nmap scan and brute force attack 
+against the honeypot system. We will target Cowrie, the 
+SSH honeypot container. 
+A. Port Scanning the Honeypot System 
+I used a Kali Linux Virtual Machine to simulate an 
+attack on the honeypot system. Information on how to 
+install 
+Kali 
+Linux 
+can 
+be 
+found 
+at 
+https://www.kali.org/docs/. Kali Linux comes with Nmap, a 
+port scanning tool, preinstalled on the system. To use this 
+tool, I opened a terminal session and ran the command 
+“nmap -p 22 140.82.3.147” as seen in Figure 8. 
+ 
+Figure 8: Nmap scan ran against the Honeypot System. 
+This command runs a port scan only checking to see if port 
+22 is open on the IP address specified. From the results 
+returned from the Nmap scan, we can see that port 22 is open 
+to be used by the SSH Service. This is because as mentioned 
+earlier, Cowrie is a high interaction SSH honeypot designed 
+to log brute force attacks and the shell interaction performed 
+by the attacker once they gain entry. 
+A. Brute-Forcing the System 
+For simplicity of the experiment and to test the 
+functionality of the honeypot system, I logged into the 
+honeypot SSH server and made the root account accessible 
+over SSH. Also, I changed the password to 12345. I then 
+went to my Kali Linux terminal and used the hydra tool to 
+crack the password for the root account to the honeypot. I 
+ran 
+the 
+command 
+“hydra 
+-l 
+root 
+-P 
+/usr/share/wordlists/Metasploit/unix_password.txt 
+-T 
+6 
+ssh://140.82.3.147” as seen below in Figure 9. 
+
+Elasticsearch
+Cockpit
+Cyberchef
+Head
+Kibana
+SecurityMeter
+Spiderfoot
+T-Pot@GitHub(kalikali)-[~
+nmap-p22140.82.3.147
+Nmapscanreportfor 140.82.3.147.vultr.com（140.82.3.147)
+Hostisup(0.20slatency).
+PORT
+STATESERVICE
+22/tcp openssh
+Nmap done:1IPaddress(1hostup)scanned in 0.o9 seconds
+(kalikali)-[~]6 
+ 
+ 
+Figure 9: Running the Hydra Brute-Force Attack. 
+The -l option in the command tells hydra to try to log in 
+with the root user. The -P option tells hydra to use the 
+password list specified in the command and the -t option 
+tells hydra how many threads to use. So, in this attack 
+scenario, we used 6 threads. Also, we told it to attack the 
+SSH service on the IP specified. As we can see in Figure 9, 
+we got the exact results that we expected. The password 
+that was cracked for the root user was 12345. Since the test 
+has concluded, I logged back into the SSH server and 
+made it not possible to login to the root user through SSH 
+like how it was prior to me modifying it. This is because 
+this is a good security practice and if it was possible to 
+login to the root user over SSH it would make the attacker 
+suspicious of the honeypot. I then changed the password 
+back to what it was prior. The original password was not 
+that difficult to crack either, but more difficult to crack 
+than 12345. This is because we do still want the attacker to 
+be able to gain access so we can gain more information on 
+what they are looking to do once they are inside the server. 
+I will now let the honeypot system run for a few weeks so it 
+can collect a large amount of data to come back and 
+analyze. 
+V. 
+ANALYZING 
+DATA 
+COLLECTED 
+FROM 
+ATTACKS 
+ON 
+THE 
+SYSTEM 
+In this section, we will analyze the data collected by 
+the honeypot system in the Kibana Dashboard. We will 
+look at the different categories of information collected to 
+see what information we can learn about the attackers. For 
+the most part, we will look at the aggregated data collected 
+by all the containers, but in some cases, we will look at 
+data collected by a specific container. 
+A. Accessing the Data 
+To access the data, we need to first go back to the T-Pot 
+Landing Page. This can be reached by entering 
+“https://<my.ip>:64297” into a web browser and logging in 
+with the credentials created. I then clicked on “Kibana” to 
+bring me to the Kibana app dashboards page. 
+ 
+Figure 10: Kibana App Main Dashboards Page. 
+ 
+As seen in Figure 10, a page opens that displays a list of 
+the different honeypot containers deployed. You can reach 
+an individual dashboard by just clicking on the name of 
+that honeypot. You will also notice that the first two titles 
+are 
+“T-Pot” and “T-Pot Live Attack Map”. T-Pot Live Attack 
+Map just shows a global map of where attacks within the 
+T- Pot Network are occurring in the current time. This is 
+not very useful to us. However, the T-Pot Dashboard is the 
+dashboard we will be mainly using because it displays the 
+visuals and data collected by all the honeypots aggregated 
+into one dashboard. Therefore, we will go ahead and click 
+on “T-Pot” to display this dashboard. 
+ 
+Figure 11: Kibana T-Pot Dashboard. 
+ 
+The dashboard displayed is shown in Figure 11. As 
+mentioned, this dashboard allows us to visualize and view 
+
+-(kalickali)-[]
+Shydraroot-P/usr/share/wordlists/netasploit/unixpasswords.txt-t6ssh://140.82.3.147
+ydrav9.2(c)2021byvanHauser/THC6DavidMaciejakPleasedonotuseinmilitaryorsecretsel
+-bindnhe*gawsndaay
+ydra（https://github.com/vanhauser-thc/thc-hydra)starting at2022-03-0518:36:55
+WARNINGRestorefile(youhave10secondstoabort...(useoption-Itoskipwaiting))fromaprev
+ore
+DATAmax6tasksper1serveroveralu6tasks,1009 Logintries(L:1/p:1009),-169 triespertas
+DATAlattacking ssh://140.82.3.147:22/
+22[ssh] host:140.82.3.147login:rootpasword12345
+1 of 1target successfully completed,1 valid password found
+ydra(https://github.com/vanhauser-thc/thc-hydra)finishedat2022-03-0518:37:06Dashboards
+Search..
+Title
+Description
+>T-Pot
+T-PotDashboard
+>T-PotLiveAttackMap
+T-PotLiveAttackMap
+Adbhoney
+AdbhoneyDashboard
+Ciscoasa
+CiscoasaDashboard
+CitrixHoneypot
+CitrixHoneypotDashboard
+Conpot
+ConpotDashboard
+Cowrie
+CowrieDashboard
+Ddospot
+Ddospot Dashboard
+Dicompot
+DicompotDashboard
+Dionaea
+Dionaea Dashboardceetineis-toto
+330.008
+198.915
+132.032
+18.613
+4.300
+3.661
+1,015
+640
+623
+520
+Covte-ltxde
+Ooge-As
+wd-Aade
+Aehony-Atste
+opy-tci
+-Ad
+Rixis
+--7 
+ 
+the data collected from all the honeypots in one place. Next, 
+we will be looking at and analyzing some of the charts and 
+data lists that are relevant to gathering intelligence about the 
+attackers. 
+B. Most Attacked Honeypots 
+At the top of the dashboard, we can see the ten 
+honeypots that experienced the most attacks ranked in 
+sequential order from the most attacked honeypot to the 
+least attacked honeypot. 
+ 
+Figure 12: Top 10 attacked honeypots 
+ 
+As we can see in Figure 12, the most attacked honeypot 
+was Cowrie, the SSH honeypot. This is not very surprising 
+since SSH is a very well-understood protocol and can be 
+used to gain remote shell access to a server. If the attacker 
+can abuse the SSH service and gain access to your server, 
+they will be able to obtain a lot of information and have 
+access to your system. From a defender’s point of view, 
+this is important information to understand because since 
+SSH is the most attacked protocol, in a live production 
+environment gaining remote access should not be very 
+easy. On top of utilizing a very secure password and SSH 
+keys, MFA should also be configured to gain remote shell 
+access. We can also see that the Dionaea is the second most 
+attacked honeypot, which uses the FTP service to capture 
+attack payloads and malware. This tells us that any way 
+that an attacker can either gain remote access to a server 
+(SSH) or be able to upload files to a server remotely (FTP) 
+is going to attract attackers. This is because the concept of 
+being able to remotely impact a server can have high 
+consequences if abused. As a defender, we can learn from 
+this to make sure that all servers that have remote protocol 
+ports open must be highly secure. 
+C. Attacks by Country 
+The next graphic we will examine is the countries where 
+the most attacks are from. I will also discuss why that 
+information may and may not be useful. 
+ 
+Figure 13: Attacks by Country Pie Chart. 
+The legend on the right of Figure 13 shows the list of 
+where the most attacks came from in descending order. As 
+we can see, the top 3 countries where the most attacks 
+came from are the United States, China, and Russia. These 
+countries tend to have more sophisticated cyber programs 
+and larger populations, so it is not very surprising. This 
+information can be useful to us because if a specific threat 
+actor is targeting an individual company, we may be able 
+to track where that threat actor is located. However, any 
+decent attacker is most likely using a VPS where they can 
+choose the location of where their IP address is located, or 
+they are using a VPN to hide their true location. That is 
+why focusing on this graphic is not very useful, but it is 
+more important to focus on the IP addresses themselves. 
+We will look at this next. 
+D. Attacker Source IP Addresses and their ASNs 
+Looking at the Source IP Addresses of where most of the 
+attacks are coming from is very useful to cyber defenders. 
+This is because most attackers have a finite number of 
+resources to work with, so their IP Space is not unlimited. 
+ 
+ 
+Figure 14: Top 10 Attacker Source IP Addresses. 
+ 
+In Figure 14, we can see the IP addresses where most of 
+the attacks came from. This information is very useful to 
+cyber defenders because these IP addresses can be 
+blacklisted from the production network. This will not allow 
+
+HoneypotAttacks-Top10
+334,592
+197,072
+132,574
+18,613
+4,308
+Cowrie-Attacks
+Dionaea-Attacks
+Honeytrap-Attacks
+Heralding-Attacks
+Adbhoney-Attacks
+3,668
+1,020
+640
+627
+526
+Rdpy-Attacks
+Tanner-Attacks
+CitrixHoneypot-Attacks
+Mailoney-Attacks
+ConPot-AttacksAttacks byCountry
+UnitedStates
+China
+Russia
+Vietnam
+India
+Canada
+Japan
+Singapore
+Hong Kong
+TurkeyAttackersourcelp-Top1o
+Source Ip
+Count
+165.22.234.121
+26,552
+2.56.56.14
+18,204
+69.171.13.237
+11,027
+140.82.156.72
+9,094
+85.100.124.175
+3,158
+109.94.179.81
+3,154
+110.227.249.142
+3,152
+190.75.220.137
+3,151
+83.52.23.252
+3,151
+103.43.77.175
+3,1498 
+ 
+these attackers to use these IP addresses on the actual 
+network. The attackers will now have to use other IP 
+addresses to gain access to your resources, therefore making 
+them exhaust their resources. If you continuously blacklist 
+the IP addresses where a lot of the attacks are coming from 
+you can deter attacks from continuing to attack your 
+network. This will help prevent a lot of the noisier attackers 
+from being able to exploit your systems, but you also must 
+be aware that more stealthy attackers will be able to remain 
+under the radar. 
+From the attacker’s IP address, we can figure out the 
+Autonomous System Number (ASN) to learn what 
+organizations are allocating the attacker’s IP address. 
+ 
+ 
+Figure 15: Top 10 ASNs used by attackers attacking the system. 
+ 
+In Figure 15 we can see that the most popular ASN 
+organization used to attack our honeypot system is 
+Digital Ocean, one of the most popular VPS providers. 
+This shows that a lot of people are abusing the benefits 
+they offer to launch cyber-attacks. There also does 
+appear to be a good number of China-based 
+organizations on the list, helping to support that there 
+might be a lot of attacks launched on the honeypot 
+from China. This information can be useful to us 
+because when we see a user utilizing an IP allocated 
+from Digital Ocean, for example, we can be more 
+cautious and possibly even raise an alert since we 
+know it is commonly used by attackers to attack our 
+systems. 
+E. Most Used OS by Attackers 
+From our honeypot system, we can gather information 
+about the most popular Operating Systems used by 
+attackers to attack the honeypot system. Kibana displays a 
+pie chart of the operating systems and displays the legend 
+in descending order from most to least used OS. 
+ 
+Figure 16: Pie Chart displaying Attacker OS Distributions. 
+ 
+From this pie chart, we can see that Windows 7/8 was 
+the most popular used OS by attackers followed by Linux 
+2.2.x-3.x. We do not get an exact version number for the 
+most part, but still, get a good idea of the OS the attacker 
+is using. I found it particularly unusual that attackers are 
+running older versions of Windows and Linux, but it can 
+be valuable information. Most common users run 
+Windows 10 as of now, so users running Windows 7/8 
+can be something to look out for when investigating 
+attackers. A lot of basic users do not run Linux, so 
+anytime there is suspicious activity coming from a user 
+running Linux it can be a red flag. 
+F. Most Common Credentials 
+In this section, we will look at the most common 
+credentials used by attackers to try to brute-force login to 
+our systems. Many of these usernames and passwords 
+come from dictionary lists of commonly used passwords. 
+ 
+Figure 17: Most Common Usernames attempted by attackers. 
+ 
+In Figure 17, we see the most common usernames 
+attackers tried to use to log in to our honeypot system. As 
+we can see, root and admin were among the most popular. 
+This makes sense because root and admin accounts tend to 
+have higher privileges. This shows us from a defensive 
+perspective why it is so important to disable remote root 
+login because it can be very dangerous if attackers manage 
+to log in. 
+ 
+
+AttackerAs/N-Top10
+AS
+ASN
+Count
+14061
+DigitalOcean,LLC
+83.743
+45899
+VNPTCorp
+25,944
+395800
+GreybeardTechnologyLLC
+18.324
+45090
+Shenzhen Tencent Computer...
+16,955
+38365
+BeijingBaiduNetcom Scienc...
+11,744
+4134
+No.31,Jin-rongStreet
+11,167
+29944
+Latisys-Ashburn,LLC
+11,027
+12389
+Rostelecom
+10,625
+174
+CogentCommunications
+9,895
+135377
+UCloud (Hk) Holdings Group...
+8,074PofoSDistribution
+Windows7or8
+Linux 2.2.x-3.x
+WindowsNTkernel
+Linux3.11andnewer
+Linux3.1-3.10
+Linux 2.2.x-3.x (barebon..
+WindowsNTkernel6.x
+WindowsNTkernel5.x
+Linux 2.2.x-3.x (no time..
+Linux 2.4.xes
+student
+zabbix
+user
+admin1
+666666
+jenkins
+minecraft service
+postgres
+Hoddns
+888888
+default
+test
+tomcat
+wwwroot
+ftp
+sa
+ubuntu
+ubnt
+root
+www-data
+administrator
+user
+phil
+Admin
+user123
+ftpuser
+testuser
+admin
+nproc
+(empty)
+mysql
+guest
+hadoop
+supervisor
+git
+oracle
+www
+server
+anonymous
+deploy
+Administrator
+web
+tech
+mother
+data
+pi
+dev9 
+ 
+ 
+Figure 18: Most Common Passwords attempted by attackers. 
+ 
+In Figure 18, we see the most common passwords 
+attackers tried to use to log in to our honeypot system. Many 
+of these are weak passwords that a basic user might make or 
+default passwords. We can learn from this that it is 
+extremely important to change default passwords and to 
+replace them with a very strong password to prevent 
+attackers from gaining easy access. Passwords should be 
+relatively long and utilize a combination of letters, 
+numbers, and special characters. 
+G. Analyzing Suricata Alerts 
+In this section, we will analyze the alert raised by 
+Suricata, an open-source IDS/IPS that feeds data into our 
+Kibana dashboard. 
+ 
+Figure 19: Suricata Alerts. 
+ 
+In Figure 19, we can see the 10 most common alerts 
+detected by Suricata. From this, we can tell that a lot of 
+attackers try to abuse TCP and manipulate the Three-Way 
+TCP Handshake. We can also tell that a lot of users were 
+using a port scanning tool like Nmap because there were a 
+lot of detections of incomplete connections. Finally, we 
+also detected some alerts that SSH and SMB sessions were 
+established. This means that some attackers did gain entry 
+inside our honeypot system, as we expected. 
+ 
+Figure 20: Suricata CVEs Detected. 
+ 
+In Figure 20, we can see the top 3 CVEs detected by 
+Suricata. As mentioned before, a CVE is a common 
+vulnerability and exposure that is known by the public and 
+is detected by a unique signature. CVE-2019-12263 is a 
+high vulnerability with a CVSS score of 8.1. This 
+vulnerability refers to a Buffer Overflow in the TCP 
+component of Wind River VxWorks 6.9.4 and vx7 [16]. 
+The next most common CVE detected is CVE-2019-0708 
+which has a CVSS score of 9.8, also very high. This 
+vulnerability is a “Remote Desktop Services Remote Code 
+Execution Vulnerability.” This means that an attacker can 
+remotely execute code on another user’s system [17]. 
+Finally, the third most popular CVE is CVE-2020-11910 
+which has a CVSS score of 5.3, a medium level severity. 
+This vulnerability is caused by the Treck TCP/IP stack 
+before version 6.0.1.66 having an ICMPv4 Out-of-bounds 
+Read [18]. This means that the system is reading packets 
+that are not in bounds. From all these CVEs detected, it can 
+teach us to make sure that our systems are patched against 
+these popular vulnerabilities. It also shows us how 
+important it is to regularly patch our systems when there 
+are security updates because attackers will try to exploit 
+our systems if they are not patched. 
+H. Cowrie Top Shell Commands Executed 
+In this section, we will look at the top commands 
+executed inside of the Cowrie honeypot. To look at this, we 
+will go back to the main T-Pot Dashboard page and select 
+“Cowrie Dashboard” to only see the Dashboard for this 
+specific honeypot. As mentioned earlier, Cowrie is a high 
+interaction SSH honeypot that tracks commands entered by 
+attackers into the command line and that data gets sent into 
+Kibana to analyze.  
+ 
+Figure 20: Top 10 Commands Executed by attackers. 
+ 
+In Figure 20, we can see the top 10 commands run by 
+
+1q2w3e4r admin1234 abc123
+1234qwer
+fucker
+666666
+54321
+111111
+1111 test
+default
+111111234567
+1 root 1234
+admin
+aquario user1
+888888
+guest
+1001chin
+(empty)123456
+system
+12345
+Win1doWs
+tech
+ivdev
+123123
+123 nproc password
+friend
+password123
+pass
+ubnt
+1234567890
+07ujMkoadmin
+00000000
+XC3511
+5up123456789
+alpine
+12345678
+vizxV
+qwerty
+ZIxx.
+1qaz2wsxSuricataAlertSignature-Top10
+Description
+Count
+SURICATASTREAMreassemblysequenceGAP--missingpacket(s)
+65,799
+ETPOLiCYSSHsessioninprogressonExpectedPort
+13,043
+SURICATASTREAMPacketwithbrokenack
+12,053
+SURICATATCPy4invalidchecksum
+5,008
+SURICATASTREAMPacketwithinvalidtimestamp
+3,609
+SURlCATAApplayerDetectprotocolonlyonedirection
+1,553
+SURICATASTREAMFINrecvbutnosession
+1,034
+ETSCANZmapUser-Agent(inbound)
+998
+SURICATASTREAMRSTrecvbutnoSession
+831
+ETINFOPotentiallyunsafeSMBv1protocolinuse
+740SuricatacVE-Top10
+CVEID
+Count
+CVE-2019-12263...
+62
+CVE-2019-0708
+14
+CVE-2020-11910
+6CowrieInput-Top1o
+CommandLineInput
+Count
+uname -a
+126
+cat/proc/cpuinfo/grepmodel/grepname
+117
+cat/proc/cpuinfo|grepname|head-n1
+a...
+117
+cat/proc/cpuinfogrepname
+WC-
+117
+crontab -l
+117
+free-m|grepMem|awk'(print$2,$3,$4,$...
+117
+Is-lh$(whichIs)
+117
+top
+117
+uname
+117
+uname -m
+11710 
+ 
+attackers inside the honeypot. As we can see, the most run 
+command is the “uname” command with the -a flag. We 
+also see many different variations of this command in the 
+top 10 list. This command reveals a lot about the system 
+such as what Linux distro is being run on the system and the 
+version of the distro. This is useful to attackers because if 
+the system is out of date there may be vulnerabilities that 
+the attacker can exploit. Therefore, it is important to update 
+and patch your systems. The next most popular command 
+that we see being run is the attacker looking for information 
+about the system’s CPU. An attacker may be interested in 
+the CPU info to see how much processing power your 
+system has. The reason they care about this is that crypto 
+mining is currently extremely popular but requires many 
+resources to drive a profit. So, if attackers can create a 
+botnet of devices that mines for crypto in the background of 
+victims’ computers, it will have no cost to the attackers, and 
+they will receive all the benefits. 
+I. 
+CONCLUSION 
+A honeypot system can be very beneficial to an 
+organization, but it must be properly implemented and 
+deployed. Honeypots need to be deceptive enough to trick 
+the attacker into thinking that they are in a real system. 
+This will cause the attackers to use up time and resources 
+when attacking the honeypot system, but it will also reveal 
+information about how the attackers are penetrating the 
+system and what they are looking for once they gain access 
+to the system. This information can be useful to the 
+defenders because we can learn how attackers operate and 
+can make sure our actual systems are secure against the 
+various attacks that are being used by attackers. We can 
+also make sure that the information attackers are looking to 
+seek is properly protected. 
+For future work, we will extend our current 
+cybersecurity framework [19-53] by integrating the 
+Honeypot system as a Cyber Deception Tactic to deceive 
+the attacker. 
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+Database, 
+09-Aug-2019. 
+[Online]. 
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+2022]. 
+[17] 
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+16-May- 
+2019. 
+[Online]. 
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+2022]. 
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+2020. 
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+2022]. 
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+5G Edge Network Using a Dynamic Hexagonal Fuzzy Method”,. 
+Sensors 2022, 22, 9. 
+[20] Hisham A. Kholidy, “Multi-Layer Attack Graph Analysis in the 
+5G Edge Network Using a Dynamic Hexagonal Fuzzy Method”, Sensor 
+Journal. Sensors 2022, 22, 9. https://doi.org/10.3390/s22010009. 
+[21] Hisham A. Kholidy, Andrew Karam, James L. Sidoran, 
+Mohammad A. Rahman, "5G Core Security in Edge Networks: A 
+Vulnerability Assessment Approach", the 26th IEEE Symposium on 
+Computers and Communications (IEEE ISCC 2021), Athens, Greece, 
+September 5-8, 2021. https://ieeexplore.ieee.org/document/9631531 
+[22] Hisham A. Kholidy, “A Triangular Fuzzy based Multicriteria 
+Decision Making Approach for Assessing Security Risks in 5G 
+Networks”, December 2021, Preprint={2112.13072}, arXiv. 
+[23] Kholidy, H.A., Fabrizio Baiardi, "CIDS: A framework for 
+Intrusion Detection in Cloud Systems", in the 9th Int. Conf. on 
+Information Technology: New Generations ITNG 2012, April 16-18, 
+Las 
+Vegas, 
+Nevada, 
+USA. 
+http://www.di.unipi.it/~hkholidy/projects/cids/  
+[24] Kholidy, H.A. (2020), "Autonomous mitigation of cyber risks in 
+the Cyber–Physical Systems", doi:10.1016/j.future.2020.09.002, Future 
+Generation Computer Systems,Volume 115, 2021, Pages 171-187, 
+ISSN 0167-739X, https://doi.org/10.1016/j.future.2020.09.002. 
+[25] Hisham A. Kholidy, Abdelkarim Erradi, Sherif Abdelwahed, 
+
+11 
+ 
+Fabrizio Baiardi, "A risk mitigation approach for autonomous cloud 
+intrusion response system", Computing Journal, Springer, DOI: 
+10.1007/s00607-016-0495-8, June 2016. (Impact factor: 2.220). 
+https://link.springer.com/article/10.1007/s00607-016-0495-8 
+[26] Hisham A. Kholidy, “Detecting impersonation attacks in cloud 
+computing environments using a centric user profiling approach”, 
+Future Generation Computer Systems, Vol 115, 17, December 13, 
+2020, ISSN 0167-739X.  
+[27] Kholidy, H.A., Baiardi, F., Hariri, S., et al.: “A hierarchical 
+cloud intrusion detection system: design and evaluation”, Int. J. Cloud 
+Comput., Serv. Archit. (IJCCSA), 2012, 2, pp. 1–24. 
+[28] Kholidy, H.A., “Detecting impersonation attacks in cloud 
+computing environments using a centric user profiling approach”, 
+Future Generation Computer Systems, Volume 115, issue 17, 
+December 
+13, 
+2020, 
+Pages 
+171-187, 
+ISSN 
+0167-739X, 
+https://doi.org/10.1016/j.future.2020.12. 
+[29] Kholidy, Hisham A.: 'Correlation-based sequence alignment 
+models for detecting masquerades in cloud computing', IET 
+Information Security, 2020, 14, (1), p.39-50, DOI: 10.1049/iet-
+ifs.2019.0409. 
+[30] Kholidy, H.A., Abdelkarim Erradi, “A Cost-Aware Model for 
+Risk Mitigation in Cloud Computing SystemsSuccessful accepted in 
+12th ACS/IEEE International Conference on Computer Systems and 
+Applications (AICCSA), Marrakech, Morocco, November, 2015.  
+[31] A H M Jakaria, Mohammad A. Rahman, Alvi A. Khalil, Hisham 
+A. Kholidy, Matthew Anderson et al “Trajectory Synthesis for a UAV 
+Swarm Based on Resilient Data Collection Objectives", IEEE 
+Transactions on Network and Service Management, November, 2022  
+doi: 10.1109/TNSM.2022.3216804. 
+[32] Hisham A. Kholidy, Andrew Karam, James Sidoran, et al. 
+“Toward Zero Trust Security in 5G Open Architecture Network Slices”, 
+the 40th IEEE Military Conference (MILCOM), San Diego, CA, USA, 
+November 29, 2022. 
+[33] Hisham A. Kholidy, Riaad Kamaludeen “An Innovative 
+Hashgraph-based Federated Learning Approach for Multi Domain 5G 
+Network Protection”, IEEE Future Networks (5G World Forum), 
+Montreal, Canada, October 2022.  
+[34] Hisham A. Kholidy, Andrew Karam, Jeffrey H. Reed, Yusuf 
+Elazzazi, "An Experimental 5G Testbed for Secure Network Slicing 
+Evaluation", IEEE Future Networks (5G World Forum), Montreal, 
+Canada, October 2022. 
+[35] Hisham A. Kholidy, Salim Hariri, Pratik Satam, Safwan Ahmed 
+Almadani “Toward an Experimental Federated 6G Testbed: A 
+Federated learning Approach”, the 13th Int. Conf. on Information and 
+Communication Technology Convergence (ICTC), Jeju Island, Korea, 
+October 9, 2022. 
+[36] NI Haque, MA Rahman, D Chen, Hisham Kholidy, “BIoTA: 
+Control-Aware Attack Analytics for Building Internet of Things”, 2021 
+18th 
+Annual 
+IEEE 
+International 
+Conference 
+on 
+Sensing, 
+Communication  
+[37] Kholidy, H.A., Ali T., Stefano I., et al, “Attacks Detection in 
+SCADA Systems Using an Improved Non-Nested Generalized 
+Exemplars Algorithm", the 12th IEEE International Conference on 
+Computer Engineering and Systems (ICCES 2017), December 19-20, 
+2017.  
+[38] Qian Chen, Kholidy, H.A., Sherif Abdelwahed, John Hamilton, 
+"Towards Realizing a Distributed Event and Intrusion Detection 
+System", the Int. Conf. on Future Network Systems and Security, 
+Florida, USA, Aug 2017.  
+[39] Hisham A. Kholidy, Abdelkarim Erradi, Sherif Abdelwahed, 
+Abdulrahman Azab, “A Finite State Hidden Markov Model for 
+Predicting Multistage Attacks in Cloud Systems", in the 12th IEEE Int. 
+Conf. on Dependable, Autonomic and Secure Computing, China, 
+August 2014. 
+[40] Ferrucci, R., & Kholidy, H. A. (2020, May). A Wireless 
+Intrusion Detection for the Next Generation (5G) Networks”, Master’s 
+Thesis, SUNY poly. 
+[41] Rahman, A., Mahmud, M., Iqbal, T., Saraireh, L., Hisham A. 
+Kholidy., et. al. (2022). Network anomaly detection in 5G networks. 
+Mathematical Modelling of Engineering Problems, Vol. 9, No. 2, pp. 
+397-404. https://doi.org/10.18280/mmep.090213 
+[42] Hisham Kholidy, 
+“State 
+Compression 
+and 
+Quantitative 
+Assessment Model for Assessing Security Risks in the Oil and Gas 
+Transmission 
+Systems”, 
+doi 
+: 
+10.48550/ARXIV.2112.14137, 
+https://arxiv.org/abs/2112.14137}, December 2021.  
+[43] Hisham A.  Kholidy, “Correlation  Based  Sequence  Alignment  
+Models  For  Detecting Masquerades in Cloud  Computing”, IET 
+Information Security Journal,  DOI: 10.1049/iet-ifs.2019.0409, Sept. 
+2019 
+(ISI 
+Impact 
+Factor(IF): 
+1.51) 
+https://digital-
+library.theiet.org/content/journals/10.1049/iet-ifs.2019.0409 
+[44] Hisham A. Kholidy, “An Intelligent Swarm based Prediction 
+Approach for Predicting Cloud Computing User Resource Needs”, the 
+Computer Communications Journal, December 19 (ISI IF: 2.766). 
+https://www.sciencedirect.com/science/article/abs/pii/S0140366419303
+329 
+[45] Hisham A. Kholidy, Abdelkarim Erradi, “VHDRA: A Vertical 
+and Horizontal Dataset Reduction Approach for Cyber-Physical Power-
+Aware 
+Intrusion 
+Detection 
+Systems”, 
+SECURITY 
+AND 
+COMMUNICATION NETWORKS Journal (ISI IF: 1.376), March 7, 
+2019. 
+vol. 
+2019, 
+Article 
+ID 
+6816943, 
+15 
+pages. https://doi.org/10.1155/2019/6816943.  
+[46] Hisham A. Kholidy, Hala Hassan, Amany Sarhan, Abdelkarim 
+Erradi, Sherif Abdelwahed, "QoS Optimization for Cloud Service 
+Composition Based on Economic Model", Book Chapter in the Internet 
+of Things. User-Centric IoT, Volume 150 of the series Lecture Notes of 
+the Institute for Computer Sciences,  Social  Informatics and  
+Telecommunications  Engineering pp  355-366,  June  2015.  
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+arXiv:2301.13438v1  [math.DG]  31 Jan 2023
+HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+Abstract. The sub-Finslerian geometry means that the metric F is defined
+only on a given subbundle of the tangent bundle, called a horizontal bundle.
+In the paper, a version of the Hopf-Rinow theorem is proved in the case of sub-
+Finslerian manifolds, which relates the properties of completeness, geodesically
+completeness, and compactness. The sub-Finsler bundle, the exponential map
+and the Legendre transformation are deeply involved in this investigation.
+1. Introduction
+In the Riemannian and Finslerian geometry, there are two concepts of complete-
+ness. The first is the completeness in the sense of metric spaces, using the Riemann-
+ian metric. Secondly, a Riemannian or Finsler manifold M is called geodesically
+complete if any geodesic γ(t) starting from x ∈ M is defined for all values of
+t ∈ R. On the other hand, the completeness in the Finsler geometry is divided
+into forward and backward geodesically completenesses, according to forward and
+backward distance metrics, resp.
+Hopf-Rinow theorem is a basic theorem of complete Riemannian manifolds,
+which connects the completeness properties with compactness, and the exponen-
+tial map. Its consequence says that any two points of a complete manifold can
+be connected by a length minimizing geodesic. In 1931, H. Hopf and W. Rinow
+showed their theorem only for surfaces, but the proof in higher dimensions is not
+significantly different. Hopf-Rinow theorem has been studied in detail in both Rie-
+mannian and Finslerian geometries in the literature, the best general references
+here are [5, 7], [10]. In the Finsler case forward geodesic completeness is involved,
+only.
+After a development of the sub-Riemannian geometry as well as its generaliza-
+tion, namely sub-Finslerian geometry, the generalization of core theorems of Rie-
+mannian geometry has been started. Relating to our issue, Strichartz [13], Rifford
+[12] and Agrachev et al. [1] gave an extension for a sub-Riemannian case, while Bao
+et al. [5] showed the Finslerian version of Hopf-Rinow theorem. It turned out that
+in sub-Riemannian geometry, for general complete sub-Riemannian structures, the
+exponential mapping is not surjective. This is due to the fact that we may have
+abnormal minimizing curves and this is the case in the sub-Finslerian context, too.
+To prove the statements of Hopf-Rinow theorem in the sub-Finsler setting, we
+need the following concepts and explanations. First, in Section 2 we review some
+of the standard facts of sub-Finsler geometry. In the third Section, we extend our
+discussion about the Legendre transformation (see [2]) to define the sub-Finsler
+2000 Mathematics Subject Classification. 53C60, 53C17, 53C22.
+Key words and phrases. sub-Finslerian geometry; sub-Hamiltonian geometry; Legendre trans-
+formation; sub-Finsler bundle; normal geodesics; Exponential map; Hopf-Rinow theorem.
+1
+
+2
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+manifold on the distribution D∗ of the cotangent space, where we look more closely
+at a sub-Hamiltonian H defined on D∗, induced by the sub-Finslerian metric F ∗.
+Afterwards, we construct a sub-Finsler bundle, which plays a major role in the for-
+malization of the sub-Hamiltonian in sub-Finsler geometry, in Section 4. Moreover,
+the sub-Finsler bundle allows an orthonormal frame for the sub-Finsler structure.
+In Section 5, we introduce the notion of an exponential map in sub-Finsler geometry.
+In the last section our main theorem is stated and proved.
+2. Definitions and some properties of sub-Finsler manifolds
+In this section we review some of the standard facts on the sub-Finsler metrics
+and set up the notations and the terminology which will play an essential role in
+this paper, for more details we refer the reader to [2, 3, 4].
+Definition 1. Let M be an n–dimensional connected manifold. A sub-Finslerian
+structure on M is a triple (D, σ, F) where:
+(1) (D, πD) is a vector bundle on M.
+(2) σ : D
+� T M is a morphism of vector bundles. In particular, the following
+diagram is commutative
+M
+π
+�
+D
+T M
+σ
+�
+M
+πD
+�❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+❄
+such that πD : D
+� M and π : T M
+� M are the canonical projections.
+(3) A function F : �D → R, where �D = D \ {0}, called a sub-Finsler metric,
+which satisfies the following properties:
+• (Positive definiteness): Fx(v) > 0 for all v ∈ �D, x ∈ M.
+• (Regularity): F is smooth, i.e. C∞ on �D.
+• (Positive homogeneity): Fx(λv) = λFx(v) for all v ∈ �Dx and λ ∈ R+.
+• (Strong convexity condition): The Hessian matrix of F 2 with respect
+to the coordinates on the fibre is positive definite.
+One can replace the strong convexity condition by the following subad-
+ditivity property (in an equivalent terminology, a triangle inequality):
+Fx(v + u) ⩽ Fx(v) + Fx(u), for all v, u ∈ �D.
+A sub-Finsler manifold is a smooth manifold M endowed with a sub-Finslerian
+structure, i.e. the triple (D, σ, F).
+Let Dx be the fiber over x ∈ M. The last condition of the sub-Finsler metric
+means that the matrix
+∂2F 2
+∂vi∂vj (x, v) is positive definite for all v = (v1, . . . , vk) ∈ Dx.
+Equivalently, the corresponding indicatrix
+Ix = {v | v ∈ Dx, Fx(v) = 1}
+is strictly convex.
+The following technique describes the association between the sub-Finsler struc-
+ture (D, σ, F) and a Finsler metric ˆF on Im(σ) ⊂ T M:
+For each u ∈ Im(σ)x ⊂ TxM and x ∈ M, we have
+ˆFx(u) = inf
+v {Fx(v)| v ∈ Dx, σ(v) = u}.
+
+HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY
+3
+From now on we suppose that D ⊂ T M,
+σ : D
+� T M is the inclusion i :
+D
+� T M and F is a sub-Finsler metric on D.
+As in the sub-Riemannian case, we call D the horizontal distribution. A piecewise
+smooth curve γ : [0, T ] → M is called horizontal, or admissible if ˙γ(t) ∈ Dγ(t) for
+all t ∈ [0, T ], that is, γ(t) is tangent to D. The length of γ is defined as usual by
+ℓ(γ) =
+� T
+0
+F(˙γ(t))dt.
+Equivalently, as in the Finslerian case, we observe that it suffices to minimize
+the energy
+E(γ) = 1
+2
+� T
+0
+F 2(˙γ(t))dt.
+instead of length ℓ(γ).
+The length induces a sub-Finslerian distance d(x, y) between two points x and
+y as in Finsler geometry:
+d(x, y) = inf{ℓ(γ) |γ : [0, T ]
+� M horizontal, γ(0) = x, γ(T ) = y},
+where we consider the infimum over all horizontal curves joining x and y. The
+distance is infinite if there is no such a horizontal curve between x and y.
+In
+addition, the horizontal curve γ : [0, T ] → M is called a length minimizing (or
+simply a minimizing) geodesic, if it realizes the distance between its end points,
+that is, ℓ(γ) = d(γ(0), γ(T )).
+Chow theorem answers to the following question: given two points x and y in a
+sub-Finsler manifold, is there a horizontal curve that joins x and y?
+In the case of an involutive distribution D the Frobenius theorem asserts that
+the set of the horizontal paths through S form a smooth immersed submanifold, the
+leaf through x, of dimension equal to the rank of distribution k. In this case, if D
+is involutive and y is not contained in the leaf through y, there is no any horizontal
+curve joining x and y.
+A positive answer is given by the Chow theorem in the case of bracket generating
+distributions, which are the ”contrary” of the involutive distributions.
+Definition 2. [9] A distribution D is said to be bracket generating if any local frame
+Xi of D, together with all of its iterated Lie brackets spans the whole tangent bundle
+T M.
+Theorem 3. (Chow’s theorem [9]) If D is a bracket generating distribution on a
+connected manifold M then any two points of M can be joined by a horizontal path.
+Remark 1. The problem of minimizing the length of a curve joining two given
+points x and y is equivalent to a time optimal problem: where the control bundle
+is (D, πD, M) and we are searching for such a curve γ(t) and a control curve v(t) ∈
+Dγ(t) minimizing the time T needed to connect x and y.
+3. Legendre transformation of sub-Finslerian geometry
+Let D∗ be a distribution of rank s on a smooth manifold M that assigns to
+each point x ∈ U ⊂ M a linear subspace D∗
+x ⊂ T ∗
+xM of dimension s, see [4]. In
+other words, D∗ of rank s is a smooth subbundle of rank s of the cotangent bundle
+
+4
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+T ∗M. Such a field of cotangent s-planes is spanned locally by s pointwise linear
+independent smooth differential 1-forms, namely,
+D∗
+x = span{α1(x), . . . , αs(x)},
+αi(x) ∈ X∗(M).
+In addition, we refer to D0
+x as the annihilator of the distribution D (isomorphic to
+D), of rank n − k, which is the set of all covectors that annihilates the vectors in
+Dx, i.e.
+D0
+x = {α ∈ T ∗
+xM : α(v) = 0 ∀ v ∈ Dx}.
+(1)
+In [2], we introduced the Legendre transformation of sub-Finsler geometry. Let
+us briefly recall it:
+The sub-Lagrange function L : D
+�R, determined by F is given in the following
+way: L = 1
+2F 2. The fiber derivative of L defines the map
+LL : D
+� D∗,
+LL(v)(w) = d
+dtLx(v + tw), where v, w ∈ Dx,
+called the Legendre transformation of (M, D, F).
+We denote by (xi) the coordinate in a neighborhood U ⊂ M with (xi, va) in
+D|U ⊂ T M, and (xi, pa) in D∗|U ⊂ T ∗M, respectively, where i = 1, . . . , n, a =
+1, . . . , k. Then the relation of the distribution D of the tangent bundle and the
+distribution D∗ of the cotangent bundle is given by the Legendre transformation in
+local coordinates as follows
+LL(xi, va) = (xi, ∂L
+∂va ).
+Then the sub-Hamiltonian is given by
+H : D∗
+� R,
+H = ιL−1
+L − L ◦ L−1
+L ,
+where ιv(p) = ⟨v, p⟩ = p(v) for any v = L−1
+L (p) ∈ D and p ∈ D∗. Moreover, locally
+given by,
+H(xi, pa) = vapa − L(xi, va), where pa = ∂L
+∂va .
+Secondly, using the fiber derivative of H, we define the Legendre transformation of
+the sub-Hamiltonian H in the following way:
+LH : D∗
+� D,
+For any p, q ∈ D∗
+x, it holds
+q(LH(p)) = d
+dtH(x, p + tq).
+This locally relates the distribution D∗ of the cotangent bundle and the distribution
+D of the tangent bundle according to the next expression:
+LH(xi, pa) = (xi, ∂H
+∂pa
+).
+Naturally, LL and LH are inverses of each other:
+LH ◦ LL = 1D,
+LL ◦ LH = 1D∗.
+
+HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY
+5
+In other hand, for every p ∈ D∗
+x, one can define the sub-Finsler metric F ∗ ∈
+�D∗ ∼ T ∗M \ D0 with help of the indicatrix Ix as follows:
+F ∗
+x(p) := sup
+w∈Ix
+p(w) =
+sup
+0̸=v∈Dx
+p[
+v
+Fx(v)].
+Observed that �D∗ is the subbundle of the cotangent bundle obtained by removing
+the zero cotangent vector from each fibre. In fact, F ∗ turns out to meet the same
+properties that mentioned in Definition 1, but on D∗ instead of D. Then
+F ∗(p) = F(v), where p = LL(v),
+and
+H := 1
+2(F ∗)2,
+see details in [5].
+4. Sub-Finsler bundle
+We define in this section a sub-Finsler vector bundle which will play a major role
+in the formalization of the sub-Hamiltonian in sub-Finsler geometry. Let us consider
+first the covector subbundle (D∗, τ, M) with the projection τ : D∗
+� M, which is
+a subbundle of rank k (= dim D∗) in the cotangent bundle of T ∗M. The illustrious
+role in our consideration will play by the pullback bundle τ ∗(τ) = (D∗×D∗, pr1, D∗)
+of τ by τ as follows:
+D∗ ×M D∗ := {(p, q) ∈ D∗ × D∗| τ(p) = τ(q)},
+pr1 : D∗ ×M D∗
+� D∗, (p, q) �→ p.
+Throughout, we call the above pullback bundle as the sub-Finsler bundle over
+D∗. Now, if p is fixed, then
+(pr1)−1(p) = {(p, q) ∈ D∗ × D∗| q ∈ D∗
+τ(q)}
+= {p} × D∗
+τ(p),
+is a fiber of the sub-Finsler bundle over p ∈ D∗.
+We can introduce a Riemannian metric g∗ on the sub-Finsler vector bundle
+induced by the sub-Hamiltonian H as follows:
+⟨q, r⟩p = g∗
+p(q, r) := ∂2H(p + tq + sr)
+∂t∂s
+|t,s=0
+for all q, r ∈ D∗
+τ(p),
+which locally means
+g∗ij =
+∂2H
+∂pi∂pj
+.
+Now the sub-Finsler bundle τ ∗(τ) allows k covector fields X1, X2, . . . , Xk which
+form an orthonormal frame with respect to the induced Riemannian metric g∗.
+Notice that Xi(p) is a covector field that depends on the position x ∈ M and the
+direction p ∈ D∗. Moreover, one can choose in a way that Xi(p) is a homogeneous of
+degree zero in p, i.e. Xi(tp) = t0Xi(p) = Xi(p). According to the above metric g∗ij
+on M which is homogeneous of degree zero, we could generate a new formalism of
+the sub-Hamiltonian function in the components pi (induces naturally by the inner
+product, see [6])
+H(x, p) = 1
+2
+n
+�
+i,j=1
+g∗ijpipj,
+(2)
+
+6
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+such that this metric defined in the extended Finsler metric which was shown in
+[2]. We can write the sub-Hamiltonian function (2) in a more useful way using the
+orthonormality of Xi as follows
+H(x, p) = 1
+2
+k
+�
+i=1
+⟨p, Xi(p)⟩2,
+p ∈ D∗
+x.
+(3)
+One can easily check the homogeneity of degree 2 in p of the sub-Hamiltonian
+function H(x, p):
+H(x, tp) = 1
+2
+k
+�
+i=1
+⟨tp, Xi(tp)⟩2 = t2
+2
+k
+�
+i=1
+⟨p, Xi(p)⟩2 = t2H(x, p).
+(4)
+The importance of H(x, p) is to define sub-Finslerian geodesics. Our function
+H(x, p) produces a system of sub-Hamiltonian differential equations, since it is
+a smooth function on D∗. Such differential equations are in terms of canonical
+coordinates (xi, pi).
+Definition 4. The generated sub-Hamiltonian differential equations
+˙xi = ∂H
+∂pi
+(x, p),
+˙pi = −∂H
+∂xi (x, p),
+i = 1, . . . , n,
+are called normal geodesic equations.
+Lemma 5. If ξ(t) := (x(t), p(t)) is a solution of the sub-Hamiltonian system for
+all t ∈ R, then there exists a constant c ∈ R such that H(x(t), p(t)) = c.
+Proof. Taking the derivative of H(x(t), p(t)) w.r.t. t, we get
+d
+dtH(x(t), p(t)) = ∂H
+∂xi (x(t), p(t)) ˙x(t) + ∂H
+∂pi
+(x(t), p(t)) ˙p(t).
+Replacing ˙x(t) and ˙p(t) by the above sub-Hamiltonian differential equations in the
+Definition 4, we obtain
+d
+dtH(x(t), p(t))) = ∂H
+∂xi (x(t), p(t))∂H
+∂pi
+(x(t), p(t)) − ∂H
+∂pi
+(x(t), p(t))∂H
+∂xi (x(t), p(t))
+= 0.
+Therefore H(x(t), p(t)) is constant.
+□
+Remark 2. From Lemma 5, it follows that any solution ξ(t) := (x(t), p(t)) of
+the sub-Hamiltonian differential equations on D∗ for a sub-Hamiltonian function
+H(p) satisfies H(x(t), p(t)) = c. Let the projection x(t) = τ(ξ(t)) ∈ M, so each
+sufficiently short subarc of x(t) is a minimizer sub-Finslerian geodesic, (see [11,
+Corollary 2.2]). In addition, this subarc is the unique minimizer joining its end
+points.
+The projection curve x(t) mentioned above is said to be the normal sub-Finslerian
+geodesics or simply the normal geodesics.
+Remark 3. In the sub-Finslerian geometry, not all the sub-Finslerian geodesics
+are normal (contrary to the Finsler geometry). This is due to the fact that the
+sub-Finslerian geodesics which are also a minimizing geodesic might not be solved
+
+HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY
+7
+the sub-Hamiltonian system. Those minimizer that are not normal geodesics called
+singular or abnormal geodesics (see [9] for more details).
+Moreover, we call the extremal pair ξ(t) = (x(t), p(t)) a normal extremal if it
+is a solution for the sub-Hamiltonian system, otherwise it is called an abnormal
+extremal.
+Turning to the relationship between the normal geodesic and the locally length-
+minimizing horizontal curves, Calin et al. proved in [6] that any normal geodesic is
+a horizontal curve and a locally length-minimizing horizontal curve. After all, by
+using (3) one can generate the system of differential equations in terms of canonical
+coordinates (x, p) as follows:
+˙xi = ∂H
+∂pi
+=
+k
+�
+j=1
+⟨p, Xj(p)⟩ (δi(Xj(p)) + ⟨p, DpiXj(p)⟩),
+(5)
+˙pi = −∂H
+∂xi = −
+k
+�
+j=1
+⟨p, Xj(p)⟩⟨p, DxiXj(p)⟩,
+(6)
+where δi is the i-th coordinate function.
+5. Exponential map in sub-Finsler geometry
+Let (M, d) be a general metric space, such that M is an n-dimensional manifold
+and the function d : M × M
+� R+ ∪ {∞}, is called a metric if have the following
+properties: for all x, y, z ∈ M,
+(i) d(x, y) = 0, with equality if and only if x = y;
+(ii) d(x, y) + d(y, z) ≤ d(x, z).
+If the function d is an asymmetric, then we can define the forward metric balls and
+forward metric spheres, with center x ∈ M and radius r > 0 as follows:
+Bx(r) = { y ∈ M : d(x, y) < r},
+Sx(r) = { y ∈ M : d(x, y) = r}.
+The cotangent balls and the cotangent spheres in D∗ are defined as follows:
+B∗
+x(r) = { p ∈ D∗ : F ∗
+x(p) < r},
+S∗
+x(r) = { p ∈ D∗ : F ∗
+x(p) = r},
+for any fix x ∈ M and radius r.
+A subset U ⊂ M is said to be open if, for each point x ∈ U, there is a forward
+metric ball about x contained in U. Then we get the topology on M and all metric
+spaces are first countable and T1-spaces. In general, we assume that the metric d
+of any metric space (M, d) is continuous with respect to the product topology on
+M × M. Thus, every backward metric ball, i.e. B−
+x (r) = { y ∈ M : d(y, x) < r},
+is open and the metric space is a Hausdorff (T2) space. Hence the compact sets in
+such a space are closed.
+As a result of the above, we immediately have the following
+Proposition 6. In a metric space (M, d) the following are equivalent:
+(i) A sequence {xk} in (M, d) converges to x ∈ M in the sense of topology.
+(ii) limk→∞ d(x, xk) = 0.
+
+8
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+Proposition 7. Let x be any point in a (reversible) sub-Finslerian manifold M,
+and ¯Bx(r) is a compact ball, for some r > 0. Then for any y ∈ Bx(r) there is a
+minimizing geodesic from x to y, that is,
+d(x, y) = min{ℓ(γ) |γ : [0, T ]
+� M horizontal, γ(0) = x, γ(T ) = y}.
+Proof. Fix y ∈ Bx(r) and suppose that γk : [0, T ]
+� M is a minimizing sequence
+of horizontal paths with unit speed from x to y and such that
+lim
+k→∞ γk(0) = x,
+lim
+k→∞ γk(T ) = y,
+lim
+k→∞ ℓ(γk) = d(x, y).
+For the reason that d(x, y) < r, we get ℓ(γk) ≤ r for all k ≥ k0 large enough.
+Proposition 6 asserts that the metric d is continuous under the topology of the
+manifold and the reversibility of F holds on a compact set.
+Consequently, any
+sequence γk of curves which have uniformly bounded lengths has an uniformly
+convergent subsequence (Ascoli—Arzela theorem), we denote this subsequence by
+the same symbol, and a Lipschitz curve γ : [0, T ]
+� M.
+From above one can assume that γk : [0, T ]
+� M is a convergent subsequence
+of length minimizers parametrized by arc length (i.e. F(˙γ(t)) = 1) on M such that
+such that γk
+� γ uniformly on [0, T ]. This gives that
+ℓ(γk) = d(γk(0), γk(T )),
+which is due to the claim that γk is a minimizing geodesic.
+The sequence γk
+converges uniformly if for every ǫ > 0 there is a natural number N such that for
+all n ≥ N and all t ∈ [0, T ] one has d(γk(t), γ(t)) < ǫ. Further, the semicontinuity
+of the length implies that if limk→∞ γk = γ then
+ℓ(γ) ≤ lim
+k→∞ inf ℓ(γk).
+Now, by continuity of the distance, we obtain
+ℓ(γ) ≤ lim
+k→∞ inf ℓ(γk) = lim
+k→∞ inf d(γk(0), γk(T )) = d(γ(0), γ(T )).
+This yields that γ is minimizing geodesic, i.e.
+ℓ(γ) = d(x, y).
+The horizontal
+property of γ follows in the same way as was done in [1], Theorem 3.41.
+□
+Next, we define the exponential map. For the general case, roughly speaking,
+if M is a smooth Finsler manifold, x a point in M and u ∈ TxM.
+Then the
+exponential map is given by
+expx : TxM
+� M,
+such that expx(u) = γu(1) for the unique geodesic γ that starts at x and has initial
+speed vector u.
+Furthermore, in the dual space the exponential map for every
+x ∈ M and p ∈ T ∗
+xM defined by
+exp∗
+x : T ∗
+xM
+� M,
+such that exp∗
+x(p) = γp(1) for the unique geodesic γ that starts at x and has initial
+speed vector u = L−1
+L (p), where L here is the Lagrangian of the Finsler manifold.
+The exponential map is an essential object in sub-Finslerian geometry, parametriz-
+ing normal extremals through their initial covectors. We are going to define the
+exponential map in both of the distribution D, D∗ of the tangent and the cotangent
+bundles respectively.
+
+HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY
+9
+Definition 8. Let Ωx ⊂ Dx be the domain of the exponential map over x ∈ M
+such that Ωx given by
+Ωx = {v ∈ Dx| ξ is defined on the interval [0, 1]} ,
+where v = LH(p) by the Legendre transformation of sub-Hamiltonian H, and ξ(t)
+is the normal extremal. Then the sub-Finsler exponential map is defined as follows
+expx : Ωx ⊂ Dx ⊂ TxM
+� M, v �→ πD(LH(ξ(1))).
+We can do the same in the distribution D∗
+x. Let Ω∗
+x ⊂ D∗
+x be the domain of the
+exponential map over x ∈ M such that Ω∗
+x given by
+Ω∗
+x = {p ∈ D∗
+x| ξ is defined on the interval [0, 1]} .
+Consequently, the sub-Hamiltonian exponential map is given by
+exp∗
+x : Ω∗
+x ⊂ D∗
+x ⊂ T ∗
+xM
+� M, p �→ τ(ξ(1)),
+where ξ(t) is the same normal extremal as above. The set Ω∗
+x contains the origin and
+star-shaped with respect to 0. Moreover, with the help of Legendre transformation
+it is fairly easy to see that
+expx(v) = exp∗
+x(p),
+where
+p = LL(v).
+(7)
+It follows that the normal sub-Finslerian geodesics x(t) = τ(ξ(t)) satisfies
+x(t) = exp∗
+x(tp),
+for all t ∈ [0, T ].
+Theorem 9. The exponential mapping exp∗
+x is a local diffeomorphism on D∗
+x ⊂
+T ∗
+xM\{0}.
+Proof. In 4, we show the homogeneity of the sub-Hamiltonian function H(x, p) with
+respect to p. So, for any constant a > 0, the curve ξ(at) : (ǫ/a, ǫ/a)
+� M is the
+same geodesic satisfying the initial conditions τ(ξp(0)) = x and ξp(0) = ap, i.e.,
+τ(ξp(at)) = τ(ξap(t)).
+Since the sub-Hamiltonian vector field
+⃗H(x, p) = gab(x, p)pb
+∂
+∂xa − 1
+2
+∂gab
+∂xk (x, p)papb
+∂
+∂pk
+,
+that introduced in [2], is smooth except for p = 0 where it is C1. Then exp∗
+x is C∞
+on D∗
+x ⊂ T ∗
+xM\{0}, while it is C1 at p = 0 and d(exp∗
+x)|0 = id. Thus, exp∗
+x is a
+local diffeomorphism.
+□
+By equation (7), one can get the following
+Corollary 10. The sub-Finsler exponential map expx is a C∞ away from the zero
+section of D and only C1 at the zero section such that for each x ∈ M
+d(expx)|0 : Ωx ⊂ Dx
+� Ωx ⊂ Dx,
+is the identity map at the origin 0 ∈ Dx.
+Remark 4. It is clear that in the case of sub-Finsler exponential map the following
+expressions holds:
+exp∗
+x[B∗
+x(r)] = Bx(r),
+exp∗
+x[S∗
+x(r)] = Sx(r),
+which are analogous to the Finslerian context, see Bao et al. [5] for more details.
+
+10
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+Remark 5. Turning to sub-Riemannian case, Strichartz in [13] stated that for
+bracket generating distributions the exponential map is a local diffeomorphism.
+This is due to the fact that the solutions of the sub-Hamiltonian system depend
+differentially on the initial data.
+But this is a difference from the Riemannian
+context, the exponential map is not a diffeomorphism at the origin just like the
+Finslerian case.
+6. Hopf-Rinow Theorem in sub-Finslerian geometry
+In the following, one can see the explanation of the terms that will be used in
+Hopf-Rinow Theorem. A sub-Finsler manifold is said to be forward complete if
+every forward Cauchy sequence converges, and it is a forward geodesically complete
+if every normal geodesic γ(t), t ∈ [0, T ) parametrized to have unit speed, can be
+extended to a geodesic for all t ∈ [0, ∞). A subset is said to be forward bounded if
+it is contained in some forward metric ball Bx(r).
+Theorem 11. Let (M, D, F) be any connected sub-Finsler manifold, where D is
+bracket generating distribution. The following conditions are equivalent:
+(i) The metric space (M, d) is forward complete.
+(ii) The sub-Finsler manifold (M, D, F) is forward geodesically complete.
+(iii) Ω∗
+x = D∗
+x, additionally, the exponential map is onto if there are no strictly
+abnormal minimizer.
+(iv) Every closed and forward bounded subset of (M, d) is compact.
+Furthermore, for any x, y ∈ M there exists a minimizing geodesic γ joining x to y,
+i.e. the length of this geodesic is equal to the distance between these points.
+Proof. (i) =⇒ (ii) Let γ(t) : [0, T )
+� M be a unit speed and maximally forward
+extended geodesic, t ∈ [0, T ). If we assume that T ̸= ∞, and choose a sequence
+{ti}
+� T in [0, T ) then γ(ti) is forward Cauchy, since
+d(γ(ti), γ(tj)) ≤ |tj − ti|, for all i ≤ j.
+Now, (i) makes it obvious that γ(ti) converges to y ∈ M. On one hand, let us
+define γ(T ) to be y. On the other hand, Lemma 4.1 in [13] told us that γ(t) can
+be extended beyond t = T . This contradicts our assumption the fact that T ̸= ∞.
+Thus, T = ∞ for sure, so we have the forward geodesically completeness.
+(ii) =⇒ (iii) It is sufficient (for first part Ω∗
+x = D∗
+x) to prove that any normal
+extremal pair ξ(t), starting from the initial conditions, is defined for all t ∈ R.
+Suppose that the normal extremal is not extendable to the some interval [0, T +δ) for
+all δ > 0 and suppose that it is defined on [0, T ). Let {ti} be any increasing sequence
+such that the limit of this sequence is T . Hence, the projection x(t) = τ(ξ(t)) is
+a curve with unit speed defined on [0, T ), therefore, the sequence {ti} is a forward
+Cauchy sequence on M, since
+d(x(ti), x(tj)) ≤ |ti − tj|.
+By completeness, it follows that the sequence x(ti) converges to some point
+y ∈ M. We suppose there are coordinates around the point y and an orthonormal
+frame X1, X2, ..., Xk in small ball B∗
+y(r) in the sub-Finsler bundle. Let us show
+that in the coordinates ξ(t) = (x(t), p(t)) the curve x(t) is uniformly bounded. This
+grants a contradiction that the normal extremal is not extendable. In fact, for every
+
+HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY
+11
+p ∈ D∗, we consider the following non-negative form (3) of the sub-Hamiltonian
+function H:
+H(x, p) = 1
+2
+k
+�
+i=1
+⟨p, Xi(p)⟩2.
+Then, the sub-Hamiltonian system has the form:
+˙xi(t) = ∂H
+∂pi
+(x(t), p(t)) =
+k
+�
+j=1
+⟨p(t), Xj(p(t))⟩(δi(Xj(p)) + ⟨p, DpiXj(p)⟩),
+˙pi(t) = −∂H
+∂xi (x(t), p(t)) = −
+k
+�
+j=1
+⟨p(t), Xj(p(t))⟩⟨p(t), DxiXj(p(t))⟩,
+for t ∈ [T − δ, T ) with δ > 0 small enough. Since Dγ(t)Xi are given in a compact
+small ball ¯B∗
+y(r), they are bounded, so there is a constant C > 0 such that
+| ˙p(t)| ≤ C|p(t)|
+∀t ∈ [T − δ, T ).
+If we apply Gronwall’s Lemma (see [12], p.122), it leads us to that |p(t)| is uniformly
+bounded on a bounded interval. This contradicts our assumption that the normal
+extremal can not be extended beyond T .
+(iii) =⇒ (iv) Assume that ¯A is a closed and forward bounded subset of (M, d).
+Applying the bracket generating assumption, for every y ∈ ¯A, Proposition 7 asserts
+that there is a minimizing geodesic exp∗
+x(tpy), 0 ≤ t ≤ T, from x to y. The set of all
+py is subset A of D∗
+x. Since F ∗
+x (py) = d(x, y), and d(x, y) ≤ r for some r due to the
+forward boundedness of ¯A, the subset A is bounded and contained in the compact
+set B∗
+x(r) ∪ S∗
+x(r). By Remark 4, exp∗
+x[B∗
+x(r) ∪ S∗
+x(r)] is compact and contained in
+the closed set ¯A, then ¯A it must be compact.
+(iv) =⇒ (i) Let {xi} be a forward Cauchy sequence in M, and by the subaddi-
+tivity it must be forward bounded. Choose A := {xi|i ∈ N}, then its closure ¯A is
+still forward bounded under the manifold topology of M. Taking into account the
+assumption (iv), ¯A should satisfy the compactness property, therefore, the sequence
+{xi} contains a convergent subsequence.
+Let {xk} be a convergent subsequence, consider it converges to some y ∈ ¯A ⊂ M.
+In other hand, we need to check that {xi} converges to y ∈ ¯A ⊂ M. To do this,
+fix ǫ > 0, since {xi} is forward Cauchy, there exist a positive number n0 such that
+j > i ≥ n0, then
+d(xi, xj) < ǫ
+2.
+At the same time {xk} converge to y. So there is a positive number n1 such that
+if k ≥ n1, then
+d(xk, y) < ǫ
+2.
+One can assume that n is greater than n0 and n1. If needed, by expanding n
+further, there is no loss of generality in assuming that n indeed equals some index
+of the convergent subsequence. Then d(xn, y) ≤ ǫ
+2, so, for i > n, we get
+d(xi, y) ≤ d(xi, xn) + d(xn, y)< ǫ
+2 + ǫ
+2 = ǫ.
+So, we have been shown that every forward Cauchy sequence is convergent. Hence
+(M, d) is forward complete.
+
+12
+LAYTH M. ALABDULSADA AND L´ASZL ´O KOZMA
+At the end, we can use the same proof of Proposition 7 to verify that for every
+x, y ∈ M there exists a length minimizing geodesic joining x and y, and it has
+to be normal geodesic by Remark 2.
+Finally, the property of compactness and
+completeness with help of Proposition 7, proves the second part of (iii).
+□
+References
+[1] A. Agrachev, D. Barilari, U. Boscain, A Comprehensive Introduction to Sub-Riemannian
+geometry. Cambridge Studies in Advanced Mathematics (2019).
+[2] L. M. Alabdulsada, L. Kozma, On the connection of sub-Finslerian geometry. Int. J. Geom.
+Methods Mod. Phys. 16, No. supp02, 1941006 (2019)
+[3] L. M. Alabdulsada, A note on the distributions in quantum mechanical systems. J. Phys.:
+Conf. Ser. 1999, (2021), 012112
+[4] L. M. Alabdulsada, Sub-Finsler geometry and nonholonomic mechanics. Submitted
+[5] D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann-Finsler geometry, Graduate Texts
+in Mathematics 200. Springer-Verlag, New York, (2000).
+[6] O. Calin, D. Chang, Subriemannian geometry, a variational approach. J. Differential Geom.
+80 (2008), no. 1, 23–43.
+[7] P. do Carmo, Riemannian geometry. Mathematics:
+Theory & Applications. Birkh¨auser
+Boston, Inc., Boston, MA, (1992).
+[8] W.-L. Chow, ¨Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung.
+Math. Ann. 117 (1939) 98-105.
+[9] R. Montgomery, A Tour of Subriemannian Geometries, their Geodesics and Applications,
+Mathematical Surveys and Monographs 91. Amer. Math. Soc., Providence, RI, (2002).
+[10] B. O’Neill, Semi-Riemannian Geometry. With applications to Relativity. Pure and Applied
+Mathematics, 103. Academic Press, Inc. New York, (1983).
+[11] C. B. Rayner, The Exponential Map for the Lagrange Problem on Differentiable Manifolds.
+Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Phys-
+ical Sciences Vol. 262, No. 1127 (Oct. 6, 1967), pp. 299-344 (46 pages) Published By: Royal
+Society
+[12] L. Rifford, Sub-Riemannian geometry and optimal transport. Springer, (2014).
+[13] R. Strichartz, Sub-Riemannian geometry. J. Differ. Geom. 24 (1986), 221-263; correction
+ibid. 30 (1989), 595-596.
+Layth M. Alabdulsada, Institute of Mathematics, University of Debrecen, H-4002
+Debrecen, P.O. Box 400, Hungary
+Email address: [email protected]
+L´aszl´o Kozma, Institute of Mathematics, University of Debrecen, H-4002 Debrecen,
+P.O. Box 400, Hungary
+Email address: [email protected]
+

99FQT4oBgHgl3EQf6TZx/content/tmp_files/load_file.txt

@@ -0,0 +1,388 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf,len=387
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='13438v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='DG]  31 Jan 2023  HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The sub-Finslerian geometry means that the metric F is defined  only on a given subbundle of the tangent bundle, called a horizontal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In the paper, a version of the Hopf-Rinow theorem is proved in the case of sub-  Finslerian manifolds, which relates the properties of completeness, geodesically  completeness, and compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The sub-Finsler bundle, the exponential map  and the Legendre transformation are deeply involved in this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Introduction  In the Riemannian and Finslerian geometry, there are two concepts of complete-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The first is the completeness in the sense of metric spaces, using the Riemann-  ian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Secondly, a Riemannian or Finsler manifold M is called geodesically  complete if any geodesic γ(t) starting from x ∈ M is defined for all values of  t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' On the other hand, the completeness in the Finsler geometry is divided  into forward and backward geodesically completenesses, according to forward and  backward distance metrics, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Hopf-Rinow theorem is a basic theorem of complete Riemannian manifolds,  which connects the completeness properties with compactness, and the exponen-  tial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Its consequence says that any two points of a complete manifold can  be connected by a length minimizing geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In 1931, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hopf and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Rinow  showed their theorem only for surfaces, but the proof in higher dimensions is not  significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hopf-Rinow theorem has been studied in detail in both Rie-  mannian and Finslerian geometries in the literature, the best general references  here are [5, 7], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In the Finsler case forward geodesic completeness is involved,  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  After a development of the sub-Riemannian geometry as well as its generaliza-  tion, namely sub-Finslerian geometry, the generalization of core theorems of Rie-  mannian geometry has been started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Relating to our issue, Strichartz [13], Rifford  [12] and Agrachev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' [1] gave an extension for a sub-Riemannian case, while Bao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' [5] showed the Finslerian version of Hopf-Rinow theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' It turned out that  in sub-Riemannian geometry, for general complete sub-Riemannian structures, the  exponential mapping is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' This is due to the fact that we may have  abnormal minimizing curves and this is the case in the sub-Finslerian context, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  To prove the statements of Hopf-Rinow theorem in the sub-Finsler setting, we  need the following concepts and explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' First, in Section 2 we review some  of the standard facts of sub-Finsler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In the third Section, we extend our  discussion about the Legendre transformation (see [2]) to define the sub-Finsler  2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' 53C60, 53C17, 53C22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' sub-Finslerian geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' sub-Hamiltonian geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Legendre trans-  formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' sub-Finsler bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' normal geodesics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Exponential map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hopf-Rinow theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  1  2  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  manifold on the distribution D∗ of the cotangent space, where we look more closely  at a sub-Hamiltonian H defined on D∗, induced by the sub-Finslerian metric F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Afterwards, we construct a sub-Finsler bundle, which plays a major role in the for-  malization of the sub-Hamiltonian in sub-Finsler geometry, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Moreover,  the sub-Finsler bundle allows an orthonormal frame for the sub-Finsler structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In Section 5, we introduce the notion of an exponential map in sub-Finsler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In the last section our main theorem is stated and proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Definitions and some properties of sub-Finsler manifolds  In this section we review some of the standard facts on the sub-Finsler metrics  and set up the notations and the terminology which will play an essential role in  this paper, for more details we refer the reader to [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let M be an n–dimensional connected manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' A sub-Finslerian  structure on M is a triple (D, σ, F) where:  (1) (D, πD) is a vector bundle on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (2) σ : D  � T M is a morphism of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In particular, the following  diagram is commutative  M  π  �  D  T M  σ  �  M  πD  �❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  such that πD : D  � M and π : T M  � M are the canonical projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (3) A function F : �D → R, where �D = D \\ {0}, called a sub-Finsler metric,  which satisfies the following properties:  (Positive definiteness): Fx(v) > 0 for all v ∈ �D, x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (Regularity): F is smooth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' C∞ on �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (Positive homogeneity): Fx(λv) = λFx(v) for all v ∈ �Dx and λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (Strong convexity condition): The Hessian matrix of F 2 with respect  to the coordinates on the fibre is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  One can replace the strong convexity condition by the following subad-  ditivity property (in an equivalent terminology, a triangle inequality):  Fx(v + u) ⩽ Fx(v) + Fx(u), for all v, u ∈ �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  A sub-Finsler manifold is a smooth manifold M endowed with a sub-Finslerian  structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' the triple (D, σ, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Let Dx be the fiber over x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The last condition of the sub-Finsler metric  means that the matrix  ∂2F 2  ∂vi∂vj (x, v) is positive definite for all v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' , vk) ∈ Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Equivalently, the corresponding indicatrix  Ix = {v | v ∈ Dx, Fx(v) = 1}  is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The following technique describes the association between the sub-Finsler struc-  ture (D, σ, F) and a Finsler metric ˆF on Im(σ) ⊂ T M:  For each u ∈ Im(σ)x ⊂ TxM and x ∈ M, we have  ˆFx(u) = inf  v {Fx(v)| v ∈ Dx, σ(v) = u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY  3  From now on we suppose that D ⊂ T M,  σ : D  � T M is the inclusion i :  D  � T M and F is a sub-Finsler metric on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  As in the sub-Riemannian case, we call D the horizontal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' A piecewise  smooth curve γ : [0, T ] → M is called horizontal, or admissible if ˙γ(t) ∈ Dγ(t) for  all t ∈ [0, T ], that is, γ(t) is tangent to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The length of γ is defined as usual by  ℓ(γ) =  � T  0  F(˙γ(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Equivalently, as in the Finslerian case, we observe that it suffices to minimize  the energy  E(γ) = 1  2  � T  0  F 2(˙γ(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  instead of length ℓ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The length induces a sub-Finslerian distance d(x, y) between two points x and  y as in Finsler geometry:  d(x, y) = inf{ℓ(γ) |γ : [0, T ]  � M horizontal, γ(0) = x, γ(T ) = y},  where we consider the infimum over all horizontal curves joining x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The  distance is infinite if there is no such a horizontal curve between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In  addition, the horizontal curve γ : [0, T ] → M is called a length minimizing (or  simply a minimizing) geodesic, if it realizes the distance between its end points,  that is, ℓ(γ) = d(γ(0), γ(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Chow theorem answers to the following question: given two points x and y in a  sub-Finsler manifold, is there a horizontal curve that joins x and y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In the case of an involutive distribution D the Frobenius theorem asserts that  the set of the horizontal paths through S form a smooth immersed submanifold, the  leaf through x, of dimension equal to the rank of distribution k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In this case, if D  is involutive and y is not contained in the leaf through y, there is no any horizontal  curve joining x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  A positive answer is given by the Chow theorem in the case of bracket generating  distributions, which are the ”contrary” of the involutive distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' [9] A distribution D is said to be bracket generating if any local frame  Xi of D, together with all of its iterated Lie brackets spans the whole tangent bundle  T M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' (Chow’s theorem [9]) If D is a bracket generating distribution on a  connected manifold M then any two points of M can be joined by a horizontal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The problem of minimizing the length of a curve joining two given  points x and y is equivalent to a time optimal problem: where the control bundle  is (D, πD, M) and we are searching for such a curve γ(t) and a control curve v(t) ∈  Dγ(t) minimizing the time T needed to connect x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Legendre transformation of sub-Finslerian geometry  Let D∗ be a distribution of rank s on a smooth manifold M that assigns to  each point x ∈ U ⊂ M a linear subspace D∗  x ⊂ T ∗  xM of dimension s, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In  other words, D∗ of rank s is a smooth subbundle of rank s of the cotangent bundle  4  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Such a field of cotangent s-planes is spanned locally by s pointwise linear  independent smooth differential 1-forms, namely,  D∗  x = span{α1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' , αs(x)},  αi(x) ∈ X∗(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In addition, we refer to D0  x as the annihilator of the distribution D (isomorphic to  D), of rank n − k, which is the set of all covectors that annihilates the vectors in  Dx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  D0  x = {α ∈ T ∗  xM : α(v) = 0 ∀ v ∈ Dx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (1)  In [2], we introduced the Legendre transformation of sub-Finsler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let  us briefly recall it:  The sub-Lagrange function L : D  �R, determined by F is given in the following  way: L = 1  2F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The fiber derivative of L defines the map  LL : D  � D∗,  LL(v)(w) = d  dtLx(v + tw), where v, w ∈ Dx,  called the Legendre transformation of (M, D, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  We denote by (xi) the coordinate in a neighborhood U ⊂ M with (xi, va) in  D|U ⊂ T M, and (xi, pa) in D∗|U ⊂ T ∗M, respectively, where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' , n, a =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then the relation of the distribution D of the tangent bundle and the  distribution D∗ of the cotangent bundle is given by the Legendre transformation in  local coordinates as follows  LL(xi, va) = (xi, ∂L  ∂va ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Then the sub-Hamiltonian is given by  H : D∗  � R,  H = ιL−1  L − L ◦ L−1  L ,  where ιv(p) = ⟨v, p⟩ = p(v) for any v = L−1  L (p) ∈ D and p ∈ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Moreover, locally  given by,  H(xi, pa) = vapa − L(xi, va), where pa = ∂L  ∂va .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Secondly, using the fiber derivative of H, we define the Legendre transformation of  the sub-Hamiltonian H in the following way:  LH : D∗  � D,  For any p, q ∈ D∗  x, it holds  q(LH(p)) = d  dtH(x, p + tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  This locally relates the distribution D∗ of the cotangent bundle and the distribution  D of the tangent bundle according to the next expression:  LH(xi, pa) = (xi, ∂H  ∂pa  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Naturally, LL and LH are inverses of each other:  LH ◦ LL = 1D,  LL ◦ LH = 1D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY  5  In other hand, for every p ∈ D∗  x, one can define the sub-Finsler metric F ∗ ∈  �D∗ ∼ T ∗M \\ D0 with help of the indicatrix Ix as follows:  F ∗  x(p) := sup  w∈Ix  p(w) =  sup  0̸=v∈Dx  p[  v  Fx(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Observed that �D∗ is the subbundle of the cotangent bundle obtained by removing  the zero cotangent vector from each fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In fact, F ∗ turns out to meet the same  properties that mentioned in Definition 1, but on D∗ instead of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then  F ∗(p) = F(v), where p = LL(v),  and  H := 1  2(F ∗)2,  see details in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Sub-Finsler bundle  We define in this section a sub-Finsler vector bundle which will play a major role  in the formalization of the sub-Hamiltonian in sub-Finsler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let us consider  first the covector subbundle (D∗, τ, M) with the projection τ : D∗  � M, which is  a subbundle of rank k (= dim D∗) in the cotangent bundle of T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The illustrious  role in our consideration will play by the pullback bundle τ ∗(τ) = (D∗×D∗, pr1, D∗)  of τ by τ as follows:  D∗ ×M D∗ := {(p, q) ∈ D∗ × D∗| τ(p) = τ(q)},  pr1 : D∗ ×M D∗  � D∗, (p, q) �→ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Throughout, we call the above pullback bundle as the sub-Finsler bundle over  D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Now, if p is fixed, then  (pr1)−1(p) = {(p, q) ∈ D∗ × D∗| q ∈ D∗  τ(q)}  = {p} × D∗  τ(p),  is a fiber of the sub-Finsler bundle over p ∈ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  We can introduce a Riemannian metric g∗ on the sub-Finsler vector bundle  induced by the sub-Hamiltonian H as follows:  ⟨q, r⟩p = g∗  p(q, r) := ∂2H(p + tq + sr)  ∂t∂s  |t,s=0  for all q, r ∈ D∗  τ(p),  which locally means  g∗ij =  ∂2H  ∂pi∂pj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Now the sub-Finsler bundle τ ∗(τ) allows k covector fields X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' , Xk which  form an orthonormal frame with respect to the induced Riemannian metric g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Notice that Xi(p) is a covector field that depends on the position x ∈ M and the  direction p ∈ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Moreover, one can choose in a way that Xi(p) is a homogeneous of  degree zero in p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Xi(tp) = t0Xi(p) = Xi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' According to the above metric g∗ij  on M which is homogeneous of degree zero, we could generate a new formalism of  the sub-Hamiltonian function in the components pi (induces naturally by the inner  product, see [6])  H(x, p) = 1  2  n  ���  i,j=1  g∗ijpipj,  (2)  6  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  such that this metric defined in the extended Finsler metric which was shown in  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' We can write the sub-Hamiltonian function (2) in a more useful way using the  orthonormality of Xi as follows  H(x, p) = 1  2  k  �  i=1  ⟨p, Xi(p)⟩2,  p ∈ D∗  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (3)  One can easily check the homogeneity of degree 2 in p of the sub-Hamiltonian  function H(x, p):  H(x, tp) = 1  2  k  �  i=1  ⟨tp, Xi(tp)⟩2 = t2  2  k  �  i=1  ⟨p, Xi(p)⟩2 = t2H(x, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (4)  The importance of H(x, p) is to define sub-Finslerian geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Our function  H(x, p) produces a system of sub-Hamiltonian differential equations, since it is  a smooth function on D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Such differential equations are in terms of canonical  coordinates (xi, pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The generated sub-Hamiltonian differential equations  ˙xi = ∂H  ∂pi  (x, p),  ˙pi = −∂H  ∂xi (x, p),  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' , n,  are called normal geodesic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' If ξ(t) := (x(t), p(t)) is a solution of the sub-Hamiltonian system for  all t ∈ R, then there exists a constant c ∈ R such that H(x(t), p(t)) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Taking the derivative of H(x(t), p(t)) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' t, we get  d  dtH(x(t), p(t)) = ∂H  ∂xi (x(t), p(t)) ˙x(t) + ∂H  ∂pi  (x(t), p(t)) ˙p(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Replacing ˙x(t) and ˙p(t) by the above sub-Hamiltonian differential equations in the  Definition 4, we obtain  d  dtH(x(t), p(t))) = ∂H  ∂xi (x(t), p(t))∂H  ∂pi  (x(t), p(t)) − ∂H  ∂pi  (x(t), p(t))∂H  ∂xi (x(t), p(t))  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Therefore H(x(t), p(t)) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' From Lemma 5, it follows that any solution ξ(t) := (x(t), p(t)) of  the sub-Hamiltonian differential equations on D∗ for a sub-Hamiltonian function  H(p) satisfies H(x(t), p(t)) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let the projection x(t) = τ(ξ(t)) ∈ M, so each  sufficiently short subarc of x(t) is a minimizer sub-Finslerian geodesic, (see [11,  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In addition, this subarc is the unique minimizer joining its end  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The projection curve x(t) mentioned above is said to be the normal sub-Finslerian  geodesics or simply the normal geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In the sub-Finslerian geometry, not all the sub-Finslerian geodesics  are normal (contrary to the Finsler geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' This is due to the fact that the  sub-Finslerian geodesics which are also a minimizing geodesic might not be solved  HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY  7  the sub-Hamiltonian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Those minimizer that are not normal geodesics called  singular or abnormal geodesics (see [9] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Moreover, we call the extremal pair ξ(t) = (x(t), p(t)) a normal extremal if it  is a solution for the sub-Hamiltonian system, otherwise it is called an abnormal  extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Turning to the relationship between the normal geodesic and the locally length-  minimizing horizontal curves, Calin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' proved in [6] that any normal geodesic is  a horizontal curve and a locally length-minimizing horizontal curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' After all, by  using (3) one can generate the system of differential equations in terms of canonical  coordinates (x, p) as follows:  ˙xi = ∂H  ∂pi  =  k  �  j=1  ⟨p, Xj(p)⟩ (δi(Xj(p)) + ⟨p, DpiXj(p)⟩),  (5)  ˙pi = −∂H  ∂xi = −  k  �  j=1  ⟨p, Xj(p)⟩⟨p, DxiXj(p)⟩,  (6)  where δi is the i-th coordinate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Exponential map in sub-Finsler geometry  Let (M, d) be a general metric space, such that M is an n-dimensional manifold  and the function d : M × M  � R+ ∪ {∞}, is called a metric if have the following  properties: for all x, y, z ∈ M,  (i) d(x, y) = 0, with equality if and only if x = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (ii) d(x, y) + d(y, z) ≤ d(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  If the function d is an asymmetric, then we can define the forward metric balls and  forward metric spheres, with center x ∈ M and radius r > 0 as follows:  Bx(r) = { y ∈ M : d(x, y) < r},  Sx(r) = { y ∈ M : d(x, y) = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The cotangent balls and the cotangent spheres in D∗ are defined as follows:  B∗  x(r) = { p ∈ D∗ : F ∗  x(p) < r},  S∗  x(r) = { p ∈ D∗ : F ∗  x(p) = r},  for any fix x ∈ M and radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  A subset U ⊂ M is said to be open if, for each point x ∈ U, there is a forward  metric ball about x contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then we get the topology on M and all metric  spaces are first countable and T1-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In general, we assume that the metric d  of any metric space (M, d) is continuous with respect to the product topology on  M × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Thus, every backward metric ball, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' B−  x (r) = { y ∈ M : d(y, x) < r},  is open and the metric space is a Hausdorff (T2) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hence the compact sets in  such a space are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  As a result of the above, we immediately have the following  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In a metric space (M, d) the following are equivalent:  (i) A sequence {xk} in (M, d) converges to x ∈ M in the sense of topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (ii) limk→∞ d(x, xk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  8  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let x be any point in a (reversible) sub-Finslerian manifold M,  and ¯Bx(r) is a compact ball, for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then for any y ∈ Bx(r) there is a  minimizing geodesic from x to y, that is,  d(x, y) = min{ℓ(γ) |γ : [0, T ]  � M horizontal, γ(0) = x, γ(T ) = y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Fix y ∈ Bx(r) and suppose that γk : [0, T ]  � M is a minimizing sequence  of horizontal paths with unit speed from x to y and such that  lim  k→∞ γk(0) = x,  lim  k→∞ γk(T ) = y,  lim  k→∞ ℓ(γk) = d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  For the reason that d(x, y) < r, we get ℓ(γk) ≤ r for all k ≥ k0 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Proposition 6 asserts that the metric d is continuous under the topology of the  manifold and the reversibility of F holds on a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Consequently, any  sequence γk of curves which have uniformly bounded lengths has an uniformly  convergent subsequence (Ascoli—Arzela theorem), we denote this subsequence by  the same symbol, and a Lipschitz curve γ : [0, T ]  � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  From above one can assume that γk : [0, T ]  � M is a convergent subsequence  of length minimizers parametrized by arc length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' F(˙γ(t)) = 1) on M such that  such that γk  � γ uniformly on [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' This gives that  ℓ(γk) = d(γk(0), γk(T )),  which is due to the claim that γk is a minimizing geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The sequence γk  converges uniformly if for every ǫ > 0 there is a natural number N such that for  all n ≥ N and all t ∈ [0, T ] one has d(γk(t), γ(t)) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Further, the semicontinuity  of the length implies that if limk→∞ γk = γ then  ℓ(γ) ≤ lim  k→∞ inf ℓ(γk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Now, by continuity of the distance, we obtain  ℓ(γ) ≤ lim  k→∞ inf ℓ(γk) = lim  k→∞ inf d(γk(0), γk(T )) = d(γ(0), γ(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  This yields that γ is minimizing geodesic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  ℓ(γ) = d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The horizontal  property of γ follows in the same way as was done in [1], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  □  Next, we define the exponential map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' For the general case, roughly speaking,  if M is a smooth Finsler manifold, x a point in M and u ∈ TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Then the  exponential map is given by  expx : TxM  � M,  such that expx(u) = γu(1) for the unique geodesic γ that starts at x and has initial  speed vector u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Furthermore, in the dual space the exponential map for every  x ∈ M and p ∈ T ∗  xM defined by  exp∗  x : T ∗  xM  � M,  such that exp∗  x(p) = γp(1) for the unique geodesic γ that starts at x and has initial  speed vector u = L−1  L (p), where L here is the Lagrangian of the Finsler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  The exponential map is an essential object in sub-Finslerian geometry, parametriz-  ing normal extremals through their initial covectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' We are going to define the  exponential map in both of the distribution D, D∗ of the tangent and the cotangent  bundles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY  9  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let Ωx ⊂ Dx be the domain of the exponential map over x ∈ M  such that Ωx given by  Ωx = {v ∈ Dx| ξ is defined on the interval [0, 1]} ,  where v = LH(p) by the Legendre transformation of sub-Hamiltonian H, and ξ(t)  is the normal extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then the sub-Finsler exponential map is defined as follows  expx : Ωx ⊂ Dx ⊂ TxM  � M, v �→ πD(LH(ξ(1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  We can do the same in the distribution D∗  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let Ω∗  x ⊂ D∗  x be the domain of the  exponential map over x ∈ M such that Ω∗  x given by  Ω∗  x = {p ∈ D∗  x| ξ is defined on the interval [0, 1]} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Consequently, the sub-Hamiltonian exponential map is given by  exp∗  x : Ω∗  x ⊂ D∗  x ⊂ T ∗  xM  � M, p �→ τ(ξ(1)),  where ξ(t) is the same normal extremal as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The set Ω∗  x contains the origin and  star-shaped with respect to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Moreover, with the help of Legendre transformation  it is fairly easy to see that  expx(v) = exp∗  x(p),  where  p = LL(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (7)  It follows that the normal sub-Finslerian geodesics x(t) = τ(ξ(t)) satisfies  x(t) = exp∗  x(tp),  for all t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The exponential mapping exp∗  x is a local diffeomorphism on D∗  x ⊂  T ∗  xM\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In 4, we show the homogeneity of the sub-Hamiltonian function H(x, p) with  respect to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' So, for any constant a > 0, the curve ξ(at) : (ǫ/a, ǫ/a)  � M is the  same geodesic satisfying the initial conditions τ(ξp(0)) = x and ξp(0) = ap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=',  τ(ξp(at)) = τ(ξap(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Since the sub-Hamiltonian vector field  ⃗H(x, p) = gab(x, p)pb  ∂  ∂xa − 1  2  ∂gab  ∂xk (x, p)papb  ∂  ∂pk  ,  that introduced in [2], is smooth except for p = 0 where it is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then exp∗  x is C∞  on D∗  x ⊂ T ∗  xM\\{0}, while it is C1 at p = 0 and d(exp∗  x)|0 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Thus, exp∗  x is a  local diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  □  By equation (7), one can get the following  Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The sub-Finsler exponential map expx is a C∞ away from the zero  section of D and only C1 at the zero section such that for each x ∈ M  d(expx)|0 : Ωx ⊂ Dx  � Ωx ⊂ Dx,  is the identity map at the origin 0 ∈ Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' It is clear that in the case of sub-Finsler exponential map the following  expressions holds:  exp∗  x[B∗  x(r)] = Bx(r),  exp∗  x[S∗  x(r)] = Sx(r),  which are analogous to the Finslerian context, see Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' [5] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  10  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Turning to sub-Riemannian case, Strichartz in [13] stated that for  bracket generating distributions the exponential map is a local diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  This is due to the fact that the solutions of the sub-Hamiltonian system depend  differentially on the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  But this is a difference from the Riemannian  context, the exponential map is not a diffeomorphism at the origin just like the  Finslerian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hopf-Rinow Theorem in sub-Finslerian geometry  In the following, one can see the explanation of the terms that will be used in  Hopf-Rinow Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' A sub-Finsler manifold is said to be forward complete if  every forward Cauchy sequence converges, and it is a forward geodesically complete  if every normal geodesic γ(t), t ∈ [0, T ) parametrized to have unit speed, can be  extended to a geodesic for all t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' A subset is said to be forward bounded if  it is contained in some forward metric ball Bx(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let (M, D, F) be any connected sub-Finsler manifold, where D is  bracket generating distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The following conditions are equivalent:  (i) The metric space (M, d) is forward complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (ii) The sub-Finsler manifold (M, D, F) is forward geodesically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (iii) Ω∗  x = D∗  x, additionally, the exponential map is onto if there are no strictly  abnormal minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (iv) Every closed and forward bounded subset of (M, d) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Furthermore, for any x, y ∈ M there exists a minimizing geodesic γ joining x to y,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' the length of this geodesic is equal to the distance between these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' (i) =⇒ (ii) Let γ(t) : [0, T )  � M be a unit speed and maximally forward  extended geodesic, t ∈ [0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' If we assume that T ̸= ∞, and choose a sequence  {ti}  � T in [0, T ) then γ(ti) is forward Cauchy, since  d(γ(ti), γ(tj)) ≤ |tj − ti|, for all i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Now, (i) makes it obvious that γ(ti) converges to y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' On one hand, let us  define γ(T ) to be y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' On the other hand, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='1 in [13] told us that γ(t) can  be extended beyond t = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' This contradicts our assumption the fact that T ̸= ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Thus, T = ∞ for sure, so we have the forward geodesically completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (ii) =⇒ (iii) It is sufficient (for first part Ω∗  x = D∗  x) to prove that any normal  extremal pair ξ(t), starting from the initial conditions, is defined for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Suppose that the normal extremal is not extendable to the some interval [0, T +δ) for  all δ > 0 and suppose that it is defined on [0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let {ti} be any increasing sequence  such that the limit of this sequence is T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hence, the projection x(t) = τ(ξ(t)) is  a curve with unit speed defined on [0, T ), therefore, the sequence {ti} is a forward  Cauchy sequence on M, since  d(x(ti), x(tj)) ≤ |ti − tj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  By completeness, it follows that the sequence x(ti) converges to some point  y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' We suppose there are coordinates around the point y and an orthonormal  frame X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=', Xk in small ball B∗  y(r) in the sub-Finsler bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Let us show  that in the coordinates ξ(t) = (x(t), p(t)) the curve x(t) is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' This  grants a contradiction that the normal extremal is not extendable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' In fact, for every  HOPF-RINOW THEOREM OF SUB-FINSLERIAN GEOMETRY  11  p ∈ D∗, we consider the following non-negative form (3) of the sub-Hamiltonian  function H:  H(x, p) = 1  2  k  �  i=1  ⟨p, Xi(p)⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Then, the sub-Hamiltonian system has the form:  ˙xi(t) = ∂H  ∂pi  (x(t), p(t)) =  k  �  j=1  ⟨p(t), Xj(p(t))⟩(δi(Xj(p)) + ⟨p, DpiXj(p)⟩),  ˙pi(t) = −∂H  ∂xi (x(t), p(t)) = −  k  �  j=1  ⟨p(t), Xj(p(t))⟩⟨p(t), DxiXj(p(t))⟩,  for t ∈ [T − δ, T ) with δ > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Since Dγ(t)Xi are given in a compact  small ball ¯B∗  y(r), they are bounded, so there is a constant C > 0 such that  | ˙p(t)| ≤ C|p(t)|  ∀t ∈ [T − δ, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  If we apply Gronwall’s Lemma (see [12], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='122), it leads us to that |p(t)| is uniformly  bounded on a bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' This contradicts our assumption that the normal  extremal can not be extended beyond T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (iii) =⇒ (iv) Assume that ¯A is a closed and forward bounded subset of (M, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Applying the bracket generating assumption, for every y ∈ ¯A, Proposition 7 asserts  that there is a minimizing geodesic exp∗  x(tpy), 0 ≤ t ≤ T, from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' The set of all  py is subset A of D∗  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Since F ∗  x (py) = d(x, y), and d(x, y) ≤ r for some r due to the  forward boundedness of ¯A, the subset A is bounded and contained in the compact  set B∗  x(r) ∪ S∗  x(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' By Remark 4, exp∗  x[B∗  x(r) ∪ S∗  x(r)] is compact and contained in  the closed set ¯A, then ¯A it must be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  (iv) =⇒ (i) Let {xi} be a forward Cauchy sequence in M, and by the subaddi-  tivity it must be forward bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Choose A := {xi|i ∈ N}, then its closure ¯A is  still forward bounded under the manifold topology of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Taking into account the  assumption (iv), ¯A should satisfy the compactness property, therefore, the sequence  {xi} contains a convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Let {xk} be a convergent subsequence, consider it converges to some y ∈ ¯A ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  In other hand, we need to check that {xi} converges to y ∈ ¯A ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' To do this,  fix ǫ > 0, since {xi} is forward Cauchy, there exist a positive number n0 such that  j > i ≥ n0, then  d(xi, xj) < ǫ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  At the same time {xk} converge to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' So there is a positive number n1 such that  if k ≥ n1, then  d(xk, y) < ǫ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  One can assume that n is greater than n0 and n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' If needed, by expanding n  further, there is no loss of generality in assuming that n indeed equals some index  of the convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Then d(xn, y) ≤ ǫ  2, so, for i > n, we get  d(xi, y) ≤ d(xi, xn) + d(xn, y)< ǫ  2 + ǫ  2 = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  So, we have been shown that every forward Cauchy sequence is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Hence  (M, d) is forward complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  12  LAYTH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' ALABDULSADA AND L´ASZL ´O KOZMA  At the end, we can use the same proof of Proposition 7 to verify that for every  x, y ∈ M there exists a length minimizing geodesic joining x and y, and it has  to be normal geodesic by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='  Finally, the property of compactness and  completeness with help of Proposition 7, proves the second part of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
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+page_content=' Box 400, Hungary  Email address: layth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='muhsin@science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='unideb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
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+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content=' Box 400, Hungary  Email address: kozma@unideb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}
+page_content='hu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQf6TZx/content/2301.13438v1.pdf'}

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+Analyzing I/O Performance of a Hierarchical
+HPC Storage System for Distributed Deep
+Learning
+Takaaki Fukai1[0000−0003−4216−4807], Kento Sato2, and Takahiro
+Hirofuchi1[0000−0002−1253−6625]
+1 National Institute of Advanced Industrial Science and Technology (AIST), Tokyo,
+Japan
+{takaaki.fukai, t.hirofuchi}@aist.go.jp
+2 RIKEN Center for Computational Science, Kobe, Japan
+[email protected]
+Abstract. Today, deep learning is an essential technology for our life.
+To solve more complex problems with deep learning, both sizes of train-
+ing datasets and neural networks are increasing. To train a model with
+large datasets and networks, distributed deep neural network (DDNN)
+training technique is necessary. For large-scale DDNN training, HPC
+clusters are a promising computation environment. In large-scale DDNN
+on HPC clusters, I/O performance is critical because it is becoming a
+bottleneck. Most flagship-class HPC clusters have hierarchical storage
+systems. For designing future HPC storage systems, it is necessary to
+quantify the performance improvement effect of the hierarchical storage
+system on the workloads. This paper demonstrates the quantitative per-
+formance analysis of the hierarchical storage system for DDNN workload
+in a flagship-class supercomputer. Our analysis shows how much perfor-
+mance improvement and volume increment of the storage will be required
+to meet the performance goal.
+Keywords: Deep neural network · Distributed deep neural network
+training · I/O performance · Hierarchical storage system · High per-
+formance computing
+1
+Introduction
+Today, the demand for large-scale deep learning has significantly increased. The
+sizes of training models and datasets for the training are expanding to meet
+the demand. For example, training models for image classification, such as Ef-
+ficientNet [19] with large datasets, such as OpenImages dataset [10] are used.
+Some computational science applications also use deep learning methods such
+as CosmoFlow [12] and DeepCam [9]. A single machine does not have enough
+computational and memory capacity for these large workloads. Therefore, dis-
+tributed deep neural network (DDNN) training technique, which allows training
+arXiv:2301.01494v1  [cs.DC]  4 Jan 2023
+
+2
+T. Fukai et al.
+models on multiple machines connected by a network, is necessary. HPC, opti-
+mized for huge and distributed workloads, are promising environments for large
+training workloads.
+I/O is becoming a bottleneck in the training workloads for the following
+reasons [14,16]. The first reason is dataset growth. The size of datasets is in-
+creasing for training higher quality models [11,5]. Large datasets that do not fit
+in memory cause training applications to issue many I/O requests. The second
+reason is expanding performance gap between computation and I/O. Although
+computation time is becoming shorter by distributed execution techniques, I/O
+performance is not improved. This expands the performance gap. Therefore the
+I/O performance of future HPC clusters is crucial.
+For designing a storage system for future HPC, it is crucial but difficult to
+find out what improvement of storage systems would achieve a performance goal
+of a target I/O intensive DDNN workload. The first reason why it is difficult is
+that most storage systems in flagship-class HPC clusters are hierarchical, which
+combines fast but small storage, and a slow but large storage system [17,20].
+Therefore, it is unclear which we should pay our cost for the throughput of the
+global or local filesystems or the volume for the local file system. The second
+reason is that tuning a DNN application for distributed execution requires several
+weeks or months for a new cluster or processor architecture. Therefore, we need a
+much cost and time to find out the I/O bottleneck in a DDNN training workload.
+This paper shows a case study on the performance analysis of a hierarchical
+storage system for DDNN workload and the estimation of necessary improve-
+ment of the storage systems to meet a performance goal. To reveal the effect
+of faster storage, we first measure the I/O operations time of a synthetic I/O
+intensive training workload with various proportions of fast and slow storage
+sizes. Then we estimate the impact of storage system improvement on the train-
+ing performance based on a result of the I/O performance analysis. The method
+can estimate the contribution of various improvement of a hierarchical storage
+system, to overall the training performance.
+The contributions of this work are: (1) A methodology to study the I/O
+bottleneck of DDNN training workloads in the hierarchical storage system; (2)
+A methodology to explore options for improvement of a storage system to achieve
+a performance goal of DDNN training workloads.
+The remainder of this paper is organized as follows. Section 2 explains the
+background. Section 3 reviews the related work. Section 4 explains our method.
+Section 5 demonstrates our methods on a flagship-class supercomputer. Section 6
+discusses the potential of the method. Conclusions are presented in Section 7.
+2
+Background
+2.1
+File access in distributed neural network workloads
+The file access pattern by DNN training applications is different from that in
+scientific computational applications. In training, stochastic gradient descent
+
+Title Suppressed Due to Excessive Length
+3
+(SGD) is a common technique to improve training speed and accuracy[3,13,22].
+In SGD, a program splits a training dataset into mini-batch and inputs a mini-
+batch to the neural network. To avoid the degradation of training accuracy due
+to a fixed input order, it shuffles the order of the files of the dataset every
+time when inputting all samples to the neural network. Therefore the program
+accesses each file once an epoch in a random order. It is hard to apply general
+cache policies because of less time and spatial locality. If the dataset is larger
+than the memory volume for page caches, the cache miss on reading file often
+occurs. It is a reason why I/O is easy to be the bottleneck.
+In distributed training, multiple compute nodes read the dataset simultane-
+ously. In data-parallel, which is one of common parallelizing techniques, each
+compute node has a part of the dataset and calculates it. There are two data
+shuffling manners called local shuffling and global shuffling. Local shuffle means
+that each process only shuffles and reads a part of the dataset. On the other
+hand, global shuffle means that the application shuffles the whole dataset and
+splits it for each computer every epoch. Local shuffling is easy to use local stor-
+age for each computer because the computer needs to access only the initial
+allocated part of the dataset. However, it reduces the training accuracy because
+it reduces the randomness of the input dataset. In some cases, the local shuffle
+approach is not suitable due to the accuracy degradation. On the other hand,
+the global shuffling does not affect the training accuracy, but replacing the part
+of the dataset for each epoch is a heavy I/O workload, especially, training with
+a large dataset and a large number of computers. Therefore, we focus on the
+global shuffling in this paper.
+2.2
+Storage system in HPC
+Recent flagship class HPC clusters provide a hierarchical storage system. Hi-
+erarchical storage systems typically consist of a small but fast storage system
+and a large but slow storage system. HPC clusters often provide the former as
+a local file system (LFS) and the latter as a global file system (GFS). Summit
+[20] provides node-local burst buffers (node-local NVMe SSD) and a parallel file
+system (IBM’s SpectrumScale GPFS™). Fugaku[17] also provides a hierarchical
+storage system that consists of the 1st level storage (an SSD for every 16 nodes)
+and the 2nd level storage (a Global storage system). We assume that DDNN
+applications in a global shuffle manner use the local storage in the hierarchical
+storage as the cache of global storage. An important question to answer to de-
+sign future storage systems in HPC for machine learning workload is the best
+balance of fast and slow storage from the viewpoint of size and performance.
+3
+Related work
+There are several works to analyze and model the DNN performance. Wang et
+al. proposed a modeling method for the DNN training workload based on the
+
+4
+T. Fukai et al.
+Roofline model [21]. They focus on the performance of the computation and
+memory accesses however the I/O performance is not considered.
+There are several works for analyzing and optimizing I/O performance for
+DDNN workloads. Several works [16,14] have analyzed the I/O performance
+for the DDNN and proposed optimization methods. Devarajan et al. proposed a
+benchmark to measure the I/O performance for DDNN and find the opportunity
+for tuning I/O parameters [7,6]. These works assume a non-hierarchical storage
+system. We focus on I/O performance of hierarchical storage systems.
+Several works [23,18,24,8] assume hierarchical storage systems in their I/O
+optimization method for DDNN workloads. They focus on application-level op-
+timization to solve the I/O bottleneck. On the other hand, our work is toward
+performance improvement of storage systems.
+Paul et al. analyzed the I/O log generated by all the jobs on Supercomputer
+Summit during a year [15]. They revealed the tendency of ML jobs and the usage
+of the storage system by them, especially, the usage of the burst buffer. In this
+work, the 23,389 ML jobs of 845,036 jobs in 2020 on Summit were analyzed. The
+analysis results suggested a rapid increment in the use of ML technologies in
+HPC, and some of the ML jobs used the burst buffer in addition to the GPFS.
+This work analyzes the comprehensive analysis of the real ML workload from the
+viewpoint of usage of the hierarchical storage system in the HPC environment.
+Our work analyzes the I/O performance of hierarchical storage systems in detail.
+4
+Methodology
+4.1
+Overview
+Our analysis method is composed of three steps, (1) measuring I/O performance,
+(2) analyzing measurement results, and (3) estimating the impact of the speed-
+up of global and local storage on training performance.
+In the measurement, we execute a DDNN benchmark with profiling the I/O
+on a hierarchical storage system. The benchmark reads a dataset from the hier-
+archical storage system and uses LFS as the cache of GFS. To reveal how LFS
+contributes to overall the I/O performance, we measure the I/O performance
+with various proportions of the size of the cached data on LFS. We expect that
+the performance characteristics depend on a performance balance of GFS and
+LFS as well as the sizes of files in a dataset. Therefore, we measure the I/O
+performance with multiple performance balances and the file sizes combination.
+In the analyzing step, we analyze the profiling data separately by file system
+and by type of I/O operations. To do this, we break down the I/O time into
+the following four I/O classes based on the I/O profiling data obtained in the
+benchmark execution. GFS-READ is a class for read operations on a GFS, GFS-
+META is a class for metadata operations (open(), close()) on a GFS, LFS-
+READ is a class for read operations on an LFS, and LFS-META is a class for
+metadata operations on an LFS. Note that file operations on the dataset in a
+DNN training are only open(), close(), and read() because applications do
+
+Title Suppressed Due to Excessive Length
+5
+not make any modifications and new samples. We target the I/O time of the
+slowest process among all parallel processes, because it is the most dominant for
+the overall training time.
+In the estimating step, we extrapolate from the above results, expected train-
+ing time enabled by the speed-up of global and local storage. We first calculate
+the expected overall I/O operation time on the assumption that the speed of an
+I/O class is improved by a given ratio. We also calculate the expected impact
+on training time by the performance improvement of multiple I/O classes and
+which combination of the improvement will satisfy the performance goal.
+4.2
+Measuring I/O performance by benchmark
+To measure the I/O performance, we use DLIO [7] benchmark, a benchmark for
+I/O performance on distributed deep neural network workloads. DLIO bench-
+mark supports distributed execution and generating the synthetic dataset for
+the benchmark. However, it does not support hierarchical storage. Therefore, we
+add the three functions to DLIO for our measurement of hierarchical storage sys-
+tems. The functions are (1) Reading the dataset from both GFS and LFS with a
+specified proportion, (2) Global shuffling, and (3) Generating the synthetic files
+on the local filesystem by each compute node.
+In the benchmark execution, the cached files are not evicted, in other words,
+the cache policy is pinning. As described in Section 2, the training application
+accesses all samples the same number of times. Therefore, the cache hit rate with
+the pinning policy is the same as the percentage of the cached file [14].
+We prepare two datasets, a small file dataset and a large file emulating Ima-
+geNet dataset and CosmoFlow dataset respectively. The small file dataset con-
+sists of 128 KiB files and the large file dataset consists of 12 MiB files. The
+numbers of files in the small and large datasets are 589824 and 6144 respec-
+tively. The total size of both datasets is 72 GiB so that whole of the dataset can
+be on the LFS and all of the processes read the same number of files. Because
+the entire dataset cannot be put on the memory of each compute node (32 GiB),
+the benchmark application reads the files from the filesystem. The file format in
+both datasets is tfrecord, and the number of samples in each file is one.
+To measure the I/O performance with multiple performance balances of the
+GFS and LFS, we measure the I/O performance with different numbers of the
+object storage targets (OSTs) of the lustre-based GFS. We can limit the number
+of OSTs to 1 by lfs command. Therefore, we measure the I/O performance with
+all provided OSTs (faster GFS) and 1 OST (slower GFS).
+To measure the I/O performance in the benchmark execution, we use Dar-
+shan [2], which is a profiling tool for I/O. Darshan can capture and record each
+file operations such as open(), close(), and read().
+4.3
+Analyzing the I/O performance
+To reveal the bottleneck in detail, we break down the I/O time of the slowest
+process into the following four I/O classes and recognize which I/O class is a
+
+6
+T. Fukai et al.
+bottleneck. We calculate the I/O time for each process and find the slowest one,
+which dominates the training performance. We analyze the I/O performance
+from the log generated by darshan using darshan-parser command [1]. We cal-
+culate for each I/O time based on POSIX_F_READ_TIME and POSIX_F_META_TIME.
+4.4
+Estimate performance by storage improvement
+To estimate impact of N% throughput improvement of a I/O class, we calcu-
+late ×
+100
+100+N of the measured time of the I/O class. The improvement may not
+directly affect the total I/O time because the improvement may change the bot-
+tleneck to another I/O class. Therefore, we calculate total I/O time for each
+process with the improvement of a class, then pick the slowest process.
+5
+Experiment results
+5.1
+Setup for experiment
+We perform the experiments on Supercomputer Fugaku[17]. Compute nodes of
+Fugaku has 48 computing cores of A64FX and 32 GiB HBM2 memory. Fugaku
+has a hierarchical filesystem comprising the 1st- and 2nd-level filesystems named
+LLIO and FEFS, respectively. In our measurement, we regard the LLIO as a local
+filesystem (LFS) and the FEFS as a global filesystem (GFS). FEFS is a lustre-
+based parallel filesystem and it has 60 OSTs in Fugaku. One per 192 compute
+nodes connected to the FEFS by InfiniBand EDR. The other compute nodes
+connected by TofuD access to FEFS via the network and the compute node.
+For LLIO, one per 16 compute nodes has an NVMe SSD and the other compute
+nodes connected access to the SSD via the network and the compute node. LLIO
+provides three areas, node temporary area, shared temporary area, and 2nd-layer
+cache area[4]. In our measurement, we only use node temporary areas, which is
+a dedicated area for a compute node, because the transparent cache does not
+allow us to control caching files of the dataset on LLIO.
+In our measurement, we run the DLIO benchmark on the 768 compute nodes
+of Supercomputer Fugaku as batch jobs. The four processes execute on every
+compute nodes, so that total number of processes is 3072. The node layout is
+8×6×16 in the TofuD torus network. We also pass an option to the job scheduler
+to strict the position of the node connected to the GFS.
+Before executing the benchmark job, we generate the datasets on the GFS.
+Because the system removes data on LFS after finishing the job, every benchmark
+job generates the same dataset on LFS as GFS. Note that the job generates the
+dataset instead of copying the dataset from GFS to reduce the setup time.
+We execute the benchmark with every 5% from 0% to 100% cache rate.
+We set the calculation time in the DLIO to zero. Therefore, the DLIO reports
+only the I/O and data processing time. The number of epochs is three to avoid
+making the darshan log files huge. The prefetch of the dataloader is enabled so
+that it is not synchronized for each iteration even if the computation threads are
+synchronized for all-reduce communication. The batch size is 12 for the small
+file dataset and 2 for the large file dataset.
+
+Title Suppressed Due to Excessive Length
+7
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Percentage of the cache(%)
+0
+10
+20
+30
+40
+Time in an epoch (seconds)
+Small file dataset with faster GFS
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Percentage of the cache(%)
+0
+20
+40
+60
+80
+100
+Small file dataset with slower GFS
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Percentage of the cache(%)
+0
+2
+4
+6
+Large file dataset with faster GFS
+Epoch
+#0 (DLIO)
+#1 (DLIO)
+#2 (DLIO)
+#0 (Darshan)
+#1 (Darshan)
+#2 (Darshan)
+Fig. 1: Execution time of DLIO benchmark and I/O time reported by Darshan
+5.2
+Measuring execution time for epochs
+In our experiments, we first measure the execution time of the DLIO benchmark
+and I/O time in the benchmark execution with various settings as mentioned
+in 4.2. We reveal the difference in the performance depending on file sizes in
+the datasets and speeds of GFS. Additionally, by comparing the execution time
+reported by the benchmark and the total I/O time reported by the I/O profiler,
+we verify that the benchmark is I/O intensive.
+Figure 1 shows the execution time and I/O time for each epoch in a job. The
+x-axes of the graphs show the percentage of the files on the LFS. The y-axes
+show the execution time of an epoch. The graph shows the results of 3 epochs
+in a benchmark execution. The lines with round markers are the execution time
+reported by the DLIO benchmark, and those with triangular markers are the
+I/O time reported by Darshan. Because the I/O time of the slowest I/O process
+is dominant, we calculate and plot them as I/O time on the graph.
+The results show that the impact of the LFS on the training performance
+depends on the file sizes and the performance balance of GFS and LFS. The
+effect of the LFS with the 1 OST of GFS is larger than that with the 60 OSTs.
+The reason is the performance difference between LFS and GFS on the 1 OST
+is larger than that on 60 OSTs. In 12 MiB files workload with 60 OST of GFS,
+the LFS does not contribute to the performance improvement, and using only
+the GFS with the 60 OST is the best.
+About the I/O time, the graph indicates that the execution time is constantly
+longer about 2 sec, but it is strongly related to the I/O time. This result indicate
+that the I/O time of the slowest process during the synchronizations among the
+processes strongly interrelates to the training performance.
+5.3
+Analyzing I/O performance
+Next, we classify the Darshan records into the four I/O classes and calculate
+the I/O time for each class. Figure 2 shows the result of the breakdown of the
+#2 epoch in the previous graphs. Each line shows the total I/O time same as
+
+8
+T. Fukai et al.
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Percentage of the cache(%)
+0
+10
+20
+30
+Time in an epoch (seconds)
+Small file dataset with faster GFS
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Percentage of the cache(%)
+0
+25
+50
+75
+100
+Small file dataset with slower GFS
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Percentage of the cache(%)
+0
+1
+2
+Large file dataset with faster GFS
+Total I/O time
+GFS-META
+GFS-READ
+LFS-META
+LFS-READ
+Fig. 2: Break down the I/O time of the slowest process for each epoch (Note:
+Range of y-axis are different)
+Figure 1. According to the result, the bottleneck is different depending on the
+setup.
+In 128 KiB files workload with the faster GFS, the bottleneck is GFS-META
+when less than 60% of data is on the LFS. On the other hand, the bottleneck is
+changed to LFS-READ with more than 60% cached data on LFS. We think
+when more than 60% of data is put on LFS, LFS throughput is saturated.
+Therefore, the read time from LFS is linearly increased with the percentage of
+the cached data. So putting more than 60% of data on the cache in the workload
+does not contribute to the training performance. For example, in training with
+ImageNet dataset whose size is almost 150 GB, almost the 90 GiB LFS for each
+compute node is enough to achieve the best I/O performance by the hierarchical
+storage system. With the 60% cached data, both GFS-META and LFS-READ
+are included in the I/O time of the slowest process.
+As compared with the faster GFS, the I/O bottleneck in the workload with
+the slower GFS is much different. The graph in the middle of Figure 2 shows
+that the bottleneck is GFS-READ instead of GFS-META with small percentages
+of the LFS (less than 80%). We think that the reason why GFS-META time
+becomes shorter is reducing the load on the metadata server of FEFS due to the
+lower throughput of the GFS.
+As compared with the small files workload, the I/O bottleneck in the large
+files workload with the faster GFS is also much different. The right side graph
+in Figure 2 shows that the bottleneck is the LFS-READ in most of the cases.
+Because the number of metadata operations is much smaller than that in the
+small file workload, the GFS fully provides its bandwidth without the bottleneck
+by the metadata operation. As a result, the total bandwidth of the GFS is higher
+than LFS. It means that the number of compute nodes is not enough to take
+advantage of the scalability of the LFS. Note that the 768 nodes are not so large
+scale as a workload in Fugaku, however from viewpoint of the machine learning
+workload, the number of nodes is large enough to lead to a large batch problem.
+From viewpoint of exploration of storage design for a performance goal, the
+result on small file dataset and faster GFS (the left side graph in Figure 2)
+
+Title Suppressed Due to Excessive Length
+9
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Percentage of the cache(%)
+0
+5
+10
+15
+20
+25
+30
+Time in an epoch (seconds)
+Epoch #2, min: 7.226sec (60 %)
+Orig: min: 8.119sec (65 %)
+Total I/O time (Orig.)
+Total I/O time (Expected)
+GFS-META
+GFS-READ
+LFS-META
+LFS-READ
+(a) Improving GFS-META
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Percentage of the cache(%)
+0
+5
+10
+15
+20
+25
+30
+Time in an epoch (seconds)
+Epoch #2, min: 6.027sec (65 %)
+Orig: min: 8.119sec (65 %)
+Total I/O time (Orig.)
+Total I/O time (Expected)
+GFS-META
+GFS-READ
+LFS-META
+LFS-READ
+(b) Improving LFS-READ
+Fig. 3: Expectation the I/O time with 50% improvement with small file dataset
+and 60 OSTs of the GFS
+is challenging situation because multiple I/O class is included in the I/O time
+in the fastest result (cache rate = 65%). This means that improving only one
+I/O class processing will not be enough to improve entire I/O performance.
+Therefore, we pick up the result to demonstrate our estimation method of the
+I/O improvement effect.
+5.4
+Estimating the impact of the storage improvement
+As mentioned in 4.4, we estimate the performance improvement by simple cal-
+culation based on the analysis result. Figure 3 shows the result of the estimation
+of the impact of a 50% improvement of GFS-META (Figure 3a) and LFS-READ
+(Figure 3b). The axes in the graph are the same as in Figure 2.
+Figure 3a shows the estimation result of improving the GFS-META by 50%.
+The best combination of the GFS and LFS is changed from 65% to 60% LFS,
+and the slowest I/O time is reduced by almost 12.8% in the best case. Figure
+3a shows the estimation result of improving the LFS-READ by 50%. The best
+cache rate is not changed, and the slowest I/O time in the best case is reduced
+by 24%.
+Next, we show the estimation of the impact of improvement of two oper-
+ations classes simultaneously. There are many parameters and values such as
+improvement rate for each operation, cache rate, and the I/O time. All of them
+are too many to put on a single graph. Again, the architect of the system needs
+knowledge of the given performance goal. Therefore, we show the estimation by
+indicating which improvement combination would meet the performance goal.
+Figure 4 shows the sufficient combinations of the performance improvement
+on two classes, GFS-META and LFS-READ, on the small files dataset and the
+faster GFS workload. The result in the graph is based on the measurement of
+
+10
+T. Fukai et al.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Improvement rate of GFS META(%)
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Improvement rate of LFS READ(%)
+0.55
+0.60
+0.65
+0.70
+0.75
+Cache rate (%)
+Fig. 4: The estimation of performance improvement of GFS-META and LFS-
+META for meeting performance goal of 4 sec / epoch (128 KiB, 60 OST GFS)
+I/O time in the #2 epoch. The x and y axes show each improvement rate. On the
+graph, the dot is plotted if the improvement combination will meet the given
+performance goal. The graph indicates a result for the performance goal of 4
+seconds I/O time in an epoch. Additionally, the colors of the dots indicate the
+minimum cache rate to meet the goal. For example, to achieve 4 seconds I/O time
+in an epoch with a 65% cache rate, at least 120% improvement of LFS-READ is
+required. In that case, a 140% improvement of the GFS-META is required. The
+architect can explore the option of the improvement choice by the plot.
+6
+Discussion
+In our evaluation, we assume the global shuffling manner to exploit GFS. How-
+ever, training applications with the local shuffle also can combine the LFS and
+GFS to put larger chunks of the dataset than that with only the LFS. In this
+case, the size of the LFS and the randomness of the shuffling are a trade-off. To
+consider how large chunks are preferred, the application user also can use our
+method to find the contributions to the performance of the LFS.
+In our evaluation, we assume a pinning cache policy on the LFS. However,
+our analysis can apply to the other cache policy if the cache hit rate can be
+calculated. You can replace the "cache rate" with "cache hit rate" in the analysis
+result because both are the same in DNN workloads with the pinning policy.
+Then you can find the required size of the LFS from the relation between the
+cache hit rate and the size of the cache in your better cache policy.
+In our evaluation, we estimate the improvement by a simple calculation.
+However, the performance characteristic may not be simple. For example, the
+estimation from the measurement results with 1 OST of the GFS with the sim-
+ple calculation does not fit that with the 60 OSTs of GFS. For more accurate
+
+Title Suppressed Due to Excessive Length
+11
+estimation, improving the calculation method is necessary by modeling the char-
+acteristic. The considerable approach is based on machine learning or queueing
+theory. Even if the calculation method will be improved, our plot method shown
+in Figure 4 is useful for the storage system architect.
+7
+Conclusion
+This paper presented a case study on the performance analysis of a hierarchical
+storage system for a DDNN workload in a flagship-class HPC cluster, discussing
+potential performance improvement enabled by the speed-up of the storage sys-
+tem. We also estimated the improvement of training performance by various
+improvement of the hierarchical filesystem. The analysis result showed that the
+I/O bottleneck in the training workload depends on performance balance be-
+tween global and local storage as well as file sizes in a dataset.
+Our estimation showed that the performance improvement of a global filesys-
+tem will contribute to reducing the necessary volume size of a local filesystem,
+and the performance improvement of the local file system will contribute to re-
+ducing fastest I/O time. Our estimation method can help architects of HPC
+filesystems to find the necessary performance and the volume size of the local
+and global filesystems to meet a given performance goal.
+Because our proposed method needs the measurement of I/O performance
+at least once, one of our future works is exploring a simpler or no measurement-
+required method. The other future work is to build the performance modeling
+of the storage system for more accurate estimation.
+References
+1. Darshan-util
+installation
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+2. Darshan – HPC I/O Characterization Tool, https://www.mcs.anl.gov/research/
+projects/darshan/
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+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf,len=487
+page_content='Analyzing I/O Performance of a Hierarchical  HPC Storage System for Distributed Deep  Learning  Takaaki Fukai1[0000−0003−4216−4807], Kento Sato2, and Takahiro  Hirofuchi1[0000−0002−1253−6625]  1 National Institute of Advanced Industrial Science and Technology (AIST), Tokyo,  Japan  {takaaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='fukai, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='hirofuchi}@aist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='jp  2 RIKEN Center for Computational Science, Kobe, Japan  kento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='sato@riken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='jp  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Today, deep learning is an essential technology for our life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  To solve more complex problems with deep learning, both sizes of train-  ing datasets and neural networks are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' To train a model with  large datasets and networks, distributed deep neural network (DDNN)  training technique is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For large-scale DDNN training, HPC  clusters are a promising computation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In large-scale DDNN  on HPC clusters, I/O performance is critical because it is becoming a  bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Most flagship-class HPC clusters have hierarchical storage  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For designing future HPC storage systems, it is necessary to  quantify the performance improvement effect of the hierarchical storage  system on the workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' This paper demonstrates the quantitative per-  formance analysis of the hierarchical storage system for DDNN workload  in a flagship-class supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Our analysis shows how much perfor-  mance improvement and volume increment of the storage will be required  to meet the performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Keywords: Deep neural network · Distributed deep neural network  training · I/O performance · Hierarchical storage system · High per-  formance computing  1  Introduction  Today, the demand for large-scale deep learning has significantly increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The  sizes of training models and datasets for the training are expanding to meet  the demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For example, training models for image classification, such as Ef-  ficientNet [19] with large datasets, such as OpenImages dataset [10] are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Some computational science applications also use deep learning methods such  as CosmoFlow [12] and DeepCam [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' A single machine does not have enough  computational and memory capacity for these large workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, dis-  tributed deep neural network (DDNN) training technique, which allows training  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='01494v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='DC]  4 Jan 2023  2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  models on multiple machines connected by a network, is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' HPC, opti-  mized for huge and distributed workloads, are promising environments for large  training workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  I/O is becoming a bottleneck in the training workloads for the following  reasons [14,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The first reason is dataset growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The size of datasets is in-  creasing for training higher quality models [11,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Large datasets that do not fit  in memory cause training applications to issue many I/O requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The second  reason is expanding performance gap between computation and I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Although  computation time is becoming shorter by distributed execution techniques, I/O  performance is not improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' This expands the performance gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore the  I/O performance of future HPC clusters is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  For designing a storage system for future HPC, it is crucial but difficult to  find out what improvement of storage systems would achieve a performance goal  of a target I/O intensive DDNN workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The first reason why it is difficult is  that most storage systems in flagship-class HPC clusters are hierarchical, which  combines fast but small storage, and a slow but large storage system [17,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Therefore, it is unclear which we should pay our cost for the throughput of the  global or local filesystems or the volume for the local file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The second  reason is that tuning a DNN application for distributed execution requires several  weeks or months for a new cluster or processor architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we need a  much cost and time to find out the I/O bottleneck in a DDNN training workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  This paper shows a case study on the performance analysis of a hierarchical  storage system for DDNN workload and the estimation of necessary improve-  ment of the storage systems to meet a performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' To reveal the effect  of faster storage, we first measure the I/O operations time of a synthetic I/O  intensive training workload with various proportions of fast and slow storage  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Then we estimate the impact of storage system improvement on the train-  ing performance based on a result of the I/O performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The method  can estimate the contribution of various improvement of a hierarchical storage  system, to overall the training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  The contributions of this work are: (1) A methodology to study the I/O  bottleneck of DDNN training workloads in the hierarchical storage system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' (2)  A methodology to explore options for improvement of a storage system to achieve  a performance goal of DDNN training workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Section 2 explains the  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Section 3 reviews the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Section 4 explains our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Section 5 demonstrates our methods on a flagship-class supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Section 6  discusses the potential of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Conclusions are presented in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='1  File access in distributed neural network workloads  The file access pattern by DNN training applications is different from that in  scientific computational applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In training, stochastic gradient descent  Title Suppressed Due to Excessive Length  3  (SGD) is a common technique to improve training speed and accuracy[3,13,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In SGD, a program splits a training dataset into mini-batch and inputs a mini-  batch to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' To avoid the degradation of training accuracy due  to a fixed input order, it shuffles the order of the files of the dataset every  time when inputting all samples to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore the program  accesses each file once an epoch in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' It is hard to apply general  cache policies because of less time and spatial locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' If the dataset is larger  than the memory volume for page caches, the cache miss on reading file often  occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' It is a reason why I/O is easy to be the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In distributed training, multiple compute nodes read the dataset simultane-  ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In data-parallel, which is one of common parallelizing techniques, each  compute node has a part of the dataset and calculates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' There are two data  shuffling manners called local shuffling and global shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Local shuffle means  that each process only shuffles and reads a part of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' On the other  hand, global shuffle means that the application shuffles the whole dataset and  splits it for each computer every epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Local shuffling is easy to use local stor-  age for each computer because the computer needs to access only the initial  allocated part of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' However, it reduces the training accuracy because  it reduces the randomness of the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In some cases, the local shuffle  approach is not suitable due to the accuracy degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' On the other hand,  the global shuffling does not affect the training accuracy, but replacing the part  of the dataset for each epoch is a heavy I/O workload, especially, training with  a large dataset and a large number of computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we focus on the  global shuffling in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='2  Storage system in HPC  Recent flagship class HPC clusters provide a hierarchical storage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Hi-  erarchical storage systems typically consist of a small but fast storage system  and a large but slow storage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' HPC clusters often provide the former as  a local file system (LFS) and the latter as a global file system (GFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Summit  [20] provides node-local burst buffers (node-local NVMe SSD) and a parallel file  system (IBM’s SpectrumScale GPFS™).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fugaku[17] also provides a hierarchical  storage system that consists of the 1st level storage (an SSD for every 16 nodes)  and the 2nd level storage (a Global storage system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We assume that DDNN  applications in a global shuffle manner use the local storage in the hierarchical  storage as the cache of global storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' An important question to answer to de-  sign future storage systems in HPC for machine learning workload is the best  balance of fast and slow storage from the viewpoint of size and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  3  Related work  There are several works to analyze and model the DNN performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Wang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' proposed a modeling method for the DNN training workload based on the  4  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Roofline model [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' They focus on the performance of the computation and  memory accesses however the I/O performance is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  There are several works for analyzing and optimizing I/O performance for  DDNN workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Several works [16,14] have analyzed the I/O performance  for the DDNN and proposed optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Devarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' proposed a  benchmark to measure the I/O performance for DDNN and find the opportunity  for tuning I/O parameters [7,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' These works assume a non-hierarchical storage  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We focus on I/O performance of hierarchical storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Several works [23,18,24,8] assume hierarchical storage systems in their I/O  optimization method for DDNN workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' They focus on application-level op-  timization to solve the I/O bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' On the other hand, our work is toward  performance improvement of storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Paul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' analyzed the I/O log generated by all the jobs on Supercomputer  Summit during a year [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' They revealed the tendency of ML jobs and the usage  of the storage system by them, especially, the usage of the burst buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In this  work, the 23,389 ML jobs of 845,036 jobs in 2020 on Summit were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The  analysis results suggested a rapid increment in the use of ML technologies in  HPC, and some of the ML jobs used the burst buffer in addition to the GPFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  This work analyzes the comprehensive analysis of the real ML workload from the  viewpoint of usage of the hierarchical storage system in the HPC environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Our work analyzes the I/O performance of hierarchical storage systems in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  4  Methodology  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='1  Overview  Our analysis method is composed of three steps, (1) measuring I/O performance,  (2) analyzing measurement results, and (3) estimating the impact of the speed-  up of global and local storage on training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In the measurement, we execute a DDNN benchmark with profiling the I/O  on a hierarchical storage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The benchmark reads a dataset from the hier-  archical storage system and uses LFS as the cache of GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' To reveal how LFS  contributes to overall the I/O performance, we measure the I/O performance  with various proportions of the size of the cached data on LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We expect that  the performance characteristics depend on a performance balance of GFS and  LFS as well as the sizes of files in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we measure the I/O  performance with multiple performance balances and the file sizes combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In the analyzing step, we analyze the profiling data separately by file system  and by type of I/O operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' To do this, we break down the I/O time into  the following four I/O classes based on the I/O profiling data obtained in the  benchmark execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' GFS-READ is a class for read operations on a GFS, GFS-  META is a class for metadata operations (open(), close()) on a GFS, LFS-  READ is a class for read operations on an LFS, and LFS-META is a class for  metadata operations on an LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Note that file operations on the dataset in a  DNN training are only open(), close(), and read() because applications do  Title Suppressed Due to Excessive Length  5  not make any modifications and new samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We target the I/O time of the  slowest process among all parallel processes, because it is the most dominant for  the overall training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In the estimating step, we extrapolate from the above results, expected train-  ing time enabled by the speed-up of global and local storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We first calculate  the expected overall I/O operation time on the assumption that the speed of an  I/O class is improved by a given ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We also calculate the expected impact  on training time by the performance improvement of multiple I/O classes and  which combination of the improvement will satisfy the performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='2  Measuring I/O performance by benchmark  To measure the I/O performance, we use DLIO [7] benchmark, a benchmark for  I/O performance on distributed deep neural network workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' DLIO bench-  mark supports distributed execution and generating the synthetic dataset for  the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' However, it does not support hierarchical storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we  add the three functions to DLIO for our measurement of hierarchical storage sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The functions are (1) Reading the dataset from both GFS and LFS with a  specified proportion, (2) Global shuffling, and (3) Generating the synthetic files  on the local filesystem by each compute node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In the benchmark execution, the cached files are not evicted, in other words,  the cache policy is pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' As described in Section 2, the training application  accesses all samples the same number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, the cache hit rate with  the pinning policy is the same as the percentage of the cached file [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  We prepare two datasets, a small file dataset and a large file emulating Ima-  geNet dataset and CosmoFlow dataset respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The small file dataset con-  sists of 128 KiB files and the large file dataset consists of 12 MiB files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The  numbers of files in the small and large datasets are 589824 and 6144 respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The total size of both datasets is 72 GiB so that whole of the dataset can  be on the LFS and all of the processes read the same number of files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Because  the entire dataset cannot be put on the memory of each compute node (32 GiB),  the benchmark application reads the files from the filesystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The file format in  both datasets is tfrecord, and the number of samples in each file is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  To measure the I/O performance with multiple performance balances of the  GFS and LFS, we measure the I/O performance with different numbers of the  object storage targets (OSTs) of the lustre-based GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We can limit the number  of OSTs to 1 by lfs command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we measure the I/O performance with  all provided OSTs (faster GFS) and 1 OST (slower GFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  To measure the I/O performance in the benchmark execution, we use Dar-  shan [2], which is a profiling tool for I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Darshan can capture and record each  file operations such as open(), close(), and read().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='3  Analyzing the I/O performance  To reveal the bottleneck in detail, we break down the I/O time of the slowest  process into the following four I/O classes and recognize which I/O class is a  6  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We calculate the I/O time for each process and find the slowest one,  which dominates the training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We analyze the I/O performance  from the log generated by darshan using darshan-parser command [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We cal-  culate for each I/O time based on POSIX_F_READ_TIME and POSIX_F_META_TIME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='4  Estimate performance by storage improvement  To estimate impact of N% throughput improvement of a I/O class, we calcu-  late ×  100  100+N of the measured time of the I/O class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The improvement may not  directly affect the total I/O time because the improvement may change the bot-  tleneck to another I/O class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we calculate total I/O time for each  process with the improvement of a class, then pick the slowest process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  5  Experiment results  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='1  Setup for experiment  We perform the experiments on Supercomputer Fugaku[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Compute nodes of  Fugaku has 48 computing cores of A64FX and 32 GiB HBM2 memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fugaku  has a hierarchical filesystem comprising the 1st- and 2nd-level filesystems named  LLIO and FEFS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In our measurement, we regard the LLIO as a local  filesystem (LFS) and the FEFS as a global filesystem (GFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' FEFS is a lustre-  based parallel filesystem and it has 60 OSTs in Fugaku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' One per 192 compute  nodes connected to the FEFS by InfiniBand EDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The other compute nodes  connected by TofuD access to FEFS via the network and the compute node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  For LLIO, one per 16 compute nodes has an NVMe SSD and the other compute  nodes connected access to the SSD via the network and the compute node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' LLIO  provides three areas, node temporary area, shared temporary area, and 2nd-layer  cache area[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In our measurement, we only use node temporary areas, which is  a dedicated area for a compute node, because the transparent cache does not  allow us to control caching files of the dataset on LLIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In our measurement, we run the DLIO benchmark on the 768 compute nodes  of Supercomputer Fugaku as batch jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The four processes execute on every  compute nodes, so that total number of processes is 3072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The node layout is  8×6×16 in the TofuD torus network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We also pass an option to the job scheduler  to strict the position of the node connected to the GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Before executing the benchmark job, we generate the datasets on the GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Because the system removes data on LFS after finishing the job, every benchmark  job generates the same dataset on LFS as GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Note that the job generates the  dataset instead of copying the dataset from GFS to reduce the setup time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  We execute the benchmark with every 5% from 0% to 100% cache rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  We set the calculation time in the DLIO to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, the DLIO reports  only the I/O and data processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The number of epochs is three to avoid  making the darshan log files huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The prefetch of the dataloader is enabled so  that it is not synchronized for each iteration even if the computation threads are  synchronized for all-reduce communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The batch size is 12 for the small  file dataset and 2 for the large file dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
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+page_content='Epoch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='#0 (DLIO)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='#1 (DLIO)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='#2 (DLIO)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='#0 (Darshan)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='#1 (Darshan)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='#2 (Darshan)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' 1: Execution time of DLIO benchmark and I/O time reported by Darshan  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='2  Measuring execution time for epochs  In our experiments, we first measure the execution time of the DLIO benchmark  and I/O time in the benchmark execution with various settings as mentioned  in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We reveal the difference in the performance depending on file sizes in  the datasets and speeds of GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Additionally, by comparing the execution time  reported by the benchmark and the total I/O time reported by the I/O profiler,  we verify that the benchmark is I/O intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Figure 1 shows the execution time and I/O time for each epoch in a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The  x-axes of the graphs show the percentage of the files on the LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The y-axes  show the execution time of an epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The graph shows the results of 3 epochs  in a benchmark execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The lines with round markers are the execution time  reported by the DLIO benchmark, and those with triangular markers are the  I/O time reported by Darshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Because the I/O time of the slowest I/O process  is dominant, we calculate and plot them as I/O time on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  The results show that the impact of the LFS on the training performance  depends on the file sizes and the performance balance of GFS and LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The  effect of the LFS with the 1 OST of GFS is larger than that with the 60 OSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  The reason is the performance difference between LFS and GFS on the 1 OST  is larger than that on 60 OSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In 12 MiB files workload with 60 OST of GFS,  the LFS does not contribute to the performance improvement, and using only  the GFS with the 60 OST is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  About the I/O time, the graph indicates that the execution time is constantly  longer about 2 sec, but it is strongly related to the I/O time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' This result indicate  that the I/O time of the slowest process during the synchronizations among the  processes strongly interrelates to the training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='3  Analyzing I/O performance  Next, we classify the Darshan records into the four I/O classes and calculate  the I/O time for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Figure 2 shows the result of the breakdown of the  #2 epoch in the previous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Each line shows the total I/O time same as  8  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  0  10  20  30  40  50  60  70  80  90  100  Percentage of the cache(%)  0  10  20  30  Time in an epoch (seconds)  Small file dataset with faster GFS  0  10  20  30  40  50  60  70  80  90  100  Percentage of the cache(%)  0  25  50  75  100  Small file dataset with slower GFS  0  10  20  30  40  50  60  70  80  90  100  Percentage of the cache(%)  0  1  2  Large file dataset with faster GFS  Total I/O time  GFS-META  GFS-READ  LFS-META  LFS-READ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' 2: Break down the I/O time of the slowest process for each epoch (Note:  Range of y-axis are different)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' According to the result, the bottleneck is different depending on the  setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In 128 KiB files workload with the faster GFS, the bottleneck is GFS-META  when less than 60% of data is on the LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' On the other hand, the bottleneck is  changed to LFS-READ with more than 60% cached data on LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We think  when more than 60% of data is put on LFS, LFS throughput is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Therefore, the read time from LFS is linearly increased with the percentage of  the cached data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' So putting more than 60% of data on the cache in the workload  does not contribute to the training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For example, in training with  ImageNet dataset whose size is almost 150 GB, almost the 90 GiB LFS for each  compute node is enough to achieve the best I/O performance by the hierarchical  storage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' With the 60% cached data, both GFS-META and LFS-READ  are included in the I/O time of the slowest process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  As compared with the faster GFS, the I/O bottleneck in the workload with  the slower GFS is much different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The graph in the middle of Figure 2 shows  that the bottleneck is GFS-READ instead of GFS-META with small percentages  of the LFS (less than 80%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We think that the reason why GFS-META time  becomes shorter is reducing the load on the metadata server of FEFS due to the  lower throughput of the GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  As compared with the small files workload, the I/O bottleneck in the large  files workload with the faster GFS is also much different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The right side graph  in Figure 2 shows that the bottleneck is the LFS-READ in most of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Because the number of metadata operations is much smaller than that in the  small file workload, the GFS fully provides its bandwidth without the bottleneck  by the metadata operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' As a result, the total bandwidth of the GFS is higher  than LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' It means that the number of compute nodes is not enough to take  advantage of the scalability of the LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Note that the 768 nodes are not so large  scale as a workload in Fugaku, however from viewpoint of the machine learning  workload, the number of nodes is large enough to lead to a large batch problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  From viewpoint of exploration of storage design for a performance goal, the  result on small file dataset and faster GFS (the left side graph in Figure 2)  Title Suppressed Due to Excessive Length  9  0  5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  85  90  95  100  Percentage of the cache(%)  0  5  10  15  20  25  30  Time in an epoch (seconds)  Epoch #2, min: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='226sec (60 %)  Orig: min: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='119sec (65 %)  Total I/O time (Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=')  Total I/O time (Expected)  GFS-META  GFS-READ  LFS-META  LFS-READ  (a) Improving GFS-META  0  5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  85  90  95  100  Percentage of the cache(%)  0  5  10  15  20  25  30  Time in an epoch (seconds)  Epoch #2, min: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='027sec (65 %)  Orig: min: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='119sec (65 %)  Total I/O time (Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=')  Total I/O time (Expected)  GFS-META  GFS-READ  LFS-META  LFS-READ  (b) Improving LFS-READ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' 3: Expectation the I/O time with 50% improvement with small file dataset  and 60 OSTs of the GFS  is challenging situation because multiple I/O class is included in the I/O time  in the fastest result (cache rate = 65%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' This means that improving only one  I/O class processing will not be enough to improve entire I/O performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Therefore, we pick up the result to demonstrate our estimation method of the  I/O improvement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='4  Estimating the impact of the storage improvement  As mentioned in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='4, we estimate the performance improvement by simple cal-  culation based on the analysis result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Figure 3 shows the result of the estimation  of the impact of a 50% improvement of GFS-META (Figure 3a) and LFS-READ  (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The axes in the graph are the same as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Figure 3a shows the estimation result of improving the GFS-META by 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  The best combination of the GFS and LFS is changed from 65% to 60% LFS,  and the slowest I/O time is reduced by almost 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='8% in the best case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Figure  3a shows the estimation result of improving the LFS-READ by 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The best  cache rate is not changed, and the slowest I/O time in the best case is reduced  by 24%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Next, we show the estimation of the impact of improvement of two oper-  ations classes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' There are many parameters and values such as  improvement rate for each operation, cache rate, and the I/O time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' All of them  are too many to put on a single graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Again, the architect of the system needs  knowledge of the given performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Therefore, we show the estimation by  indicating which improvement combination would meet the performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Figure 4 shows the sufficient combinations of the performance improvement  on two classes, GFS-META and LFS-READ, on the small files dataset and the  faster GFS workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The result in the graph is based on the measurement of  10  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Fukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  0  25  50  75  100  125  150  175  200  Improvement rate of GFS META(%)  0  25  50  75  100  125  150  175  200  Improvement rate of LFS READ(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='75  Cache rate (%)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' 4: The estimation of performance improvement of GFS-META and LFS-  META for meeting performance goal of 4 sec / epoch (128 KiB, 60 OST GFS)  I/O time in the #2 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The x and y axes show each improvement rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' On the  graph, the dot is plotted if the improvement combination will meet the given  performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The graph indicates a result for the performance goal of 4  seconds I/O time in an epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Additionally, the colors of the dots indicate the  minimum cache rate to meet the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For example, to achieve 4 seconds I/O time  in an epoch with a 65% cache rate, at least 120% improvement of LFS-READ is  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In that case, a 140% improvement of the GFS-META is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The  architect can explore the option of the improvement choice by the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  6  Discussion  In our evaluation, we assume the global shuffling manner to exploit GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' How-  ever, training applications with the local shuffle also can combine the LFS and  GFS to put larger chunks of the dataset than that with only the LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' In this  case, the size of the LFS and the randomness of the shuffling are a trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' To  consider how large chunks are preferred, the application user also can use our  method to find the contributions to the performance of the LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In our evaluation, we assume a pinning cache policy on the LFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' However,  our analysis can apply to the other cache policy if the cache hit rate can be  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' You can replace the "cache rate" with "cache hit rate" in the analysis  result because both are the same in DNN workloads with the pinning policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Then you can find the required size of the LFS from the relation between the  cache hit rate and the size of the cache in your better cache policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  In our evaluation, we estimate the improvement by a simple calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  However, the performance characteristic may not be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For example, the  estimation from the measurement results with 1 OST of the GFS with the sim-  ple calculation does not fit that with the 60 OSTs of GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' For more accurate  Title Suppressed Due to Excessive Length  11  estimation, improving the calculation method is necessary by modeling the char-  acteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The considerable approach is based on machine learning or queueing  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Even if the calculation method will be improved, our plot method shown  in Figure 4 is useful for the storage system architect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  7  Conclusion  This paper presented a case study on the performance analysis of a hierarchical  storage system for a DDNN workload in a flagship-class HPC cluster, discussing  potential performance improvement enabled by the speed-up of the storage sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' We also estimated the improvement of training performance by various  improvement of the hierarchical filesystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The analysis result showed that the  I/O bottleneck in the training workload depends on performance balance be-  tween global and local storage as well as file sizes in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Our estimation showed that the performance improvement of a global filesys-  tem will contribute to reducing the necessary volume size of a local filesystem,  and the performance improvement of the local file system will contribute to re-  ducing fastest I/O time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' Our estimation method can help architects of HPC  filesystems to find the necessary performance and the volume size of the local  and global filesystems to meet a given performance goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content='  Because our proposed method needs the measurement of I/O performance  at least once, one of our future works is exploring a simpler or no measurement-  required method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
+page_content=' The other future work is to build the performance modeling  of the storage system for more accurate estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfh_3R/content/2301.01494v1.pdf'}
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+A tutorial on the conservation of momentum in
+photonic time-varying media
+Angel Ortega-Gomez,1 Micha¨el Lobet,2, 3 J. Enrique V´azquez-Lozano,1 and I˜nigo Liberal1, ∗
+1Department of Electrical, Electronic and Communications
+Engineering, Institute of Smart Cities (ISC),
+Public University of Navarre (UPNA), 31006 Pamplona, Spain
+2John A. Paulson School of Engineering and Applied Sciences,
+Harvard University, 9 Oxford Street, Cambridge, MA 02138, USA
+3Department of Physics and Namur Institute of Structured Materials,
+University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium
+1
+arXiv:2301.03333v1  [physics.optics]  9 Jan 2023
+
+Abstract
+Time-varying media break temporal symmetries while preserving spatial symmetries intact.
+Thus, it represents an excellent conceptual framework to investigate the fundamental implications
+of Noether’s theorem for the electromagnetic field. At the same time, addressing momentum con-
+servation in time-varying media sheds light on the Abraham-Minkowski debate, where two opposing
+forms of the electromagnetic field momentum are defended. Here, we present a tutorial review on
+the conservation of momentum in time-varying media. We demonstrate that the Minkowski mo-
+mentum is a conserved quantity with three independent approaches of increasing complexity: (i)
+via the application of the boundary conditions for Maxwell equations at a temporal boundary, (ii)
+testing for constants of motion and deriving conservation laws, and (iii) applying temporal and
+spatial translations within the framework of the Lagrangian theory of the electromagnetic field.
+Each approach provides a different and complementary insight into the problem.
+I.
+INTRODUCTION
+Time-varying media are revolutionizing the fields of optics and nanophotonics by har-
+nessing time as an additional resource for controlling light-matter interactions [1–4]. Dy-
+namically modulating matter offers new possibilities for the manipulation of electromagnetic
+fields including compact and low-energy nonreciprocal devices [5], inverse prism and tempo-
+ral aiming effects [6, 7], overcoming bandwidth bounds in impedance matching [8], energy
+accumulation without a theoretical limit [9], quantum state frequency shifting [10], and
+ultra-fast switching without thermal noise amplification [10], to name a few. Time-varying
+media also empower new amplification [11] and photon generation mechanisms, such as
+directional vacuum amplification effects [12], amplified light emission from quantum emit-
+ters [13] and free electrons [14], as well as incandescent sources not constrained within the
+black-body spectrum [15].
+Because a homogeneous time-varying medium is invariant under spatial translations (see
+Fig.1), it is usually argued that time-varying media preserves the momentum of the elec-
+tromagnetic field [1–4, 16–18]. This intuition stems from Noether’s theorem [19–22], which
+more generally states that symmetries of the action of a physical system have an associated
+∗ Corresponding author: [email protected]
+2
+
+conserved quantity. However, a direct connection between invariance under spatial transla-
+tions and momentum conservation in time-varying media is not specified. In addition, the
+notion of the momentum of the electromagnetic field is quite subtle. In fact, according to the
+Abraham-Minkowski debate [23–27], there is more than one definition for the momentum of
+the electromagnetic field. On the one hand, one can define the Abraham momentum,
+PA (t) =
+�
+d3r pA (r, t) = µ0ε0
+�
+d3r E (r, t) × H (r, t)
+(1)
+where we have also defined the Abraham momentum density, which is proportional to the
+Poynting vector field, pA = µ0ε0S. On the other hand, the Minkowski momentum reads as
+PM (t) =
+�
+d3r pM (r, t) =
+�
+d3r D (r, t) × B (r, t)
+(2)
+A common simplification of those definitions for a plane wave in non-dispersive media
+is pA = ℏω/nc and pM = nℏω/c, which highlights the role of the refractive index n. In-
+terestingly, as pointed out by Leonhardt [28] one should call for the Minkowski momentum
+whenever the wave aspects dominate, for example, in experiments involving momentum re-
+coil [26, 29], while the Abraham momentum appears when the particle aspects are probed
+[23].
+A resolution of the debate was offered among others by Barnett [30, 31].
+It is sug-
+gested that the Abraham momentum is the kinetic momentum of the electromagnetic field,
+associated with energy transport. The Minkowski momentum is, however, the canonical
+momentum of the electromagnetic field, being the generator of spatial translations. Never-
+theless, certain aspects of the momentum of the electromagnetic field are still under question
+[32]. Moreover, the avenue of near-zero-index (NZI) media exacerbates the differences be-
+tween the forms of the momentum [33–35] giving rise to zero Minkowski momentum but
+nonzero Abraham momentum inside epsilon-and-mu-near-zero (EMNZ) media where both
+permittivity and permeability approach zero.
+Since time-varying media preserve spatial symmetries while breaking temporal symme-
+tries, it represents an excellent conceptual playground to illuminate the Abraham-Minkowski
+debate. Following the interpretation offered by Barnett [30], it should be expected that the
+Minkowski momentum - related to spatial translations - is a conserved quantity, while the
+Abraham momentum - related to energy transport - is not. This work aims to provide a
+3
+
+FIG. 1. Schematic depiction of time-varying media, in which both permittivity ε (t) and perme-
+ability µ (t) change with time. Thus, the systems is invariant with respect to spatial translations,
+but is not invariant with respect to temporal translations.
+tutorial review of different aspects on the conservation of the momentum of the electromag-
+netic field in time-varying media. We address three independent derivations showing that
+only the Minkowski momentum is a conserved quantity in time-varying media based on: (i)
+boundary conditions on Maxwell equations, (ii) directly evaluating constants of motion and
+deriving conservation laws, and (iii) inducing spatial translations to the Lagrangian of the
+electromagnetic field. Each approach provides a different physical insight into the problem.
+II.
+MOMENTUM CONSERVATION FROM INSPECTING MAXWELL EQUA-
+TIONS AT A TEMPORAL BOUNDARY
+Our starting point is Maxwell curl equations in time-varying media, which, in the absence
+of charges and currents, can be written as follows
+∇ × E (r, t) = −∂tB (r, t)
+(3)
+∇ × H (r, t) = ∂tD (r, t)
+(4)
+For the sake of simplicity, we assume homogeneous and instantaneous time-varying media,
+with constitutive relations
+D (r, t) = ε (t) E (r, t)
+(5)
+4
+
+(tn),μ(tn)
+...
+(t2),μ(t2)
+(ti),μ(ti)
+(to), μ(to)
+toB (r, t) = µ (t) H (r, t)
+(6)
+A more complete description of time-varying media would include the impact of dispersion
+and loss [17, 36]. However, a system with dissipation does not necessarily conserve quantities
+even in the presence of symmetries. In addition, the assumption of instantaneous media is
+widespread in the field of temporal metamaterials [2].
+Integrating Maxwell equations (3)-(4) accross a temporal boundary taking place at t0,
+where material parameters suddenly change from ε(t−
+0 ), µ(t−
+0 ) to ε(t+
+0 ), µ(t+
+0 ), gives
+� t+
+0
+t−
+0
+dt ∇ × H (r, t) =
+� t+
+0
+t−
+0
+dt ∂tD (r, t) = D
+�
+r, t+
+0
+�
+− D
+�
+r, t−
+0
+�
+(7)
+−
+� t+
+0
+t−
+0
+dt ∇ × E (r, t) =
+� t+
+0
+t−
+0
+dt ∂tB (r, t) = B
+�
+r, t+
+0
+�
+− B
+�
+r, t−
+0
+�
+(8)
+Therefore, we find that D (r, t) and B (r, t) must be continuous accross changes of the
+constitutive parameters, for finite E (r, t) and H (r, t) fields. This property is well-known
+since early works on time-varying media [16]. As a consequence, this reasoning confirms that
+the Minkowski momentum – uniquely defined as a function of D and B fields via Eq. (2) - is
+a continuous quantity across a temporal boundary, suggesting that is should be a conserved
+quantity in time-varying media. However, this approach fails at providing any insight on
+the associated conservation law and/or how it can be related to invariance under spatial
+translations. Moreover, it does not clarify the (non) conservation of Abraham momentum.
+III.
+CONSTANTS OF MOTION AND CONSERVATION LAWS
+In this section, we address the conservation of momentum in time-varying media by
+direcly testing if a given quantity is a constant of motion. To this end, one can take the time
+derivative of the quantity under question and check if it is zero, in which case it shall be a
+constant of motion/conserved quantity. Before addressing the momentum, it is instructive to
+analyze the energy of the electromagnetic field, which in time-varying media can be written
+as
+U (t) =
+�
+d3r u (r, t)
+(9)
+5
+
+with energy density
+u (r, t) = 1
+2
+�
+ε (t) E2 (r, t) + µ (t) H2 (r, t)
+�
+(10)
+Taking the time derivative of the energy and, substituting Maxwell equations (3)-(4),
+leads to the following expression
+dU
+dt = − 1
+µ0ε0
+�
+dS · pA − 1
+2
+�
+d3r
+�dε (t)
+dt E2 (r, t) + dµ (t)
+dt
+H2 (r, t)
+�
+(11)
+On the one hand, the first term in the r.h.s. of (11) is a surface term proportional to the
+E and H fields. This term physically means that the change of energy over time is partly
+due to energy either leaking out or coming into the system. It can be seen as a flux of either
+outgoing or incoming Poynting vector field, hence setting down a link with PA (Eq. (1)).
+It confirms the role of the Abraham momentum as the kinetic momentum, associated with
+energy transport. If the volume is large enough to capture the entirety of the E and H
+fields within the time interval of interest, its contribution vanishes. On the other hand, the
+second term in the r.h.s. of (11) is a volume integral directly linked to the time modulation
+of the permittivity and permeability, which results in a change of the energy of the system.
+It represents the energy that must be pumped into or retracted from the system in order to
+realize the time modulation of the material parameters. In other words, the time variation
+of the material parameters act as sources or sinks of electromagnetic energy. By contrast,
+Eq. (11) shows that for a medium with static material properties dU/dt = 0 and energy
+would be a conserved quantity.
+Eq. (11) can also be casted as a local conservation law as a function of the energy and
+momentum densities
+du (r, t)
+dt
++
+1
+µ0ε0
+∇ · pA (r, t) = −1
+2
+�dε (t)
+dt E2 (r, t) + dµ (t)
+dt
+H2 (r, t)
+�
+(12)
+where we clearly identify the source/sink at the r.h.s..
+Let us now tackle the conservation of Minkowski momentum and examine the time varia-
+tion of Abraham momentum. By introducing Maxwell equations and applying a few vector
+calculus identities, it can be found that the time derivative of the Minkowski momentum is
+given by
+dPM (t)
+dt
+= ε (t)
+� �
+p=x,y,z
+up
+�
+dS · (EpE) − 1
+2
+�
+dS (E · E)
+�
+6
+
++ µ (t)
+� �
+p=x,y,z
+up
+�
+dS · (HpH) − 1
+2
+�
+dS (H · H)
+�
+(13)
+By doing so, we find that the time derivative of the Minkowski momentum reduces to
+surface terms. Once again, if the volume of integration is taken large enough so that all the
+E and H fields are confined within its interior, all surface terms vanish. In other words,
+dPM (t) /dt = 0, proving that the Minkowski momentum is a constant of motion as expected.
+It is also instructive to note that the above equation can be written in a differential form as
+a conservation law for the momentum density:
+dpM (t)
+dt
+= ∇ · TM (r, t)
+(14)
+where we define the Minkowski stress tensor for time-varying media as
+TM (r, t) = ε (t)
+�
+E ⊗ E − 1
+2I (E · E)
+�
++ µ (t)
+�
+H ⊗ H − 1
+2I (H · H)
+�
+(15)
+with I being the identity dyadic. Conservation laws in the form of (14) can be found scattered
+in the literature, for example, in the appendix of [18].
+Proceeding similarly with the Abraham momentum reveals that in general it is not a
+conserved quantity:
+dPA
+dt
+= −
+� 1
+ε (t)
+dε (t)
+dt
++
+1
+µ (t)
+dµ (t)
+dt
+�
+PA (t)
+−ε0µ0
+µ (t)
+�
+1
+2
+�
+dS (E · E) −
+�
+p=x,y,z
+up
+�
+dS · (EpE)
+�
+− ε0µ0
+ε (t)
+�
+1
+2
+�
+dS (H · H) −
+�
+p=x,y,z
+up
+�
+dS · (HpH)
+�
+(16)
+Here again, the second and third terms are surface terms that would vanish for a suffi-
+ciently large volume. However, the first term illustrates that the Abraham momentum does
+change in time, following the change in the permittivity and permeability of the medium.
+Equation (16) can also be compactly written as a local conservation law for the momentum
+density
+dpA
+dt = ∇ · TA −
+� 1
+ε (t)
+dε (t)
+dt
++
+1
+µ (t)
+dµ (t)
+dt
+�
+pA (r, t)
+(17)
+where we define the Abraham stress tensor in time-varying media, related to the Minkowski
+stress tensor as follows
+TA =
+ε0µ0
+µ (t) ε (t) TM (r, t)
+(18)
+7
+
+In conclusion, testing for constants of motions provides an independent confirmation
+that the Minkowski momentum is indeed a conserved quantity in time-varying media. In
+addition, it provides insight in the form of the conservation law that supports its invariance.
+Furthermore, it shows that the Abraham momentum is not a constant of motion in close
+connection to energy considerations, and re-emphasizes its role as the kinetic momentum of
+the electromagnetic field. Nevertheless, writing the conservation law does not clarify the role
+of the invariance of the system under spatial translations in the conservation of momentum.
+IV.
+MOMENTUM CONSERVATION AS A CONSEQUENCE OF INVARIANCE
+UNDER SPATIAL TRANSLATIONS: A LAGRANGIAN APPROACH
+In this section we address the conservation of momentum in time-varying media from the
+perspective of the Lagrangian formalism for electromagnetic fields. Using the Lagrangian
+formalism adds an extra layer of complexity, but allows to unequivocally identify momen-
+tum conservation as a fundamental consequence of the invariance of time-varying media
+under spatial translations. We note that most works identifying the Minkowski momentum
+as the generator of spatial translations do it from a quantum description of the electro-
+magnetic field, where the Minkowski momentum appears as an operator [30]. However, it
+is important to understand that momentum conservation as a consequence of invariance
+under spatial translations is also a classical effect. Therefore, we keep here a classical La-
+grangian description of the electromagnetic fields, without introducing the quantization of
+the electromagnetic field.
+In the following, we first review the Lagrangian description of electromagnetic fields
+extended to time-varying media. Then, we derive a form of Noether’s theorem in our for-
+malism and we finally show the quantities associated with temporal and spatial translations
+for time-varying media.
+A.
+Lagrangian description of the electromagnetic field
+An in-depth review of the Lagrangian theory of the electromagnetic field can be found in
+Cohen-Tannoudji’s book [21]. Here we review it and extend it to time-varying media. From
+the perspective of Lagrangian theory, Maxwell equations are equations of motion that can
+8
+
+be derived from the principle of least (or stationary) action. This principle states that true
+path of motion corresponds to a stationary point of the action. By motion we refer to the
+values that the dynamical variables have in a given interval of time, which, when position
+is a dynamical variable, aligns with the common notion of motion. The action is defined as
+the integral of the Lagrangian between two instants of time t1 and t2:
+S (t1, t2) =
+� t2
+t1
+dt L (t)
+(19)
+with the Lagrangian
+L (t) =
+�
+d3r L (r, t)
+(20)
+and the Lagrangian density
+L (r, t) = 1
+2
+�
+d3r [ε (t) E (r, t) · E (r, t) − µ (t) H (r, t) · H (r, t)]
+(21)
+The choice of this Lagrangian density is a direct extension from the case with no time
+modulation. It is justified because Lagrange’s equation correctly recovers the equations of
+motion for the electromagnetic field, as shown below. For the Lagrangian description of
+the electromagnetic field, it is convenient to work with scalar V (r, t) and vector A (r, t)
+potentials instead of fields. For the sake of simplicity, we work in the Coulomb gauge, for
+which ∇ · A (r, t) = 0. By doing so, the scalar potential is zero in the absence of charges
+V (r, t) = 0, all the fields are transversal, and they can be simply written as a function of
+the vector potential
+D (r, t) = −ε (t) ∂tA (r, t)
+(22)
+B (r, t) = ∇ × A (r, t)
+(23)
+Then, Maxwell equations lead to the following wave equation for the components of the
+vector potential (p = x, y, z):
+∇2Ap (r, t) − µ (t) ∂t {ε (t) ∂tAp (r, t)} = 0
+(24)
+Due to field transversality, the Minkowski momentum can be compactly written as
+PM = −ε (t)
+�
+p
+�
+d3r ∂tAp (r, t) ∇Ap (r, t)
+(25)
+Similarly, the Lagrangian density reduces to
+L = 1
+2
+�
+p
+�
+ε (t) ˙A2
+p (r, t) −
+1
+µ (t) (∇ × A (r, t))2
+p
+�
+(26)
+9
+
+where we have used ˙Ap as a shorter way to write the time derivative. From this description,
+it lies that the components of the vector potential, Ap, and its time derivatives, ˙Ap, are the
+dynamical variables of the system.
+Imposing that a true path of motion is a stationary point of the action, for which δS = 0,
+leads to Lagrange’s equations
+∂L
+∂Ap
+−
+�
+q
+∂q
+�
+∂L
+∂ (∂qAp)
+�
+− d
+dt
+∂L
+∂ ˙Ap
+= 0
+(27)
+which reduces to the wave equation for Ap in (24), justifying the direct extension of the
+Lagrangian to time-varying media.
+With equation (26), we find that the conjugate momentum of each vector potential com-
+ponent, Ap, is the negative of the electric displacement field components
+Πp (r, t) = ∂L
+∂ ˙Ap
+= ε (t) ˙Ap (r, t) = −Dp (r, t)
+(28)
+This point allow us to clarify another ambiguity related to the momentum of the elec-
+tromagnetic field. For a freely moving particle of mass m with Lagrangian, L = �
+p
+1
+2 m ˙r2
+p,
+the dynamical variables are the position coordinates rp, p = x, y, z. Thus, their associated
+conjugate momenta pp = ∂L/∂ ˙rp = m ˙rp correspond to the components of the linear mo-
+mentum. The latter is also the momentum associated with the spatial translations of the
+system. However, for the electromagnetic field, position is not a dynamical variable of the
+system while the vector potential is. For this reason, one has to differentiate between the
+conjugate momentum and the momentum associated with spatial translations, as clarified
+below.
+Finally, the Hamiltonian is defined as a function of the conjugate momentum as follows
+H =
+�
+p
+�
+d3r Πp (r, t) ˙Ap (r, t) − L
+(29)
+which can be found to be fully equivalent to the form of the electromagnetic energy in
+time-varing media employed in the previous section, and given by Eqs. (9)-(10).
+B.
+Noether’s theorem in the Coulomb gauge
+In this section, we cast a form of Noether’s theorem which allows us to discern the
+conserved quantities associated with the continuous symmetries of time-varying media. To
+10
+
+FIG. 2. (a) Schematic depiction of the motion of a dynamical variable Ap (r, t) between times t1
+and t2, and an infinitesimally close motion, described by A′
+p (r, t) between times t′
+1 and t′
+2. The
+difference between both motions at time t is given by dA (r, t) = A′ (r, t) − A (r, t). The difference
+between the initial and final temporal points is given by dt1 = t′
+1−t1 and dt2 = t′
+2−t2, respectively.
+(b) Schematic depiction of trajectories for systems with (left) temporal translation symmetry, and
+(right) spatial translation symmetry.
+this end, we note that any continuous symmetry can be described as an infinitesimal variation
+of the action. Therefore, as schematically depicted in Fig. 2(a), we consider a motion between
+times t1 and t2, defined by the dynamical variables Ap (r, t), and an infinitesimally close
+motion between times t′
+1 and t′
+2, described by A′
+p (r, t).
+The variation of the dynamical
+variables at a given point of time is dAp (r, t) = A′
+p (r, t) − Ap (r, t), and the variation of the
+action can be written as
+dS = S′ − S =
+� t′
+2
+t′
+1
+dt L
+�
+A′
+p
+�
+−
+� t2
+t1
+dt L (Ap)
+=
+� t2
+t1
+dt
+�
+L
+�
+A′
+p
+�
+− L (Ap)
+�
++
+� t′
+2
+t2
+dt L
+�
+A′
+p
+�
+−
+� t′
+1
+t1
+dt L
+�
+A′
+p
+�
+(30)
+11
+
+(a)
+p (r,t)
+Ap(r,t2)
+Ap(r,t2)
+Ap(r,t1)
+Ap(r,t')
+dt2
+dti
+ti
+t2
+34
+(b)
+Temporal translation symmetry
+Spatial translation symmetry
+Ap(r -n,t2)
+Ap (r,ti)
+A"(r,t))
+Ap (r,ti)
+dAp (r,t) As(r;t2)
+Ap(r,t2)
+Ap (r,t2)
+t1
+t2
+dt
+dt
+t
+ti
+ti
+t2
+t2
+ti
+t2To first order, last two terms can be approximated by
+� t′
+2
+t2
+dt L
+�
+A′
+p
+�
+= L (Ap)|t2 dt2
+(31)
+and the equivalent expression for t1.
+Similarly, for two infinitesimally closed motions, the first term is given by
+� t2
+t1
+dt
+�
+L
+�
+A′
+p
+�
+− L (Ap)
+�
+=
+=
+� t2
+t1
+dt
+�
+p
+�
+d3r
+�
+∂L
+∂Ap (r, t) dAp (r, t) +
+∂L
+∂ ˙Ap (r, t)
+d ˙Ap (r, t)
++
+�
+q
+�
+∂L
+∂ (∂qAp (r, t))
+�
+d∂qAp (r, t)
+�
+(32)
+Similarly to the derivation of Lagrange’s equation, we integrate by parts the second term
+with respect to time and the third term with respect to r, so the variation of the action
+reduces to
+� t2
+t1
+dt
+�
+L
+�
+A′
+p
+�
+− L (Ap)
+�
+=
+=
+� t2
+t1
+dt
+�
+p
+�
+d3r
+�
+∂L
+∂Ap (r, t) − d
+dt
+∂L
+∂ ˙Ap (r, t)
+−
+�
+q
+∂q
+�
+∂L
+∂ (∂qAp (r, t))
+��
+dAp (r, t)
++
+�
+p
+�
+d3r
+∂L
+∂ ˙Ap (r, t)
+dAp (r, t)
+�����
+t2
+t1
+(33)
+Note that, in deriving the above equation we have assumed that the fields vanish at
+infinity, so that there are no surface contributions. By contrast, the fields do not need to
+vanish at the initial and final temporal boundaries, leading the contribution from the last
+term. In addition, the integrand of the first term is a solution to Lagrange’s equation (27),
+which reduces to zero. Thus, by substituting (31)-(33) into (30) we find that the variation
+of the action is given by:
+dS =
+�
+p
+�
+d3r
+�
+∂L
+∂ ˙Ap (r, t)
+dAp (r, t)
+�����
+t2
++ L (Ap)|t2 dt2
+−
+∂L
+∂ ˙Ap (r, t)
+dAp (r, t)
+�����
+t1
+− L (Ap)|t1 dt1
+�
+(34)
+If a system has a continuous symmetry, then the corresponding action remains invariant
+with respect to infinitesimal displacements, i.e., dS = 0. In addition, since dS = 0 must
+12
+
+hold for any pair of times t1 and t2 we find that the term within brackets must be a constant
+of motion. These relations correspond to Noether’s theorem applied to our formulation of
+the electromagnetic field in time-varying media in the Coulomb Gauge. Given a continuous
+symmetry, specified by the variation dAp (r, t) and the boundary condition on the Lagrangian
+L (Ap) dt, one can identify an associated conserved quantity.
+C.
+Temporal and spatial translations
+First, let us assume that the variation is produced by an infinitesimal temporal displace-
+ment dt, such that dt2 = dt1 = dt (see Fig. 2(b)). If the system is invariant with respect to
+temporal translations we can write A′
+p (r, t) = Ap (r, t − dt) ≃ Ap (r, t) − ˙Ap (r, t) dt. Then,
+we have dAp (r, t) = − ˙Ap (r, t) dt. Substituting this result in (34) and factoring out dt we
+find that the conserved quantity is
+�
+p
+�
+d3r
+�
+−
+∂L
+∂ ˙Ap (r, t)
+˙Ap (r, t) + L (Ap)
+�
+=
+= −
+�
+p
+�
+d3r 1
+2
+�
+p
+�
+ε (t) ˙A2
+p (r, t) +
+1
+µ (t) (∇ × A (r, t))2
+p
+�
+= −H
+(35)
+Therefore, it is found that invariance with respect to temporal translations implies that
+the Hamiltonian must be a conserved quantity. In time-varying media, the system is not
+invariant under temporal translations, and, consequently, the Hamiltonian manifestly de-
+pends on time. As shown in the previous section, taking its time derivative explicitly shows
+that it is not a constant of motion.
+Second, we assume that the variation is produced by an infinitesimal spatial displacement
+η (see Fig. 2(b)). Then, if the system is invariant under spatial translations we must have
+A′
+p (r, t) = Ap (r − η, t) ≃ Ap (r, t)−η·∇Ap (r, t), so that dAp (r, t) = −η·∇Ap (r, t) . Again,
+substituting this result into (34) and factoring out η we find that the conserved quantity
+must be
+P =
+�
+p
+�
+d3r
+∂L
+∂ ˙Ap (r, t)
+∇Ap (r, t) = −
+�
+p
+�
+d3r ε (t) ˙Ap (r, t) ∇Ap (r, t)
+(36)
+which equals the Minkowski momentum in (25). Therefore, we finally found that the fact
+that time-varying media are invariant under spatial translations directly enforces that the
+Minkowski momentum is a conserved quantity.
+13
+
+V.
+CONCLUDING REMARKS
+Symmetries play a fundamental role in physics. They reduce the complexity of difficult
+problems, as well as the computational cost needed to solve them. Symmetries also en-
+able the identification of conserved quantities and the formal link between both symmetries
+and conserved quantities is Noether’s theorem. One of the reasons why time-varying media
+and/or temporal metamaterials provide a fresh view on electromagnetic theory is because
+they break temporal symmetries, which are conserved in most traditional photonics systems,
+while they maintain spatial symmetries. However, the connection between spatial and tem-
+poral symmetries and the properties of time-varying media is not always explicitely stated,
+or analyzed through the point of view of Lagrangian mechanics. The present tutorial aims
+at filling this gap. Furthermore, we hope that this tutorial may clarify the subtleties of the
+conservation of the electromagnetic momentum in time-varying media, the nuances of defin-
+ing the momentum of the electromagnetic fields within the Abraham-Minkowski debate, and
+that it will foster further research on the role and significance of symmetries in temporal
+metamaterials.
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+17
+

CNE1T4oBgHgl3EQfpgW7/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf,len=411
+page_content='A tutorial on the conservation of momentum in  photonic time-varying media  Angel Ortega-Gomez,1 Micha¨el Lobet,2, 3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Enrique V´azquez-Lozano,1 and I˜nigo Liberal1, ∗  1Department of Electrical, Electronic and Communications  Engineering, Institute of Smart Cities (ISC),  Public University of Navarre (UPNA), 31006 Pamplona, Spain  2John A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Paulson School of Engineering and Applied Sciences,  Harvard University, 9 Oxford Street, Cambridge, MA 02138, USA  3Department of Physics and Namur Institute of Structured Materials,  University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='03333v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='optics]  9 Jan 2023  Abstract  Time-varying media break temporal symmetries while preserving spatial symmetries intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Thus, it represents an excellent conceptual framework to investigate the fundamental implications  of Noether’s theorem for the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' At the same time, addressing momentum con-  servation in time-varying media sheds light on the Abraham-Minkowski debate, where two opposing  forms of the electromagnetic field momentum are defended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Here, we present a tutorial review on  the conservation of momentum in time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' We demonstrate that the Minkowski mo-  mentum is a conserved quantity with three independent approaches of increasing complexity: (i)  via the application of the boundary conditions for Maxwell equations at a temporal boundary, (ii)  testing for constants of motion and deriving conservation laws, and (iii) applying temporal and  spatial translations within the framework of the Lagrangian theory of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Each approach provides a different and complementary insight into the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  INTRODUCTION  Time-varying media are revolutionizing the fields of optics and nanophotonics by har-  nessing time as an additional resource for controlling light-matter interactions [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Dy-  namically modulating matter offers new possibilities for the manipulation of electromagnetic  fields including compact and low-energy nonreciprocal devices [5], inverse prism and tempo-  ral aiming effects [6, 7], overcoming bandwidth bounds in impedance matching [8], energy  accumulation without a theoretical limit [9], quantum state frequency shifting [10], and  ultra-fast switching without thermal noise amplification [10], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Time-varying  media also empower new amplification [11] and photon generation mechanisms, such as  directional vacuum amplification effects [12], amplified light emission from quantum emit-  ters [13] and free electrons [14], as well as incandescent sources not constrained within the  black-body spectrum [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Because a homogeneous time-varying medium is invariant under spatial translations (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='1), it is usually argued that time-varying media preserves the momentum of the elec-  tromagnetic field [1–4, 16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' This intuition stems from Noether’s theorem [19–22], which  more generally states that symmetries of the action of a physical system have an associated  ∗ Corresponding author: inigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='liberal@unavarra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='es  2  conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, a direct connection between invariance under spatial transla-  tions and momentum conservation in time-varying media is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In addition, the  notion of the momentum of the electromagnetic field is quite subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In fact, according to the  Abraham-Minkowski debate [23–27], there is more than one definition for the momentum of  the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' On the one hand, one can define the Abraham momentum,  PA (t) =  �  d3r pA (r, t) = µ0ε0  �  d3r E (r, t) × H (r, t)  (1)  where we have also defined the Abraham momentum density, which is proportional to the  Poynting vector field, pA = µ0ε0S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' On the other hand, the Minkowski momentum reads as  PM (t) =  �  d3r pM (r, t) =  �  d3r D (r, t) × B (r, t)  (2)  A common simplification of those definitions for a plane wave in non-dispersive media  is pA = ℏω/nc and pM = nℏω/c, which highlights the role of the refractive index n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In-  terestingly, as pointed out by Leonhardt [28] one should call for the Minkowski momentum  whenever the wave aspects dominate, for example, in experiments involving momentum re-  coil [26, 29], while the Abraham momentum appears when the particle aspects are probed  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  A resolution of the debate was offered among others by Barnett [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  It is sug-  gested that the Abraham momentum is the kinetic momentum of the electromagnetic field,  associated with energy transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' The Minkowski momentum is, however, the canonical  momentum of the electromagnetic field, being the generator of spatial translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Never-  theless, certain aspects of the momentum of the electromagnetic field are still under question  [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Moreover, the avenue of near-zero-index (NZI) media exacerbates the differences be-  tween the forms of the momentum [33–35] giving rise to zero Minkowski momentum but  nonzero Abraham momentum inside epsilon-and-mu-near-zero (EMNZ) media where both  permittivity and permeability approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Since time-varying media preserve spatial symmetries while breaking temporal symme-  tries, it represents an excellent conceptual playground to illuminate the Abraham-Minkowski  debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Following the interpretation offered by Barnett [30], it should be expected that the  Minkowski momentum - related to spatial translations - is a conserved quantity, while the  Abraham momentum - related to energy transport - is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' This work aims to provide a  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Schematic depiction of time-varying media, in which both permittivity ε (t) and perme-  ability µ (t) change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Thus, the systems is invariant with respect to spatial translations,  but is not invariant with respect to temporal translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  tutorial review of different aspects on the conservation of the momentum of the electromag-  netic field in time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' We address three independent derivations showing that  only the Minkowski momentum is a conserved quantity in time-varying media based on: (i)  boundary conditions on Maxwell equations, (ii) directly evaluating constants of motion and  deriving conservation laws, and (iii) inducing spatial translations to the Lagrangian of the  electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Each approach provides a different physical insight into the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  MOMENTUM CONSERVATION FROM INSPECTING MAXWELL EQUA-  TIONS AT A TEMPORAL BOUNDARY  Our starting point is Maxwell curl equations in time-varying media, which, in the absence  of charges and currents, can be written as follows  ∇ × E (r, t) = −∂tB (r, t)  (3)  ∇ × H (r, t) = ∂tD (r, t)  (4)  For the sake of simplicity, we assume homogeneous and instantaneous time-varying media,  with constitutive relations  D (r, t) = ε (t) E (r, t)  (5)  4  (tn),μ(tn)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  (t2),μ(t2)  (ti),μ(ti)  (to), μ(to)  toB (r, t) = µ (t) H (r, t)  (6)  A more complete description of time-varying media would include the impact of dispersion  and loss [17, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, a system with dissipation does not necessarily conserve quantities  even in the presence of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In addition, the assumption of instantaneous media is  widespread in the field of temporal metamaterials [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Integrating Maxwell equations (3)-(4) accross a temporal boundary taking place at t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  where material parameters suddenly change from ε(t−  0 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' µ(t−  0 ) to ε(t+  0 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' µ(t+  0 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' gives  � t+  0  t−  0  dt ∇ × H (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) =  � t+  0  t−  0  dt ∂tD (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = D  �  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t+  0  �  − D  �  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t−  0  �  (7)  −  � t+  0  t−  0  dt ∇ × E (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) =  � t+  0  t−  0  dt ∂tB (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = B  �  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t+  0  �  − B  �  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t−  0  �  (8)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' we find that D (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) and B (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) must be continuous accross changes of the  constitutive parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' for finite E (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) and H (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' This property is well-known  since early works on time-varying media [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' As a consequence, this reasoning confirms that  the Minkowski momentum – uniquely defined as a function of D and B fields via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' (2) - is  a continuous quantity across a temporal boundary, suggesting that is should be a conserved  quantity in time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, this approach fails at providing any insight on  the associated conservation law and/or how it can be related to invariance under spatial  translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Moreover, it does not clarify the (non) conservation of Abraham momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  CONSTANTS OF MOTION AND CONSERVATION LAWS  In this section, we address the conservation of momentum in time-varying media by  direcly testing if a given quantity is a constant of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' To this end, one can take the time  derivative of the quantity under question and check if it is zero, in which case it shall be a  constant of motion/conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Before addressing the momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' it is instructive to  analyze the energy of the electromagnetic field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' which in time-varying media can be written  as  U (t) =  �  d3r u (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  (9)  5  with energy density  u (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = 1  2  �  ε (t) E2 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) + µ (t) H2 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  �  (10)  Taking the time derivative of the energy and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' substituting Maxwell equations (3)-(4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  leads to the following expression  dU  dt = − 1  µ0ε0  �  dS · pA − 1  2  �  d3r  �dε (t)  dt E2 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) + dµ (t)  dt  H2 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  �  (11)  On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' the first term in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' of (11) is a surface term proportional to the  E and H fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' This term physically means that the change of energy over time is partly  due to energy either leaking out or coming into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' It can be seen as a flux of either  outgoing or incoming Poynting vector field, hence setting down a link with PA (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  It confirms the role of the Abraham momentum as the kinetic momentum, associated with  energy transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' If the volume is large enough to capture the entirety of the E and H  fields within the time interval of interest, its contribution vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' On the other hand, the  second term in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' of (11) is a volume integral directly linked to the time modulation  of the permittivity and permeability, which results in a change of the energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  It represents the energy that must be pumped into or retracted from the system in order to  realize the time modulation of the material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In other words, the time variation  of the material parameters act as sources or sinks of electromagnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' By contrast,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' (11) shows that for a medium with static material properties dU/dt = 0 and energy  would be a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' (11) can also be casted as a local conservation law as a function of the energy and  momentum densities  du (r, t)  dt  +  1  µ0ε0  ∇ · pA (r, t) = −1  2  �dε (t)  dt E2 (r, t) + dµ (t)  dt  H2 (r, t)  �  (12)  where we clearly identify the source/sink at the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='.  Let us now tackle the conservation of Minkowski momentum and examine the time varia-  tion of Abraham momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' By introducing Maxwell equations and applying a few vector  calculus identities, it can be found that the time derivative of the Minkowski momentum is  given by  dPM (t)  dt  = ε (t)  � �  p=x,y,z  up  �  dS · (EpE) − 1  2  �  dS (E · E)  �  6  + µ (t)  � �  p=x,y,z  up  �  dS · (HpH) − 1  2  �  dS (H · H)  �  (13)  By doing so, we find that the time derivative of the Minkowski momentum reduces to  surface terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Once again, if the volume of integration is taken large enough so that all the  E and H fields are confined within its interior, all surface terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In other words,  dPM (t) /dt = 0, proving that the Minkowski momentum is a constant of motion as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  It is also instructive to note that the above equation can be written in a differential form as  a conservation law for the momentum density:  dpM (t)  dt  = ∇ · TM (r, t)  (14)  where we define the Minkowski stress tensor for time-varying media as  TM (r, t) = ε (t)  �  E ⊗ E − 1  2I (E · E)  �  + µ (t)  �  H ⊗ H − 1  2I (H · H)  �  (15)  with I being the identity dyadic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Conservation laws in the form of (14) can be found scattered  in the literature, for example, in the appendix of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Proceeding similarly with the Abraham momentum reveals that in general it is not a  conserved quantity:  dPA  dt  = −  � 1  ε (t)  dε (t)  dt  +  1  µ (t)  dµ (t)  dt  �  PA (t)  −ε0µ0  µ (t)  �  1  2  �  dS (E · E) −  �  p=x,y,z  up  �  dS · (EpE)  �  − ε0µ0  ε (t)  �  1  2  �  dS (H · H) −  �  p=x,y,z  up  �  dS · (HpH)  �  (16)  Here again, the second and third terms are surface terms that would vanish for a suffi-  ciently large volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, the first term illustrates that the Abraham momentum does  change in time, following the change in the permittivity and permeability of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Equation (16) can also be compactly written as a local conservation law for the momentum  density  dpA  dt = ∇ · TA −  � 1  ε (t)  dε (t)  dt  +  1  µ (t)  dµ (t)  dt  �  pA (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  (17)  where we define the Abraham stress tensor in time-varying media,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' related to the Minkowski  stress tensor as follows  TA =  ε0µ0  µ (t) ε (t) TM (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  (18)  7  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' testing for constants of motions provides an independent confirmation  that the Minkowski momentum is indeed a conserved quantity in time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In  addition, it provides insight in the form of the conservation law that supports its invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Furthermore, it shows that the Abraham momentum is not a constant of motion in close  connection to energy considerations, and re-emphasizes its role as the kinetic momentum of  the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Nevertheless, writing the conservation law does not clarify the role  of the invariance of the system under spatial translations in the conservation of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  MOMENTUM CONSERVATION AS A CONSEQUENCE OF INVARIANCE  UNDER SPATIAL TRANSLATIONS: A LAGRANGIAN APPROACH  In this section we address the conservation of momentum in time-varying media from the  perspective of the Lagrangian formalism for electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Using the Lagrangian  formalism adds an extra layer of complexity, but allows to unequivocally identify momen-  tum conservation as a fundamental consequence of the invariance of time-varying media  under spatial translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' We note that most works identifying the Minkowski momentum  as the generator of spatial translations do it from a quantum description of the electro-  magnetic field, where the Minkowski momentum appears as an operator [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, it  is important to understand that momentum conservation as a consequence of invariance  under spatial translations is also a classical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Therefore, we keep here a classical La-  grangian description of the electromagnetic fields, without introducing the quantization of  the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  In the following, we first review the Lagrangian description of electromagnetic fields  extended to time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Then, we derive a form of Noether’s theorem in our for-  malism and we finally show the quantities associated with temporal and spatial translations  for time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Lagrangian description of the electromagnetic field  An in-depth review of the Lagrangian theory of the electromagnetic field can be found in  Cohen-Tannoudji’s book [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Here we review it and extend it to time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' From  the perspective of Lagrangian theory, Maxwell equations are equations of motion that can  8  be derived from the principle of least (or stationary) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' This principle states that true  path of motion corresponds to a stationary point of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' By motion we refer to the  values that the dynamical variables have in a given interval of time, which, when position  is a dynamical variable, aligns with the common notion of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' The action is defined as  the integral of the Lagrangian between two instants of time t1 and t2:  S (t1, t2) =  � t2  t1  dt L (t)  (19)  with the Lagrangian  L (t) =  �  d3r L (r, t)  (20)  and the Lagrangian density  L (r, t) = 1  2  �  d3r [ε (t) E (r, t) · E (r, t) − µ (t) H (r, t) · H (r, t)]  (21)  The choice of this Lagrangian density is a direct extension from the case with no time  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' It is justified because Lagrange’s equation correctly recovers the equations of  motion for the electromagnetic field, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' For the Lagrangian description of  the electromagnetic field, it is convenient to work with scalar V (r, t) and vector A (r, t)  potentials instead of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' For the sake of simplicity, we work in the Coulomb gauge, for  which ∇ · A (r, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' By doing so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' the scalar potential is zero in the absence of charges  V (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' all the fields are transversal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' and they can be simply written as a function of  the vector potential  D (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = −ε (t) ∂tA (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  (22)  B (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = ∇ × A (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  (23)  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Maxwell equations lead to the following wave equation for the components of the  vector potential (p = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' z):  ∇2Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) − µ (t) ∂t {ε (t) ∂tAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)} = 0  (24)  Due to field transversality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' the Minkowski momentum can be compactly written as  PM = −ε (t)  �  p  �  d3r ∂tAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) ∇Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  (25)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' the Lagrangian density reduces to  L = 1  2  �  p  �  ε (t) ˙A2  p (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) −  1  µ (t) (∇ × A (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t))2  p  �  (26)  9  where we have used ˙Ap as a shorter way to write the time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' From this description,  it lies that the components of the vector potential, Ap, and its time derivatives, ˙Ap, are the  dynamical variables of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Imposing that a true path of motion is a stationary point of the action, for which δS = 0,  leads to Lagrange’s equations  ∂L  ∂Ap  −  �  q  ∂q  �  ∂L  ∂ (∂qAp)  �  − d  dt  ∂L  ∂ ˙Ap  = 0  (27)  which reduces to the wave equation for Ap in (24), justifying the direct extension of the  Lagrangian to time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  With equation (26), we find that the conjugate momentum of each vector potential com-  ponent, Ap, is the negative of the electric displacement field components  Πp (r, t) = ∂L  ∂ ˙Ap  = ε (t) ˙Ap (r, t) = −Dp (r, t)  (28)  This point allow us to clarify another ambiguity related to the momentum of the elec-  tromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' For a freely moving particle of mass m with Lagrangian, L = �  p  1  2 m ˙r2  p,  the dynamical variables are the position coordinates rp, p = x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Thus, their associated  conjugate momenta pp = ∂L/∂ ˙rp = m ˙rp correspond to the components of the linear mo-  mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' The latter is also the momentum associated with the spatial translations of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, for the electromagnetic field, position is not a dynamical variable of the  system while the vector potential is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' For this reason, one has to differentiate between the  conjugate momentum and the momentum associated with spatial translations, as clarified  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Finally, the Hamiltonian is defined as a function of the conjugate momentum as follows  H =  �  p  �  d3r Πp (r, t) ˙Ap (r, t) − L  (29)  which can be found to be fully equivalent to the form of the electromagnetic energy in  time-varing media employed in the previous section, and given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' (9)-(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Noether’s theorem in the Coulomb gauge  In this section, we cast a form of Noether’s theorem which allows us to discern the  conserved quantities associated with the continuous symmetries of time-varying media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' To  10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' (a) Schematic depiction of the motion of a dynamical variable Ap (r, t) between times t1  and t2, and an infinitesimally close motion, described by A′  p (r, t) between times t′  1 and t′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' The  difference between both motions at time t is given by dA (r, t) = A′ (r, t) − A (r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' The difference  between the initial and final temporal points is given by dt1 = t′  1−t1 and dt2 = t′  2−t2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  (b) Schematic depiction of trajectories for systems with (left) temporal translation symmetry, and  (right) spatial translation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  this end, we note that any continuous symmetry can be described as an infinitesimal variation  of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Therefore, as schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' 2(a), we consider a motion between  times t1 and t2, defined by the dynamical variables Ap (r, t), and an infinitesimally close  motion between times t′  1 and t′  2, described by A′  p (r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  The variation of the dynamical  variables at a given point of time is dAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) = A′  p (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) − Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' and the variation of the  action can be written as  dS = S′ − S =  � t′  2  t′  1  dt L  �  A′  p  �  −  � t2  t1  dt L (Ap)  =  � t2  t1  dt  �  L  �  A′  p  �  − L (Ap)  �  +  � t′  2  t2  dt L  �  A′  p  �  −  � t′  1  t1  dt L  �  A′  p  �  (30)  11  (a)  p (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t)  Ap(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t2)  Ap(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t2)  Ap(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t1)  Ap(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content="t')  dt2  dti  ti  t2  34  (b)  Temporal translation symmetry  Spatial translation symmetry  Ap(r -n," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t2)  Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='ti)  A"(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t))  Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='ti)  dAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t) As(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='t2)  Ap(r,t2)  Ap (r,t2)  t1  t2  dt  dt  t  ti  ti  t2  t2  ti  t2To first order, last two terms can be approximated by  � t′  2  t2  dt L  �  A′  p  �  = L (Ap)|t2 dt2  (31)  and the equivalent expression for t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' for two infinitesimally closed motions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' the first term is given by  � t2  t1  dt  �  L  �  A′  p  �  − L (Ap)  �  =  =  � t2  t1  dt  �  p  �  d3r  �  ∂L  ∂Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) dAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) +  ∂L  ∂ ˙Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  d ˙Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  +  �  q  �  ∂L  ∂ (∂qAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t))  �  d∂qAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  �  (32)  Similarly to the derivation of Lagrange’s equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' we integrate by parts the second term  with respect to time and the third term with respect to r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' so the variation of the action  reduces to  � t2  t1  dt  �  L  �  A′  p  �  − L (Ap)  �  =  =  � t2  t1  dt  �  p  �  d3r  �  ∂L  ∂Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t) − d  dt  ∂L  ∂ ˙Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  −  �  q  ∂q  �  ∂L  ∂ (∂qAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t))  ��  dAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  +  �  p  �  d3r  ∂L  ∂ ˙Ap (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  dAp (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' t)  �����  t2  t1  (33)  Note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' in deriving the above equation we have assumed that the fields vanish at  infinity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' so that there are no surface contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' By contrast, the fields do not need to  vanish at the initial and final temporal boundaries, leading the contribution from the last  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In addition, the integrand of the first term is a solution to Lagrange’s equation (27),  which reduces to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Thus, by substituting (31)-(33) into (30) we find that the variation  of the action is given by:  dS =  �  p  �  d3r  �  ∂L  ∂ ˙Ap (r, t)  dAp (r, t)  �����  t2  + L (Ap)|t2 dt2  −  ∂L  ∂ ˙Ap (r, t)  dAp (r, t)  �����  t1  − L (Ap)|t1 dt1  �  (34)  If a system has a continuous symmetry, then the corresponding action remains invariant  with respect to infinitesimal displacements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=', dS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In addition, since dS = 0 must  12  hold for any pair of times t1 and t2 we find that the term within brackets must be a constant  of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' These relations correspond to Noether’s theorem applied to our formulation of  the electromagnetic field in time-varying media in the Coulomb Gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Given a continuous  symmetry, specified by the variation dAp (r, t) and the boundary condition on the Lagrangian  L (Ap) dt, one can identify an associated conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Temporal and spatial translations  First, let us assume that the variation is produced by an infinitesimal temporal displace-  ment dt, such that dt2 = dt1 = dt (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' If the system is invariant with respect to  temporal translations we can write A′  p (r, t) = Ap (r, t − dt) ≃ Ap (r, t) − ˙Ap (r, t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Then,  we have dAp (r, t) = − ˙Ap (r, t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Substituting this result in (34) and factoring out dt we  find that the conserved quantity is  �  p  �  d3r  �  −  ∂L  ∂ ˙Ap (r, t)  ˙Ap (r, t) + L (Ap)  �  =  = −  �  p  �  d3r 1  2  �  p  �  ε (t) ˙A2  p (r, t) +  1  µ (t) (∇ × A (r, t))2  p  �  = −H  (35)  Therefore, it is found that invariance with respect to temporal translations implies that  the Hamiltonian must be a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' In time-varying media, the system is not  invariant under temporal translations, and, consequently, the Hamiltonian manifestly de-  pends on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' As shown in the previous section, taking its time derivative explicitly shows  that it is not a constant of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  Second, we assume that the variation is produced by an infinitesimal spatial displacement  η (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Then, if the system is invariant under spatial translations we must have  A′  p (r, t) = Ap (r − η, t) ≃ Ap (r, t)−η·∇Ap (r, t), so that dAp (r, t) = −η·∇Ap (r, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Again,  substituting this result into (34) and factoring out η we find that the conserved quantity  must be  P =  �  p  �  d3r  ∂L  ∂ ˙Ap (r, t)  ∇Ap (r, t) = −  �  p  �  d3r ε (t) ˙Ap (r, t) ∇Ap (r, t)  (36)  which equals the Minkowski momentum in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Therefore, we finally found that the fact  that time-varying media are invariant under spatial translations directly enforces that the  Minkowski momentum is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  13  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content='  CONCLUDING REMARKS  Symmetries play a fundamental role in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' They reduce the complexity of difficult  problems, as well as the computational cost needed to solve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Symmetries also en-  able the identification of conserved quantities and the formal link between both symmetries  and conserved quantities is Noether’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' One of the reasons why time-varying media  and/or temporal metamaterials provide a fresh view on electromagnetic theory is because  they break temporal symmetries, which are conserved in most traditional photonics systems,  while they maintain spatial symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' However, the connection between spatial and tem-  poral symmetries and the properties of time-varying media is not always explicitely stated,  or analyzed through the point of view of Lagrangian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' The present tutorial aims  at filling this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
+page_content=' Furthermore, we hope that this tutorial may clarify the subtleties of the  conservation of the electromagnetic momentum in time-varying media, the nuances of defin-  ing the momentum of the electromagnetic fields within the Abraham-Minkowski debate, and  that it will foster further research on the role and significance of symmetries in temporal  metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfpgW7/content/2301.03333v1.pdf'}
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FNFRT4oBgHgl3EQfCDcd/content/tmp_files/2301.13467v1.pdf.txt

@@ -0,0 +1,594 @@
+Research OR Education of Physics
+Revista Mexicana de Física ?? (*?*) ???–???
+MES? AÑO?
+Perfect QCD – a new Universal approach to soft QCD
+P. Christiansen
+Division of Particle Physics, Lund University, Sweden
+The ideas presented in this proceeding aims to be a first step towards a description of hadronic collisions where all soft processes are
+fundamentally strongly coupled and the same Universal strongly coupled physics drives both initial and final-state interactions.
+As it is not currently possible to derive such a picture from first principles, instead, an attempt to generalize the perfect liquid observation to
+a “perfect QCD” guiding principle is presented, focusing on implications for particle production in small systems. The first steps towards
+a microscopic model is taken by arguing that “perfect QCD” suggests that the screening in the initial state is so large that multi-parton
+interactions are of little or no importance. Instead, a target and projectile remnant is coherently excited and particle production is mainly
+driven by radiation in a qualitative similar manner as e+e− → q¯q.
+Finally, some of the possible implications of this “excited remnant model” are presented. It is argued that the time ordering of soft and hard
+physics can explain the absence of jet quenching in small systems and that the coherence scale of the projectile and target provides insights
+into what small systems will exhibit flow.
+Keywords:
+1
+Introduction
+The goal of this proceeding for the Winter Workshop 2022 is
+to present a new picture for hadronic collisions. To be pre-
+cise, the focus in this paper is only on non-diffractive inelastic
+collisions and only the soft physics1, which is expected to be
+responsible for bulk particle production. When hadronic col-
+lisions are mentioned in the following it always refers to this
+type of collision unless another type is explicitly mentioned.
+The motivation for doing this is the observation of several
+phenomena in small systems2 that has traditionally been as-
+sociated with the formation of a quark-gluon plasma (QGP)
+in large systems, see, e.g., Refs. [3,4] for an overview. These
+new phenomena can all be explained by the presence of large
+final-state interactions in small system and many excellent
+ideas have been presented for describing this with weakly
+coupled physics, see e.g., [5], but what seems to the author to
+be a fundamental flaw in these models is that a weakly cou-
+pled interaction leads to a non-vanishing mean free path so
+that the QGP-like effects will build up as the system grows
+and first dominate at a certain system size [5]. This means
+that QGP-like effects do not in a natural way extend down
+to the smallest systems, even if there is no indication in data
+of an onset [3, 4]. At the same time, a non-vanishing mean
+free path will introduce diffusion and dissipation effects that
+will supposedly modify the initial-state correlations, which
+the author is unaware of experimental evidence for, see e.g.
+C. A. Pruneau’s contribution to these proceedings [6].
+FIGURE 1. Illustration of how the initial soft scatterings are de-
+scribed in different models/pictures. Top: in Pythia, soft and hard
+interactions are modeled in the same way as leading-order perturba-
+tive processes and multiple interactions occur in most pp collisions.
+The strings forming between color charges are not shown. Bottom:
+in the “perfect QCD” picture a remnant of each nucleon is excited
+as a whole, as if there was only a single interaction, and the picture
+is therefore denoted the “excited remnant model”.
+1 Meaning that momentum transfers are small so that perturbative cal-
+culations are inaccurate. As, for example, the Pythia generator [1, 2] for
+proton-proton collisions treats all interactions perturbatively, this is not a
+unique definition but part of the motivation for exploring a completely dif-
+ferent picture in this paper.
+2 Small systems are taken to mean proton-proton, proton-nuclei and
+ultra-peripheral nuclei-nuclei collisions.
+arXiv:2301.13467v1  [hep-ph]  31 Jan 2023
+
+QQQQQQQQQ
+ooooopooooopooTarget nucleon
+Projectile nucleon
+Target "sea"
+Excited
+remnant
+Color neutral
+Projectile"sea"
+spectator remnant2
+RMF EDITORIAL TEAM
+In this paper, the decision has been to take a fresh look at
+things from the perspective offered by the new measurements
+and try to bring forth a picture that is fundamentally strongly
+coupled with a vanishing mean free path so that large final-
+state effects are present in all systems and do not introduce
+diffusion or dissipation (are essentially reversible) thereby
+hopefully preserving correlations such as those introduced by
+string breakings or similar processes. In traditional pictures,
+“soft” can have two very different meanings:
+1. The extrapolation from high-momentum transfers to
+low momentum transfer, e.g., using leading-order per-
+turbative cross sections even for situations where next-
+to-leading order correlations are large
+2. Phenomenological physics such as the Lund string
+model [7]
+The approach in this paper is to claim that point 1 does not
+work, meaning that next-to-leading order corrections distorts
+the leading-order picture, and the proposal is instead that
+“perfect QCD” is a Universal version of point 2 and can pro-
+vide guidance in that way. This means that any time where
+soft is mentioned in the text one should in principle be able
+to apply the “perfect QCD” principle.
+To help convince
+the reader that this leads to fundamentally different physics
+from that found in existing models, one of the main find-
+ings will be already discussed here and illustrated in Fig. 1.
+In pp event generators, such as Pythia [1, 2], one typically
+treats the initial stages of pp collisions as two interacting par-
+ton gases where the scattering of each parton-parton inter-
+action is motivated by perturbative (weakly coupled) QCD,
+Fig. 1 top. In the Color-Glass Condensate (CGC) model, not
+shown, one instead considers it as a weakly coupled interac-
+tion between dense gluon fields [8] that produce longitudinal
+Glasma tubes, Fig. 1. In both models the collision can involve
+one or more interactions and the number of interactions is the
+main driving mechanism of the final-state multiplicity. In the
+picture motivated in this paper, one considers a strongly cou-
+pled scenario where the color field of each projectile parton is
+neutralized by the target partons. It is argued that this results
+instead in that the remnant of the projectile and the target is
+coherently excited, corresponding essentially to a single soft
+interaction. This gives rise to two semi-independent color
+fields, Fig. 1 bottom, which would mean that most of the par-
+ticle production is driven by final-state radiation from the col-
+ored target and projectile remnants, similar to e+e− → q¯q.
+Concretely, the idea of this paper is to extend the experi-
+mental observation that the QGP behaves like a perfect liquid
+to a “perfect QCD” principle that can guide our understand-
+ing of particle production in general. The goal is not to come
+up with a full model, but to demonstrate that it is possible us-
+ing the proposed “perfect QCD” principle to obtain surpris-
+ing insights into particle production where the physics and
+the explanations for observed phenomena are very different
+from those found in existing models, such as Pythia and the
+CGC.
+2
+Perfect QCD
+One of the most remarkable discoveries of the heavy-ion pro-
+gram at RHIC and LHC is that the Quark-Gluon Plasma
+(QGP) behaves as a perfect liquid [9–15].
+The shear-
+viscosity-to-entropy density (η/s) is as low as possible [16].
+This means that the build up of flow is almost deterministic,
+which has enabled the precise measurement of fluctuations
+in the initial distribution of matter, e.g., Refs. [17,18]. At the
+Winter Workshop it was further shown how the same mini-
+mal η/s is also obtained when analyzing balance functions
+and momentum correlations, see C. A. Pruneau’s contribu-
+tion to these proceedings [6].
+The perfect nature of the liquid seems to indicate that it
+is very fundamental and since it is observed in all hadronic
+collisional systems (pp, p–Pb, and Pb–Pb collisions), see for
+example Refs. [19,20] for small systems, one could hope that
+it provides a deep insight into QCD.
+Based on the characteristics of the perfect liquid it is pro-
+posed that “perfect QCD” has to have the following two char-
+acteristics:
+• Strongly interacting
+• Minimal entropy production
+The minimal entropy production comes from the observation
+that the hydrodynamic description of the QGP is as close to
+ideal (reversible) as it can be and means that dissipation and
+diffusion can play no significant role in the description of the
+system.
+3
+The Perfect QCD Picture of Particle Produc-
+tion
+It might seem impossible to derive a microscopic picture
+from a strongly interacting soft QCD model because one
+looses the perturbative guidance but the surprise is that the
+proposed picture is extremely simple. The “perfect QCD”
+principle dictates that the entropy production during the ini-
+tial collisions should be as small as possible, yet strongly in-
+teracting, and this suggests that all that happens is the ex-
+change of a single soft gluon so that only color and essen-
+tially no momentum is exchanged. As the interacting hadrons
+are of course made up of partons, this would require that the
+screening in the initial state is so strong for the initial inter-
+actions that the soft parton-parton (and/or CGC equivalent)
+interactions are suppressed to a degree where they can be ne-
+glected. One will of course have parton-parton interactions
+for very large momentum transfers but they are not of inter-
+est here where the focus is on bulk production.
+Let us first treat the rest of the collision, ignoring pos-
+sible radiation, using the Lund string model [7], which, as
+it is derived from the confining long-range part of the QCD
+potential, is a strongly coupled model. In the Lund string
+model, strings will form between colors and anti-colors that
+Rev. Mex. Fis. ?? (*?*) (????) ???–???
+
+A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF
+3
+eventually breaks, producing hadrons uniformly in rapidity.
+In this case, two strings will form as the gluon carries both
+a color and anti-color. Let us assume that all the energy of
+each proton is carried by the color and the anti-color sys-
+tems. If both have half the energy, the total string length will
+be ≈4(ybeam−log 2) while if one color (or anti-color) has all
+the energy one can supposedly form a string of length 2ybeam
+(this must be the minimal length for the color field to stretch
+between the target and projectile). As the average number of
+particles produced a by a string is proportional to the string
+length [7], the “perfect QCD” principle tells us that nature
+will take the 2nd solution. This means that instead of hav-
+ing two remnants with a similar amount of energy, one will
+have a “valence”-like remnant with almost all the energy and
+a “sea”-like remnant with almost no energy. This is reminis-
+cent of the BGK picture [21], and so it is naturally to propose
+that the “valence” remnant in one proton is color-coupled to
+the “sea” remnant in the other proton, and vice versa, so that
+one in some sense has two semi-independent systems carry-
+ing approximately half the total initial energy each.
+Let us finally try to give a partonic picture of how the “per-
+fect QCD” picture can be understood. As the two nucleon
+penetrate at high energy the partons inside them are interact-
+ing strongly but the claim is that they interact in a way that
+screens the partonic interactions. However, this screening can
+only happen in a certain regime. If x denotes the usual four
+momentum fraction then one can maximally “organize” the
+nucleon into n ≈ 1/x constituents. Screening will be impos-
+sible when the four momentum transfer, Q2, is very large be-
+cause one can resolve individual partons (the hard scattering
+limit), or when x is large so that the number of constituents is
+small. The latter argument is why nucleon remnants will be
+excited as a whole.
+In the current picture, dNch/dη at η = 0 would be inde-
+pendent of √s as all the energy will go to extend the strings
+in rapidity. What has been ignored is radiation: the color
+charge carrying most of the energy is, as QCD is strongly in-
+teracting, very likely to emit soft or collinear radiation. How
+to calculate this radiation is not trivial, but one can at least
+note that one qualitatively get a system very similar to what
+one has for e+e− → q¯q (denoted e+e− in the following).
+Comparing particle production in e+e− collisions to that of
+pp collisions, one finds that the former produces more parti-
+cles on the average [11]. The common understanding is that
+it is possible for part of the proton to escape as a color neutral
+object, taking away around 50% of the energy [22]. Based
+on the observed particle production in e+e−, it is concluded
+that there is no fundamental reason one should not be able to
+create the observed particle production via radiation also in
+pp, pA, and AA collisions.
+To recap, the general microscopic “perfect QCD” pic-
+ture of pp, pA, and AA collisions will be that the soft initial
+interactions will excite a remnant of each nucleon in a “pro-
+jectile” coherently and that the main particle production at
+high energy collisions is driven by final-state radiation. For
+this reason the picture will be denoted the “excited remnant
+model”. This might sound like the Dual Parton Model but
+it is important to note that the Dual Parton Model contains
+MPIs [23].
+In the limit that particle production is dominated by radi-
+ation, the color-connections to the “sea” systems in the “tar-
+get” can be ignored and one can therefore factorize the soft
+particle production into Npart semi-independent terms. Semi-
+independent, because there must be some dependence on the
+nucleon-nucleon impact parameter to explain the slightly in-
+creased particle production per participant in AA collisions.
+3.1
+An Illustration of Particle Production in pp Colli-
+sions
+5
+−
+4
+−
+3
+−
+2
+−
+1
+−
+0
+η
+ 
+0
+2
+4
+6
+8
+10
+η
+/d
+ch
+N
+ d
+ n
+≤
+52 
+ 50
+≤
+ n 
+≤
+42 
+ 40
+≤
+ n 
+≤
+32 
+ 30
+≤
+ n 
+≤
+22 
+ 20
+≤
+ n 
+≤
+12 
+ 10
+≤
+ n 
+≤
+2 
+NSD
+FIGURE 2. dNch/dη measured in √s = 200 GeV/c pp collisions by
+UA5 for NSD events and for events with different final-state charged
+particle multiplicities, n. The data have been read off from the pub-
+lished figures [24]. As the figure is just meant to illustrate a trend,
+the statistical uncertainties have not been included for clarity.
+The main goal here is to discuss small systems. In these sys-
+tems, e.g., pp collisions, the full “perfect QCD” picture of a
+collision is:
+1. the initial interactions produce up to three semi-
+independent systems:
+• coherently excited target and projectile remnants
+• possible color-neutral target and projectile rem-
+nants that act as spectators (escape with energy
+along beam direction)
+• possible hard parton-parton scatterings
+2. the excited remnants radiate gluons
+3. the color fields decay into partons
+Rev. Mex. Fis. ?? (*?*) (????) ???–???
+
+4
+RMF EDITORIAL TEAM
+4. final-state partonic interactions: flow, strangeness en-
+hancement
+5. hadronization
+6. possible final-state hadronic rescattering
+One could in principle try to implement a generator along
+these lines but the goal here is to illustrate the picture using
+UA5 data [24]. Fig. 2 shows the dNch/dη measured by UA5
+for NSD events as well as for multiplicity selected events. In
+low-multiplicity events, dNch/dη is flat as one would expect
+for a single long string. As the multiplicity grows, one ob-
+serves a narrowing of dNch/dη, which in the perfect QCD
+picture should be caused by the radiation adding shorter and
+shorter (less energetic) strings. In this way the “excited rem-
+nant model” is at least qualitatively consistent with the ob-
+served trends by UA5.
+4
+Insights and Predictions for Small Systems
+In this section, the hope is to demonstrate for the reader that
+the perfect-QCD picture of particle production can provide
+many new insights and predictions.
+4.1
+A Simple Explanation for the Absence of Jet
+Quenching in Small Systems
+One can immediately notice that, if the time scales involved
+with the hard interactions are shorter than the formation time
+for step 2 (“the excited remnants radiate gluons”) as one
+would imagine from the scales of the momentum transfers
+involved, then one can understand why there is no jet quench-
+ing in small systems even if there is a relation between flow
+and jet quenching in a large system. The medium simply has
+not been produced yet when the jet propagates. This seems
+very attractive to the author as this is in line with experimental
+findings, see, for example, Ref. [25], and it is hard to explain
+in most existing models.
+4.2
+Flow in pp Collisions ≫ Flow in e+e− Collisions
+It should be clear from the way the “excited remnant model”
+work that it “postdicts” that the particle production in pp col-
+lisions and e+e− collisions should be very similar because
+in this model, and unlike traditional MPI-based models, the
+growth with √s is in both cases driven by radiation. Indeed
+this surprising similarity have been noted and discussed much
+in the past by experimental collaborations [11, 26], even it
+was never theoretically understood.
+It can therefore be surprising that while one observes
+strong flow in pp collisions, one does not observe it for e+e−
+collisions [27]. However, there could be a simple explana-
+tion for that. As the “excited remnant model” postulates that
+for each nucleon a single “valence” remnant is excited as a
+whole, then it is clear that the radiation in step 2 will have
+to have very low transverse momentum, pT < 1/R, where R
+is the size of the excited remnant. As the pT is so low, the
+color fields will have to stack and so one will naturally get a
+quite dense system of parallel color fields with a large energy
+density. In the “perfect QCD” picture these color fields will
+be strongly interacting and so they will immediately start to
+build up collective flow. This makes a big difference when
+comparing to e+e−, where all the energy is located with a
+single parton and so the radiated gluons can and will typi-
+cally have very large pT. This means that most energy will be
+radiated away from the initial color field and so there is little
+time where system is dense and can build up collective flow.
+4.3
+How to Control Flow in Ultra-Small Systems
+In the previous subsection it was argued that for small sys-
+tems, the size of the excited remnant determines the flow that
+can be built up in the final system. This then is naturally in
+line with the observation of flow in Ultra-Peripheral Colli-
+sions (UPCs), where the photon field of one nuclei interacts
+with the other nuclei, because in this case the photon field
+has a long wavelength since it is emitted coherently by the
+protons in the nuclei. Recall that photons can interact as a
+“hadronic” system by fluctuating into a q¯q pair, which will
+have a size that reflects the photon four momentum (Q2). AT-
+LAS has observed flow in UPC Pb–Pb events [28] and CMS
+has reported non-zero v2{2} in p–Pb events [29], which is in
+line with the ideas presented here.
+By going to electron-proton or electron-ion collisions one
+can in principle measure the wavelength of the photon from
+the change in electron four momentum. One can in this way
+select different sizes of the excited remnants and if the pic-
+ture is true, control the pT radiation and switch on (low Q2)
+and off (high Q2) flow. ZEUS and H1 has reanalyzed old
+data both for low and high Q2 but neither ZEUS [30, 31]
+nor H1 [32] observes any signatures of collective flow. This
+clearly goes against the ideas presented here. However, it
+seems that if there is flow in UPCs at LHC then there would
+also likely be flow in low Q2 ep collisions at HERA and
+vice verse. On the other hand, one knows that flow in small
+systems is very hard to detect. Looking from the outside, it
+would be good if one could resolve the situation so that one is
+as certain as possible that similar procedures have been used
+before one concludes too strongly on the current results.
+5
+Conclusions
+An attempt to generalize the perfect-liquid nature from flow
+to particle production has been presented.
+The “perfect
+QCD” principle has been proposed to be a Universal principle
+for soft QCD that applies both in the initial and final state of
+hadronic collisions. Using the idea of minimal entropy pro-
+duction, a microscopic picture, the “excited remnant model”,
+has been presented. In the microscopic picture, the screening
+as the two hadronic systems penetrate is so large that sub-
+collisions between constituents does not occur, in contrast to
+most existing pictures, e.g., MPI and CGC based ones.
+Rev. Mex. Fis. ?? (*?*) (????) ???–???
+
+A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF
+5
+No attempt has been done to prove the “perfect QCD”
+principle in this paper but several surprising insights have
+been provided, such as simple arguments for why jet quench-
+ing is absent in small systems and which collisional systems
+will exhibit flow. The hope is that the principle can be used to
+provide novel insights into a wide range of topics, for exam-
+ple, jet quenching in large systems and the relation between
+diffractive and non-diffractive physics.
+6
+Acknowledgements
+The author would like to thank Adrian Nassirpour for
+many valuable comments on earlier versions of similar
+manuscripts.
+7
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+

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+page_content='Research OR Education of Physics  Revista Mexicana de Física ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' (*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
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+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  MES?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' AÑO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Perfect QCD – a new Universal approach to soft QCD  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Christiansen  Division of Particle Physics, Lund University, Sweden  The ideas presented in this proceeding aims to be a first step towards a description of hadronic collisions where all soft processes are  fundamentally strongly coupled and the same Universal strongly coupled physics drives both initial and final-state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  As it is not currently possible to derive such a picture from first principles, instead, an attempt to generalize the perfect liquid observation to  a “perfect QCD” guiding principle is presented, focusing on implications for particle production in small systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The first steps towards  a microscopic model is taken by arguing that “perfect QCD” suggests that the screening in the initial state is so large that multi-parton  interactions are of little or no importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Instead, a target and projectile remnant is coherently excited and particle production is mainly  driven by radiation in a qualitative similar manner as e+e− → q¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Finally, some of the possible implications of this “excited remnant model” are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' It is argued that the time ordering of soft and hard  physics can explain the absence of jet quenching in small systems and that the coherence scale of the projectile and target provides insights  into what small systems will exhibit flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Keywords:  1  Introduction  The goal of this proceeding for the Winter Workshop 2022 is  to present a new picture for hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' To be pre-  cise, the focus in this paper is only on non-diffractive inelastic  collisions and only the soft physics1, which is expected to be  responsible for bulk particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' When hadronic col-  lisions are mentioned in the following it always refers to this  type of collision unless another type is explicitly mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  The motivation for doing this is the observation of several  phenomena in small systems2 that has traditionally been as-  sociated with the formation of a quark-gluon plasma (QGP)  in large systems, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' [3,4] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' These  new phenomena can all be explained by the presence of large  final-state interactions in small system and many excellent  ideas have been presented for describing this with weakly  coupled physics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', [5], but what seems to the author to  be a fundamental flaw in these models is that a weakly cou-  pled interaction leads to a non-vanishing mean free path so  that the QGP-like effects will build up as the system grows  and first dominate at a certain system size [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This means  that QGP-like effects do not in a natural way extend down  to the smallest systems, even if there is no indication in data  of an onset [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' At the same time, a non-vanishing mean  free path will introduce diffusion and dissipation effects that  will supposedly modify the initial-state correlations, which  the author is unaware of experimental evidence for, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Pruneau’s contribution to these proceedings [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Illustration of how the initial soft scatterings are de-  scribed in different models/pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Top: in Pythia, soft and hard  interactions are modeled in the same way as leading-order perturba-  tive processes and multiple interactions occur in most pp collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  The strings forming between color charges are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Bottom:  in the “perfect QCD” picture a remnant of each nucleon is excited  as a whole, as if there was only a single interaction, and the picture  is therefore denoted the “excited remnant model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  1 Meaning that momentum transfers are small so that perturbative cal-  culations are inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As, for example, the Pythia generator [1, 2] for  proton-proton collisions treats all interactions perturbatively, this is not a  unique definition but part of the motivation for exploring a completely dif-  ferent picture in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  2 Small systems are taken to mean proton-proton, proton-nuclei and  ultra-peripheral nuclei-nuclei collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='13467v1  [hep-ph]  31 Jan 2023  QQQQQQQQQ  ooooopooooopooTarget nucleon  Projectile nucleon  Target "sea"  Excited  remnant  Color neutral  Projectile"sea"  spectator remnant2  RMF EDITORIAL TEAM  In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' the decision has been to take a fresh look at  things from the perspective offered by the new measurements  and try to bring forth a picture that is fundamentally strongly  coupled with a vanishing mean free path so that large final-  state effects are present in all systems and do not introduce  diffusion or dissipation (are essentially reversible) thereby  hopefully preserving correlations such as those introduced by  string breakings or similar processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In traditional pictures,  “soft” can have two very different meanings:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The extrapolation from high-momentum transfers to  low momentum transfer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', using leading-order per-  turbative cross sections even for situations where next-  to-leading order correlations are large  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Phenomenological physics such as the Lund string  model [7]  The approach in this paper is to claim that point 1 does not  work, meaning that next-to-leading order corrections distorts  the leading-order picture, and the proposal is instead that  “perfect QCD” is a Universal version of point 2 and can pro-  vide guidance in that way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This means that any time where  soft is mentioned in the text one should in principle be able  to apply the “perfect QCD” principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  To help convince  the reader that this leads to fundamentally different physics  from that found in existing models, one of the main find-  ings will be already discussed here and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  In pp event generators, such as Pythia [1, 2], one typically  treats the initial stages of pp collisions as two interacting par-  ton gases where the scattering of each parton-parton inter-  action is motivated by perturbative (weakly coupled) QCD,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' 1 top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In the Color-Glass Condensate (CGC) model, not  shown, one instead considers it as a weakly coupled interac-  tion between dense gluon fields [8] that produce longitudinal  Glasma tubes, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In both models the collision can involve  one or more interactions and the number of interactions is the  main driving mechanism of the final-state multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In the  picture motivated in this paper, one considers a strongly cou-  pled scenario where the color field of each projectile parton is  neutralized by the target partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' It is argued that this results  instead in that the remnant of the projectile and the target is  coherently excited, corresponding essentially to a single soft  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This gives rise to two semi-independent color  fields, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' 1 bottom, which would mean that most of the par-  ticle production is driven by final-state radiation from the col-  ored target and projectile remnants, similar to e+e− → q¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Concretely, the idea of this paper is to extend the experi-  mental observation that the QGP behaves like a perfect liquid  to a “perfect QCD” principle that can guide our understand-  ing of particle production in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The goal is not to come  up with a full model, but to demonstrate that it is possible us-  ing the proposed “perfect QCD” principle to obtain surpris-  ing insights into particle production where the physics and  the explanations for observed phenomena are very different  from those found in existing models, such as Pythia and the  CGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  2  Perfect QCD  One of the most remarkable discoveries of the heavy-ion pro-  gram at RHIC and LHC is that the Quark-Gluon Plasma  (QGP) behaves as a perfect liquid [9–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  The shear-  viscosity-to-entropy density (η/s) is as low as possible [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  This means that the build up of flow is almost deterministic,  which has enabled the precise measurement of fluctuations  in the initial distribution of matter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' At the  Winter Workshop it was further shown how the same mini-  mal η/s is also obtained when analyzing balance functions  and momentum correlations, see C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Pruneau’s contribu-  tion to these proceedings [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  The perfect nature of the liquid seems to indicate that it  is very fundamental and since it is observed in all hadronic  collisional systems (pp, p–Pb, and Pb–Pb collisions), see for  example Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' [19,20] for small systems, one could hope that  it provides a deep insight into QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Based on the characteristics of the perfect liquid it is pro-  posed that “perfect QCD” has to have the following two char-  acteristics:  Strongly interacting  Minimal entropy production  The minimal entropy production comes from the observation  that the hydrodynamic description of the QGP is as close to  ideal (reversible) as it can be and means that dissipation and  diffusion can play no significant role in the description of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  3  The Perfect QCD Picture of Particle Produc-  tion  It might seem impossible to derive a microscopic picture  from a strongly interacting soft QCD model because one  looses the perturbative guidance but the surprise is that the  proposed picture is extremely simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The “perfect QCD”  principle dictates that the entropy production during the ini-  tial collisions should be as small as possible, yet strongly in-  teracting, and this suggests that all that happens is the ex-  change of a single soft gluon so that only color and essen-  tially no momentum is exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the interacting hadrons  are of course made up of partons, this would require that the  screening in the initial state is so strong for the initial inter-  actions that the soft parton-parton (and/or CGC equivalent)  interactions are suppressed to a degree where they can be ne-  glected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' One will of course have parton-parton interactions  for very large momentum transfers but they are not of inter-  est here where the focus is on bulk production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Let us first treat the rest of the collision, ignoring pos-  sible radiation, using the Lund string model [7], which, as  it is derived from the confining long-range part of the QCD  potential, is a strongly coupled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In the Lund string  model, strings will form between colors and anti-colors that  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' (*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' *) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=') ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF  3  eventually breaks, producing hadrons uniformly in rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  In this case, two strings will form as the gluon carries both  a color and anti-color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Let us assume that all the energy of  each proton is carried by the color and the anti-color sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' If both have half the energy, the total string length will  be ≈4(ybeam−log 2) while if one color (or anti-color) has all  the energy one can supposedly form a string of length 2ybeam  (this must be the minimal length for the color field to stretch  between the target and projectile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the average number of  particles produced a by a string is proportional to the string  length [7], the “perfect QCD” principle tells us that nature  will take the 2nd solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This means that instead of hav-  ing two remnants with a similar amount of energy, one will  have a “valence”-like remnant with almost all the energy and  a “sea”-like remnant with almost no energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This is reminis-  cent of the BGK picture [21], and so it is naturally to propose  that the “valence” remnant in one proton is color-coupled to  the “sea” remnant in the other proton, and vice versa, so that  one in some sense has two semi-independent systems carry-  ing approximately half the total initial energy each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Let us finally try to give a partonic picture of how the “per-  fect QCD” picture can be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the two nucleon  penetrate at high energy the partons inside them are interact-  ing strongly but the claim is that they interact in a way that  screens the partonic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' However, this screening can  only happen in a certain regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' If x denotes the usual four  momentum fraction then one can maximally “organize” the  nucleon into n ≈ 1/x constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Screening will be impos-  sible when the four momentum transfer, Q2, is very large be-  cause one can resolve individual partons (the hard scattering  limit), or when x is large so that the number of constituents is  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The latter argument is why nucleon remnants will be  excited as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  In the current picture, dNch/dη at η = 0 would be inde-  pendent of √s as all the energy will go to extend the strings  in rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' What has been ignored is radiation: the color  charge carrying most of the energy is, as QCD is strongly in-  teracting, very likely to emit soft or collinear radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' How  to calculate this radiation is not trivial, but one can at least  note that one qualitatively get a system very similar to what  one has for e+e− → q¯q (denoted e+e− in the following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Comparing particle production in e+e− collisions to that of  pp collisions, one finds that the former produces more parti-  cles on the average [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The common understanding is that  it is possible for part of the proton to escape as a color neutral  object, taking away around 50% of the energy [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Based  on the observed particle production in e+e−, it is concluded  that there is no fundamental reason one should not be able to  create the observed particle production via radiation also in  pp, pA, and AA collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  To recap, the general microscopic “perfect QCD” pic-  ture of pp, pA, and AA collisions will be that the soft initial  interactions will excite a remnant of each nucleon in a “pro-  jectile” coherently and that the main particle production at  high energy collisions is driven by final-state radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' For  this reason the picture will be denoted the “excited remnant  model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This might sound like the Dual Parton Model but  it is important to note that the Dual Parton Model contains  MPIs [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  In the limit that particle production is dominated by radi-  ation, the color-connections to the “sea” systems in the “tar-  get” can be ignored and one can therefore factorize the soft  particle production into Npart semi-independent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Semi-  independent, because there must be some dependence on the  nucleon-nucleon impact parameter to explain the slightly in-  creased particle production per participant in AA collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='1  An Illustration of Particle Production in pp Colli-  sions  5  −  4  −  3  −  2  −  1  −  0  η  0  2  4  6  8  10  η  /d  ch  N  d  n  ≤  52  50  ≤  n  ≤  42  40  ≤  n  ≤  32  30  ≤  n  ≤  22  20  ≤  n  ≤  12  10  ≤  n  ≤  2  NSD  FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' dNch/dη measured in √s = 200 GeV/c pp collisions by  UA5 for NSD events and for events with different final-state charged  particle multiplicities, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The data have been read off from the pub-  lished figures [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the figure is just meant to illustrate a trend,  the statistical uncertainties have not been included for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  The main goal here is to discuss small systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In these sys-  tems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', pp collisions, the full “perfect QCD” picture of a  collision is:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' the initial interactions produce up to three semi-  independent systems:  coherently excited target and projectile remnants  possible color-neutral target and projectile rem-  nants that act as spectators (escape with energy  along beam direction)  possible hard parton-parton scatterings  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' the excited remnants radiate gluons  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' the color fields decay into partons  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' (*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' *) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=') ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  4  RMF EDITORIAL TEAM  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' final-state partonic interactions: flow, strangeness en-  hancement  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' hadronization  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' possible final-state hadronic rescattering  One could in principle try to implement a generator along  these lines but the goal here is to illustrate the picture using  UA5 data [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' 2 shows the dNch/dη measured by UA5  for NSD events as well as for multiplicity selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In  low-multiplicity events, dNch/dη is flat as one would expect  for a single long string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the multiplicity grows, one ob-  serves a narrowing of dNch/dη, which in the perfect QCD  picture should be caused by the radiation adding shorter and  shorter (less energetic) strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In this way the “excited rem-  nant model” is at least qualitatively consistent with the ob-  served trends by UA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  4  Insights and Predictions for Small Systems  In this section, the hope is to demonstrate for the reader that  the perfect-QCD picture of particle production can provide  many new insights and predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='1  A Simple Explanation for the Absence of Jet  Quenching in Small Systems  One can immediately notice that, if the time scales involved  with the hard interactions are shorter than the formation time  for step 2 (“the excited remnants radiate gluons”) as one  would imagine from the scales of the momentum transfers  involved, then one can understand why there is no jet quench-  ing in small systems even if there is a relation between flow  and jet quenching in a large system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The medium simply has  not been produced yet when the jet propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This seems  very attractive to the author as this is in line with experimental  findings, see, for example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' [25], and it is hard to explain  in most existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='2  Flow in pp Collisions ≫ Flow in e+e− Collisions  It should be clear from the way the “excited remnant model”  work that it “postdicts” that the particle production in pp col-  lisions and e+e− collisions should be very similar because  in this model, and unlike traditional MPI-based models, the  growth with √s is in both cases driven by radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Indeed  this surprising similarity have been noted and discussed much  in the past by experimental collaborations [11, 26], even it  was never theoretically understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  It can therefore be surprising that while one observes  strong flow in pp collisions, one does not observe it for e+e−  collisions [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' However, there could be a simple explana-  tion for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the “excited remnant model” postulates that  for each nucleon a single “valence” remnant is excited as a  whole, then it is clear that the radiation in step 2 will have  to have very low transverse momentum, pT < 1/R, where R  is the size of the excited remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' As the pT is so low, the  color fields will have to stack and so one will naturally get a  quite dense system of parallel color fields with a large energy  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In the “perfect QCD” picture these color fields will  be strongly interacting and so they will immediately start to  build up collective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This makes a big difference when  comparing to e+e−, where all the energy is located with a  single parton and so the radiated gluons can and will typi-  cally have very large pT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This means that most energy will be  radiated away from the initial color field and so there is little  time where system is dense and can build up collective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='3  How to Control Flow in Ultra-Small Systems  In the previous subsection it was argued that for small sys-  tems, the size of the excited remnant determines the flow that  can be built up in the final system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This then is naturally in  line with the observation of flow in Ultra-Peripheral Colli-  sions (UPCs), where the photon field of one nuclei interacts  with the other nuclei, because in this case the photon field  has a long wavelength since it is emitted coherently by the  protons in the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Recall that photons can interact as a  “hadronic” system by fluctuating into a q¯q pair, which will  have a size that reflects the photon four momentum (Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' AT-  LAS has observed flow in UPC Pb–Pb events [28] and CMS  has reported non-zero v2{2} in p–Pb events [29], which is in  line with the ideas presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  By going to electron-proton or electron-ion collisions one  can in principle measure the wavelength of the photon from  the change in electron four momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' One can in this way  select different sizes of the excited remnants and if the pic-  ture is true, control the pT radiation and switch on (low Q2)  and off (high Q2) flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ZEUS and H1 has reanalyzed old  data both for low and high Q2 but neither ZEUS [30, 31]  nor H1 [32] observes any signatures of collective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' This  clearly goes against the ideas presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' However, it  seems that if there is flow in UPCs at LHC then there would  also likely be flow in low Q2 ep collisions at HERA and  vice verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' On the other hand, one knows that flow in small  systems is very hard to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Looking from the outside, it  would be good if one could resolve the situation so that one is  as certain as possible that similar procedures have been used  before one concludes too strongly on the current results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  5  Conclusions  An attempt to generalize the perfect-liquid nature from flow  to particle production has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  The “perfect  QCD” principle has been proposed to be a Universal principle  for soft QCD that applies both in the initial and final state of  hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Using the idea of minimal entropy pro-  duction, a microscopic picture, the “excited remnant model”,  has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' In the microscopic picture, the screening  as the two hadronic systems penetrate is so large that sub-  collisions between constituents does not occur, in contrast to  most existing pictures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', MPI and CGC based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' (*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' *) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=') ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  A LATEXTEMPLATE FOR THE RMF, RMF-E, SRMF  5  No attempt has been done to prove the “perfect QCD”  principle in this paper but several surprising insights have  been provided, such as simple arguments for why jet quench-  ing is absent in small systems and which collisional systems  will exhibit flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' The hope is that the principle can be used to  provide novel insights into a wide range of topics, for exam-  ple, jet quenching in large systems and the relation between  diffractive and non-diffractive physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  6  Acknowledgements  The author would like to thank Adrian Nassirpour for  many valuable comments on earlier versions of similar  manuscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  7  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Sjöstrand, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=', An Introduction to PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='2, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' 191 (2015) 159, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='cpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='024  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Sjöstrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Mrenna, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Skands, PYTHIA 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='4 Physics  and Manual, JHEP 05 (2006) 026, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='1088/1126-6708/  2006/05/026  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Loizides, Experimental overview on small collision systems  at the LHC, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' A 956 (2016) 200, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  nuclphysa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='022  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Nagle  and  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content='  Zajc,  Small  System  Col-  lectivity in Relativistic Hadronic and Nuclear Collisions,  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfCDcd/content/2301.13467v1.pdf'}
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+A Framework for Large Scale Particle Filters Validated
+with Data Assimilation for Weather Simulation
+Sebastian Friedemanna, Kai Kellerb, Yen-Sen Luc, Bruno Raffina,∗,
+Leonardo Bautista-Gomezb
+aUniv. Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000, Grenoble, France
+bBarcelona Supercomputing Center, Barcelona, 08034, Spain
+cForschungzentrum Juelich, Juelich, 52428, Germany
+Abstract
+Particle filters are a group of algorithms to solve inverse problems through
+statistical Bayesian methods when the model does not comply with the lin-
+ear and Gaussian hypothesis. Particle filters are used in domains like data
+assimilation, probabilistic programming, neural network optimization, local-
+ization and navigation. Particle filters estimate the probability distribution
+of model states by running a large number of model instances, the so called
+particles. The ability to handle a very large number of particles is critical
+for high dimensional models. This paper proposes a novel paradigm to run
+very large ensembles of parallel model instances on supercomputers. The
+approach combines an elastic and fault tolerant runner/server model min-
+imizing data movements while enabling dynamic load balancing. Particle
+weights are computed locally on each runner and transmitted when available
+to a server that normalizes them, resamples new particles based on their
+weight, and redistributes dynamically the work to runners to react to load
+imbalance. Our approach relies on a an asynchronously managed distributed
+particle cache permitting particles to move from one runner to another in the
+background while particle propagation goes on. This also enables the num-
+ber of runners to vary during the execution either in reaction to failures and
+∗Corresponding author
+Email addresses: [email protected] (Sebastian Friedemann),
+[email protected] (Kai Keller), [email protected] (Yen-Sen Lu),
+[email protected] (Bruno Raffin), [email protected] (Leonardo
+Bautista-Gomez)
+arXiv:2301.02668v1  [cs.DC]  6 Jan 2023
+
+restarts, or to adapt to changing resource availability dictated by external de-
+cision processes. The approach is experimented with the Weather Research
+and Forecasting (WRF) model, to assess its performance for probabilistic
+weather forecasting. Up to 2,555 particles on 20,442 compute cores are used
+to assimilate cloud cover observations into short–range weather forecasts over
+Europe.
+Keywords:
+ls Data Assimilation, Particle Filter, Ensemble Run, Resilience,
+Elasticity
+1. introduction
+Given an output and a transformation function, finding the input states
+represents a so called inverse problem. A wide range of approaches to ad-
+dress this central problem exist. Statistical Bayesian methods stand out as
+they provide uncertainty measures of the proposed input in form of probabil-
+ity density functions. In this paper, we consider particle filters, a statistical
+Bayesian method combining uncertainties of both the dynamical system and
+observations, to estimate the system state. Several realizations of the dy-
+namical system, called particles, with differently perturbed internal states,
+are propagated up to a time where observation data are available. These
+particles are then weighted corresponding to their distance to the obser-
+vations. The weights are used to generate a new sample of particles that
+better matches the observations. This process repeats while observations are
+available.
+Particle filters are used for several purposes, like Data Assimilation (DA) [1],
+probabilistic programming [2, 3, 4], neural network optimization [5], local-
+ization and navigation[6]. Particle filters stands by their ability to work with
+nonlinear and/or non-Gaussian state space models in opposition to technics
+like Ensemble Kalman Filtering (EnKF). But this ability comes with a need
+to run larger number of particles. If the dynamical system is an advanced
+parallel high-dimensional numerical model solver, as for geoscience applica-
+tions, thousands of particles may be necessary to avoid undersampling and
+degeneracy. While high-dimensional large-scale solvers are compute intense
+already, the execution of several thousands of instances adds orders of magni-
+tude of calculations. Large scale DA with particle filters is for instance used
+for geoscience applications such as weather forecasting [7]. Supercomputers,
+reaching today Exascale, have the compute power to support very large scale
+2
+
+particle filters. But using such resources efficiently, time and energy wise, is
+challenging. Applications need to limit the use of the Parallel File System
+(PFS), a classical supercomputer bottleneck, and favor instead in situ data
+processing as well as local data storage to reduce data movements, asyn-
+chronism to overlap tasks whenever possible. Applications also need to be
+flexible to adapt to changes during execution, requiring support for resilience,
+elasticity and dynamics load balancing.
+Existing large scale approaches can be divided into two types: online
+and offline approaches. Offline approaches use temporary files to exchange
+data. To propagate one particle, one model instance starts, loads the particle
+from a file, propagates it up to a given time, stores the resulting particle back
+to a file and shuts down. This approach is flexible, fault tolerance is easy
+to support, but performance, especially at scale is impaired by the heavy
+use of the file system and the cost of starting a new model instance for each
+propagation. Online approaches bypass the file system by running a large
+MPI application that encompasses the full workflow, where the particles are
+distributed to the different model instances through the network via MPI
+communications. While saving I/O overheads, this approach loses flexibility.
+For instance, a fault during a single particle propagation stops the entire
+application.
+Thus existing online approaches, as will be detailled in the
+related work section (Section 6), usually do not support fault tolerance or
+dynamic load balancing.
+In this paper we develop an alternate approach that leads to a high ef-
+ficient yet flexible framework. The key to achieve this goal is the virtual-
+ization of particle propagations. We turn a numerical model solver instance
+into a runner capable of propagating several particles one after the other
+with low overheads and idle times. Each runner is coupled with a node-local
+distributed state cache enabling fast loads and stores of particles. The caches
+are asynchronously persisted to the file system for checkpointing and load bal-
+ancing between runners. Asynchronous prefetching of particles into the cache
+enables overlapping particle loads with the particle propagation. A server
+organizes the work distribution to the runners and performs the centralized
+tasks of the particle filter update and (re-)sampling. Runners and server are
+each executed as independent executables to support elasticity and facilitate
+fault tolerance.
+The association of these different features complemented
+with a fault tolerance protocol, leads to an elastic and resilient framework,
+minimizing data movements while enabling dynamic load balancing. Parti-
+cle virtualization enables to decouple resource allocation from the number
+3
+
+Figure 1: Initially particles are uniformly sampled. They are propagated to T1 where
+they are weighted taking into account observation data. Resampling leads to discard some
+particles with low weights (top and bottom), while others with high weights become parent
+of several ones (3 here).
+of particles. The number of runners can vary during the execution either
+in reaction to failures and restarts, or to adapt to changing resource avail-
+ability dictated by external decision processes. The proposed architecture
+is designed for running at extreme scale, leveraging deep storage hierarchies
+and heterogeneous cluster designs of current and future supercomputers.
+We strain our proposed particle filter framework with a realistic use-case,
+interfacing with the Weather Research and Forecasting (WRF, version 3.7.1)
+model [8]. WRF is a widely used weather model for operational forecasting
+and research. Using our particle filter, we are able to run 2,555 particles on
+20,442 compute cores for WRF simulations on a European domain with 87 %
+efficiency.
+The rest of the paper is structured as follows: Section 2 reviews the
+principles of particle filters and the associated workflow. Section 3 presents
+the architecture of our proposed approach, while Section 4 is dedicated to
+experiments and Section 5 to discussion. The papers ends with related work
+in Section 6 and a conclusion in Section 7.
+2. Particle Filters
+In this section, we give a brief introduction on the particle filter formalism,
+focusing on properties that we exploit in our proposal. For a comprehensive
+4
+
+weighting
+resampling
+weighting
+initial
+To
+T1
+Ti
+T2
+Assimilation Cycle
+Assimilation Cycleintroduction, we refer to [1, 9]. Let M be a numerical model, that propagates
+a particle p from state xp,t−1 at time t − 1 to state xp,t at time t:
+xp,t = M(xp,t−1) + βt
+1
+Where β is a random forcing representing errors in the model. Let H be the
+projection operator from the state space to the observation space:
+y = H(x) + ϵt
+2
+Where ϵ is a random vector, representing the measurement errors.
+The bootstrap particle filter formalism can be derived using Bayes’ the-
+orem:
+p(xt|yt) = p(yt|xt) p(xt)
+p(yt)
+3
+Where p(xt|yt) is the posterior Probability Density Function (PDF), p(xt) is
+the prior PDF, p(yt|xt) is the likelihood of observing yt if xt would represent
+the true state, and p(yt) is the evidence available. The goal of the filtering
+formalism is to derive the posterior p(xt|yt), which describes the PDF of the
+state xt taking into account the evidence yt.
+In the bootstrap particle filter, the prior p(xt) is estimated via sampling
+an ensemble of P particles xp,t representing different model states
+p(xt) = 1
+P
+P−1
+�
+p=0
+δ(xt − xp,t),
+4
+The likelihood p(yt|xt) is assumed to be known, estimated when calibrating
+the sensor. It is derived from the PDF of ϵ applied to the distance between
+state and observation yt − H(xt):
+p(yt|xt) = pϵ(yt − H(xt))
+5
+The evidence p(yt) can be computed by:
+p(yt) =
+�
+p(yt|xt)p(xt) dxt
+6
+=
+P−1
+�
+p=0
+1
+P p(yt|xp,t)
+7
+8
+5
+
+Putting all together and replacing the expressions in Bayes’ theorem (Equa-
+tion 3) we arrive to the expression for the posterior [1]:
+p(xt|yt) ≈
+P−1
+�
+p=0
+ˆwp,t δ(xt − xp,t)
+9
+With ˆwp,t being the normalized particle weights:
+ˆwp,t =
+p(yt|xp,t)
+�P−1
+q=0 p(yt|xq,t)
+=
+wp,t
+�P−1
+q=0 wq,t
+10
+and wp,t being the unnormalized particle weights:
+wp,t = p(yt|xp,t) wp,t−1
+11
+Note that the initial weights are set equal to wp,0 = 1/P.
+Especially for high dimensional models, particle filters tend to suffer from
+weight degeneration, i.e., one normalized weight is close to one and all the
+others are close to zero. A classical approach against ensemble degeneration
+is Sequential Importance Resampling (SIR) [10, 11]. The procedure of SIR
+consists in resampling particles from the posterior (Equation 9) at the end of
+the propagation step; P particles are randomly drawn, resampled, from the
+existing particles, each with a probability wp,t. Low weighted particles be-
+come discarded, while high weighted particles can become the starting point
+of multiple particle propagations (Figure 1). More precisely, the resampling
+leads to the multiset P defined by the ordered pair (Q, α). Where Q is the
+set of unique particles q in P, and αq the number of the occurrences of q in
+P. The particles q are hereinafter called parent particles.
+The resampled particles are all assigned the same weight of wp,t = 1/P
+again. Particles departing from the same parent may need to become stochas-
+tically perturbed if the model does not contain a stochastic component itself.
+Otherwise, the trajectories of those particles would be identical.
+Different flavors of SIR and resampling algorithms, like Residual Resam-
+pling, exist [12]. Some perform a resampling step after each propagation
+phase, while others make this dependent on criteria like the variance of the
+weights. In this paper we rely on SIR with resampling after each propagation
+phase.
+6
+
+Box 1
+(a) The propagation of particle p depends only on the associated state
+xp,t and can be performed independently of other particles.
+(b) Weights wp,t depend only on the associated particle p and observa-
+tion vector yt, and can be computed independently of other weights
+and particles.
+(c) The filter update only depends on the weights wp,t, and not on the
+particles and associated states.
+(d) The states xp,t associated to the particles p remain unchanged during
+the filter update.
+Box 1 lists the properties of particle filters that are the basis for our
+implementation.
+We exploit property (d): In contrast to other DA techniques, such as
+EnKF, particle states remain unchanged during the filter update.
+Parti-
+cles that have departed too much from the observations (low weights) are
+discarded, and the sample set is narrowed around the best particles (high
+weights). The associated states, however, are not changed. Property (a)
+follows directly from Equation 11.
+Property (b) results from decoupling
+the weight calculation from the filter update (decentralization). The update
+itself, only consist of the weight normalization and particle resampling. Fi-
+nally, property (c) is an intrinsic property of the bootstrap particle filter,
+since particles are either withdrawn or selected, but not changed. In the fol-
+lowing sections, we will show how we can exploit those properties to improve
+efficiency of and resilience particle filter implementations.
+3. Architecture
+In this section we detail the proposed architecture to run a large number
+of particles with parallel numerical models. The algorithm, as presented in
+Section 2, is a sequence of two main steps:
+1. A first compute intensive massively parallel step where particles can be
+processed concurrently to:
+(a) propagate each sate: xp,t = M(xp,t−1),
+7
+
+Batch Scheduler
+Server: 
+- schedule propagations and cache evictions
+- gather weights and performs resampling
+Shared Particle Store 
+(PFS) 
+Launcher: 
+- submit and monitor jobs
+- manage recovery on server or   
+  runner fault
+Checkpoints
+Observations
+Submit jobs
+Run Jobs
+Job status
+Monitoring
+Weights
+Cache evictions 
+and 
+particle propagations 
+ 
+Parallel Runner: 
+- propagates particles
+- calculates weights
+- sends weights to server
+Parallel Runner: 
+- propagates particles
+- calculates weights
+- sends weights to server
+Parallel Runner: 
+- propagate particles
+- calculate weights
+- send weights to server
+Local
+Particle
+Cache
+Particle states
+Figure 2: Architecture overview.
+(b) compute each unormalized weight from each state and observation
+data:
+wp,t = pϵ(yt − H(xp,t))
+12
+2. A second lightweight step that requires to gather all unormalized weights
+wp,t, usually one double per weight, for normalization and resampling.
+We attribute the first step work to runners and the second step to a server.
+A runner is designed to propagate several particles one after the other with
+low overheads and idle times (Figure 2). Each one is coupled with a node-
+local distributed cache enabling fast loads and stores of particles. The caches
+are asynchronously persisted to the global file system for checkpointing and
+dynamic load balancing (i.e., ensure global availability of the particles). Be-
+cause resampling can lead to discard some particles, or duplicate others orig-
+inating from the same parent (with a local perturbation if needed), states
+need to be dynamically redistributed to runners to keep them evenly busy.
+The server drives the dynamic distribution of particle propagation tasks to
+runners. Runners use the distributed cache to load from the file system the
+8
+
+missing states. This design ensures low communications between the server
+and runners, and reduced state movements. The runners and the server run
+as independent executables, enabling to have a dynamically changing number
+of runners. This is a key feature used for fault tolerance and elasticity. Elas-
+ticity (sometimes also called maleability) is the ability to run under changing
+resource availability, here varying number of runners.
+In the following we detail this design: the runners (Section 3.1), the
+server (Section 3.2), the distributed cache (Section 3.3), the workflow be-
+tween these components (Section 3.4), the particle propagation scheduling
+(Section 3.5), the jobs monitoring (Section 3.6), and the fault tolerance pro-
+tocol (Section 3.7) before ending with additional implementation details (Sec-
+tion 3.8).
+3.1. Runners
+Runners are built from the simulation code, often an advanced parallel
+code or even a coupling of several parallel codes, with significant start-up
+times to load and build the different internal data structures.
+To avoid
+paying the cost of a restart for each particle propagation, we augment the
+simulation code with a mechanism to store and load particle states. This is
+the base of particle virtualization: a runner can load a particle, propagate
+it, store the result, and repeat this as many times as necessary. Runners are
+associated with a distributed cache to accelerate state loads and stores as
+detailled in Section 3.3. Runners also compute the associated weights wp,t.
+Hence, each runner also needs to load the observations yt once per cycle.
+Notice that the size of the observations is typically much smaller than the
+size of the states xp,t.
+3.2. Server
+The server is entrusted with multiple tasks. First, it is responsible for
+scheduling the particle propagations to the runners (Section 3.5). Second,
+it gathers the weights from the runners and performs the resampling at the
+end of each assimilation cycle. Third, it controls the content of a distributed
+particle cache (Section 3.3). To collect the weights wp,t, the server is mes-
+saged from the runners after each propagation. If there are still particles
+to propagate in the current cycle, the server responds to the message with
+an id uniquely defining a particle (hereinafter called particle-id) for the next
+propagation. If not, the server performs the resampling and starts the new
+cycle by scheduling the sampled particles to the runners. Very little data is
+9
+
+      Runner 1
+Server
+Node 1
+Node n
+MPI Communicator
+Model process (master)
+Model process
+Helper process
+Helper process (master)
+Node local cache
+MPI communication
+ZMQ communication
+File transfer
+      Runner M
+Node 1
+Node n
+PFS
+Figure 3: Internal runner architecture and interactions with the server and global storage
+(PFS). Communications with the server combine MPI and ZMQ data exchanges.
+exchanged between a runner and server. The runners send the particle-id
+(a single int) and the corresponding weight (a single float), and the server
+responds with the particle-id next to propagate.
+3.3. Distributed Particle Cache
+To allow multiple propagations of one particle on different runners, it is
+necessary to make them globally available. A straight forward approach is to
+store particles on global storage. However, on supercomputers global storage
+is subject to large throughput variability due to the high workload and the
+limited bandwidth. Node-local storage, on the other hand, is only used by the
+processes that run on the nodes, and the bandwidth can be stacked. Storing
+the particles locally results in scalable I/O performance, scaling linearly with
+the number of nodes.
+To leverage node-local storage while still providing the particles globally,
+runners rely on a distributed particle cache. Each runner executes helper
+processes (one per node) in addition to the model processes, where both
+groups of processes are associated with its own MPI communicator (Figure 3).
+The model processes propagate the particles and store the associated states
+locally on the nodes allocated to the runner (RAM disk or other node-local
+storage when available). The helper processes then stage the states from local
+10
+
+Init
+Propagate x4
+Store
+State x4
+Calculate
+weight w4
+Load State x5
+Calculate
+weight w5
+Load
+State x6
+Init
+App process 0
+Helper process
+Init
+Load
+State x4
+Propagate x4
+Calculate
+weight w4
+Calculate
+weight w5
+App process 1
+Propagate x5
+Propagate x5
+Request new state
+from server
+Request new state
+from server
+Send w4 to server
+Request new state
+from server
+Time
+Init
+Load
+State x1
+Propagate x1
+Store
+State x1
+Calculate
+weight w1
+Load State
+x2
+Calculate
+weight w2
+Load
+State x3
+Init
+App process 0
+Helper process
+Init
+Propagate x1
+Calculate
+weight w1
+Calculate
+weight w2
+App process 1
+Propagate x2
+Propagate x2
+Request new state
+from server
+Request new state
+from server
+Send w1 to server
+Request new state
+from server
+Parallel File System
+Runner 1
+Runner 2
+Figure 4: Schematic Gantt diagram showing the activity of two runners (initialization
+followed by two assimilation cycles). Focus on how the helper process asynchronous loads
+and stores enables to shadow the parallel file system accesses. For sake of simplicity no
+cache is used here.
+11
+
+to global storage asynchronously, enabling to overlap the associated I/O costs
+(Figure 4). Also notice that persisting particles to global storage acts as a
+particle checkpoint used by the fault tolerance protocol (Section 3.7).
+We allow keeping a number of particles in each of the runner caches
+to exploit property (d) from Box 1: resampling does not change the parti-
+cle states. Hence, keeping propagated particles in the cache, increases the
+probability to find a particle locally for future propagations (i.e., during the
+next cycle). If available in its local cache, a runner can propagate a parti-
+cle without loading it from global storage. To further minimize cache loads,
+runners implement an optimized cache eviction strategy. The eviction strat-
+egy becomes especially important if the cache capacity is exceeded by the
+accumulated size of the particles propagated during one cycle. Because the
+runners have no knowledge about the status of the particle filtering (propa-
+gations, resampling), the evictions are controlled by the server and directed
+to the runners.
+As explained in Section 3.4, each time a particle has been stored in the
+cache by the model processes upon successful propagation, the helper pro-
+cesses copy it in the background to global storage. Hence, all propagated
+particle states can be selected for eviction, since they are safely stored on
+global storage. When an eviction is required, the server selects a particle
+from the cache in the following order:
+1. A particle from the previous cycle discarded by resampling;
+2. A parent particle from the current cycle for which all associated prop-
+agations have been performed, and all weights received;
+3. The particle with the lowest weight propagated during the current cy-
+cle;
+4. A randomly selected particle.
+The particle states for cases 1 and 2 can safely be removed from cache, since
+those particle are not needed anymore for future propagations. In case 3, we
+select the particle state with the lowest weight, as it has the lowest probability
+of being selected for future cycles during the resampling.
+3.4. Runners/Server Workflow
+Once a runner job has started, it dynamically connects to the server and
+requests a particle to propagate from it. The server selects the particle fol-
+12
+
+lowing a scheduling policy described in Section 3.5. The model checks the
+location of the particle. If already located inside the local cache, the prop-
+agation starts. Otherwise, the model processes request the helper processes
+to load the state into the cache. The model processes block until the helper
+processes fetched the particle into the cache, and afterwards start the prop-
+agation.
+Once a particle propagation finishes, the model computes the associated
+weight wp and stores the particle into the cache. Further, the weight and
+particle-id are sent to the helper processes and a new particle is requested for
+propagation. The helper processes, after receiving the weight from the model
+processes, stage the particle from the cache to global storage and afterwards
+sends the weight and particle-id to the server. This order ensures that the
+server receives a weight only after the corresponding particle is propagated
+and successfully stored on global storage.
+The helper processes further prefetch particles in parallel to the propa-
+gations (Figure 4). The goal is to avoid blocking the model processes while
+waiting for a particle load from global storage (cache miss). Each time helper
+processes send a weight to the server, they also request the next-to-next
+particle-id to propagate. This particle is prefetched into the cache to become
+locally available for the next to follow propagation. Prefetching is suspended
+at the end of each propagation cycle, as propagation work for the next cycle
+becomes only known after the server has performed the resampling of all par-
+ticles. Notice that a helper may need to cancel prefetching if the prefetched
+particle was in the meantime assigned to another runner, making idle the
+model process while waiting for the next particle to propagate. When the
+server makes such a decision to better balance the work load, it also takes
+care of ensuring coherency between runners. Globally prefetching proved to
+be very efficient for overlapping particle state loads with propagation (Sec-
+tion 4.2).
+3.5. Particle Propagation Scheduling
+In this section, we present the scheduling algorithm implemented on the
+server to distribute the particle propagations to the runners. The algorithm
+aims to ensure an even load balancing between runners and minimizing the
+global particle loads, i.e. transfers of particles from global to local storage.
+13
+
+Compulsory state load
+Extra state load due to parallelization
+Runner work lists
+Figure 5: Two possible schedules of 24 propagation tasks of equal duration on 4 runners.
+All particles propagated from the same parent particle state have the same color (9 parents
+here). Top schedule is optimal with 9 compulsory loads (one per parent), and one for the
+dark blue parent that cannot fit in one runner. The bottom schedule, with 2 more sate
+loads, is a possible one that our on-line scheduling algorithm can produce. This is not
+optimal but still below the general Q+R−1 bound as the algorithm ensures that no more
+than R − 1 ”color cuts” occur and avoids the same runner loads more than once a given
+parent particle state.
+3.5.1. Static Scheduling
+Let R be the number of runners. Let P be the number of particles to
+propagate. Resampling may lead to have some parent particles drawn to be
+propagated several times. Let Q the number of parent particles q, and αq the
+number of times the particle q needs to be propagated. The total number of
+particles to propagate is:
+P =
+�
+0≤q<Q
+αq
+13
+To assess the performance of our scheduling algorithm, we first derive a
+lower and upper bound of the minimum number of particle loads c∗ for the
+static case, where: (i) runners do not cache states, (ii) the number of runners
+is constant, and (iii) all particle propagations take the same amount of time.
+Under these conditions, each runner propagates P
+R particles. Without local
+cache, each parent particle q needs to be loaded at least once. Therefore,
+the number of compulsory particle loads is Q. If αq = 1 for all q, that is,
+every particle is drawn only once, then c∗ = P. Otherwise, parallelizing the
+propagation on R runners may require some particles to be loaded by more
+than one runner, accounting for extra particle loads beyond the compulsory
+ones. Indeed, each particle q needs to be provided at least on sq runners,
+14
+
+where
+sq =
+�
+αq
+P
+R
+�
+.
+14
+Distributed to R runners, the list of P particles is cut R − 1 times. Con-
+sequently, the extra particle loads are at most R − 1.
+This is visualized
+in Figure 5. This upper bound occurs if all particles are propagated from a
+single parent(Q = 1). Thus, the minimum number of particle loads is tightly
+bound by
+Q ≤ c∗ ≤ Q + R − 1.
+15
+We can apply a static schedule that respects the upper bound: distribute
+P
+R particles per runner, where each parent particle q is given to no more than
+sq runners, and by imposing that runners do not switch to the next particle
+before completing all propagations associated to the current one.
+3.5.2. Dynamics List Scheduling
+However, we target a more general case. We soften the initial assump-
+tions now considering that the number of runners can vary, and the time
+to propagate particles may vary significantly and is not known beforehand
+(but we still have no cache). In this context we propose to rely on the clas-
+sical dynamic list scheduling algorithm: when idle, a runner requests work
+from the server that returns a particle-id to propagate. In the general case
+the list scheduling algorithm guarantees to be at worst twice as long as the
+optimal schedule that requires to know the particle propagation time in ad-
+vance [13, 14]. Instead of blindly selecting the next particle to propagate,
+we adapt the static scheduling strategy for particle selection with the goal of
+limiting the number of particle loads. The scheduling is based on the split
+factor sq (Equation 14). However, we adapt the static scheduling to the dy-
+namic case by recomputing sq each time with the updated values of αq, P,
+and R. To implement this algorithm on the server, we need a bookkeeping of
+the number of runners Rq currently propagating particle q, and the number
+αq of remaining propagations for particle q. Let r be the runner requesting a
+particle for propagation, the particle distribution algorithm works as follows:
+1: If αq > 0 for particle q last propagated by r, decrement αq and assign q
+again. If αq = 0 continue with (2);
+2: Select a different particle q′ with αq′ > 0;
+15
+
+3: Compute split factor sq′. If Rq′ < sq′ assign q′, increment Rq′, and decre-
+ment αq′. If Rq′ = sq′ continue with (2).
+Notice that when the server recognizes the loss of one runner, it needs to
+update the bookkeeping to reintegrate the particle that this runner was prop-
+agating.
+In conditions of even propagation time and a static number of runners,
+this algorithm leads to the same distribution as for the static schedule and
+respects the upper bound of Equation 15.
+3.5.3. Cache Aware Scheduling
+We now remove the last assumption to propose a scheduling strategy that
+takes into consideration the particle cache. This is a heuristic build upon the
+previous strategy and validated though several experiments. The particle
+selection strategy is:
+1. Select a parent particle pi already loaded in the runner cache (cache
+hit);
+2. Select a parent particle pi that is in no runner cache (cache miss);
+3. Select a particle pi fulfilling the split factor criterion (cache miss);
+4. Select a parent particle pi with maximal split factor si (even if voids
+the split factor) (cache miss).
+The three first items comply with the scheduling proposed in Section 3.5.2.
+The first item gives priority to particles already in the cache, before they may
+be evicted to provide space for a particle load. The next two items pursue
+with the strategy of Section 3.5.2, favoring particles with no previous propa-
+gation. The rational is to start as soon as possible with new parent particles
+and, once in a cache, propagate them has often as required, and intend to
+reduce the need for splitting. The last item departs from the strategy of
+Section 3.5.2, but its addition proved efficient by our experiments. This case
+occurs when reaching the end of a cycle. It proved to be an efficient strat-
+egy to keep runners busy, even at the cost of extra loads, to improve load
+balancing and so completion time.
+16
+
+3.6. Job Submission and Monitoring
+The workflow is controlled by the launcher. The launcher is the user
+entry point to configure and start the application. The launcher starts first
+and is responsible to start and monitor the runner and server instances,
+that all run in separate executables/jobs. The launcher is also in charge of
+killing and restarting the runners or server as requested by the fault tolerance
+protocol (Section 3.7), or for elasticity purpose.
+The launcher tightly interacts with the job scheduler (Slurm or OAR
+for instance) of the machine.
+The launcher can be configured to submit
+one job per runner and server to the batch scheduler. This strategy offers
+the maximum flexibility for the batch scheduler to optimize the machine
+ressource allocation, but the execution progress becomes very dependent on
+the machine availability. The user may need more control on the number of
+concurrently running runners. In that case the launcher can be set to request
+to the batch scheduler one or several large resource allocations and fit several
+runner instances in each one. To support this feature the launcher relies on
+a combination of Slurm salloc/srun [15], or OAR containers[16]. For even
+more flexible schemes, we plan to support workflow pilot-based schedulers
+like Radical-Pilot [17] or QCG-PilotJob [18].
+3.7. Fault Tolerance
+The fault tolerance relies on the fast identification of failures from run-
+ner and server instances. Runner failures are detected in two different ways.
+Runner crashes are recognized by the launcher, which is monitoring their
+execution using the cluster scheduler. Unresponsive runners are identified
+by the server relying on timeouts for the particle propagations. If propaga-
+tions exceed the timeout, the server requests the launcher to terminate the
+respective runner. In both cases, the launcher eventually starts a new runner
+instance. The new runner connects to the server and requests work. If a
+runner fails, the server cancels the on-going propagation, and the time spent
+in the propagation plus the time to recognize the runner failure is lost.
+Server failures are detected similarly, either directly if the server crashes,
+or if the server exceeds a timeout. The timeout is mediated by a periodical
+exchange of signals between launcher and server (heartbeats). If the server
+fails, the launcher terminates all runner instances and restarts the framework
+as a whole. The server frequently stores the status of the propagations in
+checkpoints, and in case of failures, the framework can recover to the point
+of the last successful propagations.
+17
+
+Finally, a launcher failure is detected by the server monitoring the heart-
+beat connection between launcher and server. In case of a missing heartbeat,
+the server checkpoints the current particle state and shuts down. In parallel,
+the runners detect the server crash and shut down, again using timeouts.
+While runner or server failures lead to an automatic restart, the framework
+needs to be restarted manually if the launcher fails.
+3.8. Implementation Details
+The launcher and server are developed in Python.
+The runner relies
+on the simulation code instrumented with our framework API, supporting
+C/C++, Fortran and Python.
+The implementation reuses some software
+components, like the launcher, from the framework developed for EnKF
+DA [19]. The distributed cache implementation relies on the Fault Tolerance
+Interface (FTI) [20]. FTI is a multilevel checkpoint-restart library supporting
+asynchronous checkpointing to global storage. One of the main modification
+performed to FTI is related to its event loop. Events are triggered in form of
+MPI communication between the application and FTI processes. The events
+are identified by tags. To extend this mechanism, we enabled to register a
+callback function. This callback function is called inside the event loop and
+can trigger user defined events using unique tags. With this, it becomes pos-
+sible to use the application checkpointing into all available levels of reliability
+FTI provides, and to implement the cache mangement on the dedicated FTI
+processes.
+The communication between helper and model processes relies on asyn-
+chronous MPI messages. Communications with the server are implemented
+in two steps for efficiency purpose. Only rank 0 (master) of the application
+(i.e., model) communicator and the rank 0 (master) of the helper process
+communicator communicate with the server. As a dynamic connection is
+needed, each master connects to the server using a socket through the ZMQ
+library. Information that needs to be propagated between helper or model
+processes relies on MPI collective communications in the associated commu-
+nicator (Figure 3).
+The framework code is available at https://gitlab.inria.fr/melissa/
+melissa-da.
+18
+
+4. Experiments
+4.1. WRF Use Case
+Experiments rely on an established Numerical Weather Prediction (NWP)
+system; the Weather Research and Forecasting Model (WRF) (V3.7.1)[8].
+The core of WRF is based on solving fully compressible non-hydrostatic equa-
+tions with complete Coriolis and curvature terms, and a large set of physics
+options. The simulation domain covers most of Europe (See Figure 6) and is
+composed of 220 by 220 grid cells with a horizontal resolution of 15 km and
+49 vertical levels with uneven thickness. We perform one day-ahead weather
+forecasting (24 hours of initial time plus 48 hours of production time) of an
+arbitrary date (2018-10-12) with 24-seconds or 100-seconds time steps. The
+model employs the WSM6 microphysics, MYNN2 boundary layer physics,
+Grell-3 cumulus parameterization, Eta Monin-Obukhov similarity surface
+layer processes, and RUC land surface model.
+Also non-hydrostatics are
+activated to provide more details in simulated clouds and precipitation. The
+input, initial, and boundary conditions are based on the reanalyzed ERA5
+dataset from the European Center for Medium-Range Weather Forecasts
+(ECMWF), which is updated every 3 hours. Data assimilation is performed
+using the cloud fraction (CFRACT). The particle weights are determined by
+comparison against the observed cloud mask obtained from the EUMETSAT
+Level-2 satellite data of the cloud mask. The simulated cloud fraction is con-
+verted into cloud mask and the observed cloud mask data is upscaling to the
+size of the model gridcells for the further applications. The data is hourly
+available, thus, one assimilation cycle comprises 150 (36) model time steps
+(150 × 24 s �= 1 h or 36 × 100 s �= 1 h).
+The experiments presented in this article leverage our proposed Particle
+Filtering (PF) implementation with a sample size of up to 2,555 particles
+on the European domain. In that case, we utilize 20,442 compute cores on
+512 Nodes of the Jean-Zay supercomputer. The compute nodes are equipped
+with 2 Intel Cascade Lake 6,248 processors, summing up to 40 cores with
+2.5 GHz and 192 GiB RAM per node. Intel Omni-Path (100 GB/s) connects
+the compute nodes, and the global file system is an IBM Spectrum Scale
+(ex-GPFS) parallel file system with SSD disks (GridScaler GS18K SSD). For
+all experiments the node-local caches were stored on RAM disk. In Table 1
+we list the parameters of our main experiments.
+The meteorological state of the European domain associated to one par-
+ticle comprises 2.5 GiB of data. Hence, the data from 2,555 particles for the
+19
+
+Figure 6: The topography of the target domain of Europe for the simulation.
+full simulation period of 48h (48 time steps) correspond to an aggregated
+size of about 300 TiB. The experiments performed for this article, including
+small beta-stage experiments, account for about 900,000 CPU hours split
+between the JUWELS, Jean-Zay and Marenostrum supercomputers.
+4.2. Runner Activity
+The benefit of the local cache in combination with the cache-aware schedul-
+ing leads to a drastic reduction in transfers from global to local file system
+layers. The cache hit ratio, i.e., the ratio of particles found inside the cache
+to the total number of particle loads, depends on the cache size and the ratio
+of particles per runner. Figure 7 shows how the cache misses develop for
+different cache sizes. Our experiments demonstrate a cache hit ratio of 88 %
+for 128 particles, 32 runners, and a cache size of 9 particles. This translates
+to a saving of 88 % in transfers from global to local storage. The pattern of
+cache hits and misses is visualized in Figure 8. The initial phase is dominated
+by starting up the runners, and all the particles are fetched from the global
+storage (cache warm up). But beginning with the next assimilation cycle,
+the low transfer ratio from global to local storage starts to establish.
+Runners are designed to separate I/O operations to the PFS from other
+tasks: model processes only perform local I/O operations. We observe in
+our experiments that this leads to a high computational efficiency. The local
+I/O accesses are negligible compared to the computational tasks (< 0.1 s
+20
+
+20°W
+10°W
+00
+10°E
+20°E
+30°E
+40°E
+3000
+2500
+60°N
+20°W
+2000
+Latitude
+Elevation
+1500
+50°N
+1000
+500
+40°N
+10°W
+10°E
+20°E
+Longitude3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+cache size
+0.2
+0.3
+0.4
+0.5
+cache miss ratio
+Figure 7: Cache miss ratio for different cache sizes on each runner. In total 128 particles
+run on 32 runners. First and last assimilation cycles were disregarded to remove warm up
+effects and not fully recorded cycles.
+21
+
+Experimental Setup
+Particles
+315
+635
+1,275
+2,555
+Number of runners
+63
+127
+255
+511
+Number of nodes
+64
+128
+256
+512
+Model processes
+2,457
+4,953
+9,945
+19,929
+Particles per runner (avg.)
+5
+5
+5
+5
+Particle state size (GiB)
+2.5
+2.5
+2.5
+2.5
+Performance Data
+Scaling efficiency
+92%
+91%
+92%
+87%
+Resampling (ms)
+2.21
+4.06
+8.16
+16.37
+Assimilation cycle (s)
+136
+138
+139
+146
+Propagation (s)
+25.1
+25.2
+25.1
+25.0
+Load particle state
+from PFS to cache (s)
+2.1
+2.1
+2.4
+4.1
+Write particle state
+from cache to PFS (s)
+1.4
+1.6
+1.8
+2.3
+Writes to PFS per cycle (TiB)
+0.77
+1.55
+3.11
+6.24
+Reads from PFS per cycle (TiB)
+0.30-0.40
+0.64-0.79
+1.27-1.79
+2.54-3.82
+Table 1: Experimental setting and performance overview at 4 different scales. The times
+are given as average in all cases. Model time steps were set to 100 seconds.
+compared to up to 6 s). Some general idle periods can be observed between
+assimilation cycles when runners are waiting for the last propagations of one
+cycle to finish so that the server can normalize weights, resample and start to
+distribute work again. This is illustrated in Figure 9 where we show a trace
+recorded from the execution of an arbitrary runner. The trace illustrates the
+efficiency of the runners in performing the actual tasks of the simulation,
+particle propagation and weight calculation, while the I/O tasks are moved
+to the background.
+A global parallelization of the computational tasks is achieved by dynami-
+cally distributing the particle propagations to the available runners. The fully
+parallelized case corresponds to R = P, i.e., the number of runners matches
+the number of particles. The sequential case corresponds to R = 1, i.e., all
+propagations are performed by only one runner. However, The best parallel
+efficiency is achieved at values between those limits. Because WRF propa-
+gates particles with very low time variability (maximum variation of 10%),
+we observe an even distribution of propagations to runners when R divides P
+(Figure 8). A single-particle propagation takes between 24 and 26.5 seconds,
+22
+
+0
+1000
+2000
+3000
+4000
+5000
+Time (s)
+0
+10
+20
+30
+Runner ID
+state load
+Cache hit
+Cache miss
+Assimilation cycle
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+cachesize: 9, cache hit ratio (cycles 2-14): 0.88
+Figure 8:
+Gantt chart of particle propagations executed by the 32 runners over 15
+assimilation cycles. Tasks are green if the associated parent particle state was already
+present in the runner cache and did not require a load from the PFS (red otherwise).
+globally making from 87% to 92% of an average assimilation cycle. Calcu-
+lating weights takes 1% of the time and communicating with the server from
+7%to 12% including the idle time at the end of each cycle (Table 1 – Perfor-
+mance Data). The extra resources for helper processes, one core per runner
+node, and the server, 1 node, comprise only 2.7% for our largest experiment
+at 512 nodes. On the other hand, leveraging the runner’s particle cache, and
+the cache aware dynamic scheduling on the server, move > 97% of the state
+loads completely into the background. Loading and writing particle states
+synchronously would otherwise add about 6.4 seconds to each single-particle
+propagation corresponding to 14% of the average propagation time (Table 1
+– 2,555 particles).
+Note that in contrast to the traditional offline approach, we start-up the
+numerical simulation code only once for several particle propagations. The
+setup involves the request and allocation of the runner job and initializing
+the simulation code.
+On the other hand, the traditional offline approach
+associates each particle propagation with a different job, and the start-up
+has to be performed anew for each particle. Starting up the WRF model on
+23
+
+Figure 9: Trace detailing the activity of a runner over the course of an assimilation cycle.
+Helper processes enable to keep model processes busy with particle propagation, except at
+the end of assimilation cycles when they wait for the server to finish particle resampling
+(dark blue). Some activities are so thin that they are not visible here (state copies from
+cache to model).
+the European domain on one node until the first model propagation begins
+takes 3-4 s, excluding the provisioning of the job allocation via the cluster
+scheduler.
+4.3. Server Activity
+We further measured the server responsivity to runner requests.
+The
+response time is always in the order of a few hundred microseconds, except
+for some job requests that take up to a few seconds (Figure 10). However,
+these are outliers at the end of the assimilation cycle, resulting from idle times
+due to the load balancing and the particle filter update. During our largest
+experiments with 511 runners, the server processes 676 requests per second
+at maximum load. This shows that the server is fast enough to support this
+scale, even though it is a sequential python code. Simple optimizations are
+at reach if the server needs to be accelerated (e.g., adding parallelization).
+24
+
+Assimilation cycle
+Weight calculation
+processes
+Model
+Request job from server-
+Propagation -
+Load state from cache
+Load state from PFS into cache-
+processes
+Helper
+Write state from cache into PFS -
+300
+400
+500
+600
+Time (s)Delete request
+Job request
+Prefetch request
+Push weight to server
+1
+1e2
+1e4
+Duration (ms)
+Figure 10: Server response times on runner requests.
+4.4. State Transfers To/From PFS
+Next, we take a closer look at the particle loads from the PFS (Figure 11).
+With a sample size of 1024 particles, leveraging 256 runners, and a local cache
+size of 9 particles, between 121 and 248 particles are loaded to the cache
+during each cycle. The number of distinct parent particles Q propagated per
+cycle lies between 813 and 889. Each one is propagated at most 5 times to
+sum to a total of 1024 particles. The cache enables to achieve significantly
+less loads than the Q + R − 1 upper bound expected with static scheduling
+and no cache (Equation 15).
+The access times to the Parallel File System (PFS) (load/store) vary sig-
+nificantly and increase with the number of runners (Figure 12), showing that
+25
+
+Q
+Q+R-1
+Figure 11: Number of parent particles Q, particles loads from the PFS to the cache,
+and Q + R − 1 upper bound from Equation 15 for different ensemble sizes, a cache size
+of 9 particles with 4 particles per runner.
+26
+
+our application alone can stress the PFS 1. Each particle is associated with
+2.5 GiB of data. During each assimilation cycle, all the propagated parti-
+cles are written to the PFS for supporting fault-tolerance and dynamic load
+balancing. For our experiments at 512 nodes with 2,555 particles, this accu-
+mulates to about 6.2 TiB of data each cycle (compare Table 1). However, our
+experiments on the Jean-Zay and JUWELS supercomputer demonstrate that
+our framework performs most of those transfers asynchronously (Section 4.2).
+In less than 2% of the cases, the model processes wait more than 0.1 seconds
+for a particle to be loaded corresponding to cases where helper processes do
+not (entirely) finish prefetching. Time to perform the local loads and stores
+from the cache shows a constant average independently on the number of
+runners (Figure 13).
+63
+127
+255
+511
+Number of runners
+1000
+2000
+3000
+4000
+5000
+6000
+Duration (ms)
+Load state from
+PFS into cache
+Write state from
+cache into PFS
+Figure 12: Mean time to load or store particle states of 2.5 GiB from / to the PFS with
+different numbers of runners.
+4.5. Fault Tolerance, Elasticity and Load Balancing
+Fault tolerance relies on 1) persisting the particle to the PFS 2) the
+framework elasticity enabling to adjust dynamically the number of runners.
+1these numbers may also be impacted by other jobs on the cluster
+27
+
+128
+256
+512
+1024
+members
+1e-4
+1e-2
+1
+duration [s]
+Load from local cache
+Store to local cache
+Figure 13: Box plot of the time spent for loads and stores from/to the local cache with
+different numbers of particles.
+If a runner fails, the launcher requests the execution of a new one, so as to
+maintain a constant number of runners. Once this new runner connects to
+the server, it asks for a particle to propagate to the server, assigned according
+to the load balancing algorithm.
+We tested the fault tolerance and elasticity on an experiment with 63
+runners provoking the crash of 2 runners (Figure 14). First, notice that the
+fault tolerance algorithm reacts appropriately as it restarts a new runner after
+each crash. The first crash (runner #53) occurs in the worst-case situation:
+just when propagating the last particle of the current cycle, leading to a
+significant idle period. The idle period is caused first, because the server
+needs to wait for the timeout (set to 60 s) to acknowledge that runner #53
+is unresponsive and second, there is no work left except the particle that
+runner #53 was propagating, which is re-assigned to runner #44. Meanwhile
+all other runners stay idle until the beginning of the next cycle. If the crash
+happens earlier during a cycle, smaller idle periods appear.
+This can be
+observed during the second crash (runner #48), as the other runners are
+kept busy with propagation work.
+We generally observe an efficient load
+balancing, as the work load is kept well distributed amongst runners, even
+when their number varies.
+28
+
+0
+250
+500
+750
+1000
+1250
+Time (s)
+0
+20
+40
+60
+Runner ID
+Initial propagation
+Assimilation cycle
+1
+2
+3
+4
+5
+6
+7
+8
+Figure 14:
+Gantt chart of particle propagations executed by the 63 runners over 8
+assimilation cycles.
+After runners #48 and #53 crashed (black cross), two new ones
+restarted (top 2 runners #63 and #64).
+4.6. Scaling
+We evaluated the performance of the particle filter in a strong scaling
+scenario, constant number of runners while increasing the number of particles,
+and a weak scaling scenario, constant particle-to-runner ratio while increasing
+the number of runners. In the strong scaling scenario we observe that the
+runner utilization shows an upwards trend when increasing the number of
+particles per runner, with a plateau at about 96% (Figure 15). As global
+I/O operations are almost completely shadowed, thanks to the asynchronous
+prefetching, increasing the number of particles per runner mainly enables to
+better amortize the cost of the synchronization associated with resampling.
+We observe an almost constant time for the assimilation cycle, demonstrating
+a desirable weak scaling behavior. The time for the cycles increase only by
+8% from 63 to 511 runners, indicating an efficient scaling behavior of the
+29
+
+framework up to production scale (Figure 16). Particle filtering with WRF
+on a European domain for short-range weather prediction at this scale is
+an important advancement of the previous work done by Berndt et. al. [21].
+Moreover, besides assimilating at a higher frequency, our proposal offers fault
+tolerance, automatic load balancing and elasticity while minimizing the I/O
+cost and time to calculate weights.
+63 particles  
+(1 per runner)
+315 particles  
+(5 per runner)
+630 particles  
+(10 per runner)
+945 particles  
+(15 per runner)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Scaling efficiency
+Figure 15: Scaling efficiency using different numbers of particles with 63 runners. One
+runner sets the reference case.
+4.7. Comparison to a File-based Approach
+Melissa
+ESIAS-met
+ESIAS-met/Melissa
+part.
+cores
+time (s)
+core.s/part.
+cores
+time(s)
+core.s/part.
+resource usage ratio
+128
+384
+1062
+3186
+1536
+267
+3204
+1.01
+256
+768
+1062
+3186
+3072
+317
+3804
+1.19
+512
+1536
+1068
+3204
+6144
+422
+5064
+1.58
+1024
+3072
+1071
+3213
+12288
+761
+9132
+2.84
+Table 2: Comparing the resource usage (core.second/particle) per cycle for Melissa and
+ESIAS-met (file-based) runs.
+We compare Melissa with the file-based approach ESIAS-met [22] using
+the same simulation code WRF (V3.7.1) and the same data set. For the
+same number of particle, both approaches use a very different amount of
+30
+
+63
+(315)
+127
+(635)
+255
+(1275)
+511
+(2555)
+Runners
+(particles)
+0
+50
+100
+150
+Assimilation cycle
+duration (s)
+5 particles per runner
+2522
+5082
+10202
+20442
+Cores
+Figure 16: Weak scaling performance test: assimilation cycle duration for different num-
+bers of runners, but always 5 particles per runner.
+cores (Table 2). With ESIAS-met each particle propagation requires to start
+a dedicated instance of WRF. Each time it includes the cost from loading
+and storing the particle state from/to a file. At 1024 particles ESIAS-met
+uses 12288 cores while Melissa just needs 3072 cores as runners propagate
+several particles each. ESIAS-met execution time is thus shorter as highly
+parallelized, but the resource usage (core.second/particle/cycle) is 2.84 times
+improved for Melissa due to the combined strategies developed to improve
+efficiency. The gain increases with the number of particles, showing that the
+Melissa approach is particularly beneficial when targeting the large ensemble
+size.
+5. Discussion
+in Section 4.1 we derived that the total amount of data resulting from
+48 time-steps of particle filtering on the european domain with 2,555 parti-
+cles accumulates to about 300 TiB. Post processing this amount of data is
+challenging. Our framework could be extended using in situ data processing
+techniques as presented in Terraz et. al. [23].
+We only considered the case where the propagation time is longer than
+the time for loading states from the PFS; while applications where propaga-
+tions are shorter than loading the states would limit our proposal efficiency,
+as we cannot further hide the I/O cost in that case. On the other hand,
+we already have short propagation times in the WRF context as we per-
+31
+
+form hourly resampling.
+We chose this frequency primarily to stress our
+proposal. Production runs usually do not require such a high frequency, and
+rather have even longer propagation times as in our experiments. However,
+to minimize transfer times further, we are evaluating approaches leveraging
+node-local persistent storage as globally shared storage layer. Solutions for
+this are readily available in form of distributed ad hoc file-systems [24] such
+as BeeOND, GekkoFS, and BurstFS. We have also experimented with con-
+necting the runners, establishing a peer to peer network, where runners can
+exchange directly the required states between each other.
+The particle propagation time in our experiments with WRF is relatively
+even, showing at most a 10% variability. Situations with more variability are
+possible using different physics in WRF, with other simulation codes, or, if
+runners execute on heterogeneous resources, some runners propagating faster
+than others by leveraging GPUs for instance. Also use cases from other con-
+texts such as Simulation Based Inference (SIB) and ensemble classification,
+which can be performed using our framework, might lead to vastly different
+propagation times. Therefore, testing our framework under such conditions
+is an important future work.
+Our proposal currently relies on filters that do not compute internal mem-
+ber state corrections.
+Extending our approach to such particle filters [1]
+would possibly require aggregating more than just the particle weights to the
+server. Exploring the requirements to align our framework to such cases is
+the goal of a future implementation of the particle filter that we propose. We
+validated our proposal with the SIR particle filter, but many variations exist
+and are active research topics [25, 26]. One challenge is the exponentially
+growing required particle number with the dimension of the problem [27, 28].
+This is particularly acute for geoscience use cases that, as in this paper, work
+in high dimensional spaces. The survey [1] gives an extensive overlook of DA
+by particle filters for geoscience and ways to cope with dimensionality issues.
+Particle filters, as used here, require a synchronization point at the end
+of each assimilation cycle. For our framework, this is the major remaining
+source of inefficiency. Loosening this requirement needs revisiting the particle
+filtering algorithm, which constitutes an active topic of research [29, 30, 31].
+6. Related Work
+The DA domain encompasses a large variety of techniques and algorithms,
+like nudging [32], kriging [33], ensemble Kalman Filter [34], ensemble max-
+32
+
+imum likelihood filter [35], or particle filter [36]. For an overview, we refer
+to [37, 38]. We focus here on statistical DA relying on an ensemble run of
+the model to compute a statistical estimator (co-variance matrix for EnKF,
+PDF for particle filters).
+To aggregate the data produced by all members (i.e., particles) two main
+groups of approaches are used. Either the data is stored to files and then
+processed in a second step (off-line mode), or the data is processed on-line
+usually within a large MPI code in charge of running the members and data
+processing. Frameworks relying on the off-line mode include EnTK [39], with
+the largest published DA use cases reaching 4,096 members for a molecular
+dynamics application with an EnKF filter [40]. OpenDA also follows this
+model, using NetCDF for data exchange with the NEMO code [41]. DART
+supports both [42], with reports of large scale DA in off-line mode in [43]
+(about 1,000 members with an oceanic code), or [44, 45] (1,024 member,
+LETKF filter, 6 M Fugaku cores). File based approaches have the benefit of
+their simplicity, providing fault tolerance and elasticity. But these solutions
+do not support member virtualization, state caching and prefetching.
+So
+starting or restarting a member requires to request a new resource allocation
+launching a new instance of the model code with all the associated start-up
+costs. Node-local persistent storage capabilities, for instance with SSDs, can
+store intermediate files, avoiding the PFS to loosen the I/O bottleneck. They
+are used for member state storage in [44], but without specific fault tolerance
+mechanism. So if a node fails and the node-local storage becomes unavailable,
+the lost member states need to be recomputed. Besides leveraging the node
+storage for the distributed cache, using node-storage rather than the parallel
+file system as a globally shared file system layer is one of our future goals.
+The on-line mode avoids the I/O bottleneck. PDAF [46], which supports
+both modes, has for instance been used on-line for the assimilation of ob-
+servations into the regional earth system model TerrSysMP. DA was based
+on EnKF with up to 256 members [47]. ESIAS uses on-line DA via particle
+filters with up to 4,096 particles on a wind power simulation on Europe [21].
+Notice that we work with the same WRF component of ESIAS in this paper,
+using a configuration on a similar domain but at higher spatial resolution and
+with more advanced and more time consuming physics. We also find ad hoc
+MPI codes for on-line DA as in [48] (atmospheric model, 10,240 members,
+Local ENKF filter, 4,608 compute nodes). But all these MPI approaches
+lead to monolithic code without support for fault tolerance, elasticity or load
+balancing. In [49], the authors analyze various particle propagation schedul-
+33
+
+ing but at limited scale (6 compute nodes and 300 particles). We performed
+experiments on a similar architecture as our proposed one, but for EnKF in-
+stead of PF [19]. We demonstrated fault-tolerance, elasticity and scalability
+for experiments using up to 16 k members, 16 k cores for DA with EnKF for
+the hydrology code Parflow. In contrast to our novel proposal, EnKF re-
+quires a centralized filter update, gathering the full ensemble of states at the
+central instance for the assimilation of observations. In our novel proposal
+for PF, we exploit certain properties of particle filters to suppress the server
+bottleneck and significantly reduce data movements.
+7. Conclusion
+In this article we proposed an architecture for handling very large en-
+sembles for particle filters.
+The architecture was designed to address the
+challenge of exascale computing that will allow massive ensemble runs [50].
+The architecture is based on a server/runner model where runners support a
+distributed cache and virtualization of particle propagation, while the server
+aggregates the weights computed by the runners and ensures the dynamic
+balancing of the work load. Particle propagation is virtualized so the re-
+quired number of runners is decoupled from the particle number. With the
+addition of a distributed checkpointing mechanism, the architecture supports
+dynamic changes in the number of runners during execution for fault toler-
+ance and elasticity. Experiments with the WRF weather simulation code
+show that our framework can run at least 2,555 particles on 20,442 cores
+with a 87% scaling efficiency. Dynamic particle-propagation scheduling and
+caching enable to avoid 88% of the global I/O operations. Compared to the
+ESIAS file based approach, Melissa improves resource usage 2.83 times at
+1024 particles.
+Future work includes experimenting with adaptive or localized particle
+filters as well as combining particle and Kalman filter.
+We also plan to
+extend the distributed cache and fault tolerance algorithm to fully avoid the
+centralized file system and only rely on node-local SSDs for particle storage.
+Acknowledgement
+This project has received funding from the European Union’s Horizon
+2020 research and innovation program under grant agreement No 824158
+(EoCoE-2). This work was granted access to the HPC resources of IDRIS
+34
+
+under the allocation 2020-A8 A0080610366 attributed by GENCI (Grand
+Equipement National de Calcul Intensif). The authors gratefully acknowl-
+edge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for
+funding this project by providing computing time through the John von Neu-
+mann Institute for Computing (NIC) on the GCS Supercomputer JUWELS
+at J¨ulich Supercomputing Centre (JSC). We acknowledge the access to the
+meteorological input data from the Meteocloud of SDL Climate Science, JSC.
+We also acknowledge PRACE for awarding us access to JUWELS at J¨ulich
+Supercomputing Centre (JSC), Germany.
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+Artificial neural network as an effective tool to calculate parameters
+of positron annihilation lifetime spectra
+M. Pietrow∗1 and A. Miaskowski2
+1Institute of Physics, M. Curie-Skłodowska University, Pl. M. Curie-Skłodowskiej 1, 20-031
+Lublin, Poland
+2Faculty of Production Engineering, University of Life Sciences, Akademicka 13, 20-950
+Lublin, Poland
+January 5, 2023
+Abstract
+The paper presents the application of the multi-layer perceptron regressor model for predicting the
+parameters of positron annihilation lifetime spectra using the example of alkanes in the solid phase.
+A
+good agreement of calculation results was found when comparing with the commonly used methods. The
+presented method can be used as an alternative quick and accurate tool for decomposition of PALS spectra
+in general. The advantages and disadvantages of the new method are discussed.
+1
+Introduction
+Positron Annihilation Lifetime Spectroscopy (PALS) is one of the useful experimental methods using positrons
+for studying structural details in a wide spectrum of materials, in particular in the solid state [1]. This method is
+based on the annihilation of positrons where their lifetime and annihilation intensity in the sample is dependent
+on some properties of the material in the nano scale, including local electron density, bound electron energy,
+and the density and size of free volumes in the sample.
+Depending on the material, besides the process of direct annihilation, a positron can form a meta-stable atomic
+state with an electron, called a positronium (Ps), which can exist in two spin states referred to as para- and
+ortho-Ps differing in properties (especially, their lifetimes differ in vacuum by three orders of magnitude) [2]. A
+number of conditions must be met for the Ps to be formed in matter. One of them is that free volumes of a
+sufficiently large size must be present. For these materials, a Ps is extremely useful in material science since its
+lifetime can be related to the size of free volumes [3]. Depending on the structure of the sample, there is possibly
+a variety of Ps components which annihilate with characteristic lifetimes. All these populations give their own
+account to the positron annihilation spectrum measured experimentally. PALS spectra require decomposition
+in the post-measuring procedure of decomposition resulting in both the calculation of lifetimes for particular
+species of positrons and the relative amplitudes for these processes (so-called spectrum inversion problem) [4].
+Many algorithms used for data processing require assuming an exponential character of positron decay. They
+also require fixing the number of components used during the decomposition. For example, the method used by
+one of the adequate software, the LT programme [5] or PALSfit [6], consists in fitting the PALS experimental
+spectrum to a sum of a given number of exponential functions usually convoluted with the (multi) gaussian
+apparatus resolution curve.
+The PALS spectra used here were measured for normal alkanes (n-alkanes), i.e. the simplest organic molecules
+where carbon atoms form a straight chain of the molecule and are saturated by hydrogen atoms. The n-alkanes
+with a different number n of carbon atoms in the molecule form a homologous series described by the general
+chemical formula CnH2n+2 (Cn is used as an abbreviation). Alkanes in the solid phase form molecular crystals
+where the trains of elongated molecules are separated by gaps called the inter-lamellar gaps. Ps can be formed
+∗Corresponding Author: [email protected]
+1
+arXiv:2301.01521v1  [physics.atom-ph]  4 Jan 2023
+
+Figure 1: Schematic view of the MLP applied. The PALS data from consecutive channels of the MCA are
+transferred as the amplitudes of the consecutive input neurons Ini. hi denote neurons in the i-th hidden layer
+whereas Outi denote output neurons returning chosen PALS decomposition parameters.
+in the free volumes made by both the gaps and the spaces generated by changes in the conformation with
+temperature [3]. Using the PALS technique, the size of these free volumes can be determined from the lifetime
+and the relation between both being given by the Tao-Eldrup formula or its modification [7]. According to our
+previous analysis of alkanes carried out with the use of the PALS technique, the best results of the spectrum
+decomposition are achieved assuming only one population of ortho- and para-Ps, whereas the ratio of the ortho
+to para intensity is fixed at 3/1.
+Tools of machine learning like genetic algorithms or artificial neural networks have been used to perform nu-
+merical calculations in a variety of aspects in positron science [8, 9, 8, 10, 11, 12]. They have also been used for
+unfolding the lifetimes and intensities from PALS spectra [13, 14, 15, 16]. Possibly due to the low computing
+power of the hardware and the low time resolution of PALS spectrometers at the time when the neural network
+algorithms for decomposition of PALS spectra were proposed, most of the spectra used in these calculations are
+simulated by the software but not measured directly. For the same reason, the neural network architecture used
+there does not allow changing parameters as much as is allowed by algorithms developed today. Furthermore,
+no procedure allowing application for the same calculations of spectra registered for different time constants per
+channel has been presented since then. Thus, the preferred software used for spectrum decomposition is still
+based on non-linear fitting algorithms which do not include a possibility of establishing the result based on a
+multi-spectra set at the same time.
+Here, we present an approach to analysis of PALS spectra based on the multi-layer perceptron (MLP) model,
+which is one of the tools of machine learning [17]. The model assumes a network of inter-connected neurons
+grouped in the input layer (Ini), the hidden neurons layers (hk
+i ) and the output neurons layer (Outi), where i
+goes over the neurons in a given layer and k numbers the hidden layers. A graphical diagram of the network
+used is shown in fig. 1. The numbers of In and Out neurons are determined by the amount of the independent
+input data introduced to the network and the data defined to be the results of calculation in a given problem,
+respectively. The number of hidden layers and the number of neurons within these layers are set experimentally
+to optimise the network to give required results. To each layer (excluding the output layer), one bias neuron is
+attached for technical reasons [18]. The tool assumes the learning process first, where the In neurons are fed
+with the data for which the result of the Out neurons is known in advance. During this process, the weight
+coefficients for pairs of inter-connected neurons are adjusted by an algorithm, so that the output of the MLP
+can give results most similar to the expected ones. The MLP becomes to be trained after a number of iterations
+of training. Once the MLP results of learning are satisfied, the MLP can be used to calculate the output for
+the input data never used in the training process.
+2
+
+Chanel (ns)
+MCA times values followed by log of
+normed PALS spectra amplitudes
+Input
+Input
+Input I
+T
+2
+z
+000
+000
+000
+000
+000
+Output
+indno
+T
+3
+PALS output parameters
+I2, T2, T3The MLP type of network can be applied to solve the problem of both classification and regression. For the
+first group of problems, it is required from the MLP to ascribe the values of the output parameters in the form
+of well separated categories. These so-called labels can always be parametrised by a discrete set of numbers.
+The problem described in this paper is classified as rather a regression problem (MLPR) where the values of
+the output at each Out neuron are characterised by a continuous set of values. Consequently, the output may
+contain values approaching these appearing during the learning process but may not necessarily be exactly of
+the same value. The internal algorithms of the MLPR allows regarding the learning process as a way of finding
+the quasi-continuous output function of input parameters. In our case, based on the data from the PALS spectra
+applied as the input values of the perceptron, the MLPR is used for solving the regression problem of finding
+the values of key PALS parameters on the output.
+2
+Method
+The scikit-learn library was used to estimate the PALS parameters for alkanes [19]. In our case, the MLP
+regressor class (called MLPRegressor), which belongs to one of the supervised neural network models, was
+implemented. In this class, the output is a set of continuous values. It uses the square error as the loss function.
+This model optimises the squared error using the Broyden–Fletcher–Goldfarb–Shanno algorithm (LBFGS) [20],
+which belongs to quasi-Newton methods. Some MLPRegressor parameters playing a key role are mentioned
+below. Their values require to be tuned, especially the alpha hyper-parameter, which helps in avoiding over-
+fitting by penalising weights with large magnitudes. A full list of parameters of MLPRegressor is defined in [21].
+In the learning process here, we used spectra collected for years from an analog spectrometer for several alkanes
+(in the range of C6 – C40) measured at several temperatures (-142◦C – 100◦C). Irrespective of both the goal of
+the particular experiment and the length of the alkane chain used as a sample, the initial assumptions made for
+starting the analysis of the spectra made by the LT programme [5] were the same. Each measurement resulting
+in the spectra used was performed with a sample prepared in a similar way, i.e. the sample was degassed, and the
+rate of cooling or heating was the same. In each case, the measurement at constant temperature took place for
+at least one hour which gave some hundreds of thousands of annihilations (the strength of the radioactive source
+was similar in each case). During some experiments the temperature was changed stepswise but each spectrum
+was collected at constant temperature. The most important issue here is that the post-experimental analysis of
+the spectra was conducted under the same general assumptions every time. Especially, for the decomposition of
+these spectra, we used LT supposing that the time resolution curve can be approximated by one-gaussian curve.
+Every time it was assumed that the annihilation process in the Kapton envelope accounted for 10% (so-called
+source correction). Additionally, only one component was always assumed for para- and ortho-Ps, whereas their
+intensity ratio was fixed at the value 3/1 (see [22] for details of the experimental procedure).
+Taking into account these assumptions, each spectrum was decomposed into three exponential curves for which
+the intensities (I) and lifetimes (τ) were calculated for the following sub-populations of positrons: the free
+positron annihilation (I2, τ2), para (I1, τ1), and ortho-Ps (I3, τ3)1. The database collected in this way contained
+7973 PALS spectra, wherein about 75% were used in the neural network training process and the rest were used
+as a testing set for checking the accuracy of the results given by the learned network.
+The number of input neurons is determined by the number of channels of the Multi-channel Analyser (MCA)
+module of the PALS spectrometer recording PALS spectra. Furthermore, the number of the output neurons
+is related in this model to the number of PALS parameters, which are supposed to be predicted for further
+studies of physical processes in the sample. The decomposition of the PALS spectrum made by commonly used
+programs, like LT, allows determining (I,τ) pairs for all assumed components of a given spectrum. However,
+often, not all these parameters are needed for further analysis. Furthermore, some of these parameters are
+inter-dependent. For example, in the case of PALS spectra for the alkanes discussed here, one assumes that
+the spectrum is built up by events from the three populations of positrons mentioned above (τ1 – τ3, I1 –
+I3 parameters). However, from the practical view point, only τ2, I2, τ3, and I3 are then used for studying
+physical processes and the structure of the sample. Furthermore, in this case, Ii are inter-dependent and fulfil
+the following relations I1+I2+I3=100%2 and I3/I1=3. Thus, effectively, the parameters considered as the Out
+1Numbering of the indices is related to the length of τ. The increasing values of the indices correspond to the rising length of
+lifetime.
+2Annihilation in Kapton was subtracted in advance.
+3
+
+Figure 2: Number of spectra (horizontal axis) with a given value of the time constant per channel ∆ (vertical
+axis) used as a data set in the presented calculations.
+parameters of MLPR are only I2, τ2, and τ3. According to this, we declared in our modelling only three output
+neurons for receiving values for these three parameters.
+3
+Preparation of input and output data
+During the PALS measurements, the time constant per channel (∆) varied, depending on the internal properties
+and settings of the spectrometer. Most of the data used here were collected with ∆=11.9 ps; however, some
+spectra were measured with ∆=11.2 ps, 13.2 ps, 11.6 ps, and 19.5 ps (fig. 2). Therefore, it is important for the
+In neurons to code the PALS amplitude samples not in the relation to the channel numbers but in the scale of
+time. Hence, in addition to the spectrum amplitudes, the regressor has to learn the times associated with these
+amplitudes. Thus, one half of the In neurons is fed with time values for consecutive channels of a spectrum,
+whereas the second half is fed with the values of their amplitude. The advantage of the regression approach
+applied here is the ability to test spectra measured even for a time sequence that has never appeared in an
+extreme case in the training process.
+This method requires setting correctly a common zero-time for each spectrum. To achieve this, the original
+data from the left slope of the spectrum peak (and only a few points to its right) were used to interpolate the
+resolution curve, which is assumed to be in the gaussian form. The position of this peak defines a zero-time
+for a spectrum. One-gaussian interpolation is compatible with previous LT analysis assumptions. Based on
+the common starting position for all spectra established in this way, the values of time for each channel on the
+right to the peak were re-calibrated for each spectrum depending on ∆ for which the spectrum was measured.
+Finally, for further analysis, we took the same N number of consecutive channels for each spectrum on the right
+to its peak (points pi in fig. 3). The δ parameter shown in fig. 3 denotes the distance (in time units) between
+the first point on the right to the peak and the calculated time position of the peak. The number N taken
+for further analysis was established experimentally. Finally, the spectrum data for the MLPR input are the N
+points pi with their two values: the re-calibrated number of counts in a given channel (see below) and their
+re-calibrated times of annihilation.
+Then, to minimise errors, the original input data were transformed before application. Each original spectrum
+was stored in 8192 channels of MCA. Firstly, starting from the first channel on the right to the spectrum
+maximum (p1 in fig. 3), 2k channels were taken from the original spectrum.
+This means that the spectra
+were truncated at about 25 ns of the registration time (varying to some extent, depending on the ∆ for a
+given spectrum). Secondly, to smooth random fluctuations, the data were smoothed in most cases. One of the
+examples of smoothing is averaging over five consecutive channels. In this case, the number of samples in each
+spectrum shrank from the original 2k channels to the amount of 400. Since the In neurons transfer information
+about the pair of values – times (t-part) and amplitudes (A-part), 800 input neurons that fed the MLPR with
+the data in this case were declared. Thirdly, to standardise the range of the input data values, the set of the
+PALS amplitudes was normalised to the maximum value of the amplitude and then logarithmised. According
+to these transformations, the A-part data covered the numerical range [-9,0] – fig. 4. Furthermore, to adjust
+the range of the values in the t-sector, the values of time were divided by -2.5. As a result, all data transferred
+4
+
+(sd)
+ constant (
+18
+16
+Time calibration
+14
+12
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+Number of spectraFigure 3: Schematic view of a peak region of the PALS spectrum. The bullets indicate (t, log(A)) pairs saved
+in the MCA channels, whereas the star indicates a position of a peak calculated assuming a gaussian shape
+of an apparatus distribution function. Only the points to the right of the star (p1, p2, ...) are taken as data
+introduced to MLPR. The δ parameter denotes a time distance between the calculated peak and the first point,
+whereas ∆ is the time distance between two points.
+Figure 4: Input data directed to In neurons can be divided into two sub-sets: t-part which is a set of time
+values for points p1, p2, ... (see fig. 3) and A-part coding the log function of their normalised amplitudes. In
+special cases, these data are smoothed or compressed before use in MLPR.
+to the In neurons were in the range of [-10,0].
+Additionally, we applied some transformation of the original values for the Out neurons in order to have their
+values at each neuron scaled to the same range. Initially, the first output neuron is related to I2, whereas its
+original value range is typically tenths (in % units). The second neuron transfers the information related to
+τ2 whose original values are of the order of 0.1 (of ns), whereas the order of τ3 related to the third neuron is
+originally 1 (of ns). In order to have the uniform order of numerical values on all Out neurons, the data that
+finally feed with them are [I2/10, τ2·10, τ3].
+The criterion of acceptance of training the network was the best value of the score validation function defined
+for this regressor as
+S = 1 −
+�
+N(Otrue − Opred)2
+�
+N O2
+true
+,
+(1)
+where Otrue, Opred – expected (known) and calculated (predicted) values of the result, respectively [19, 21]. S
+is calculated for both the learning and testing sets separately. N here denotes the number of spectra in the
+trained or tested set. The optimum value of S is ≈1.
+5
+
+0
+P
++2△0
+t-part
+A-part
+-2 
+-4 
+a.u.
+-6 
+-8 
+0
+100
+200
+300
+400
+500
+600
+700
+800
+neurons4
+Results
+The MLPRegressor used in these calculations requires establishing some key parameters [21] influencing the
+ability to learn and a speed of the learning process. We performed some tests trying to optimise these param-
+eters. The best results we obtained by the settings shown in tab. 1. Both the names and the meaning of the
+Table 1: Values of the MLPRegressor parameters applied for producing the final MPLR results.
+Parameter
+value
+hidden_layer_sizes =
+7×150
+activation =
+relu
+solver =
+lbfgs
+alpha =
+0.01
+learning_rate =
+invscaling
+power_t =
+0.5
+max_iter =
+5e+9
+random_state =
+None
+tol =
+0.0001
+warm_start =
+True
+max_fun =
+15000
+technical parameters shown in the table are identical to these defined in the routine description [21]. Once
+the key parameters of the MLPR were established (especially the solver), we performed tests of credibility
+of the network changing the number of hidden layers, the number of neurons within (hidden_layer_sizes
+parameter), and the alpha parameter. The results in tab. 2 show examples of the results. For these networks,
+we specified the mean validation score parameter for both the training ⟨Str⟩ and testing ⟨Ste⟩ sets separately
+with their variation δS. Averaging was made over the results of ten runs of the training process for identical
+networks differing by initially random weights. We did not notice any rule giving a ratio of the numbers of
+neurons that should be declared in the consecutive hidden layers (especially as the number of neurons should
+decrease proportionally in the consecutive layers). A few initial examples shown here suggest that the accuracy
+of results increases when both the number of hidden layers and the number of neurons inside increase. However,
+the last two rows of the table show that a further increase in these parameters does not give better results.
+Finally, the network that gave a nearly best result was chosen (marked in bold ⟨Str⟩ in the table). It was checked
+for this network that an increase in the iterations of training (max_iter parameter) beyond about 5·109 did
+not improve ⟨S⟩.
+Table 2: Valuation score for chosen values of some MLPR parameters. S values are averages over 10 runs with
+random initial neuron weights. A nearly optimum case of parameters is placed in a row with S marked in bold.
+hidden_layer_sizes
+max_iter
+alpha
+⟨Str⟩
+δStr
+⟨Ste⟩
+δSte
+30 × 25 × 15
+106
+0.7
+0.950
+0.003
+0.942
+0.004
+3 × 100
+108
+0.7
+0.969
+0.005
+0.965
+0.008
+3 × 100
+108
+0.1
+0.974
+0.005
+0.968
+0.008
+3 × 100
+5· 108
+0.1
+0.975
+0.003
+0.976
+0.003
+4 × 100
+108
+0.1
+0.978
+0.004
+0.975
+0.006
+500 × 400 × 300 × 200
+5·109
+0.01
+0.977
+0.003
+0.974
+0.007
+7 × 150
+5·108
+0.01
+0.985
+0.002
+0.975
+0.013
+500 × 500 × 400 × 400×
+×300 × 300 × 200 × 200
+5·109
+0.01
+0.978
+0.005
+0.977
+0.008
+8 × 500
+5·109
+0.01
+0.982
+0.004
+0.980
+0.005
+For several finally tested networks, the spectrum of the magnitude of inter-neurons weights was checked. It
+is expected that weights that differ significantly from the average range of values may affect the stability of
+the results. In this case, the range of weight values seems to be quite narrow. As shown in fig. 5, the weight
+magnitude order (exponent of weights) for the chosen network ranges from 10−5 to 100, while the relative
+number of cases in these subsets changes exponentially. The lack of values outside the narrow set of values
+6
+
+suggests that self-cleaning of the resultant weights is performed by the MLPRegressor algorithm itself.
+Figure 5: Number of cases (log scale) of exponents of weights for a network with 7×100 hidden layers. For this
+network, the key parameters are: solver=lbfgs, max_iter=5·1011, alpha=0.005, learning_rate=invscaling,
+and activation=relu.
+The number of all PALS spectra used as a database for the network was 7973, and 6500 were used to learn
+the output values (training set) by the network, while the rest were used for checking the results of learning
+(testing set). Tab. 3 shows a few examples of randomly taken results given by one of the networks finally used.
+The results given by the trained network were compared to the expected values known from the LT analysis.
+Table 3: Examples of a few randomly taken results of calculations (prediction) of the I2, τ2, and τ3 parameters
+compared to the expected values calculated by LT. Here, hidden_layer_size=7×150, S=0.985 for both
+training and testing sets.
+Example
+I2 [%]
+τ2 [ns]
+τ3 [ns]
+expected
+predicted
+expected
+predicted
+expected
+predicted
+1
+68.0
+68.8
+0.27
+0.28
+1.21
+1.20
+2
+47.2
+47.6
+0.38
+0.39
+3.23
+3.18
+3
+65.4
+65.5
+0.30
+0.29
+1.35
+1.38
+4
+78.3
+78.3
+0.23
+0.24
+1.11
+1.12
+5
+52.4
+52.4
+0.35
+0.34
+2.93
+2.93
+6
+60.5
+60.4
+0.31
+0.30
+1.25
+1.20
+7
+59.3
+58.8
+0.29
+0.29
+1.19
+1.22
+8
+61.0
+62.3
+0.30
+0.31
+1.91
+1.91
+9
+70.2
+69.5
+0.21
+0.21
+1.06
+1.10
+10
+39.9
+38.8
+0.23
+0.22
+1.15
+1.13
+Although S for both the trained and tested sets in this case is not the highest one obtained in our tests, the
+result of the use of this network is satisfactory in a practical sense because the deviation of the predicted and
+expected result is in the range of deviation given by LT itself.
+The problem of pre-preparation of spectra for calculations by MLPR is worth mentioning. The main problems
+are where the spectrum should be cut and to what extent it is acceptable to smooth the spectra by averaging
+their consecutive values. As for the first problem, it was determined by series of runs for which the spectra were
+cut at other that mentioned limit of 2k channels that this number of channels was almost the best choice. ⟨S⟩
+was found to worsen in the case of a shorter cut (say, 1.5k channels), and did not improve significantly in the
+case of the longer ones (e.g. 3k channels) (but it took longer to compute the result because of an increase in
+the number of In neurons).
+7
+
+amount of weights
+104
+1000
+100
+10
+-5
+-3
+-2
+-4
+1
+0
+exponent of weightThe accuracy of the prediction increases when the learning and testing processes are limited to one only ∆ with
+all parameters of the network kept constant. In this case, the set of values in the t-part for every spectrum
+varies in a much narrower range (only δ changes). In this case, the training process is more effective even if
+the size of the training set is reduced. To show this, we separated the set of spectra measured for only one
+∆=11.9 ps. Consequently, the whole set of samples under consideration shrank to 4116 members, and 3000 of
+them were used for training after the transformations described above. The score S obtained in this case was
+much greater than S for an identical network applied to spectra with all possible ∆. The comparison of these
+two cases is shown in tab. 4 (the last row of the table). Here, to have the result reliable for networks with
+different (random) initial weights, the score was averaged over 30 runs.
+Table 4: Comparison of MLPR validation score S for different formats of the input data. Comparison of the
+results for ’raw’ data (log of normalised and adjusted data according to the procedure described in section 3)
+and data on which the moving average and compressing average (by each 3 and 5 separate spectrum points)
+are applied. The result for the network fed with the data collected for one chosen ∆ is added in the last row of
+the table.
+MLPR and spectra parameters
+⟨Str⟩
+⟨Ste⟩
+solver=lbfgs
+activation=relu
+learning_rate=invscaling
+alpha=0.01, In=800
+hidden_layer_sizes=7×150
+max_iter=5×109
+averaged over 30 trials
+Several ∆s
+Ntr=6500 (∼ 80%)
+Nte=1473
+unsmoothed data
+0.984±0.005
+0.963±0.006
+moving average
+0.984±0.005
+0.977±0.007
+k=3
+0.981±0.005
+0.976±0.007
+k=5
+0.981±0.006
+0.974±0.010
+Fixed ∆=11.9 ps
+Ntr=3000 (∼73%)
+Nte=1116
+k=5
+0.993±0.002
+0.989±0.003
+The validation score S is sensitive to smoothing the spectrum which reduces to some extent the information
+given by the PALS spectrum. In tab. 4, two cases are compared where each 3- and 5-tuples of points of the
+spectrum (forming non-overlapping windows) were taken to calculate their average amplitude. For example,
+N=3500 points of the initial spectrum are reduced to 700 points when averaging over k=5 points; when the
+remainder of the division of N by k is not zero, an integer quotient is taken. ⟨S⟩ calculated for these two cases
+shows that both of them give the same results statistically. However, further shrinking the spectrum by setting
+k=6 or more produces worse S.
+Tab. 4 also shows the S parameter when the moving average is applied during preparation of spectra. The
+sampling window applied here is 10. The comparison of this result to the result of calculation with unsmoothed
+data shows that the application of the moving average does improve predictions for the testing data set.
+5
+Conclusions
+We have shown in this paper that the easy-to-reach machine learning MLPRegressor tool enhanced with some
+programming in Python making some preparation of data, can be used as an alternative method of solving
+the problem of inversion of PALS spectra. The main disadvantage of the presented method is the need of
+decomposition of training spectra by other software to have Out values for training. Once the training set is
+collected and the network is trained, the algorithm works very quickly, giving the result for the tested spectrum.
+The training process used here is based on results given by LT, i.e. a method producing results with some
+uncertainty itself. The uncertainty produced by the LT is caused by the use of numerical methods to compute
+the fit in particular cases. On the other hand, since the MLPR prediction bases on information from a large
+set of spectra, this approach seems to be less sensitive to the specific shape of a given spectrum and may
+be more accurate in predicting parameters. Furthermore, the presented method seems to be faster than the
+referenced ones, since calculations made by a trained network are reduced to simple transformations of matrices
+and vectors, which is not demanding computationally and less sensitive to numerical problems.
+Although the model presented here is similar to that described in [14] (and repeated in [15]), there are significant
+differences indicated in tab. 5. Our experimental data are collected by spectrometers differing in functional
+properties, especially differing in time resolution. Even for one spectrometer, this parameter should be re-
+calibrated periodically due to changes in experimental conditions, especially temperature. In the algorithm
+8
+
+Table 5: Comparison of key parameters and results of the MLPR modelling applied in this study (skLearn) and
+a three-component spectrum analysis published previously (presented in [14] and [15]).
+Pázsit [14]
+An [15]
+skLearn
+Type of training spectra
+simulated
+simulated
+real (alkanes)
+No. of training spectra
+575
+920
+7973
+Type of spectra tested
+simulated
+simulated, silicon
+alkanes
+No. of test spectra
+50
+100 (30)
+1473
+Type of network
+one-layer perc.
+one-layer perc.
+multi-layer perc.
+Channel width [ps]
+23.2
+24.5
+some (11.2-19.5)
+No. of MCA/taken channels
+1500/1500
+1024/1024
+8192/3500
+Approx. no. of counts in spec.
+10M
+10M
+∼400k
+Solver
+backward error prop.
+backward error prop.
+some
+No. of hidden layers
+1
+1
+some
+I2, τ3 average error [%]
+on tested simulated spectra
+7.3, 1.0
+1.07-3.52, 0.55-1.21
+-, -
+I2, τ3 average error [%]
+on tested real spectra
+-, -
+-, -
+1.03, 1.70
+presented in [14], the same resolution curve for all spectra is assumed. In our data preparation procedure, the
+parameters of the resolution curve are interpolated for each case. Based on this, the δ parameter is calculated
+and the value of the shift in time is established for consecutive channels. Although one-gaussian resolution
+curve was assumed here, it is possible to extend this algorithm for much more complicated cases where the
+distribution curve consisted in a sum of gaussians, for example. As already mentioned in [14], in that case,
+a possibility of recognising a distribution function would give compatibility to MELT [23]. Such an extension
+requires extending the calculations by applying another neural network, working in advance, which returns the
+parameters of the resolution curve in a given case. This problem has been solved by application of a Hopfield
+neural network [16]. Taking into account our collection of spectra, it was checked with the use of LT and
+(occasionally) with MELT that the apparatus resolution curve is one-gaussian for our spectra. Hence, they do
+not allow testing such an extended model.
+Furthermore, the MCA module of spectrometers may differ in the time constant per channel ∆. Thus, spectra
+used as a training data set may be collected for different channel widths. Taking into account the method
+presented in [14] for fixed ∆, the training result is of little use for spectra collected with another ∆. Oppositely,
+we have shown the possibility of application of an improved algorithm to data collected for different ∆s. The
+data collected from many spectrometers may contribute to a large training data set, which allows solving the
+inversion problem for any PALS spectrum and, thus, may be a universal tool that can be used in different
+laboratories. Although the set of ∆ used here is small, the accuracy of the results is quite good. To use this
+tool to determine the real-world spectrum parameters, the training process should be extended by adding the
+spectra measured for a wider range of ∆.
+For greater generalisation, it is possible in principle to attach spectra collected for other compounds to the
+training data set. For consistency, it suffices for a training database to keep the same number of components
+(three here) in spectrum decomposition. However, in practice, some incompatibilities of the spectra for different
+compounds may arise because decomposition into a few exponential processes is probably always a simplification
+of a real case where some distribution of the size and shape of free volumes should be taken into account as well
+as other Ps formation details.
+Although the approach presented here is reduced to the analysis of alkanes solely, the algorithm can be applied
+in calculation of PALS parameters of other types of samples as well.
+References
+[1] D. Manara, A. Seibert, et al. Positron annihilation spectroscopy. In M.H.A. Piro, editor, Advances in
+Nuclear Fuel Chemistry, chapter 2.3.5, pages 126–131. Elsevier Ltd., Woodhead Publishing, 2020.
+[2] W. Greiner and J. Reinhardt. Field Quantization. Springer, 2013.
+9
+
+[3] T. Goworek. Positronium as a probe of small free volumes in crystals, polymers and porous media. Ann.
+Univ. Mariae Curie Sklodowska, sectio AA – Chemia, LXIX:1–110, 2014.
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