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+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain
+Decomposition Methods
+Ali Taghibakhshi 1 Nicolas Nytko 2 Tareq Uz Zaman 3 Scott MacLachlan 4 Luke N. Olson 2 Matt West 1
+Abstract
+Domain decomposition methods (DDMs) are pop-
+ular solvers for discretized systems of partial dif-
+ferential equations (PDEs), with one-level and
+multilevel variants. These solvers rely on several
+algorithmic and mathematical parameters, pre-
+scribing overlap, subdomain boundary conditions,
+and other properties of the DDM. While some
+work has been done on optimizing these param-
+eters, it has mostly focused on the one-level set-
+ting or special cases such as structured-grid dis-
+cretizations with regular subdomain construction.
+In this paper, we propose multigrid graph neural
+networks (MG-GNN), a novel GNN architecture
+for learning optimized parameters in two-level
+DDMs. We train MG-GNN using a new unsuper-
+vised loss function, enabling effective training on
+small problems that yields robust performance on
+unstructured grids that are orders of magnitude
+larger than those in the training set. We show
+that MG-GNN outperforms popular hierarchical
+graph network architectures for this optimization
+and that our proposed loss function is critical to
+achieving this improved performance.
+1. Introduction
+Among numerical methods for solving the systems of equa-
+tions obtained from discretization of partial differential equa-
+tions (PDEs), domain decomposition methods (DDMs) are
+a popular approach (Toselli & Widlund, 2005; Quarteroni
+& Valli, 1999; Dolean et al., 2015). They have been exten-
+sively studied and applied to elliptic boundary value prob-
+lems, but are also considered for time-dependent problems.
+1Department of Mechanical Science and Engineering, Univer-
+sity of Illinois at Urbana-Champaign 2Department of Computer
+Science, University of Illinois at Urbana-Champaign 3Scientific
+Computing Program, Memorial University of Newfoundland
+4Department of Mathematics and Statistics, Memorial Univer-
+sity of Newfoundland. Correspondence to: Ali Taghibakhshi
+<[email protected]>.
+Schwarz methods are among the simplest and most popular
+types of DDM, and map well to MPI-style parallelism, with
+both one-level and multilevel variants. One-level methods
+decompose the global problem into multiple subproblems
+(subdomains), which are obtained either by discretizing
+the same PDE over a physical subdomain or by projection
+onto a discrete basis, using subproblem solutions to form a
+preconditioner for the global problem. Classical Schwarz
+methods generally consider Dirichlet or Neumann bound-
+ary conditions between the subdomains, while Optimized
+Schwarz methods (OSM) (Gander et al., 2000) consider
+a combination of Dirichlet and Neumann boundary condi-
+tions, known as Robin-type boundary conditions, to improve
+the convergence of the method. Restricted additive Schwarz
+(RAS) methods (Cai & Sarkis, 1999) are a common form
+of Schwarz methods, and optimized versions of one-level
+RAS has been theoretically studied by St-Cyr et al. (2007).
+Two-level methods extend one-level approaches by adding
+a (global) coarse-grid correction step to the preconditioner,
+generally improving performance but at an added cost.
+In recent years, there has been a growing focus on using
+machine learning (ML) methods to learn optimized parame-
+ters for iterative PDE solvers, including DDM and algebraic
+multigrid (AMG). In Greenfeld et al. (2019) convolutional
+neural networks (CNNs) are used to learn the interpola-
+tion operator in AMG on structured problems, and in a
+following study (Luz et al., 2020), graph neural networks
+(GNNs) are used to extend the results to arbitrary unstruc-
+tured grids. In a different fashion, reinforcement learning
+methods along with GNNs are used to learn coarse-gird
+selection in reduction-based AMG in Taghibakhshi et al.
+(2021). As mentioned in Heinlein et al. (2021), when com-
+bining ML methods with DDM, approaches can be catego-
+rized into two main families, namely using ML within a clas-
+sical DDM framework to obtain improved convergence and
+using deep neural networks as the main solver or discretiza-
+tion module for DDMs. In a recent study (Taghibakhshi
+et al., 2022), GNNs are used to learn interface conditions
+in optimized Schwarz DDMs that can be applied to many
+subdomain problems, but their study is limited to one-level
+solvers. Two-level domain decomposition methods often
+converge significantly faster than one-level methods since
+they include coarse-grid correction, but obtaining optimized
+arXiv:2301.11378v1  [cs.LG]  26 Jan 2023
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+multilevel DDMs for general unstructured problems with
+arbitrary subdomains remains an open challenge.
+Graph neural networks (GNNs) extend learning based meth-
+ods and convolution operators to unstructured data. Similar
+to structured problems, such as computer vision tasks, many
+graph-based problems require information sharing beyond
+just a limited local neighborhood in a given graph. How-
+ever, unlike in CNNs, where often deep CNNs are used with
+residual skip connections to achieve long range information
+passing, GNNs dramatically suffer depth limitations. Stack-
+ing too many GNN layers results in oversmoothing, which
+is due to close relation of graph convolution operators to
+Laplacian smoothing (Li et al., 2018; Oono & Suzuki, 2019).
+Oversmoothing essentially results in indistinguishable node
+representations after too many GNN layers, due to informa-
+tion aggregation in a large local neighborhood. Inspired by
+the Unet architecture in CNNs (Ronneberger et al., 2015),
+graph U-nets (Gao & Ji, 2019) were introduced as a rem-
+edy for longe-range information sharing in graphs without
+using too many GNN layers. Similar to their CNN counter-
+parts, graph-Unet architectures apply down-sampling layers
+(pooling) to aggregate node information to a coarser repre-
+sentation of the problem with fewer nodes. This is followed
+by up-sampling layers (unpooling, with the same number
+of layers as pooling) to reconstruct finer representations of
+the problem and allow information to flow back to the finer
+levels from the coarser ones.
+As mentioned in Ke et al. (2017), U-net and graph-Unet
+architectures suffer from a handful of problems and non-
+optimalities. In these architectures, scale and abstraction are
+combined, meaning earlier, finer layers cannot access the in-
+formation of the coarse layers. In other words, initial layers
+learn deep features only based on a local neighborhood with-
+out considering the larger picture of the problem. Moreover,
+finer levels do not benefit from information updates until
+the information flow reaches the coarsest level and flows
+back to the finer levels. That is, the information flow has
+to complete a full (graph) U-net cycle to update the finest
+level information, which could potentially require multiple
+conventional layers, leading to oversmoothing in the case of
+graph U-nets. More recently, there has been similar hierar-
+chical GNN architectures utilized for solving PDEs, such
+as those proposed by Fortunato et al. (2022) and Li et al.
+(2020). In each case, the architecture is similar to a U-net, in
+terms in terms of information flow (from the finest to coars-
+est graph and back), and there is no cross-scale information
+sharing, making them prone to the aforementioned U-net
+type problems.
+To fully unlock the ability of GNNs to learn optimal DDM
+operators, and to mitigate the shortcomings of graph U-
+nets mentioned above, we introduce here a novel GNN
+architecture, multigrid graph neural networks (MG-GNN).
+MG-GNN information flow is parallel at all scales, mean-
+ing every MG-GNN layer processes information from both
+coarse and fine scale graphs. We employ this MG-GNN
+architecture to advance DDM-based solvers by developing
+a learning-based approach for two-level optimized Schwarz
+methods. Specifically, we learn the Robin-type subdomain
+boundary conditions needed in OSM as well as the overall
+coarse-to-fine interpolation operator. We also develop a
+novel loss function essential for achieving superior perfor-
+mance compared to previous two-level optimized RAS. The
+summary of contributions of this paper is as follows:
+• Introduce a multigrid graph neural network (MG-
+GNN) architecture that outperforms existing hierarchi-
+cal GNN architectures and scales linearly with problem
+size;
+• Improve the loss function with theoretical guarantees
+essential for training two-level Schwarz methods;
+• Enforce scalability, leading to effective training on
+small problems and generalization to problems that are
+orders of magnitude larger; and
+• Outperform classical two-level RAS, both as station-
+ary algorithm and as a preconditioner for the flexible
+generalized minimum residual (FGMRES) iteration.
+2. Background
+In this section, we review one and two-level DDMs. Let Ω
+be an open set in R2, and consider the Poisson equation:
+−∆Φ = f,
+(1)
+where ∆ is the Laplace operator and f(x, y) and Φ(x, y)
+are real-valued functions. Alongside (1), we consider inho-
+mogeneous Dirichlet conditions on the boundary of Ω, ∂Ω,
+and use a piecewise linear finite-element (FE) discretization
+on arbitrary triangulations of Ω. In the linear FE discretiza-
+tion, every node in the obtained graph corresponds to a
+degree of freedom (DoF) in the discretization, and the set of
+all nodes is denoted by D. The set D is decomposed into
+S non-overlapping subdomains {D0
+1, D0
+2, . . . , D0
+S} (where
+the superscript in the notation indicates the amount of over-
+lap; hence, the superscript zero for the non-overlapping
+decomposition). The union of the subdomains covers the
+set of all DoFs, D = ∪D0
+i , so that each node in D is con-
+tained in exactly one D0
+i . Denote the restriction operator
+for discrete DoFs onto those in D0
+i by R0
+i and the corre-
+sponding extension from D0
+i to D by (R0
+i )T . Following
+the FE discretization of problem, we obtain a linear system
+to solve, Ax = b, where A is the global stiffness matrix.
+For every D0
+i , we obtain the subdomain stiffness matrix
+as A0
+i = R0
+i A(R0
+i )T . In the OSM setting, alternative def-
+initions to this Galerkin projection for A0
+i are possible as
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+noted below. To obtain the coarse-level representation of
+the problem, let P ∈ RS×|D| be the piecewise-constant
+interpolation operator that assigns every node in D0
+i to the
+i-th coarse node. The coarse-level operator is then obtained
+as AC = P T AP.
+The restricted additive Schwarz method (RAS) (Cai &
+Sarkis, 1999) is an important extension to the Schwarz
+methodology for the case of overlapping subdomains, where
+some nodes in D belong to more than one subdomain. De-
+noting the overlap amount by δ ∈ N, we define the sub-
+domains Dδ
+i for δ > 0 by recursion, as Dδ
+i = Dδ−1
+i
+∪
+{j | akj ̸= 0 for k ∈ Dδ−1
+i
+}. For the coarse-grid interpo-
+lation operator, P, each of the overlapping nodes is now
+associated with multiple columns of P, which is typically
+chosen as a partition of unity, with rows of P having equal
+non-zero weights (that can be interpreted as the probability
+of assigning a fine node to a given subdomain). The con-
+ventional two-level RAS preconditioner is then defined by
+considering the fine-level operator, MRAS, and the coarse-
+level correction operator, C2-RAS, given by
+MRAS =
+S
+�
+i=1
+( ˜Rδ
+i )T (Aδ
+i )−1Rδ
+i ,
+(2)
+C2-RAS = P(P T AP)−1P T ,
+(3)
+where Aδ
+i = (Rδ
+i )
+T ARδ
+i . The operator Rδ
+i denotes restric-
+tion for DoFs in D to those in Dδ
+i while ˜Rδ
+i is a modified
+restriction from D to Dδ
+i that takes nonzero values only for
+DoFs in D0
+i . The two-level RAS preconditioner is given
+as M2-RAS = C2-RAS + MRAS − C2-RASAMRAS, with the
+property that I − M2-RASA = (I − C2-RASA)(I − MRASA).
+In the case of optimized Schwarz, the subdomain systems
+(fine-level Aδ
+i ) are modified by imposing a Robin bound-
+ary condition between subdomains, writing ˜Aδ
+i = Aδ
+i + Li,
+where Li is the term resulting from the Robin-type condi-
+tion:
+αu + ⃗n · ∇u = g(x),
+(4)
+where g denotes inhomogeneous data and ⃗n is the outward
+unit normal to the boundary. The fine-level operator for
+optimized Schwarz is then given by
+MORAS =
+S
+�
+i=1
+�
+˜Rδ
+i
+�T �
+˜Aδ
+i
+�−1
+Rδ
+i ,
+(5)
+where the choice of weight, α, in the subdomain Robin
+boundary condition is a parameter for optimization. Simi-
+larly, the method can be improved by optimizing the choice
+of coarse-level interpolation operator, P, but this has not
+been fully explored in the OSM literature. Similarly to
+with RAS, we define the two-level ORAS preconditioner as
+M2-ORAS = C2-RAS + MORAS − C2-RASAMORAS.
+The work of Taghibakhshi et al. (2022) suggests a method
+to learn Li for one-level ORAS. Here, we learn both Li
+and P for two-level methods since, as later shown in Fig-
+ure 7, the two-level methods are significantly more robust.
+Furthermore, as we show in Section 5.2, while learning
+both ingredients improves the performance, learning the
+interpolation operator, P, is significantly more important
+than learning Li’s in order to obtain a two-level solver that
+outperforms classical two-level RAS.
+3. Multigrid graph neural network
+The multigrid neural architecture (Ke et al., 2017) is an ar-
+chitecture for CNNs that extracts higher level information in
+an image more efficiently by cross-scale information shar-
+ing, in contrast to other CNN architectures, such as U-nets,
+where abstraction is combined with scale. That is, in one
+multigrid layer, the information is passed between different
+scales of the problem, removing the necessity of using deep
+CNNs or having multilevel U-net architectures. Inspired
+by Ke et al. (2017), we develop a multigrid architecture for
+GNNs, enabling cross-scale message (information) passing
+without making the GNN deeper; we call our architecture
+Multigrid GNN, or MG-GNN. Figure 1 shows one layer of
+the MG-GNN with two levels (a fine and a coarse level).
+The input data to one layer of an MG-GNN has L different
+graphs, from fine to coarse, denoted by G(ℓ) = (X(ℓ), A(ℓ)),
+where A(ℓ) ∈ Rnℓ×nℓ and X(ℓ) ∈ Rnℓ×d are adjacency and
+node feature matrices, respectively, and nℓ and d are the
+number of nodes and node feature dimension in ℓ-th graph
+for ℓ ∈ {0, 1, . . . , L − 1}, with ℓ = 0 denoting the finest
+level. If the input graph does not have multiple levels, we
+obtain the coarser levels recursively by considering a node
+assignment matrix (clustering operator) R(ℓ) ∈ Rnℓ+1×nℓ,
+for ℓ ∈ {0, 1, . . . , L − 2}:
+X(ℓ+1) = R(ℓ)X(ℓ),
+(6)
+A(ℓ+1) = R(ℓ)A(ℓ)(R(ℓ))T .
+(7)
+We note that, in general, the assignment matrix Rℓ could
+be any pooling/clustering operator, such as k-means clus-
+tering, learnable pooling, etc. We denote R(ℓ→k) to be the
+assignment matrix of graph level ℓ to level k (with ℓ < k),
+which is constructed through R(ℓ→k) =
+k−1
+�
+j=ℓ
+R(j) (down-
+sampling). To complement this terminology, we also define
+R(k→ℓ) = (R(ℓ→k))T for ℓ > k (up-sampling), and for
+the case of ℓ = k, the assignment matrix is simply the
+identity matrix of dimension nℓ. The mathematical for-
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+𝐶!
+"
+𝐶#
+"
+𝐶!
+𝐶#
+Figure 1. One layer of MG-GNN. ci and c′
+i denote the feature
+dimensions of different levels before and after passing through an
+MG-GNN layer, respectively.
+malism of the m-th layer of the MG-GNN with L levels
+is as follows: given all graphs feature matrices, X(ℓ)
+m , for
+ℓ ∈ {0, 1, . . . , L − 1}:
+˙Xℓ→k = F ℓ→k(X(ℓ)
+m , X(k)
+m , R(ℓ→k))
+(8)
+˜X(ℓ)
+m = [ ˙X0→ℓ∥ ˙X1→ℓ∥ . . . ∥ ˙Xk−1→ℓ]
+(9)
+X(ℓ)
+m+1 = GNN(ℓ)( ˜X(ℓ)
+m , A(ℓ))
+(10)
+where ∥ denotes concatenation, and GNN(ℓ) and F ℓ→ℓ
+could be any homogeneous and heterogeneous GNNs, re-
+spectively. For the case of ℓ ̸= k, we consider F ℓ→k to be
+a heterogeneous message passing scheme between levels ℓ
+and k, which is defined as follows. Consider any node v in
+G(ℓ) and denote the row in X(ℓ)
+m corresponding to the feature
+vector of node v by xv. Then, F ℓ→k(X(ℓ)
+m , X(k)
+m , R(ℓ→k))
+is given by
+mv = gℓ→k
+�
+□
+ω∈N (v)f ℓ→k(xv, xω, evω), xv
+�
+(11)
+where evω is the feature vector of the edge (if any) connect-
+ing v and ω, □ is any permutation invariant operator such
+as sum, max, min, etc., and f ℓ→k and gℓ→k are learnable
+multilayer perceptrons (MLPs). See Figure 2 for visualiza-
+tion of up-sampling and down-sampling in MG-GNN. In
+this study, we consider a two-level MG-GNN (see Figure 1)
+and, for the clustering, we consider a k-means-based clus-
+tering algorithm (best known as Lloyd’s algorithm) which
+has O(n) time complexity and guarantees that every node
+will be assigned to a subdomain (Bell, 2008; Lloyd, 1982))
+in a connected graph. As mentioned earlier, the MG-GNN
+architecture could alternatively use any pooling/clustering
+method such as DiffPool (Ying et al., 2018), top-K pool-
+ing (Gao & Ji, 2019), ASAP (Ranjan et al., 2020), SAG-
+Pool (Lee et al., 2019), to name but a few. However, for
+the case of this paper, since RAS (and therefore, ORAS)
+necessitates every node in the fine grid be assigned to a
+subdomain, we do not consider the aforementioned pooling
+(clustering) methods.
+≡
+≡
+≡
+Figure 2. Upsampling and downsampling in MG-GNN.
+4. Optimization problem and loss function
+In this section, we denote the ℓ2 norm of a matrix or vector
+by ∥ · ∥ and the spectral radius of matrix T by ρ(T). Our
+objective is to minimize the asymptotic convergence factor
+of the two-level ORAS method, defined as minimizing ρ(T),
+where T = I−M2-ORASA = (I−C2-ORASA)(I−MORASA)
+is the error propagation operator of the method. Since T
+is not necessarily symmetric, ρ(T) is formally defined as
+the extremal eigenvalue of T T T. As discussed in Wang
+et al. (2019), numerical unsuitability of backpropagation
+of an eigendecomposition makes it infeasible to directly
+minimize ρ(T). To this end, Luz et al. (2020) relax the
+spectral radius to the Frobenius norm (which is an upper
+bound for it), and minimize that instead. However, for
+the case of optimizing one-level DDM methods, the work
+in Taghibakhshi et al. (2022) highlights that the Frobenius
+norm is not a “tight” upper bound for ρ(T), and considers
+minimizing a relaxation of ρ(T) inspired by Gelfand’s for-
+mula, ∀K∈N
+ρ(T) ≤ ∥T K∥
+1
+K = supx:∥x∥=1(∥T Kx∥)
+1
+K .
+We present a modified version of the loss function intro-
+duced by Taghibakhshi et al. (2022) and, in Section 5.3, we
+show the necessity of this modification for improving the
+two-level RAS results.
+Consider the discretized problem with DoF set D of size
+n, decomposed into S subdomains, Dδ
+1, Dδ
+2, . . . , Dδ
+S with
+overlap δ. The GNN takes D, its decomposition, and a
+sparsity pattern for the interface values and that of the inter-
+polation operator as inputs and its outputs are the learned
+interface values and interpolation operator (see Appendix B
+for more discussion on inputs and outputs of the network):
+P (θ), L(θ)
+1 , L(θ)
+2 , . . . , L(θ)
+S
+← ψ(θ)(D).
+(12)
+where ψθ denotes the GNN, and θ represents the learnable
+parameters in the GNN.
+We obtain the modified two-level ORAS (Optimized
+Restricted Additive Schwarz) operator by using the
+learned coarse grid correction operator, Cθ
+2-ORAS
+=
+P (θ) �
+(P (θ))T AP (θ)�−1 �
+P (θ)�T , and the fine grid oper-
+ator, M (θ)
+ORAS from (12). The associated 2-level error propa-
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+gation operator is then given by T (θ) = (I−C(θ)
+2-ORASA)(I−
+M (θ)
+ORASA).
+In order to obtain an approximate measure of ρ(T (θ)) while
+avoiding eigendecomposition of the error propagation ma-
+trix, similar to Taghibakhshi et al. (2022), we use stochas-
+tic sampling of
+���
+�
+T (θ)�K���, generated by the sample set
+X ∈ Rn×m for some m ∈ N, given as
+X = [x1, x2, . . . , xm], ∀j xj ∼ Rn uniformly, ∥xj∥ = 1,
+(13)
+where each xj is sampled uniformly randomly on a unit
+sphere in Rn using the method introduced in Box (1958).
+We then define
+Y (θ)
+K
+=
+�����
+�
+T (θ)�K
+x1
+���� ,
+����
+�
+T (θ)�K
+x2
+���� , . . . ,
+����
+�
+T (θ)�K
+xm
+����
+�
+.
+(14)
+Note that
+���
+�
+T (θ)�K xj
+��� is a lower bound for
+���
+�
+T (θ)�K���.
+Taghibakhshi et al. (2022) use L(θ) = max(Y (θ)
+K ) as a
+practical loss function. However, for large values of K, this
+loss function suffers from vanishing gradients. Moreover,
+as we show in Section 5.3, employing this loss function
+results in inferior performance of the learned method in
+comparison to two-level RAS. To overcome these issues,
+we define Z(θ)
+k
+= max((Y (θ)
+k
+)
+1
+k ) for 1 ≤ k ≤ K to arrive
+at a new loss function,
+L(θ) = ⟨softmax(Z(θ)), Z(θ)⟩ + γtr
+�
+(P (θ))T AP (θ)�
+,
+(15)
+where Z(θ) =
+�
+Z(θ)
+1 , Z(θ)
+2 , . . . , Z(θ)
+K
+�
+, 0 < γ is an ad-
+justable constant, and tr(M) is the trace of matrix M.
+Adding the term tr((P (θ))T AP (θ)) is inspired by energy
+minimization principles, to obtain optimal interpolation op-
+erators in theoretical analysis of multilevel solvers (Xu,
+1992; Wan et al., 1999). In Section 5.3, we show the signifi-
+cance of this term in the overall performance of our model.
+Nevertheless, for the first part of the new loss function (15),
+we prove that it convergence to the spectral radius of the er-
+ror propagation matrix in a suitable limit. First, we include
+two lemmas:
+Lemma 1. For x, y ∈ R, with 0 ≤ y ≤ x and any K ∈ N,
+x
+1
+K − y
+1
+K ≤ (x − y)
+1
+K
+Proof. See Lemma 3 from Taghibakhshi et al. (2022).
+Lemma 2. For any nonzero square matrix T ∈ Rn×n,
+k ∈ N, ϵ, ξ > 0, and 0 < δ < 1, there exists M ∈ N
+such that for any m ≥ M, if we choose x1, x2, . . . , xm
+uniformly random from {x ∈ Rn | ∥x∥ = 1}, and Z =
+max{∥T kx1∥
+1
+k , ∥T kx2∥
+1
+k , . . . , ∥T kxm∥
+1
+k } then, with a
+probability of at least (1 − δ), the following hold:
+0 ≤ ∥T k∥
+1
+k − Z ≤ ϵ,
+(16)
+ρ(T) − ξ ≤ Z.
+(17)
+Proof. The left side of the first inequality is achieved by
+considering the definition of matrix norm, i.e. for any 1 ≤
+i ≤ m, ∥T kxi∥ ≤
+sup
+∥x∥=1
+∥T kx∥ = ∥T k∥, then taking
+the kth root of both sides. For the right side of the first
+inequality, consider the point x∗ ∈ {x ∈ Rn | ||x|| = 1}
+such that ∥T kx∗∥ = sup
+∥x∥=1
+∥T kx∥ (such a point exists since
+Rn is finite dimensional). Let S be the total volume of the
+surface of an n dimensional unit sphere around the origin,
+and denote by ˜S the volume on this surface within distance
+˜ϵ of the point x∗ in the ℓ2 measure, for ˜ϵ =
+ϵ
+∥T k∥
+1
+k . Let
+m ≥ M1 >
+log(δ)
+log
+�
+1− ˜
+S
+S
+�, then, since 0 < δ < 1, we have:
+P(∥x∗ − xi∥ > ˜ϵ, ∀i) =
+�
+1 −
+˜S
+S
+�m
+≤ δ
+(18)
+Therefore, with probability of at least (1−δ), there is one xi
+within the ˜ϵ neighborhood of x∗ on the unit sphere. Without
+loss of generality, let x1 be that point. Using Lemma 1 and
+the reverse triangle inequality, we have
+��T kx∗��
+1
+k −
+��T kx1
+��
+1
+k ≤
+���T kx∗�� −
+��T kx1
+��� 1
+k
+≤
+��T k��
+1
+k ∥x∗ − x1∥
+1
+k ≤ ∥T k∥
+1
+k ˜ϵ = ϵ
+(19)
+which finishes the proof for the right side of the first inequal-
+ity.
+For the second inequality, since ρ(T) ≤ ∥T k∥
+1
+k , choose
+M2 such that, with probability 1 − δ, (16) holds for ϵ =
+∥T k∥
+1
+k − ρ(T) + ξ > 0. Rearranging (16) then yields (17)
+for any m ≥ M = max{M1, M2}.
+We next state the main result on optimality.
+Theorem 3. For any nonzero matrix T, ϵ > 0, and
+δ
+<
+1, there exist M, K
+∈
+N such that for any
+m
+>
+M, if one chooses m points, xj, uniformly
+at random from {x
+∈
+Rn | ∥x∥
+=
+1} and defines
+Zk = max{∥T kx1∥
+1
+k , ∥T kx2∥
+1
+k , . . . , ∥T kxm∥
+1
+k }, then
+Z = (Z1, Z2, . . . , ZK) satisfies:
+P (|⟨softmax(Z), Z⟩ − ρ(T)| ≤ ϵ) > 1 − δ.
+(20)
+Proof. Since
+ρ(T)
+≤
+∥T k∥
+1
+k
+for
+any
+k
+and
+limk→∞ ∥T k∥
+1
+k = ρ(T), for any 0 < α, there exists
+K∗ ∈ N such that for any k > K∗, 0 ≤ ∥T k∥
+1
+k −ρ(T) < α.
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+Take 0
+<
+α
+<
+min{ ϵ
+2, log( e−ϵ(ϵ+ρ(T ))
+ϵ
+2 +ρ(T )
+)}, let u
+=
+max{ max
+1≤k≤K∗{∥T k∥
+1
+k }+α, ρ(T)+2α}, ˜δ = 1−(1−δ)
+1
+K ,
+and take K > max{
+K∗(ueu−(ρ(T )+α)eρ(T )+α)
+eρ(T )−ϵ(ϵ+ρ(T )−(ρ(T )+α)eα+ϵ), K∗}.
+Note that, by the choice of α, we have ρ(T)+α < ρ(T)+ ϵ
+2
+and eα+ϵ <
+ρ(T )+ϵ
+ρ(T )+ ϵ
+2 , which (along with the choice of u)
+guarantees a positive K. By Lemma 2, for any 1 ≤ i ≤ K∗
+and K∗ < j ≤ K, there exists ni, nj ∈ N such that:
+P(ρ(T) − ϵ ≤ Zi ≤ u) > 1 − ˜δ
+for m > ni,
+(21)
+P(ρ(T) − ϵ ≤ Zj ≤ ρ(T) + α) > 1 − ˜δ
+for m > nj.
+(22)
+For any 1 ≤ k ≤ K, take nk independent points on unit
+sphere so that the above inequalities are satisfied for all
+k. Note that this can be achieved by taking M =
+K
+�
+k=1
+nk.
+Since the points for satisfying equations (21) and (22) are
+chosen independently, for any m > M, with probability of
+at least
+K�
+k=1
+(1 − ˜δ) = 1 − δ, we have ρ(T) − ϵ ≤ Zk for all
+1 ≤ k ≤ K. Consequently:
+− ϵ =
+(ρ(T) − ϵ)
+K
+�
+i=1
+eZi
+K
+�
+i=1
+eZi
+− ρ(T) ≤
+K
+�
+i=1
+ZieZi
+K
+�
+i=1
+eZi
+− ρ(T)
+(23)
+= ⟨softmax(Z), Z⟩ − ρ(T)
+(24)
+≤ uK∗eu + (K − K∗)(ρ(T) + α)eρ(T )+α
+Keρ(T )−ϵ
+− ρ(T)
+(25)
+= K∗(ueu − (ρ(T) + α)eρ(T )+α)
+Keρ(T )−ϵ
++ (ρ(T) + α)eα+ϵ − ρ(T) ≤ ϵ,
+(26)
+where the last inequality is obtained by the choice of K.
+In addition to these properties of the loss function, we now
+show that obtaining the learned parameters using our MG-
+GNN architecture scales linearly with the problem size.
+Theorem 4. The time complexity to obtain the optimized
+interface values and interpolation operator using our MG-
+GNN is O(n), where n is the number of nodes in the grid.
+Proof. Every in/cross-level graph convolution of the MG-
+GNN has linear complexity. This must be the case when the
+graph convolution is a message passing scheme due to the
+sparsity in finite-element triangulations. For the case that
+the graph convolution is a TAGConv layer, we have y =
+�L
+ℓ=1 Gℓxℓ + b1n, where xℓ ∈ Rn are the node features,
+L is the node feature dimension, b is a learnable bias, and
+Gℓ ∈ Rn×n is the graph filter. In TAGConv layers, the
+graph filter is given as Gℓ = �J
+j=0 gℓ,jM j, where M is
+the adjacency matrix, J is a constant, and gℓ,j are the filter
+polynomial coefficients. In other words, the graph filter it
+is a polynomial in the adjacency matrix M of the graph.
+Moreover, the matrix M is sparse, hence obtaining M j has
+O(n) computation cost, resulting in full TAGConv O(n)
+time complexity. Moreover, for both the interface value
+head and the interpolation head of the network, the cost of
+calculating edge feature and the feature networks are O(n),
+resulting in overall O(n) cost of MG-GNN.
+5. Experiments
+5.1. Training
+We train each model on 1000 grids of sizes ranging from
+800–1000 nodes. The grids are generated randomly as a
+convex polygon and using PyGMSH (Schl¨omer, 2021) for
+meshing its interior. The subdomains are generated using
+Lloyd clustering on the graph (Bell, 2008), the subdomain
+overlap is set to one, and the weights of the edges along
+the boundary determine the interface value operators, L(θ)
+i .
+As shown in the interpolation head of the network in Ap-
+pendix B Figure 11, the weight of the edges connecting the
+coarse and fine grids determine the interpolation operator.
+In our case, the edges between the coarse and fine grids
+connect every fine node to the coarse node corresponding to
+its own subdomain and its neighboring subdomains. Alter-
+natively, every fine node could connect only to the coarse
+node corresponding to its subdomain but, as we discuss in
+Section 5.3, this significantly impacts the performance of
+the model. Moreover, each row of the interpolation operator,
+P (θ), is scaled to have sum of one, as would be the case
+for classical interpolation operators. Figure 3 shows several
+example training grids.
+The model is trained for 20 epochs with batch size of 10
+using the ADAM optimizer (Kingma & Ba, 2014) with a
+fixed learning rate of 5 × 10−4. For the full discussion on
+model architecture, see Appendix B and Figure 11. For the
+loss function parameters introduced in Section 4, we use
+K = 10 iterations and m = 100 samples. We developed
+our code1 using PyAMG (Bell et al., 2022), NetworkX (Hag-
+berg et al., 2008), and PyTorch Geometric (Fey & Lenssen,
+2019). All training is executed on an i9 Macbook Pro CPU
+with 8 cores. In the training procedure, we aim to mini-
+mize the convergence of the stationary algorithm and, as
+described in Section 4, we develop a loss function to achieve
+this goal by numerically minimizing the spectral radius of
+1All code and data for this paper is at https://github.
+com/JRD971000/Code-Multilevel-MLORAS/ (MIT li-
+censed).
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+Figure 3. Training grid examples with about 1k nodes.
+Figure 4. Test grid example with about 7.4k nodes.
+the error propagation matrix. In practice, optimized RAS
+methods are often used as preconditioners for Krylov meth-
+ods such as FGMRES; as shown in Appendix A, the trained
+models using this procedure also outperform other baselines
+when used as preconditioners for FGMRES. Directly train-
+ing to minimize FGMRES iterations would require using
+FGMRES in the training loop and backpropagation through
+sparse-sparse matrix multiplication (Nytko et al., 2022),
+which is left for future studies.
+We evaluate the model on test grids that are generated in
+the same fashion as the training grids, but are larger in size,
+ranging from 800 to 60k DoFs. An example of a test grid is
+shown in Figure 4.
+5.2. Interface values and interpolation operator
+As mentioned in the Section 4, to optimize two-level RAS,
+one could optimize the parameters in the interface condi-
+tions (2) and/or the interpolation operator (3). For one-level
+RAS, on the other hand, there is no interpolation operator
+(since there is no coarse grid), leaving only the interface
+values to optimize, as was explored in Taghibakhshi et al.
+(2022) and St-Cyr et al. (2007). To compare the importance
+of these two ingredients in the two-level RAS optimization,
+we compare three different models. Each of these models
+is trained as described in Section 5.1; however, one of the
+models (labeled “interface”) is trained by only learning the
+interface values (ignoring the interpolation head of the net-
+work), and using classical RAS interpolation to construct
+T (θ). Another model, which we label “interpolation”, only
+learns the interpolation operator weights, and uses zeros for
+interface matrices L(θ)
+i
+to construct T (θ). The other model
+uses both training heads (see Figure 11), learning the inter-
+103
+104
+Grid size
+25
+50
+75
+100
+125
+Stationary iterations
+Interface
+RAS
+Interpolation
+Ours (both)
+Figure 5. Effect of learning interface values, interpolation operator,
+or both on stationary iterations.
+face values and the interpolation operator. We compare the
+performance of these models with classical RAS in Figure 5
+as a stationary algorithm, and in Figure 9 as a preconditioner
+for a Krylov method, FGMRES.
+The results show that learning the interpolation operator is
+more important in optimizing the 2-level RAS. Intuitively,
+the coarse-grid correction process in (3) plays an important
+role in scaling performance to large problems, due to its
+global coupling of the discrete DOFs. The interpolation
+operator is critical in achieving effective coarse-grid cor-
+rection. On the other hand, the interface values are local
+modifications to the subdomains (see (2)), that cannot (by
+themselves) make up for a poor coarse-grid correction pro-
+cess. Learning both operators clearly results in the best
+performance in Figures 5 and 9, where the interpolation
+operator can be adapted to best complement the effects of
+the learned interface values.
+5.3. Loss function and sparsity variants
+We first compare five variants of our method with the RAS
+baseline, as shown in Figures 6 and 9. The main model
+is trained as described in Section 5.1 with the loss func-
+tion from Section 4. All but one of the variants only dif-
+fer in their loss function, and share the rest of the details.
+The variant labeled “Max loss” is trained with the loss
+function from Taghibakhshi et al. (2022). The variant la-
+beled “Max+Trace loss” is trained with the loss function
+from Taghibakhshi et al. (2022) plus the γtr(P T AP) term.
+Similarly, the variant labeled “Softmax loss” is trained by re-
+moving the γtr(P T AP) part from the loss function in (15).
+For the last variant, we restrict the sparsity of the interpola-
+tion operator to that obtained by only connecting every fine
+node to its corresponding coarse node, labelled “DDM stan-
+dard sparsity”, and trained using the loss function from (15).
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+RAS
+Softmax+Trace loss (ours)
+Max loss
+Max+Trace loss
+Softmax loss
+DDM standard sparsity
+40
+60
+80
+Avg stationary iterations
+Figure 6. Effect of every ingredient in the model on average station-
+ary iterations. All three variants outperforming the RAS baseline
+are utilizing modifications introduced in this paper (see (15)) com-
+pared to the “Max loss” from Taghibakhshi et al. (2022).
+As shown in Figure 6, the learned operator using this variant
+achieves worse performance than the baseline RAS. This
+is partly because, for this variant, the constraint on unit row
+sums of the interpolation operator effectively removes most
+of the learned values, since many rows of interpolation have
+only one nonzero entry in this sparsity pattern.
+To show the effectiveness of the coarse-grid correction and
+the learned operator, we also compare two-level RAS and
+our two-level learned RAS (MLORAS 2-level) with one-
+level RAS and one-level optimized RAS from Taghibakhshi
+et al. (2022) in Figures 7 and 10.
+5.4. Comparison to Graph U-net and number of layers
+In this section, the performance of Graph U-net and MG-
+GNN with different numbers of layers is studied. Figures 8
+and 10 show the performance of each of the models as
+stationary iterations and preconditioners for FGMRES, re-
+spectively. For a fair comparison, the MG-GNN and graph
+U-nets that share the same number of layers also have the
+same number of trainable parameters. As shown here, the
+best performance is achieved with 4 layers of MG-GNN, and
+MG-GNN strictly outperforms the graph U-net architecture
+with the same number of layers.
+6. Conclusion
+In this study, we proposed a novel graph neural network
+architecture, which we call multigrid graph neural network
+(MG-GNN), to learn two-level optimized restricted additive
+Schwarz (optimized RAS or ORAS) preconditioners. This
+103
+104
+Grid size
+102
+103
+Stationary iterations
+RAS 1-level
+MLORAS 1-level
+RAS 2-level
+MLORAS 2-level (ours)
+Figure 7. Comparison of stationary iterations of 2-level methods
+with 1-level methods from Taghibakhshi et al. (2022).
+2
+4
+6
+8
+Number of layers
+45
+50
+55
+Avg stationary iterations
+Graph U-net
+MG-GNN (ours)
+Figure 8. Average stationary iterations of graph U-net and MG-
+GNN with different number of layers on the test set.
+new MG-GNN ensures cross-scale information sharing at
+every layer, eliminating the need to use multiple graph con-
+volutions for long range information passing, which was a
+shortcoming of prior graph network architectures. More-
+over, MG-GNN scales linearly with problem size, enabling
+its use for large graph problems. We also introduce a novel
+unsupervised loss function, which is essential to obtain im-
+proved results compared to classical two-level RAS. We
+train our method using relatively small graphs, but we test
+it on graphs which are orders of magnitude larger than the
+training set, and we show our method consistently outper-
+forms the classical approach, both as a stationary algorithm
+and as an FGMRES preconditioner.
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+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+A. FGMRES plots
+In Section 5, in Figures 5 to Figure 8, the performance of the methods was evaluated by considering the convergence
+of stationary iterations. Here, we present another possible evaluation criterion, assessing the number of iterations to
+convergence for the preconditioned systems using FGMRES, a standard Krylov method. The following figures are analogous
+to those provided in the main paper, and demonstrate that our method also achieves superior results compared to other
+methods, and that the MG-GNN architecture outperforms graph U-nets.
+103
+104
+Grid size
+30
+40
+50
+FGMRES iterations
+Interface
+RAS
+Interpolation
+Ours (both)
+RAS
+Softmax+Trace loss (ours)
+Max loss
+Max+Trace loss
+Softmax loss
+DDM standard sparsity
+30.0
+32.5
+35.0
+37.5
+40.0
+42.5
+45.0
+Avg FGMRES iterations
+Figure 9. Left: Effect of learning interface values, interpolation operator, or both on FGMRES iterations. All three variants outperforming
+the RAS baseline are utilizing modifications introduced in this paper (15) compared to the “Max loss” from Taghibakhshi et al. (2022).
+Right: Effect of every ingredient in the model on average FGMRES iterations.
+103
+104
+Grid size
+50
+100
+150
+FGMRES iterations
+RAS 1-level
+MLORAS 1-level
+RAS 2-level
+MLORAS 2-level (ours)
+2
+4
+6
+8
+Number of layers
+35.5
+36.0
+36.5
+37.0
+37.5
+38.0
+Avg FGMRES iterations
+Graph U-net
+MG-GNN (ours)
+Figure 10. Left: Comparison of FGMRES iterations of 2-level methods with 1-level methods from Taghibakhshi et al. (2022). Right:
+Average FGMRES iterations of graph U-net and MG-GNN with different number of layers on the test set.
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+B. Model architecture
+Inputs and outputs:
+The model takes any unstructured grid as its input, which consists of the node features, edge features,
+and adjacency matrix of both the fine and coarse grids. Every node on the fine level has a binary feature, indicating whether
+it lies on the boundary of a subdomain. Fine level edge features are obtained from the discretization of the underlying PDE,
+A, and the adjacency matrix of the fine level is simply the sparsity of A. Similar attributes for the coarse level are obtained
+as described in Section 3, Equations (6) and (7), and Lloyd aggregation has been used for obtaining subdomains throughout.
+The outputs of the model are the learned interface values and the interpolation operator.
+We use node and edge preprocessing (3 fully connected layers of dimension 128, followed by ReLU activations, in the
+node and feature space, respectively) followed by 4 layers of MG-GNN. For GNN(ℓ) in (10) and F ℓ→ℓ in (8), we use a
+TAGConv layer (Du et al., 2017) and, for F ℓ→k with ℓ ̸= k, we use a heterogeneous message passing GNN as shown in
+Equation (11). Specifically, we choose summation as the permutation invariant operator in (11) and, for the MLPs, we use
+two fully connected layers of size 128 with ReLU nonlinearity for f ℓ→k and gℓ→k(x, y) = x.
+Following the MG-GNN layers, the network will split in two heads, each having a stack layer (which essentially concatenates
+the features of nodes on each side of every edge) and an edge feature post-processing (see Figure 12 for details). The edge
+weights between the coarse and fine level are the learned interpolation operator weights, and the edge values along the
+subdomains in the fine level are the learned interface values. The upper head of the network has a masking block at the end,
+which masks the edge values that are not along the boundary, hence only outputting the learned interface values. The overall
+GNN architecture for learning the interpolation operator and the interface values is shown in Figure 11.
+ Edge Feature Post-
+processing
+Learned interface  
+edges
+Dnode : Node features
+Dedge : Edge features
+Masking
+Stack 
+MG-GNN
+MG-GNN
+4 Blocks
+ Edge Feature Pre-
+processing
+Coarse Grid
+Fine Grid
+ Edge Feature Post-
+processing
+Learned interpolation  
+weights
+Stack 
+Edge feature flow
+Edge feature
+Fine node feature
+Edge
+Fine node
+Node feature flow
+Coarse node
+Coarse node feature
+ Node Feature Pre-
+processing
+Figure 11. GNN architecture used in this study.
+
+MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
+����������������
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+��������
+�������
+�������
+�������
+������
+���������������
+�������������
+ Edge Feature
+Postprocessing
+Learned interface  
+edges
+Dnode : Node features
+Dedge : Edge features
+Masking
+Stack 
+HGNN
+HGNN
+4 Blocks
+ Edge Feature
+Preprocessing
+Coarse Grid
+Fine Grid
+ Edge Feature
+Postprocessing
+Learned interpolation  
+weights
+Stack 
+Edge feature flow
+Edge feature
+Fine node feature
+Edge
+Fine node
+Node feature flow
+Coarse node
+Coarse node feature
+Figure 12. Edge feature post-processing block.
+

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+Learning to Perceive in Deep Model-Free Reinforcement
+Learning
+Gonçalo Querido
+Instituto Superior Técnico
+Lisbon, Portugal
+[email protected]
+Alberto Sardinha
+INESC-ID & Instituto Superior
+Técnico & PUC-Rio
+Lisbon, Portugal
+[email protected]
+Francisco Melo
+INESC-ID & Instituto Superior
+Técnico
+Lisbon, Portugal
+[email protected]
+ABSTRACT
+This work proposes a novel model-free Reinforcement Learning
+(RL) agent that is able to learn how to complete an unknown task
+having access to only a part of the input observation. We take inspi-
+ration from the concepts of visual attention and active perception
+that are characteristic of humans and tried to apply them to our
+agent, creating a hard attention mechanism. In this mechanism,
+the model decides first which region of the input image it should
+look at, and only after that it has access to the pixels of that region.
+Current RL agents do not follow this principle and we have not
+seen these mechanisms applied to the same purpose as this work.
+In our architecture, we adapt an existing model called recurrent
+attention model (RAM) and combine it with the proximal policy op-
+timization (PPO) algorithm. We investigate whether a model with
+these characteristics is capable of achieving similar performance
+to state-of-the-art model-free RL agents that access the full input
+observation. This analysis is made in two Atari games, Pong and
+SpaceInvaders, which have a discrete action space, and in CarRac-
+ing, which has a continuous action space. Besides assessing its
+performance, we also analyze the movement of the attention of
+our model and compare it with what would be an example of the
+human behavior. Even with such visual limitation, we show that
+our model matches the performance of PPO+LSTM in two of the
+three games tested.
+KEYWORDS
+Reinforcement Learning, Model-Free, Attention Mechanism, Hard
+Attention, Active Perception, Visual Attention
+1
+INTRODUCTION
+In our everyday lives, even though we are constantly being flooded
+with visual stimuli, we do not give the same importance to ev-
+erything in our field of view. Instead, we focus on small regions
+that attract us the most. In those moments, we take advantage of a
+cognitive process called visual attention [4]. Unconsciously, we in-
+terpret those regions and extract meaning from them using another
+mental process named perception [15]. The combination of both
+these processes allows us to solve complex tasks because, from all
+the visual information we receive, we filter the most important to
+perform our activities and not pay attention to irrelevant elements
+in our surroundings.
+Current Reinforcement Learning (RL) models, even though they
+achieve excellent performance in a broad range of tasks, do not
+follow this behavior typical of humans. For example, when learning
+to play a video game, RL algorithms typically process the whole
+input image, giving the same importance to every region of the
+input game frame. Such design results in models that rely on large
+convolutional neural networks (CNNs) that process a large number
+of pixels, making the model take too long to train, requiring high
+computational power, and potentially limiting their applicability [8].
+To keep the training time reasonable, images are often preprocessed
+to reduce the size of the input, losing some of its details. Using these
+low-resolution images can hamper the models from completely
+understanding what is present in their input, which lowers their
+performance [19].
+To overcome these limitations, in this paper we contribute the
+first RL architecture that implements an attention mechanism sim-
+ilar to the one humans have. Applying such a mechanism allows
+the model to only process the pixels it perceives as the most useful,
+which makes it much more computationally efficient and able to
+use the original images without resizing them.
+Attention mechanisms such as the one described above recently
+started appearing in the literature, but none of them were applied to
+the same purpose as this work. The closest to our work is, perhaps,
+the model proposed by Mnih et al., called recurrent attention model
+(RAM) [12], which implements the same attention mechanism we
+use in our work in the context of image classification. The authors
+introduce the concept of glimpse, a retina-like representation of a
+portion of an image centered around a location 𝑙. The region of the
+image around 𝑙 has high resolution; regions further away from 𝑙
+have increasingly lower resolution. Such representation is crucial
+to the performance of the agent and is what makes the complexity
+of the model not dependent on the size of the input images. Instead,
+it depends on the size of the glimpses. The RAM paper uses four
+networks: one responsible for extracting the glimpses from an im-
+age and learning features from them; a Long Short Term Memory
+(LSTM) network that builds an internal representation of the envi-
+ronment combining the information from all the previous glimpses;
+one network responsible for the classification of the image received,
+and another that chooses the coordinates of the next glimpse.
+There is also a number of other models that apply different types
+of attention mechanisms. For example, the work of Tang et al. [18]
+is an example of an RL agent that is capable of achieving top perfor-
+mance in games such as CarRacing [9] and DoomTakeCover [14].
+Their agent starts by dividing the input video frame into patches,
+assigning a level of importance to each one of them. Then, it selects
+the 𝐾 most important patches and extracts features from them. This
+information is later used by a controller which selects the actions to
+take. Although this model only has around 3600 parameters, since
+it is trained using a computational intensive evolutionary strategy
+called Covariance-Matrix Adaptation Evolution Strategy (CMA-ES),
+it takes a long time to train.
+arXiv:2301.03730v1  [cs.LG]  10 Jan 2023
+
+In this paper, we address the following research questions:
+(1) Is it possible to attain state-of-the-art performance in com-
+plex control tasks with limited (but active) perception?
+(2) Is there any similarity between the attention movements of
+the model and human behavior?
+To address these questions, the paper proposes a novel architecture
+that combines a glimpse-based attention mechanism with a model-
+free reinforcement learning algorithm (PPO). Our results show our
+model can match the performance of PPO+LSTM in two of the
+three games tested while processing a significantly smaller number
+of pixels from the input images. Our model has fewer training
+parameters than its vanilla counterparts and is, therefore, more
+efficient than existing models that apply attention mechanisms.
+In the remainder of this paper, we go over the modifications we
+have made to the RAM [12] architecture, showing how our model
+compares against multiple versions of the PPO algorithm when
+playing video games such as Pong, SpaceInvaders, and CarRacing.
+In the end, we analyze the movement of the attention of our model
+and compare it with what would be an example of human behavior.
+2
+BACKGROUND
+This section introduces the core concepts and notation used in the
+remainder of the document.
+2.1
+Active Perception and Attention
+The concept of active perception was defined by Bajcsy as the intel-
+ligent acquisition of information about an environment in order to
+understand it better [2]. In comparison to a passive perception agent
+that statically senses its environment and takes actions accordingly,
+an active perception agent can improve its performance by actively
+moving its sensors to better reason about its environment [10].
+In artificial intelligence, active perception models can be imple-
+mented using deep learning methods such as CNNs [10]. Since the
+selection of what an agent should sense can take inspiration from
+the human visual attention process, an attention mechanism can
+be used to replicate such behavior. In machine learning, attention
+is a technique that consists in choosing which parts of the input
+are the most important, resulting in more computational power
+being allocated to them. In the literature [6, 20, 21], we can find
+two categories of attention mechanisms:
+• Soft attention: is a mechanism that splits the input into
+multiple parts, assigning a weight to each one of them. The
+weights represent the importance associated with each por-
+tion of the image, and since this model is differentiable, they
+can be updated using backpropagation.
+• Hard attention: is a mechanism where the model does
+not go through some parts of its input because the neu-
+ral network itself stochastically decides which are the parts
+it should pay attention to. However, this mechanism is not
+differentiable but can be trained using RL.
+The model from Tang et al. [18] used a soft attention mechanism,
+while in our work, we use a hard attention mechanism.
+2.2
+Reinforcement Learning
+In RL, an agent interacts with its environment and learns, by trial
+and error, an action-selection rule (a policy) that maximizes the
+agent’s reward over time.
+In our problem, the agent does not have full observability of
+the environment, so the best mathematical framework to formally
+represent this problem is a Partially Observable Markov Decision
+Process (POMDP). A POMDP is defined as a tuple (S, A, Z, P, O,𝑟),
+where S is a discrete set of states, A is a discrete set of actions,
+Z is a discrete set of observations, P(𝑠′ | 𝑠,𝑎) represents the tran-
+sition probability of going from state 𝑠 to 𝑠′ performing action 𝑎,
+O(𝑧 | 𝑠′,𝑎) represent the probability of receiving the observation 𝑧,
+while being in state 𝑠′ after taking action 𝑎, and 𝑟 (𝑠,𝑎) is the reward
+function 𝑟 : S × A → R.
+In our work, we decided to propose a model-free RL architecture
+instead of a model-based. We did not choose the latter because
+we decided to understand first if an RL model was capable of hav-
+ing good performance with such visual restriction while having
+a simpler, model-free approach. Building a complete model of the
+environment while just seeing a region of it, is another challenge
+that we leave for future work. Since we made that decision, our
+agent is not able to know either the transition probabilities P or
+the reward function 𝑟 of the environment. Therefore, it has to learn,
+explicitly by trial and error, the optimal policy 𝜋 : H → Δ(A),
+which is a mapping from the history of past observations to a dis-
+tribution over actions. One way of learning that policy is using an
+actor-critic policy-gradient method. This algorithm learns a param-
+eterized policy (an actor) and estimates a value function (a critic).
+For our problem, we need two actors: one to learn the action policy
+𝜋𝜃 (𝑎 | 𝑧) (Actor), and the other to learn the policy 𝜋𝜇 (𝑙 | 𝑧) that
+chooses the coordinates of the image to look at (Locator).
+The specific actor-critic method used in this work was the PPO
+algorithm, presented by Schulman et al. [17]. PPO is simpler than
+other policy gradient methods, well-studied, and was already tried
+alongside the RAM architecture [22]. For these reasons, we decided
+to have it as the base of our model.
+In PPO, we alternate between interacting with the environment
+to get data and updating the policy using stochastic gradient ascent.
+In every policy update, this policy gradient method guarantees that
+the difference between the new policy and the old policy is small,
+which prevents the algorithm from having a high variance during
+training. Having the probability ratio
+𝑟𝑡 (𝜃) = 𝜋𝜃 (𝑎𝑡 | 𝑠𝑡)
+𝜋𝜃old (𝑎𝑡 | 𝑠𝑡)
+(1)
+and ˆ𝐴𝑡, an estimator of the advantage function at timestep 𝑡,
+PPO maximizes the following surrogate objective:
+𝐿𝐶𝐿𝐼𝑃 (𝜃) = ˆE𝑡
+�
+min(𝑟𝑡 (𝜃) ˆ𝐴𝑡, clip(𝑟𝑡 (𝜃), 1 − 𝜖, 1 + 𝜖) ˆ𝐴𝑡)
+�
+(2)
+where 𝜖 is a hyperparameter. The clipping prevents large policy
+updates and penalizes the probability ratio 𝑟𝑡 (𝜃) when it tries to
+move far away from 1.
+2
+
+3
+GLIMPSE-BASED ACTOR-CRITIC (GBAC)
+In this section, we introduce a novel model called Glimpse-Based
+Actor-Critic (GBAC) that combines a hard attention mechanism
+with a model-free RL algorithm. When compared to other RL mod-
+els, GBAC processes much fewer pixels, and its training parameters
+do not depend on the size of the input, which makes it more effi-
+cient.
+In our problem, the game environment 𝐸 gives, at each timestep,
+a frame 𝑠𝑡 of the game. Since our model cannot have access to
+all the information of 𝑠𝑡, it has to select only a portion of it to
+be its observation 𝑧𝑡. That observation has the coordinates 𝑙𝑡−1 =
+(𝑥𝑡−1,𝑦𝑡−1) that were chosen by the model in the previous timestep.
+During its interaction with 𝐸, our agent has to learn the best policy
+𝜋𝜃 (𝑎𝑡 | 𝑧𝑡) that selects the action 𝑎𝑡 to be performed in the game.
+While doing this, the model also has to understand which regions of
+𝑠𝑡 have the most valuable information and learn a policy 𝜋𝜇 (𝑙𝑡 | 𝑧𝑡)
+to choose the set of coordinates (𝑥𝑡,𝑦𝑡) to be taken in the next
+timestep.
+As we have seen in Section 2.2, this problem can be seen as an
+instance of a POMDP. In our case, the observations the agent takes
+are represented by the glimpses, and the history of past interactions
+with the environment is stored in the hidden state of two LSTMs.
+When combining the memory, the current glimpse, and the previ-
+ously chosen location, the LSTMs have all the information needed
+to learn the policies that select the actions and the locations to take
+next.
+3.1
+Architecture
+The architecture of RAM [12] was the basis for our proposal, as a
+result, we present it in Figure 1 and will briefly describe it next.
+Action  
+Network
+Location
+Network
+Core 
+Network
++
+Glimpse Network
+Figure 1: Architecture of RAM. Image adapted from the orig-
+inal paper of Mnih et al. [12]
+Every timestep, RAM receives the game frame 𝑠𝑡 and a set of
+coordinates 𝑙𝑡−1. Its Glimpse Network takes a glimpse 𝑧𝑡 centered
+at 𝑙𝑡−1 and extracts features, not only from 𝑧𝑡, which are repre-
+sented in Figure 1 by 𝑘𝑡, but also from 𝑙𝑡−1. Using fully connected
+layers the two features are merged into the vector 𝑔𝑡. Next, this
+vector is fed to an LSTM, the Core Network, which stores all the
+previous information from the glimpses and the coordinates chosen,
+building its internal memory ℎ𝑡. After that, ℎ𝑡 is the input for two
+other networks, the Action Network and the Location Network,
+which are also fully connected layers that select the next action 𝑎𝑡
+and coordinates 𝑙𝑡, respectively. Since the Location Network was
+non-differentiable, it was trained using the policy gradient method
+REINFORCE, while the other components used backpropagation.
+After the presentation of RAM, some papers proposing improve-
+ments to the image classification capabilities of the model were
+written, such as the publications of Ba et al. [1] and Zuur [22]. After
+the presentation of our architecture, we will describe which refine-
+ments we took into consideration and how we have adapted them
+to our problem.
+Glimpse
+Network
+Action  
+Network
+Location
+Network
+Figure 2: Overview of the architecture of the Glimpse-Based
+Actor-Critic
+Figure 2 presents an overview of the GBAC architecture. We
+start by receiving a game frame 𝑠𝑡 and the set of coordinates 𝑙𝑡−1
+our agent chose at the end of the previous timestep. The Glimpse
+Network saves into 𝑘𝑡 the features extracted just from the glimpse
+taken from 𝑠𝑡 and centered at 𝑙𝑡−1. After that, the vector 𝑘𝑡 is used
+as the input of the Action Network. This network outputs not only
+the action 𝑎𝑡 the agent will take next, but also an estimate 𝑣𝑡 of
+the value function. 𝑘𝑡 is also used by the Location Network, which
+merges it with the features extracted from the location 𝑙𝑡−1 to select
+the next location coordinates 𝑙𝑡. In opposition to RAM, instead of
+doing this merging for the input of both networks, we only do it
+for the Location Network.
+We now present the key refinements of RAM that we took into
+consideration to address our problem. To start, we followed Ba et
+al. [1] and added a CNN to the Glimpse Network (represented in
+Figure 3) because it was necessary for the model to extract better
+features from the video frames. As one of the experiments done by
+Zuur [22] suggested, we also found that it was beneficial to provide
+separate inputs for the Action and Location Networks (Figure 2).
+Therefore, the configuration which performed best was the one
+that fed to the Action Network only the features (𝑘𝑡) extracted
+from the glimpse, and to the Location Network the vector resultant
+from the merge between 𝑘𝑡 and the features extracted from 𝑙𝑡−1.
+Still with this idea of separating the processes of choosing the next
+action and the next glimpse coordinates, we added an extra LSTM,
+making the Action Network and the Location Network use their
+separate LSTM (Figures 4 and 5). This separation destroyed the
+need for having an explicit Core Network like in RAM, avoiding
+the mix of information that is necessary for each task and allowing
+us to fine-tune the parameters for each LSTM. The last change we
+made was the selection of PPO instead of REINFORCE to train the
+model since it achieves better results in harder environments [22]
+and we found it simpler to train. This change meant that in each
+timestep, our Action Network also needed to make a prediction 𝑣𝑡
+about the quality of performing the action 𝑎𝑡 in the current state
+of the environment.
+3.1.1
+Glimpse Network. The Glimpse Network, represented in Fig-
+ure 3, is the module responsible for extracting from the game frame,
+3
+
+====the region the agent chose to focus its attention. With the image
+coordinates 𝑙𝑡−1 chosen in the previous timestep, this network ex-
+tracts an observation 𝑧𝑡 from 𝑠𝑡, which is called a glimpse. This
+glimpse can have multiple patches, each having double the size
+of the previous. For example, Figure 3 presents a glimpse with
+three patches. However, to simulate peripheral vision, all the larger
+patches are downscaled to the dimension of the smallest. That
+smallest patch will have the highest resolution, making it the focal
+point. Regarding the other patches, the larger they are, the further
+away they are from the focus point, so the lower their resolution
+is. When compared to the original image, this process results in a
+vector with fewer pixels.
+After rescaling, the resultant vector passes through a set of con-
+volutional layers and a fully connected layer to extract a vector of
+features 𝑘𝑡. The number of convolutional layers and their respective
+kernel sizes and stride are changed depending on the size of the
+glimpse.
+CNN
+Figure 3: Detailed diagram of the Glimpse Network of GBAC
+During this work, we assumed that the glimpses are always
+squares. If the coordinates of 𝑙𝑡 make a patch catch pixels that are
+out of the bounds of 𝑠𝑡, that patch will be moved in order to fit
+inside the frame. This mechanism proved to achieve better results
+than simply filling the pixels out of bounds with a value.
+3.1.2
+Action Network. The Action Network, depicted in Figure 4, is
+responsible to choose the game action 𝑎𝑡 the agent should perform
+in each timestep. Since we chose an actor-critic algorithm to train
+GBAC, the Action Network also estimates the value function, which
+helps the agent to understand if it is performing well or not.
+In this network, its LSTM saves the information 𝑧𝑡 extracted pre-
+viously from the glimpse and combines it with its hidden memory
+ℎ𝑔
+𝑡−1, which stores all the information gathered in earlier timesteps
+to build an internal representation of the game environment. The
+new hidden state ℎ𝑔
+�� is fed to two fully connected layers, the Actor
+and the Critic, that output the action 𝑎𝑡 and the value 𝑣𝑡, respec-
+tively.
+3.1.3
+Location Network. The Location Network, illustrated in Fig-
+ure 5, is the module behind the hard attention mechanism and is
+responsible to choose the image coordinates 𝑙𝑡 where the agent
+should look in the next timestep. Those coordinates are sampled
+from a truncated normal distribution whose mean is given by this
+network, being the standard deviation a fixed value.
+In order to choose the mean value, the Location Network has
+a neural network that extracts features from the coordinates 𝑙𝑡−1,
+LSTM
+Actor
+Critic
+128
+128
+Figure 4: Detailed diagram of the Action Network of GBAC
+merging them with the features 𝑘𝑡. The resultant vector 𝑔𝑡 is then
+used to update the internal state of an LSTM. Its internal state ℎ𝑙
+𝑡 is
+fed to the Locator, which is a neural network with two fully con-
+nected layers and ReLU and Tanh activation functions, respectively.
+The Locator chooses the mean value used in the truncated normal
+distribution from which the next set of coordinates 𝑙𝑡 is extracted.
+LSTM
+Locator
+256
+640
+Figure 5: Detailed diagram of the Location Network of GBAC
+Since this model only uses a small portion of an image, its com-
+plexity is not dependent on the size of the input image but rather
+on the size of its glimpses. This can be an advantage if the model
+has to handle large images.
+3.2
+Training
+The training process of our model is very similar to the one pre-
+sented by Schulman et al. in the PPO paper [17]. We follow the
+suggestions they proposed, and the only difference is that we have
+two policy losses, instead of just one. Like PPO, our model alternates
+between interacting with the environment to get new information
+and updating both its policies with that new data.
+When exploring the environment, in each timestep, the model
+saves the action and coordinates of the glimpse it chose, the value
+function estimated by the critic, the reward received from the game,
+and a flag that indicates if the current episode has finished.
+After a predetermined number of timesteps, the model uses the
+collected data to update its policies. For both policies, we use the
+"surrogate" objective already presented in Section 2.2:
+𝐿𝐶𝐿𝐼𝑃𝜋 (𝜃) = ˆE𝑡
+�
+min(𝑟𝜋
+𝑡 (𝜃) ˆ𝐴𝑡, clip(𝑟𝜋
+𝑡 (𝜃), 1 − 𝜖, 1 + 𝜖) ˆ𝐴𝑡)
+�
+(3)
+where 𝜖 is a hyperparameter and the advantage function ˆ𝐴𝑡 is
+computed recurring to Generalized Advantage Estimation [16].
+4
+
+==Besides the two policy losses, the objective has two more terms:
+an entropy bonus 𝐵 that promotes the exploration of the environ-
+ment, and the squared-error loss of the value function, 𝐿𝑉 𝐹
+𝑡
+. With
+these additions, the objective’s formula is the following:
+𝐿𝑡 (𝜃) = ˆE𝑡
+�
+𝐿𝐶𝐿𝐼𝑃a
+𝑡
+(𝜃) + 𝐿𝐶𝐿𝐼𝑃g
+𝑡
+(𝜃) − 𝛼𝐿𝑉 𝐹
+𝑡
+(𝜃) + 𝛽𝐵[𝜋𝑎](𝑠𝑡)
+�
+(4)
+where 𝛼 and 𝛽 are coefficients. Note that the entropy bonus is
+only calculated for the action policy, not for both policies. Adding
+the same bonus for the location policy did not seem to improve the
+results, thus we decided to keep the objective simpler.
+4
+EXPERIMENTAL EVALUATION
+In this chapter, we present all the experiments that enabled us
+to test our model, establishing which base PPO algorithm is the
+fairest choice to compare our model with, and discovering which
+size of the glimpses gives better results. In the end, we have all the
+information necessary to answer the questions in Section 1.
+4.1
+Evaluation Process
+4.1.1
+Game Environments. In order to measure the performance
+of GBAC and compare it against the state-of-the-art RL agents, we
+selected three game environments with different characteristics.
+The first two are both games from the Atari 2600 and have a discrete
+action space, while the third, is CarRacing [9] from OpenAI Gym [3]
+and has a continuous action space. We tried to select three games
+that were not the easiest ones available and that required the agents
+to learn different sets of skills, such that we could see how they
+were capable to adapt to each type of task.
+In order to obtain the best performance in the Atari games, we
+used the same environment modifications proposed by Mnih et
+al. [13] and Machado et al. [11], thus leading to better results for
+these games. Those modifications are well accepted and used in the
+literature. For example, we resize the original frame from 210x160
+to 84x84 pixels, clip the rewards, and scale the pixel values to [0, 1].
+4.1.2
+Comparison Models. We decided to compare GBAC with
+three different versions of PPO. The first one is the original PPO
+agent presented by Schulman et al. [17] because it was used as
+the base for our model. However, since our implementation uses
+LSTMs in its architecture and the version of PPO with an LSTM
+is also quite common in the RL literature, we decided to choose
+the PPO+LSTM agent for comparison as well. The third version of
+PPO we used is a modification of the PPO+LSTM algorithm where
+the restriction of only using a small portion of the game frame
+was imposed. But, different from GBAC, the coordinates where the
+agent looks are chosen completely randomly. This allows us to test
+if the perception mechanism implemented in our model is better
+than one that makes its choices randomly.
+The PPO implementation used in this work was not the original
+one provided by Schulman et al. [17] in OpenAI Baselines [5], but
+rather a revised implementation presented by Shengyi et al. [7] that
+closely follows the performance of the original. This implementa-
+tion provides many versions of PPO including one with an LSTM.
+To make the PPO agents capable of playing the selected games, we
+added to their architecture a CNN with the same layout as the one
+presented in the DQN paper [13].
+By comparing GBAC with the first two versions of PPO, we can
+discover if it is possible to achieve state-of-the-art performance
+playing video games, despite having a limited (but active) percep-
+tion of the environment, which was our first question.
+4.1.3
+Evaluation Metrics. Evaluating RL models is never a straight-
+forward task due to their high variance. For this reason, during
+training and testing, we average the episodic return each agent
+achieved over the last 100 episodes.
+In training, Pong, SpaceInvaders, and CarRacing learned dur-
+ing 15 million, 20 million, and 5 million timesteps, respectively.
+Then, the agent that achieved the best performance over the last
+100 episodes is tested for another 100 episodes. For each configura-
+tion, the results are always the average over three runs and their
+respective standard deviations are also presented.
+Seeding is another aspect that can have an impact on the per-
+formance of the agent. Certain seeds can make the agent perform
+significantly better or worse. Thus, in each run, the seed used in
+the environment is chosen arbitrarily.
+Regarding the policy that chooses the locations of the glimpses,
+in order to understand if our model learns a behavior that resembles
+the human vision, we analyze the evolution of the policy throughout
+the training phase and present a visual representation of the results.
+This is the information we need to answer our second question.
+Now that the entire evaluation process is detailed, we present,
+in the next sections, the results we obtained.
+4.2
+Base Models Selection
+In contrast to one of the Atari optimizations proposed by Mnih et
+al. [13] and Machado et al. [11], our model does not perform better
+when the original image is resized. In fact, it never learned a way to
+beat its adversary in Pong when a glimpse smaller than the resized
+size of 84x84 was used. This meant that with that setup, we could
+not take advantage of the glimpse’s structure because the model
+needed the entire image to perform well, which does not follow the
+restriction of our problem. An explanation for this result can be the
+fact that since the glimpses taken by GBAC have multiple patches
+that end up being resized to a lower resolution, and the input frame
+fed to the model was also resized, the loss in information might be
+so much that our agent is not capable of solving the game.
+Therefore, to make the comparisons fair, we decided that the
+base PPO models should also use the entire image as input. In order
+to discover how much this decision could impact the performance
+of the base agents when playing the two Atari games, we studied
+the difference in performance between using the original 210x160
+image and the resized 84x84 input. In CarRacing, since the original
+image is just 96x96, we decided not to compare the base models with
+a resized version of the input. In addition, since GBAC uses LSTMs,
+we compared PPO with PPO+LSTM to find out if they perform any
+differently.
+In Table 1, we show the training and testing performances that
+PPO and PPO+LSTM had when using either the full image frame
+or the resized input. The results shown do not allow us to conclude
+with absolute certainty that one way is better than the other. In
+Pong, there are not any significant differences in performance either
+in regular PPO or in PPO+LSTM. In SpaceInvaders, for the regular
+PPO, the agent that uses the full image achieves average returns that
+5
+
+PongNoFrameskip-v4
+SpaceInvadersNoFrameskip-v4
+CarRacing-v0
+Model
+Full Img.
+Max. Train Avg.
+Test Avg.
+Max. Train Avg.
+Test Avg.
+Max. Train Avg.
+Test Avg.
+PPO
+No
+21.00 ± 0.01
+20.98 ± 0.02
+2090.00 ± 254.16
+2013.07 ± 244.56
+–
+-
+PPO
+Yes
+20.91 ± 0.11
+20.83 ± 0.13
+2261.90 ± 295.87
+2221.62 ± 201.31
+867.16 ± 6.64
+824.31 ± 8.04
+PPO + LSTM
+No
+20.11 ± 0.25
+20.00 ± 0.31
+1182.57 ± 259.03
+1077.63 ± 245.82
+-
+-
+PPO + LSTM
+Yes
+20.03 ± 0.23
+19.85 ± 0.39
+900.20 ± 79.16
+812.58 ± 111.51
+783.62 ± 11.58
+659.73 ± 24.42
+Table 1: Comparison between the training and testing performance of PPO and PPO+LSTM, using a resized frame (84x84) of
+the input and also the full game frame (210x160)
+are around 200 points better than the agent that resizes the input.
+This result indicates that, for this game, some useful information
+is lost during the resizing of the image. However, for the model
+that uses PPO+LSTM, the opposite is verified. Since the size of the
+LSTMs was the same in both cases, this suggests that for the full
+image, the LSTM needed to be bigger because it could extract better
+data from fewer pixels.
+From the same table, when comparing the PPO agent against
+PPO+LSTM for the same type of input, we can see that the use of an
+LSTM deteriorates the performance of the agent. Having in mind
+that in the PPO game, a match ends when a player reaches 21 points
+and the reward of the agent is the difference between the points
+scored and scored against, the difference between the two models is
+marginal. It only means that, on average, the PPO+LSTM agent let
+the opponent score one point, while PPO did not. In SpaceInvaders,
+the performance drops by half, which is a significant reduction.
+In CarRacing, there is also a drop in performance, even though it
+is less accentuated than in SpaceInvaders. The major difference
+between the PPO and PPO+LSTM models is that the former uses
+a stack of four frames as input, while the latter receives just one
+frame at a time, counting on its LSTM to discover and store the
+information that is useful to the agent. Therefore, in SpaceInvaders
+and in CarRacing, the LSTM is not able to store all the information
+needed, like the velocity and direction of the objects from the game,
+which would allow the agent to perform better.
+In short, from these results, we cannot conclude that using the
+resized frame is better than using the full frame, and since our
+model utilizes the full image, in order to make the comparisons
+fairer, with as many equal variables as possible, from here onwards,
+we will be referring to the version that uses an LSTM and the entire
+frame as input when mentioning the base model.
+4.3
+GBAC Performance Analysis
+In this section, we study not only the impact that different sizes of
+glimpses and different numbers of patches have on the performance
+of our agent but also how the best configuration performs against
+the three versions of PPO.
+In our architecture, each glimpse can have one or more patches.
+Since we stipulated that each new patch has double the size of the
+previous, increasing the number of patches results in glimpses with
+a smaller focus region. This means that if we want glimpses with
+two patches, the largest possible size for the smallest patch is 80x80,
+with three patches it will be 40x40, and with four patches 20x20.
+The higher the number of patches, the larger the "peripheral vision"
+of our model. Nonetheless, this increase in information comes with
+the price of it not being as detailed as the portions of the image
+closer to the focal point.
+With this in mind, besides their architectures, using glimpses
+with just one patch makes our model no different from the base
+PPO agents. Therefore, the results from the agents that use glimpses
+with one patch are just useful to compare the two architectures,
+and to discover how large the input image has to be, in order for
+the agent to maintain its performance. Therefore, the results that
+are relevant to understand the performance of our model are the
+ones given by configurations that use more than one patch.
+4.3.1
+Pong. In relation to the Atari game Pong, Table 2 presents
+the maximum training average and the testing performance of the
+three PPO versions, as well as the best glimpse size for each number
+of patches of our agent. Since the returns for this game are bounded
+between -21 and +21, when analyzing the performance of all the
+agents, we can see that one of three scenarios occurred: the agent
+learned how to beat its adversary and ended up with scores close to
++20; the agent did not manage to learn anything useful, resulting in
+scores around -19; or the timesteps were not enough for the agent
+to learn a good policy so it achieved a score between the previous
+two.
+PongNoFrameskip-v4
+Model
+Glimpse
+Max. Train Avg.
+Test Avg.
+PPO
+-
+20.91 ± 0.11
+20.83 ± 0.13
+PPO+LSTM
+-
+20.03 ± 0.23
+19.85 ± 0.39
+GBAC
+1p, 160x160
+7.61 ± 10.42
+6.95 ± 10.89
+GBAC
+2p, 80x80
+7.15 ± 20.80
+6.64 ± 21.14
+GBAC
+3p, 40x40
+20.06 ± 0.44
+19.82 ± 0.88
+GBAC
+4p, 20x20
+-14.86 ± 5.39
+-15.77 ± 4.82
+PPO Random
+3p, 40x40
+-19.87 ± 0.05
+-20.16 ± 0.11
+Table 2: Training and testing performance in Pong
+Regarding the glimpses with one patch, our agent did not manage
+to achieve scores near +20 with the larger glimpse consistently
+because, in two of the three runs, the agent was still improving its
+performance when the timesteps finished. Therefore in this game,
+our agent was not capable of matching the performances of PPO.
+With two patches, the results were slightly better than the previous
+because the standard deviation is much higher. The model achieved
+a score of around +19 in two of the three runs. Using three patches,
+GBAC was capable of matching the performance of the PPO+LSTM
+agent, consistently beating its opponent by 21-1. The difference to
+the regular PPO agent, only means that our agents let the opponent
+6
+
+score a point, while the other did not. After seeing the results until
+this point, we might think that the performance keeps improving
+while we increase the number of patches of each glimpse. However,
+this is not the case when we look at the scores achieved using four
+patches. The size of the patches was so small that our model was
+not capable of achieving a performance of +20 in, at least, one of
+the three runs, which was true in any of the previous results.
+Regarding the PPO agent that took glimpses at random locations,
+we see that it performed very poorly, not being able to learn an
+optimal policy to play Pong. Therefore, we can say that, in this
+game, choosing good glimpse locations matters.
+4.3.2
+SpaceInvaders. Table 3 shows the training and testing results
+of all the agents for the SpaceInvaders game.
+SpaceInvadersNoFrameskip-v4
+Model
+Glimpse
+Max. Train Avg.
+Test Avg.
+PPO
+-
+2261.90 ± 295.87
+2221.62 ± 201.31
+PPO+LSTM
+-
+900.20 ± 79.16
+812.58 ± 111.51
+GBAC
+1p, 160x160
+741.45 ± 392.54
+607.42 ± 287.84
+GBAC
+2p, 80x80
+444.18 ± 40.78
+341.47 ± 25.71
+GBAC
+3p, 40x40
+596.50 ± 182.35
+544.43 ± 166.79
+GBAC
+4p, 20x20
+439.72 ± 26.27
+378.00 ± 28.70
+PPO Random
+3p, 40x40
+516.62 ± 60.59
+467.38 ± 50.53
+Table 3: Training and testing performance in SpaceInvaders
+Starting with the results of our model for one patch, we found
+that this time, the average score of GBAC was closer to the one
+achieved by the PPO+LSTM agent. However, when considering the
+results for the glimpses with two patches, the performance dropped
+almost by half. With three patches, GBAC achieves the best perfor-
+mance when considering the use of more than one patch. Nonethe-
+less, the model is not able to match the performance achieved when
+it used the bigger glimpse with just one patch. With four patches,
+we have the same decrease in performance, already seen in Pong.
+However, this time it was not as severe and slightly beat the results
+from two patches while having the smallest standard deviation in
+both training and testing of any of the experiments.
+In this game, our model was not capable of matching the per-
+formance of the PPO+LSTM agent, however, we still consider it
+an interesting result, considering the viewing restrictions of our
+problem. While processing 86% fewer pixels than PPO+LSTM (4.800
+vs. 33.600) in each timestep, GBAC only had a performance drop of
+33%.
+In this game, the random agent has a performance that is on par
+with our model, which possibly means that in SpaceInvaders the
+location of a glimpse is not that important. One possible reason
+behind this proximity could be the fact that, in this game, GBAC
+can pay attention to a lot more things that can improve the return
+received from the environment. As opposed to Pong, where our
+agent only had three objects (two paddles and one ball) to keep
+track of.
+4.3.3
+CarRacing. Lastly, Table 4 shows the performance that GBAC
+and the other agents achieved during training and testing, when
+playing the CarRacing game, which has a game frame of size 96x96.
+CarRacing-v0
+Model
+Glimpse
+Max. Train Avg.
+Test Avg.
+PPO
+-
+867.16 ± 6.64
+824.31 ± 8.04
+PPO+LSTM
+-
+783.62 ± 11.58
+659.73 ± 24.42
+GBAC
+1p, 96x96
+815.12 ± 5.75
+660.02 ± 69.38
+GBAC
+2p, 40x40
+694.50 ± 107.94
+641.11 ± 57.42
+GBAC
+3p, 20x20
+676.82 ± 84.89
+564.00 ± 56.42
+PPO Random
+2p, 40x40
+622.41 ± 17.89
+589.87 ± 13.19
+Table 4: Training and testing performance in CarRacing
+Like in the other two games, here, the largest glimpse size is
+also the one that achieves the better result for a glimpse with one
+patch. In addition, this time our model was capable of matching
+the performance of the PPO+LSTM agent in testing and beating it
+during training. For two patches, the achieved results maintained
+the performance seen in testing when using one patch, even though
+the score in training decreased and had a much higher standard de-
+viation. With three patches, which in CarRacing was the maximum
+number we tested, we start to see the performance drop slightly.
+In this game, GBAC was capable of maintaining its performance
+across all number of patches, which is a good result since the scores
+closely match the PPO+LSTM agent.
+Regarding the PPO agent with random glimpses, the perfor-
+mance difference is again not that far behind our model. Since the
+generated track occupies a good portion of the image, it may be eas-
+ier for the random model to select good actions from every glimpse
+it receives.
+From this study, we can conclude that even when using the entire
+frame (glimpses with one patch) the performance of our model
+is capable of matching the performance of PPO+LSTM, although
+not consistently. This shows our architecture could compete with
+PPO+LSTM if we did not impose our restrictions. Additionally,
+we discovered that the performance of GBAC does not increase
+linearly with the number of patches. It reaches a point where the
+information lost with the reduction of the glimpse size is more
+significant than the information gained with the addition of another
+patch. The optimal number of patches is three for the Atari games
+and two for CarRacing. Those numbers of patches proved to be
+the right balance between having a patch size that discarded the
+irrelevant information, allowing the model to just focus on the most
+important, and not being too small such that after rescaling the
+large patches, it was still possible to understand what was present
+in the "peripheral vision" of the agent. Finally, we saw that in most
+cases, the largest glimpse size possible for each number of patches
+is the one that produces the best results. In CarRacing, for two and
+three patches this was not the case, and the sizes 48x48 and 24x24,
+respectively, did not give the best results, even though they were
+very close.
+An important fact we should mention is that, since the com-
+plexity of our model is independent of the size of the input, we
+can achieve these results with a model that, in the Atari games,
+has 15% fewer total training parameters (∼1.7M vs. ∼2.0M) than
+the PPO+LSTM model, and almost the same number has PPO. In
+CarRacing, since the actions are continuous, the model is bigger,
+but it still has 10% fewer parameters than PPO+LSTM (∼2.2M vs.
+7
+
+(a) Pong 40x40 Glimpse
+(b) Pong Heatmap
+(c) SpaceInvaders 40x40 Glimpse
+(d) SpaceInvaders heatmap
+Figure 6: Frames with glimpses of 3 patches and heatmaps representing the number of times during an episode that the agent
+chose a specific set of image coordinates to be the center of the glimpse, for the Pong and SpaceInvaders games respectively
+∼2.5M) and only more 70k than PPO. If we use bigger environments
+having frames with many more pixels, this difference between the
+number of training parameters required will only keep increasing.
+4.4
+Glimpse Movements Analysis
+After discovering how GBAC performs against the other models, it
+is also important to understand how the location of its glimpses is
+evolving throughout training and which behavior the model finds
+outs to be the best. While making this analysis, we also compare
+the agent’s decisions with the choices we would consider when
+playing the games.
+Regarding the regions that our best agents chose to look at during
+the episodes, we can see in Figure 6b and Figure 6d that, for the
+Atari games, their distribution is relatively similar in both games.
+In SpaceInvaders, our agent keeps its focus near the center of the
+image, while in Pong, it chooses locations slightly upwards from
+the center. In general, the distribution of locations in both games
+has more choices closer to the center and becomes more sparse the
+further they are from it.
+Having the focus point almost near the center of the image, as
+we can see in Figure 6b, means that, in Pong, the agent gets the
+location of each paddle from its "peripheral vision" (Figure 6a). We
+think that this choice is different from what a human would select
+to focus on in this particular game. We believe that a human would
+pay more attention to the position of its own paddle and the ball.
+In relation to SpaceInvaders, in our opinion, the choices are much
+closer to what a human would do because the agent keeps its focus
+on the lower rows of enemies (Figure 6c), which are the ones that
+need to be destroyed first.
+Regarding Car Racing, our model presents quite unusual be-
+havior during the training process. It starts similarly to the other
+two games with a circular normal distribution for the location co-
+ordinates (Figure 7a). However, after a few training epochs, that
+distribution starts dispersing over multiple parts of the entire image,
+ending up with the region shown in Figure 7b, creating some kind
+of borders, which constrain the choice of coordinates, and that do
+not correspond to the limits of the image. This last behavior is the
+one that is present in the best solution at the end of training.
+In our opinion, even though in two of the three games, our model
+does not follow exactly what we think is the human vision behavior
+when playing those video games, we would need a more systematic
+(a) Beginning of training
+(b) Middle of training
+Figure 7: Heatmaps representing the evolution of glimpse
+locations in the CarRacing game
+way of analyzing the regions that we select to focus our attention,
+using, for example, an eye-tracking device, to better compare the
+agent’s behavior in relation to humans.
+5
+CONCLUSION
+This work proposed a solution for the problem of an agent that
+has limited vision, and for that reason, besides deciding which
+action it has to take in the environment, it also has to choose which
+part of the environment it should look at. To solve this problem,
+we proposed GBAC, a model that combines a glimpse-based hard
+attention mechanism with a model-free RL algorithm.
+We started by proving that, for some games like Pong and CarRac-
+ing, our model is already capable of achieving similar performance
+to the PPO version that more closely resembles our model, that is,
+the variant that also uses an LSTM. On other games like SpaceIn-
+vaders, a drop in performance is verified, with means, there still
+is room for improvement. We finished concluding that our model
+does not necessarily choose the same regions of the image that we
+selected to look at when we played the video games ourselves. Only
+in SpaceInvaders we considered this was verified. In addition, in
+CarRacing, we are not able to explain the reason behind the unusual
+behavior of our model.
+ACKNOWLEDGMENTS
+This work was partially supported by Portuguese national funds
+through Fundação para a Ciência e a Tecnologia (FCT) under projects
+8
+
+4
+20
+40
+3
+60
+2
+80
+0
+20
+40
+60
+808
+20
+6
+5
+40
+4
+60
+m
+2
+80
+0
+20
+40
+60
+80
+0口U
+50
+5
+4
+100
+3
+150
+2
+200
+0
+50
+100
+150分然4
+50
+100
+2
+150
+200
+0
+50
+100
+150UIDB/50021/2020 (INESC-ID multi-annual funding), PTDC/CCI-
+COM/5060/2021 (RELEvaNT), PTDC/CCI-COM/7203/2020 (HOTSPOT).
+In addition, this research was partially supported by the Air Force
+Office of Scientific Research under award number FA9550-22-1-0475
+and an EU Horizon 2020 project (TAILOR) under GA No. 952215.
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+9
+
+A
+EXTENDED RESULTS
+PongNoFrameskip-v4
+SpaceInvadersNoFrameskip-v4
+No. Patches
+Glimpse Size
+Max. Train Avg.
+Test Avg.
+Max. Train Avg.
+Test Avg.
+1
+160
+7.61 ± 10.42
+6.95 ± 10.89
+741.45 ± 392.54
+607.42 ± 287.84
+1
+150
+-18.96 ± 1.67
+-19.32 ± 1.38
+384.43 ± 72.19
+338.45 ± 59.50
+1
+140
+-18.93 ± 1.80
+-19.36 ± 1.47
+386.50 ± 75.94
+347.08 ± 99.27
+1
+130
+-19.92 ± 0.04
+-20.19 ± 0.03
+330.76 ± 77.16
+265.00 ± 53.71
+1
+120
+-19.86 ± 0.04
+-20.06 ± 0.05
+343.98 ± 137.35
+273.00 ± 113.99
+1
+110
+-19.89 ± 0.11
+-20.22 ± 0.08
+381.66 ± 85.91
+340.12 ± 77.90
+1
+100
+-19.89 ± 0.05
+-20.24 ± 0.11
+321.91 ± 51.51
+256.78 ± 77.16
+1
+90
+-19.65 ± 0.42
+-19.90 ± 0.36
+285.76 ± 61.01
+214.35 ± 23.69
+1
+80
+-19.90 ± 0.01
+-20.28 ± 0.05
+244.53 ± 7.94
+190.83 ± 26.97
+1
+70
+-18.47 ± 1.30
+-18.88 ± 1.08
+243.08 ± 23.85
+186.83 ± 19.25
+1
+60
+-19.89 ± 0.05
+-20.16 ± 0.09
+232.95 ± 17.10
+214.58 ± 13.55
+1
+50
+-19.63 ± 0.54
+-19.80 ± 0.56
+251.46 ± 33.61
+190.62 ± 22.40
+1
+40
+-19.93 ± 0.04
+-20.19 ± 0.11
+218.78 ± 2.93
+163.42 ± 14.12
+1
+30
+-19.95 ± 0.03
+-20.17 ± 0.17
+221.73 ± 4.20
+170.33 ± 21.35
+1
+20
+-19.75 ± 0.32
+-20.11 ± 0.25
+244.51 ± 4.90
+203.97 ± 23.26
+1
+10
+-19.87 ± 0.07
+-20.23 ± 0.06
+238.90± 15.14
+188.23 ± 32.58
+2
+80
+7.15 ± 20.80
+6.64 ± 21.14
+444.18 ± 40.78
+341.47 ± 25.71
+2
+70
+-6.68 ± 22.92
+-6.99 ± 22.80
+371.68 ± 58.30
+296.10 ± 36.34
+2
+60
+-19.86 ± 0.05
+-20.27 ± 0.18
+371.21 ± 6.30
+350.15 ± 10.13
+2
+50
+-16.72 ± 5.48
+-17.17 ± 5.20
+283.81 ± 76.94
+217.58± 62.06
+2
+40
+-19.91 ± 0.03
+-20.19 ± 0.07
+252.58 ± 22.31
+194.52 ± 26.93
+2
+30
+-17.24 ± 2.34
+-17.52 ± 2.43
+248.98 ± 5.78
+208.03 ± 7.23
+2
+20
+-19.87 ± 0.07
+-20.17 ± 0.16
+241.50 ± 10.81
+190.52 ± 31.40
+2
+10
+-19.91 ± 0.06
+-20.24 ± 0.09
+243.51 ± 5.16
+209.43 ± 32.67
+3
+40
+20.06 ± 0.44
+19.82 ± 0.88
+596.50 ± 182.35
+544.43 ± 166.79
+3
+30
+-10.99 ± 10.42
+-11.80 ± 9.05
+274.38 ± 5.24
+212.07 ± 34.62
+3
+20
+-19.92 ± 0.01
+-20.12 ± 0.10
+293.12 ± 28.46
+228.63 ± 28.06
+3
+10
+-19.93 ± 0.03
+-20.35 ± 0.39
+251.60 ± 12.27
+194.13 ± 31.28
+4
+20
+-14.86 ± 5.39
+-15.77 ± 4.82
+439.72 ± 26.27
+378.00 ± 28.70
+4
+10
+-19.86 ± 0.05
+-20.23 ± 0.08
+297.45 ± 9.98
+252.77 ± 13.72
+Table 5: Training and testing performance of every glimpse size for every number of patches tested in Pong and SpaceInvaders.
+10
+
+CarRacing-v0
+No. Patches
+Glimpse Size
+Max. Train Avg.
+Test Avg.
+1
+96
+815.12 ± 5.75
+660.02 ± 69.38
+1
+90
+675.12 ± 84.89
+607.46 ± 39.81
+1
+80
+741.03 ± 112.73
+534.56 ± 72.77
+1
+70
+747.12 ± 103.91
+258.70 ± 274.03
+1
+60
+692.40 ± 121.21
+546.35 ± 40.01
+1
+50
+559.05 ± 19.45
+361.30 ± 73.27
+1
+40
+539.68 ± 11.35
+485.85 ± 9.80
+1
+30
+187.67 ± 136.04
+32.64 ± 81.82
+1
+20
+155.15 ± 77.05
+152.99 ± 75.43
+1
+10
+138.51 ± 88.40
+128.90 ± 94.17
+2
+48
+748.70 ± 74.90
+544.58 ± 132.04
+2
+40
+694.50 ± 107.94
+641.11 ± 57.42
+2
+30
+616.74 ± 84.43
+347.20 ± 243.31
+2
+20
+465.76 ± 74.51
+375.30 ± 164.83
+2
+10
+161.44 ± 36.96
+145.80 ± 48.40
+3
+24
+700.53 ± 28.06
+472.35 ± 295.73
+3
+20
+676.82 ± 84.89
+564.00 ± 56.42
+3
+10
+545.18 ± 92.57
+509.85 ± 70.74
+Table 6: Training and testing performance of every glimpse size for every number of patches tested in CarRacing.
+B
+ALGORITHMS HYPERPARAMETERS
+Hyperparameter
+Value(s)
+PPO
+Advantage normalization
+True
+Annealing learning rate
+True
+Batch size
+[1024, 2048]
+Clipping coefficient - action
+[0.1, 0.2]
+Clipping coefficient - location
+0.2
+Clipped value loss
+True
+Entropy coefficient
+[0.01, 0]
+GAE lambda
+0.95
+Gamma
+0.99
+Grayscale
+True
+Learning rate - action
+[2.5e-4, 3e-4]
+Learning rate - location
+3e-5
+Locator normal distrib. variance
+0.1
+Maximum gradient clipping norm
+0.5
+Minibatch size
+[256, 64]
+No. environments
+[8, 1]
+No. minibatches
+[4, 32]
+No. steps
+[128, 2048]
+Optimizer
+Adam
+Update k epochs
+[4, 10]
+Value function coefficient
+0.5
+Glimpse
+Glimpse scale
+2
+Glimpse FC layer size
+[384, 512]
+Location FC layer size
+256
+LSTM size
+128
+No. glimpses
+1
+Table 7: List of parameters used in GBAC, in the Atari games and CarRacing, respectively
+11
+

8NAyT4oBgHgl3EQfQvar/content/tmp_files/2301.00053v1.pdf.txt

@@ -0,0 +1,1060 @@
+arXiv:2301.00053v1  [gr-qc]  30 Dec 2022
+MNRAS 000, 1–8 (2022)
+Preprint 3 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Time delay induced by plasma in strong lens systems
+Gennady S. Bisnovatyi-Kogan1,2★ and Oleg Yu. Tsupko1†
+1Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia
+2National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31,
+Moscow 115409, Russia
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+If the gravitational lens is surrounded by non-homoheneousplasma, in addition to the vacuum
+gravitational deflection, chromatic refraction occurs. Also, the speed of signal propagation
+decreases compared to vacuum. In this article, we investigate analytically the time delay in
+the case of gravitational lensing in plasma, focusing on strong lens systems. We take into
+account the following contributions: geometric delay due to trajectory bending in the presence
+of both gravity and plasma; potential delay of the ray in the gravitational field of the lens;
+dispersion delay in the plasma due to decrease of speed of light signal in the medium. We
+consider singular isothermal sphere as a model of gravitational lens, and arbitrary spherically
+symmetric distribution of surrounding plasma. For this scenario, plasma corrections for the
+time delay between two images are found in compact analytical form convenient for estimates.
+We discuss also the possible influence of the plasma on the value of the Hubble constant,
+determined from observations of the time delay in strong lens systems.
+Key words: gravitational lensing: strong – plasmas
+1
+INTRODUCTION
+As one of its effects, gravitational lensing results in a time delay of
+the rays compared to unlensed propagation. In strong lens systems
+with multiple images, different images have different delays, and
+the time delay between images can be measured. Observations of
+the time delay make it possible to determine the Hubble constant.
+If the gravitational lens is surrounded by non-homogeneous
+plasma, chromatic refraction occurs, in addition to the vacuum grav-
+itational deflection.1 As a result, various chromatic effects can be
+expected (see next Section for a description of the state of research).
+Also, the speed of signal propagation decreases compared to vac-
+uum. This manifests itself, for example, in the measured time delay
+of the signal at different radio frequencies from pulsars in the inter-
+stellar medium.
+The time delay between images in presence of plasma around a
+★ E-mail: [email protected] (GSBK); ORCID iD: https://orcid.org/0000-
+0002-2981-664X
+† E-mail: [email protected], [email protected] (OYuT); ORCID iD:
+https://orcid.org/0000-0002-2159-8350
+1 It should be noted that even if the plasma is homogeneous, i.e., there
+is no refraction, the deflection of light will be different from the vac-
+uum and chromatic. This issue has been discussed in detail in our pa-
+pers (Bisnovatyi-Kogan & Tsupko 2009, 2010; Tsupko & Bisnovatyi-Kogan
+2013), and the corresponding corrections have been calculated. A similar
+effect will be observed in other media with dispersion (Tsupko 2021). Usu-
+ally these corrections are less than the corrections connected with refraction,
+and in the present paper we neglect them. For more details, see our review
+(Bisnovatyi-Kogan & Tsupko 2017).
+gravitational lens (that is, with the simultaneous influence of gravity
+and the plasma medium) has not yet been widely discussed in the
+literature. Combined effect of plasma and weak gravitational field
+on propagation of signals was considered in Muhleman & Johnston
+(1966). They discussed the signal delay at radio frequencies in solar-
+corona plasma, in relation to earlier suggested Shapiro’s experiment
+(Shapiro 1964). See also Muhleman, Ekers & Fomalont (1970).
+Much later, already after creation of the developed gravita-
+tional lens theory, the time delay in the case of gravitational lensing
+in plasma has been briefly discussed by Er & Mao (2014). They have
+modeled the strong lens system and, in particular, numerically esti-
+mated the time delay between images and considered its influence
+on measuring of the Hubble constant value. A related discussion is
+presented in the article of Er, Yang & Rogers (2020). The authors
+consider the time delay for the light propagation in the plasma, i.e.
+do not consider gravitational effects.2 They take into account both
+2 To avoid confusion, we would like to remind that three different concepts
+should be distinguished: (i) conventional vacuum gravitational lensing, in
+which the deflection angle is determined by the gravitational deflection of
+rays by massive objects in vacuum, for example, Einstein’s formula for small
+angles; (ii) diverging plasma lensing, in which the deflection is caused by
+refractionoflightrays oncompact plasmainhomogeneties (nogravity effects
+here); (iii) gravitational lensing in presence of plasma, in which the weak
+or strong deflection of light due to gravity is combined with the chromatic
+deflection in the plasma. Effects of gravity and plasma can be completely
+separated only for weak deflection case. For strong gravity regime (not
+considered in this article), self-consistent approach should be applied, see
+Section 2 for more details.
+© 2022 The Authors
+
+2
+G. S. Bisnovatyi-Kogan and O. Yu. Tsupko
+the dispersion delay and plasma geometric delay. A detailed com-
+parison of vacuum gravitational lensing and plasma lensing (notion
+used for refraction on compact plasma inhomogeneities, Clegg et al
+(1998); Tuntsov et al (2016); Vedantham et al (2017); Dong et al
+(2018); Er, Yang & Rogers (2020)) is presented in the article of
+Wagner & Er (2020). Just recently, the effects of plasma on the time
+delay for strongly lensed fast radio bursts has been investigated by
+Er & Mao (2022), where both gravitational and plasma effects are
+taken into account; see also Kumar & Beniamini (2022).
+In this article, we investigate the time delay in the case of grav-
+itational lensing in plasma, focusing on strong lens systems. We
+consider this subject completely analytically and find compact ex-
+pressions for plasma corrections which are convenient for making
+estimates. By ’strong lens system’ and ’strong lensing’, we mean
+here a observational situation where a gravitational lens produces
+multiple images. To model such systems found in observations, it is
+always sufficient to use the deflection angle in the weak deflection
+approximation. Therefore, the notion ’strong lensing’ should not
+be confused with ’strong deflection lensing’ or ’strong field lens-
+ing’ by compact objects where the deflection angles are large and
+can produce higher-order images (e.g., Tsupko & Bisnovatyi-Kogan
+2013).
+In a weak deflection approximation, the total deflection angle
+can be calculated as the sum of vacuum gravitational deflection, and
+contribution of the refractive plasma deflection. In this approxima-
+tion, the time delay of lensed ray (relative to the undeflected straight
+line ray) can be presented as a sum of the following contributions:
+geometric delay due to trajectory bending in the presence of both
+gravity and plasma; potential delay of the ray in the gravitational
+field of the lens; dispersion delay in the plasma due to decrease of
+speed of light signal in the medium; see also Er & Mao (2022).
+We calculate the time delay between two images in the follow-
+ing scenario: gravitational lens is given by the singular isothermal
+sphere (SIS) model, while the plasma component is given by arbi-
+trary spherically symmetric distribution. Plasma corrections to the
+image positions and the time delay are found in a compact analytical
+form. We discuss also a possible influence of plasma effects on the
+value of the Hubble constant, determined from observations of the
+time delay in strong lens systems.
+The paper is organized as follows. Next Section is a brief
+overview of the current state of research of gravitational lensing in
+plasma. In Section 3 we present the general expression for the time
+delay under considered approximations. In Section 4, the plasma
+corrections for the image positions in case of SIS lens are calculated.
+In Section 5, the plasma corrections for the time delay for SIS lens
+are found. In Section 6 we present the example of application of our
+formulas. Section 7 is our Conclusions.
+2
+STUDIES OF GRAVITATIONAL LENSING IN
+PLASMA: BRIEF REVIEW OF RESEARCH
+The simplest approach for considering gravitational lensing in
+plasma is to use the linearized approximation, when both vacuum
+gravitational deflection and refraction in an inhomogeneous plasma
+aresmall, andtwodeflectionanglesarewrittencompletelyseparated
+from each other. Such method was considered in Bliokh & Minakov
+(1989). The same way of deflection angle calculation was used for
+numerical modelling of different effects in strong lens systems by
+Er & Mao (2014) and for investigation of microlensing effect in
+plasma by Tsupko & Bisnovatyi-Kogan (2020); Sun, Er & Tsupko
+(2022). The discussion of different scenarios is also presented in se-
+ries of papers of Bisnovatyi-Kogan & Tsupko (2009, 2010, 2015).
+In the current paper we are also working in such approximation for
+calculation of the deflection angle.
+If a more accurate account of plasma effects is neces-
+sary, due to physical conditions of the problem, a self-consistent
+approach should be used. This means that the total deflec-
+tion angle (determined by gravity and plasma) should be de-
+rived from the same theory. The propagation of rays in the
+curved space in presence of a medium was considered by Synge
+(1960) and Perlick (2000), see also Bičák & Hadrava (1975) and
+Kulsrud & Loeb (1992). On the basis of Synge’s equations, the
+deflection angle of ray in presence of both gravity and plasma
+has been investigated in the weak deflection approximation by
+Bisnovatyi-Kogan & Tsupko (2009, 2010, 2015). In particular, it
+has been shown that even in a homogeneous plasma where there
+is no refraction, the gravitational deflection angle differs from the
+vacuum one. Calculation of the deflection angle of light rays in
+plasma media up to higher order terms in expansion by plasma
+and gravity have been made in the works of Crisnejo and Gallo
+with coathours (Crisnejo & Gallo 2018; Crisnejo, Gallo & Rogers
+2019; Crisnejo, Gallo & Jusufi 2019). For a weak deflection in
+the Kerr metric, see Morozova, Ahmedov & Tursunov (2013) and
+Crisnejo, Gallo & Jusufi (2019).
+The exact expression for the deflection angle for a ray propaga-
+tion in a gravitational field in the presence of plasma, was first found
+in the monograph of Perlick (2000). The expression was derived for
+the motion in the equatorial plane of Kerr black hole. In the arti-
+cle of Tsupko & Bisnovatyi-Kogan (2013), the positions of higher-
+order images (Darwin 1959; Virbhadra & Ellis 2000; Bozza et al
+2001; Perlick 2004; Bisnovatyi-Kogan & Tsupko 2008, 2017) were
+calculated analytically when lensed by the Schwarzschild black
+hole in a homogeneous plasma. An important contribution to the
+subject has been made in a series of articles by Rogers (2015,
+2016, 2017a,b), where various effects near compact objects have
+been considered. Recently, spatial dispersion of light rays prop-
+agating through a plasma in Kerr space-time was investigated
+in Kimpson, Wu & Zane (2019a), see also Kimpson, Wu & Zane
+(2019b). The exact expression for the deflection angle in gravita-
+tional field in presence of arbitrary medium with a spherical sym-
+metry was found by Tsupko (2021).
+Recent
+studies
+of
+gravitational
+lensing
+in
+plasmas
+in
+the
+strong
+gravity
+regime
+are
+also
+focused
+on
+the
+shadow (Falcke, Melia & Agol 2000; Cunha & Herdeiro 2018;
+Perlick & Tsupko 2022) of black holes. The influence of the
+plasma medium on the propagation of rays and on the size
+and shape of the black hole shadow was calculated for the
+Schwarzschild black hole in Perlick, Tsupko & Bisnovatyi-Kogan
+(2015),
+and
+for
+the
+Kerr
+black
+hole
+in
+Perlick & Tsupko
+(2017), see also subsequent papers Huang, Dong & Liu (2018);
+Yan (2019); Babar et al (2020); Badía & Eiroa (2021, 2022);
+Chowdhuri & Bhattacharyya (2021); Li et al (2022); Zhang et al
+(2022); Briozzo, Gallo & Mädler (2022). The light propagation in
+plasma (incl. separability of the Hamilton-Jacobi equation and black
+hole shadow) was considered in a general case of axially symmetric
+and stationary spacetime by Bezděková, Perlick & Bičák (2022).
+Interesting results are obtained by Briozzo & Gallo (2022)
+who have generalized the approximate formula of Beloborodov
+(2002) for gravitational bending of light near compact objects by
+including plasma corrections. Diffraction of the light by the gravity
+of the Sun and the solar corona is discussed by Turyshev & Toth
+(2019a,b). For propagation of light through magnetized plasma in
+presence of gravity see Breuer & Ehlers (1980), Breuer & Ehlers
+MNRAS 000, 1–8 (2022)
+
+Time delay induced by plasma in strong lens systems
+3
+(1981a),
+Breuer & Ehlers
+(1981b),
+Broderick & Blandford
+(2003a), Broderick & Blandford (2003b). For some other re-
+cent
+studies
+see
+Schulze-Koops, Perlick & Schwarz
+(2017),
+Javed, Abbas & Övgün
+(2019),
+Sárený and Balek
+(2019),
+Matsuno (2021), Chainakun, Watcharangkool & Young (2022);
+Guerrieri & Novello (2022). Review of different plasma ef-
+fects, incl. strong lens systems and shadow is presented by
+Bisnovatyi-Kogan & Tsupko
+(2017).
+Recently,
+Crisnejo et al
+(2022); Crisnejo (2022); Ulla (2022) have considered strong lens
+systems given by singular isothermal ellipsoid and have studied the
+plasma effects analytically using perturbative approach.
+Plasma lensing studies have been also carried out in Clegg et al
+(1998), Tuntsov et al (2016), Cordes et al (2017), Vedantham et al
+(2017), Dong et al (2018). In a series of papers of Er and
+Rogers (Er & Rogers 2018; Rogers & Er 2019; Er & Rogers 2019;
+Er, Yang & Rogers 2020), the formalism of gravitational lensing has
+been successfully applied to consider different models of plasma
+lenses. Interesting discussion and comparison of vacuum gravita-
+tional lensing and plasma lensing can be found in the paper of
+Wagner & Er (2020).
+3
+TIME DELAY IN THE CASE OF SIMULTANEOUS
+PRESENCE OF GRAVITY AND PLASMA: GENERAL
+EXPRESSION
+In this Section, we will consider the time delay in presence of both
+gravitational lens and plasma. We write the total deflection angle of
+light ray as the sum of vacuum gravitational deflection and refractive
+plasma deflection:
+ˆ훼 = ˆ훼푔푟푎푣 + ˆ훼푟푒 푓 푟 ;
+ˆ훼푔푟푎푣, ˆ훼푟푒 푓 푟 ≪ 1 .
+(1)
+Both angles are assumed to be small and independent on each other.
+Effects of gravity and plasma are taken into account in the linear
+order only, all mixed and higher-order terms are neglected here.
+Working in the same order of approximation, we take into
+account the following contributions in time delay, as compared to
+straight-line propagation in vacuum in absence of gravity:
+(i) the geometric delay Δ푡푔푒표푚 associated with additional path
+length due to the bending of the trajectory in the presence of both
+gravity and plasma;
+(ii) the potential delay Δ푡 푝표푡 of the ray caused by time re-
+tardation of the ray while moving in the gravitational field of the
+lens;
+(iii) the dispersion delay Δ푡푑푖푠푝 in the plasma associated with
+a decrease of the signal velocity in the medium.
+This approach can be compared with the vacuum gravitational
+lensing case (when geometrical and potential delays are taken into
+accound) and plasma lensing case (when geometrical and dispersive
+delay are used).
+We assume spherical symmetry of both the lens and the sur-
+rounding plasma. Therefore, one-dimensional variables can be used
+in the lens equation and the time delay expressions. Note that in
+general case, positions of source and images are described by two
+dimensional quantities in the source plane and lens plane, corre-
+spondingly. However, for axially symmetric cases, it is possible to
+switch to one-dimensional values in the lens equation which will be
+now defined in the plane where all rays forming images are located
+(plane of picture in Fig.1).
+The geometrical delay is (e.g., Schneider, Ehlers & Falco
+1992;
+Schneider, Kochanek & Wambsganss
+2006;
+Figure 1. (COLOR ONLINE)Standard scheme of gravitational lensing. The
+light ray from the distant source at the angular position 훽 is deflected by the
+lens at the angle ˆ훼 and comes to the observer at the angle 휃. Primary image
+is denoted as 휃+, and secondary image as 휃−. 퐷푑 is the distance between
+the lens and the observer, 퐷푠 defines the distance between the distant source
+and the observer, and 퐷푑푠 is the distance between the source and the lens.
+Congdon & Keeton 2018; Dodelson 2017):
+Δ푡푔푒표푚(휃) = 1 + 푧푑
+푐
+퐷푑퐷푠
+퐷푑푠
+(휃 − 훽)2
+2
+.
+(2)
+Here 훽 is the angular position of the source, 휃 is the position of
+the image, 푧푑 is the redshift of the lens, 퐷푖 are angular diameter
+distances (Fig.1). The form of expression (2) is universal in the
+sense that it can be used for deflection caused by any physical rea-
+son. It is used not only in vacuum gravitational lensing but also
+in plasma lensing (Er, Yang & Rogers 2020; Wagner & Er 2020).
+The importance of the geometric delay in comparison with poten-
+tial delay was discussed in our paper (Tsupko et al 2020), see also
+Hackmann & Dhani (2019).
+The potential delay is (e.g., Schneider, Ehlers & Falco 1992;
+Schneider, Kochanek & Wambsganss
+2006;
+Congdon & Keeton
+2018; Dodelson 2017):
+Δ푡 푝표푡 (휃) = −1 + 푧푑
+푐
+퐷푑퐷푠
+퐷푑푠
+휓(휃) + const .
+(3)
+Here 휓(휃) is the deflection potential (Schneider, Ehlers & Falco
+1992; Schneider, Kochanek & Wambsganss 2006) (lens potential,
+Congdon & Keeton (2018)) that depends on the mass distribution
+in the lens and is defined so that its gradient gives the deflection
+angle:
+휶 = ∇휓,
+where
+휶 = 퐷푑푠
+퐷푠
+ˆ휶 .
+(4)
+The dispersive delay is considered here for cold non-
+magnetized plasma with the refractive index 푛:
+푛2 = 1 −
+휔2푝
+휔2 ,
+휔2
+푝 = 4휋푒2
+푚푒
+푁푒 ,
+(5)
+where 휔푝 is the plasma electron frequency, 휔 is the photon
+frequency (locally measured), 푚푒 and 푒 are the electron mass
+MNRAS 000, 1–8 (2022)
+
+4
+G. S. Bisnovatyi-Kogan and O. Yu. Tsupko
+and charge, 푁푒 is the electron number density in plasma. Work-
+ing in the linearized approximation with separated gravitational
+and plasma terms, we can neglect the change of photon fre-
+quency due to the gravitational field (gravitational redshift) in
+the terms containing plasma. Note that this cannot be neglected
+if it is necessary to calculate the plasma effects more precisely,
+for example, when finding corrections to the vacuum gravita-
+tional deflection due to the presence of homogeneous plasma
+(Bisnovatyi-Kogan & Tsupko 2009, 2010), or in case of black hole
+shadow calculation (Perlick, Tsupko & Bisnovatyi-Kogan 2015;
+Perlick & Tsupko 2017, 2022). In addition, if one is interested in
+the frequency of observation 휔0, the following relation must be
+taken into account: 휔 = (1 + 푧푑) 휔0, where 푧푑 is the lens redshift,
+see, e.g., Crisnejo et al (2022); Cordes et al (2017).
+With usual assumption, 휔 ≫ 휔푝, one finds:
+푛 ≃ 1 −
+휔2푝
+2휔2 = 1 − 2휋푒2
+푚푒휔2 푁푒 ,
+(6)
+and for delay of ray in plasma in comparison with vacuum propa-
+gation we write:
+Δ푡 = 1
+푐
+∫
+� 1
+푛 − 1
+�
+푑푙 ≃
+2휋푒2
+푐푚푒휔2 푁푖푛푡 ,
+(7)
+where
+푁푖푛푡 =
+∫
+푁푒푑푙 .
+(8)
+Here 푁푖푛푡 is the projected electron density along the line of sight,
+usually referred to as the dispersion measure DM. Together with the
+cosmological factor we find finally:
+Δ푡푑푖푠푝 = 1 + 푧푑
+푐
+2휋푒2
+푚푒휔2 푁푖푛푡 .
+(9)
+As a result, the formula for the time delay becomes (see also
+Er, Yang & Rogers (2020), Er & Mao (2022)):
+Δ푡(휃) = 퐷Δ푡
+푐
+� 1
+2 (휃 − 훽)2 − 휓(휃)
+�
++ 1 + 푧푑
+푐
+퐾푒
+2휔2 푁푖푛푡 (|휃|) ,
+(10)
+where we use a notation 퐾푒 ≡ 4휋푒2/푚푒. The variable
+퐷Δ푡 = (1 + 푧푑) 퐷푑퐷푠
+퐷푑푠
+(11)
+is known as time-delay distance (Suyu et al 2010, 2018).
+The formula (10) is general in the sense that any distribution
+of gravitating mass in the lens can be considered (which defines
+the function 휓(휃) together with the deflection angle, according
+to eq.(4)) and any spherically symmetric distribution of surround-
+ing plasma given by 푁푒 and 푁푖푛푡 (|휃|) can be used. We also note
+that the dispersive term is present both in homogeneous and not-
+homogeneous plasma, but it is not related to correction to gravita-
+tional deflection in homogeneous plasma discussed in the Introduc-
+tion.
+With the simultaneous presence of gravity and plasma, the
+geometric delay has a curious feature. Gravity and plasma compete
+with each other, acting in opposite directions. Therefore, if the angle
+of gravitational deflection and the angle of refraction exactly cancel
+each other for some image, the trajectory becomes straight and the
+geometric delay for this image can be equal to zero.
+4
+PLASMA CORRECTIONS TO THE IMAGE
+POSITIONS IN THE CASE OF SIS LENS
+In this Section, we analytically calculate the plasma corrections to
+image positions in strong lens system.
+The simplest model suitable for description of lensing by a
+galaxy or cluster is the singular isothermal sphere (SIS). For this lens
+model we have the following density profile 휌푔푟푎푣, the deflection
+angle ˆ훼푔푟푎푣, the Einstein radius 휃퐸 and the deflection potential
+휓(휃):
+휌푔푟푎푣 (푟) =
+휎2
+2휋퐺푟2 ,
+ˆ훼푔푟푎푣 = 4휋 휎2
+푐2 ,
+(12)
+휃퐸 = 4휋
+� 휎
+푐
+�2 퐷푑푠
+퐷푠
+,
+휓(휃) = 휃퐸 |휃| .
+(13)
+Here
+휎
+is
+the
+velocity
+dispersion.
+For
+more
+de-
+tails,
+see,
+e.g.,
+Schneider, Ehlers & Falco
+(1992),
+Schneider, Kochanek & Wambsganss (2006), Dodelson (2017),
+Congdon & Keeton (2018). For analytical studies of plasma effects
+in strong lens systems with singular isothermal ellipsoid lens, we
+refer to Crisnejo et al (2022); Crisnejo (2022); Ulla (2022).
+We assume also that the gravitating lens is surrounded by
+non-homogeneous spherically symmetric plasma, which leads to
+the refractive deflection ˆ훼푟푒 푓 푟 of light ray. Our subsequent cal-
+culations are appropriate for arbitrary law of plasma distribution.
+In such an approach it becomes possible to: neglect the mass of
+plasma particles, or take into account the gravity created by them.
+In Sec.6 we will consider the example when the gravitating mass is
+given by dark matter particles together with plasma particles, and
+there is the additional refractive deflection on this plasma. Alterna-
+tively, in frame of our approach, one could completely neglect the
+contribution of plasma particles to total gravitating mass.
+Having the deflection angle ˆ훼 in (1) as the function of impact
+parameter 푏, we write 푏 = 퐷푑 ·|휃| and introduce the reduced angle
+as
+훼(|휃|) = 퐷푑푠
+퐷푠
+ˆ훼(퐷푑·|휃|) =
+(14)
+= 퐷푑푠
+퐷푠
+ˆ훼푔푟푎푣 (퐷푑·|휃|) + 퐷푑푠
+퐷푠
+ˆ훼푟푒 푓 푟 (퐷푑·|휃|) .
+Here the expression 퐷푑 ·|휃| is the argument of the functions.
+With the gravitational deflection angle ˆ훼푔푟푎푣 from (12) and
+the Einstein angular radius 휃퐸 from (13), we find:
+훼(|휃|) = 휃퐸 − 퐵휔(|휃|) .
+(15)
+Here, for convenience, we have introduced the function
+퐵휔(|휃|) = − 퐷푑푠
+퐷푠
+훼푟푒 푓 푟 (퐷푑·|휃|) ,
+(16)
+which is positive for density profiles falling with radius and diverg-
+ing refractive deflection. By physical meaning, 퐵휔(|휃|) is the re-
+fractive deflection angle normalized by the distances ratio 퐷푑푠/퐷푠
+and taken with opposite sign. Similar notation was used in our
+previous paper (Tsupko & Bisnovatyi-Kogan 2020).
+To correctly describe negative 휃, we write the lens equation
+as (e.g., Suyu 2012; Schneider, Kochanek & Wambsganss 2006;
+Tsupko & Bisnovatyi-Kogan 2020):
+훽 = 휃 − 훼(|휃|) 휃
+|휃| .
+(17)
+MNRAS 000, 1–8 (2022)
+
+Time delay induced by plasma in strong lens systems
+5
+Finally, we have the lens equation with both gravitational and
+refractive contributions as
+훽 = 휃 − 휃퐸
+휃
+|휃| + 퐵휔(|휃|) 휃
+|휃| .
+(18)
+In order to solve the lens equation completely analytically, we
+will further also assume that the plasma effect is small compared to
+gravity:
+| ˆ훼푟푒 푓 푟 | ≪ | ˆ훼푔푟푎푣 | .
+(19)
+This leads to condition:
+퐵휔(|휃|) ≪ 휃퐸 .
+(20)
+With the condition (20), the equation (18) can be solved per-
+turbatively. To do this, we introduce a bookkeeping parameter 휀
+associated with plasma terms, expand on it in a series, and at the
+end we will put it equal to unity. The equation (18) can be written
+as
+훽 = 휃 − 휃퐸
+휃
+|휃| + 휀퐵휔(|휃|) 휃
+|휃| ,
+(21)
+and we look for a solution in the form of
+휃 = 휃 (0) + 휀 휃 (1) .
+(22)
+The zero-order term is a vacuum solution, and the first-order term
+describes a linear correction due to plasma presence.
+For 휃 > 0, the equation (21) becomes:
+훽 = 휃 − 휃퐸 + 휀퐵휔(휃) .
+(23)
+Substituting (22) into (23), we find for angular position 휃+ of the
+primary image:
+휃 (0)
++
+= 휃퐸 + 훽 ,
+(24)
+휃 (1)
++
+= −퐵+ ,
+where 퐵+ ≡ 퐵휔(휃 (0)
++ ) = 퐵휔(휃퐸 + 훽) ,
+(25)
+휃+ = 휃퐸 + 훽 − 퐵+ .
+(26)
+Similarly, for the secondary image (휃− < 0) we find:
+휃 (0)
+−
+= −휃퐸 + 훽 ,
+(27)
+휃 (1)
+−
+= 퐵− ,
+where 퐵− ≡ 퐵휔(|휃 (0)
+− |) = 퐵휔(휃퐸 − 훽) ,
+(28)
+휃− = −휃퐸 + 훽 + 퐵− .
+(29)
+Solutions
+of
+zero-order
+(vacuum)
+are
+known
+from
+the
+literature
+(Schneider, Ehlers & Falco
+1992;
+Schneider, Kochanek & Wambsganss
+2006;
+Dodelson
+2017;
+Congdon & Keeton 2018). Here we find first-order plasma correc-
+tions analytically. The secondary image exists only for 훽 < 휃퐸, as
+can be seen from Eq.(27). With 훽 → 휃퐸, magnification factor of
+secondary image goes to zero, and it disappears after 훽 becomes
+bigger than 휃퐸, see, e.g., Dodelson (2017) for SIS lens in vacuum.
+5
+TIME DELAY BETWEEN TWO IMAGES IN CASE OF
+SIS LENS IN PRESENCE OF PLASMA
+In this Section, we analytically calculate the time delay between
+primary and secondary images for SIS lens surrounded by plasma.
+With substitutions of Eq.(13), the general expression (10) for the
+time delay becomes:
+Δ푡(휃) = 퐷Δ푡
+푐
+� 1
+2 (휃 − 훽)2 − 휃퐸 |휃|
+�
++ 1 + 푧푑
+푐
+퐾푒
+2휔2 푁푖푛푡 (|휃|) .
+(30)
+In order to find expressions for the time delay for a particu-
+lar image, we must substitute the expressions (26) and (29) found
+earlier. After substitution into (30), we keep terms of the first order
+in plasma. By this way, we find the plasma corrections to all three
+terms in (30). By coincidence, for this model of lens, linear plasma
+corrections for geometrical and potential terms of time delay ex-
+actly cancel each other. This happens because with the inclusion
+of the plasma, the light deflection becomes smaller and ray goes
+closer to the lens, so the geometric delay decreases but the potential
+delay increases, and for this particular distribution of matter two
+corrections are of the same magnitude but opposite signs. Finally,
+for primary image we have:
+Δ푡+ = 퐷Δ푡
+푐
+�
+−1
+2휃2
+퐸 − 휃퐸 훽
+�
++ 1 + 푧푑
+푐
+퐾푒
+2휔2 푁푖푛푡 (휃퐸 + 훽) ,
+(31)
+whereas for secondary image:
+Δ푡− = 퐷Δ푡
+푐
+�
+−1
+2휃2
+퐸 + 휃퐸 훽
+�
++ 1 + 푧푑
+푐
+퐾푒
+2휔2 푁푖푛푡 (휃퐸 − 훽) .
+(32)
+Time delay between images is:
+Δ푡+ − Δ푡− = 퐷Δ푡
+푐
+(−2휃퐸 훽) +
+(33)
++ 1 + 푧푑
+푐
+퐾푒
+2휔2 [푁푖푛푡 (휃퐸 + 훽) − 푁푖푛푡 (휃퐸 − 훽)] .
+Until now, we have only assumed that 훽 < 휃퐸 (in order for the
+SIS lens to produce a secondary image). Let us now additionally
+assume that 훽 ≪ 휃퐸. This will allow us to significantly simplify
+the formulas.
+First, we write approximately:
+Δ푁푖푛푡 (휃퐸 + 훽) − Δ푁푖푛푡 (휃퐸 − 훽) ≃ 2훽 푑푁푖푛푡
+푑휃
+����휃=휃퐸
+.
+(34)
+Second, we remind that the deflection angle for the pho-
+ton
+with
+unperturbed
+motion
+along
+푧-axis
+is
+written
+as
+(Bisnovatyi-Kogan & Tsupko 2009, 2010, 2015):
+ˆ훼푟푒 푓 푟 (푏) = 퐾푒
+2휔2
+∞
+∫
+−∞
+휕푁
+휕푏 푑푧 .
+(35)
+Then, for positive 휃, we write 푏 = 퐷푑휃, and make the following
+transformation:
+1 + 푧푑
+푐
+퐾푒
+2휔2 2훽 푑푁푖푛푡
+푑휃
+= 1 + 푧푑
+푐
+2훽 퐷푑 ˆ훼푟푒 푓 푟 =
+(36)
+= 1 + 푧푑
+푐
+퐷푑퐷푠
+퐷푑푠
+퐷푑푠
+퐷푠
+ˆ훼푟푒 푓 푟 2훽 = − 퐷Δ푡
+푐
+퐵(|휃|) 2훽 .
+Finally, we find the compact expression for time delay between
+primary and secondary images of SIS lens surrounded by plasma:
+Δ푡+ − Δ푡− = − 퐷Δ푡
+푐
+2훽 [휃퐸 + 퐵휔(휃퐸)] .
+(37)
+MNRAS 000, 1–8 (2022)
+
+6
+G. S. Bisnovatyi-Kogan and O. Yu. Tsupko
+Here 퐷Δ푡 is given by Eq.(11), the variable 휃퐸 is the vacuum Einstein
+radius of SIS lens (13), and the function 퐵휔 is defined in Eq.(16),
+where ˆ훼푟푒 푓 푟 is supposed to be known as the function of 푏.
+Measuring the time delay between images allows one to
+determine the Hubble constant. Lens parameters from right-
+hand side of Eq.(37) are found from observations and mod-
+elling of the lens system. Time-delay distance (11) is in-
+versely proportional to Hubble constant (Refsdal 1964; Schneider
+1985; Schneider, Kochanek & Wambsganss 2006; Suyu et al 2010;
+Wong et al 2020):
+퐷Δ푡 ∝ 1
+퐻0
+.
+(38)
+As a result, the value of Hubble constant estimated from time de-
+lay measurements becomes a bit bigger if the plasma presence is
+taken into account. It should be noted however that results may vary,
+because here the simple model is considered, where plasma correc-
+tions to geometrical and potential terms are exactly cancelled. In
+more realistic models, the result will depend on signs and contribu-
+tions of plasma corrections in all three terms of time delay.
+In order to determine the Hubble constant, we need to measure
+the time delay (Δ푡+ −Δ푡−) between images. All values (except 퐷Δ푡)
+in the right-hand side of the eq.(37) are found from modeling the
+mass distribution in the lens based on observational data. Thus,
+from eq.(37), the time delay distance 퐷Δ푡 can be calculated, and,
+correspondingly, we are able to find the Hubble constant 퐻0. See,
+e.g., Suyu et al (2018).
+6
+EXAMPLE OF CALCULATION
+Let us write the corresponding formulas for non-homogeneous
+plasma with power-law number density:
+푁(푟) = 푁0
+�푟0
+푟
+� 푘
+,
+(39)
+where
+푁0 = const, 푟0 = const, 푘 = const > 1 .
+(40)
+In order to calculate the refractive deflection ˆ훼푟푒 푓 푟 (푏) by for-
+mula (35), one needs to substitute 푟 = (푏2 + 푧2)1/2, take a
+partial derivative by 푏 and then integrate on 푧. It results in
+(Bisnovatyi-Kogan & Tsupko 2009, 2010, 2015; Bliokh & Minakov
+1989):
+ˆ훼푟푒 푓 푟 (푏) = − 퐾푒
+휔2
+√휋 Γ
+�
+푘
+2 + 1
+2
+�
+Γ
+�
+푘
+2
+�
+푁0
+�푟0
+푏
+� 푘
+,
+(41)
+with Γ-function
+Γ(푥) =
+∞
+∫
+0
+푡푥−1푒−푡 푑푡 .
+(42)
+Correspondingly, the function 퐵휔(|휃|) which characterizes the
+plasma influence is:
+퐵휔(|휃|) = 퐷푑푠
+퐷푠
+퐾푒
+휔2
+√휋 Γ
+�
+푘
+2 + 1
+2
+�
+Γ
+�
+푘
+2
+�
+푁0
+�
+푟0
+퐷푑 ·|휃|
+� 푘
+.
+(43)
+For example, for 푘 = 2 we have:
+ˆ훼푟푒 푓 푟 (푏) = − 퐾푒휋
+2휔2 푁0
+�푟0
+푏
+�2
+,
+(44)
+퐵휔(|휃|) = 퐷푑푠
+퐷푠
+퐾푒휋
+2휔2 푁0
+푟2
+0
+퐷2
+푑|휃|2 .
+(45)
+As a numerical example, let us consider SIS lens with density
+distribution (12) at redshift 푧푑 = 0.5 and the source at some further
+redshift. Corresponding value of 퐷푑 is calculated with the following
+cosmological parameters: 퐻0 = 70 (km/s)/Mpc, Ω푚0 = 0.3, ΩΛ0 =
+0.7. Distances 퐷푠 and 퐷푑푠 are not specified. With typical value
+휎 = 200 km/s we find:
+휃퐸 = 1.15′′ 퐷푑푠
+퐷푠
+.
+(46)
+Let us consider the simplest case. The distribution of gravi-
+tating matter is given by (12), and we assume that the plasma has
+the same (up to constant coefficient) distribution as the rest of the
+matter, see Bisnovatyi-Kogan & Tsupko (2010):
+푁(푟) = 휌푔푟푎푣 (푟)
+휅푚푝
+.
+(47)
+Here coefficient 휅 ≃ 6 characterizes the contribution of plasma
+particles in comparison with other components of matter; 푚푝 is
+the proton mass. For more details, see Eq.(61) and Eq.(62) in
+Bisnovatyi-Kogan & Tsupko (2010).
+Refractive deflection produced by inhomogeneous plasma with
+number density (47) equals to (Bisnovatyi-Kogan & Tsupko 2010):
+ˆ훼푟푒 푓 푟 (푏) = −1
+4
+퐾푒
+휅푚푝
+휎2
+퐺휔2푏2 .
+(48)
+Correspondingly, according to Eq.(16), the function 퐵휔(|휃|)
+which characterizes the plasma influence is:
+퐵휔(|휃|) = 1
+4
+퐷푑푠
+퐷푠퐷2
+푑
+퐾푒
+휅푚푝
+휎2
+퐺휔2|휃|2 .
+(49)
+With the frequency of the observation 휈0 = 327 × 106 Hz, we
+write 휔0 = 2휋휈0 and 휔 = (1 + 푧푑) 휔0, and find:
+퐵휔(휃퐸) = 0.000054′′ 퐷푑푠
+퐷푠
+.
+(50)
+The rays forming the images have impact parameter of the
+order of 푏 = 휃퐸 퐷푑. At given values of other parameters, we obtain
+that near the lens the ray passes through the number density 푁푒 of
+about 0.5 cm−3. For bigger plasma densities, the bigger values of
+plasma influence can be expected. See, e.g., Er & Mao (2014) who
+have considered the number densities about 10 cm−3.
+7
+CONCLUSIONS
+(i) The time delay in strong lens systems surrounded by plasma envi-
+ronment is investigated in analytical way. We take into account three
+components of the time delay, as compared to straight-line propaga-
+tion: the geometrical delay, the potential delay in the gravitational
+field, dispersion delay in the plasma. See Eq.(10).
+(ii) Strong lens system is modelled by the singular isothermal
+sphere model. Plasma refractive deflection is taken into account
+in the form of small correction to gravitational deflection. Plasma
+corrections to image positions are found analytically, see Eqs. (26),
+(29). The effect is negligibly small in the optical range, but can be
+more significant in the radio range.
+MNRAS 000, 1–8 (2022)
+
+Time delay induced by plasma in strong lens systems
+7
+(iii) Analytical expression is derived for the time delay be-
+tween two images in case of SIS lens surrounded by arbitrarily
+distributed spherically symmetric plasma, see Eq.(37). This allows
+one to estimate simply the plasma effects for the particular lens.
+(iv) If the difference in image positions in different bands
+is observable for lens under consideration, this indicates that the
+plasma effects need to be taken into account in the lens modeling
+and further applications.
+ACKNOWLEDGEMENTS
+This article is partially supported by the Russian Foundation for
+Basic Research, project No. 20-52-12053.
+DATA AVAILABILITY
+Data availability is not applicable to this article as no new data were
+created or analysed in this study.
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+

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@@ -0,0 +1,1421 @@
+Capturing near-field circular dichroism enhancements from far-field measurements
+Jorge Olmos-Trigo,1, 2, ∗ Jon Lasa-Alonso,1, 2 Iker G´omez-Viloria,2 Gabriel Molina-Terriza,1, 2, 3 and Aitzol Garc´ıa-Etxarri1, 3
+1Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain.
+2Centro de F´ısica de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-San Sebastian, Spain.
+3IKERBASQUE, Basque Foundation for Science, Mar´ıa D´ıaz de Haro 3, 48013 Bilbao, Spain.
+Molecular Circular dichroism (CD) spectroscopy faces significant limitations due to the inherent weakness
+of chiroptical light-matter interactions. In this view, resonant optical antennas constitute a promising solution
+to this problem since they can be tuned to increase the CD enhancement factor, fCD, a magnitude describing
+the electromagnetic near-field enhancement of scatterers associated with a given helicity. Here, we derive an
+exact multipolar expansion of fCD, which is valid to deduce the integrated near-field CD enhancements of chiral
+molecules in the presence of scatterers of any size and shape under general illumination conditions. Based on
+our analytical findings, we show that the near-field fCD factor can be related to magnitudes that can be computed
+in the far-field, i.e., the scattering cross-section and the helicity expectation value. Moreover, we show that in
+the case of lossless cylindrically symmetric samples, the near-field fCD factor can be inferred experimentally
+only from two far-field measurements at specific scattering angles. Our contribution paves the way for the
+experimental characterization of devices capable of enhancing molecular CD spectroscopy.
+Chirality is a geometrical property of objects which are not
+superimposable with its mirror image. Chiral objects are ubiq-
+uitous in nature. Many organic molecules, such as glucose and
+most biological amino acids, are chiral. Not to mention the
+DNA double helix, which, in its standard form, always twists
+like a right-handed screw [1].
+In the pharmaceutical industry, chiral specificity is criti-
+cal because opposite enantiomers, i.e., mirror pairs of chiral
+molecules, can have beneficial or detrimental biological ef-
+fects on our organism depending on their handedness. Perhaps
+the most representative case of this is the so-called “Thalido-
+mide tragedy” which took place during the late 50s [2].
+Thalidomide was extensively prescribed to pregnant women
+due to its benefits in alleviating nausea. Instead, this drug
+prevented the proper growth of the fetus [3]. The outcome
+was that thousands of infants were born with severe congenital
+malformations. That fatality occurred because Thalidomide is
+a chiral molecule that was marketed as a 50/50 mixture of R
+and S enantiomers. While the S enantiomer is effective in alle-
+viating nausea, the R enantiomer is toxic. Thus, drugs consist-
+ing of chiral molecules may indeed have different therapeutic
+or toxicological effects.
+Even if they share the same atomic composition, enan-
+tiomer pairs are indistinguishable when measuring their scalar
+molecular properties.
+Their chiral nature is revealed only
+when interacting with other chiral entities.
+In electromag-
+netism, the most common chiral observable is helicity [4].
+Chiral molecules show a preferential absorption for fields of
+opposite helicities (left or right-handed polarized waves with
+helicity eigenvalues of σ = ±1, respectively). In a conven-
+tional Circular dichroism (CD) spectroscopy setup, a chiral
+molecular solution is sequentially illuminated by fields of op-
+posite helicities, and the total transmitted power is recorded
+for each case. The CD signal is then computed by taking the
+difference between these two power measurements in trans-
+mission. However, the inherent weakness of chiroptical re-
+sponses strongly limits the sensitivity of CD spectroscopy.
+∗ [email protected]
+Optical antennas, designed to control the properties of
+light [5], are promising candidates to enhance the spectro-
+scopic signals of chiral samples. The underlying phenomenon
+is that optical antennas can be engineered to enhance elec-
+tromagnetic fields while preserving the electromagnetic helic-
+ity [6–8]. Many works have explored optical antennas [9–11]
+and metasurfaces, namely, flat planar arrays of optical anten-
+nas, made of metallic [12–14] or/and high refractive index
+materials [15, 16] for enhanced chiral sensing [17, 18]. Ex-
+amples of such works can be found, for instance, in the ultra-
+violet [19], visible [20–26] or near -and far-infrared spectral
+range [27–33]. Moreover and quite recently, optical cavities
+have been also proposed as efficient platforms for enhanced
+chiral sensing [34–36].
+However, researchers rely exclusively on numerical meth-
+ods to design enhanced chiral sensing devices, as capturing
+the vector character of the near-field contribution can be chal-
+lenging. Here, we derive an exact multipolar expansion of fCD,
+which is valid to deduce the integrated near-field CD enhance-
+ments of chiral molecules in the presence of scatterers of any
+size and shape under general illumination conditions. From
+our analytical results, we show that fCD is proportional to
+the scattering cross-section and the helicity expectation value,
+which are experimentally measurable magnitudes in the far-
+field. In addition to this, we also show that for lossless and
+cylindrically symmetric scatterers, it is possible to infer the
+fCD factor only with two far-field measurements: the extinc-
+tion cross-section and the helicity density at specific scattering
+angles.
+To describe the excitation of chiral molecules, we adopt the
+formalism introduced by Tang and Cohen [37]. The CD signal
+of a chiral molecule under the illumination of a well-defined
+helicity field (σ = ±1) can be computed in vacuum as
+CDinc(r) = −4
+ε Im{G}Cσ
+inc(r),
+(1)
+where G is the chiral polarizability of the molecule and Cσ
+inc(r)
+is the incident local density of optical chirality [38],
+Cσ
+inc(r) = kε
+2 Im{Eσ
+inc(r)·ZHσ
+inc
+∗(r)}.
+(2)
+arXiv:2301.01248v1  [physics.optics]  3 Jan 2023
+
+2
+Here, k is the radiation wavevector, ε is the electric per-
+mittivity of the medium and Z =
+�
+µ/ε its electromagnetic
+impedance. Moreover, Eσ
+inc(r) and Hσ
+inc(r) refer to incident
+electromagnetic fields with well-defined helicity [39] (see Ap-
+pendix A 1 for more details).
+In the presence of achiral antennas (see Fig. 1) and in
+the helicity basis [40], the total electromagnetic fields can
+be generally written as Eσσ′
+tot (r) = Eσσ′
+sca (r) + Eσ
+inc(r)δσσ′.
+Here Eσσ′
+sca (r) is the scattered electromagnetic field written in
+terms of the electromagnetic modes with well defined helic-
+ity σ′ = ±1 (see Appendix A 2 for more details). In analogy
+with Eq. (1), we can express the total CD signal of a chiral
+molecule in the presence of an achiral optical antenna as [41]
+CDtot(r) = −4
+ε Im{G}Cσ
+tot(r) = k
+2Im{G} ∑
+σ′=±1
+σ′|Eσσ′
+tot (r)|2.
+(3)
+We seek to maximize CDtot(r) with Eq. (3). However, such
+enhancements cannot be achieved through G since it is a fixed
+chiral molecular parameter that cannot be engineered upon
+illumination. In contrast, the total density of optical chiral-
+ity, Cσ
+tot(r), can be tuned to enhance light-matter interactions
+and thus, increase the sensitivity of CD spectroscopy [42].
+To get a deeper insight into the total density of optical chi-
+rality, let us split Cσ
+tot(r) into three contributions; namely,
+Cσ
+tot(r) = Cσ
+inc(r)+Cσ
+sca(r)+Cσ
+int(r), with
+Cσ
+inc(r) = σ|Eσ
+inc(r)|2,
+(4)
+Cσ
+sca(r) = |Eσ+
+sca (r)|2 −|Eσ−
+sca (r)|2,
+(5)
+Cσ
+int(r) = 2σRe{Eσ
+inc
+∗(r)·Eσσ
+sca (r)}.
+(6)
+It is experimentally challenging to place enantiomers at
+will, namely, at a desired spatial coordinate r. Accordingly,
+it is more convenient to introduce an averaged expression of
+the local CD enhancement factor that gives insight into how
+efficient the nanoantenna might be at enhancing CD spec-
+troscopy [31]. To that end, let us first integrate both Eq. (1)
+and Eq. (3) over an imaginary sphere of radius r surrounding
+the achiral antenna to calculate then the ratio between these
+integrals, namely,
+fCD =
+� CDtot(r)dS
+� CDinc(r)dS = 1+
+� Cσ
+sca(r)dS+
+� Cσ
+int(r)dS
+� Cσ
+inc(r)dS
+,
+(7)
+where dS = r2 sinθdϕdθ. To compute Eq. (7), we need the
+orthogonality relations that the incident and scattered elec-
+tromagnetic fields satisfy when written in terms of the mul-
+tipoles with well-defined helicity. Fortunately, these can be
+derived from Jackson’s book in its third edition [43] (see Ap-
+pendix B 1 for the explicit derivation). Now, by considering
+these relations, and after some algebra (see Appendix B 2 for
+more details), we arrive at
+fCD = 1+
+˜Cσ
+sca + ˜Cσ
+int
+˜Cσ
+inc
+,
+(8)
+FIG. 1.
+Scattering process in which an incident field with well-
+defined helicity (red beam with σ = +1) impinges on an achiral an-
+tenna, represented by a glossy cube. Both R-and-S Thalomide enan-
+tiomers are also depicted close to the achiral antenna.
+where
+˜Cσ
+inc = σ ∑
+lm
+|Cσ
+lm|2Gjl jl,
+(9)
+˜Cσ
+sca = σ ∑
+lm
+|Cσ
+lm|2Re{almblm∗}Ghlhl,
+(10)
+˜Cσ
+int = σ ∑
+lm
+|Cσ
+lm|2Re{(alm +blm)G jlhl}.
+(11)
+Here ˜C denotes the average of C over a spherical surface, alm
+and blm are the so-called electric and magnetic scattering coef-
+ficients of an arbitrary sample while Cσ
+lm denotes the incident
+coefficients characterizing the nature of the wave. Moreover,
+G flgl = 1
+2
+�
+2 f ∗
+l (u)gl(u)+ 1
+u2
+∂
+∂u
+�
+uf ∗
+l (u) ∂
+∂u (ugl(u))
+��
+.
+(12)
+Here G flgl is a scalar function that depends on spherical Bessel
+and Hankel functions. In particular, we may have {fl,gl} =
+{jl, jl},{ jl,hl},{hl,hl}, jl and hl being the spherical Bessel
+and Hankel functions, respetively [43]. Moreover, notice that
+u = kr is the radius of the imaginary sphere, normalized by
+λ/2π, surrounding the object where the averaging integral is
+performed. For more details, please check Appendix B 1.
+Equation (8), together with Eqs. (9)-(12), is the first main
+result of this paper. These equations describe the integrated
+CD enhancement in the presence of scatterers of any size and
+shape under the excitation of fields with well-defined helicity.
+Our results overcome previous approximations, such as the
+widely used system of a circularly-polarized plane-wave illu-
+minating dipolar objects [22–33]. Thus, our findings may find
+applications in chiral sensing and chiral spectroscopy tech-
+niques beyond the current state-of-the-art.
+At this point and for didactic purposes, we provide the steps
+to find fCD around any optical antenna, which can be organized
+as follows:
+1. First, we need the solution of the electromagnetic fields
+under the illumination of a well-defined helicity beam.
+These can be obtained by any Maxwell’s solver [44].
+
+AAL3
+FIG. 2. Circular dichroism enhancements for a multipolar sphere
+under the illumination of a circularly polarized plane-wave. a) fCD
+(exact) and b) fCD − ˜Cσ
+int/ ˜Cσ
+inc (scattering). Here u = x + δx, with
+δx ≪ 1, being x = ka the optical size and m the index contrast.
+2. Then, we project in the far field the exact solution of the
+electromagnetic fields scattered by the object to obtain
+the scattering coefficients (see Eq. (9.123) in Jackson’s
+book in its third edition [43]). For convenience, we pro-
+vide in Appendix C the conversion between our scatter-
+ing coefficients alm and blm to the ones employed in
+Jackson’s book. Also notice that for spherical objects,
+we might use Mie’s theory to directly jump to step 3.
+3. Finally, we can compute fCD via Eq. (8), together with
+Eqs. (9)-(12).
+As an illustrative example of the abovementioned recipe,
+we depict the exact expression of fCD for a sphere sustain-
+ing several multipoles under plane-wave illumination (see
+Fig. 2a)). Moreover, we also depict in Fig. 2b) the scattering
+contribution to fCD. That is, fCD − ˜Cσ
+int/ ˜Cσ
+inc. In fact, we infer,
+by comparing Fig. 2a) with Fig. 2b), that the exact solution of
+fCD can be fairly approximated to just scattering in near-field,
+namely, fCD ∼ 1+ ˜Cσ
+sca/ ˜Cσ
+inc. Now, we understand the validity
+of this approximation based on the following facts:
+• On mathematical grounds, and according to Eq. (B3),
+it can be checked that Ghlhl ≫ |G jlhl| for u < l. That
+is, fundamental properties of spherical Bessel functions
+dictate that the scattering contribution dominates over
+the interference term for u < l.
+• On physical grounds, we notice that there are no strong
+resonances of the l-th multipole for u > l. For example,
+dipolar and quadrupolar resonances typically arise for
+u < 1 and 1 < u < 2, respectively [45].
+At this point, we will delve into Eq. (10), which approxi-
+mates the efficiency of an optical antenna to enhance the fCD
+factor, as it has just been previously explained. In particu-
+lar, let us examine its physical meaning when just one mul-
+tipole order (electric and magnetic) contributes to the opti-
+cal response of the antenna. In this regard, it is essential to
+note that the one multipole approximation includes the most
+studied scenario by the nanophotonic community devoted to
+enhanced chiral sensing: a circularly polarized plane wave in-
+cident on dipolar objects [22–33]. Now, when the scattering
+can be described by just one multipole order, l, Eq. (10) reads
+˜Cσ
+sca = σGhlhl
+m=l
+∑
+m=−l
+|Cσ
+lm|2Re{almblm∗},
+(13)
+By inspecting Eq. (13), we notice that ˜Cσ
+sca is proportional
+to the interference between the electric and magnetic scatter-
+ing coefficients Re{almblm∗}. Now, let us introduce the nor-
+malized (and unit-less) electromagnetic helicity expectation
+value, which reads [46–48]
+⟨Λ⟩ =
+�
+Ω
+�
+|Eσ+
+sca |2 −|Eσ−
+sca |2�
+dΩ
+�
+Ω
+�
+|Eσ+
+sca |2 +|Eσ−
+sca |2�
+dΩ.
+(14)
+Computing ⟨Λ⟩ in the case in which the optical response can
+be described by a single multipolar order l [46–48], we obtain
+⟨Λ⟩ = 2σ
+∑m |Cσ
+lm|2Re{almblm∗}
+∑m |Cσ
+lm|2 (|alm|2 +|blm|2) = σ ∑m |Cσ
+lm|Re{almblm∗}
+k2σsca
+,
+(15)
+where σsca is the scattering cross section [49]. At this point,
+we notice that the expression for ˜Cσ
+sca resembles ⟨Λ⟩.
+In
+fact and without loss of generality, we can write ˜Cσ
+sca =
+Ghlhl⟨Λ⟩k2σsca. As a result, fCD yields
+fCD ∼ 1+
+˜Cσ
+sca
+˜Cσ
+inc
+= 1+ Ghlhl⟨Λ⟩k2σsca
+˜Cσ
+inc
+.
+(16)
+This is another significant result of the present work.
+We
+have linked the averaged optical chirality associated with scat-
+tered fields, ˜Cσ
+sca, which is usually computed in the near-field,
+with quantities that can be evaluated or measured in far-field,
+i.e.
+the helicity expectation value, ⟨Λ⟩ and the scattering
+cross-section, σsca. In addition, and for helicity-preserving
+objects, ⟨Λ⟩ ∼ 1, the averaged optical chirality associated
+with scattered fields is simply given by the scattering cross-
+section. That is, ˜Cσ
+sca ∼ Ghlhlk2σsca. It is also essential to
+notice that, for both lossless and helicity-preserving objects,
+˜Cσ
+sca ∼ Ghlhlk2σext, with σext = σsca, σext being the extinction
+cross-section [49]. That is, the CD enhancement captured by
+the fCD factor is related to a single measurement of the ex-
+tincted power in the forward direction whenever ⟨Λ⟩ ∼ 1. The
+relation between fCD and σext for helicity-preserving achiral
+objects greatly reduces eventual experimental and numerical
+calculations devoted to enhanced chiral sensing.
+
+1.0
+0.9
+X 0.8
+0.7
+0.6
+2.5
+3.5
+4.5
+m4.5
+4.5
+= 3.5
+E 3.5
+2.5
+2.5
+0.5
+1.0
+1.5
+2.0
+u4.5
+4.5
+= 3.5
+E 3.5
+2.5
+2.5
+0.5
+1.0
+1.5
+2.0
+u4
+TABLE I. Analytic expressions for the CD enhancement factor, fCD, depending on the interaction between an incident plane-wave and an
+achiral scatterer. Here alm and blm denote the electric and magnetic scattering coefficients and {Ghlhl,G jlhl} can be computed from Eq. (B3);
+⟨Λ⟩ denotes the electromagnetic helicity expectation value and Λθ the helicity density; σsca and σext are the scattering and extinction cross-
+sections; and λ the radiation wavelength.
+Approximation in the calculation of fCD
+Plane wave illumination
+Exact multipolar expansion
+fCD = 1+∑lm (2l +1)
+�
+Re{almblm∗}Ghlhl +Re{(alm +blm)G jlhl}
+�
+Scattering approximation
+fCD ∼ 1+∑lm (2l +1)Re{almblm∗}Ghlhl
+Scattering approximation for arbitrary samples
+well-described by a single multipolar order l
+fCD ∼ 1+ πGhlhl
+λ 2
+⟨Λ⟩σsca −→
+����
+Lossless
+fCD ∼ 1+ πGhlhl
+λ 2
+⟨Λ⟩σext
+Scattering approximation for cylindrical samples
+well-described by a single multipolar order l
+fCD ∼ 1+ πGhlhl
+λ 2
+Λθσsca −→
+����
+Lossless
+fCD ∼ 1+ πGhlhl
+λ 2
+Λθσext
+So far, we have shown an alternative way to infer local CD
+enhancements in the near-field limit by computing far-field
+magnitudes such as the helicity expectation value, the scatter-
+ing cross-section, or the extinction cross-section. In particular,
+a scenario of major interest for the purpose of enhanced chi-
+ral detection occurs when the antenna preserves the helicity
+of the incident illumination, i.e. whenever ⟨Λ⟩ ∼ 1. These
+objects satisfy |Eσσ′
+sca (r)| ∼ 0 for σ ̸= σ′. This phenomenon
+is desirable since the local sign of optical chirality is pre-
+served. Experimentally, identifying helicity-preserving scat-
+terers requires measuring the polarization of all the scattered
+field components, something which is not feasible in practice.
+Thus, our question is: can we infer the helicity expectation
+value, ⟨Λ⟩, from a single measurement in the far-field? In
+the last part of this work, we will discuss scenarios in which
+the helicity density at a given scattering angle can be identical
+to its expected value. In particular, we will focus on cylin-
+drically symmetric scatterers which preserve the total angular
+momentum in the incident direction (m = m′) and whose op-
+tical response can be well-described by a single multipolar
+order (l = l′), e.g., nanodisks at normal incidence or spherical
+particles under the illumination of tightly-focused Laguerre-
+Gaussian beams [50].
+Mathematically, we can express the aforementioned condi-
+tion as ⟨Λ⟩ = Λθ, where Λθ denotes the helicity density at an
+angle θ, where θ is the scattering angle. After some algebra
+(see Appendix D), we obtain:
+Pm
+l (cosθ)∂Pm
+l (cosθ)
+∂ cosθ
+= 0
+=⇒
+⟨Λ⟩ = Λθ,
+(17)
+where Pm
+l (cosθ) are the associated Legendre Polynomi-
+als [43]. This is another key result of the present work. The
+helicity expectation value can indeed be computed from a sin-
+gle measurement of the helicity density at a specific angle θ.
+Our result implies that for a cylindrically symmetric scatterer
+whose response can be well-described by a single multipo-
+lar order, l, if we excite it with an illumination with a fixed
+total angular momentum, m, Eq. (17) specifies the angle at
+which the helicity density is equal to its expected value. For
+instance, if we consider the typical case of a cylindrically sym-
+metric dipolar target (l = 1) under a circularly polarized plane-
+wave illumination (m = ±1), Eq. (17) yields that the angle we
+should look at is θ = π/2. That is, the expectation value of
+the electromagnetic helicity can be inferred from a single mea-
+surement at the right angle. Thus, and according to Eq. (16),
+for lossless and cylindrically symmetric targets, we can infer
+the fCD factor by just considering two far-field measurements:
+extinction cross-section, in the forward direction, and helicity
+density, at an angle θ specified by Eq. (17) [51].
+Table I resumes the main results of this work, particularized
+for plane-wave illumination, as it is the most common exter-
+nal excitation for enhanced chiral sensing to date. In short, we
+have derived an exact multipole expansion of the integrated
+CD enhancement factor, fCD, for scatterers of any form and
+shape under general illumination conditions. In addition, we
+have established a roadmap to infer local CD enhancements
+from far-field measurements. That is, fCD can be extracted
+by calculating the helicity expectation value and the scatter-
+ing cross section, two Stokes parameters that can be evaluated
+in the far-field limit. Finally, we have shown an even more
+practical route to deduce fCD for cylindrically symmetric ob-
+jects by means of measuring the extinction cross-section and
+the local density of electromagnetic helicity at specific an-
+gles. Our results pave the way for experimental verification
+and characterization of building blocks for CD enhancement
+from far-field measurements, and thus, may give rise to novel
+developments in the field of chiral light-matter interactions.
+
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+exact solution of the electromagnetic fields under the illumina-
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+tor, Moreover, imaginary integrating spheres surrounding the
+object would be needed at each calculation of the fields in order
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+portional to the s0 parameter and helicity density is the ratio
+s3/s0 [48].
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+bridge university press, 1995).
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+Journal of Physics 12, 184 (1991).
+
+7
+Appendix A: Multipolar electromagnetic fields in a well-defined helicity basis
+1.
+Incident electromagnetic fields
+To start with the calculation of the surface-enhanced circular dichroism (CD), we need first to write down the electromagnetic
+fields. The most generic expression of the incident electromagnetic fields are given by
+Einc(r) =
+∞
+∑
+l=0
++l
+∑
+m=−l
+ge
+lmNNN j
+lm(r)+gm
+lmM
+M
+M j
+lm(r),
+(A1)
+iZHinc(r) =
+∞
+∑
+l=0
++l
+∑
+m=−l
+ge
+lmM
+M
+M j
+lm(r)+gm
+lmNNN j
+lm(r),
+(A2)
+where ge
+lm and gm
+lm stands for the incident electric and magnetic beam’s shape coefficients, respectively, and
+M
+M
+M j
+lm(r) = jl(kr)XXXlm(r),
+NNN j
+lm(r) = ∇∇∇×M
+M
+M j
+lm(r)
+k
+,
+XXXlm(r) = LYlm(θ,ϕ)
+�
+l(l +1)
+.
+(A3)
+Here M
+M
+M j
+lm(r) and NNN j
+lm(r) are (incident) Hansen’s multipoles [43], jl(kr) are the spherical Bessel functions, k being the radia-
+tion wavelength, and r the observation point. Moreover, Ylm(θ,ϕ) are the spherical harmonics, θ and ϕ being the polar and
+azimuthal angles, and L = {−ir×∇∇∇} is the total angular momentum operator. At this point, let us consider an arbitrary incident
+electromagnetic field with well-defined helicity, σ = ±1. Mathematically, we can write this well-defined helicity field as
+Eσ
+inc(r) = Einc(r)+σiZHinc(r)
+2
+=
+∞
+∑
+l=0
++l
+∑
+m=−l
+�ge
+lm +σgm
+lm
+√
+2
+��
+NNN j
+lm(r)+σM
+M
+M j
+lm(r)
+√
+2
+�
+=
+∞
+∑
+l=0
++l
+∑
+m=−l
+Cσ
+lmΨΨΨσ
+lm(r),
+(A4)
+where
+Cσ
+lm = ge
+lm +σgm
+lm
+√
+2
+,
+and
+ΨΨΨσ
+lm(r) = NNN j
+lm(r)+σM
+M
+M j
+lm(r)
+√
+2
+.
+(A5)
+Let us recall that the multipoles ΨΨΨσ
+lm(r) are eigenvectors of the squared angular momentum L2, the projection of the angular
+momentum on one direction, Lz, and helicity ΛΛΛ = (1/k)∇× [52] with eigenvalues l(l +1), m, σ, respectively.
+2.
+Scattered and total electromagnetic fields
+At this point, let us consider the scattered electromagnetic fields. The most generic expression of these fields is given by
+Eσ
+sca(r) =
+∞
+∑
+l=0
++l
+∑
+m=−l
+aσ
+lmNNNh
+lm(r)+bσ
+lmM
+M
+Mh
+lm(r),
+(A6)
+iZHσ
+sca(r) =
+∞
+∑
+l=0
++l
+∑
+m=−l
+aσ
+lmM
+M
+Mh
+lm(r)+bσ
+lmNNNh
+lm(r),
+(A7)
+where aσ
+lm and bσ
+lm stand for the electric and magnetic scattering coefficients, respectively. Notice that aσ
+lm and bσ
+lm depend on the
+incident illumination. As a result, we have explicitly indicated the σ-dependency. Moreover,
+M
+M
+Mh
+lm(r) = hl(kr)XXXlm(r),
+NNNh
+lm(r) = ∇∇∇×M
+M
+Mh
+lm(r)
+k
+,
+(A8)
+where M
+M
+Mh
+lm(r) and NNNh
+lm(r) are (scattered) Hansen’s multipoles [43] and hl(kr) are the spherical Hankel functions. Following the
+steps done in Eq. (A4), the electric field reads as
+Eσσ′
+sca (r) = Eσ
+sca(r)+σ���iZHσ
+sca(r)
+2
+=
+∞
+∑
+l=0
++l
+∑
+m=−l
+�aσ
+lm +σ′bσ
+lm
+√
+2
+��
+NNNh
+lm(r)+σ′M
+M
+Mh
+lm(r)
+√
+2
+�
+=
+∞
+∑
+l=0
++l
+∑
+m=−l
+Dσσ′
+lm ΦΦΦσ′
+lm(r),
+(A9)
+where
+Dσσ′
+lm = aσ
+lm +σ′bσ
+lm
+√
+2
+,
+and
+ΦΦΦσ′
+lm(r) = NNNh
+lm(r)+σ′M
+M
+Mh
+lm(r)
+√
+2
+.
+(A10)
+
+8
+At this stage, let us write down the relation between the incident and scattering amplitudes. These are given by Cramer’s rule of
+the tangential Maxwell boundary conditions [43], aσ
+lm = almge
+lm and bσ
+lm = blmgm
+lm where alm and blm are the so-called electric and
+magnetic scattering coefficients, respectively. Notice alm and blm do not depend on the incident illumination but on the optical
+properties and geometry of the target, e.g., the electric and magnetic Mie coefficients. Now, and since we are dealing with a
+well-defined helicity field, we can notice from Eq. (A4) that ge
+lm = σgm
+lm. Accordingly, we can write
+aσ
+lm = alm
+�Cσ
+lm
+√
+2
+�
+,
+bσ
+lm = σblm
+�Cσ
+lm
+√
+2
+�
+.
+(A11)
+Now, by inserting Eq. (A11) into the left side of Eq. (A10), we arrive to
+Dσσ′
+lm = Cσ
+lm
+(alm +σσ′blm)
+2
+.
+(A12)
+From now on, this will be our choice for the representation of the scattered coefficients. To conclude Appendix A, let us write
+the total electromagnetic fields. These are given by the additive sum of the incident and scattered electromagnetic fields. Hence,
+by taking into account both Eq. (A4) and Eq. (A9), we can straightforwardly write,
+Eσ
+tot(r) = ∑
+σ′=±1
+Eσσ′
+tot (r),
+iZHσ
+tot(r) = ∑
+σ′=±1
+σ′Eσσ′
+tot (r),
+(A13)
+with
+Eσσ′
+tot (r) = Eσσ′
+sca (r)+Eσ
+inc(r)δσσ′,
+(A14)
+where δσσ′ is a Kronecker delta. Next, we will use the orthogonality expressions that satisfy both the incident and scattered
+electromagnetic fields to compute the exact multipolar expansion of the CD enhancement factor.
+Appendix B: An exact multipolar expansion of fCD beyond the plane-wave picture
+1.
+An exact multipolar expansion of fCD: Orthogonality relations of well-defined helicity multipoles
+To derive an exact multipolar expansion of the CD enhancement factor, fCD, beyond the plane-wave picture, we need to know
+the orthogonality relations that satisfy both incident and scattered electromagnetic fields over an integrating sphere surrounding
+the object under illumination. To that end, we need first to calculate the set of orthogonality relations that fulfill the multipoles
+with well-defined helicity. That is, we need the orthogonality relations between incident {ΨΨΨσ
+lm,ΨΨΨσ
+l′m′}, interference {ΨΨΨσ
+lm,ΦΦΦσ
+l′m′},
+and scattering {ΦΦΦσ
+lm,ΦΦΦσ
+l′m′} terms, according to Eq. (7). Fortunately, all these relations can be derived from Jackson’s third edition
+book. Let us start this section by transcribing Eq. 10.48 that can be found on page 472 of Ref. [43]:
+�
+Ω N f
+lm
+∗ ·Ng
+l′m′dΩ = δll′δmm′
+�
+f ∗
+l (u)gl(u)+ 1
+u2
+∂
+∂u
+�
+uf ∗
+l (u) ∂
+∂u (ugl(u))
+��
+,
+�
+Ω M f
+lm
+∗ ·Mg
+l′m′dΩ = f ∗
+l (u)gl(u)δll′δmm′.
+(B1)
+Here u = kr denotes the optical radius of the integrating sphere and {δll′,δmm′} are Kronecker deltas. Notice that { fl(u),gl(u)}
+denote either Bessel or Hankel spherical functions, namely, jl(u) and hl(u) [43], depending on the nature of the field: incident
+or scattered, respectively.
+At this point, we have already learned that multipoles with well-defined helicity {ΨΨΨσ
+lm,ΦΦΦσ
+lm} are constructed by a linear
+combination of the Hansel multipoles (see the right side of Eq. (A5) and Eq. (A10) . As a result, we can write from Eq. (B1)
+�
+Ω(ΨΨΨσ
+l′m′)∗ ·ΨΨΨσ
+lm dΩ = G jl jlδll′δmm′,
+�
+Ω(ΦΦΦσ
+l′m′)∗ ·ΦΦΦσ
+lm dΩ = Ghlhlδll′δmm′,
+�
+Ω(ΨΨΨσ
+l′m′)∗ ·ΦΦΦσ
+lm dΩ = G jlhlδll′δmm′,
+(B2)
+with
+G flgl = 1
+2
+�
+2 f ∗
+l (u)gl(u)+ 1
+u2
+∂
+∂u
+�
+uf ∗
+l (u) ∂
+∂u (ugl(u))
+��
+.
+(B3)
+Now, we can re-write Eq. (B3) to get rid of second derivatives by making use of fundamental properties of the Ricatti-Bessel
+functions [53]. In particular, we can write,
+G jl jl = 1
+2
+�
+j2
+l (u)
+�
+1+ l(l +1)
+u2
+�
++ 1
+u2
+� ∂
+∂u (ujl(u)),
+�2�
+Ghlhl = 1
+2
+�
+|hl(u)|2
+�
+1+ l(l +1)
+u2
+�
++ 1
+u2
+����
+∂
+∂u (uhl(u))
+����
+2�
+,
+(B4)
+G jlhl = 1
+2
+�
+jl(u)hl(u)
+�
+1+ l(l +1)
+u2
+�
++ 1
+u2
+� ∂
+∂u (ujl(u)) ∂
+∂u (uhl(u))
+��
+,
+(B5)
+where ∂
+∂u (u fl(u))) = ufl−1(u)−l fl(u) is satisfied for f(u) = { jl(u),hl(u)}.
+
+9
+2.
+An exact multipolar expansion of fCD: From the helicity basis to the standard electric and magnetic multipolar expansion
+At this stage, we have all ingredients to calculate the exact multipolar expansion of fCD. The starting point of this section will
+be Eq. (7). According to Eq. (7), we need the orthogonality relations that satisfy the Rieman-Silberstein representation of the
+incident and scattered electromagnetic fields:
+˜Cσ
+inc =
+�
+Ω Cσ
+inc(r)dΩ =
+�
+Ω σ|Eσ
+inc(r)|2dΩ,
+(B6)
+˜Cσ
+sca =
+�
+Ω Cσ
+sca(r)dΩ =
+�
+Ω |Eσ+
+sca (r)|2 −|Eσ−
+sca (r)|2dΩ,
+(B7)
+Cσ
+int =
+�
+Ω
+˜Cσ
+int(r)dΩ = 2σ
+�
+Ω Re{Eσ
+inc
+∗(r)·Eσσ
+sca (r)}dΩ.
+(B8)
+These orthogonality relations can be computed by combining Eq. (A4) and Eq. (A9) with Eq. (B2). In fact and after some
+algebraic manipulations, it can be shown that
+˜Cσ
+inc = σ ∑
+lm
+|Cσ
+lm|2G jl jl,
+˜Cσ
+sca = ∑
+lm
+�
+|Dσ+
+lm |2 −|Dσ−
+lm |2�
+Ghlhl,
+˜Cσ
+int = 2σ ∑
+lm
+Re{Cσ
+lm
+∗Dσσ
+lm Gjlhl}.
+(B9)
+Now, let us insert Eq. (A12) into Eq. (B9) to obtain a closed-relation of the optical chirality enhancements in terms of the electric
+and magnetic scattering coefficients. After some algebra, we arrive to
+fCD = 1+
+˜Cσ
+sca + ˜Cσ
+int
+˜Cσ
+inc
+,
+(B10)
+with
+˜Cσ
+inc = σ ∑
+lm
+|Cσ
+lm|2Gjl jl,
+˜Cσ
+sca = σ ∑
+lm
+|Cσ
+lm|2Re{almblm∗}Ghlhl,
+˜Cσ
+int = σ ∑
+lm
+|Cσ
+lm|2Re{(alm +blm)Gjlhl}.
+(B11)
+Appendix C: Conversion from our conventions to those in Jackson’s book
+The multipolar expansion of the scattered electromagnetic fields provided in Jackson’s book reads as [43]
+EJack
+sca = Z
+∞
+∑
+l=0
++l
+∑
+m=−l
+iaE(l,m)NNNh
+lm(r)+aM(l,m)M
+M
+Mh
+lm(r),
+(C1)
+iZHJack
+sca = Z
+∞
+∑
+l=0
++l
+∑
+m=−l
+iaE(l,m)M
+M
+Mh
+lm(r)+aM(l,m)NNNh
+lm(r).
+(C2)
+Now, by inspecting Eqs. (C1)-(C2), we notice that the conversion from our conventions to those in Jackson’s book are given by
+alm =
+√
+2iZ
+�aE(l,m)
+Cσ
+lm
+�
+,
+blm =
+√
+2σZ
+�aM(l,m)
+Cσ
+lm
+�
+.
+(C3)
+where Z = µ/ε is the medium impedance. Notice that the electric and magnetic coefficients, provided in Jackson’s book, can
+be computed by conventional far-field projections of the scattered electromagnetic fields (see Eq. (9.123) in Ref. [43]). For
+completeness, we transcribe these expressions,
+ZaE(l,m)hl(kr) = −
+k
+�
+l(l +1)
+�
+Y ∗
+lm(θ,ϕ)r·EJack
+sca ,
+aM(l,m)hl(kr) =
+k
+�
+l(l +1)
+�
+Y ∗
+lm(θ,ϕ)r·HJack
+sca .
+(C4)
+Appendix D: Extracting the helicity expectation value from a single measurement of its local density
+In this Appendix, we derive the condition given in Eq. (17), that relates the local density of helicity at a certain angle, Λθ,φ,
+with the helicity expectation value, ⟨Λ⟩. For that aim, we should first define the local density of helicity, which we consider to
+be given in far-field by:
+Λθ,ϕ ≡ lim
+kr→∞
+|Eσ+
+sca (r,θ,ϕ)|2 −|Eσ−
+sca (r,θ,ϕ)|2
+|Eσ+
+sca (r,θ,ϕ)|2 +|Eσ−
+sca (r,θ,ϕ)|2 ,
+(D1)
+
+10
+where Eσσ′
+sca (r,θ,ϕ) is the scattered field written in terms of electromagnetic modes with well-defined helicity. From Eq. (D1),
+we notice that we require the asymptotic behavior of Eσσ′
+sca (r,θ,ϕ) in the far-field limit. For that aim, we need first to calculate
+how Hansel multipoles behave in the far-field limit. After some algebra, we arrive to
+lim
+kr→∞NNNh
+lm(r) = −eikr
+kr
+(−i)l+1
+�
+l(l +1
+ξξξ lm(θ,ϕ),
+lim
+kr→∞M
+M
+Mh
+lm(r) = −eikr
+kr
+(−i)l+1
+�
+l(l +1
+iηηηlm(θ,ϕ),
+(D2)
+where ξξξ lm(θ,ϕ) = r∇Ylm(θ,ϕ) and ηηηlm(θ,ϕ) = ˆr × ξξξ lm(θ,ϕ). Now, the far-field expression of the electromagnetic field
+scattered by an arbitrary sample can be computed [54]:
+Eσσ′
+sca (r,θ,ϕ) = −eikr
+kr ∑
+lm
+(−i)l+1
+Cσ
+lm
+�
+2l(l +1)
+�alm +σσ′blm
+√
+2
+��ξξξ lm +iσ′ηηηlm
+√
+2
+�
+.
+(D3)
+Substituting the scattered field in Eq. (D3) into the expression of the helicity density given by Eq. (D1), we obtain for fixed l
+and m values:
+Λθ = 2σ Re(almblm∗)
+�
+|ξξξ lm|2 +|ηηηlm|2�
+−
+�
+|alm|2 +|blm|2�
+Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�
+[|alm|2 +|blm|2][|ξξξ lm|2 +|ηηηlm|2]−4Re(almblm∗)Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�.
+(D4)
+The type of scatterers which may be described by fixed l and m values are cylindrically symmetric particles, illuminated by a
+beam with a well-defined angular momentum, m, and with a non multipolar response. Due to the cylindrical symmetry of the
+scatterers, helicity density cannot depend on ϕ variable. This is the reason why we have chosen to write helicity density as Λθ in
+Eq. (D4). Crucially, it can be checked that whenever Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�
+= 0, one recovers the expression of the helicity expectation
+value, i.e.
+Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�
+= 0
+=⇒
+Λθ = ⟨Λ⟩ = 2σ Re(almblm∗)
+|alm|2 +|blm|2 .
+(D5)
+Importantly, the condition Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�
+= 0 is purely geometrical, i.e. does not depend on the particular response of the
+scatterer. This fact makes the expression completely general and applicable to any type of cylindrical sample whose response is
+well-described by a fixed l. Thus, for this type of scatterers, there are locations in the far-field at which the helicity density is
+equal to the helicity expectation value.
+The specific sites at which ⟨Λ⟩ = Λθ are obtained by finding the solutions to the equation Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�
+= 0. More explicitly,
+we have that the vector and scalar spherical harmonics are written as:
+ξξξ lm(θ,ϕ) = ∂Ylm(θ,ϕ)
+∂θ
+ˆuθ +
+1
+sinθ
+∂Ylm(θ,ϕ)
+∂ϕ
+ˆuϕ
+(D6)
+ηηηlm(θ,ϕ) = ∂Ylm(θ,ϕ)
+∂θ
+ˆuϕ −
+1
+sinθ
+∂Ylm(θ,ϕ)
+∂ϕ
+ˆuθ
+(D7)
+Ylm(θ,ϕ) =
+�
+2l +1
+4π
+(l −m)!
+(l +m)!Pm
+l (cosθ)eimϕ,
+(D8)
+where Pm
+l (cosθ) are the associated Legendre polynomials. With the definitions above it is straightforward to check that
+Pm
+l (cosθ)∂Pm
+l (cosθ)
+∂ cosθ
+= 0
+=⇒
+Im
+�
+ξξξ ∗
+lm ·ηηηlm
+�
+= 0.
+(D9)
+In conclusion, for fixed values of l and m, there is an angle θ, given by the transcendental equation above, at which the helicity
+density, Λθ, is equal to the helicity expectation value, ⟨Λ⟩.
+

C9FRT4oBgHgl3EQfwjj-/content/tmp_files/2301.13639v1.pdf.txt

@@ -0,0 +1,861 @@
+Title:  
+Behavioural predictors of math anxiety 
+Authors:  
+Name: Chen Mun Yip Kinnard 
+Institution: Nanyang Technological University 
+Email: [email protected] 
+ 
+Name: Azilawati Jamaludin 
+Institution: National Institute of Education, Nanyang Technological University 
+Email: [email protected] 
+ 
+Name: Aik Lim Tan 
+Institution: National Institute of Education, Singapore 
+Email: [email protected] 
+ORCiD: https://orcid.org/0000-0001-5910-7003 
+Corresponding author: 
+Name: Aik Lim, TAN 
+Institution: National Institute of Education, Singapore (1 Nanyang Walk, Singapore 637616) 
+Email: [email protected] 
+ 
+Declarations of Interest: The authors declare no conflict of interest. 
+Funding 
+This research was funded by the Singapore National Research Foundation (NRF) under the Science 
+of Learning Initiative (NRF2016-SOL002-003). 
+ 
+ 
+ 
+
+Behavioural Predictors of Math Anxiety 
+Abstract 
+Math anxiety (MA) is a highly prevalent problem in education that has consistently shown to lead to 
+poorer math performance. This study sought to investigate whether certain behaviours are predictive 
+of MA among students. The study involved elementary school students (n=124) who were low-
+progressing in math and is part of an educational intervention program to improve their mathematical 
+abilities through a neural-informed math game. Ten classification types of behavioural indicators were 
+identified, such as counting out loud. A multiple linear regression was conducted where students’ 
+anxiety scores were regressed on these behavioural observations, along with their gender, mood, and 
+math profile. Three behavioural observations were positively and significantly associated with their 
+math anxiety. Implications and limitations of the study are discussed. 
+Keywords – Math Anxiety, Low-progress math, Behavioural Predictors 
+INTRODUCTION 
+Math anxiety (MA) refers to a state of fear and apprehension when one is engaging with math 
+(Ashcraft & Krause, 2007). It is a highly prevalent problem in many educational settings and persists 
+across many different levels of education (Gunderson et al., 2018; Jain & Dowson, 2009), beginning 
+in primary school (Harari et al., 2013) and even up until university education (Røykenes & Larsen, 
+2010). Furthermore, MA has been shown to lead to poorer math performance. For instance, a meta-
+analysis by Zhang et al. (2019) found that MA was robustly and negatively associated with math 
+performance. In other words, higher MA tend to lead students to perform more poorly in math. 
+Researchers have also found that MA can have an adverse impact on one’s initial learning of 
+mathematics and subsequently, a poorer acquisition of math skills and advanced mathematical 
+concepts as well (Krinzinger et al., 2009). For instance, MA has been shown to adversely affect basic 
+numerical operations such as simple addition and subtraction – operations that serve as the 
+foundation for more complex mathematical abstraction (Ashcraft, 2002; Maloney et al., 2010). Indeed, 
+Ashcraft and Moore (2009) found that individuals with MA tend to find it more difficult to engage in 
+complex mathematical calculations. 
+It should also be noted that MA can both be the cause and the result of poorer math performance 
+among students (Foley et al., 2017). This means that MA does not only lead to poorer math 
+performance, but it can also occur due to having performed poorly in a standardized math test 
+(Ashcraft & Krause, 2007). In the long-term, this can result in a feedback loop whereby after having 
+performed poorly in a math test, students get MA and start to avoid math lessons (Ashcraft & Moore, 
+2009). Subsequently, they might also develop negative beliefs about their math abilities, which further 
+exacerbates their MA and avoidance of math classes (Lent et al., 1991). For example, MA is often 
+regarded as one of the primary reasons for the low enrolment in math courses among female 
+students (Meece et al., 1982). This avoidance can also extend to mathematics-related professions, 
+which further limits one’s career opportunities (Eispino et al., 2017).  
+Similarly, a recent meta-analysis by Barroso et al. (2021) found that MA was negatively associated 
+with math achievement, and that this negative association was present from childhood through to 
+adulthood. It should be noted that poorer math performance also lead to negative outcomes in 
+adulthood such as poorer financial planning (McKenna and Nickols, 1988) and lower confidence 
+among math educators (Olango and Assefa, 2013).  
+Given that MA can lead to poorer math learning and performance both in the short-term and long-
+term, it is important to help students manage or ameliorate their MA. A systematic review by Balt et al. 
+(2022) suggests that interventions that help manage one’s MA also serve to improve one’s math 
+performance. It is therefore important for the early identification of students with higher levels of MA, 
+in order to help them reduce their feelings of anxiety, and subsequently, improve their math 
+performance. 
+
+Presently, the predominant way of identifying MA in students is through self-report measures (Carey 
+et al., 2017; Luttenberger et al., 2018). However, this method of identifying MA among students in 
+actual educational settings might not be very practical as time and manpower are needed to 
+administer and collate the results of these measures (Wu et al., 2012). Furthermore, given that one’s 
+level of MA can change over time (Ashcraft & Krause, 2007), there might be a need for regular testing 
+to obtain a more accurate measure of the students’ MA. This suggests that there is a need for a more 
+cost-effective way of identifying MA among students and across time. 
+One way of identifying MA might be through the identification of behaviours that are symptomatic of 
+one’s underlying MA. Anxiety can manifest itself in a variety of behaviours such as fidgeting and 
+restlessness (Yadav et al., 2017). This suggests that it might be possible to identify MA in students 
+through their behaviour when engaging with math-related tasks. This is important as any behaviour 
+that is predictive of MA can help educators to easily identify students with high MA and, consequently 
+help reduce their MA by either managing their emotions or improving their math performance (Balt et 
+al., 2022). Thus, this study sought to investigate whether certain behaviours are predictive of MA 
+among students. 
+LITERATURE REVIEW 
+Math Anxiety 
+The concept of MA was first introduced as a topic of study in academic research by Dreger and Alken 
+(1957) to describe students’ anxiety toward numbers. Presently, MA commonly refers to a state of 
+fear and apprehension when one is engaging with math (Ashcraft & Krause, 2007) and is regarded as 
+a primarily emotional response (Mammarella et al., 2015). It should be acknowledged that although 
+most studies conceptualize MA as a single factor, some researchers suggest that MA consists of two 
+dimensions: cognitive and affective (Wigfield & Meece, 1988). The cognitive dimension broadly refers 
+to one’s thoughts and concerns about math performance while the affective dimension includes one’s 
+emotions like nervousness regarding math testing (Wigfield & Meece, 1988). 
+MA can also be differentiated from test anxiety or general anxiety (Ashcraft & Ridley, 2005) For 
+instance, different measures of MA are more highly correlated with one another as compared to test 
+and general anxiety (Ashcraft & Ridley, 2005). In other words, MA has been shown to be sufficiently 
+different from other forms of test or general anxiety, and can thus be considered to be an independent 
+construct (Dowker et al., 2016).  
+There are many factors that can affect one’s level of MA and they can be differentiated into three 
+main categories: environmental, cognitive and personal factors (Rubinsten & Tannock, 2010). 
+Environmental factors include the experiences of math learning in the classroom or at home. For 
+instance, Mutodi and Ngirande (2014) found that negative experiences of math learning in both at 
+home and in the classroom can lead to MA. Similarly, stressful teaching strategies like time testing 
+(Ashcraft & Moore, 2009) and assigning math-related tasks as a form of punishment have been found 
+to increase one’s MA (Oberlin, 1982). Conversely, McNaught and Grouws (2007) suggests that 
+teachers and parents should focus on creating a more positive environment for students when they 
+are learning mathematics. They also suggested that parents should focus on exploring any fear and 
+anxieties that the child might have towards learning mathematics, especially during the earlier stages 
+of math learning. Furthermore, parents’ own MA can have a negative intergenerational effect on their 
+children’s experiences of math learning, leading them to have poorer math learning and higher levels 
+of MA (Maloney et al., 2015). The culture in which one is brought up could also influence one’s level 
+of MA, whereby students in countries with a greater emphasis on doing well in examinations could 
+experience higher levels of MA (Dowker et al., 2016; Tan & Yates, 2011).  
+Personal factors include low self-efficacy (Kesici & Erdoğan, 2010; Maloney et al., 2011), low self-
+esteem, previous negative experiences (Mutodi & Ngirande, 2014; Rubinsten & Tannock, 2010) and 
+even genetic vulnerabilities (Wang et al., 2014). For instance, Wang et al. (2014) found that one’s 
+genetic vulnerability to general anxiety also increases one’s likelihood of developing MA. Cognitive 
+factors involve having low intelligence and poor cognitive abilities in mathematics (Rubinsten & 
+Tannock, 2010). 
+Assessments of Math Anxiety 
+
+Math anxiety is assessed almost exclusively with self-report questionnaires, across all age groups, in 
+both educational and research settings (Carey et al., 2017; Luttenberger et al., 2018). For adults, 
+common measures include the Mathematics Anxiety Rating Scale (MARS; Richardson & Suinn, 1972) 
+and the Revised Mathematics Anxiety Rating Scale (R-MARS; Baloğlu & Zelhart, 2007). There are 
+also questionnaires developed for younger populations such as primary school children, involving 
+more pictorial rating scales and greater focus on more concrete math situations. Examples include the 
+Scale for Early Mathematics Anxiety (SEMA; Wu et al., 2012), and the Children's Attitude to Math 
+Scale (James, 2013).  
+As with most self-report measures, the accuracy of these questionnaires are subjected to the 
+participants’ truthfulness and the accuracy of their own self-perceptions (Dowker et al., 2016). To 
+counteract this problem, some studies utilized physiological measures to determine the individual’s 
+level of MA, such as the level of cortisol secretion (Mattarella-Micke et al., 2011). Neurological 
+measures such as EEG recordings (Núñez-Peña & Suárez-Pellicioni, 2015) and functional MRI scans 
+have also been used to measure and map out MA (Pletzer et al., 2015). However, such measures 
+might not be very practical in actual educational settings as time and manpower are needed to 
+administer and collate the results of these measures (Pletzer et al., 2015; Wu et al., 2012). Given that 
+one’s level of MA can change over time (Ashcraft & Krause, 2007), regular testing might be required 
+to obtain a more accurate measure of students’ MA. This suggests that there is a need for a more 
+cost-effective and adaptive way of measuring one’s level of MA among students and across time.  
+Behavioural Predictor of Math Anxiety 
+One possible indicator of MA could be the behaviour of the individual while he or she is engaging with 
+math-related tasks. MA can affect the individual and manifest itself at various levels, such as the 
+physiological level (Hunt et al., 2017), the psychological or cognitive level (Hunt et al., 2014), and the 
+emotional level (Papousek et al., 2012).  
+MA can also affect individuals on a behavioural level (Pizzie & Kraemer, 2017). For instance, 
+individuals who have higher levels of MA tend to display higher instances of disengagement, 
+exemplified by the avoidance of mathematical stimuli such as math classes or math-related work 
+(Preis & Biggs, 2001; Pizzie & Kraemer, 2017). Similarly, some researchers assert that MA invokes 
+three types of reactions: affective, cognitive, and behavioural responses (Olango, 2016; Ashcraft, 
+2019). Olango (2016) states that these behavioural responses fall on a continuum with approach and 
+avoidance behaviours as the continuum extremes. Olango (2016) also describes how the three 
+domains can interact with one another in a reinforcing way. This could produce either a negative 
+cycle, whereby avoidance behaviour reciprocally interacts feelings of anxiety and worry, or whereby 
+one’s approach behaviour reciprocally interacts with thoughts of math achievement and feelings of 
+confidence, creating a positive and self-reinforcing cycle.  
+Similarly, Pries and Biggs (2001) outlines how the negative cycle of avoidance of mathematical stimuli 
+occurs in four phases and how it is self-reinforcing. In the first phase, the individual has a negative 
+experience with mathematics or mathematics situations. This leads to the second phase, whereby the 
+individual starts to avoid mathematics due to the negative experience previously. For the third phase, 
+after avoiding mathematics, the individual has less opportunities to improve their mathematics skills, 
+consequently becoming less proficient in and prepared for mathematics (Dowker et al., 2016; Pries & 
+Biggs, 2001). In the fourth phase, as a consequence of being less prepared for mathematics, the 
+individual has poorer mathematics performance. This creates a negative experience with 
+mathematics, leading the individual back to phase one and developing a vicious cycle of mathematics 
+avoidance and MA (Pries & Biggs, 2001; Ramirez et al., 2018). Such avoidance behavior is so 
+common that Ashcraft and Moore (2009) describe it as “an overriding characteristic of math-anxious 
+individuals” (p. 201). 
+This suggests that certain behaviours could be a manifestation of one’s MA and consequently be 
+used to predict the presence of it. This is consistent with the symptomology of anxiety in clinical 
+research, whereby certain behaviours have been shown to be symptomatic of one’s anxiety 
+(American Psychiatric Association, 2013; Beck et al., 1988; Spitzer et al., 2006). For instance, the 
+Beck Anxiety Inventory (BAI) – a self-report measure of general anxiety symptomatology – indicates 
+that certain behaviours such as being “shaky”, “hands trembling” and “wobbliness in legs” are 
+common symptoms of anxiety (Beck et al., 1988). Similarly, the Generalised Anxiety Disorder-7 scale 
+
+(GAD-7) by Spitzer et al. (2006) – a common self-report tool to determine the severity of anxiety 
+symptoms – includes the behavioural symptom of “being so restless that it is hard to sit still”. The 
+clinical literature therefore suggests that the presence of certain behaviours is predictive of one’s 
+anxiety, signifying that certain behaviours might also be predictive of one’s MA. 
+METHOD 
+Participants 
+A total of 124 Singapore primary school students – aged 6 to 7 years old – were recruited from 
+primary schools in Singapore for this study. Of the 124 students, 48.39% were females (n = 60) and 
+51.61% were males (n = 64). The students had different math profiles; they were labelled as either 
+Typically Developing (TD; 15.32%; n = 19), Low-Progress Learners (LP; 41.13%; n = 51), or at-risk of 
+Developmental Dyscalculia (DD; 43.55%; n = 54). The students were categorised into the three 
+groups based on a national numeracy screening assessment administered by the schools when 
+students first enter primary school. The students were grouped according to four levels based on their 
+scores, ranging from 0 to 3. Students with scores 2 or below are considered low-progressing and will 
+need to attend a math pull-out program aimed at providing more support for these students. Students 
+who score a ‘3’ are considered Typically Developing. In this particular study, the students also 
+completed the Dyscalculia Screener (Butterworth, 2003), which identified students who may be at-risk 
+of developmental dyscalculia. 
+The study was conducted in accordance with the Declaration of Helsinki, and the protocol was 
+approved by the NTU Institutional Review Board (Reference Number: IRB-2017-10-030). 
+Measures 
+Math Anxiety 
+To measure the students’ math anxiety, the Scale for Early Mathematics Anxiety (SEMA) by Wu et al. 
+(2012) was used. The SEMA scale is a 20-item self-report measure that requires participants to 
+indicate how anxious they would feel in certain situations involving mathematics. It utilizes a 5-point 
+Likert scale, ranging from 1 (not nervous at all) to 5 (very, very nervous). Each of the 5 points on this 
+Likert scale are accompanied by a drawing of an anxious face – each with different gradations of 
+anxiety based on the number (or the intensity of the anxiety) they represent. The accompaniment of 
+the anxious faces helps the children to better identify and choose the number that best represent their 
+levels of anxiety (Wu et al., 2012). 
+The measure can be differentiated into its two factors: 1) Numerical Processing Anxiety (NPA) factor 
+and 2) Situational and Performance Anxiety (SPA) factor. The NPA items require respondents to 
+indicate how anxious they would feel if they had to solve certain math questions (e.g., “Is this right? 9 
++ 7 = 18”), while the SPA factor focuses on situations where mathematics is involved (e.g., “You are 
+in math class and your teacher is about to teach something new”). Each factor consists of 10 items. 
+The total SEMA score is calculated by summing all 20 item scores while the total score of its factors 
+(e.g., NPA and SPA) is calculated by summing the scores of their corresponding 10 items. Higher 
+scores represent higher levels of MA. 
+Math Game – ‘Number Beads’ 
+Number Beads is a digital game that was developed by Butterworth & Laurillard (2017), that sought to 
+incorporate and apply the findings from neuroscience and cognitive science research to help low 
+attaining learners (e.g., individuals with dyscalculia) in learning mathematics. 
+In the game, there is a target set of beads at the top of the screen (grouped with the word ‘Target’; 
+refer to Figure 1), with the main objective of the game being to construct the target set by either 
+splitting or joining the different colour-coded beads available in the play area. More detailed 
+information about the game ‘Number Beads’ can be found in Laurillard (2016). 
+
+ 
+Figure 1. A screenshot of the Number Beads math game from Laurillard (2016). 
+Behavioural Observations 
+Given limited research on behaviours predictive of MA, this study took a thematic analysis approach 
+towards the formulation of the different categories of behaviours that would be investigated. More 
+specifically, the researchers observing the students were instructed to write down any notable 
+behaviours that the students exhibited while engaging in the math game. 
+Based on these notable behaviours, they were then clustered and categorised into 10 different 
+behavioural observations (BO; e.g., BO1 to BO10): 
+1. Counting out loud (BO1) 
+This involves counting by saying the numbers out loud one by one (e.g., “three, four, five…”). 
+2. Verbalization of thought process (BO2) 
+This refers to the students’ verbalization of their thought process as to how they would carry out the 
+arithmetical calculations (e.g., “I can get 6 by taking 2 from 8”). 
+3. Utterance of other comments or sounds (BO3) 
+Any other verbal utterances that do not fall into the above two behavioural observations would be 
+included in this behavioural observation. Examples include humming, singing of songs, making 
+nonsensical sounds or utterances that bear no meaning or relation to the arithmetical tasks at hand. 
+4. Use of fingers to assist in counting (BO4) 
+This includes using fingers to point to the ‘beads’ on the computer screen while counting and/or using 
+one’s fingers as representations of digits to assist in arithmetic calculations. 
+5. Checking on and interacting with peers (BO5) 
+This behavioural observation broadly refers to any of the students’ interactions with their peers. 
+Examples include looking at their classmates, asking how they are doing and/or competing with one 
+another to achieve the highest scores. 
+6. Gross hand and leg movements (BO6) 
+This involves any gross hand and leg movements such as fidgeting, shaking of one’s legs or playing 
+with stationery. 
+7. Asking questions when in doubt (BO7) 
+
+Menu
+Bank
+Target:Any act of clarification regarding how to play the math game or asking for assistance to complete a 
+particular set by the students would be included in this behavioural category. 
+8. Celebrating success (BO8) 
+This involves any behaviour that signifies a celebration of having successfully completed a level or a 
+task. Examples of this behavioural observation include punching the air and saying “Yes! I did it!” or 
+raising both hands in the air and cheering.  
+9. Looking elsewhere or being distracted (BO9) 
+This refers to any behaviour that is suggestive of a lack of attention or being distracted such as 
+looking elsewhere. If the student is looking at their peers, it will not fall under this behavioural 
+observation and will instead fall under the behavioural observation of ‘Checking on and interacting 
+with peers’. 
+10. Random splitting and joining of Number Beads (BO10) 
+This refers to the beads in the math game that participants manipulate (e.g., separating a group of 3 
+beads from a larger group of 8 beads to do the calculation for ‘8-3=5’). Even though this is not strictly 
+a physical behaviour, there is a strong parallel between this behaviour and the actual manipulation of 
+physical objects or what is termed as ‘manipulatives’ to do arithmetic (Jones & Tiller, 2017). 
+Mood & Math Profile 
+Based on the observations by the research team, each student’s mood while engaging in the math 
+game was labelled as one of four moods: 1) happy, 2) neutral, 3) bored or 4) stressed. 
+The math profile of the students – as either TD, LP or DD – was determined by both school’s criteria 
+(using the Ministry of Education’s Early Numeracy Indicator) and the Dyscalculia Screener 
+(Butterworth, 2003). 
+Procedure 
+The students were first instructed to complete the SEMA measure. Aligned with Wu et al. (2012), a 
+trained researcher assisted the students in the completion of this measure by helping to read the 
+questions aloud and asking the students to point to the face that best represented how anxious they 
+felt. The students were also encouraged to ask questions and clarify the meaning of the questions if 
+they did not understand the scale’s items. 
+Students were subsequently invited to play the neural-informed math game – Number Beads – on a 
+designated laptop. During the game, the researchers took note of the students’ mood and any notable 
+behaviours that they exhibited. 
+The administration of SEMA and game play took place in school on separate sessions where the 
+game was played for about 2-4 hours per student. 
+Data Analysis 
+Firstly, descriptive analyses of the following variables were conducted: 1) mood; and 2) SEMA scores. 
+Secondly, the Chi-square test was employed to investigate whether gender, affect and the 
+behavioural observations were significantly related to the students’ math profiles.  
+Thirdly, multiple linear regression analyses were conducted where the SEMA scores were regressed 
+on the 1) behavioural observations, and the students’ 2) gender, 3) mood and 4) math profile. Given 
+that there is no research on such behavioural indicators of math anxiety among Singapore learners, a 
+stepwise regression was chosen as it would allow for the selection of a model that provides the most 
+
+efficient prediction of MA (Aiken & West, 1991). For the stepwise regression analysis, the PIN will be 
+set at .05 and POUT will be set at .10. 
+As SEMA scale can be broken down into two factors: NPA & SPA factors. Thus, the stepwise 
+regression analyses were conducted thrice; each with a different dependent variable (DV). For the 
+first model, the DV is the total SEMA score. The second model’s DV is the total NPA score, while the 
+third model’s DV is the total SPA score. 
+RESULTS 
+Descriptive Analyses 
+Mood  
+Based on the researchers’ observations, 51.61% of the students were happy (n = 64), 39.52% were 
+of neutral mood (n = 49), 8.06% were bored (n = 10) and 0.81% was stressed (n = 1).  
+SEMA scores 
+With regards to the total SEMA score, the mean score across all participants was 23.5 (possible 
+range of 0 – 80), with a standard deviation (SD) of 15.91. For each item, the average score was 1.18 
+(out of 4), with an SD of 0.80.  
+The SEMA scores can be differentiated into their two factors: the NPA factor and the SPA factor. For 
+the NPA factor, the mean score across all participants was 13.24 (possible range of 0 – 40), with a 
+SD of 8.60. For each item within the NPA factor, the average score was 1.32 (out of 4), with an SD of 
+0.86.For the SPA factor, the mean score across all participants was 10.26 (possible range of 0 – 40), 
+with a SD of 8.76. For each item within the SPA factor, the average score was 1.03 (out of 4), with an 
+SD of 0.88. 
+Math Profile and Gender, Affect & Behavioral Observations 
+The chi-square tests revealed that affect, (X2 (2, N = 124) = 5.696, p = .458), and gender were not 
+significantly related to math profile (X2 (2, N = 124) = 0.611, p = .737).  
+Math profile was only significantly related to one of the behavioral observations, namely BO5 - 
+Checking on and interacting with peers (X2 (2, N = 124) = 8.540, p = .014). More specifically, those 
+with development dyscalculia were more likely to exhibit the behavior of ‘checking on and interacting 
+with peers’ when engaging with math, as compared to those who were ‘low-progress learners’ and 
+‘typically developing’. It was a small to medium effect size (V = .262; Cohen, 1988).  
+First regression model: Total SEMA score 
+The first regression model regressed the students’ 1) affect, 2) gender, 3) math profile and 4) 
+behavioural observations on the total SEMA scores, using a stepwise approach.  
+The first predictor added into the regression model was BO1 – Counting out loud. It had the largest 
+coefficient of determination (R2 = .061, F(1, 122) = 7.863, p < 0.01). After BO1 was present in the 
+model, BO10 – Random splitting and joining of Number Beads – had the next biggest change in the 
+coefficient of determination (R2 = .050, F(1, 121) = 6.825, p < 0.05), and was thus added into the 
+model.  
+The concluding model revealed BO1 and BO10 to be significant factors in affecting total SEMA scores 
+(refer to Table 1). This set of predictors explained 11.1% of the variance in total SEMA scores (R2 = 
+.111, F(2, 121) = 7.531, p < 0.001).  
+Table 1. Summary of the Multiple Regression Analysis for the First Model on Total SEMA scores.  
+
+ 
+Unstandardised 
+Coefficients 
+ 
+Standardised 
+Coefficients 
+ 
+ 
+ 
+Variable 
+B 
+Std. Error 
+ 
+β 
+ 
+t 
+Sig. 
+Constant 
+19.078 
+1.777 
+ 
+ 
+ 
+10.737 
+.000 
+ 
+BO1 – Counting 
+out loud 
+ 
+7.807 
+ 
+2.934 
+ 
+ 
+.229 
+ 
+ 
+2.660 
+ 
+.009 
+ 
+B010 – Random 
+splitting and 
+joining of Number 
+Beads 
+ 
+8.409 
+ 
+3.219 
+ 
+ 
+.225 
+ 
+ 
+2.612 
+ 
+.010 
+ 
+Second regression model: Total NPA score 
+Similarly, the second regression model regressed the students’ 1) affect, 2) gender, 3) math profile 
+and 4) behavioural observations on the total NPA scores, using a stepwise approach.  
+The behavioural observation – BO1 (Counting out loud) – was the only predictor added in this 
+regression model (R2 = .047, F(1, 122) = 5.983, p < 0.05). In other words, this model showed that 
+BO1 was the only factor that was significantly associated with the total NPA scores (refer to Table 2). 
+This model explained 4.7% of the variance in total SEMA scores (R2 = .047, F(1, 122) = 5.983, p < 
+0.05).  
+Table 2. Summary of the Multiple Regression Analysis for the Second Model on Total NPA scores. 
+ 
+Unstandardised 
+Coefficients 
+ 
+Standardised 
+Coefficients 
+ 
+ 
+ 
+Variable 
+B 
+Std. Error 
+ 
+β 
+ 
+t 
+Sig. 
+Constant 
+11.988 
+.914 
+ 
+ 
+ 
+13.117 
+.000 
+ 
+BO1 – Counting out 
+loud 
+ 
+3.986 
+ 
+1.630 
+ 
+ 
+.216 
+ 
+ 
+2.446 
+ 
+.016 
+ 
+Third regression model: Total SPA score 
+Lastly, the third regression model regressed the same four classes of variables on the total SPA 
+scores, using a stepwise approach. 
+The first predictor added into the regression model was BO10 – Random splitting and joining of 
+Number Beads (R2 = .075, F(1, 122) = 9.906, p < 0.01). BO1 – Counting out loud – was then 
+subsequently added into the model (R2 = .046, F(1, 121) = 6.319, p < 0.05). Lastly, the model then 
+included BO9 – Looking elsewhere or being distracted (R2 = .033, F(1, 120) = 4.739, p < 0.05). 
+The concluding model showed that BO1, BO9 and BO10 were significant factors in affecting total SPA 
+scores (refer to Table 3). This set of predictors explained 15.4% of the variance in total SEMA scores 
+(R2 = .154, F(1, 120) = 7.304, p < 0.001). 
+Table 3. Summary of the Multiple Regression Analysis for the Third Model on Total SPA scores. 
+
+ 
+Unstandardised 
+Coefficients 
+ 
+Standardised 
+Coefficients 
+ 
+ 
+ 
+Variable 
+B 
+Std. Error 
+ 
+β 
+ 
+t 
+Sig. 
+Constant 
+6.818 
+1.049 
+ 
+ 
+ 
+6.501 
+.000 
+ 
+BO10 – Random 
+splitting and joining 
+of Number Beads 
+ 
+4.564 
+ 
+1.769 
+ 
+ 
+.221 
+ 
+ 
+2.580 
+ 
+.011 
+B01 – Counting out 
+loud 
+4.419 
+1.592 
+ 
+.235 
+ 
+2.776 
+.006 
+BO9 – Looking 
+elsewhere or being 
+distracted 
+3.694 
+1.697 
+ 
+.187 
+ 
+2.177 
+.031 
+ 
+Discussion 
+This study sought to investigate whether affect, gender, math profile and certain behavioural 
+observations are predictive of MA. This study concluded with 3 different regression models: 1) total 
+SEMA scores, 2) total NPA scores and 3) total SPA scores. 
+Is gender related to MA? 
+The findings revealed that there was no significant relationship between gender and MA for all three 
+regression models. This is consistent with many previous studies which found that there were no 
+gender differences in MA for elementary school students (i.e., students aged 5 to 10; Erturan & 
+Jansen, 2015; Harari et al., 2013; Kucian et al., 2018; Schleepen & Van Mier, 2016). 
+Is affect (mood) related to MA? 
+There was no significant association found between affect and MA for all three regression models. In 
+other words, the findings suggest that one’s affect is not related to one’s level of MA. This insignificant 
+association could be due to the students’ differential abilities in emotional regulation. One form of 
+emotional regulation is expressive suppression, which refers to the attempt to conceal one’s feelings 
+and physiological state (Cohen et al., 2021). Prior studies found that an individual’s ability to engage 
+in emotional regulation is linked to MA (Brooks, 2014; Pizzie & Kraemer, 2021). This suggests that the 
+student’s differential abilities in emotional regulation could be a moderating factor in the relationship 
+between affect and MA, which might explain the lack of a significant association between these two 
+variables.  
+Are there differential associations of MA with math learning profiles? 
+The results suggest that there is no significant relationship between math profile and MA for all 
+regression models. That is, there was no significant differences in the MA scores among students with 
+different math profiles. The lack of a significant relationship between math profile and MA in the 
+current study could be due to differences in the testing contexts. Unlike previous studies which 
+measured the students’ MA in traditional educational contexts (e.g., before and after a standardized 
+math test; Zhang et al., 2019), the current study measured the students’ MA in their mathematics 
+classrooms and prior to the gameplay. 
+Are there behavioural predictors of MA? 
+
+The findings revealed that three behavioural observations were significantly and positively associated 
+with MA: namely, 1) the act of counting out loud, 2) randomly splitting and joining of the Number 
+Beads, and 3) looking elsewhere or being distracted. 
+BO1 – counting out loud – was consistently included in all three regression models. This suggests 
+that the behaviour of counting out loud could be a predictor of a student with MA. This finding seems 
+to be consistent with the current literature. Past findings suggest that individuals with MA tend to have 
+smaller work memory capacity when engaging in arithmetic tasks as they often have intrusive anxious 
+thoughts that take up working memory resources (Ashcraft & Krause, 2007; Maloney et al., 2010; Shi 
+& Liu, 2016). It was also found that one’s working memory capacity mediates the relationship between 
+MA and visual enumeration (Maloney et al., 2010). Given that saying material out loud has been 
+shown to improve one’s memory of it (Forrin & MacLeod, 2018), the act of counting out loud could 
+similarly be a compensatory strategy to aid one’s reduced working memory capacity. 
+BO10 – random splitting and joining of Number Beads was included in two regression models – 
+where the predictors were regressed on SEMA scores and SPA scores. This suggests a positive 
+relation between student’s overall MA and his/her situational and performance anxiety. This is 
+consistent with current literature on the negative relationship between MA and math performance 
+(Zhang et al., 2019). This behavioural observation (BO) of the random management of the digital 
+manipulatives suggests that the student may not know how to procedurally solve the arithmetic task, 
+suggesting underlying math struggles manifested by poor math performance. Furthermore, MA can 
+both be the cause and the result of poorer math performance (Foley et al., 2017). Taken together, 
+BO10 could be symptomatic of one’s poorer math performance and or MA. However, given that this 
+behavioural observation was not significantly associated with NPA scores, it suggests that this 
+behaviour only occurs for individuals who are anxious about being evaluated on their ability to do 
+math problems, and not for those who feel anxious in nonevaluative situations. 
+BO9 – looking elsewhere or being distracted – was only included in one regression model, where it 
+was focused on SPA scores. This suggests that this behaviour is positively related to one’s situational 
+and performance anxiety. This is consistent with prior studies on the behavioural responses that are 
+commonly associated with math anxiety, such as behavioural disengagement (Pizzie & Kraemer, 
+2017). Similarly, Eysenck et al. (2007) also showed that anxiety decreases one’s focus on the task at 
+hand and makes one more vulnerable to distraction. An inability to focus on the task at hand and 
+being distracted by other stimuli (e.g., one’s intrusive thoughts) are common characteristics of those 
+with high levels of anxiety (Ramirez et al., 2016). The behavioural observation of looking elsewhere or 
+being distracted seems to be symptomatic of this tendency towards disengagement and distraction. 
+However, since BO9 was not significantly related to SEMA and NPA scores, this suggests that this 
+avoidance behaviour only occurs for individuals who feel anxious when they are being evaluated on 
+their ability to do math problems, and not for those who feel anxious in nonevaluative situations. 
+ Theoretical and practical implications 
+Given that there has been no study investigating the relationship between behavioural observations 
+and MA particularly in the Singapore education context, this study helped to bridge this literature gap. 
+The current study therefore provides researchers with preliminary insights into MA through an 
+ethnographic lens, affording opportunities for future research to be done in this area. The findings 
+provide some practical implications in the context of math education.  
+Instead of relying on self-report measures, educators can tap on their behavioural observations of 
+classroom manifestations to anticipate quickly and cost-effectively socio-emotional issues related to 
+math learning such as that of Math Anxiety. It would also be less onerous for educators to rely on 
+such BO instead of having students complete interim self-report measures of MA. Such behavioural 
+observations could also be used for educational contexts in which students are learning to count by 
+manipulating physical objects (Jones & Tiller, 2017). The findings suggest that the random 
+manipulation of physical objects (e.g., beads) could also be used as an indicator of one’s MA. Such 
+observations are therefore able to provide teachers with early indicators to identify students with 
+possible MA, thereby enabling them to intervene early and provide the necessary support for these 
+students.  
+Limitations and Future research 
+
+This study has two main limitations. Firstly, this study was conducted in the context of a math game. 
+More specifically, the students that were being observed in the present study were playing a math 
+game. Such educational games tend to be less anxiety-inducing than traditional educational settings 
+due to the lower stakes and expectations (Marshall et al., 2016; Rocha & Dondio, 2021; Taylor & 
+Mohr, 2001). Thus, the behaviours that were predictive of MA in the current study’s context might not 
+be generalizable to traditional educational settings. More research will be needed to ascertain the 
+utility of such behavioural predictors in real-world educational settings. 
+Secondly, this study was conducted in Singapore, an East Asian country with a high level of math 
+proficiency among its students (Programme for International Student Assessment, 2018). Many 
+studies have shown that emotional suppression – which refers to the concealment of one’s true 
+emotions (Schouten et al., 2020) – is more common in Asian countries like Singapore and Japan as 
+compared to Western countries like Belgium and the United States (Butler et al., 2007; Cheung & 
+Park, 2010; Soto et al., 2011). This suggests that students from different countries may have differing 
+levels of emotional suppression or regulatory abilities when engaging in math-related activities and 
+consequently differing expressions of MA. 
+CONCLUSION 
+This study sought to investigate the relationship between behavioural observations and MA. The 
+findings suggest that behavioural observations like counting out loud, randomly splitting and joining 
+Number Beads (or any physical objects) and looking elsewhere or being distracted can be predictive 
+of one’s MA. It suggests that educators can use such behavioural observations as a quick gauge of 
+whether a student is at risk of related socio-emotional struggles such as experiencing anxiety when 
+learning math. While current findings might be limited in their generalisability to traditional educational 
+settings and western countries, future studies should look to investigating the variables in other 
+contexts. 
+ 
+  
+ 
+ 
+
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+ 
+

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+ 
+ 
+1 
+Deterministic multi-level spin orbit torque switching  
+using focused He+ ion beam irradiation 
+Jinu Kurian1, Aleena Joseph1, Salia Cherifi-Hertel1, Ciaran Fowley2, Gregor Hlawacek2, Peter Dunne1, 
+Michelangelo Romeo1, Gwenaël Atcheson3, J. M. D. Coey3, Bernard Doudin1* 
+1Université de Strasbourg, CNRS, IPCMS UMR 7504, 23 rue du Loess, F-67034 Strasbourg, France 
+2Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden - Rossendorf, 
+Bautzner Landstraße 400, 01328 Dresden, Germany 
+3AMBER and School of Physics, Trinity College, Dublin 2, Ireland 
+* Corresponding author; email: [email protected] 
+ 
+KEYWORDS spintronics, spin orbit torque switching, nanomagnetism, ion beam irradiation, Hall bars 
+ 
+He+ ion irradiation is used to pattern multiple areas of Pt/Co/W films with different 
+irradiation doses in Hall bars. The resulting perpendicular magnetic anisotropy landscape 
+enables selective multilevel current-induced switching, with full deterministic control of the 
+position and order of the individual switching elements. Key pattern design parameters are 
+specified, opening a way to scalable multilevel switching devices.  
+ 
+Spin orbit torque (SOT) magnetic memory devices have unique advantages in terms of low power 
+consumption, fast operation capabilities and simplified circuitry design, placing them at the forefront 
+of information technology development.1–3 Their current-induced switching properties rely on the 
+perpendicular magnetic anisotropy (PMA) of their thin magnetic layers, which is readily modified by 
+light ion irradiation.4 We have previously shown the benefit of combining nanometer-scale 
+irradiation in a He+ microscope with in situ electronic transport property measurements to reveal 
+how the local PMA evolves under irradiation.5 Reduction of the PMA in specific locations makes 
+selective SOT switching of magnetic bits possible. Our purpose here is to show how this technique 
+can be extended to multiple irradiation zones in the same device, taking advantage of fine tuning the 
+spatial distribution of PMA calibration, to achieve multi-level switching in devices. 
+Multi-level storage is a promising approach to increase memory density6 while avoiding the 
+addressability problems associated with size reduction. Increasing the number of available states by 
+a factor N leads to an N-fold improvement in the density of memory without any change to the 
+device geometry. Multi-level SOT devices can also be exploited for bio-inspired computing 
+architectures, such as neuromorphic computing based on spintronics7,8, and multi-level switching 
+
+ 
+ 
+2 
+has been achieved in PMA heterostructures in many ways. These include stabilizing intermediate 
+multidomain states by means of multi-ferromagnetic layer stacks,9–12 wedged magnetic layers,13,14 
+ferrimagnetic15,16, ferromagnetic/antiferromagnetic structures17, or with interleaved interfaces with 
+different spin Hall angles.18 A different approach defines the multiple levels as a sequence of 
+pinning-depinning states in SOT-driven reversal19,20, which has been shown in both irradiated and 
+non-irradiated single magnetic films21. Here, several unique states are accessible using shaped 
+current pulses22, but the exact magnetic configuration is highly dependent on sample fabrication and 
+reproducibility.23 Our aim is to create well-defined, reproducible, deterministic states, that are 
+addressable by SOT switching, and avoid the limitations of sample reproducibility. 
+Thin film stacks of Ta (5.0)/Pt (2.0)/Co (1.2)/W (1.5)/Pt (1.5)/Ta (1.5) (thickness in nm) were grown 
+on thermally-oxidized Si substrates by DC magnetron sputtering at room temperature. The Ta (5.0) 
+seed layer ensure good adhesion and (111) texture of Pt that promotes perpendicular anisotropy of 
+the Co layer.24 The heterostructure of Co sandwiched between Pt and W was chosen to optimize SOT 
+switching efficiency due to opposite signs of the spin Hall angles of Pt and W. The PMA of the cobalt 
+was confirmed by magnetometry. Films were patterned into Hall bar structures of width 10 µm and 
+length 180 µm by UV lithography. The Hall crosses were irradiated using an Orion Nanofab helium 
+ion microscope system while monitoring the evolution of the anomalous Hall resistance (RAHE) in-
+situ.  (Figure 1 a,b)5. For uniformly irradiated Hall crosses, RAHE gradually decreases with irradiation 
+until there is a sharp drop at a critical dose of 42.5 ± 2 ions/nm2, where the easy axis of 
+Figure 1. (a) Schematic of SOT setup of a patterned Hall bar undergoing He+ irradiation with the directions of current and magnetic 
+field indicated. (b) Evolution of in-situ anomalous Hall resistance with irradiation, the letters P,Q indicate the two-zone irradiation 
+doses shown in Fig. 2a, and A-D the four-zone doses shown in Fig. 3a. The magnetic anisotropy field (orange squares) as a function of 
+irradiation, deduced from Hall measurement on ex-situ samples, is also indicated. (d)  Comparison of SOT induced magnetization 
+switching before and after irradiating with 35 ions/nm2 under a 100 mT bias field. 
+ 
+
+0.0
+1.0
+-0.2.
+0.8
+-0.4
+0.6
+-0.6.
+0.4
+-0.8
+0.2 
+ 
+3 
+magnetization flips from out-of-plane to in-plane. The critical dose is reproducible within 5% from 
+sample-to-sample.  
+The magnetic anisotropy field (Fig. 1b) was measured ex-situ by Hall effect as a function of applied 
+field. Hall measurements were performed under 1mA bias current. SOT-induced switching 
+experiments were performed with 1 ms current pulses in an interval of 1 s, under 100 mT bias field 
+applied along the current direction (Fig. 1c) . An average current density values of approximately 8 
+MA/cm2 corresponds to a 10 mA pulse current. Our experimental procedure does not give 
+indications of long-term sample Joule heating if the current stress remains below typically 35 mA. 
+One can imagine Joule heating during the 1 ms current pulse, but we found that the current 
+hysteresis curves remain fully reproducible (within a few percent) if the current sweeps does not 
+exceed 35 mA, providing confidence that the current stress does not impact the Co thin film and its 
+interface down the atomic level. The observed trends in the critical dose, fall in anisotropy and 
+reduced critical current are consistent with our previous work.5 Fig. 1c compares the SOT switching 
+hysteresis curve of a non-irradiated cross and one irradiated, illustrating the reduction of switching 
+current after irradiation with 35 ions nm-2Further characterisation and calibration of both the 
+magnetic anisotropy as well as SOT switching currents was carried out for several doses below the 
+critical dose. In this work, we focus on Hall crosses which were individual well-defined zones were 
+irradiated with different doses.  
+Figure 2. (a) Schematic of a two-zone irradiated sample (doses P,Q Fig. 1b) with and without a gap between irradiated zones, and two 
+boundary zones Q0 to minimize domain wall propagation during SOT switching (see text). (b) SOT induced magnetization switching of two 
+junctions with and without a gap between two irradiated zones with dose 23 ions/nm2 and 35 ions/nm2 at a bias field of 125 mT. (c) MOKE 
+images of the SOT induced magnetization switching when sweeping the current from +21 mA to -21 mA, and then (below broken line) 
+back to +21 mA. Intermediate currents (21 to -5 mA, and -21 to 5 mA) where the magnetization does not switch are not shown.  
+ 
+
+P
+- 
+ 
+4 
+Fig. 2 details the simplest case of a two-zone irradiation device, with one zone irradiated with 23 
+ions nm-2 (dose denoted ‘P’ in Fig. 1b) and the other with 35 ions nm-2 (‘Q’). For a direct comparison 
+we irradiated neighbouring Hall crosses, labelled 1 and 2 in Fig. 2a. In cross 1 the irradiated zones are 
+adjacent to each other, whereas in cross 2 they are separated by approximately 1 µm. In both 
+crosses, we detect additional resistance states due to the different magnetic anisotropies of the two 
+zones P & Q, denoted as ↑↑, ↑↓, ↓↑ and ↓↓ (Fig. 2b). These states have unique RAHE values and are 
+accessible by cycling the applied current pulses. We found that measured Hall resistance values were 
+stable within 1% over many hours. The resulting states are non-volatile, holding their resistance 
+values for at least two days.  The switching currents of the irradiated areas are also reproducible 
+within a few percent when repeating the current sweeps.  Magneto-optical Kerr effect (MOKE) 
+microscopy confirms the correspondence between the Hall resistance data and the switching of each 
+irradiated area, allowing us to visualize the resulting magnetic configurations (Fig. 2). The boundary 
+of non-irradiated and irradiated zones are known to pin propagating domain walls during magnetic 
+field driven reversal25,26, and are found to play such a role in the SOT induced switching by Kerr 
+microscopy visualisation (not shown here). We found that it was crucial to irradiate magnetic 
+boundary zones on either side of the Hall cross, marked ‘Q0’ in Fig. 2a, using a near-critical dose so 
+that these zones switch at the smallest currents. This inhibits propagation of magnetization reversal 
+from far away in the main bar into the cross, which would interfere with the Hall measurement. 
+Essentially, this additional irradiation acts to isolate the magnetic switching behaviour of a cross, 
+without severing its electrical contacts. Unfortunately, we found that reproducibility of the switching 
+current of unirradiated zones surrounded by Q0 irradiated areas was quite poor, but nevertheless 
+sufficient to avoid a nucleation of reversal far from the sample of interest at lower current values 
+than those switching irradiated areas.   
+Fig. 2 also illustrates how independent switching of the two zones P and Q can be improved by 
+leaving a non-irradiated strip between them. A width of 1 µm was chosen to allow adequate 
+resolution in the Kerr microscope. Under a bias field of 125 mT we find a clear independent SOT 
+switching of P and Q in both the measured Hall resistance (Fig. 2b) and MOKE imaging (Fig. 2c) only 
+when there is an irradiation gap (P2, Q2). When the current is swept from 21 to -21mA, the zones 
+irradiated with 35 ions nm-2, Q1,Q2 , selectively switch at -5 mA and the adjacent irradiated zone, P1, 
+reverses its magnetization continuously between -7 mA and -8 mA. In contrast, the zone P2, 
+separated by the non-irradiated gap, switches at -11 mA. This larger critical current results from 
+domain wall pinning in the unirradiated zone which retains the larger initial PMA, preventing the 
+reversal of one zone to expand to the neighbouring one. Note that for this sample, the unirradiated 
+areas switch at current values similar to Q2, which explains why the antiparallel state does not 
+correspond to zero RAHE. 
+Irradiation of multiple zones within a Hall cross illustrate the scalability of this process.  Four zones, 
+A–D, separated by 1 µm gaps and irradiated with different doses, 35, 27, 20 and 13 ions nm-2, are 
+shown in Fig. 3a. As with the sample shown in Fig. 2, magnetic isolation bands, Q0, were irradiated 
+either size of the junction with a dose of 35 ions nm-2, protecting the unirradiated zone against 
+switching up to 20 mA current. The SOT induced switching of the device under a bias field of 125 mT 
+in Fig. 3b shows the three intermediate steps corresponding to the independent switching of the 
+four zones upon sweeping applied current. MOKE microscopy confirms this interpretation of the AHE 
+data (Fig. 3c), displaying a sequential switch of each area from lowest to highest PMA, i.e. from 
+highest to lowest ion dose. This demonstrates that our approach can be extended to irradiate N 
+
+ 
+ 
+5 
+zones with different doses, across different areas. We expect the domain wall width to define the 
+scale of the minimum bit size, leading to a separation wall of a few tens of nm for individual cells 
+below the 100 nm size range. This is well within the resolution limits of the irradiation process but is 
+challenging for magnetic imaging and electrical detection and is beyond the scope of this paper.  
+ 
+In summary, magnetic anisotropy engineering by appropriate multi-zone He-ion irradiation makes 
+multi-level switching of magnetic configurations robust and predictable. Our experimental design 
+was chosen for maximum simplicity, to demonstrate that we can independently switch the magnetic 
+state of individually irradiated areas in a single Hall cross. Moreover, by taking advantage of the 
+hysteresis properties of the current-induced switching, many macrospin configurations become 
+accessible (green arrows in Fig. 2b), and one can imagine switching any combination of the 
+macrospin presented in Fig. 2. This would allow the number of distinct electrical states to be much 
+larger than the number of irradiated zones N. More sophisticated designs are also possible on a 
+smaller scale, for example where a switching of a given area influences its neighbour via its magnetic 
+stray field. Furthermore, ion irradiation is compatible with other approaches for realizing multistate 
+memories without altering their physical design, such as more sophisticated magnetic stacks, wedge 
+layers or lithographic patterns.8-16 We believe that the strategy we have outlined proposes a 
+versatile strategy to realize high-density and low-power-consumption SOT-based memory devices 
+and is relevant for future types of computing architecture. 
+Author Contributions 
+P.D., C.F. and J.K. initially designed the experiment, J.K. and A.J. performed most experiments, G.A. 
+and J.M.D.C. helped for the materials fabrication, G.H. C.F. for the ion irradiation, S.C. and M.R. for 
+Kerr measurements. B.D. and J.M.D.C. supervised the experiments. J.K., A.J. and B.D. wrote the 
+manuscript, where all authors contributed to its improvement.  
+Acknowledgements 
+We thank Fabien Chevrier and the staff of the STnano nanofabrication facility for daily support. This 
+project has received funding from the European Union’s Horizon 2020 research and innovation 
+Figure 3. (a) Schematic of the four zones of a junction irradiated with doses 13 ions nm-2, 20 ions nm-2, 27 ions nm-2 and 35 ions nm-2. (b) 
+MOKE acquired SOT induced magnetization switching of the junction under a bias field of 125 mT. (c) MOKE images of the SOT induced 
+magnetization switching of the junction when sweeping from 21 to – 25 mA.  
+ 
+
+B
+D
+A 
+ 
+6 
+programme under the Marie Skłodowska-Curie grant agreement MaMi No. 766007 and QUSTEC No. 
+847471, the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021 2028 program of the 
+University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR 10 IDEX 0002), and 
+by SFRI STRAT'US project (ANR 20 SFRI 0012) and EUR QMAT ANR-17-EURE-0024 under the 
+framework of the French Investments for the Future Program. 
+Conflicts of interest 
+There are no conflicts of interest 
+ 
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Coey3, Bernard Doudin1* 1Université de Strasbourg, CNRS, IPCMS UMR 7504, 23 rue du Loess, F-67034 Strasbourg, France 2Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany 3AMBER and School of Physics, Trinity College, Dublin 2, Ireland * Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' email: bernard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='doudin@ipcms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='fr  KEYWORDS spintronics, spin orbit torque switching, nanomagnetism, ion beam irradiation, Hall bars  He+ ion irradiation is used to pattern multiple areas of Pt/Co/W films with different irradiation doses in Hall bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The resulting perpendicular magnetic anisotropy landscape enables selective multilevel current-induced switching, with full deterministic control of the position and order of the individual switching elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Key pattern design parameters are specified, opening a way to scalable multilevel switching devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  Spin orbit torque (SOT) magnetic memory devices have unique advantages in terms of low power consumption, fast operation capabilities and simplified circuitry design, placing them at the forefront of information technology development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='1–3 Their current-induced switching properties rely on the perpendicular magnetic anisotropy (PMA) of their thin magnetic layers, which is readily modified by light ion irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='4 We have previously shown the benefit of combining nanometer-scale irradiation in a He+ microscope with in situ electronic transport property measurements to reveal how the local PMA evolves under irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='5 Reduction of the PMA in specific locations makes selective SOT switching of magnetic bits possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Our purpose here is to show how this technique can be extended to multiple irradiation zones in the same device, taking advantage of fine tuning the spatial distribution of PMA calibration, to achieve multi-level switching in devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Multi-level storage is a promising approach to increase memory density6 while avoiding the addressability problems associated with size reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Increasing the number of available states by a factor N leads to an N-fold improvement in the density of memory without any change to the device geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Multi-level SOT devices can also be exploited for bio-inspired computing architectures, such as neuromorphic computing based on spintronics7,8, and multi-level switching  2 has been achieved in PMA heterostructures in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' These include stabilizing intermediate multidomain states by means of multi-ferromagnetic layer stacks,9–12 wedged magnetic layers,13,14 ferrimagnetic15,16, ferromagnetic/antiferromagnetic structures17, or with interleaved interfaces with different spin Hall angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='18 A different approach defines the multiple levels as a sequence of pinning-depinning states in SOT-driven reversal19,20, which has been shown in both irradiated and non-irradiated single magnetic films21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Here, several unique states are accessible using shaped current pulses22, but the exact magnetic configuration is highly dependent on sample fabrication and reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='23 Our aim is to create well-defined, reproducible, deterministic states, that are addressable by SOT switching, and avoid the limitations of sample reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Thin film stacks of Ta (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='0)/Pt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='0)/Co (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='2)/W (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='5)/Pt (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='5)/Ta (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='5) (thickness in nm) were grown on thermally-oxidized Si substrates by DC magnetron sputtering at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The Ta (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='0) seed layer ensure good adhesion and (111) texture of Pt that promotes perpendicular anisotropy of the Co layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='24 The heterostructure of Co sandwiched between Pt and W was chosen to optimize SOT switching efficiency due to opposite signs of the spin Hall angles of Pt and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The PMA of the cobalt was confirmed by magnetometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Films were patterned into Hall bar structures of width 10 µm and length 180 µm by UV lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  The Hall crosses were irradiated using an Orion Nanofab helium ion microscope system while monitoring the evolution of the anomalous Hall resistance (RAHE) in- situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (Figure 1 a,b)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' For uniformly irradiated Hall crosses, RAHE gradually decreases with irradiation until there is a sharp drop at a critical dose of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='5 ± 2 ions/nm2, where the easy axis of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (a) Schematic of SOT setup of a patterned Hall bar undergoing He+ irradiation with the directions of current and magnetic field indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (b) Evolution of in-situ anomalous Hall resistance with irradiation, the letters P,Q indicate the two-zone irradiation doses shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2a, and A-D the four-zone doses shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The magnetic anisotropy field (orange squares) as a function of irradiation, deduced from Hall measurement on ex-situ samples, is also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (d)  Comparison of SOT induced magnetization switching before and after irradiating with 35 ions/nm2 under a 100 mT bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='2  3 magnetization flips from out-of-plane to in-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The critical dose is reproducible within 5% from sample-to-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The magnetic anisotropy field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 1b) was measured ex-situ by Hall effect as a function of applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Hall measurements were performed under 1mA bias current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' SOT-induced switching experiments were performed with 1 ms current pulses in an interval of 1 s, under 100 mT bias field applied along the current direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 1c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' An average current density values of approximately 8 MA/cm2 corresponds to a 10 mA pulse current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Our experimental procedure does not give indications of long-term sample Joule heating if the current stress remains below typically 35 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' One can imagine Joule heating during the 1 ms current pulse, but we found that the current hysteresis curves remain fully reproducible (within a few percent) if the current sweeps does not exceed 35 mA, providing confidence that the current stress does not impact the Co thin film and its interface down the atomic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The observed trends in the critical dose, fall in anisotropy and reduced critical current are consistent with our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 1c compares the SOT switching hysteresis curve of a non-irradiated cross and one irradiated, illustrating the reduction of switching current after irradiation with 35 ions nm-2Further characterisation and calibration of both the magnetic anisotropy as well as SOT switching currents was carried out for several doses below the critical dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  In this work, we focus on Hall crosses which were individual well-defined zones were irradiated with different doses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (a) Schematic of a two-zone irradiated sample (doses P,Q Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 1b) with and without a gap between irradiated zones, and two boundary zones Q0 to minimize domain wall propagation during SOT switching (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (b) SOT induced magnetization switching of two junctions with and without a gap between two irradiated zones with dose 23 ions/nm2 and 35 ions/nm2 at a bias field of 125 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (c) MOKE images of the SOT induced magnetization switching when sweeping the current from +21 mA to -21 mA, and then (below broken line) back to +21 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Intermediate currents (21 to -5 mA, and -21 to 5 mA) where the magnetization does not switch are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  P 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2 details the simplest case of a two-zone irradiation device, with one zone irradiated with 23 ions nm-2 (dose denoted ‘P’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 1b) and the other with 35 ions nm-2 (‘Q’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' For a direct comparison we irradiated neighbouring Hall crosses, labelled 1 and 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' In cross 1 the irradiated zones are adjacent to each other, whereas in cross 2 they are separated by approximately 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' In both crosses, we detect additional resistance states due to the different magnetic anisotropies of the two zones P & Q, denoted as ↑↑, ↑↓, ↓↑ and ↓↓ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' These states have unique RAHE values and are accessible by cycling the applied current pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' We found that measured Hall resistance values were stable within 1% over many hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The resulting states are non-volatile, holding their resistance values for at least two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The switching currents of the irradiated areas are also reproducible within a few percent when repeating the current sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Magneto-optical Kerr effect (MOKE) microscopy confirms the correspondence between the Hall resistance data and the switching of each irradiated area, allowing us to visualize the resulting magnetic configurations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The boundary of non-irradiated and irradiated zones are known to pin propagating domain walls during magnetic field driven reversal25,26, and are found to play such a role in the SOT induced switching by Kerr microscopy visualisation (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  We found that it was crucial to irradiate magnetic boundary zones on either side of the Hall cross, marked ‘Q0’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2a, using a near-critical dose so that these zones switch at the smallest currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' This inhibits propagation of magnetization reversal from far away in the main bar into the cross, which would interfere with the Hall measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Essentially, this additional irradiation acts to isolate the magnetic switching behaviour of a cross, without severing its electrical contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Unfortunately, we found that reproducibility of the switching current of unirradiated zones surrounded by Q0 irradiated areas was quite poor, but nevertheless sufficient to avoid a nucleation of reversal far from the sample of interest at lower current values than those switching irradiated areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2 also illustrates how independent switching of the two zones P and Q can be improved by leaving a non-irradiated strip between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' A width of 1 µm was chosen to allow adequate resolution in the Kerr microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Under a bias field of 125 mT we find a clear independent SOT switching of P and Q in both the measured Hall resistance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2b) and MOKE imaging (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2c) only when there is an irradiation gap (P2, Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' When the current is swept from 21 to -21mA, the zones irradiated with 35 ions nm-2, Q1,Q2 , selectively switch at -5 mA and the adjacent irradiated zone, P1, reverses its magnetization continuously between -7 mA and -8 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  In contrast, the zone P2, separated by the non-irradiated gap, switches at -11 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' This larger critical current results from domain wall pinning in the unirradiated zone which retains the larger initial PMA, preventing the reversal of one zone to expand to the neighbouring one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Note that for this sample, the unirradiated areas switch at current values similar to Q2, which explains why the antiparallel state does not correspond to zero RAHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Irradiation of multiple zones within a Hall cross illustrate the scalability of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Four zones, A–D, separated by 1 µm gaps and irradiated with different doses, 35, 27, 20 and 13 ions nm-2, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' As with the sample shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2, magnetic isolation bands, Q0, were irradiated either size of the junction with a dose of 35 ions nm-2, protecting the unirradiated zone against switching up to 20 mA current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' The SOT induced switching of the device under a bias field of 125 mT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 3b shows the three intermediate steps corresponding to the independent switching of the four zones upon sweeping applied current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' MOKE microscopy confirms this interpretation of the AHE data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 3c), displaying a sequential switch of each area from lowest to highest PMA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' from highest to lowest ion dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' This demonstrates that our approach can be extended to irradiate N  5 zones with different doses, across different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' We expect the domain wall width to define the scale of the minimum bit size, leading to a separation wall of a few tens of nm for individual cells below the 100 nm size range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' This is well within the resolution limits of the irradiation process but is challenging for magnetic imaging and electrical detection and is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  In summary, magnetic anisotropy engineering by appropriate multi-zone He-ion irradiation makes multi-level switching of magnetic configurations robust and predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Our experimental design was chosen for maximum simplicity, to demonstrate that we can independently switch the magnetic state of individually irradiated areas in a single Hall cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Moreover, by taking advantage of the hysteresis properties of the current-induced switching, many macrospin configurations become accessible (green arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2b), and one can imagine switching any combination of the macrospin presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' This would allow the number of distinct electrical states to be much larger than the number of irradiated zones N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' More sophisticated designs are also possible on a smaller scale, for example where a switching of a given area influences its neighbour via its magnetic stray field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Furthermore, ion irradiation is compatible with other approaches for realizing multistate memories without altering their physical design, such as more sophisticated magnetic stacks, wedge layers or lithographic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='8-16 We believe that the strategy we have outlined proposes a versatile strategy to realize high-density and low-power-consumption SOT-based memory devices and is relevant for future types of computing architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Author Contributions P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' initially designed the experiment, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' performed most experiments, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' helped for the materials fabrication, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' for the ion irradiation, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' for Kerr measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' supervised the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' wrote the manuscript, where all authors contributed to its improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Acknowledgements We thank Fabien Chevrier and the staff of the STnano nanofabrication facility for daily support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' This project has received funding from the European Union’s Horizon 2020 research and innovation Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (a) Schematic of the four zones of a junction irradiated with doses 13 ions nm-2, 20 ions nm-2, 27 ions nm-2 and 35 ions nm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (b) MOKE acquired SOT induced magnetization switching of the junction under a bias field of 125 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' (c) MOKE images of the SOT induced magnetization switching of the junction when sweeping from 21 to – 25 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='  B  D  A  6 programme under the Marie Skłodowska-Curie grant agreement MaMi No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 766007 and QUSTEC No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=" 847471, the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR 10 IDEX 0002), and by SFRI STRAT'US project (ANR 20 SFRI 0012) and EUR QMAT ANR-17-EURE-0024 under the framework of the French Investments for the Future Program." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Conflicts of interest There are no conflicts of interest  References 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Yasin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Couet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Souriau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Swerts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Rao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Van Beek, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Kim, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Kundu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Tsvetanova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Croes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Jossart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Grimaldi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Baumgartner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Crotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Fumémont, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Gambardella, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Kar, in 2018 IEEE Symposium on VLSI Circuits (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 81–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Fassbender, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Ravelosona, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Fowley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Hlawacek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Kurian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Atcheson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Colis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Teichert, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Kundys, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Venkatesan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Doudin, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Aidala, Journal of Applied Physics 115, 17D511 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' 18 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' 19 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' 116, 242401 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' 15, 054013 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' 21 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Su, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Tian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Hong, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' You, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Chong, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' Chen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Cambril, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Bernas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Jamet, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Ferré, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content=' 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Franken, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Hoeijmakers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Lavrijsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Swagten, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Koopmans, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' van Veldhoven, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}
+page_content=' Maas, Journal of Applied Physics 109, 07D504 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE0T4oBgHgl3EQfQADn/content/2301.02188v1.pdf'}

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+ 
+1 
+Photochemical and RadiatiOn Transport model for Extensive USe 
+(PROTEUS) 
+ 
+Yuki Nakamura1,2, Naoki Terada1, Shungo Koyama1, Tatsuya Yoshida1, Hiroki Karyu1, Kaori 
+Terada1, Takeshi Kuroda1, Arihiro Kamada1, Isao Murata1, Shotaro Sakai1, Yuhei Suzuki1, Mirai 
+Kobayashi1, and François Leblanc2 
+ 
+1 Graduate School of Science, Tohoku University, Sendai, Japan 
+2 LATMOS, Sorbonne Université, Paris, France 
+ 
+Corresponding author: Yuki Nakamura ([email protected]) 
+ 
+ 
+Abstract 
+We introduce a new flexible one-dimensional photochemical model named Photochemical and 
+RadiatiOn Transport model for Extensive USe (PROTEUS), which consists of a Python graphical 
+user interface (GUI) program and Fortran 90 modules. PROTEUS is designed for adaptability to 
+many planetary atmospheres, for flexibility to deal with thousands of or more chemical reactions 
+with high efficiency, and for intuitive operation with GUI. Chemical reactions can be easily 
+implemented into the Python GUI program in a simple string format, and users can intuitively 
+select a planet and chemical reactions on GUI. Chemical reactions selected on GUI are 
+automatically analyzed by string parsing functions in the Python GUI program, then applied to 
+the Fortran 90 modules to simulate with the selected chemical reactions on a selected planet. 
+PROTEUS can significantly save the time for those who need to develop a new photochemical 
+model; users just need to write chemical reactions in the Python GUI program and just select them 
+on GUI to run a new photochemical model.  
+ 
+Keywords: Photochemical model, Graphical user interface, PROTEUS 
+ 
+ 
+Introduction 
+Photochemical models are essential for investigating the vertical chemical structure of planetary 
+atmospheres and their evolution throughout the history of the planets. They solve continuity-
+transport equations considering production and loss of each atmospheric species by numerous 
+chemical reactions including photolysis. So far, plenty of photochemical models have been 
+developed for various planetary atmospheres (e.g., Kasting et al., 1979; Nair et al., 1994; Kim 
+
+ 
+2 
+and Fox, 1994; Fox and Sung, 2001; Krasnopolsky, 2009; Krasnopolsky, 2012; Chaffin et al., 
+2017). As the mass and spectral resolutions of measurements for detecting chemical species in 
+planetary atmospheres increase and the theory of chemical kinetic systems become more complex, 
+the need for photochemical models with hundreds or thousands of chemical reactions increase.  
+ 
+There are roughly three approaches to develop a numerical code for solving a lot of chemical 
+reactions (Damian et al., 2002). The first approach is a hard-coding approach, in which the 
+developer analyzes the chemical reactions, derives all the production and loss rate terms for each 
+chemical species, and codes them into a program by hand. This approach is easy to develop when 
+the number of chemical reactions is smaller than a hundred, however, it takes a lot of time to 
+develop a code when the number of reactions becomes larger than hundreds or more. It is also 
+difficult to add new chemical reactions into an already hard-coded program.  
+ 
+The second approach is a totally integrated approach, in which the chemical reactions are listed 
+in a specific file in a certain format and they are parsed by a program and stored in a memory 
+when it is run. This approach is flexible in adding new chemical reactions after the development 
+of the core program, and easy to deal with hundreds of chemical reactions without developer’s 
+manual derivation. This approach was used in the Atmos model in Fortran language for instance, 
+which was originally developed by Kasting et al. (1979), updated by Zahnle et al. (2006) and 
+recently described in Arney et al. (2016). Recently an integrated Martian photochemical model 
+was developed by Chaffin et al. (2017) in Julia language with a more flexible way of describing 
+reactions and rate coefficients.  
+ 
+The third approach is a preprocessing approach, in which the chemical reactions are listed in a 
+specific file like the totally integrated approach, but are parsed by a preprocessor to generate a 
+hard-coded program in a high-level language such as Fortran or C language (Damian et al., 2002). 
+This approach is also flexible in implementing hundreds of chemical reactions and is as efficient 
+as the hard-coding approach. This approach was used in the kinetic preprocessor (KPP) originally 
+developed by Damian et al. (2002), which has been widely used for chemical kinetic models for 
+Earth’s atmosphere. 
+ 
+In this paper, we present a new integrated photochemical model named Photochemical and 
+RadiatiOn Transport model for Extensive USe (PROTEUS), with a totally integrated approach. 
+PROTEUS couples Python and Fortran modules, which is designed for adaptability to many 
+planetary atmospheres, for flexibility to deal with thousands of or more chemical reactions with 
+high efficiency, and for intuitive operation with a graphical user interface (GUI). A Python GUI 
+
+ 
+3 
+program integrates a list of chemical reactions, GUI functions controlling the behavior of GUI 
+operation, and a string parsing functions analyzing chemical reactions that output a Fortran 90 
+module. Fortran 90 modules solve differential equations numerically. Chemical reactions are 
+written in a simple and flexible string format in the Python GUI program, making it easy to add 
+new chemical reactions into the Python GUI program. The feature of PROTEUS that the Python 
+GUI program outputs a Fortran 90 module is similar to the preprocessing approach, leading to a 
+high efficiency, however, the Fortran modules in PROTEUS are not hard-coded but is rather 
+generic.  
+ 
+PROTEUS has been newly developed and independent of other photochemical models or KPPs 
+that have been developed so far. 
+ 
+ 
+Model description 
+Equations 
+PROTEUS is a one-dimensional photochemical model that solves a system of continuity 
+equations for each species as follows: 
+ 
+𝜕𝑛!
+𝜕𝑡 = 𝑃! − 𝐿! − 𝜕Φ!
+𝜕𝑧 ,								(1) 
+ 
+where 𝑛! is the number density of 𝑖th species, 𝑃! is the production rate of 𝑖th species, 𝐿! is 
+the loss rate of 𝑖th species, 𝑧 is the altitude and Φ! is the vertical flux of 𝑖th species. The 
+vertical flux Φ! for both neutral and ionized species can be expressed as follows: 
+ 
+Φ! = −𝑛!𝐷! 1	 1
+𝑛!
+𝜕𝑛!
+𝜕𝑧 + 1
+𝐻!
++ 𝑞!
+𝑞"
+𝑇"/𝑇!
+𝑃"
+𝜕𝑃"
+𝜕𝑧 + 1 + 𝛼!
+𝑇!
+𝜕𝑇!
+𝜕𝑧 8 − 𝑛!𝐾 1 1
+𝑛!
+𝑑𝑛!
+𝑑𝑧 + 1
+𝐻 + 1
+𝑇
+𝑑𝑇
+𝑑𝑧8,							(2) 
+ 
+where 𝐷! is the binary diffusion coefficient between 𝑖th species and the background atmosphere, 
+𝐻!=𝑘#𝑇!/𝑚!𝑔 is the scale height of 𝑖th species, 𝑚! is the mass of 𝑖th species, 𝑘# is the 
+Boltzmann constant, 𝑔 is the gravitational acceleration, 𝑞! is the charge of 𝑖th species, 𝑞" is 
+the elementary charge, 𝑇" and 𝑇! are the temperatures of electrons and 𝑖th species, respectively, 
+𝑃" =𝑛"𝑘#𝑇"  is the electron pressure, 𝑛"  is the electron number density, 𝛼!  is the thermal 
+diffusion coefficient, 𝐾 is the eddy diffusion coefficient, 𝐻 =𝑘#𝑇/𝑚𝑔 is the mean scale height 
+of the background atmosphere, 𝑚 is the mean molecular mass of the atmosphere, and 𝑇 is the 
+neutral temperature. The temperature profiles are assumed to be stationary in time. The third term 
+
+ 
+4 
+in Equation (2) is the ambipolar diffusion term, which is applied only to charged species. The 
+altitude-dependent gravitational acceleration 𝑔 is calculated by using the mass and radius of the 
+planet. The basic equations are stiff equations in which some of the variables such as number 
+densities of short-lived species change more quickly than others. Thus, PROTEUS applied an 
+implicit method solving the differential equations as follows:  
+ 
+𝒙$%& = 𝒙$ + 1 𝐈
+Δ𝑡 − 𝓙8
+'𝟏
+𝑭(𝒙$)					(3) 
+ 
+where 𝒙$ and 𝒙$%& are vector forms of the number density of chemical species at time step 𝑛 
+and 𝑛 + 1, respectively, 𝐈 is the unit matrix, Δ𝑡 is the timestep size, 𝑭(𝒙$) is the right-hand 
+side of equation (1) in the vector form, and 𝓙 is the sparse Jacobian matrix defined as follows: 
+ 
+𝓙 ≡ 𝜕𝑭(𝒙$)
+𝜕𝒙$
+=
+⎝
+⎜
+⎜
+⎜
+⎜
+⎛
+𝜕𝐹&
+𝜕𝑥&
+𝜕𝐹&
+𝜕𝑥)
+𝜕𝐹)
+𝜕𝑥&
+𝜕𝐹)
+𝜕𝑥)
+⋯
+𝜕𝐹&
+𝜕𝑥*
+…
+𝜕𝐹)
+𝜕𝑥*
+⋮
+⋮
+𝜕𝐹*
+𝜕𝑥&
+𝜕𝐹*
+𝜕𝑥)
+⋱
+⋮
+…
+𝜕𝐹*
+𝜕𝑥*⎠
+⎟
+⎟
+⎟
+⎟
+⎞
+					(4) 
+ 
+The time step size can gradually increase from an initial value 10-8 sec to ideally more than 1014 
+sec, allowing us to investigate from a sporadic event response in several minutes to the evolution 
+of planetary atmospheres in a billion-year time scale.  
+ 
+PROTEUS is basically a one-dimensional photochemical model and currently does not solve 
+horizontal transportation, but considers the rotation of a planet as options to obtain a simplified 
+global distribution. PROTEUS has four options for the simulation geometry: (1) one-dimensional 
+simulation at a given latitude at noon, (2) two-dimensional simulation at noon at each latitude 
+from the north pole to the south pole, (3) two-dimensional simulation at a given latitude with 
+rotation, and (4) three-dimensional simulation at all latitudes with rotation. The three-dimensional 
+simulation has already been applied by Nakamura et al. (2022) for the Jovian ionosphere. 
+ 
+Radiative transfer 
+PROTEUS uses the solar EUV irradiance model for aeronomic calculations (EUVAC) (Richards 
+et al., 1994) for the reference irradiance spectrum of solar extreme ultraviolet (EUV) flux to 
+calculate the photoionization rates of atmospheric species. EUVAC model provides the solar EUV 
+
+ 
+5 
+flux in 37 wavelength bins ranging from 5 nm to 105 nm. EUVAC model requires the input of 
+F10.7 value and its 81 days running average value. High resolution solar EUV reference flux 
+model such as the high-resolution version of EUVAC (HEUVAC) (Richards et al., 2006) and the 
+Flare Irradiance Spectral Model-Version 2 (FISM2) (Chamberlin et al., 2020) will be 
+implemented into PROTEUS in the future. For calculating the photodissociation rates of 
+atmospheric species, we used the reference irradiance spectrum of the solar flux in the wavelength 
+range 0.05-2499.5 nm taken from Woods et al. (2009). Adopting the solar flux taken from EUVAC 
+model and Woods et al. (2009), the radiative transfer is solved by considering the absorption of 
+the solar irradiation by atmospheric species. In PROTEUS, users can flexibly change the 
+wavelength bin size for the solar irradiance of Woods et al. (2009) and absorption/dissociation 
+cross sections of chemical species at each wavelength. The solar flux and cross section data can 
+be provided in any wavelength bins, which are automatically interpolated and binned to the 
+wavelength bin given by the user. The automatically binning algorithm is especially useful when 
+users need high resolution wavelength bins in a limited wavelength range. For example, if users 
+need to resolve the Schumann-Runge bands of the oxygen molecule, the user can set a 0.01 nm 
+resolution at 176-192.6 nm and 1 nm at other wavelength range, which could reduce the 
+computational cost in solving radiation transfer and dissociation rate of atmospheric species even 
+fully resolving the structured Schumann-Runge bands of the oxygen molecule. This algorithm is 
+also useful for resolving a slight difference in absorption cross sections between isotopes (Yoshida, 
+et al., 2022 submitted).   
+ 
+The reference solar irradiance spectra are then divided by the square of the distance 𝑟 in the unit 
+AU between the planet and the Sun. The distance 𝑟 between the planet and the Sun at a given 
+solar longitude 𝐿+ is given by  
+ 
+𝑟 = 𝑟,
+1 − 𝑒)
+1 + 𝑒 cosX𝐿+ − 𝐿+,.Y					(5) 
+ 
+where 𝑟, is the mean distance in unit AU between the planet and the Sun, 𝑒 is the eccentricity 
+of the planetary orbit, and 𝐿+,. is the solar longitude at perihelion. The solar zenith angle at 
+latitude 𝜃 and an hour angle 𝜂 is given by 
+ 
+cos 𝜒 = sin 𝜃 sin 𝛿 + cos 𝜃 cos 𝛿 cos 𝜂					(6) 
+ 
+where 𝛿 is solar declination. 𝛿 and 𝜂 are given by 
+ 
+
+ 
+6 
+sin 𝛿 = sin 𝜀 sin 𝐿+					(7) 
+𝜂 = 2𝜋𝑡/
+𝑇0
+					(8) 
+ 
+where 𝜀 is the tilt angle of the rotational axis, 𝑡/	is the time in second measured from the local 
+noon, and 𝑇0 is the rotational period of the planet.  
+ 
+The absorption, ionization, and dissociation cross sections implemented into PROTEUS are 
+described in the following sections and Appendix. The Rayleigh scattering cross section 𝜎1 is 
+given by (Liou, 2002): 
+𝜎1 = 128𝜋2
+3𝜆3
+𝛼0)					(9)
+ 
+where 𝜆 and 𝛼0  are the wavelength in nm and polarizability in nm3 of gaseous species, 
+respectively. The polarizability 𝛼0  of gaseous species are taken from the Computational 
+Chemistry Comparison and Benchmark DataBase (https://cccbdb.nist.gov). 
+ 
+ 
+Structure of PROTEUS 
+PROTEUS consists of a Python GUI program and of Fortran 90 modules. The Python GUI 
+program contains a list of chemical reactions for each planet, string parsing functions that parse 
+the reactions and reaction rate coefficients selected in GUI, and GUI functions that control the 
+behavior of GUI. Python language was adopted because of its flexibility in parsing strings and its 
+capability in operating GUI. Since Python language is not efficient for numerical calculation, 
+Fortran language was adopted to solve differential equations numerically. The structure of 
+PROTEUS is illustrated in Figure 1. Chemical reactions listed in the Python file are first read by 
+GUI functions (arrow 1 in Figure 1). Then, users select reactions on GUI (arrows 2 in Figure 1), 
+which will be analyzed by string parsing functions (arrow 3 in Figure 1) to output a Fortran 90 
+module named “v__in.f90” and data files (in directories “PLJ_list” and “settings”) including 
+information of the selected chemical reactions (arrow 4 in Figure 1). The Fortran 90 module 
+“v__in.f90” is the only module that includes information of chemical reactions, and other Fortran 
+90 modules are independent of chemical reactions to be used in the simulation. Datafiles of initial 
+density profiles and temperature profiles are read by “v__in.f90” (arrow 5 in Figure 1). The 
+selected reactions, settings such as temperature profile and initial density profiles are applied to 
+the Fortran 90 model when the main Fortran 90 routine “e__main.f90” call subroutines in 
+“v__in.f90” (arrow 6 in Figure 1). Users can then run the Fortran model by compiling all the 
+Fortran 90 modules (indicated as “7” in Figure 1).  
+
+ 
+7 
+ 
+ 
+ 
+Figure 1 Schematic illustration of the structure of PROTEUS.  
+ 
+ 
+The structure of the Fortran codes is illustrated in Figure 2. It should be noted that each Fortran 
+90 file contains several modules with distinct functions, but only the Fortran 90 file is indicated 
+for simplicity. The main routine “e__main.f90” consists of three parts, (1) initialization, (2) 
+calculation, and (3) finalization. The description of each parts and each Fortran 90 files are as 
+follows. 
+ 
+(1) All variables are defined in “v__tdec.f90”. Information of the chemical reactions, boundary 
+conditions, and calculation settings are defined in “v__in.f90”. Physical constants and parameters 
+of the planetary orbit are given in “c__prm.f90”. Production rates calculated by other models (e.g., 
+ionization rate calculated by a meteoroid model (Nakamura et al., 2022)) can be input in 
+“p__Mars.f90” and “p__Jupiter.f90”. The solar EUV flux is calculated by the EUVAC model and 
+the absorption and ionization cross sections are defined in “p__EUVAC.f90”. The solar flux of 
+Woods et al. (2009) is defined and absorption and dissociation cross sections are calculated 
+in ”p__UV.f90”.  
+ 
+Python GUI program
+Chemical reaction list 
+String parsing functions
+GUI functions
+Generic Fortran 90 modules 
+(independent of reactions)
+v__in.f90 PLJ_list settings
+Reaction analysis information
+1
+3
+4
+e__main.f90
+subroutines
+Analyze selected reactions
+Display chemical reactions 
+6
+7 Compile F90 modules and run the model
+Input files
+Initial 
+density
+Tempera-
+ture
+5
+Select reactions on GUI
+2
+
+省 
+8 
+(2) 
+The 
+radiative 
+transfer 
+is 
+solved 
+and 
+the 
+optical 
+depth 
+is 
+calculated 
+in 
+“p__photochem_opticaldepth.f90”. Ionization and dissociation rates, reaction rate coefficients 
+and production and loss rates of each species are calculated in “p__photochem_rate.f90”. The 
+vertical diffusion flux is calculated in “p__photochem_transport.f90”. The eddy and binary 
+diffusion coefficients are defined in “p__eddy_diffusion.f90” and “p__molecular_diffusion.f90”, 
+respectively. “p__photochem_scheme.f90” calculates the Jacobian matrix and advances the 
+timestep using the implicit method.  
+ 
+(3) At last, “p__io.f90” outputs the calculated simulation results.  
+ 
+ 
+ 
+Figure 2 Schematic illustration of the structure of Fortran codes.  
+ 
+ 
+Graphical user interface 
+The Python GUI program uses the tkinter package, a standard library of Python to use the Tcl/Tk 
+GUI toolkit (https://docs.python.org/3/library/tkinter.html). The Python GUI allows users to 
+easily and intuitively select a planet, chemical reactions of interest, and run the simulation. An 
+example of GUI is shown in Figure 3, and the operation of GUI is as follows. Once the user runs 
+the Python GUI program, one can select a planet as indicated (“1” in the upper panel of Figure 
+3). After selecting a planet, one can create a new project directory, or select or rename a project 
+directory that still exists as indicated (“2” in the upper panel of Figure 3). Then the chemical 
+e__main.f90
+v_ _in.f90
+Chemical reaction info
+Boundary conditions
+Calculation settings
+c_ _prm.f90
+Physical constants
+Planetary orbit parameters
+p_ _Mars.f90, p_ _Jupiter.f90
+Special reactions for each planet
+e.g.) Input production rates calculated by   
+other models
+p_ _photochem_rate.f90
+Calculating ionization & dissociation rates
+Calculating reaction rate coefficients
+Calculating production & loss rates
+p_ _photochem_transport.f90
+Calculating vertical diffusion flux
+p_ _photochem_scheme.f90
+Advancing time step using an implicit method
+p_ _photochem_opticaldepth.f90
+Solving radiative transfer
+Calculating optical depth 
+p_ _molecular_diffusion.f90
+Binary diffusion coefficient
+(defined for each planet) 
+p_ _eddy_diffusion.f90
+Eddy diffusion coefficient
+(defined for each planet) 
+p_ _io.f90
+Output results
+v_ _tdec.f90
+Declaration of variables
+Initialization
+Calculation
+Finalization
+p_ _EUVAC.f90
+Solar EUV flux (EUVAC model)
+Absorption & ionization cross sections
+p_ _UV.f90
+Solar flux of Woods et al. (2009)
+Absorption & dissociation cross sections 
+e_ _: executable
+v_ _: variables
+c_ _: parameters
+p_ _: packages
+
+ 
+9 
+reaction list for the selected planet appears in the window (the lower panel of Figure 3). Chemical 
+reactions and their rate coefficients written in the Python GUI program are automatically 
+converted into Unicode and displayed, making them easy to read. One can select or clear reactions 
+by clicking on the checkbox at each reaction (“3” in the lower panel of Figure 3). By inserting 
+chemical species, reference or label into a search box, only related reactions appear in the window. 
+One can set upper and lower boundary conditions, initial density profiles, vertical grid size, and 
+other calculation settings such as dimension of the simulation, season, latitude, integration time, 
+maximum time step size (“4” in the lower panel of Figure 3). After all the settings are done, one 
+can press “Output f90 module”, then the following files are generated in the selected project 
+directory: a Fortran 90 module named “v__in.f90”, setting files stored in the directory “settings”, 
+and information about production and loss reactions of each chemical species, rate coefficient 
+labels, and Jacobian matrix analyzed by the string parsing function stored in the directory 
+“PLJ_list”. The list of chemical species, the number of chemical species, indices, mass, and 
+charge of each chemical species are automatically determined and written in “v__in.f90” at this 
+time. Those text files are read by the Fortran 90 module “v__in.f90”. One can also output those 
+files, compile and run the Fortran codes by pressing “Output f90 module & Run model”, which 
+requires the installation of an open source software CMake into user’s computer (“5” in the lower 
+panel of Figure 3). All the settings and selected reactions are saved, and users can use the same 
+settings and selected reactions the next time they run GUI. At the end of the simulation, users can 
+quickly plot the density profiles by pressing “Plot setting” button (“6” in the lower panel of Figure 
+3).  
+ 
+
+ 
+10 
+ 
+Figure 3 Overview of GUI and instruction of the operation. 
+ 
+ 
+Format of chemical reaction list  
+The main feature of PROTEUS is the simple format of chemical reaction list in the Python GUI 
+program and on the GUI. Format of chemical reaction list in the Python GUI program and some 
+examples are illustrated in Figure 4.  
+1. Select a planet
+2. Create / select / rename a project
+1
+2
+3. Select chemical reactions
+4. Set initial density profile data, boundary 
+conditions, vertical grid, and calculation settings
+5. Output F90 modules and run the model
+6. Plot density profiles after the simulation
+3
+5
+4
+6
+
+Select Project
+Mars
+Select or Create Project
+# If you want to create new directory, please enter the name of new project (directory) and press "Create New".
+# If you want to load a directory and save as new project (directory),
+please enter the name of new project (directory) name and press "Save as".
+# Location of the project (directory) is: ./Mars/"project (directory) name"
+Create New
+Project (directory) list:
+Select
+Project_1
+Save as
+Select
+Project_2
+Save as！！
+GUI for Photochemical model
+Choose Planet
+Help
+Exit
+Venus
+Earth
+Mars
+Jupiter..
+GUI for Photochemical model
+Mars I Curent project: ./Mars/Project1
+Version: 1.01
+Search reaction
+Set input species
+Set z Grid
+[Help
+[Exit
+Update list
+Boundary Condition
+Calculation settings
+Plot setting
+Chemical Reactions
+Rates
+Label
+Reference
+0+0D
+CO2 + O+(4S)
+9.60 × 10-11
+R1b in Fox and Sung [20011
+Fehsenfeld et al. [19701
+CO2 + 02*
+5.50 × 10-11 × (300/Ti)0.82 for T = ~ 1500 [K)
+1.50 × 10-11 × (1500/Ti)-0,75 for T = 1500 ~ [K]
+R2 in Fox and Sung [20011
+Anicich [1993a].Ferguson et
+ CO2* + NO
+→
+NO+ + CO2
+1.23 × 10-10
+R3 in Fox and Sung [20011
+Anicich [1993a]
+CO2* + N
+→
+NO + CO+
+3.40 × 10-10
+R4 in Fox and Sung [20011
+Scott et al. [19981
+ CO2* + N(2D)
+N* + CO2
+2.00 × 10-10
+R5 in Fox and Sung [2001]
+estimated, see Fox [1982al
+H +O0
+→
+HCO2* + H
+8.70 × 10-10
+R6 in Fox and Sung [20011
+Scott et al. 119971
+CO2* + H
+HCO+ + O
+4.46 × 10-10
+R7a in Fox and Sung [20011
+Scott et al. 119971
+ CO2* + H
+→
+H+ + CO2
+2.35 × 10-11
+R7b Fox and Sung [20011
+Scott et al. [19971
+CO* + 0
+CO + O*(4S)
+1.40 × 10-10
+R8inFoxand Sung[2001]
+Fehsenfeld and Ferguson [19
+CO* + NO
+→
+CO + NO+
+4.20 × 10-10
+R9 in Fox and Sung [20011
+Anicich [1993al
+ CO+ + 02
+00 +0
+1.50 × 10-10 × (300/Ti)1.1
+R10 in Fox and Sung I20011
+Anicich [1993al., Miller et al.
+ CO+ + CO2
+→
+CO2* + CO
+1.10 × 10-9
+R11 in Fox and Sung /20011
+Anicich [1993al
+H ++00
+HCO+ + H
+7.50 × 10-10
+R12a in Fox and Sung [20011
+Scott et al. [19971
+CO* + H
+H+ + CO
+4.00 × 10-10
+R13 in Fox and Sung /20011
+Scott et al. [19971
+ CO+ + N
+→
+NO+ + C
+8.20 × 10-11
+R14 in Fox and Sung [20011
+Scott et al. [19981
+ O2+ + N
+→
+NO+ + O
+1.00 × 10-10
+R15 in Fox and Sung [20011
+Scott et al. 119981
+ O2+ + N(2D)
+→
+NO+ + O
+1.80 × 10-10
+R16a in Fox and Sung [20011
+Goldan et al. 119661
+ O2+ + N(°D)
+→
+ZO + +N
+8.65 × 10-11
+R16b in Fox and Sung [20011
+reverse of (R31b) O'Keefe et
+ 02* + NO
+→
+ZO + +ON
+4.50 × 10-10
+R17 in Fox and Sung 120011
+Midey and Viggiano [19991
+All Select
+All Clear
+Undo
+[Redo]
+[Reaction Analysis
+Output f90 module
+Outputf90 module
+& Run model 
+11 
+ 
+	
+Figure 4 Format of the chemical reaction list in the Python GUI program. 
+	
+Any reactions and their rate coefficients are described in the following string format in the Python 
+GUI program. Reaction and rate coefficient are separated by a colon “:”, and left- and right-hand 
+side of the reaction are separated by an arrow “->”. Chemical species can be written simply as 
+string. For instance, ionized species “N2+”, “CO2+”, and “H+(H2O)4” are simply described as 
+“N2+”, “CO2+”, and “H+(H2O)4”, respectively, and electron is described as “e-”. Isotope 
+species such as “13CO2” can be written as “^13CO2”. Each chemical species and an addition 
+operator “+” or an arrow “->” should be separated by at least one space. PROTEUS also deals 
+with three-body reactions with the expression “M” describing the total atmospheric number 
+density. Temperature-dependent rate coefficient equation can be simply described as string in 
+infix notation. Addition operator “+”, subtraction operator “-”, multiplication operator “*”, 
+division operator “/”, exponentiation operator “^” or “**”, exponential function “exp()” square 
+root function ”sqrt()”, neutral, ion and electron temperatures “Tn”, “Ti”, and “Te”, respectively, 
+altitude in km “h”, decimal fraction values such as “1.16”, integer values such as “300”, and 
+values in E notation such as “4.9e-11” can be used in the rate coefficient equation. If there is a 
+temperature range T1-T2 [K] in which the rate coefficient is valid, one can describe the 
+temperature range by “for	T	=	T1	~	T2	[K]”.  
+ 
+Reactions and their rate coefficients selected on GUI are parsed by the string parsing functions in 
+the Python GUI program. Index for each chemical species is automatically determined by the 
+string parsing function, and mass and charge of each chemical species are also automatically 
+identified by the string parsing function. String of each species are automatically divided into 
+constituent elements and mass is calculated by the sum of mass of all the elements, and charge is 
+Examples of chemical reaction list in Python code
+# Basic format
+reaction_rate_list.append(" Reactants  -> Products : Rate coefficient  for T = T1 ~ T2 [K] @ Reference # Label ")
+# Photo-ionization and dissociation reactions
+reaction_rate_list.append(" CO2  +  hv ->  CO2+  +  e- : Photoionization ")
+reaction_rate_list.append(" CO2  +  hv ->  CO 
++  O : Photodissociation ")
+# Normal two-body reactions
+reaction_rate_list.append(" O(1D)  +  N2O  ->  N2  
++  O2 : 4.9e-11        
+# R75 in Nair et al. [1994] ")
+reaction_rate_list.append(" NO      +  O3     ->  NO2  +  O2 : 2.0e-12 * exp(-1400/Tn)  # R76 in Nair et al. [1994] ")
+# Reaction with several expression of rate coefficient at different temperature ranges
+reaction_rate_list.append(" N2+  +  O2  ->  N2  +  O2+  : 5.10e-11 * (300/Ti)^1.16        for T = ~ 1000 [K]   
+&& ∖
+1.26e-11 * (1000/Ti)^(-0.67)  for T = 1000 ~ 2000 [K]   && ∖
+2.39e-11     
+for T = 2000 ~ [K]  @ Scott et al. [1999], Dotan et al. [1997] # R23 in Fox and Sung [2001] ")
+# Pressure-dependent three-body reaction
+reaction_rate_list.append(" H  +  O2  +  M  ->  HO2  +  M  : k0   = 8.8e-32 * (300/Tn)^1.3      && ∖
+kinf = 7.5e-11 * (300/Tn)^(-0.2)   # Chaffin et al. [2017] ")
+# Reaction with unusual rate coefficient equation
+reaction_rate_list.append(" N2+  +  N2  +  M  -> N4+  +  M : 6.8e-29 * (300/Tn)^2.23 * (1-0.00824*(300/Tn)^0.89) @ Troe [2005] # R31 in Pavlov [2014] ")
+# Cluster ion reactions
+reaction_rate_list.append(" H+(H2O)4   +  H2O  +  M  ->  H+(H2O)5   + M 
+: 4.6e-28 * (300/Tn)^14    # R41 in Verronen et al. [2016] ")
+reaction_rate_list.append(" H+(H2O)4   +  CO3-(H2O)2 ->  H   +  6H2O  + O  + CO2    : 6.0e-8 * (300/Tn)^0.5     # R9 in Verronen et al. [2016] ")
+hv
+M
+e-
+Tn
+Ti
+Te
+K0
+kinf
+: Photon
+: Total atmospheric number density
+: Electron
+: Neutral temperature
+: Ion temperature
+: Electron temperature
+: Low-pressure-limit rate coefficient
+: High-pressure-limit rate coefficient
+Reaction rates are calculated by using 
+local photon flux and cross sections
+
+ 
+12 
+calculated by counting the number of “+” and “-” in the string of each species. The string parsing 
+function analyzes which reaction produces or lose each chemical species. The rate coefficient 
+expressions written in infix notation are first separated into tokens. Then the order of tokens in 
+infix notation are converted into reverse Polish notation (i.e., postfix notation) and automatically 
+labeled. The Fortran 90 modules calculates the reaction rate coefficient using the labeled tokens 
+arranged in reverse Polish notation. This method allows PROTEUS to process a variety of 
+expression of temperature- and altitude-dependent rate coefficients (as seen in Figure 4) at high 
+computational speed. All the information needed to calculate production rate, loss rate and 
+Jacobian matrix are output as text files, which will be read by Fortran 90 module to apply the 
+information about chemical reactions selected. The number of chemical species and reactions, 
+mass and charge of chemical species, rate coefficient of each reaction and contribution of each 
+reaction to production/loss of each species are automatically applied to Fortran 90 modules by 
+reading those text files. Those features make PROTEUS a flexible photochemical model that can 
+be applied to many planetary atmospheres with different set of chemical reactions.  
+ 
+ 
+Application to planetary atmospheres 
+Mars     
+For the application to the Martian atmosphere, parameters of Mars and its orbit are implemented 
+into PROTEUS; The mean distance between Mars and the sun is 𝑟,=1.524 AU, the eccentricity 
+is 𝑒= 0.0934, the solar longitude at perihelion is 𝐿+,.=250°, tilt angle of the rotational axis is 
+𝜀=25.2°, the rotational period is 𝑇0=88775 sec, the mass of Mars is 6.417×1023 kg, and the mean 
+radius of Mars is 3389.5 km (Patel et al., 2002; Williams, 2021).  
+ 
+The cross sections implemented into PROTEUS for the application to Mars are as follows. 
+Ionization cross sections of CO2, CO, O2, N2, and O are taken from Schunk and Nagy (2009). 
+Absorption cross sections and quantum yields for calculating dissociation rates of atmospheric 
+molecules are listed in Table A.1. In order to validate PROTEUS, we compared with one-
+dimensional Martian photochemical model by Chaffin et al. (2017) (hereafter called as C17 
+model), using the same chemical reactions and their rate coefficients, boundary conditions, 
+temperature and water vapor profiles, and binary and eddy diffusion coefficient profiles. The 
+neutral density profiles simulated by PROTEUS and C17 model are shown in Figure 5. 
+PROTEUS and C17 model are in good agreement except for small differences for O3, OH, HO2 
+and H2O2. Those differences could be due to the difference in the photo-absorption and 
+dissociation cross sections used in the two models.  
+ 
+
+ 
+13 
+ 
+ 
+Figure 5 Vertical profiles of neutral density simulated by PROTEUS (solid) and 
+one-dimensional Martian photochemical model by Chaffin et al. (2017) 
+(dashed). The same boundary conditions, chemical reactions and their rate 
+coefficient, binary and eddy diffusion coefficients, and temperature profile, 
+were used in both simulations for validation. 
+ 
+ 
+Jupiter     
+For the application to the Jovian atmosphere, parameters of Jupiter and its orbit are implemented 
+into PROTEUS; The mean distance between Jupiter and the sun is 𝑟,=5.2 AU, the eccentricity 
+𝑒, the solar longitude at perihelion 𝐿+,., and tilt angle of the rotational axis 𝜀 are set to zero for 
+simplicity, the rotational period is assumed to be the System III period related to the period of 
+radio burst 𝑇0= 35729.71 sec, the mass of Jupiter is 1.898×1027 kg, and the equatorial radius of 
+Jupiter is 71492 km (Williams, 2021; Russell et al., 2001).  
+ 
+Ionization cross sections of hydrogen molecule and atom, helium atom, hydrocarbon molecules 
+(CH4, C2H2, C2H4, and C2H6) and metallic atoms (Fe, Mg, Si, and Na) implemented into 
+PROTEUS for the application to the Jovian ionosphere are found in Appendix of Nakamura et al. 
+(2022) and references therein.  
+ 
+PROTEUS has recently been applied to the Jovian ionosphere by Nakamura et al. (2022). 
+Chemical reactions regarding hydrocarbon ion chemistry used in the simulation are described in 
+ 0
+ 50
+ 100
+ 150
+ 200
+100
+102
+104
+106
+108 1010 1012 1014 1016 1018
+Altitude [km]
+Density [cm-3]
+CO2
+CO
+O2
+O3
+O
+O(1D)
+H2O
+OH
+HO2
+H2O2
+H2
+H
+
+ 
+14 
+Nakamura et al. (2022). Ion density profiles calculated by PROTEUS with 218 reactions are 
+shown in Figure 6. Simulated ion density profiles are in good agreement with Kim and Fox (1994), 
+as discussed in Nakamura et al. (2022). Slight differences seen in the shape of profiles of 
+hydrocarbon ions could result from the difference in the initial density profiles of hydrocarbon 
+molecules, which are not indicated in Kim and Fox (1994).  
+ 
+ 
+ 
+Figure 6 Ion density profiles of the Jovian ionosphere simulated by PROTEUS. 
+Profiles are the same as Figure 3(a) in Nakamura et al. (2022) that used 
+PROTEUS.  
+ 
+Summary     
+We have newly developed a flexible one-dimensional photochemical model named PROTEUS, 
+which consists of a Python GUI program and of Fortran 90 modules. Chemical reactions can be 
+easily implemented into Python code as a simple string format, and users can intuitively select a 
+planet and chemical reactions to be considered in their calculation on GUI. Chemical reactions 
+selected on GUI are automatically analyzed by a string parsing code written in Python, which will 
+be applied to Fortran 90 modules to simulate with selected chemical reactions on a selected planet. 
+This paper presents examples of PROTEUS application to the Martian atmosphere and the Jovian 
+ionosphere, which are in good agreement with previous numerical models. PROTEUS can 
+significantly save time for those who need to develop a new photochemical model; they just need 
+to add chemical reactions in the Python code and just select them on GUI to run a new 
+photochemical model. PROTEUS can be easily extended to other planets and satellites, e.g., 
+Venus, Earth, Titan, and exoplanets in the future. 
+
+ 
+15 
+ 
+ 
+Appendix 
+ 
+Table A.1 List of cross sections and quantum yields implemented into 
+PROTEUS.  
+ 
+Species or reactions 
+Wavelength range 
+References 
+𝜎! 
+ 
+𝜙 
+𝜎" 
+CO2 (absorption) 
+ 
+CO2 + hν → CO + O 
+CO2 + hν → CO + O(1D) 
+0.1254-138.8869 nm 
+138.8913 - 212.7660 nm 
+138.8913 - 212.7660 nm 
+0.1 - 138 nm 
+Huestis and Berkowitz (2011)a 
+Schmidt et al. (2013) 
+(Assumed to be 1.0) 
+Huebner and Mukherjee (2015)b 
+𝜎! 
+13CO2 (absorption) 
+138.8913 - 212.7660nm 
+Schmidt et al. (2013) 
+𝜎! 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+𝜙 
+𝜙 
+O2 (absorption) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+O2 + hν → O + O 
+O2 + hν → O + O(1D) 
+0.99 - 43.5 nm 
+49.043646 - 103.066357 nm 
+103.62 - 107.74 nm 
+108.75 - 114.95 nm 
+115 - 130.02 nm 
+130.04 - 175.24 nm 
+175.4 - 204 nm 
+193 - 245 nm  
+103 - 242 nm 
+103 - 175 nm 
+Huffman (1969)a 
+Holland et al. (1993)a 
+Lee (1955)a 
+Ogawa and Ogawa (1975)a 
+Lu et al. (2010)a 
+Yoshino et al. (2005)a 
+Minschwaner et al. (1992)a 
+Yoshino et al. (1992)a 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+𝜎! 
+ 
+ 
+ 
+ 
+ 
+𝜙 
+𝜙 
+H2O (absorption) 
+ 
+ 
+ 
+ 
+ 
+H2O + hν → H + OH 
+H2O + hν → H2 + O(1D) 
+6.2 - 59.04 nm 
+60.01 - 114.58 nm 
+114.80 - 120.35 nm 
+120.38 - 139.99 nm 
+140.00 - 196.00 nm 
+196.031 - 230.413 nm 
+105 nm -  
+105 - 145 nm 
+Chan et al. (1993)a 
+Gürtler et al. (1977)a 
+Mota et al. (2005)a 
+Yoshino et al. (1996, 1997)a 
+Chung et al. (2001)a 
+Ranjan et al. (2020)a 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+𝜎! 
+ 
+ 
+𝜙 
+𝜙 
+O3 (absorption) 
+ 
+ 
+O3 + hν → O2 + O(1D) 
+O3 + hν → O2 + O 
+0.06 - 210 nm 
+213.330 - 1100 nm 
+ 
+220 - 340 nm 
+220 - 340 nm 
+Huebner and Mukherjee (2015)b 
+Gorshelev et al. (2014) 
+Serdyuchenko et al. (2014) 
+Matsumi et al. (2002)a 
+(Assumed to be 1 − 𝜙(O3→O(1D))) 
+𝜎! 
+HO2 (absorption) 
+190 - 260 nm 
+Burkholder et al. (2015) 
+
+ 
+16 
+𝜙 
+HO2 + hν → OH + O 
+190 - 260 nm 
+Burkholder et al. (2015) 
+𝜎! 
+ 
+𝜙 
+𝜙 
+H2O2 (absorption) 
+ 
+H2O2 + hν → HO2 + H 
+H2O2 + hν → OH + OH 
+121.33 - 189.70 nm 
+190.00 - 255.00 nm 
+121 - 230 nm 
+121 - 340 nm 
+Schürgers and Welge (1968)a 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+𝜎! 
+𝜎" 
+𝜎" 
+OH (absorption) 
+OH + hν → H + O 
+OH + hν → H + O(1D) 
+0.06 - 282.3 nm 
+124.5 - 261.65 nm 
+93 - 511.4 nm 
+Huebner and Mukherjee (2015)b 
+Huebner and Mukherjee (2015)b 
+Huebner and Mukherjee (2015)b 
+𝜎! 
+𝜎" 
+H2 (absorption) 
+H2 + hν → H + H 
+0.1 - 110.86 nm 
+84.48 - 110.86 nm 
+Huebner and Mukherjee (2015)b 
+Huebner and Mukherjee (2015)b 
+𝜎! 
+𝜎" 
+N2 (absorption) 
+N2 + hν → N + N 
+0.1 - 103.8 nm 
+51.96 - 103.8 nm 
+Huebner and Mukherjee (2015)b 
+Huebner and Mukherjee (2015)b 
+𝜎! 
+𝜎" 
+NO (absorption) 
+NO + hν → N + O 
+0.1 - 191 nm 
+0.1 - 191 nm 
+Huebner and Mukherjee (2015)b 
+Huebner and Mukherjee (2015)b 
+𝜎! 
+ 
+𝜎" 
+𝜙 
+ 
+NO2 (absorption) 
+ 
+NO2 + hν → NO + O(1D) 
+NO2 + hν → NO + O 
+0.06 - 238 nm 
+238.08219 - 666.57808 nm 
+108 - 243.88 nm 
+108 - 238 nm 
+239 - 300 nm 
+300 - 422 nm 
+Huebner and Mukherjee (2015)b 
+Vandaele et al. (1998)a 
+Huebner and Mukherjee (2015)b 
+Huebner and Mukherjee (2015)b 
+(Assumed to be 1) 
+Burkholder et al. (2015) 
+𝜎! 
+𝜙 
+𝜙 
+NO3 (absorption) 
+NO3 + hν → NO2 + O 
+NO3 + hν → NO + O2 
+400 - 691 nm 
+400 - 640 nm 
+586 - 640 nm  
+Wayne et al. (1991)a 
+Johnston et al. (1996)a 
+Johnston et al. (1996)a 
+𝜎! 
+ 
+ 
+ 
+ 
+𝜙 
+N2O (absorption) 
+ 
+ 
+ 
+ 
+N2O + hν → N2 + O(1D) 
+16.8 - 59.0 nm 
+60.0 - 99.9 nm 
+108.20 - 122.18 nm 
+122.25 - 172.88 nm 
+173 - 210 nm 
+140 - 230 nm 
+Hitchcock et al. (1980)a 
+Cook et al. (1968)a 
+Zelikoff et al. (1953)a 
+Rabalais et al. (1971)a 
+Selwyn et al. (1977)a 
+Burkholder et al. (2015) 
+𝜎! 
+ 
+ 
+𝜙 
+𝜙 
+N2O5 (absorption) 
+ 
+ 
+N2O5 + hν → NO3 + NO2 
+N2O5 + hν → NO3 + NO + O 
+152 - 198 nm 
+200 - 260 nm 
+260 - 410 nm 
+248 - 410 nm 
+152 - 289 nm 
+Osborne et al. (2000)a 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+𝜎! 
+𝜙 
+HNO2 (absorption) 
+HNO2 + hν → NO + OH 
+184 - 396 nm 
+All 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+
+ 
+17 
+𝜎! 
+𝜙 
+𝜙 
+𝜙 
+HNO3 (absorption) 
+HNO3 + hν → HNO2 + O 
+HNO3 + hν → HNO2 + O(1D) 
+HNO3 + hν → OH + NO2 
+192 - 350 nm 
+193 - 260 nm 
+193 - 222 nm 
+193 - 350 nm 
+Burkholder et al. (2015) 
+Estimatedc 
+Estimatedc 
+Estimatedc 
+𝜎! 
+ 
+𝜙 
+𝜙 
+HO2NO2 (absorption) 
+ 
+HO2NO2 + hν → HO2 + NO2 
+HO2NO2 + hν → OH + NO3 
+190 - 280 nm 
+280 - 350 nm 
+190 - 350 nm 
+190 - 350 nm 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+𝜎! 
+𝜙 
+𝜙 
+H2CO (absorption) 
+H2CO + hν → H2 + CO 
+H2CO + hν → H + HCO 
+224.56 - 376 nm 
+250 - 360 nm 
+250 - 360 nm 
+Meller and Moortgat (2000)a 
+Burkholder et al. (2015) 
+Burkholder et al. (2015) 
+ 
+𝜎4: Absorption cross section, 𝜎5: dissociation cross section, 𝜙: quantum yield, 
+a: data files are taken from The MPI-Mainz UV/VIS Spectral Atlas (Keller-
+Rudek et al., 2013), b: data files are taken from PHIDRATES (Huebner and 
+Mukherjee, 2015), c: quantum yields for each photolysis reaction of HNO3 were 
+estimated by quantum yield of each product (OH, O, and O(1D)) obtained by 
+Johnston et al. (1974), Turnipseed et al. (1992), and Margitan and Watson 
+(1982). 
+ 
+ 
+ 
+ 
+Acknowledgements 
+ 
+ 
+Reference 
+Arney, G., Domagal-Goldman, S. D., Meadows, V. S., Wolf, E. T., Schwieterman, E., Charnay, 
+B., Claire, M., Hébrard, E., and Trainer, M. G. (2016). The Pale Orange Dot: The Spectrum 
+and 
+Habitability 
+of 
+Hazy 
+Archean 
+Earth, 
+Astrobiology, 
+16:11, 
+873-899 
+doi:10.1089/ast.2015.1422. 
+ 
+Burkholder, J. B., Sander, S. P., Abbatt, J., Barker, J. R., Huie, R. E., Kolb, C. E., Kurylo, M. J., 
+Orkin, V. L., Wilmouth, D. M., and Wine, P. H. (2015). Chemical Kinetics and 
+Photochemical Data for Use in Atmospheric Studies, Evaluation No. 18, JPL Publication 
+15-10, Jet Propulsion Laboratory, Pasadena, http://jpldataeval.jpl.nasa.gov. 
+
+ 
+18 
+ 
+Chaffin, M. S., Deighan, J., Schneider, N. M., and Stewart, A. I. F. (2017). Elevated atmospheric 
+escape of atomic hydrogen from Mars induced by high-altitude water, Nature Geoscience, 
+10, 174–178, doi:10.1038/ngeo2887. 
+ 
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+Simpler and faster algorithms for detours in planar digraphs ∗
+Meike Hatzel†
+Konrad Majewski‡
+Micha�l Pilipczuk§
+Marek Soko�lowski¶
+Abstract
+In the Directed Detour problem one is given a digraph G and a pair of vertices
+s and t, and the task is to decide whether there is a directed simple path from s to t
+in G whose length is larger than distG(s, t). The more general parameterized variant,
+Directed Long Detour, asks for a simple s-to-t path of length at least distG(s, t) + k,
+for a given parameter k. Surprisingly, it is still unknown whether Directed Detour
+is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and
+Wang [Networks, ’15] proposed an O(n3)-time algorithm for Directed Detour, while
+Fomin et al. [STACS 2022] gave a 2O(k)·nO(1)-time fpt algorithm for Directed Long De-
+tour. The algorithm of Wu and Wang relies on a nontrivial analysis of how short detours
+may look like in a plane embedding, while the algorithm of Fomin et al. is based on a reduc-
+tion to the 3-Disjoint Paths problem on planar digraphs. This latter problem is solvable
+in polynomial time using the algebraic machinery of Schrijver [SIAM J. Comp., ’94], but
+the degree of the obtained polynomial factor is huge.
+In this paper we propose two simple algorithms: we show how to solve, in planar
+digraphs, Directed Detour in time O(n2) and Directed Long Detour in time
+2O(k) · n4 log n. In both cases, the idea is to reduce to the 2-Disjoint Paths problem
+in a planar digraph, and to observe that the obtained instances of this problem have a
+certain topological structure that makes them amenable to a direct greedy strategy.
+∗This work is a part of project BOBR (KM, MP, MS) that has received funding from the European Research
+Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement
+No. 948057). M. Hatzel was supported by the Federal Ministry of Education and Research (BMBF) and by a
+fellowship within the IFI programme of the German Academic Exchange Service (DAAD).
+†National Institute of Informatics, Tokyo, Japan ([email protected])
+‡Institute of Informatics, University of Warsaw, Poland ([email protected])
+§Institute of Informatics, University of Warsaw, Poland ([email protected])
+¶Institute of Informatics, University of Warsaw, Poland ([email protected])
+arXiv:2301.02421v1  [cs.DM]  6 Jan 2023
+
+erc
+European Research Council
+Established by the European CommissionSPONSOREDBYTHE
+Federal Ministry
+of Education
+and Research1
+Introduction
+The complexity status of Directed Detour is arguably one of the most tantalizing open
+questions within the area of graph algorithms. The problem asks to decide, for a given digraph
+G and two terminals s and t, whether there is a simple path in G from s to t that is not
+the shortest — has length strictly larger than distG(s, t). It is still unknown whether this
+problem is polynomial-time solvable on general digraphs. See the work of Fomin et al. [3] for
+a discussion of relevant literature.
+Given this state of the affairs, it is interesting to study Directed Detour on restricted
+classes of digraphs, in hope for finding more positive results or useful insight. In this vein,
+a particularly well-motivated idea is to consider the class of planar digraphs. The reason for
+this is that in planar digraphs, the k-Disjoint Paths problem — decide the existence of
+disjoint directed paths linking given k pairs of terminals — is polynomial-time solvable for
+every fixed k [8], and even fixed-parameter tractable when parameterized by k [2]. This is
+not the case in general digraphs, where the problem is NP-hard already for k = 2 [5]. This
+can be used for Directed Detour. Namely, Fomin et al. [3] showed that the more general
+parameterized version of the problem — Directed Long Detour, where we look for a
+simple s-to-t path of length at least distG(s, t) + k, for a given parameter k — can be reduced
+(with a 2O(k) · nO(1) multiplicative overhead in the complexity) to 3-Disjoint Paths, which
+can be solved in polynomial time in planar digraph using the algorithm of Schrijver [8]. This
+of course applies also to the basic Directed Detour problem by setting k = 1, but even
+earlier Wu and Wang [9] gave a direct O(n3)-time algorithm for this case.
+While the reduction of Fomin et al. [3] is actually quite simple, the algorithm of Schri-
+jver [8] for 3-Disjoint Paths in planar digraphs is not, as it relies on an involved algebraic
+framework. In particular, the degree of the polynomial bounding the running time is at least
+a two-digit number. On the other hand, the cubic algorithm of Wu and Wang [9] is also quite
+complicated and relies on an analysis of different planar configurations that may occur.
+In this work we propose two simple algorithms: one for Directed Detour and one for
+Directed Long Detour, both in planar digraphs. These are summarized below.
+Theorem 1.1. Directed Detour in planar digraphs can be solved in time O(n2).
+Theorem 1.2. Directed Long Detour in planar digraphs can be solved in time 2O(k) ·n4
+by a Monte Carlo algorithm, or deterministically in time 2O(k) · n4 log n.
+The main idea in the proof of Theorem 1.1 is to perform a reduction to the 2-Disjoint
+Paths problem, roughly similarly as in the work of Fomin et al. [3]. However, we observe that
+if this reduction is performed carefully, then one essentially obtains an instance of 2-Disjoint
+Paths where three out of four terminals lie on one face, say the outer face. Such instances
+can be solved very easily: one of the paths — the one with both terminals on the outer face
+— can be chosen greedily so that it leaves the maximum possible space for the other path.
+Then the other path can be found by a simple reachability check.
+For Theorem 1.2, as in Fomin et al. [3], we use the result of Bez´akov´a et al. [1] that the
+Exact Directed Long Detour problem — finding a shortest s-to-t path of length exactly
+distG(s, t) + k — can be solved in time 2O(k) · n2, even on general digraphs. This allows us
+to assume, when solving Directed Long Detour, that there are no s-to-t paths of length
+between distG(s, t)+k and distG(s, t)+3k. Having this assumption, the proof of Theorem 1.2
+proceeds by expanding the basic idea behind Theorem 1.1 with color coding.
+1
+
+We believe that compared to [3,9], our algorithms provide a simpler explanation for the
+tractability of Directed (Long) Detour in planar digraphs. While we do not expect that
+the gained insight will be directly applicable to the case of general digraphs, we hope that
+it might be a better starting point for generalizations to less restrictive topological graph
+classes, for instance to digraphs of bounded genus or digraphs whose underlying undirected
+graphs exclude a fixed minor.
+2
+Preliminaries
+Graphs.
+In this paper we consider planar directed graphs G = (V (G), E(G)).
+For the
+purposes of our problem, we can assume that the input graphs are simple, that is, they do
+not have self-loops or multiple arcs connecting two vertices in the same direction. Moreover,
+we assume that G is weakly connected, that is, the underlying undirected graph is connected.
+For convenience, we set n = |V (G)| and m = |E(G)|. Since G is planar and simple, we have
+that m ∈ O(n).
+Given an arc e = uv ∈ E(G), we say that u is the tail of e and v is the head of e. Here,
+we consider e incident to both u and v. A sequence v1v2 . . . vk of vertices is called a walk in
+G if for each i ∈ {1, . . . , k − 1}, vivi+1 is an arc in G. The vertex v1 is called the origin of
+the walk, while vk is its destination. If the vertices of a walk are pairwise different, the walk
+is a path. Given two walks W1, W2, if the destination of W1 coincides with the origin of W2,
+then we define W1 ◦ W2 as the concatenation of both walks. The length of a path P, denoted
+length(P), is the number of arcs it contains. Given a path P and two of its vertices x and
+y, we denote by P[x → y] the subpath of P which starts at x, goes along the arcs of P, and
+ends in y.
+Given a pair of vertices u, v ∈ V (G), the distance from u to v in G is the length of the
+shortest u-to-v-path in G and is denoted by distG(u, v). If there is no directed path from u
+to v in G, we put distG(u, v) = +∞. If the graph G is known from the context, we may omit
+G from the notation and simply write dist(u, v).
+Plane embeddings.
+Given a planar graph G, its embedding into the plane is a mapping
+of the vertices of G to pairwise distinct points in the plane and the arcs of G to plane curves,
+so that:
+• each arc uv ∈ E(G) is mapped to a plane curve whose endpoints coincide with the
+images of u and v, and such that no other vertex of G is mapped to a point on the
+curve; and
+• the images of the edges of G are pairwise internally disjoint.
+A plane graph is a planar graph together with a fixed embedding of the graph into the plane.
+Such an embedding splits the plane in a number of regions, called faces. One of the faces
+is unbounded and called the outer face. Given a face F, its boundary ∂F can be described
+as a cyclic sequence of arcs bounding F (assuming G is connected). If ∂F is isomorphic to
+a simple polygon (ignoring the orientations of the arcs), we say that F is simple.
+For algorithmic purposes, we represent planar embeddings combinatorially: each vertex v
+of the graph stores the anti-clockwise ordering of all arcs of G incident to v. Given a planar
+graph G, its combinatorial embedding can be computed in time linear with respect to the
+2
+
+F
+v
+w
+P1
+P2
+P3
+(a)
+F
+v
+w
+P
+(b)
+Figure 1: (a) Three (v, w)-grounded paths. We have P1 ≺lex P2 ≺lex P3, P1 ≺top P2, P1 ≺top
+P3, but P2 ̸≺top P3. (b) A (v, w)-grounded path P and its left area ΓL(P) (filled in blue).
+size of the graph [6]. Note that the combinatorial embedding uniquely determines the faces
+of the planar graph, however, it does not designate the outer face of the embedding. Hence,
+in our algorithms, we may elect any face to be the outer face.
+Comparing paths in plane graphs.
+Let G be a plane graph and F be its outer face.
+Assume that F is simple. We say that a path P in G is v-grounded (with respect to F) if its
+origin v is a vertex of F. Moreover, we say that P is (v, w)-grounded (with respect to F) if
+both its origin v and its destination w are vertices of F. We now present two ways to compare
+(doubly) grounded paths in G. We note that the following definitions are standard.
+Definition 2.1. Assume that P1 and P2 are two different v-grounded paths with respect to a
+face F in a plane graph G. We say that P1 is lexicographically left of P2 (denoted P1 ≺lex P2)
+if one of the following conditions holds:
+• P1 is a prefix of P2; or
+• consider a plane graph Gv created by taking G, together with its planar embedding, and
+adding a fresh vertex v′ and a fresh arc v′v, mapped to the plane so that the image of
+v′ is placed outside of F (i.e., so that v′ is not enclosed by the boundary of F). Let P ′
+1
+and P ′
+2 be paths in Gv, constructed by prepending v′ to P1 and P2, respectively. Let q be
+the last vertex of the longest common prefix of P ′
+1 and P ′
+2, p be the vertex preceding q on
+the common prefix, and r1 ̸= r2 be the vertices succeeding q on P ′
+1 and P ′
+2, respectively.
+Then, the arcs pq, qr1 and qr2 are embedded clockwise in this order around q.
+Intuitively, P1 ≺lex P2 if at the end of the common prefix of P1 and P2, the path P1
+terminates or branches off left of P2 (Figure 1(a)). Observe that ≺lex induces a linear order
+on all paths grounded at a vertex v ∈ V (F). Thus, given any vertex w ∈ V (G) reachable
+3
+
+from v in G, we define Lvw, the lexicographically leftmost path from v to w, as the unique
+minimal path from v to w with respect to ≺lex. Given a combinatorial embedding of G, we
+can compute the lexicographically leftmost path from v to w in time O(n): it suffices to run
+a depth-first search of G in which each vertex considers all its neighbors in the left-to-right
+order.
+We follow with a more restrictive way of comparing paths in G. Given a (v, w)-grounded
+path P, we define the area left of P, denoted ΓL(P), as the region of the plane whose anti-
+clockwise boundary is the closed walk defined as the concatenation of P and the anti-clockwise
+segment of the boundary of F from w to v (Figure 1(b)). Note that the interior of ΓL(P)
+may be disconnected if P internally intersects the boundary of F.
+Definition 2.2. Given two (v, w)-grounded paths P1 and P2, we say that P1 is topologically
+left of P2 (denoted P1 ≺top P2) if ΓL(P1) ⊊ ΓL(P2).
+Note that if P1 ≺top P2, then P1 ≺lex P2. Thus, ≺top is a partial order on the set of
+all (v, w)-grounded paths, and ≺lex restricted to those paths is a linear extension of ≺top.
+It is straightforward to observe that the lexicographically leftmost path from v to w — the
+minimum (v, w)-grounded path in ≺lex — is also the unique minimum (v, w)-grounded path
+in ≺top. So if both v and w lie on F, we call Lvw simply the leftmost path from v to w.
+Analogously, we define the (lexicographically) rightmost path from v to w, denoted Rvw,
+and the area right of a (v, w)-grounded path P, denoted ΓR(P).
+Note that each (v, w)-
+grounded path P splits the graph into two parts: the area ΓL(P) left of P and the area ΓR(P)
+right of P, with both areas intersecting only at P.
+Simplifying the outer face.
+Both the lexicographical and the topological comparisons
+of grounded paths require the outer face F of a plane graph G to be simple. However, this
+precondition might not be satisfied in the setting of our problem. To alleviate this issue, we
+show how to simplify the outer face by adding two directed arcs to the plane graph.
+Let u and v be two vertices of F. The (u, v)-simplification of the outer face in G, denoted
+Simplify(G, u, v), is the digraph obtained from G by adding two arcs f1, f2, each from u to v.
+Both arcs are embedded in the outer face F of G, so that the outer face of Simplify(G, u, v) is
+bounded by f1 and f2 (see Figure 2). Thus, the outer face of Simplify(G, u, v) is simple and
+contains two vertices, u and v. We remark that Simplify(G, u, v) is not necessarily a simple
+digraph; however, this does not pose a problem since after the simplification, the number of
+edges of the graph is still linearly bounded in the number of vertices of the graph.
+Cutting arcs of plane graphs.
+We finally describe an operation on plane graphs that
+introduces new vertices to the outer face F of the embedding. Assume that F is simple. Take
+two vertices u ∈ V (F), v /∈ V (F), both incident to an arc e ∈ E(G). An operation of cutting
+the graph along e produces a new graph Ge which is the result of the following process (see
+Figure 3):
+1. Enumerate all arcs incident to u in anti-clockwise order: e1, e2, . . . , ek, where e1, ek ∈
+E(F).
+2. Let ℓ ∈ {2, 3, . . . , k − 1} be such that eℓ = e.
+3. Remove u from the graph, along with all arcs incident to u.
+4
+
+u
+v
+u
+v
+f1
+f2
+Figure 2: An example plane digraph (left) and its (u, v)-simplification (right).
+v
+u
+e
+P
+(a)
+v
+u2u1
+(b)
+(c)
+Figure 3: (a) A directed plane graph G. (b) The result of cutting G along e. (c) The result
+of cutting G along P.
+4. Introduce two new vertices u1, u2 to the graph. For each i ∈ {1, 2, . . . , ℓ}, add to the
+graph a new arc e1
+i which is obtained from the arc ei by replacing the endpoint u with u1.
+Similarly, for each i ∈ {ℓ, ℓ + 1, . . . , k}, add to the graph a new arc e2
+i which is obtained
+from the arc ei by replacing the endpoint u with u2.
+Intuitively, Ge is produced by drawing the graph G on a piece of paper and cutting the
+piece of paper along e.
+Note that after this operation, the resulting graph is still planar
+(Figure 3).
+Moreover, the outer face Fe of Ge is simple, and V (F) \ V (Fe) = {u} and
+V (Fe) \ V (F) = {u1, v, u2}. That is, v now lies on the outer face of the new graph.
+We can generalize this procedure to paths instead of just edges.
+Assume that P =
+v1v2 . . . vk is a v1-grounded path which is disjoint with V (F) except for v1. Then, cutting the
+graph along P entails cutting the graph along the arcs v1v2, v2v3, . . . , vk−1vk, in this order.
+This way we obtain a plane graph where the destination of the path lies on the outer face.
+See Figure 3 for an illustration.
+It is immediate that given the combinatorial embedding of a plane graph G, one can
+compute a plane embedding of the graph obtained by cutting G along a path P in time linear
+with respect to the size of the graph. Thus we again obtain a plane graph.
+5
+
+3
+Directed Detour
+In this section we prove Theorem 1.1.
+Let G be a plane digraph, and let s, t ∈ V (G) be a pair of vertices. Recall that we assume
+G to be weakly connected and we fix the plane embedding of G. Therefore, from now on
+for simplicity we identify features in G (vertices, edges, paths, etc.) with their images under
+the embedding. Our goal is to decide whether there is an s-to-t path in G of length at least
+distG(s, t) + 1. We are going to reduce this question to solving a set of instances of the 2-
+Disjoint Paths problem. The reduction is similar to the algorithm of Fomin et al. [3] for
+Directed Long Detour.
+First, we run breadth-first search (BFS) starting from s in G. Let Li denote the i-th layer
+of this BFS, that is, Li = {v ∈ V (G) | distG(s, v) = i} for i ∈ {0, 1, . . . , n}.
+Suppose now that the instance (G, s, t) is a yes-instance, and let P be an s-to-t path
+witnessing this fact.
+Since P is a non-shortest path from s to t, there is an index p ∈
+{0, 1, . . . , n} such that the layer Lp contains at least two vertices of P. Let us choose p to
+be the smallest such an index. Let G⩾p be the plane digraph obtained from G by removing
+all the vertices in L0 ∪ L1 ∪ . . . ∪ Lp−1. We may assume that G⩾p is weakly connected, for
+otherwise we discard all of its weakly connected components that do not contain the vertex t.
+Claim 1. All vertices of Lp lie on one face of G⩾p.
+Proof. Consider any vertex v of Lp and any shortest path Q from s to v in G. Observe that
+all vertices of Q except for v lie in layers L0, L1, . . . , Lp−1, hence they are removed when
+constructing G⩾p from G. We conclude that v lies on the boundary of the (unique) face of
+G⩾p that contains s.
+Denote by x and y, respectively, the first and the second vertex on the path P that lie in
+the layer Lp. In our algorithm we iterate over all possible choices for the vertex y. Note that
+the choice of y determines the value of p, because y ∈ Lp. Observe that if we guess the vertex
+y correctly, then in order to find a non-shortest path from s to t it is enough to find a vertex
+x ∈ Lp (x ̸= y) and three paths Pstart, Pmiddle and Pend such that
+• Pstart is a shortest path from s to x in G; and
+• Pmiddle, Pend are two internally vertex-disjoint paths in G⩾p going respectively from x
+to y and from y to t.
+Note that y might be equal to t and then Pend is a trivial path only consisting of the single
+vertex t.
+On one hand, vertices x and y divide P into subpaths Pstart, Pmiddle, and Pend satisfying the
+properties stated above. On the other hand, if we find a vertex x and paths Pstart, Pmiddle, Pend
+satisfying the above, then their concatenation forms a valid solution. This is because the
+path Pstart goes consecutively through layers L0, L1, . . . , Lp due to being a shortest path,
+and thus it is internally vertex-disjoint from Pmiddle and Pend. Also, the concatenation of
+Pstart, Pmiddle, Pend is a non-shortest path from s to t due to containing at least two different
+vertices from layer Lp.
+It would be natural to also iterate through all possible choices for x, but for the sake of
+optimizing the running time we do the following instead. Construct a digraph H from the
+digraph G⩾p by adding to it a vertex xsuper together with arcs xsuperu for all u ∈ Lp \ {y}.
+6
+
+Claim 2. H is a planar digraph and its implicit embedding can be obtained from the implicit
+embedding of G⩾p in linear time.
+Proof. It suffices to extend the implicit embedding of G⩾p by embedding xsuper anywhere in
+the unique face whose boundary contains all vertices of Lp (which exists by Claim 1) and
+draw arcs connecting xsuper with vertices of Lp \ {y} within this face. It is straightforward to
+see that this can be done in the setting of implicit embeddings in linear time.
+Observe that pairs (P1, P2) of internally vertex-disjoint paths going respectively from xsuper
+to y in H and from y to t in H correspond one-to-one to pairs (Pmiddle, Pend) of internally
+vertex-disjoint paths going respectively from some vertex x ∈ Lp (x ̸= y) to y in G⩾p and
+from y to t in G⩾p. We may additionally assume that the face of H containing both xsuper
+and y is the outer face of H. This way we reduced our original problem to a set of at most n
+instances (one instance per each choice of the vertex y) of the following problem:
+Problem A
+Input: A plane digraph H and three vertices x, y, t ∈ V (H), where x and y lie on the
+outer face of H.
+Question: Are there two internally vertex-disjoint paths P1 and P2 such that P1 is
+an x-to-y path and P2 is a y-to-t path in H?
+So to conclude Theorem 1.1 it is enough to show the following lemma.
+Lemma 3.1. Problem A can be solved in time O(n).
+Proof. Let H⋆ be a (y, x)-simplification of the outer face of H. Naturally, the outer face of H⋆
+is simple. Moreover, (H⋆, x, y, t) is a yes-instance if and only if (H, x, y, t) is a yes-instance:
+H⋆ is exactly the graph H with two additional yx arcs, which plainly cannot be a part of the
+sought solution.
+Suppose now that (H⋆, x, y, t) is a yes-instance, and let (P1, P2) be a pair of paths in H⋆
+that form a solution to Problem A on H⋆. P1 connects x to y which lie on the outer face
+FO of H⋆ (which is a simple cycle), thus it is (x, y)-grounded with respect to FO. Therefore,
+it splits the whole digraph H into two parts, the area ΓL(P1) left of P1 and the area ΓR(P1)
+right of P1, with both areas intersecting only at P1. Since P2 is internally disjoint with P1, it
+must be entirely contained within ΓL(P1) or ΓR(P1); without loss of generality assume that
+P2 is contained in the latter. Then, let Lx,y be the leftmost path from x to y in H⋆.
+Claim 3. (Lx,y, P2) is also a valid solution to Problem A.
+Proof. By the definition of the leftmost path we know that ΓL(Lx,y) ⊆ ΓL(P1), that is, Lx,y
+cannot use any vertex lying on the right side of P1. Since all vertices of P2 (apart from y) lie
+on the right side of P1 we conclude the claim.
+An analogous argument shows that if P2 is contained in ΓL(P1), then (Rx,y, P2) is a valid
+solution to Problem A, where Rx,y is the rightmost path from x to y in H⋆.
+By the
+observation above, we know that in order to verify whether (H⋆, x, y, t) is a yes-instance it is
+enough to find Lx,y and Rx,y as candidates for P1, and to check for each candidate whether
+the vertex t is reachable from y in H⋆ − (V (P1) \ {y}). All these checks can be done in time
+O(n) by running a proper depth-first search algorithm.
+7
+
+4
+Directed Long Detour
+In this section we prove Theorem 1.2.
+We are given a plane digraph G, two vertices s, t ∈ V (G) and an integer k ∈ N. As in
+Section 3, we assume that G is weakly connected, and we fix the plane embedding of G. We
+need to decide whether there is an s-to-t path in G of length at least distG(s, t) + k. We are
+going to reduce this question to solving a set of instances of the 2-Disjoint Paths problem
+(with some additional requirements).
+We begin with running the algorithm of Bez´akov´a et al. [1] to check whether there is
+an s-to-t path of length distG(s, t) + l for some l ∈ {k, k + 1, . . . , 3k − 1}. From now on, we
+assume there is no such s-to-t path.
+As in Section 3, we run a BFS starting from s in G and set Li = {v ∈ V (G) | distG(s, v) =
+i} for i = 0, 1, . . . , n. For i = 1, 2, . . . , n, let G⩾i be the plane digraph obtained from G by
+removing all the vertices in L0 ∪ . . . ∪ Li−1. Again, without loss of generality, we assume that
+G⩾i is weakly connected.
+Now, we iterate through all values for p = 1, 2, . . . , n, and through all choices for x, y ∈ Lp,
+where x ̸= y (there are at most n2 choices for (p, x, y)). Recall from Section 3 (Section 1)
+that both x and y lie on the outer face of G⩾p. Next, let Gx,y
+⩾p be the (y, x)-simplification of
+the outer face of G⩾p. We are now going to search for three paths Pstart, Pmiddle and Pend
+satisfying the conditions:
+(a) Pstart is a shortest s-to-x path in G;
+(b) Pmiddle is an (x, y)-grounded path in Gx,y
+⩾p of length at least 2k;
+(c) Pend is a y-to-t path in Gx,y
+⩾p;
+(d) paths Pmiddle and Pend are internally vertex-disjoint; and
+(e) if Pend ⊆ ΓR(Pmiddle), then there is no (x, y)-grounded path P ′
+middle in Gx,y
+⩾p of length at
+least k such that P ′
+middle ≺top Pmiddle; and if Pend ⊆ ΓL(Pmiddle), then there is no such
+path with Pmiddle ≺top P ′
+middle.
+We remark that the two yx arcs added in the process of (y, x)-simplification of G⩾p cannot
+be a part of any of the paths Pstart, Pmiddle, Pend and hence their existence can be safely ignored
+in the following series of claims.
+First, we show that the procedure described above is actually equivalent to solving the
+Directed Long Detour problem on the instance (G, s, t, k).
+Claim 4. Assume that the paths Pstart, Pmiddle and Pend satisfy the properties (a) – (d) above.
+Then the concatenation
+P = Pstart ◦ Pmiddle ◦ Pend
+forms a valid solution for the Directed Long Detour problem.
+Proof. First, P is a simple s-to-t path. Indeed, Pmiddle and Pend are internally vertex-disjoint
+by property (d). Moreover, since the path Pstart is a shortest s-to-x path in G, it does not use
+any vertex of Lp ∪. . .∪Ln = V (G⩾p) apart from x, and thus Pstart is internally vertex-disjoint
+from Pmiddle and Pend as well.
+8
+
+By property (b) the length of P is at least
+distG(s, x) + 2k + distG(y, t) = distG(s, y) + 2k + distG(y, t) ⩾ distG(s, t) + 2k,
+which finishes the proof.
+Claim 5. If (G, s, t, k) is a yes-instance, then there exist an integer p ∈ N and vertices
+x, y ∈ Lp for which there are paths Pstart, Pmiddle and Pend satisfying the properties (a) – (e).
+Proof. Assume that (G, s, t, k) is a yes-instance. Let P be an s-to-t path witnessing this fact.
+If there are many such paths, we choose P to be a shortest one. As we established that there
+is no s-to-t path of length distG(s, t) + l for any l ∈ {k, k + 1, . . . , 3k − 1}, we may assume
+that the path P is of length at least distG(s, t) + 3k. Since P is a non-shortest s-to-t path,
+we may define p ∈ {1, 2, . . . , n} to be the smallest index such that Lp contains at least two
+vertices of the path P. Let x and y be, respectively, the first and the second vertex on the
+path P that lie in the layer Lp.
+By definition of p, the subpath P[s → x] is a shortest s-to-x path, and thus we may set
+Pstart := P[s → x]. We also set Pend := P[y → t]. We will choose Pmiddle later in the course
+of the proof.
+Let us observe that the length of the subpath P[x → y] is at least 2k: otherwise, we
+consider the path P ′ being the concatenation of a shortest s-to-y path in G and the subpath
+P[y → t]. As in Claim 4, we argue that P ′ is a simple s-to-t path in G, and the length of P ′
+is
+distG(s, y) + length(P[y → t]) = distG(s, x) + length(P[y → t])
+= length(P) − length(P[x → y])
+> (distG(s, t) + 3k) − 2k = distG(s, t) + k.
+Hence, P ′ is also a valid solution for our instance (G, s, t, k) that is shorter than P, which is
+a contradiction.
+Since P[x → y] is an (x, y)-grounded path in Gx,y
+⩾p and is internally disjoint from P[y → t],
+we may assume that P[y → t] ⊆ ΓR(P[x → y]) in Gx,y
+⩾p; the case P[y → t] ⊆ ΓL(P[x → y]) is
+symmetric.
+Observe now, that if we set Pmiddle := P[x → y], then the properties (a) – (d) are satisfied.
+If the condition (e) holds as well, then we are done. Otherwise, we define Pmiddle as a minimal,
+with respect to ≺top, (x, y)-grounded path in Gx,y
+⩾p of length at least 2k. Then, Pmiddle ≺top
+P[x → y], and consequently Pmiddle is internally vertex disjoint from Pend = P[y → t] as we
+have P[y → t] ⊆ ΓR(P[x → y]).
+It remains to show that the pair (Pmiddle, Pend) satisfies the property (e). By the definition
+of Pmiddle we know there is no (x, y)-grounded path of length at least 2k which is topologically
+left of Pmiddle. Suppose now that there exists an (x, y)-grounded path P ′
+middle in Gx,y
+⩾p such
+that P ′
+middle ≺top Pmiddle and length(P ′
+middle) ∈ [k, 2k − 1]. Consider the following walk P ′:
+P ′ = P[s → x] ◦ P ′
+middle ◦ P[y → t].
+P ′ is a simple path as in G⩾p we have P ′
+middle ≺top Pmiddle ≺top P[x → y] and P[y → t] ⊆
+ΓR(P[x → y]), therefore P ′
+middle is internally vertex-disjoint from both P[s → x] and P[y → t].
+Moreover, the length of P ′ is at least
+distG(s, x) + length
+�
+P ′
+middle
+�
++ distG(y, t) =
+distG(s, y) + distG(y, t) + length
+�
+P ′
+middle
+�
+⩾ distG(s, t) + length
+�
+P ′
+middle
+�
+⩾ distG(s, t) + k.
+9
+
+Consequently, P ′ is a valid solution for the instance (G, s, t, k). Finally, since length(P ′
+middle) <
+2k ⩽ length(P[x → y]), the path P ′ is shorter than the path P which is a contradiction.
+It remains to show how we can find paths Pstart, Pmiddle and Pend satisfying the above-
+mentioned conditions. Let us assume that we guess the values of p, x and y correctly. By
+Claim 4, we know that Pstart can be set to any shortest s-to-x path, and we see that in order
+to find the desired paths Pmiddle and Pend we can restrict ourselves to the digraph Gx,y
+⩾p. Let
+us call a pair of paths (Pmiddle, Pend) special for (Gx,y
+⩾p, x, y, t, k) if they satisfy the properties
+(b) – (e) stated above. This reduces our instance of the Directed Long Detour problem
+to solving the set of at most n2 instances (one for each choice of x and y) of the following
+problem (with H := Gx,y
+⩾p).
+Problem B
+Input: A plane digraph H, three vertices x, y, t ∈ V (H), and an integer k ∈ N, where
+x and y lie on the outer face of H. The outer face of H is simple and contains only the
+vertices x and y.
+Question: Does there exist a special pair of paths (P1, P2) for (H, x, y, t, k)?
+To finish the proof of Claim 1.2 it is enough to show the following lemma.
+Lemma 4.1. Problem B can be solved in time 2O(k) · n2 by a Monte Carlo algorithm, or
+deterministically in time 2O(k) · n2 log n.
+Proof. We use a variant of the standard color coding technique [4]. Let us color independently
+every vertex of H with one of two colors, say green and blue, each with probability 1
+2.
+Assume for now that (H, x, y, t, k) is a yes-instance, and let (P1, P2) be a special pair of
+paths for our instance. As per our previous considerations, we know that P2 ⊆ ΓL(P1) or
+P2 ⊆ ΓR(P1). Without loss of generality assume that P2 ⊆ ΓR(P1). Recall that length(P1) ⩾
+2k. Therefore, we may define x′ to be the k-th vertex on P1 and y′ to be the k-th vertex
+from the end of P1. Then, length(P1[x → x′]) = length(P1[y′ → y]) = k − 1 and the paths
+P1[x → x′] and P1[y′ → y] are vertex-disjoint.
+Suppose that the colors are assigned in such a way that all vertices of the path P1[x → x′]
+are green, and all vertices of the path P1[y′ → y] are blue. The probability of such an event
+occurring is (1/2)2k. Let G be the set of green vertices of H in the random coloring. Note
+that we assume that V (P1[x → x′]) ⊆ G.
+Claim 6. P1[x → x′] is the lexicographically leftmost x-to-x′ path in H entirely contained
+in G.
+Proof. Suppose that there is an x-to-x′ path Q in H, entirely contained in G, such that
+Q ≺lex P1[x → x′]. Let q ∈ V (H) be the vertex for which path P1[x → q] is the longest
+common prefix of P1 and Q. Also, let r be the first vertex on Q[q → x′], not including q,
+such that r ∈ V (P1). Here, r is well-defined as the vertex x′ belongs to both Q[q → x′] and
+P1. Since r lies on Q, we necessarily have that r is green. Thus, r ̸∈ V (P1[y′ → y]) since
+P1[y′ → y] consists only of blue vertices. Hence, the following walk P ′
+1:
+P ′
+1 = Q[x → r] ◦ P1[r → y]
+10
+
+is an (x, y)-grounded path in H of length at least
+length
+�
+P1[y′ → y])
+�
++ 1 = k.
+Recall that Q ≺lex P1 in H. By the definition of r we observe that Q[x → r] ≺top P1[x → r].
+Hence, P ′
+1 ≺top P1. We conclude that the path P ′
+1 has length at least k and is left of P1,
+which contradicts the property (e) of the pair (P1, P2).
+Note that the path P1[x → x′] is disjoint from the set of vertices on the outer face of H,
+apart from its first vertex x. This allows us to define Hcut as the plane digraph obtained from
+H by cutting along the path P1[x → x′]. Next, let Z ⊆ V (Hcut) be the set of new vertices
+introduced to H′ by this operation. Let H′ be a (y, x′)-simplification of the outer face of
+Hcut.
+Claim 7. P1[x′ → y] is the leftmost path in H′ among all the paths between x′ and y which
+do not contain any vertex of Z.
+Proof. Suppose that there is an x′-to-y path Q in H′ which avoids Z and satisfies Q ≺lex
+P1[x′ → y]. Let q ∈ V (H) be the vertex of H such that Q[x′ → q] is the longest common
+prefix of Q and P1[x′ → y]. Let r be the first vertex on Q[q → y] (not including q) such that
+r ∈ V (P1).
+Analogously as in the proof of Claim 6, we show that the walk P ′
+1 defined in H as follows:
+P ′
+1 = P1[x → x′] ◦ Q[x′ → r] ◦ P1[r → y]
+is a simple path of length at least k that lies lexicographically left of P1 in H. This leads to
+a contradiction to property (e) of (P1, P2).
+The observations above lead to an algorithm for Problem B. First, consider the case
+where in the solution (P1, P2), we have that P2 ⊆ ΓR(P1). After the random assignment of
+colors to vertices of H, we iterate through all vertices u ∈ V (H) \ {x, y, t} as candidates for
+the vertex x′. Then, we find the leftmost path S from x to u in H whose all vertices are
+green. Next, we construct Hcut and H′, and find the leftmost path T from u to y in H′ that
+avoids any new vertices introduced to H′ by the split. Finally, we set P1 = S ◦ T. Then it
+only remains to verify whether there exists any path P2 from y to t in H that is internally
+disjoint with P1. All these checks can be done by running proper depth-first searches in total
+linear time.
+Next, we consider the case where P2 ⊆ ΓL(P1). Hence, we repeat all the above steps with
+any leftmost paths in H replaced with the corresponding rightmost paths. To obtain the
+constant probability of error, we repeat the entire process above 2O(k) times.
+To derandomize this algorithm, instead of assigning the colors to vertices at random, we
+construct and use an (n, 2k)-universal set [7]. In our case this universal set is a family U of
+subsets of V (H) such that for any subset A of V (H) of size 2k the family {A ∩ U | U ∈ U}
+contains all 22k subsets of A. We interpret each element of U as a single coloring, that is, a
+subset A ∈ U corresponds to the coloring c which assigns color green to the vertices of A, and
+blue to the rest.
+We know that there is such family U of size 2O(k) · log n and it can be constructed in
+time 2O(k) · n log n [7]. We construct it once in the algorithm, and instead of drawing random
+colorings, we iterate through all elements of the constructed universal set U. By the property
+11
+
+of U there is an element U ∈ U such that V (P1[x → x′]) ⊆ U and V (P1[y′ → y]) ∩ U = ∅
+because |V (P1[x → x′]) ∪ V (P1[y′ → y])| = 2k.
+This guarantees the correctness of our
+deterministic algorithm.
+Acknowledgement.
+The authors thank Olek �Lukasiewicz for pointing us to the work of
+Wu and Wang [9].
+References
+[1] Ivona Bez´akov´a, Radu Curticapean, Holger Dell, and Fedor V. Fomin. Finding detours is
+fixed-parameter tractable. SIAM J. Discret. Math., 33(4):2326–2345, 2019.
+[2] Marek Cygan, D´aniel Marx, Marcin Pilipczuk, and Micha�l Pilipczuk. The Planar Directed
+k-Vertex-Disjoint Paths problem is fixed-parameter tractable. In 54th Annual IEEE Sym-
+posium on Foundations of Computer Science, FOCS 2013, pages 197–206. IEEE Computer
+Society, 2013.
+[3] Fedor V. Fomin, Petr A. Golovach, William Lochet, Danil Sagunov, Kirill Simonov, and
+Saket Saurabh. Detours in Directed Graphs. In 39th International Symposium on Theoret-
+ical Aspects of Computer Science, STACS 2022, volume 219 of LIPIcs, pages 29:1–29:16.
+Schloss Dagstuhl — Leibniz-Zentrum f¨ur Informatik, 2022.
+[4] Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi.
+Long directed (s, t)-path: FPT algorithm. Inf. Process. Lett., 140:8–12, 2018.
+[5] Steven Fortune, John E. Hopcroft, and James Wyllie. The directed subgraph homeomor-
+phism problem. Theor. Comput. Sci., 10:111–121, 1980.
+[6] John E. Hopcroft and Robert Endre Tarjan.
+Efficient Planarity Testing.
+J. ACM,
+21(4):549–568, 1974.
+[7] Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and near-optimal
+derandomization. In 36th Annual Symposium on Foundations of Computer Science, Mil-
+waukee, Wisconsin, USA, 23-25 October 1995, pages 182–191. IEEE Computer Society,
+1995.
+[8] Alexander Schrijver. Finding k disjoint paths in a directed planar graph. SIAM J. Com-
+put., 23(4):780–788, 1994.
+[9] Bang Ye Wu and Hung-Lung Wang. The next-to-shortest path problem on directed graphs
+with positive edge weights. Networks, 65(3):205–211, 2015.
+12
+

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+Automatic Generation of German Drama Texts
+Using Fine Tuned GPT-2 Models
+Mariam Bangura, Kristina Barabashova, Anna Karnysheva, Sarah Semczuk, Yifan Wang
+Universität des Saarlandes
+{maba00008, krba00001, anka00001, s8sasemc, yiwa00003}@stud.uni-saarland.de
+7009604, 7023878, 7010958, 2573377, 7023035
+Abstract
+This study is devoted to the automatic gen-
+eration of German drama texts. We suggest
+an approach consisting of two key steps: fine-
+tuning a GPT-2 model (the outline model) to
+generate outlines of scenes based on keywords
+and fine-tuning a second model (the generation
+model) to generate scenes from the scene out-
+line. The input for the neural model comprises
+two datasets: the German Drama Corpus (Ger-
+DraCor) and German Text Archive (Deutsches
+Textarchiv or DTA). In order to estimate the ef-
+fectiveness of the proposed method, our mod-
+els are compared with baseline GPT-2 models.
+Our models perform well according to auto-
+matic quantitative evaluation, but, conversely,
+manual qualitative analysis reveals a poor qual-
+ity of generated texts. This may be due to the
+quality of the dataset or training inputs.
+1
+Introduction
+Text generation is a subarea of natural language pro-
+cessing (NLP), appearing in the 1970s (Goldman,
+1974). Its main purpose is the automatic genera-
+tion of natural language texts, which can satisfy
+particular communicative requirements (Liu and
+Özsu, 2009). Text generation can be a constituent
+of AI-based tools related to machine translation, di-
+alogue systems, etc. Computational generation of
+stories is specifically challenging task, as it refers
+to the problem of selecting a sequence of events or
+actions that meet a set of criteria and can be told as
+a story (Alhussain and Azmi, 2021). Many studies
+focus on automatic story generation (Cheong and
+Young, 2014), however, a limited number of them
+emphasize drama generation (Rosa et al., 2020).
+Dramatic texts differ from other genres by hav-
+ing dialogues of acting characters, authorial notes,
+scenes, and other specific elements, usually written
+for the purpose of being performed on stage (Leth-
+bridge and Mildorf, 2004). Therefore, the methods
+described in research devoted to generation of nar-
+ratives or poetry is not always applicable for the
+drama generation. The approaches considered in
+our study are mentioned in Section 2.
+Nowadays, some of the most advanced methods
+for text generation comprise transformer decoder or
+encoder-decoder architecture pre-trained on large-
+scale unsupervised texts. In previous study refer-
+ring to drama generation, GPT-2 is applied (Rosa
+et al., 2021). In the current study, we propose an
+approach to the generation of drama texts in Ger-
+man, based on the production of outlines (Fan et al.,
+2018; Yao et al., 2018), and compare it with two
+baseline GPT-2 models. The detailed information
+about these models and their comparison can be
+found in Section 4. The datasets, used as training
+materials for the system, are described in Section
+3.
+In order to analyze the performance of story gen-
+eration models, various evaluation metrics can be
+involved (Alabdulkarim et al., 2021). For the mod-
+els represented in the current study, we propose
+automatic quantitative evaluation along with man-
+ual qualitative analysis, described in Section 5. The
+main challenges and limitation referring to the pro-
+posed approach and ideas for further improvement
+of drama generation are discussed in Section 6.
+2
+Related Work
+Automatic text generation has long been a task
+of research interests, and various approaches have
+been proposed to improve the quality of generated
+outputs. Among all genres, story generation sees
+the most innovation and progress. Before the era of
+deep learning, some structural and planning-based
+models have been applied to perform story gener-
+ation. The prevalence of RNN (Rumelhart et al.,
+1986) and LSTM (Hochreiter and Schmidhuber,
+1997) motivated researches to introduce deep learn-
+ing to the field of text generation, which results
+in higher model capacity and better performance.
+Leveraging language models with more complex
+architecture and pre-trained on large scale datasets
+arXiv:2301.03119v1  [cs.CL]  8 Jan 2023
+
+further improved the generation quality by a con-
+siderable margin. (Alabdulkarim et al., 2021)
+In addition to the increasing complexity of
+model architecture, researchers are also committed
+to proposing innovative generation schemes. Peng
+et al. attempted to steer generation by adding con-
+trol factors. They extracted control factors from ex-
+isting corpora and trained a model conditioned on
+them, so that users can control the generation pro-
+cess by selecting different control factors. Fan et al.
+(2018, 2019) explored the possibility of a hierarchi-
+cal story generation process, where an intermediate
+stage expands the given prompt and simplifies the
+following generation process by conditioning it on
+expanded prompts. Similarly, Wang et al. (2020)
+also applied a two-stage story generation scheme,
+where the system additional generates a story out-
+line as a guideline for the second stage. It is shown
+that the hierarchical generation scheme effectively
+enhances the consistency and coherency of outputs.
+Despite the similarities with story generation,
+drama generation faces some extra challenges.
+Firstly, a drama play is usually longer than the
+upper limit of pre-trained language models, thus an
+iterative generative process is necessary. Secondly,
+the lack of prompt-output data makes it impossible
+to adopt the same approaches as in story generation,
+and the model must learn to generate plays from
+nothing. The inherent difficulty of drama genera-
+tion task discourages researches in this field. To our
+best knowledge, the only drama generation model
+is THEaiTRE project (Rosa et al., 2020, 2021). The
+system leverages a GPT-2 model to generate each
+scene step by step conditioned on both local and
+remote contexts. However, the generative model is
+not fine-tuned on any drama texts, and the genera-
+tion process requires intensive human interference,
+which compromise usability of the model and is
+not suitable for amateur users.
+3
+Drama Preprocessing
+3.1
+Corpora
+The input for the neural model were dramas from
+the German Drama Corpus (GerDraCor) developed
+by the Drama Corpora Project (DraCor) (Fischer
+et al., 2019) and German Text Archive (Deutsches
+Textarchiv or DTA) (Deutsches Textarchiv, 2022).
+GerDraCor consists of 591 German dramas, with
+the earliest written in the 1640s and the latest in the
+1940s. 46 dramas appeared to be the same with the
+ones in DTA and were removed, resulting in 545
+dramas used from GerDraCor. In the corpus, speak-
+ers, stages and sets1, scenes and acts are annotated.
+There is also metadata available for the whole cor-
+pus and containing information about number of
+speakers and their sex, number of acts and words,
+etc.
+DTA, hosted by the CLARIN service center at
+the Berlin-Brandenburg Academy of Sciences and
+Humanities, is the largest single corpus of his-
+torical New High German that contains around
+1500 cross-genre texts from the early 16th to the
+early 20th century. 92 drama texts with an ortho-
+graphic normalization of historical spelling were
+extracted from the corpus. One of them was ex-
+cluded, as it was a poem. All historical spellings
+are adopted true to the original, i.e., they are not
+implicitly modernized. However, modern or oth-
+erwise normalized equivalents of historical writ-
+ings may be noted with the tags <orig> (histori-
+cal spelling) and <reg> (modernized/normalized
+spelling) (Deutsches Textarchiv, 2022).
+The standard GerDraCor format and DTA basic
+format (DTABf), which were used in this work,
+follow the P5 guidelines of the Text Encoding Ini-
+tiative (TEI), which are specified for the annotation
+of historical printed works in a corpus (Deutsches
+Textarchiv, 2022; Fischer et al., 2019). The TEI
+Guidelines for Electronic Text Encoding and In-
+terchange determines and document markup lan-
+guages for the representation of the structural and
+conceptual text features. They refer to a modu-
+lar, extensible XML schema, consisting of a set
+of markers (or tags) and accompanied by detailed
+documentation, and they are published under an
+open-source license2.
+The following sections describe how dramas
+from aforementioned sources were parsed and pre-
+processed in Python.
+3.2
+Drama Parsing
+Parsing of dramas in XML format was performed
+with XMLHandler class inheriting from Con-
+tentHandler class from “xml.sax” module. This
+class reads xml-tags and operates with their param-
+eters and/or content between starting and closing
+tags. The class contains methods that were over-
+written in order to suit the task of parsing dramas
+from both GerDraCor and DTA (Table 1).
+1Stages and sets are texts describing the setting (decora-
+tions, position of characters) or commenting on characters’
+actions and manner of speech.
+2https://tei-c.org/
+
+Method
+Parameters
+Functionality
+__init__
+output: the empty dictionary that is filled with the
+data from processed XML file
+- initializes instant variables used for the XML tags
+and processed text
+- assigns the empty “output” dictionary to the in-
+stance variable
+startElement
+xml_tag: the start xml-tag (of the <tag> form) which
+is passed to the method from the file
+- stores xml-tag and its attributes in instance variables
+attrs: attributes of the tag
+endElement
+xml_tag: the end xml-tag (of the </tag> form) which
+is passed to the method from the file
+- stores the text processed between start and end tags
+into a specific instance variable
+characters
+content: the text between start and end xml-tags
+- processes the text by skipping empty lines, tokeniz-
+ing text into words at spaces
+- normalizes words spelling if needed (in GerDraCor
+only)
+- stores processed words by adding them into a list
+Table 1: XMLHandler Class Structure
+The tag passed to “startElement” and “endEle-
+ment” defined how the content between tags should
+be stored. For example, if “startElement” read
+<TEI> tag, then the value of the “xml:id” tag was
+stored from that as drama id; if a tag “</text> was
+passed to the “endElement”, then it signaled of
+the end of the drama, and stored all the previously
+parsed text in a dictionary under the drama id as a
+key. The text itself was the content read and written
+in “characters” method and could be the speech of
+a particular character between specific opening and
+closing “speech tags”, or, similarly, a description
+of a stage or a set. Additionally, inside “charac-
+ters”, text was orthographically normalized: histor-
+ical spelling of words was replaced with modern
+spelling, which was looked up in a file containing
+obsolete-modern spelling pairs and was produced
+earlier with a File Comparator (described in detail
+in Section 3.3). That was done for GerDraCor ex-
+clusively, as DTA already contained normalized
+versions of dramas. In general, XMLHandler was
+designed to go through each drama, and extract
+all the drama text, excluding the front page and
+the cast list. Further, parsed dramas were conse-
+quently written into a single text file. In order to
+separate dramas and their parts from each other,
+specific tags were introduced: “$” as opening tag
+and “@” as a closing tag, which were followed by
+the attribute name or value without a blank space.
+For example, at the start/end of each drama a line
+with an opening/closing tag and drama id was writ-
+ten (e.g., “$id_ ger000569” at the beginning and
+“@id_ger000569” at the end) (Table 2).
+The function for writing parsed drama allowed
+to produce two different outputs: dramas with the
+whole text parsed or only characters’ speeches (sep-
+arated by scenes as well) without sets or stages.
+Eventually, the latter version was used for the fur-
+ther model training. Figure 1 shows an example of
+a drama parsed from GerDraCor with characters’
+speeches alone.
+$id_ger000066
+... a
+$scene
+b
+$sp_#dalton
+Ein abscheuliches Unglück – ich kann es nicht
+erzählen – dieser Tag ist der letzte dieses
+Hauses.
+@sp_#dalton
+$sp_#frau_von_wichmann
+Dalton – ist es –
+@sp_#frau_von_wichmann
+$sp_#dalton
+Belmont –
+@sp_#dalton
+$sp_#frau_von_wichmann
+Ach – lebt meine arme Julie noch?
+@sp_#frau_von_wichmann
+...
+@scene
+...
+@id_ger000066
+a ”. . . ” replaces the text skipped in this example.
+b Blank lines are added for the convenience of reading the
+example.
+Figure 1: A Shortened Example of a Drama Parsed
+Only with Speeches
+
+Attribute name
+Text following the “$” or “@” tag
+Text enclosed between tags
+Example of opening/closing tag
+Drama id
+id_dramaid
+Parsed drama
+$id_ger000569 / @id_ger000569
+Set / stagea
+A set / a stage
+$/@
+Scene/actb
+scene
+A scene / an act
+$scene / @scene
+Speaker id
+sp_#speakername
+A speech of a particular character
+$sp_#detlev / @sp_#detlev
+a There was no text following “$” and “@” signs for sets and stage, and the text was enclosed just between those signs.
+b 126 dramas in GerDraCor and 15 dramas in DTA did not contain scenes and were separated by acts or equivalent text
+delimiters, which were marked with a “scene” tag.
+Table 2: Tags Used in Parsed Dramas with Examples
+transliterated
+Hinweg
+sie
+nah’n
+Dort
+sind
+wir
+sicher
+normalized
+Hinweg
+sie
+nah
+‘n
+Dort
+sind
+wir
+sicher
+Table 3: Example of Erroneously Added Blank Space After Normalization
+3.3
+File Comparator
+Since it was undesirable for generated dramas to
+contain antiquated spellings and characters, the
+version of DTA texts used for training the model
+was the normalized version offered by the resource.
+GerDraCor did not offer normalized versions of
+their drama texts, though. To mitigate the influence
+of historical spelling on the training of the model,
+an effort was made to normalize GerDraCor texts
+by using DTA texts.
+The DTA offers different versions of each of
+their drama texts, two of which were important for
+the File Comparator.
+1. transliterated: A character-normalized ver-
+sion with transliterated orthography. Given
+the age of many of the dramas, the original
+texts included characters outside the Latin-
+1 encoding, as for example the ’langes s’
+(U+017F) or the elevated ’e’(U+0364) for
+marking umlauts.
+2. normalized: A version standardized with
+regard to spelling, as well as transliterated
+orthography.
+Historical spellings such as
+"Erk
+eandtnuß." and "weißheyt" are transferred
+to their modern equivalents "Erkenntnis" and
+"Weisheit".
+Therefore, a collection of word pairs was created,
+by comparing the transliterated and the normalized
+versions of the DTA drama texts (Table 4). Punc-
+tuation and other unwanted characters (e.g., “%”,
+“(“, “/”) were cleaned from the strings before com-
+parison. Each word pair consists of the old spelling
+of a word, as well as its modern equivalent. Using
+this list of word pairs, words in GerDraCor with
+the old spelling could be changed into their new
+form.
+Since the normalized version resolved hyphen-
+ation at the page and line break and sometimes
+replaced one word with two words, or connected
+two words into one, the word pairs could not be col-
+lected by simply comparing each line word by word
+in both version. Sometimes, it was indicated in the
+DTA normalized version, if words were previously
+merged (e.g., “wie_es” in the normalized version
+corresponded to "wie’s" in the original text). How-
+ever, such indication was not done consistently:
+“thu’s” for example was normalized into "tu es"
+without an underscore, and therefore, could be
+treated by the algorithm as two words rather than a
+single unit.
+Issues like these could be easily solved by check-
+ing for a specific pattern. The algorithm detects
+words ending with “‘s” in the transliterated ver-
+sion and tests whether the corresponding word in
+the transliterated version is followed by an “es”,
+and if this is the case, then the normalized ver-
+sion likely contains two words (e.g., transliterated
+“thu’s” is correctly paired with the normalized "tu
+es"). But sometimes the normalized version added
+spaces between words, which could not be pre-
+dicted and caused wrong indexing, meaning two
+different words in the line to be compared to each
+other, as shown in the example in Table 3. Added
+space in the normalized version (“nah ‘n”) causes
+the algorithm to combine wrong words in pairs,
+e.g., “Dort – ‘n”, meaning that “’n” is considered a
+normalized version of “Dort”.
+In order to exclude wrong pairs, where two
+different words were treated as normalized and
+transliterated versions of the same word, an al-
+gorithm to compare the similarity of words was
+implemented. If the normalized version was too
+different from the transliterated version, the word
+pair was considered faulty (consisting of two dif-
+
+ferent words). Firstly, Levenshtein Distance was
+used to find possibly faulty word pairs. With using
+similarity threshold of 3, which appeared to be the
+most optimal threshold, this method excluded 576
+word pairs, but many of them seemed to be correct
+edits of old spellings (Table 5).
+For that reason, it was decided to try another
+method and estimate word similarity in each pair
+with the SequenceMatcher class from the “difflib”
+module. SequenceMatcher uses “Gestalt Pattern
+Matching” algorithm for string matching. In case,
+similarity ratio between words in a pair was less
+than 0.53, this word pair was deleted from a list
+of transliterated-normalized pairs. As getting rid
+of wrong pairs was the priority, the 0.5 threshold
+allowed us to exclude as many as possible faulty
+pairs at the cost of losing a few correct ones. Al-
+though, this method excluded 712 pairs (more than
+Levenshtein distance), more of them looked like
+real faulty pairs (Table 6).
+Thus, the final version of File Comparator nor-
+malizes words by using word pairs left after exclud-
+ing faulty word pairs with SequenceMatcher.
+While parsing GerDraCor, if the word from
+drama was found in the dictionary of word pairs, it
+was lowered, changed to its normalized version and
+restored with regards to its original capitalization.
+Transliterated
+Normalized
+Ueberraschungen
+Überraschungen
+Medicinerei
+Medizinerei
+practicieren
+praktizieren
+Caffeegeschirr
+Kaffeegeschirr
+Cigarettentasche
+Zigarettentasche
+Hausflurthür
+Hausflurtür
+Nachtheil
+Nachteil
+Legirung
+Legierung
+legirt
+legiert
+Gratulire
+Gratuliere
+nothwendigerweise
+notwendigerweise
+adressirt
+adressiert
+cuvertiert
+kuvertiert
+todtgeboren
+totgeboren
+Table 4: Examples of Pairs Collected from Transliter-
+ated and Normalized Versions of DTA Drama Texts
+4
+The Proposed Approach
+Inspired by the two-stage story generation ap-
+proaches employed by (Fan et al., 2018; Yao et al.,
+2018), we also decided to divide the drama scene
+generation process into two stages. However, while
+3Ratio varies from 0 to 1, where 0 means no commonalities
+and 1 means identical strings.
+Transliterated
+Normalized
+Wohlhäbige
+Wohlhabende
+Verlaubst
+Laubest
+Thu’s
+tue es
+daß’s
+dass es
+hoamgangen
+heimgegangen
+Zen
+Zähne
+veracht’
+Acht
+Table 5:
+Examples of Word Pairs Excluded After
+Checking for Faulty Word Pairs with the Levenshtein
+Distance Algorithm
+Transliterated
+Normalized
+Hizt
+Jetzt
+nachi
+nage
+itz
+Jets
+Creyß
+Kreis
+Flick
+Flügge
+Vehd
+Fett
+dy
+die
+Table 6:
+Examples of Word Pairs Excluded After
+Checking for Faulty Word Pairs with the Sequence-
+Matcher Algorithm
+Fan et al. first generate a storyline, which is subse-
+quently used as input to the model that generates
+the story, we train a model to produce outlines,
+which become part of the input prompt in the sec-
+ond stage. Likewise, our approach is different from
+Yao et al.’s in that it uses just 10 keywords instead
+of one keyword per sentence in the story. With this
+approach, we aim to guide the generation process
+of the model by providing it with the keywords sum-
+marizing the most important parts of each scene.
+Our second goal is to reduce the workload of the
+user by allowing them to provide only 10 keywords
+and let the hierarchical model do the rest of the
+work.
+First, we fine-tune a GPT-2 model (the outline
+model) to generate outlines of scenes based on an
+input of keywords extracted from the text. In the
+second step, we fine-tune a second model (the gen-
+eration model) to generate scenes based on input
+which consists of the outline of the scene, a sum-
+mary of the remote context as well as that of the
+local context.
+4.1
+GPT-2
+GPT-2 (Radford et al., 2019) has been demon-
+strated to achieve state-of-the-art results in a range
+of NLP tasks such as natural language inference,
+semantic similarity, text classification as well as
+question answering. Moreover, GPT-2 has success-
+fully been used for story generation (Wang et al.,
+
+2020; See et al., 2019).
+GPT-2, introduced by Radford et al., is an auto-
+regressive transformer consisting of 12, 24, 36
+or 48 decoder blocks, depending on the size of
+the model. In contrast to BERT (Devlin et al.,
+2018), which consists of encoder blocks only, GPT-
+2 stacks decoder blocks. Furthermore, an important
+property of GPT-2 is its autoregressivity, i.e. the
+model conditions the next token on the previous
+token thus allowing text generation.
+According to Radford et al., an additional key
+feature of GPT-2 is its ability to learn a downstream
+task in a zero-shot manner, i.e. without any need
+for parameter tweaking or modifications to the ar-
+chitecture of the model.
+GPT-2 was trained with a slightly modified lan-
+guage modeling objective: instead of estimating the
+conditional distribution P(output|input), GPT-2
+estimates P(output|input, task). But, instead of
+separately modeling this at the architectural level,
+the task can be prepended to the input sequence.
+As there is no official GPT2 model for Ger-
+man, we use the German GPT2 model4 uploaded
+to Huggingface. It uses the 12-block setting, result-
+ing in a 117M parameter model. The model was
+trained on a 16GB and 2,350,234,427 tokens data
+set consisting of data from the Wikipedia dump, EU
+Bookshop corpus, Open Subtitles, CommonCrawl,
+ParaCrawl and News Crawl.
+4.2
+Pre-processing & Train/Dev/Test Split
+First, we pre-process the Dracor dataset, generating
+training instances needed for training both models.
+As both the outline and the generation models use
+scenes and gold standard outlines as input, we gen-
+erate those first.
+For both scenes and outlines we create two ver-
+sions: one with speakers left in the text and one
+without speakers. The first version serves as in-
+put to both models, while the latter is only used
+once during the keyword extraction process. In the
+first version, each utterance starts with a <speaker
+name:> and is followed by a newline character,
+so that the actual utterance is on a separate line.
+For the version without speakers, we simply make
+sure each utterance is on a separate line. For sen-
+tence boundary detection, we employ the NLTK
+tokenizer for sentences from the NLTK Tokenizer
+package 5.
+4https://huggingface.co/dbmdz/german-gpt2
+5https://www.nltk.org/api/nltk.tokenize.html
+In addition, as there are no real outlines available
+for the plays, we experiment with two summariza-
+tion algorithms to get the gold standard outlines.
+First, following Wang et al.’s approach we employ
+TextRank, an extractive text summarization algo-
+rithm, to extract the outlines from scenes. We also
+try abstractive summarization with a BERT2BERT
+model6 trained on MLSUM, a dataset of 1.5M on-
+line news articles. Upon inspection, we found that
+the BERT2BERT model’s output was unsatisfac-
+tory: most of the time the summary consisted of 2-3
+sentences and was often truncated. Furthermore,
+as the format of a play presupposes some form of
+a dialogue, it is quite different from that of a nor-
+mal text written in prose. We hypothesize that the
+strange output of the model is due to it having been
+trained on news articles. Thus, we proceed with
+utilizing the TextRank algorithm for outline gen-
+eration. Prior to performing summarization with
+TextRank, we remove the speakers, and add them
+back in to each sentence in the outline.
+4.3
+The Outline Generation Model
+As the data set we use does not have gold standard
+outlines, we decided to follow Wang et al.’s ap-
+proach, in which they use Textrank (add citation to
+extract the outline of the story (or the scene in our
+case). We then utilize these outlines as the ground
+truth output for our model. As input to the outline
+model, we use keywords extracted from the scenes
+and their outlines.
+4.3.1
+Keyword extraction
+In the search for a keyword extraction algorithm
+which could yield a good set of keywords for each
+scene/outline, we have experimented with 6 differ-
+ent algorithms: Yake (Campos et al., 2020), Rake
+(Rose et al., 2010), MultiRake 7, KeyBert (Grooten-
+dorst, 2020), TextRank(Mihalcea and Tarau, 2004)
+and tf-idf.
+Keyword Extraction Algorithms
+RAKE first
+generates a set of candidate keywords for the doc-
+ument subsequently creating a co-occurrence ma-
+trix from those. In the next step, for each can-
+didate a score, defined as the sum of its member
+word scores, is calculated. The word scores are
+calculated using word frequency (freq(w)), word
+degree (deg(w)),and (3) ratio of degree to frequency
+(deg(w)/freq(w)).
+6https://huggingface.co/mrm8488/bert2bert_shared-
+german-finetuned-summarization
+7https://pypi.org/project/multi-rake/
+
+MultiRake is simply the multilingual version of
+the RAKE algorithm which has some additional pa-
+rameters such the addition of one’s own stopwords
+or the possibility to vary the length and number of
+keywords.
+KeyBert is based on creating BERT embeddings
+for both the individual tokens in a document as well
+as the document itself. Then, the cosine similarity
+of the embedding of each word and the document in
+which the word appears is calculated. Those words
+that have the highest cosine similarity with the doc-
+ument embedding are identified as the keywords of
+the document.
+TextRank is a graph-based ranking model which
+takes into account the co-occurrence of words in a
+window of N words, adding edges between those
+nodes and then applying applying a ranking algo-
+rithm until convergence.
+In contrast to the algorithms mentioned above,
+tf-idf not only quantifies the importance of a term
+to a specific document in a collection of documents
+but also off-setts it by the number of occurrences
+of this term in other documents in the set. This al-
+lows to mitigate the effect of highly frequent terms
+occurring in a large number of documents on the
+final score.
+tf(t, d) =
+f(t, d)
+�
+t′∈d
+f(t′, d)
+(1)
+YAKE differs from the other algorithms in that
+it relies on a set of features which are supposed
+to characterize each term. These include casing,
+the position of the word in the document, word fre-
+quency, word relatedness to context and frequency
+of word in different sentences. Finally, these fea-
+tures are combined into a single score which repre-
+sents the word (Sw).
+S(kw) =
+�
+w∈kw S(w)
+TF(kw) ∗ (1 + �
+w∈kw S(w))
+(2)
+Algorithm Comparision
+In order to select the
+most suitable algorithm for this task, we performed
+a qualitative evaluation of the keyword extraction
+results. We used a small set of 5 randomly chosen
+scenes. Upon inspection of the extracted keywords,
+we observed that only the keywords obtained us-
+ing tf-idf and TextRank actually yielded acceptable
+results. For example, Rake, MultiRake and YAKE
+return quite a few repeating keyword or keywords
+or keywords that differ only in the grammatical
+case. Furthermore, some keywords were simply
+a concatenation of neighboring tokens which do
+not make much sense when put together, especially
+if the preceding tokens are missing. In addition,
+RAKE and MultiRAKE return lowercased version
+of the keywords, which can be problematic for Ger-
+man text, as casing signals the POS of a word and
+thus serves an important function, distinguishing
+nouns from other parts of speech. As GPT-2 uses
+byte-pair-encoding, the starting vocabulary, i.e. the
+set of all individual characters, consists of both
+lower and upper case characters. This means that
+when the BPE algorithm learns to merge adjacent
+characters, it treats AB and ab as different tokens.
+In light of our observations, we decided to ex-
+tract keywords using tf-idf and TextRank and train
+two outline models.
+Keyword extraction
+As a large number of
+scenes are quite long and the keyword extraction al-
+gorithms often return phrases that are only uttered
+once by the speaker, we decided to try out keyword
+extraction from both whole scenes and outlines of
+scenes. We have noticed that keywords extracted
+from outlines are often more relevant to the outline.
+As a result, our models are trained on keywords
+extracted from outlines, where the outline version
+is that without speakers.
+Another important parameter for our keywords
+input is the number of keywords (k) to be extracted
+from the scenes. Our experiments have shown that
+when k > 10, many of the terms in the lower half
+of the keyword list are extremely random and unre-
+lated to the outline. As a result, we chose k=10 for
+both tf-idf and TextRank.
+Tf-idf
+Despite using existing implementations of
+tf-idf 8 and TextRank 9, we had to apply some pre-
+processing steps. In the case of tf-idf, we first apply
+a SpaCy10 POS tagger with the de_core_news_sm
+German model in order to exclude auxiliary verbs,
+particles, adpositions and adverbs. In addition, any
+tokens appearing in NLTK’s stopword list for Ger-
+man are dropped.
+TextRank
+Similarly, we only keep keywords that
+are not part of the list of German stopwords. In
+addition, as TextRank extracts sequences of tokens,
+not individual tokens, repetitions containing tokens
+8https://scikit-learn.org/stable/modules/generated/sklearn.
+feature_extraction.text.TfidfVectorizer.html
+9https://pypi.org/project/pytextrank/
+10https://spacy.io/
+
+that only differ by grammatical case are inevitable.
+In this case, we discard repeated keywords. For
+instance, in the case of die gute Oma and der guten
+Oma we only keep the lemmatized version of the
+first occurrence of the keyword.
+4.3.2
+Model training
+To fine-tune the German GPT-2 model to produce
+outlines given keywords as input, we concatenate
+the keywords K and the corresponding outline O
+extracted using TextRank and separate them with
+the <SEP> token. In addition, the concatenated
+sequence C is prepended with a <BOS> token and
+a <EOS> token is attached to the end of the con-
+catenated input.
+The model is trained for 3 epochs with a training
+batch size of 4 and a test batch size of 8. The
+default optimizer AdamW is used and the number
+of warm up steps for the learning rate scheduler is
+set to 500. The model is evaluated every 400 steps.
+During training, we compute the cross-entropy of
+the tokens in C.
+At test time, the model is fed the sequence
+<BOS> + K + <SEP> and is expected to gener-
+ate the outline tokens. Generation stops once the
+<EOS> token is generated. We use top p sam-
+pling, wherein the next token to be generated is
+selected from the vocabulary items that make up
+70% (top_p=0.9) of the probability mass. In addi-
+tion, repetition_penalty is set to 2.0.
+As has been mentioned before, we have trained
+two versions of the outline model (using the same
+settings): one in which the keywords are extracted
+using tf-idf and the other using TextRank. The two
+models are evaluated with respect to their perfor-
+mance on the downstream task of scene generation,
+discussed in Section 5.2.
+4.4
+The Generation Model
+In the second part of our system, another fine-tuned
+GPT-2 model is leveraged to generate a drama
+scene from given start and outline. The genera-
+tion model can be characterized by the following
+three aspects:
+1. Iterative generation: As many drama scenes
+are longer than the upper length limit of GPT
+model (1024 tokens), it is not possible to gen-
+erate a whole scene at once. Therefore, we
+adopt an iterative generation strategy: in each
+iteration, the model only generates 100 tokens,
+and all generated tokens are then fed into a
+summarizer to produce the prompt for next
+iteration.
+2. Dynamic prompt: In our system, the prompt is
+split into three individual parts: outline, sum-
+mary of remote context and local context. The
+outline is either drawn from the original play
+or generated by the first part of the system,
+and remains unchanged in all generative itera-
+tion. When the outline and generated outputs
+are longer than 924 tokens, only the nearest
+250 tokens are preserved, and the remote con-
+text is summarized by a TextRank model. The
+three parts are concatenated with a <SEP> to-
+ken to form the prompt of each generation
+step. In this way, our model can maintain
+local coherency as well as memorizing impor-
+tant information even if it is mentioned far
+ahead of the current position. The introduc-
+tion of outlines provides a guideline for the
+plot and guarantees global consistency. Thus,
+the model is provided with dynamic prompt
+with different information in each iteration. A
+figure describing the model structure can be
+found below. (Figure ??)
+Figure 2: Model architecture
+3. Automatic post-editing: Despite the improved
+performance of our fine-tuned GPT-2 model,
+it still fails to produce drama as human ex-
+perts. This can be attributed to the inherent
+difficulty of drama generation and the diverse
+writing styles and formats of our collected
+training corpora. To address some recurring
+format problems, we apply a few automatic
+post-editing methods. In particular, we have
+resolved the following issues:
+• Repetitiveness: As the input information is rel-
+atively little at the beginning of generation, the
+model tends to repeat sentences from prompt
+or generated lines. To counter repetitiveness,
+
+Output (100 tokens)
+Fine-tunedGPT-2model
+Local
+<BOS>
+Outline
+<SEP>
+Summary
+<SEP>
+<EOS>
+contextwe set the repetition penalty to 1.01 and for-
+bid repeated 4-grams during generation. We
+also discard any new lines (excluding charac-
+ter names) that have already been generated.
+Since we have a strict penalty for repetition,
+it can occur that the model cannot generate
+a valid line in an iteration. To prevent these
+cases, the model returns 10 sequences each
+time for the post-editing module to select, and
+it is backed off to generation without outline
+when none of the 10 returns include any valid
+lines.
+• Bad character names: In most cases the model
+is able to identify characters in the play and
+continues the dialogue with their names. How-
+ever, it sometimes misspells names or adds
+new characters abruptly, which harms the plot
+consistency. In our system we identify mis-
+spelling by its edit distance from any given
+character name. If the edit distance is small
+(less than or equal to 2), it is considered as
+misspelling and the wrong name is corrected
+to a given character name. Otherwise, the new
+name is seen as an invalid character and will
+be removed along with its speech.
+• Empty speeches: The model may output char-
+acter names at the start of a new line but does
+not assign any speech to them. We manage to
+resolve this problem by identifying character
+names followed immediately by another name
+and discarding the lines.
+5
+Experiments and Results
+To study the effectiveness of our proposed ap-
+proach, we compare our models with baseline GPT-
+2 models. In particular, we have two baseline mod-
+els: a not fine-tuned GPT-2 model and a GPT-2
+model fine-tuned on the same training set but with
+no outline or summarization (-dynamic prompt).
+All models are based on a German-language GPT-
+2 model named german-gpt2 from HuggingFace.11
+For generation model training, we first extract
+prompt-generated output pairs from collected cor-
+pora and fine-tune our model on them. In particular,
+the outline part in the prompt is extracted from the
+original scene using TextRank. We run the training
+for 3 epochs with a batch size of 8, evaluating on
+the dev set every 400 steps. A default optimizer is
+used with 500 warm-up steps and the checkpoint
+with the lowest perplexity on dev set is chosen,
+11https://huggingface.co/dbmdz/german-gpt2
+with an early stopping patience of 10. The fine-
+tuned baseline model is trained on the raw drama
+scripts directly with the same set of hyperparame-
+ters as generation model training, except that it is
+trained for 10 epochs, as there are fewer optimiza-
+tion steps in each epoch compared to generation
+model fine-tuning.
+5.1
+Evaluation Metrics
+To evaluate the performance of our approach, we
+adopt several automatic quantitative evaluation met-
+rics as well as a manual qualitative analysis. 100
+scenes from test set are generated by each model
+given start of the scene (approximately 100 tokens)
+as well as an outline (approximately 200 tokens,
+set to NULL for baseline models) and their perfor-
+mance is measured and compared by the following
+metrics.
+• Average number of sentences per speech:
+In general, drama is comprised of conversa-
+tions, which means each character is supposed
+to take turns to give their speeches. Thus, it
+is important that the model should not gener-
+ate a text where only one or two characters
+give very long speeches. Average number
+of sentences per speech is a metric reflecting
+how well the generated plays resemble a hu-
+man written play in format. Abnormally high
+value in this metric indicate that model fails
+to capture the format features of drama.
+• Average sentence length: Average sentence
+length is a simple yet effective measurement
+of performance of generative models(Kincaid
+et al., 1975; Roemmele et al., 2017). While
+too long sentences might harm readability, too
+short sentences are more likely to be incor-
+rect or illogical in the context. In our experi-
+ment, we compare the average sentence length
+of generated texts to that of human written
+scripts, to evaluate and compare how each
+model performs in generating fluent and read-
+able sentences.
+• Perplexity: We also measure the perplexity
+score of the generated scenes from each model
+(including human written plays) using german-
+gpt2. Perplexity is usually assigned by a to-
+be-evaluated language model on a real text
+(Jelinek et al., 1977), while in our case we
+reverse the process and leverage a pre-trained
+language model to evaluate the fluency and
+
+Models
+w/o fine-tuning
+w/o outline
+w extracted outline
+w TextRank outline
+w TF-IDF outline
+Human
+Sentences per speech
+64.74
+5.40
+4.47
+5.97
+6.04
+3.21
+Sentence length
+4.60
+6.89
+6.16
+6.82
+6.60
+8.82
+Perplexity
+20.58
+18.90
+19.18
+18.82
+19.46
+17.13
+1-gram overlap
+0.11
+0.19
+0.20
+0.17
+0.18
+0.10
+2-gram overlap
+0.012
+0.028
+0.040
+0.020
+0.022
+0.009
+3-gram overlap
+0.0007
+0.0053
+0.0109
+0.0024
+0.0032
+0.0013
+Topic drift (2-gram)
+-8.33%
+34.3%
+18.2%
+27.3%
+42.3%
+20.0%
+Topic drift (3-gram)
+28.6%
+66.3%
+23.0%
+34.6%
+79.2%
+57.1%
+Distinct-1
+0.503
+0.433
+0.463
+0.438
+0.443
+0.576
+Distinct-2
+0.880
+0.842
+0.860
+0.846
+0.846
+0.921
+Distinct-3
+0.969
+0.963
+0.966
+0.962
+0.960
+0.982
+Table 7: Automatic evaluation results on 100 test set.
+coherency of generated texts. In particular, to
+balance the evaluation efficiency and accuracy,
+we use a stride of 100. Lower perplexity score
+indicates better coherency.
+• N-gram overlap: For n=1,2,3, we measure
+the F1 score of n-gram overlap between the
+start of scene and generated text. Low value
+means lower similarity between the generated
+text and start of the scene.
+• Topic drift: In addition to n-gram overlap,
+we also calculate the overlap for the first half
+and second half of generated texts separately,
+and measure the proportion of decrease in F1
+as a metric for topic drift. We assume that, if
+a story is globally consistent, the topic drift
+should be relatively small, while a story lack-
+ing plot consistency tends to have larger topic
+drift.
+• Distinct-n: To examine the model’s ability of
+producing diverse words and phrases, we also
+compute the average proportion of unique n-
+gram in the whole generated text (Li et al.,
+2015). Higher proportion of unique n-grams
+reflects that the model is highly capable of
+generating unseen words or phrases, either by
+rephrasing existing expressions or introducing
+new contents.
+5.2
+Quantitative Results
+Table 7 shows the results of our automatic evalu-
+ation. It is obvious that the model without fine-
+tuning fails to produce texts that are formally sim-
+ilar to drama: each speech consists of on average
+64.74 sentences and each sentence is composed
+of only 4.6 words, indicating it just start a speech
+randomly and many sentences are only one phrase
+or even one word. For this reason, it will not be
+analyzed later in this section. All other models
+perform reasonably well in average number of sen-
+tences per speech. The human-written scripts have
+the lowest value.
+Similar patterns can be observed in average sen-
+tence length and perplexity. Human-written scripts
+demonstrate the best readability and coherency.
+Among the machine generation approaches, despite
+the gap being trivial, the model with no outline
+and the model with outline generated by keywords
+(TextRank) display superiority to the model with
+extracted outlines in terms of fluency.
+When it comes to n-gram overlap, the model
+with extracted outlines has by far the highest over-
+lap with the given start of the scene, followed by the
+model with no outlines. The models with generated
+outlines do not reach a decent result, probably be-
+cause of the poorer quality of outlines. It is worth
+mentioning that the real drama scripts have the
+lowest overlap score. We attribute this to the out-
+standing ability of human experts of rephrasing and
+controlling the flow of plot, thus it is not directly
+comparable to machine generation approaches.
+Besides, we notice that introducing extra out-
+line information indeed contributes to a better
+global consistency: models using both extracted
+outlines and outlines generated by TextRank key-
+words show competitive or even better performance
+than the human-written plays in topic drift, while
+model that do not leverage such information suffer
+severely from topic drift.
+Finally, no significant difference is observed in
+the ability of using diverse vocabulary among ma-
+chine generation models. Human playwrights, as
+we have expected, show their irreplaceable advan-
+tage in diction.
+5.3
+Qualitative Analysis
+5.3.1
+Qualitative Evaluation of Outlines
+Firstly, in most of the cases, only the first line of the
+outline contains a speaker. Naturally, this makes
+
+it impossible for the subsequent generation model
+not to come up with random characters that do not
+appear in the outline. Furthermore, after the first
+couple of sentences, the generated outline quite
+often consists of direct speech followed by a report-
+ing clause (i.e. "sagte der Mann" - "a man said",
+"fragte er" - "he said"), as can be seen in both gener-
+ated outlines in Table A1 in the Appendix. This is
+quite surprising, considering that the gold standard
+outlines do not contain any such text, as all of the
+drama pieces are in dialogue format. A possible
+explanation for this could be that the amount of
+drama texts used for training is insignificant com-
+pared to the large amounts of news data the model
+was pre-trained on.
+5.3.2
+Qualitative Evaluation of drama texts
+Manual evaluation reveals that none of the models
+were able to produce coherent and meaningful texts.
+On average the texts created by the model with no
+outline are shorter compared to the texts from other
+models, which mostly end after a maximum itera-
+tion number is reached. Though all of the models
+produced texts that ended with the repetition of
+mildly changed words or phrases, the model using
+an extracted outline did so more frequently. This
+can be seen quite well in the generated example
+text found in Table A2 Part 3/3 given in the ap-
+pendix and might explain the extremely low topic
+drift values for this model. The two models using
+generated outlines did not introduce as many new
+characters and did not switch between speakers as
+often as the other two models, creating mostly long
+monologues instead of dialogues. All of the models
+overused ’»’ and ’«’ in normal dialog and started
+a lot of sentences with a hyphen. Since this is a
+problem occurring in all of the models, it can be
+assumed that the varying formalization across dif-
+ferent dramas used in the training process caused
+this issue. In multiple drama texts two or more fol-
+lowing hyphen were used to mark pauses in speech.
+One example of an excessive use of hyphen in the
+original drama texts can be found in the excerpt
+from ’Die Pietisterey im Fischbein-Rocke’ given
+in Figure A1 in the appendix. The models tend to
+overuse hyphens in a way, that hinders meaningful
+text generation instead.
+6
+Discussion and Outlook
+Our proposed method described above is able to
+handle some known issues like lack of global infor-
+mation. However, drama generation/completion is
+a challenging task even for human experts, and in
+our work, there are still some problems remaining
+unresolved:
+1. Abrupt ending: Although a special token
+<endoftext> is added to the end of each scene
+and used for training, we notice that in most
+cases the generation only stops when a max-
+imum iteration number is reached. This will
+lead to an abrupt ending problem. A better
+method should be explored to provide more
+control over the story ending without dramati-
+cally harming the conciseness of drama.
+2. Non-uniform format: Despite an extra post-
+editing process during generation, some in-
+consistency in format is still not completely
+avoided. Some bad names are not detected as
+well and are thus kept in the text and compro-
+mise readability.
+3. Instability: While some previous works (Rosa
+et al., 2020, 2021) rely on manual interven-
+tion to detect and prevent unsatisfactory gen-
+eration results, we decided to adopt a more
+convenient fully automatic approach, which
+inevitably induces accumulated errors and re-
+sults in instability in generation.
+4. Incompetence of generating a whole play: The
+proposed model can only generate one scene
+at a time and cannot produce a whole play. Fu-
+ture work can focus on this more challenging
+task, for example by introducing an additional
+layer to the hierarchy, aiming to generate out-
+lines for each scene based on a outline of the
+whole play and summary of previous scenes.
+7
+Conclusion
+This paper compares the quantitative results of dif-
+ferent models attempting the generation of German
+drama texts. Furthermore it explores the oppor-
+tunity of generating German drama texts with ex-
+tracted outlines. While the quantitative results of
+the models suggested sensible outcomes, qualita-
+tive analysis of the generated texts found them to
+be lacking in regards of coherency, meaning and
+form. A lot of issues can be hypothesized to stem
+from the varying formalization in the drama texts
+used in the training of the models and the poor qual-
+ity of the generated outlines. A bigger and cleaner
+dataset of German drama texts would be desirable
+for further testing of generational models.
+
+References
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+ACM
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+
+A
+Appendix
+id
+model
+Generated outline
+id_ger000018_5
+textrank
+Liddy: Ich muß sagen – ich bin entzückt..?« »Do is doch was zu do«, sagte er
+zum Käthchel und wies auf die große Verantwortung hin; Herr Littmann
+meinte so schön es im Scherz ang’scheckt wie eine Maus in einer Mädl’n oder
+Frauenzimmerin des Herrn von der Hellenen-Pensionärsfeuerwehre mit dem
+Namen Ihnen ’nen Kauz auszulachen«. „Mein Vater war ein alter Knabe aus
+dem Dorf Der hat sich für einen Berufsstart als Tischler entschieden Aber da
+kommt man ja nicht mehr drauf zurück...“ Nun das ist wahrhaftig kein
+Wortschatzfehler sondern daß mir auch noch ganz ungebildete Leute zuhören
+können Als ob mich mein Sohn nur um seine gute Laune gebeten hätte wenn
+meine Tochter einmal über ihm wäre...
+id_ger000018_5
+tf-idf
+Valeria: Ich wollte dich nicht verlassen. – Ich habe dir es noch gesagt; ich bin
+dein Freund und deine Dienerin gewesen!« »Und du hast mich verraten?«,
+sagte er zu mir mit dem Augenschmalze auf den Lippen in der Sprache seiner
+eigenen Muttersprache so leise als möglich.« Nun komme endlich nach Hause
+zurück von diesem Abend hier im Garten meiner Schwester Liddy oder dieser
+Ihen Tochter Molly... wenns kein Irrtum ist gegen das Schicksal meines Vaters
+für die gute Sache aufzuwarten?’ laß ihn allein sprechen!». ’tschuldigung
+Vater, aber da war ein Brief an Sie bei Ihrer Ankunft am Bahnhof : Er kam aus
+Wien ohne Ihre Erlaubnis herüber ins Haus Ihres Sohnes Marquis de La Salle
+. . . Und nun kommen sie alle wieder hierher um sich ihren Spaß daran gönnen
+; denn wie gewöhnlich haben wir beide uns schon einmal unterhalten über
+einige Dinge zwischen Ihnen beiden.... man muß doch sagen daß diese
+Begegnung eine gewisse Wahr heit enthält!!’ Ach was soll nur sein – lassen Sie
+mal hören!’.Das wird ja wohl niemand sehen wollen ob Ihr Sohn wirklich
+einen Fehler begangen hat unter seinem besten Namen.Herr Baronin, erlauben
+Herr Graf Renz-Honau, bitte recht freundlichst unsere Hand deines Mannes
+küssend anzubezeigen....binde Fräulein Gräfin Marie wird alles verzeihen
+müssen meine Frage wäre jetzt keine Antwort nötig öffne deinen Mund weit
+genug.” / So sagt mein Bruder zum Doktor – das tut weh!’, fragt er scherzhaft
+nach einer Bemerkung des Herrn Kommissars... vielleicht sei der Mann zu
+jung dafür geworden, weil er nicht anders kann tun als seine Frau lieben Zu
+machen ihm seinen Kummer verständlich darzustellen als auch seine
+Leidenschaften zu zeigen.
+Table A1: Examples of outlines generated by the two outline generation models (tf-idf and textrank)
+
+id_ger000275_37.txt
+START_OF_SCENE
+Rosamunde: Ja er ist es! Gelobt sei Gott!
+Flodoardo: Eure Durchlaucht...
+Doge: Seid willkommen, Edler Ritter, tausendmal... Habt Ihr Euer Wort gelöst?
+Flodoardo: Alles, wie Ihr mir geboten, Ist erfüllt; die Republik Frei von jeglicher Gefahr; Und der
+Himmel lächelt heiter.
+Doge: Und der Mörder unserer Freunde?
+Flodoardo:
+w/o outline
+Flodoardo: Wir sind gerettet.
+Doge: Der Tag wird kommen, Daß er sich uns naht.
+Flodoardo: Stimme: Euer Tod ist unser Tod!
+Doge: Gelobt ist Gott! ...
+Flodoardo: Das Glück ruft nach uns! – Der Ritter Des Todes, Der die Stadt umringt, Der Ritter von
+Florenz, Der den Kaiser stürzen will! – Hört mich an!
+Flodoardo: O Gott!
+Doge: Und er, der den Kaiser stürzen wollen –?
+Flodoardo: Er soll sich erheben, Und der Herr ist frei! Weh ihm, daß er es erfährt! Ich bitte Euch,
+laßt ihn, er soll nicht mehr Euch, Ihm allein, sprechen. Fluch und Haß und Tod sind für uns Leben.
+w extracted outline
+Flodoardo: Auch dieser Mensch, der mit uns ins Gefängnis kam, Der auf diesen Felsen gesprungen
+ist, Und sich vor mir entblößte, so als der Retter der Republik?
+Flodoardo: Er war der letzte, der hier saß, bis zum letztenmal. Ich bin, wer er? Dogt mir.
+Flodoardo:
+Doge: Wer war der erste? O wie Ihr ihn nennt, den ersten.
+Flodoardo: Wen nannte der König?
+Doge: Mein Herzog. Dogt nicht das Gerücht.
+Doge: Die Republik hat ihn ausgerufen.
+Flodoardo: Ein Fürst.
+Doge: Ein Mann. – Ihr, mein edler Herr Abbate; Ihr, hochfürstlicher Herr Follo liegt bereits in den
+Eisen.
+Flodoardo: Was wird aus Eurer Freiheit? – Contarini, Euer Oberst Frosso liegt schon In den Eisen;
+Es wird von der Nacht so viel geredet, Die ihn verderben und aufhetzen.
+Parozzi: seht, daß Ihr, das schwöre ich, Den Prinzen Valerio vor euch schützt.
+Flodoardo: Wünschst du, daß der Herzog nicht? –
+Doge:
+2-3: Wünschest du, daß ich es nicht könne? – Alle anderen? – Die ganze Republik? – – Wir?
+Flodoardo: Denselben, den ihr im Dunklen habt?
+Doge: So wie sie sind, so muß ich auch das sagen. Wir müssen ihn selbst holen.
+Flodoardo: Und der Richter von Florenz?
+Doge: Der Herzog kommt mir zu nah.
+Flodoardo: Ist die Flucht so ein Glück?
+Flodoardo: der Herzog von Parozzi.
+Doge: Er wird mich mit seinen Kindern einsperren.
+Table A2: Example texts generated by the different models.
+Part 1/3. Continued on the next page.
+
+id_ger000275_37.txtr
+w TextRank outline
+Flodoardo: Sie sind so frei und liebenswürdig?
+Flodoardo: Und unsere ganze Menschheit? Und die Menschheit...
+Flodoardo: Und die? ...
+Doge: Und was noch wichtiger ist! Denn in der ganzen Welt, die auch noch so verwegen, und die so
+viel Geld bringt, die ganze Menschheit noch so viel Geld macht!
+Flodoardo: Die Menschheit ist ein ganz Anderer! Ihr macht es doch noch schlimmer, wenn Ihr mir
+die Hand reichen wollt.
+Doge: Wenn Ihr Euch damit nicht zufrieden macht, so muß ich Euch doch den Kopf zerbrechen lassen.
+Flodoardo: Ihr solltet Euch denn nicht mehr ärgern, wenn Euch Euer Leben nicht mehr wert ist.
+Doge: So könnt Ihr Eure Seele zum Tode schlagen.
+Flodoardo: Wie!
+Doge: Sie sind so frei...
+Flodoardo: So, meine Herren, mit Euch muß man nicht mehr um das Leben tanzen, daß man doch
+gleich zu Grunde kommen will.
+Doge: Und Euer einziges Recht – so!
+Flodoardo: Das Recht –
+Doge: Denkt Ihr?
+Flodoardo: Ja wenn Ihr mich nicht wollt. Wir sind doch beide nicht frei. Ihr werdet noch alle Eurer
+Liebe, Euren Haß, Euren Eifer, Euren Schmerz verlieren. Ihr seid mein Freund!
+Doge: Und Ihr wollt leben, wo Ihr es wagt.
+Flodoardo: Es gibt Euch doch wohl in der Welt nichts was mehr ein Mann wagen würde.
+Doge: und Ihr seid auch ein Weib!
+Flodoardo: Ich! – Ihr seid noch nicht frei?
+Doge: Ihr könnt auch für Euren Mann leben.
+Doge: Wenn ich mir so in Eure Hände lerne... wenn Ihr mir nun nicht mehr traut.
+Flodoardo:
+Flodoardo: ich schwöre Euch auf Euer Recht. Ihr seid es, Ihr habt es, so will ich es! – Auch Euch ist
+es nicht anders möglich?
+Flodoardo: Nein, Euer Herz ist zu schwach. Es war ja der einzige Grund, Euch hierher zu schicken.
+Doge: ja was hast du da zu tun?
+Flodoardo: Wie?
+Doge: Das ist ganz der Mann in dieser Welt, der ich sein will.
+w TF-IDF outline
+Flodoardo: Er hat den Mann, der ihn umgebracht hat, umgebracht... – O, die Herrschaft hat mehr als
+uns geboten! Wenn er uns nicht den Mann genommen hat, den er umgebracht hat, und den Leichnam
+hat er sich selbst um den Hals geworfen. O nein, man wird uns doch nicht einmal mit dieser Schmach
+zu helfen wagen.
+Flodoardo: Der Mord ist doch nicht eine Sache, die wir vor der Welt tun könnten... man müßte es
+doch wohl auch tun! – und daß er sich hier umgebracht hat! – daß wir uns mit ihm verbanden? – daß
+die Unschuld uns nicht mehr gebiet. – daß er sich auch in dem Mord einen Mord an einem anderen
+schuldig gemacht hat? – er hat uns nicht mit einem Mord gebieten, den der uns nicht mehr wollte. –
+daß der Mörder uns erbarmen hat, ist ein Undank. – weil er keinen mehr in seinem Leben hat? – daß
+er eine Mörderbande an den Mörder hat!
+Flodoardo: Das muß ich sagen... So soll er sich doch nicht schuldig machen... – und daß alle in dem
+Leben eines anderen sind! – warum sollten sie dem anderen noch was zutrauen! – – daß sie uns
+doch nur noch eine Freude machen könnten! – er ist doch gar nicht einmal auf dem Wege! – aber er
+hat doch etwas, was wir uns zutrauen haben! – oder daß er sich von einer Stunde an selber schuldig
+macht... – daß keiner den Mörder hat, der sich ihm den Mord hat nehmen lassen, und der dem Mörder
+noch ein Mensch ist! – Und dann wird er sich in die Arme von diesem Mörder gelegt haben. – dass
+wir nichts zu tun hätten! – ob wir denn von unserem Mord wüssten?
+Doge: Und daß diese Mörder von ihrem eigenen Blut sein werden? – und daß wir es wirklich nicht
+mehr sein können, dass er uns die Schuld gibt! – weil uns das Gesetz sagt. – und daß ich den Mord
+nicht will! – – und daß wir uns das nicht selber antun! – daß er uns nicht will! ? – dass er den Mord
+nicht kann, wenn wir ihn nicht kriegen! – dass sein Blut nicht weiß! – was er uns tut! – denn wenn er
+das nicht kann! – uns beide! – in den Tod! – wenn er sich selbst umgebracht hat! – nicht mit dem
+Blut der Kinder? – wenn sein Herz nicht in dem Blute ist! – und wir!
+Continuation of Table A2. Example texts generated by the different models.
+Part 2/3. Continued on the next page.
+
+id_ger000200_4.txtr
+START_OF_SCENE
+Kauz: Schön gute Morgen, Frau Nièce!
+Frau Von Erbsenstein: Der Morgen kann gut und schön sein, ich bin aber bös und wild!
+Kauz: Bös, das kann sein, aber wild –? Im Gegenteil, ich finde, daß dieser Anzug –
+Frau Von Erbsenstein:
+w/o outline
+Frau Von Erbsenstein: Er ist noch einmal ein Anderer – er ist ein besserer Herr als du.
+Kauz: Ist er aber nicht! Ich bin nicht in der Stimmung, daß ich mir einbilde. – Die Nacht ist schön,
+da kommt es mir vor, als sollte ich schlafen.
+Kauz: Du, Marie, wie ist das möglich?
+Kauz: Ich liebe Sie, aber nicht so sehr, als wenn ich mit Ihnen schlafen wollte. Der Herr von
+Boskowitz ist ein Mann, was weiß ich! Und ich soll mit Ihnen schlafen?
+Kauz: Ich werde heiraten.
+Kauz: Das kann ich nicht sagen. Ich habe noch nie gesagt, daß ich so jung heiraten will.
+Marie – Ich auch nicht, aber ich sage es ganz offen: ich will nicht verheiratet sein.
+w extracted outline
+Frau Von Erbsenstein: Es is wieder Zeit, das Gute zu sagen, das zu sagen, was zu sagen – wie ich
+sagen will.
+Kauz:
+Frau Von Erbsensteins Sohn: Ja, richtig, das weiß man im übrigen schon, den Onkel, der hat auch
+den Titel schon bekommen.
+Herr von Erbsenstein.
+Kauz: Keizer.
+Kauz: »Wild«: Ach! Frau von Erbsensteins Tochter! Keizers Tochter?
+Kauz: Nun, dann wäre die Zeit gekommen, daß ich auf die andere Seite hinübergehe, und michFrau
+Von Rätzer, Frau Von Erbsen! ...
+Kauz: Ich bin ja schon so früh dran, die Nacht werde ich nicht vergessen, wenn ich mich schon auf
+die Seite setze.
+Kauz: Das geht nicht, Frau Von Rätzern...
+Kauz: Ja so. Keizer!
+FrauVon Erbsenstein! Kiezer! Kiezer?
+Kauz: Da. Kriezer! Da sind ja die anderen. Frau Und da sind es die anderen.
+Kauz: Kaizer! Was geht uns das alles an!
+w Textrank outline
+Frau Von Erbsenstein: Ich glaube sie hat ihn ganz richtig, Frau Von Erbsensteins –
+Kauz: Ja so, ja so.
+Kauz: Die ist ja ganz schwarz und weiß wie die Haare von Hunden! Aber ich finde die Haare doch
+noch so dunkel! Ich muß mal schaun! Brauchst dich nicht zu drehen, Frau Von Zedlitz, das hat nix
+mit mir zu tun. Der Herr hat gesagt es wäre ja ein kleiner Mensch von dem ich glaube ich muß es
+doch nicht sein? Der Herr hat denn auch gesagt das ist nichts als ein ungebärdriges Ding, das kann
+doch ein nicht wahr! Das heißt, du meinst Herr von Zedlitz? Ja so – was denn? Das is ein wenig zu
+viel der Anstand und wie viel Herr von Zadl ist, aber das is eine andere Geschichte. Aber ich habe
+ja immer gesagt das is was für die Leute –
+Kauz: die haben gesagt das is nix von Mensch! – Das muß doch net anders sein als mit mir – Was is
+denn? Bös! Der Herr kommt schon zurück. Ich hab es ja schon immer gesagt das war was er! Ich
+habe das so satt mit Herrn von Zedl. Ja das is halt mein Herr! So! Du gehst doch zum Teufel, ob ich
+mich net bewerbe! Da kann ich nicht widersprechen!
+Kauz: Also du? Na wenn du meinst nicht? Du gehst ja! der is nur ’n bißchen auf deine Finger
+gekriegt! Bösl! Der Bursche is ja schon wieder! Wenn du dir nachher den Teufel zum Maul hält,
+wie die anderen. Ich glaube du solltest der Kerl sein den ich mir ausgedacht habe!
+Kauz: Das is wirklich gut!
+w TF-IDF outline
+Frau Von Erbsenstein: Ich muß mit Ihnen ehrlich und herzlich über unsere Lage sprechen, wir
+habens immer vermieden, daß es zu einer Entscheidung kommt. Jetzt weiß ich schon, daß der Alte
+nicht ein ehrlicher Mensch ist, sondern ein Schuft! Was ist die Frau? Ich bin auch da! Frau von
+Erbsenstein, Sie müssen kommen! »Ein Herr hat mir erzählt, daß Sie in Amerika sind, wie Sie
+so schön heißen? »Ein junger Herr aus der Stadt ist hier von der Börse gegangen. Und ein so ein
+alter Mann wie Sie – wie? ...« Im Gegenteil sind wir doch alle drei ein hübsches Paar! Frau... Im
+Gegenteil!
+Kauz: Und die Herrn! Sind drei Jahre lang im Ausland – das freut mich! Frau Soehne! Frau! ! Was
+sagen Sie? Haben Sie denn das ganze Jahr wohl noch nicht bemerkt? Es ist doch was anderes! Ein
+Mensch! »Was steht drin?« Haben Sie das ganze Jahr noch nicht bemerkt. »Was wissen Sie noch
+von dem Geld? Ich hätte es Ihnen sagen müssen, daß ich den alten Herrn aus Amerika bin, den ich
+noch nicht kenne. »Und nun? »Was tun Sie, Frau?« »Was geht mich das an?« »Mir bleibt nichts
+erspart, ich kann jetzt nur lachen! »Wie steht das Geld?« »Was sagen Sie, Frau, Sie haben es wohl
+nicht gelesen?« »Woher »?«« »Was verstehen Sie von diesen Büchern?« »Warum »Warum nicht
+gelesen? Warum nicht gelesen, und nicht gelesen?«. »Was geht mich denn das an? Lesen Sie sich
+das alles! Kaufen Sie sich über die Bücher? »Ich lese alles was sie mir so sagen, aber ich bin ja eine
+ganz vernünftige Frau!« »Was steht darin geschrieben?« Ich bin gar nicht zufrieden! »Und was steht
+drin? »Wie stehen Sie da?« »Wer weiß, was Sie über die Bücher gelesen haben.« »Warum nicht
+die Menschen?« Und mit »Der Mensch! Der Mensch!« »Der Mensch? »Den Menschen?« »Den
+Mensch! Sie können lachen! Haben Sie sich je an mich geliehen? »Haben Sie Ihr Buch im Ausland
+gelesen? »Hat er das Buch geschrieben?« »Hat er das Bücher geschrieben, Herr Professor? »Und
+hat er das Buch gelesen?« Wie gesagt!
+Continuation of Table A2. Example texts generated by the different models.
+Part 3/3
+
+Herr Scheinfromm.
+Madam Glaubeleichten, was sagen sie?
+Frau Glaubeleichten.
+Die Wiedergeburt ist das süße Quell-Wasser des Herzens sage ich, welches aus der Sophia urständet,
+und das himmlische Weltwesen gebühret.
+Herr Scheinfromm,
+( nachdenklich.)
+Das süß - - se Quell - Was - - sehr des - - Her - - sens - - das ist ziemlich deutlich. Wel - - ches - - aus -
+- der - - So - - phi - - a - - ur - - stän - - det, - - und - - das - - himm - - li - - sche - - Welt - - wesen ge - -
+bieh - - ret. Das ist sehr schön und deutlich erklärt. Und sie Madame?
+Frau Zanckenheimin.
+Ich sage, es ist die Erbohrenwerdung der himmlischen Wesenheit aus der Selbstheit der animalischen
+Seele in dem Centro des irdischen Menschen, und windet sich einwärts wie ein Rad.
+Herr Scheinfromm.
+Die - - Er - - boh - - ren - - wer - - dung - - der - - hemm - - li - - schen - - We - - sein - - heit - - In
+Wahrheit! das ist sehr schön gesagt! Und sie Madame?
+Frau Seufzerin.
+Es ist eine himmlische Tinktur, wodurch die neue Seele das vegetabilische Leben der vier Elementen
+wegwirft, und die magische Seele, als die
+Gottheit in seiner Gleichheit, nach dem Modell der Weisheit in alle Dinge einbildet.
+Herr Scheinfromm.
+Pots tausend! das ist hoch! Eine himmlische Tinktur, wodurch die vegetabilische Seele - - -
+Frau Seufzerin.
+Nein! die neue Seele - - -
+Herr Scheinfromm.
+Schon gut! es ist einerlei. Aber die Erklärung gefällt mir sehr.
+Figure A1: Excerpt from the drama ’Die Pietisterey im Fischbein Rocke’(1736) by Luise Adelgunde Victorie
+Gottsched, which was one of the dramas used for training.
+

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+ 
+ 
+ 
+Wealth Redistribution and Mutual Aid: Comparison 
+using Equivalent/Nonequivalent Exchange Models 
+of Econophysics 
+Takeshi Kato 1,* 
+1 Hitachi Kyoto University Laboratory, Open Innovation Institute, Kyoto University, Kyoto 606-8501, Japan 
+* Correspondence: [email protected] 
+Abstract: Given the wealth inequality worldwide, there is an urgent need to identify the mode of 
+wealth exchange through which it arises. To address the research gap regarding models that 
+combine equivalent exchange and redistribution, this study compares an equivalent market 
+exchange with redistribution based on power centers and a nonequivalent exchange with mutual 
+aid using the Polanyi, Graeber, and Karatani modes of exchange. Two new exchange models based 
+on multi-agent interactions are reconstructed following an econophysics approach for evaluating 
+the Gini index (inequality) and total exchange (economic flow). Exchange simulations indicate that 
+the evaluation parameter of the total exchange divided by the Gini index can be expressed by the 
+same saturated curvilinear approximate equation using the wealth transfer rate and time period of 
+redistribution and the surplus contribution rate of the wealthy and the saving rate. However, 
+considering the coercion of taxes and its associated costs and independence based on the morality 
+of mutual aid, a nonequivalent exchange without return obligation is preferred. This is oriented 
+toward Graeber's baseline communism and Karatani's mode of exchange D, with implications for 
+alternatives to the capitalist economy. 
+Keywords: inequalities and wealth redistribution; taxes and redistribution; mutual aid; equivalent 
+exchange; nonequivalent exchange; markets; economic flow; econophysics;  
+ 
+1. Introduction 
+Wealth inequality is a major social problem in various countries worldwide [1]. 
+Survey results indicate that the global Gini index is approximately 0.7, indicating 
+widespread inequality [2]. A Gini index of 0.4 represents a warning level for social unrest 
+[3], and some countries far exceed this level, including South Africa, Namibia, and 
+Suriname [4]. Higher social unrest leads to lower production and further inequality, 
+which in turn leads to social unrest again, creating a vicious cycle [5]. 
+The United Nations Sustainable Development Goals has prioritized Goal 10—which 
+aims to “reduce income inequalities,” “promote universal social, economic and political 
+inclusion,” and “adopt fiscal and social policies that promote equality,” among other 
+targets—along with Goals 1, 2, 8, and 16 (no poverty, zero hunger, inclusive economic 
+growth, and justice and inclusive institutions, respectively) [6]. Moreover, the United 
+Nations University studies the impact of inequality on economic growth, human 
+development, and governance, with inequality as a core concern [7]. Thus, there is a need 
+to identify which types of economic relations—that is, which modes of exchange of 
+wealth—result in inequality. For this, different definitions of the modes of exchange must 
+be considered. 
+The economist Polanyi identified three modes of economic exchange: reciprocity, 
+redistribution, and market exchange [8]. Reciprocity includes the transfer of goods 
+
+ 
+2 of 17 
+ 
+ 
+through gifts with the obligation to provide returns in nonhierarchical relationships; 
+redistribution indicates the transfer of goods through the collection and refund of taxes 
+based on the centrality of power; and market exchange represents the equivalent transfer 
+of goods based on money prices in the market. In other words, reciprocity is a 
+nonequivalent exchange with the obligation to return, market is an equivalent exchange, 
+and redistribution is an equivalent exchange coordinated by a power center. 
+The anthropologist Graeber presented baseline communism, exchange, and 
+hierarchy as the three moral principles of economic relations [9]. Baseline communism is 
+a mutual-aid human relationship wherein each person contributes based on their ability 
+and is provided a return according to need; exchange is a process toward equivalence, an 
+inhuman relationship that can be dissolved through profit and loss; and hierarchy 
+represents a relationship bound and controlled by custom and precedent, with no 
+tendency to operate through reciprocity. Therefore, baseline communism is a 
+nonequivalent exchange without the obligation to return, exchange is an exactly 
+equivalent exchange, and hierarchy is a specific form of redistribution with tribute 
+imposed on proteges and alms posed as protection of a power center. 
+The philosopher Karatani presented four modes of exchange as the various stages of 
+the world system [10]. Modes of exchange A, B, C, and D represent reciprocity in civil 
+society (gift and return), plunder and redistribution in the empire (domination and 
+protection), commodity exchange in the capitalist economy (money and commodities), 
+and restoration of the reciprocal-mutual aid relationship of A to a higher level in the world 
+republic idealized by Kant, respectively. Mode of exchange A is thus a nonequivalent 
+exchange with the obligation to return, B is a form of redistribution, C is an equivalent 
+exchange, and D is a nonequivalent exchange without the obligation to return. 
+A comparison of the three typologies above show that the following definitions 
+correspond with each other, as shown in Table 1: Polanyi’s reciprocity and Karatani’s 
+mode of exchange A with nonequivalence and return; Polanyi’s redistribution, Graeber's 
+hierarchy, and Karatani’s mode of exchange B with centrality of power; Polanyi’s market 
+exchange, Graeber's exchange, and Karatani’s mode of exchange C with equivalence; and 
+Graeber's baseline communism and Karatani’s mode of exchange D with nonequivalence 
+and without return.  
+Table 1. Comparison of economic typologies by Polanyi, Graeber and Karatani. 
+Typology 
+Polanyi 
+Graeber 
+Karatani 
+Nonequivalent exchange 
+with obligation to return 
+Reciprocity 
+— 
+Mode of exchange A 
+Redistribution 
+by power center 
+Redistribution 
+Hierarchy 
+B 
+Equivalent exchange 
+in market 
+Market exchange 
+Exchange 
+C 
+Nonequivalent exchange 
+without obligation to return 
+— 
+Baseline communism 
+D 
+The contemporary capitalist economy and social security comprise a hybrid of 
+equivalent market exchange (C) and redistribution by power center (B). In contrast, 
+alternatives to the capitalist economy can be considered as a mutual-aid baseline 
+communism and mode of exchange D, which sublimates mode of exchange A. Therefore, 
+it is necessary to identify whether a hybrid of equivalent exchange and redistribution (B 
+and C) or that of a mutual-aid nonequivalent exchange without obligation to return (D) 
+would be preferable to suppress wealth inequality. 
+Econophysics uses a statistical physics approach for examining wealth exchange and 
+distribution and the mechanisms of redistribution (see, for example, the comprehensive 
+reviews by Chakrabarti A. S. and Chakrabarti B. K., Rosser, and Ribeiro, respectively [11–
+13]). Champernowne explained Pareto's law based on time series changes in income 
+
+ 
+3 of 17 
+ 
+ 
+distribution through stochastic processes [14]. Sociologist Angle showed that the gamma 
+distribution arises through economic agents’ stochastic processes [15]. Furthermore, 
+Dragulescu and Yakovenko illustrated that the monetary distribution follows an 
+exponential Boltzmann–Gibbs distribution based on the analogy of energy conservation 
+[16], and Chakraborti and Hayes demonstrated that a delta distribution arises when 
+applying a model of random wealth transfer to one of the poor and the wealthy based on 
+the analogy of kinetic energy exchange in collisions of ideal gas particles [17,18]. 
+Chatterjee and Chakrabarti extended these models and showed that an exponential 
+distribution can be obtained using a model in which wealth is randomly divided among 
+agents [19]. Furthermore, Chakraborti and Chakrabarti indicated that gamma and power 
+distributions can be obtained using a model in which agents follow a nonequivalent 
+exchange except for savings [20]. Kato et al. showed that a delta distribution can be 
+obtained using a model in which wealth is exchanged equivalently according to the poor 
+[21]. In addition, Guala used a nonequivalent exchange model combining exchange and 
+tax redistribution for obtaining exponential and gamma distributions based on the tax rate 
+[22], and Chakrabarti A. S. and Chakrabarti B. K. used a model combining nonequivalent 
+exchange and redistribution by insurance to obtain insurance rate-based exponential, 
+gamma, and delta distributions [23].  
+Furthermore, Kato and Hiroi used a nonequivalent exchange model in which the 
+wealthy contribute surplus stock to obtain delta and gamma-like distributions based on 
+the contribution rate; they showed that the contribution of surplus stock by the wealthy 
+is necessary for activating economic flow and reducing inequality [24]. Kato further used 
+an exchange model combining interest, profit and loss, and redistribution to obtain delta 
+and gamma-like distributions and demonstrated that the prohibition of interest, fair 
+distribution of profit and loss, and redistribution based to the quintile axiom in welfare 
+economics are required for reducing inequality [25].  
+Elsewhere, Iglesias showed that inequality, as measured by the Gini index, is 
+dramatically reduced by extremally modeling the collection of a tax proportional to the 
+wealth difference from local or global agents around the poorest agent and the 
+redistribution of the tax to the poorest agent [26]. Moreover, Lima et al. showed that a 
+combination of win/lose equivalent transaction based on the wealth of the poor, power-
+law tax that is more burdensome on the wealthy, and tax exemption for the poor can result 
+in bimodal or flat wealth distributions, and that tax exemptions do not necessarily reduce 
+inequality, as assessed using the Gini index [27]. These Iglesias and Lima models are 
+effectively nonequivalent exchanges, since each exchange is taxed according to wealth. 
+The abovementioned studies do not use models that combine an equivalent exchange 
+with a redistribution separated from it by a certain time period, however. In this study, I 
+aim to reconstruct an exchange model that represents a hybrid of equivalent exchange 
+and redistribution (modes of exchange B and C) and a mutual-aid nonequivalent 
+exchange without obligation to return (mode D) based on the abovementioned exchange 
+model of econophysics. I also compare redistribution and mutual aid in terms of wealth 
+distribution, inequality, and economic flow to provide guidelines for alternative 
+capitalism. This study is novel in that it compares redistribution with mutual-aid 
+nonequivalent exchange. Furthermore, it describes new relationships between the 
+following phenomena: economic flow and inequality; wealth transfer, time period, and 
+redistribution; and surplus contribution of the wealthy, saving, and mutual aid. Based on 
+the comparison, I provide new insights into alternatives to the capitalist economy. 
+In the present model (hybrid of equivalent exchange and redistribution), I combine 
+the equivalent exchange model of Kato et al. [21] to represent Polanyi's market exchange, 
+Graeber's exchange, and Karatani's mode of exchange C and Kato's redistribution model 
+[25] to represent Polanyi's redistribution, Graeber's hierarchy, and Karatani's mode of 
+exchange B. To model mutual-aid nonequivalent exchange, I adopt Kato and Hiroi's 
+surplus stock contribution model [24] to represent Graeber's baseline communism and 
+
+ 
+4 of 17 
+ 
+ 
+Karatani's mode of exchange D, repositioning the surplus stock contribution of the 
+wealthy as a mutual aid without the obligation of return.  
+The remainder of this paper is organized as follows. The Methods section (Section 2) 
+presents models of equivalent exchange and redistribution and mutual-aid nonequivalent 
+exchange models, as well as the methods for calculating the Gini index and total exchange 
+to assess wealth inequality and economic flow. The Results section (Section 3) compares 
+the simulation results of wealth distributions and the Gini index and total exchange 
+calculations for the two models to illustrate their relationship. The Discussion section 
+(Section 4) examines the contemporary significance of mutual aid for redistribution 
+considering these results and presents discussions on the nature of mutual aid for 
+alternatives to the capitalist economy. The Conclusion section (Section 5) presents the key 
+conclusions and future challenges. 
+2. Methods 
+2.1. Exchange Models 
+Figure 1 visualizes different exchange models that can be used to measure and 
+understand inequality, to be explained in detail below. 
+Figure 1. Exchange models. 𝑚𝑖 and 𝑚𝑗 represent the wealth of agents 𝑖 and 𝑗, respectively, at 
+times 𝑡, 𝑡 + 1 and 𝑡 + ∆. 𝜆 represents the common savings rate, and 𝜀 represents the random 
+division probability. (a) Basic exchange model; (b) Equivalent exchange model (EX) with 
+redistribution rate 𝜉 and time period 𝑡𝑝; (c) Nonequivalent exchange model (NX) with surplus 
+contribution rate 𝛾. 
+2.1.1 Basic Exchange Model 
+First, I present the basic wealth exchange model proposed by Chakraborti and 
+Chakrabarti [20]. Two agents 𝑖,𝑗 (= 1,2, ⋯,𝑁) are selected randomly from among 𝑁 
+economic agents. Let the wealth of agents 𝑖  and 𝑗 at time 𝑡 be 𝑚𝑖(𝑡) and 𝑚𝑗(𝑡), 
+respectively, with a common saving rate 𝜆 for both. Figure 1 (a) shows that the two 
+agents 𝑖 and 𝑗 save part of their wealth at time 𝑡 with a savings rate 𝜆 and exchange 
+the remaining wealth (1 − 𝜆) ∙ (𝑚𝑖(𝑡) + 𝑚𝑗(𝑡)) , excluding savings, with a random 
+division probability 𝜀, which is a uniform random number defined in the range 0 ≤ 𝜀 ≤
+1. This basic model is a nonequivalent exchange model wherein the poor and the wealthy 
+
+(a) Basic exchange model
+(c) NX model (nonequivalent exchange with mutual aid)
+(1 - ) · y · (mj - mi)
+(1 - ^) ·mj
+(1 - ) ·mi
+(1 - ) : mi
+(1 -) · mi
+^· mj
+· mj
+A·mi
+入·mi
+m;(t)
+m;(t)
+m;(t + 1) m;(t + 1)
+(b)EXmodel(equivalentexchangewithredistribution)
+tp period
+(1 - ) · mi
+(1 -~) · mi
+Λ·mi
+Λ· mj
+m;(t + 1) m;(t + 1)
+m,(t + ) mz(t + △)
+m;(t + △)
+m;(t + △)
+mv(t + △) 
+5 of 17 
+ 
+ 
+offer all their wealth (except their savings) in exchange. The wealth 𝑚𝑖(𝑡 + 1) and 𝑚𝑗(𝑡+
+1) of the two agents 𝑖 and 𝑗, respectively, at time 𝑡 + 1 are expressed as  
+𝑚𝑖(𝑡 + 1) = 𝜆 ∙ 𝑚𝑖(𝑡) + 𝜀 ∙ (1 − 𝜆) ∙ (𝑚𝑖(𝑡) + 𝑚𝑗(𝑡));  
+(1a) 
+𝑚𝑗(𝑡+ 1) = 𝜆 ∙ 𝑚𝑗(𝑡) + (1 − 𝜀) ∙ (1 − 𝜆) ∙ (𝑚𝑖(𝑡) + 𝑚𝑗(𝑡)).  
+(1b) 
+2.1.2. Equivalent Exchange Model 
+The equivalent exchange model that matches the wealth of the poor (hereafter, the 
+EX model) proposed by Kato et al. [21] is based on the nonequivalent exchange model, as 
+presented in Equations (1a) and (1b). As indicated in Figure 1 (b), the EX model 
+determines the amount of exchange based on the wealth Min(𝑚𝑖(𝑡), 𝑚𝑗(𝑡)) of the poorer 
+of the two agents, 𝑖 and 𝑗. The exchange amount presented by the wealthy and the poor 
+is exchanged with a random division probability 𝜀, which is a uniform random number 
+in the range 0 ≤ 𝜀 ≤ 1. Wealth 𝑚𝑖(𝑡+ 1) and 𝑚𝑗(𝑡+ 1) at time 𝑡 + 1 are expressed as  
+𝑚𝑖𝑛 = Min(𝑚𝑖(𝑡), 𝑚𝑗(𝑡)),   
+(2a) 
+𝑚𝑖(𝑡 + 1) = 𝑚𝑖(𝑡) − (1 − 𝜆) ∙ 𝑚𝑖𝑛 + 2 ∙ 𝜀 ∙ (1 − 𝜆) ∙ 𝑚𝑖𝑛; 
+(2b) 
+𝑚𝑗(𝑡 + 1) = 𝑚𝑗(𝑡) − (1 − 𝜆) ∙ 𝑚𝑖𝑛 + 2 ∙ (1 − 𝜀) ∙ (1 − 𝜆) ∙ 𝑚𝑖𝑛.  
+(2c) 
+Repeating the exchange process in the EX model yields a delta distribution in which 
+all wealth is concentrated in one agent’s hands, as shown in the literature [21]. 
+Furthermore, in the EX model, redistribution is newly combined with the equivalent 
+exchange shown in Equations (2a)–(2c). For the redistribution, I use the model proposed 
+by Kato [25]. In this model, the wealth transfer rate 𝜉  and the time period 𝑡𝑝 for 
+redistribution are set, and 𝑁  agents simultaneously distribute the wealth 𝜉 ∙ 𝑚𝑖(𝑡) 
+corresponding to the transfer rate 𝜉 to all others equally in every period 𝑡𝑝 (Figure 1 (b)). 
+This is because establishing an average period and an average amount of redistribution 
+when assessing the effectiveness of redistribution in reducing inequality is considered 
+sufficient. The wealth 𝑚𝑖(𝑡 + ∆) of agent 𝑖 at time 𝑡 + ∆ immediately after period 𝑡𝑝 is 
+expressed as  
+𝑚𝑖(𝑡 + ∆) = (1 − 𝜉) ∙ 𝑚𝑖(𝑡) + 𝜉 ∙ ∑
+𝑚𝑗(𝑡)
+𝑗≠𝑖
+𝑁 − 1
+.  
+(3) 
+2.1.3 Nonequivalent Exchange Model 
+I use the model proposed by Kato and Hiroi [24] as a mutual-aid nonequivalent 
+exchange model without obligation to return (hereafter, the NX model). The NX model is 
+a compromise between the nonequivalent and equivalent exchange models presented in 
+Equations (1a), (1b) and (2a)–(2c), respectively. In the first, the wealthy contribute all 
+surplus wealth except savings, which is not realistic in exchange, that is, economic 
+transactions. In the second, the wealthy only contribute wealth equivalent to that of the 
+poor; in the absence of redistribution, extreme inequality, such as a delta distribution, is 
+likely. Thus, Kato and Hiroi set up a model in which the wealthy contribute a portion of 
+their surplus wealth over that of the poor to control inequality to a practical extent. 
+As shown in Figure 1 (c), in the NX model, the wealth of the poor and the wealthy 
+are 𝑚𝑖𝑛 = Min(𝑚𝑖(𝑡), 𝑚𝑗(𝑡))  and 𝑚𝑎𝑥 = Max (𝑚𝑖(𝑡), 𝑚𝑗(𝑡)) , respectively; the poor 
+take surplus wealth (1 − 𝜆) ∙ 𝑚𝑖𝑛 as the exchange amount. The wealthy’s exchange 
+amount is the wealth (1 − 𝜆) ∙ (𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛)); this is the sum of the poor’s 
+surplus wealth (1 − 𝜆) ∙ 𝑚𝑖𝑛  and the wealth (1 − 𝜆) ∙ 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛) , which is the 
+amount of the wealthy’s surplus wealth (1 − 𝜆) ∙ 𝑚𝑎𝑥 less (1 − 𝜆) ∙ 𝑚𝑖𝑛 multiplied by 
+the surplus contribution rate γ.  
+
+ 
+6 of 17 
+ 
+ 
+The poor and wealthy then exchange the amounts mutually proposed with a random 
+division probability 𝜀, which is a uniform random number defined in the range 0 ≤ 𝜀 ≤
+1. Graeber’s baseline communism is a mutual-aid relationship in which each person 
+contributes based on their ability and each person is given according to their need, 
+without obligation to return. Although the contribution of surplus wealth from the 
+wealthy to the poor inherently varies based on need, the surplus contribution rate 𝛾 is 
+set as a constant parameter to observe the general trends. The wealth 𝑚𝑖(𝑡 + 1) and 
+𝑚𝑗(𝑡+ 1) of two agents 𝑖 and 𝑗, respectively, are expressed as  
+𝑚𝑖𝑛 = Min(𝑚𝑖(𝑡), 𝑚𝑗(𝑡)),   
+(4a) 
+𝑚𝑎𝑥 = Max (𝑚𝑖(𝑡), 𝑚𝑗(𝑡)),  
+(4b) 
+𝑖𝑓   𝑚𝑖(𝑡+ 1) ≤ 𝑚𝑗(𝑡 + 1),  
+𝑚𝑖(𝑡 + 1) = 𝑚𝑖(𝑡) − (1 − 𝜆) ∙ 𝑚𝑖𝑛  
++𝜀 ∙ (1 − 𝜆) ∙ (2 ∙ 𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛));  
+(4c) 
+𝑚𝑗(𝑡 + 1) = 𝑚𝑗(𝑡) − (1 − 𝜆) ∙ (𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛))   
+ +(1 − 𝜀) ∙ (1 − 𝜆) ∙ (2 ∙ 𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛)).  
+(4d) 
+𝑖𝑓   𝑚𝑖(𝑡+ 1) > 𝑚𝑗(𝑡 + 1),  
+𝑚𝑖(𝑡 + 1) = 𝑚𝑖(𝑡) − (1 − 𝜆) ∙ (𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛))  
++𝜀 ∙ (1 − 𝜆) ∙ (2 ∙ 𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛)); 
+(4e) 
+����𝑗(𝑡 + 1) = 𝑚𝑗(𝑡) − (1 − 𝜆) ∙ 𝑚𝑖𝑛  
+ +(1 − 𝜀) ∙ (1 − 𝜆) ∙ (2 ∙ 𝑚𝑖𝑛 + 𝛾 ∙ (𝑚𝑎𝑥 − 𝑚𝑖𝑛)).  
+(4f) 
+The NX model equals the nonequivalent exchange model shown in Equations (1a) and 
+(1b) when the surplus contribution rate 𝛾 = 1 and the equivalent exchange model 
+shown in Equations (2a)–(2c) when 𝛾 = 0. 
+2.2. Evaluation Indices 
+2.2.1 Gini Index 
+The Gini index 𝑔, used as a parameter for evaluating wealth inequality [28], is 
+obtained by drawing the Lorenz curve and equal distribution line [29]. Various proposed 
+inequality indices are calculated from Lorenz curves [30], but the Gini index is used here 
+because it is most common. Mathematically, the wealth 𝑚𝑖(𝑡) of the 𝑁 agents at time 𝑡 
+is ordered from the smallest to the largest, and the Gini index 𝑔 is calculated as  
+𝑟𝑖(𝑡) = Sort(𝑚𝑖(𝑡)), 
+(5a) 
+𝑔 = 2 ∙ ∑
+𝑖 ∙
+𝑁
+𝑖=1
+𝑟𝑖(𝑡)
+𝑁 ∙ ∑
+𝑟𝑖(𝑡)
+𝑁
+𝑖=1
+− 𝑁 + 1
+𝑁
+. 
+(5b) 
+When the wealth of 𝑁 agents is perfectly equal (uniform distribution), the Gini index 
+𝑔 = 0; when all wealth is concentrated in a single agent’s hands (delta distribution), 𝑔 =
+1. In other words, 𝑔 ranges from 0 to 1. The greater the inequality, the larger the value 
+of 𝑔. 
+ 
+
+ 
+7 of 17 
+ 
+ 
+2.2.2 Total Exchange 
+The total exchange amount 𝑓 is used to evaluate economic flow [24]. The total 
+exchange 𝑓 is the sum of the exchanges of the wealthy and poor (1 − 𝜆) ∙ (2 ∙ min (𝑡) +
+𝛾 ∙ (𝑚𝑎𝑥(𝑡) − 𝑚𝑖𝑛(𝑡))) at time 𝑡 from time 𝑡 = 1 to 𝑡 = 𝑡𝑚𝑎𝑥.  
+𝑓 =
+∑
+(1 − 𝜆) ∙ (2 ∙ 𝑚𝑖𝑛(𝑡) + 𝛾 ∙ (𝑚𝑎𝑥(𝑡) − 𝑚𝑖𝑛(𝑡)))
+𝑡𝑚𝑎𝑥
+𝑡=1
+2 ∙ 𝑡𝑚𝑎𝑥
+. 
+(6) 
+Furthermore, Equation (6) applies to Equations (2a)–(2c) if 𝛾 = 1. The denominator 
+in Equation (6), intended for normalization, is the total amount exchanged between the 
+two agents from time 𝑡 = 1 to 𝑡 = 𝑡𝑚𝑎𝑥, when the two agents exchange one amount each. 
+The larger the total exchange 𝑓, the more active the exchange of wealth, that is, the 
+economic flows are large and the market is active. 
+3. Results 
+I first examine wealth distributions for the EX model of equivalent exchange and 
+redistribution represented by Equations (2a)–(2c) and (3) and the NX model of 
+nonequivalent exchange represented by Equations (4a)–(4f). Figure 2 shows a 
+representative example of the simulated wealth distribution results. I set a savings rate of 
+𝜆 = 0.25 because the average global savings rate relative to the gross domestic product 
+(GDP) is approximately 0.25 [31], and a transfer rate of 𝜉 = 0.5 in the EX model because 
+the highest inheritance tax rate in the Organisation for Economic Co-operation and 
+Development (OECD) countries is approximately 0.5 [32,33].  
+ 
+Figure 2. Wealth distribution. (a1) and (a2) represent EX models, and (b1) and (b2) represent NX 
+models. In all models, the number of agents is 𝑁 = 1,000, the initial values of wealth at time 𝑡 = 0 
+are 𝑚𝑖(0) = 1 (𝑖 =  1, 2,⋯ ,𝑁), and the savings rate is 𝜆 = 0.25. In the EX model, the transfer rate 
+is 𝜉 = 0.5, and the time period is 𝑡𝑝 = 104,105. In the NX model, the surplus contribution rate is 
+𝛾 = 0.1, 0.5 . To determine the changes in wealth distribution, the time (number of exchange 
+repetitions) is 𝑡 = 103, 3 × 103,106. 
+
+(a1) EX model
+(b1) NX model
+N = 1,000,^ = 0.25, = 0.5,t, = 104
+200
+N=1,000,A=0.25.y=0.1
+300
+250
+t = 106
+150
+Frequency
+t = 106
+200
+t = 3 × 103
+t = 3 × 103
+100
+150
+t = 103
+t = 103
+100
+50
+50
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Wealth m; (t)
+Wealth m; (t)
+(a2) EX model
+(b2) NX model
+N = 1,000,^ = 0.25, = 0.5,tp = 105
+200
+N= 1,000,^= 0.25,y= 0.5
+800
+t = 106
+150
+600
+Frequency
+t = 106
+t = 3× 103
+t = 3 × 103
+400
+100
+t = 103
+t = 103
+200
+50
+0
+oF
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Wealth m; (t)
+Wealth m;(t) 
+8 of 17 
+ 
+ 
+A comparison of Figures 2 (a1) and (a2) reveals that as the wealth distribution in the 
+EX model approaches a power distribution, a delta distribution with an increase in the 
+redistribution period 𝑡𝑝 = 104 to 105 occurs—that is, inequality increases. This implies 
+that some form of redistribution must be conducted because only equivalent exchange 
+leads to extreme inequality, as suggested by the literature [21] with respect to regional 
+inequality. A comparison of Figures 2 (b1) and (b2) shows that the wealth distribution 
+approaches a gamma-like distribution from an exponential distribution in the NX model 
+when the wealthy’s surplus contribution rate increases from 𝛾 = 0.1 to 0.5, that is, the 
+inequality narrows. This suggests that inequality can be controlled if considerable mutual 
+aid is provided in a nonequivalent exchange. 
+Next, I examine the change in the Gini index (inequality) 𝑔 over time (number of 
+exchanges) 𝑡 for the EX and NX models by Equations (5a) and (5b). Figure 3 shows the 
+results of these simulations.  
+ 
+Figure 3. Gini index on time passage. The number of agents is 𝑁 = 1,000, initial values of wealth at 
+time 𝑡 = 0 are 𝑚𝑖(0) = 1 (𝑖 =  1,2,⋯ ,𝑁), and the savings rate is 𝜆 = 0.25. In the EX model, the 
+transfer rate is 𝜉 = 0 and 0.5, and the time period is 𝑡𝑝 = 103,104, 105. In the NX model, the 
+surplus contribution rate is 𝛾 = 0, 0.1, 0.5,1. 
+In Figure 3, cases 𝜉 = 0 (i.e., no redistribution in the EX model) and 𝛾 = 0 (i.e., no 
+mutual aid in the NX model) are identical; as time 𝑡 passes, the Gini index approaches 
+𝑔 = 1, and all wealth is concentrated in one agent’s hands. In other words, in an equivalent 
+market exchange, inequality can only be maximized. In the EX model with 𝜉 = 0.5, the 
+redistribution period 𝑡𝑝 = 105 to 103 is shortened. In the NX model, the Gini index 𝑔 
+decreases and inequality is suppressed when the rate of surplus contribution from the rich 
+to the poor increases from 𝛾 = 0 to 𝛾 = 0.5; however, 𝛾 = 0.5 and 𝛾 = 1 show little 
+difference. The reason the Gini index saturates with respect to 𝛾 is presumably because 
+the shape of the Lorenz curve itself, which calculates the Gini index 𝑔, does not change, 
+although the wealthy and poor switch as 𝛾 increases, as discussed in the literature [24] 
+regarding the rank correlation coefficient. 
+In Figures 2 and 3, the savings rate 𝜆 = 0.25  is held constant. Subsequently, I 
+examine the Gini index (inequality) 𝑔 by Equations (5a) and (5b) and total exchange 
+(economic flow) 𝑓  by Equation (6) for the savings rate 𝜆  and the redistribution 
+parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ of the EX model and for the savings rate 𝜆 and the surplus 
+contribution rate (mutual aid) 𝛾 of the NX model. 𝜉 𝑡𝑝 × 10−3
+⁄
+ is introduced because the 
+same inequality suppression effect is expected for an increase in the transfer rate 𝜉 and a 
+decrease in the period 𝑡𝑝; the × 10−3 is used for adjusting the computational orders of 
+magnitude. Figure 4 shows the results of these simulations. The time (number of 
+exchanges) 𝑡 is set to 106, at which the Gini index 𝑔 is almost stable, as shown in Figure 
+3. 
+
+1.0 [
+EX:=0,NX:y=0
+N=1,000,入=0.25
+EX:  = 0.5,tp = 105
+0.8
+6
+EX:  = 0.5,tp = 104
+Gini index g
+0.6
+NX:= 0.1
+NX:Y=0.5
+0.4
+NX: =1
+0.2
+EX:  = 0.5,tp = 103
+103
+104
+105
+106
+Time(exchangecounts)t 
+9 of 17 
+ 
+ 
+ 
+Figure 4. Three-dimensional graphs of Gini index 𝑔 and total exchange 𝑓 for saving rate 𝜆 and 
+redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ or mutual aid 𝛾. (a) EX model and (b) NX model.  
+The comparison of Figures 4 (a) and (b) reveals the same trend for both EX and NX 
+models. In the EX model, the larger the savings rate 𝜆, the smaller is the Gini index 𝑔 
+(inequality is suppressed) and the smaller is the total exchange 𝑓 (economic flow is 
+reduced). Furthermore, the larger the redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+, the smaller 
+is the 𝑔 (inequality is suppressed) but the larger is the total exchange 𝑓 (economic flow 
+is activated). Figure 4 (b) shows that in the NX model, the larger the savings rate 𝜆, the 
+smaller are the 𝑔 (inequality is suppressed) and 𝑓 (economic flow becomes stagnant). 
+Moreover, the larger the mutual aid 𝛾, the smaller is the 𝑔 (inequality is suppressed) and 
+the larger is the 𝑓  (economic flow is activated). In other words, inequality 𝑔 and 
+economic flows 𝑓  are inversely related with respect to the redistribution parameter 
+𝜉 𝑡𝑝 × 10−3
+⁄
+ in the EX model and mutual aid 𝛾 in the NX model. 
+As specific values are difficult to read in Figure 4, I examine the Gini index 𝑔 for the 
+redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ of the EX model and the surplus contribution rate 
+(mutual aid) 𝛾 of the NX model. Figure 5 shows the results of these simulations based on 
+Figures 2–4.  
+Figure 5. Relationship of Gini index 𝑔 for the redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ or mutual aid 
+𝛾. (a) EX model and (b) NX model. In both models, dotted lines represent approximate curves. 
+Figures 5 (a) and (b) show that, as in Figure 4, the larger the redistribution parameter 
+𝜉 𝑡𝑝 × 10−3
+⁄
+ in the EX model and the mutual aid 𝛾 in the NX model, the smaller is the 
+Gini index 𝑔 (inequality is reduced). In addition, both plots are accurately approximated 
+by the saturation curve (dotted line in the figure) because the coefficient of determination 
+𝑅2 is sufficiently large. At a global average savings rate 𝜆 = 0.25 [31], the redistribution 
+parameter and the mutual aid must be as follows: 𝜉 𝑡𝑝 × 10−3
+⁄
+≥ 0.2 in the EX model and 
+𝛾 ≥ 0.2 in the NX model, respectively, to avoid exceeding the warning level 𝑔 = 0.4 [3]. 
+In other words, Figure 5 suggests that without a certain degree of redistribution or mutual 
+
+(a) EX model
+(b) NX model
+1.01
+N=1,000,t=106,a=0.25
+1.0
+N=1,000.t=106a=0.25
+0.8
+0.8
+-
+9
+6
+g~1-0.643
+g~1- 0.595(1-e-19.2y)
+Giniindex
+0.6
+Gini index
+0.6
+R2=0.994
+R²=0.995
+0.4
+0.4
+0.2
+0.2
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+5
+Redistribution
+Mutual aid y
+tp×10-3(a) EX model
+(b) NX model
+N=1,000,t=106
+Totalexchangef
+Totalexchangef
+1.0
+1.0
+1.0
+1.0
+Value0.5
+Gini index g
+Value 0.5
+0.0
+0.0F
+Apie
+0.0
+0.5
+0.0
+0.5
+Redistribution
+Mutual
+0.5
+0.5
+Saving^
+Saving^
+0.0
+0.0
+1.0
+1.0 
+10 of 17 
+ 
+ 
+aid, social unrest and disturbance will be triggered and Goal 10 of the Sustainable 
+Development Goals to reduce wealth inequality [6] will not be achieved. 
+Finally, based on the inversely proportional relationship between the Gini index 𝑔 
+and the total exchange 𝑓 in Figure 4, I introduce the parameter 𝑓 𝑔
+⁄ . Then, I examine the 
+relationship of 𝑓 𝑔
+⁄  to the redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ in the EX model and to 
+the parameter (1 − 𝜆) ∙ 𝛾, comprising the savings rate 𝜆 and the surplus contribution 
+rate (mutual aid) 𝛾, in the NX model. I introduce the parameter (1 − 𝜆) ∙ 𝛾 in the NX 
+model because reducing the savings rate 𝜆 and increasing the surplus contribution rate 
+𝛾 are believed to increase the unitary exchange and mutual aid per exchange. In contrast, 
+in the EX model, the transfer rate 𝜉 is multiplied by the entire wealth, including savings, 
+in every period 𝑡𝑝; thus, the effect of redistribution is considered independent of the 
+savings rate 𝜆. Figure 6 shows these simulation results.  
+Figure 6. Relationship of the 𝑓 𝑔
+⁄  parameter for the redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ or 
+mutual aid (1 − 𝜆) ∙ 𝛾. (a) EX model and (b) NX model. In both models, dotted lines represent 
+approximate curves. 
+Figures 6 (a) and (b) show that the parameter 𝑓 𝑔
+⁄  increases as 𝜉 𝑡𝑝 × 10−3
+⁄
+ and 
+(1 − 𝜆) ∙ 𝛾 are increased for the EX and NX models, respectively. Furthermore, both plots 
+are accurately approximated by the saturation curves (dotted lines in the figure) because 
+the coefficient of determination 𝑅2 are larger than 0.9. The EX and NX models yield 
+𝑓 𝑔
+⁄ ~0.241 ln𝜉 𝑡𝑝 × 10−3
+⁄
++ 1.48 (𝑅2 = 0.779) and 𝑓 𝑔
+⁄ ~0.403 ln(1 − 𝜆) ∙ 𝛾 + 1.92 (𝑅2 =
+0.937), respectively, when approximated by logarithmic curves. The NX model results 
+indicate that the logarithmic curves can be approximated with adequate accuracy, which 
+is consistent with the view in the literature [24]. In Figure 6, I compare the EX and NX 
+models using saturation curves that can be accurately approximated because both have 
+sufficiently large 𝑅2. It is safe to say that both approximations are isomorphic and that  
+𝑓
+𝑔 ~2(1 − 𝑒−5𝑥), 
+(7a) 
+𝑥~
+𝜉
+𝑡𝑝 × 10−3 ~(1 − 𝜆) ∙ 𝛾. 
+(7b) 
+holds. Therefore, the redistribution parameter 𝜉 𝑡𝑝 × 10−3
+⁄
+ in an equivalent exchange 
+and the mutual aid (1 − 𝜆) ∙ 𝛾 that considers savings in a nonequivalent exchange yield 
+roughly the same result with respect to the parameter 𝑓 𝑔
+⁄ . The approximate equations 
+shown in Figure 6 and Equations (7a) and (7b) imply that if the right side has a constant 
+value, the Gini index (inequality) 𝑔 and the total exchange (economic flow) 𝑓 on the left 
+side are inversely proportional, that is, activating economic flow will increase inequality. 
+Additionally, it is necessary to increase 𝜉 𝑡𝑝 × 10−3
+⁄
+ and (1 − 𝜆) ∙ 𝛾 on the right side for 
+the EX and NX models, respectively, to increase 𝑓 𝑔
+⁄  on the left side (i.e., to increase the 
+total exchange 𝑓 while decreasing the Gini index 𝑔). Moreover, redistribution must 
+
+(a) EX model
+(b) NX model
+N = 1,000, t = 106
+N=1,000,t=106
+2.0
+2.0
+1.5
+1.5
+f-g
+f
+92
+1.93(1 - e-5.59(1-)-)
+1.0
+g
+1.0
+6
+R2 = 0.968
+R2 = 0.989
+0.5
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Sr
+(1 - a) ·
+tp × 10-3 
+11 of 17 
+ 
+ 
+either occur with a high transfer rate 𝜉 and a short period 𝑡𝑝 or with a low saving rate 
+𝜆 and considerable mutual aid 𝛾 to simultaneously reduce inequality and stimulate 
+economic flow. 
+The numerical values presented in Figure 6 (a) indicate that the redistribution 
+parameters 𝜉 𝑡𝑝 × 10−3
+⁄
+~1 and 𝑓 𝑔
+⁄ ~2 are at the saturation point of the EX model. At 
+this point, the periods 𝑡𝑝~1,000, 𝑡𝑝~800, and 𝑡𝑝~500 should be set for transfer rates 
+𝜉~1, 𝜉~0.8, and 𝜉~0.5, respectively. Given the results in Figure 3, this is tantamount to 
+redistributing wealth before wealth distribution occurs, which is not realistic. If the target 
+is 𝜉 𝑡𝑝 × 10−3
+⁄
+~0.2, where 𝑓 𝑔
+⁄  does not drop considerably on the saturation curve, 
+𝑡𝑝~5,000 for 𝜉~1, 𝑡𝑝~3,000 for 𝜉~0.6, and 𝑡𝑝~2,000 for 𝜉~0.4; this seems feasible 
+within the range of the latter two, that is, 𝜉~0.5 and 𝑡𝑝~2,500. 
+Based on the numerical values presented in Figure 6 (b), the saturation point of the 
+NX model is (1 − 𝜆) ∙ 𝛾 = 1  and 𝑓 𝑔
+⁄
+~2. The savings rate 𝜆 = 0  and the surplus 
+contribution rate 𝛾 = 1 should be set at this point; however, it is unrealistic for the 
+wealthy to always contribute the entirety of their surplus wealth, and for the poor and the 
+wealthy to always save no wealth, respectively. The latter is because they must save to 
+maintain long-term future reserves and meet contingent expenditures attributable to 
+disasters. If the target is (1 − 𝜆) ∙ 𝛾~0.2, where 𝑓 𝑔
+⁄  does not drop considerably on the 
+saturation curve, 𝛾~1 for 𝜆~0.8, 𝛾~0.33 for 𝜆~0.4, and 𝛾~0.25 for 𝜆~0.2; it would be 
+feasible to achieve 𝜆~0.3 and 𝛾~0.28 within the range of the last two considering the 
+global average savings rate of 0.25 [31]. 
+Figure 7 shows the relationship between redistribution and mutual aid based on 
+Equations (7a) and (7b). The circle represents the tentative target. Lengthening the period 
+of redistribution from 𝑡𝑝 = 2,500  to 5,000  results in a transfer ratio 𝜉~1 , that is, 
+transferring all assets and further lengthening the period would no longer maintain the 
+same 𝑓 𝑔
+⁄  as the mutual aid, and this would lead to economic stagnation or widening 
+inequality. Conversely, if the redistribution period is shortened from 𝑡𝑝 = 2,500  to 
+1,250,625 , the transfer rate decreases to 𝜉~ 0.25, 0.125, which necessitates frequent 
+redistributions. 
+ 
+Figure 7. Relationship between redistribution parameter 𝜉 and mutual aid 𝛾. The savings rate is 
+𝜆 = 0.25, and the time period of redistribution is 𝑡𝑝 = 625,1,250,2,500,5,000. 
+4. Discussion 
+This study compared a model combining equivalent exchange and redistribution 
+(Polanyi's market exchange, Graeber's exchange, and Karatani's mode of exchange C 
+combined with Polanyi's redistribution, Graeber's hierarchy, and Karatani's mode of 
+exchange B) and a mutual-aid nonequivalent exchange model (Graeber's baseline 
+communism and Karatani's mode of exchange D). This comparison reveals that both 
+produce the same computational interpretation of the results for wealth inequality and 
+economic flow. Reducing inequality and stimulating economic flow requires either 
+
+tp = 5,000
+tp = 2,500
+1.0
+tp = 1,250
+Redistribution (transfer rate) 
+0.8
+0.6
+tp = 625
+0.4
+0.2
+入= 0.25
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Mutual aid (surplus contribution rate)  
+12 of 17 
+ 
+ 
+power-centered collection and redistribution at a high tax rate and frequency in an 
+equivalent market exchange or a mutual-aid nonequivalent exchange without obligation 
+of return, in which savings are kept low and the wealthy’s rate of surplus wealth 
+contribution is high. 
+What does the computational similarity of authoritative redistribution and 
+nonauthoritative mutual aid imply? With respect to time 𝑡 in these exchange models, a 
+human lifetime would be considered equivalent to approximately 104  order of 
+magnitude (~365 days x 100 years). Therefore, a redistribution target of 𝜉 𝑡𝑝 × 10−3
+⁄
+~0.2 
+would mean that a tax of ~50% is levied once every few decades on all assets and not the 
+income. The maximum inheritance tax rate in OECD countries (i.e., once in a lifetime) is 
+50% [32,33], which means that collection and redistribution should be conducted more 
+frequently. Expenses to the government are ~30% of GDP [34], and the collection and 
+redistribution of taxes by the power center is extra costly; furthermore, the institutional 
+design creates redistribution bias, that is, inequality.  
+In contrast, a mutual aid target of (1 − 𝜆) ∙ 𝛾~0.2  implies that the wealthy 
+voluntarily give ~30% of their surplus stock to the poor in a single exchange, without 
+obligation to return, assuming an average saving rate 𝜆 = 0.25  [31]. Although a 
+prescribed surplus contribution rate 𝛾  is specified when modeling a mutual-aid 
+nonequivalent exchange, the original baseline communism or mode of exchange D only 
+requires that mutual aid be provided as required. In addition, even if redistribution and 
+mutual aid are “computationally” similar, they are “qualitatively” different in that 
+redistribution is coercion-based and driven by the centrality of power, whereas mutual 
+aid is a voluntary choice based on noncentrism and morality. Extrapersonal altruism and 
+compassion, as opposed to coercion, are believed to result in wellbeing [35]. Therefore, it 
+is evident that a mutual-aid nonequivalent exchange without obligation to return (the 
+alternative human economy) is preferable to redistribution by power centers in an 
+equivalent market exchange (capitalist economy and social security). 
+Here, examining the mechanism of the Islamic economy is instructive. As a legal 
+system, the Islamic economy encompasses politics, economics, and society and prohibits 
+interest (riba) and speculation (gharar), which lead to inequality. Furthermore, it also 
+successfully balances selfishness as the pursuit of self-interest through joint ventures 
+(mudaraba), consensual contracts (murabaha), and futures trading (salam) and altruism as 
+mutual aid through donation (waqf), alms (sadaqah), and charity (zakat) in an equal and 
+noncentered community (ummah) under God [36–38]. Redistribution through various 
+institutions according to the Islamic legal system, rather than coercion by power centers, 
+is more like a nonequivalent exchange of mutual aid. 
+According to Graeber, history over the past five millennia has alternated between 
+cycles of bullion-based monetary economies and virtual money-based credit economies 
+[9]. The monetary economic period is generally characterized by interest-bearing debt, 
+war, and slavery, whereas the credit economic period has witnessed a morally peaceful 
+society. In the Middle Ages, a credit economy era that predated the modern era, moral 
+and financial innovations emerged from the Islamic world. As the modern era transitions 
+from a monetary economy to a credit economy, the Islamic economy could, once again, 
+provide an alternative to the capitalist economy [39–41]. 
+Kato compares the Islamic and capitalist economies from the econophysics 
+perspective; he proposes a return to a “real transaction-based economy” rooted in nature 
+and local communities, the promotion of a “face-to-face association economy,” and the 
+revival of an “economy embedded in the morality of mutual aid” as guidelines for a credit 
+economy as an alternative to capitalism [25]. He then states that the challenge in the non-
+Islamic world lies not in redistribution through taxes collected under centralized power 
+but in mutual aid through one’s free choice under the community’s noncentrality and in 
+the rebuilding of the morality of mutual aid, that is, without a specific religion. 
+These guidelines can be considered to be oriented toward anarchism. Anarchism is 
+an ideology wherein individual freedom and communal solidarity are not contradictory. 
+
+ 
+13 of 17 
+ 
+ 
+It seeks to build a free and equal society through mutual agreement. Graeber and Grubacic 
+define anarchism in terms of four qualities: noncentrality, voluntary association, mutual 
+aid, and the network model [42]. Graeber's baseline communism and Karatani's mode of 
+exchange D, which are represented in this nonequivalent exchange model, are oriented 
+toward anarchism as they both aim for a human economy in which free exchange occurs 
+while incorporating the morality of mutual aid [43]. 
+Philosopher Deguchi describes the East Asian view of the self, “Self-as-We,” which 
+is connected to the lineage of Laozhuang and Zen thought, as opposed to the Western 
+view of self, “Self-as-I” [44–46]. According to Deguchi, human beings have a 
+“fundamental incapability” to live alone, and “Self-as-We” is a network of multi-agents—
+including “I”—who entrust themselves to each other. The “mixed-life society” in which 
+“we” live is one in which different self-nomadic people interact, mingle, and remain in 
+contact, recognizing each other's “fundamental incapability” and sublimating it into 
+solidarity. Deguchi's ideology also underlies Graeber's baseline communism, Karatani's 
+mode D of exchange, and the face-to-face association economy based on real transactions 
+in the morality of mutual aid. 
+Another perspective is the triangle “state (public agencies)–community–market 
+(private firms)” presented by political scientist Pestoff [47]; the “public (state)–common 
+(community)–private (market)" framework presented by policy scholar Hiroi [48,49]; and 
+the three pillars “state, community, market” presented by economist Rajan [50]. The state 
+corresponds to Polanyi’s redistribution, Graeber’s hierarchy, and Karatani’s mode of 
+exchange B; the community corresponds to Polanyi’s reciprocity and Karatani’s mode of 
+exchange A; and the market corresponds to Polanyi’s market exchange, Graeber’s 
+exchange, and Karatani’s mode of exchange C. Thus, Pestoff's association at the center of 
+the triangle, Hiroi's synthesis of “public–community–private” and the departure from the 
+local level, and the balance between Rajan's three pillars is oriented toward Graeber's 
+baseline communism and Karatani's mode of exchange D. 
+In recent work, Karatani notes that the mode of exchange D has emerged repeatedly 
+through the return of mode of exchange A (reciprocity and return) at a higher level, not 
+as a world religion such as a monotheistic religion supporting the empire but as a 
+universal religion emerging on the periphery in defiance of the empire. He also states that 
+because of the crises of war and depression induced by modes of exchange B (imperial 
+plunder and redistribution) and C (money and commodity exchange), mode of exchange 
+D will arrive “from beyond” human will and planning [51].  
+Historian Sheidel states that human history has witnessed wars, revolutions, collapse 
+of states, and epidemics decrease economic inequality [52]. Currently, the world is 
+suffering from the COVID-19 pandemic, war in Ukraine, and natural disasters and 
+conflicts caused by the effects of global warming. Although these crises are unfortunate, 
+they may hasten the arrival of mode of exchange D and facilitate the transition from a 
+capitalist economy to alternatives as suggested by Graeber and Karatani. 
+This study is limited in that it compares general trends in redistribution and mutual 
+aid, that the same transfer rate 𝜉 and period 𝑡𝑝 is set for all agents in the EX model, and 
+that the same surplus contribution rate 𝛾 is set for all agents in the NX model. Future 
+analytical studies should be conducted in more detail, for example, by setting the transfer 
+rate 𝜉 and period 𝑡𝑝 in the EX model based on various social security programs and by 
+choosing the surplus contribution rate 𝛾 in the NX model according to the ability of the 
+wealthy and the needs of the poor. Moreover, empirical studies are needed that use real-
+world evidence to examine the relationship between economic flow and Gini index with 
+respect to tax rate and frequency for the EX model, and with respect to stock and surplus 
+contribution of the wealthy for the NX model. 
+In addition, this study uses a conservative model for aggregate wealth that deals only 
+with exchange. Therefore, it does not deal with production and consumption, or interest 
+and profit/loss in the real-world economy [13]. With respect to interest and profit/loss, 
+there is a non-conservative model introduced by Kato in comparison of Islamic and 
+
+ 
+14 of 17 
+ 
+ 
+capitalist economies [25]. In a future study, the redistribution or mutual aid of interest and 
+profit/loss for the wealthy and the poor can be considered in such a non-conservative 
+model.  
+It should be added that, although the present study used a model based on the kinetic 
+energy exchange analogy, there is another model that uses potential function to compute 
+probability distributions for income and expenditure [53], and a model that uses 
+population dynamics to compute time developments for growth and inequality [54]. 
+Future research could thus include such models that take into account the finiteness of 
+earth resources and the sustainability of economy. Such non-conservative models are 
+subject to the constraint of resource limits, however, and eventually researchers may wish 
+to revert to a conservative model that is primarily based on exchange. 
+5. Conclusions 
+In this study, I develop econophysical exchange models for a hybrid of a market-
+based equivalent exchange (EX) and power-centered redistribution and a mutual-aid 
+nonequivalent exchange (NX). I also compare redistribution and mutual aid in terms of 
+wealth inequality and economic flow.  
+Simulations conducted using these exchange models to evaluate the Gini index 
+(inequality) 𝑔 and total exchange (economic flow) 𝑓 show that in both the EX and NX 
+models, the larger the savings rate 𝜆, the more the inequality is suppressed and economic 
+flows stagnate. Furthermore, the larger the synthetic parameters 𝜉 𝑡𝑝 × 10−3
+⁄
+ and 
+(1 − 𝜆) ∙ 𝛾 in the EX and NX models, respectively, the more the inequality is suppressed 
+and economic flows are activated. I show that the EX and NX models have the same 
+saturated curvilinear approximation equations 𝑓 𝑔
+⁄ ~2 ∙ (1 − 𝑒−5𝑥),𝑥~𝜉 𝑡𝑝 × 10−3
+⁄
+~(1 −
+𝜆) ∙ 𝛾 for these relationships. This approximate expression indicates that inequality and 
+economic flows are inversely proportional and that the parameter 𝑥 must be large to 
+achieve both. 
+Although the EX and NX models are “computationally” isomorphic approximations, 
+the NX model of mutual-aid nonequivalent exchange, is “qualitatively” preferable to the 
+EX model, a hybrid of market equivalence exchange and power redistribution. This is 
+indicative of Graeber’s baseline communism, Karatani’s mode D of exchange, a face-to-
+face association economy based on real transactions as learned from the Islamic economy, 
+and the ideals of anarchism. 
+Notwithstanding the fact that mutual aid is “qualitatively” preferable to 
+redistribution, there remain issues that are beyond the scope of this study’s econophysical 
+approach: the reconstruction of a moral system in the non-Islamic world that is not based 
+on any particular religion; the realization of a “mixed-life society” of “We” with 
+“fundamental incapability;” and the incorporation of Graeber's stated capitalist economic 
+alternative and Karatani’s mode of exchange D. Future social practice activities based on 
+philosophy, economics, and sociology should focus on addressing these issues.  
+Specifically, in order to shift steadily from redistribution toward mutual aid—that is, 
+toward Pestoff's association and Hiroi's synthesis of "public-community-private" 
+described in the Discussion section—mutual-aid communities could be built through 
+cooperatives [55] and social enterprises [56,57] using environmental, social, and 
+governance investing [58] as well as social impact bonds [59]. Such cooperatives and social 
+enterprises will require governmental policies that provide them with preferential 
+taxation and financial resources. They will also need to be administrated in a way that 
+allows for the delegation of authority and lateral support. Still, though the progress 
+toward social innovation will always be confronted by various social challenges [60], we 
+must nevertheless reduce inequalities. This may be achieved in the future through the 
+fusion of human society and information systems, such as platform democracy [61], 
+platform cooperatives [62], and cyber-human social cooperating systems [63]. 
+
+ 
+15 of 17 
+ 
+ 
+Funding: This work was supported by JSPS Topic-Setting Program to Advance Cutting-Edge 
+Humanities and Social Sciences Research Grant Number JPJS00122679495. 
+Institutional Review Board Statement: Not applicable. 
+Data Availability Statement: Not applicable. 
+Acknowledgments: I am deeply grateful to Professor Yasuo Deguchi of the Graduate School of 
+Letters, Kyoto University, whose ideas of “Self-as-We,” “fundamental incapability,” and “mixed-
+life society” directly motivated this study. I would also like to express my deepest gratitude to 
+Professor Yoshinori Hiroi of the Institute for the Future of Human Society, Kyoto University, for his 
+useful recommendations on post-capitalism, sustainable welfare society, and the economy of 
+mutual aid. Furthermore, I would also like to thank my colleagues at the Hitachi Kyoto University 
+Laboratory of the Kyoto University Open Innovation Institute for their ongoing cooperation, and 
+thank Dr. Bismark Addai and Editage (www.editage.com) for English language editing. 
+Conflicts of Interest: The author declares no conflict of interest. 
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+

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+Strain Induced Enhanced Photocatalytic Activities
+in Layered Two Dimensional C2N/MoS2
+Heterostructure: A Meta-GGA Study
+Soumendra Kumar Das1, Lokanath Patra2, Prasanjit Samal1,
+Pratap K Sahoo1
+1School of Physical Sciences, National Institute of Science Education and Research
+(NISER) Bhubaneswar, HBNI, Jatni, Khurda-752050, Odisha, India.
+2Department of Mechanical Engineering, University of California Santa Barbara, CA,
+93106, USA.
+E-mail: [email protected], [email protected]
+Abstract.
+The improved photocatalytic water splitting using 2D materials has
+technological importance for economically viable renewable energy. The present study
+focuses on the effect of uniaxial, biaxial, and vertical strain on the energy gap and band
+edge positions of C2N/MoS2 van der Waals heterostructures through first-principles
+density functional theory using PBE and SCAN functionals. The calculations establish
+that SCAN functional provides comparatively much better results as compared to the
+PBE for the band gap and band alignment study. The heterostructure exhibits a type-
+II band alignment which is beneficial for the efficient separation of charge carriers. For a
+good photocatalyst, the band edge positions should straddle the water redox potentials.
+It is observed that for both compressive and tensile vertical strain, the water redox
+potential values lie within the valence band maximum (VBM) and conduction band
+minimum (CBM) of the heterostructure. On the other hand, for uniaxial and biaxial
+strain, the system can be used as a useful photocatalyst only for larger compressive
+strain, whereas for tensile strain, the energy gap between VBM and CBM keeps on
+decreasing and lie within the water oxidation/reduction potential.
+Our study also
+establishes that the meta-GGA SCAN functional shows similar results as compared
+to the computationally expensive hybrid HSE functionals. The present work can be
+extremely useful for experimentalists to design artificial heterostructure devices for
+better performance in photocatalytic water splitting.
+Keywords: Photocatalytic water splitting, DFT, Van der Waals heterostructures, strain,
+Type-II Band alignment.
+Submitted to: 2D Mater.
+arXiv:2301.03809v1  [cond-mat.mtrl-sci]  10 Jan 2023
+
+2
+1. Introduction
+The emergence of photocatalytic water splitting has been a successful technology to
+meet the demand for the energy crisis and environmental pollution created by our fast-
+growing economy. The development of high-performance photo-catalytic materials to
+create hydrogen by using solar energy has been a serious focus of research for many
+years [1, 2]. The key factor for achieving highly efficient photocatalysts (PCs) is that
+the band gap should be larger than the water redox potentials. More specifically, the
+conduction band minimum (CBM) of the PCs should be above the H+/H2 potential and
+the valence band maximum (VBM) should be below H2O/O2 potential simultaneously,
+thus requiring a minimum band gap of 1.23 eV [3]. In addition, literature reports have
+established the importance of co-catalysts for boosting the electron-hole separation
+and improving the reaction kinetics [4]. Under such circumstances, two-dimensional
+materials like graphene, hexagonal boron nitride (h-BN) mono layers, transition metal
+dichalcogenides (TMDCs), C3N4, C2N, etc, have created a lot of interest, in meeting
+the demand, because of their novel electronic, thermal and optoelectronic properties.
+In particular, MoS2 has a direct band gap (2 eV), and high carrier mobility in the
+form of a single monolayer, which makes it an important candidate for photocatalytic
+and photovoltaic applications [5, 6].
+Similarly, the porous C2N monolayer is found
+to be a direct band gap semiconductor with a gap of 1.96 eV [7]. Zhao et al. have
+adopted a 2D/2D polymeric Z-scheme heterostructure by using a pair of ultrathin
+g-C3N4 nanosheets in order to provide H2- and O2- evolving photocatalysts through
+the strategy of electrostatic self-assembly. Using Pt and Co(OH)2 as co-catalysts, the
+heterostructure achieved a solar-to-hydrogen efficiency of 1.16 % which originates due to
+the formation of direct Z-scheme charge transfer pathway through the interface between
+H2- and O2- evolving components [8]. It has been suggested that the use of C2N and
+MoS2 can be highly efficient for photocatalytic study and also can be complementary
+to the use of graphene and h-BN [9].
+Despite the extensive use of C2N and MoS2, there are some challenges as well for the
+application of these materials for photocatalytic study. The charge distribution of the
+valence band maximum (VBM) and conduction band minimum (CBM) states for these
+systems are not well separated in space resulting in reduced light absorbing efficiency
+because of the recombination of the photoinduced electrons and holes [7, 10]. Therefore,
+attempts have been made to use van der Waals heterostructures to fix the issues. The
+electronic properties of C2N/InSe heterostructure are found to be greatly affected by
+vertical strain and electric field. Without any electric field, the heterostructure possesses
+a type-II band alignment with an indirect band gap of 1.34 eV at an equilibrium
+interlayer distance of 3.325 ˚A. Application of an electric field or a change in interlayer
+distance results in a transition from type-II to type-I band alignment and indirect to
+direct band gap in this heterostructure [11].
+Band gap and band offset engineering at C2N/MSe2 (M = Mo, W) interface have
+established that the heterostructure possesses a narrow indirect band gap with type-II
+
+3
+band alignment which is favourable for the photogenerated electron-hole pairs. The
+application of vertical strain and electric field strongly modulate the magnitude of
+band gap values and band offsets, but the type-II band alignment nature remains
+preserved [12].
+Strain engineering plays a significant role in tuning the electronic
+properties and photocatalytic performance of 2D heterostructures. Wang et al. have
+studied the effect of strain on the electronic structure of C2N/MTe (M= Ga, In)
+through DFT calculations which exhibit excellent optical properties with good structural
+stability. It was observed that for C2N/GaTe heterostructure, the exciton-Bohr radius
+remains unaffected by the application of strain. On the other hand, compressive strain
+reduces the exciton-Bohr radius in C2N/InTe system. The power conversion efficiency
+shows an increase up to 22.1% for C2N/GaTe with 4% strain and 19.8% for C2N/GaTe
+heterostructure with 6% strain [13]. Han et al. reported that the catalytic efficiency
+of C2N/SiH heterojunction can be effectively adjusted by the application of -2% and
++4% biaxial strain for the hydrogen evolution reaction (HER) and oxygen evolution
+reaction (OER), respectively [14]. The Cs3Bi2I9/C2N heterostructure exhibits a charge
+distribution across the whole structure due to the difference in the work function between
+the two monolayers and a charge transfer at the interface due to the formation of an
+internal electric field [15]. The photocatalytic study in MoS2/ZnO heterostructure shows
+an indirect band gap with type-II band alignment, a significant built-in potential of
+7.42 eV, and a valence band offset of 1.23 eV across the interface. The photogenerated
+carriers are localized in different layers and can effectively generate hydrogen energy.
+In contrast, the MoSe2/ZnO heterostructure possesses a type-I band alignment with a
+direct band gap of 1.80 eV, built-in potential around 3.64 eV, and a valence band offset
+of 0.34 eV [16].
+Despite the extensive studies on C2N based van der Waals heterostructures,
+significant progress has not been achieved both theoretically and experimentally. As a
+typical case, reports on the C2N/MoS2 heterostructures are very scarce and a systematic
+investigation of the physical properties of this system is still lacking. In this article,
+we report the effect of vertical, uniaxial, and biaxial strain on the photocatalytic
+water splitting performance of C2N/MoS2 van der Waals heterostructures through first-
+principles electronic structure calculations.
+2. Results and Discussion
+The schematic of the crystal structure of C2N monolayer, MoS2 layer, and C2N/MoS2
+heterostructure is illustrated in Fig.
+1.
+In C2N monolayer, twelve C-N bonds are
+alternately connected to the six C-C bonds in such a way that the periodic holes are
+present without any atom in the middle of the lattice. The in-plane bond length is
+estimated to be around 1.43 and 1.47 ˚A for the C-C bond and 1.34 ˚A for the C-N bond,
+respectively which is also close to the previously reported values [17]. The optimized
+lattice constant using PBE functional is estimated to be 8.33 ˚A and 3.19 ˚A for the C2N
+and MoS2 monolayer, respectively which is in reasonable agreement with the reported
+
+4
+experimental and theoretical results [18, 19].
+The heterostructure is constructed by
+making a supercell of (3×3×1) for C2N and (8×8×1) for MoS2 layer, respectively. The
+corresponding lattice mismatch between the two monolayers is around 1.9% which is
+within the acceptable range.
+The heterostructure after ionic relaxation results in a
+corrugated structure (Fig. 1d), very similar to silicene. The initial van der Waals gap
+(3.5 ˚A) between the layers reduces to 3.373 ˚A with an optimized lattice constant of
+25.47 ˚A (Fig. 1c, d).
+Figure 1. Schematic representation of the top view of isolated (a) C2N monolayer,
+(b) MoS2 monolayer, (c) side view and (d) top view of C2N/MoS2 heterostructure.
+The schematic symbols of the C, N, Mo, and S atoms are indicated as blue, red, green,
+and brown colors, respectively.
+The calculated electronic structure of free-standing C2N monolayer, MoS2 using
+the PBE functional are given in Fig. 2a,b. The band dispersion for both the material
+indicates the presence of valence band maximum (VBM) and conduction band minimum
+(CBM) at the same k value in the Brillouin zone. In addition, for both structures, the
+VBM remains close to the Fermi level as compared to the CBM. This confirms that
+both C2N and MoS2 monolayers are p-type direct band gap semiconductors and the
+calculated energy gap values are estimated to be 1.735 eV and 1.645 eV, respectively.
+It is interesting to note that the energy band dispersion in C2N monolayer is relatively
+flat where as MoS2 shows highly dispersive bands both in VB and CB regions. Hence,
+it is expected that the MoS2 system will have a relatively low electron and hole effective
+mass along the high symmetry path (Γ-M-K-Γ) as compared to that of C2N monolayer.
+Therefore, the electron and hole mobilities of MoS2 would be larger than that of
+the C2N monolayer, since the effective mass is inversely proportional to the carrier
+
+5
+mobility. Figure 2e illustrates the electronic structure of C2N/MoS2 heterostructure
+which preserves the direct band gap behaviour of the isolated monolayers. The VBM
+is situated close to the Fermi level suggesting that the charge carriers are of p-type.
+However, the value of the energy gap is comparatively reduced and becomes 1.353 eV.
+We note that the ideal band gap for a semiconductor to use more visible light is around
+1.5 eV [20].
+Figure 2. Electronic band structure of (a) C2N, (b) MoS2 monolayer, (c) C2N/MoS2
+heterostructure, (d-f) The total and projected density of states for the corresponding
+system calculated using PBE functional. The dashed line in each figure indicates the
+Fermi level.
+It is well known that the PBE functional severely underestimates the energy gap
+of the system. Again calculations involving the hybrid density functional like HSE-06,
+B3LYP, etc are highly computationally expensive which is beyond the computational
+resources available to us. Therefore we have further investigated the systems with SCAN
+meta-GGA functional. The obtained results are consistent with the experiment as well
+as previously reported simulated results. The contribution of different orbitals at the
+band edges is evident from the total and orbital projected density of states analysis as
+given in Fig. 2b-f. From Fig. 2b, it is clear that the valence band of C2N is dominated
+by the C ‘2p’ states with a significant contribution from N ‘2p’ states as well. The CBM
+minimum is also populated by the hybridization of the ‘p’ states of C and N atoms.
+The partial DOS analysis of MoS2 establishes that the VBM is highly dominated by Mo
+‘4d’ states with a relatively less population than S ‘3p’ states. On the other hand, the
+CBM is mainly populated by Mo ‘4d’ states. The projected DOS for the heterostructure
+
+3
+(a)
+3
+1
+(eV)
+(e)
+(c)
+F
+0
+0
+3-3
+0
+3
+3
+M
+r
+K
+M
+r
+M
+K
+M
+r
+K
+M
+(b)
+total dos
+16
+((d):
+total dos .
+DOS (StatesleV)
+C_2P
+Mo_4d
+S_3p
+N_2P
+(f)
+8
+90
+0
+0
+0
+2
+0
+2
+-2
+0
+2
+-2
+0
+2
+E-E. (eV)
+E-E. (eV)
+E-E. (eV)6
+indicates that the VBM is populated by the states from MoS2 and the CBM is dominated
+by the C2N.
+It has been established that strain plays a significant role in efficiently tuning the
+electronic, optical, and photocatalytic properties of van der Waals heterojunctions. In
+this section, we studied the effect of vertical strain by changing the interlayer distance
+between the C2N and MoS2 monolayers.
+However, we would like to mention that
+experimentally, the interlayer distance can be effectively varied by changing pressure
+with a scanning tunneling microscopy tip [21], through vacuum thermal annealing [22],
+inserting a dielectric BN layer inside the van der Waals gap of the heterostructure [23],
+using diamond anvil cells [24].
+The vertical strain (∆D) can be defined as ∆D =
+d−d0
+d0
+× 100, where d0 and d are the interlayer separation between C2N and MoS2 under
+equilibrium and strained configurations, respectively. The effect of vertical strain on
+the band gap of the heterostructure is given in Fig. 3a. It is observed that with an
+increase in tensile strain along the ‘z’ direction, the band gap value shows an increasing
+trend and exhibits a linear variation. Instead, when the compressive strain between
+the layer is increased, the band gap decreases and follows a linear relationship with the
+direction of applied strain. The band evolution is further analysed through the strain-
+dependent projected density of states (PDOS) as shown in supporting information S1.
+It is clear that the compressive strain increases the population of Mo ‘4d’ states for the
+VBM near the Fermi level. The CBM is mainly composed of a mixture of ‘p’ states
+of C2N and MoS2. With an increase in tensile strain both Mo ‘4d’ and C2N, MoS2 ‘p’
+states move away from the Fermi level resulting in a larger band gap. The band gap
+calculation is further analysed using the meta-GGA SCAN functional which indicates
+an enhanced band gap value as compared to the PBE result for the entire range of
+compressive and tensile strains considered. The band gap variation for the compressive
+strain shows a nearly linear behaviour consistent with the PBE result. Similarly, for the
+tensile configuration, the energy gap exhibits an increase in value up to 3% strain and
+then remains constant with further expansion in interlayer separation.
+Figure 3.
+Band gap tuning of C2N/MoS2 heterostructure under (a) vertical, (b)
+uniaxial, and (c) biaxial strain using PBE and SCAN functional
+
+1.5
+1.5
+(a)
+(eV)
+(q)
+-O-PBE
+1.2
+1.2
+.2
+O-SCAN
+Vertical strain
+(c)
+0.9
+0.9
+1.0
+0.6
+Uniaxial strain
+0.6
+Biaxialstrain
+-5
+0
+5
+-5
+0
+5
+-5
+0
+5
+Strain (%)
+Strain (%)
+Strain (%)7
+The present heterostructure is also investigated with the application of uniaxial
+and biaxial strain in order to study the influence of in-plane orbital overlapping. The
+PBE result for the band gap evolution as a function of uniaxial strain is represented
+in Fig. 3b. It is interesting to note that the band gap remains nearly linear for larger
+compressive strain. As the compressive strain decreases from -5% to -1% the band gap
+shows a mild increase in value.
+The application of tensile strain along the uniaxial
+direction shows a different trend as compared to the compressive one. The band gap
+shows a systematic decrease in value with the increase in tensile strain from 0 to +5%.
+The reduction in band gap may be due to the decrease in orbital overlapping between
+the ‘p’ states of C and N and the ‘4d’ states of Mo. The corresponding partial density of
+state (See supporting information Figure S2a, b) suggests that tensile strain increases
+the DOS near the Fermi level for Mo ‘4d’ and C, N ‘p’ states. Similarly, the band
+gap study was also executed by applying biaxial strain from -5% to +5% range. The
+system shows a completely different behaviour as compared to the result for vertical
+strain. The linear variation of the band gap for the compressive strain region remains
+preserved as similar to the case of uniaxial strain with a linear variation with a slowly
+decreasing trend. However, if we apply tensile strain then the band gap decreases much
+more rapidly. The corresponding PDOS analysis indicates that for -5% compressive
+strain, the VBM is mainly composed of C and N ‘2p’ states, and CBM is populated
+by ‘p’ states of N atoms. When the strain increases from -5% to +5%, Mo ‘4d’ states
+become the highest occupied band near the Fermi level. The CBM comes closer to the
+Fermi level and resulting in the reduced band gap. The band gap calculation is further
+analysed using SCAN functional for the uniaxial and biaxial strain configuration which
+provides an enhanced band gap value as compared to the PBE functional. However, the
+trend in variation of the band gap remains almost the same (Fig. 3b, c).
+The essential requirement for any material for efficient photocatalytic water
+splitting application is to identify appropriate band edge positions relative to the
+hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) potentials of
+the water. The CBM of the material should lie above the reduction reaction potential
+and the VBM should lie below the oxidation reaction potential of water. To estimate
+the band edge positions relative to the oxidation and reduction potential, the absolute
+energy position of the CBM and VBM are calculated with respect to the vacuum level.
+The vacuum level for the material is obtained by calculating the local potential and
+then taking the average of the constant potential region. In the present study, we have
+investigated the photocatalytic activity of the C2N/MoS2 heterostructure by applying
+strain in the uniaxial, biaxial, and vertical directions.
+We note that the reduction
+potential (EH+/H2) and oxidation potential (EO2/H2O) of water with respect to vacuum
+level are -4.44 and -5.67 eV, respectively [25]. The band alignment is then calculated
+by plotting the appropriate band edge position with respect to the vacuum level as a
+function of strain.
+The calculated band edge position of the CBM and VBM under the influence of
+vertical strain, using PBE functional is presented in Fig. 4a. It was observed that
+
+8
+Figure 4. Band edge position of C2N/MoS2 heterostructure as a function of vertical
+strain with respect to vacuum potential using (a) PBE and (b) SCAN functional. The
+horizontal solid lines represent the water redox potentials
+without applying any strain, the heterostructure exhibits band edge positions within
+the water redox potentials of the water.
+This result indicates that the sample is
+not useful to carry out photocatalytic water splitting.
+The sample is subjected to
+compressive (tensile) vertical strain by reducing (increasing) the interlayer separation
+between the C2N and MoS2 monolayers, respectively. It was observed that by applying
+compressive strain, the band edge positions got improved as compared to the zero
+strain case. However, both the CBM and VBM lie within the water redox potential
+values. Applying tensile strain, the VBM and CBM move closer to the water redox
+potential values but still lie below them. Therefore the PBE result indicates that the
+application of -5 to +5% vertical strain does not enhance the band edge positions to be
+useful for photocatalytic water splitting. However, we would like to mention that PBE
+functional severely underestimates the band gap and band edge positions. Therefore
+the calculations are further executed using meta-GGA SCAN functional (Fig. 4b). The
+result indicates a significant improvement as compared to that of the PBE functional.
+It was observed that, without applying any strain, the VBM comes below the oxidation
+reaction potential, whereas the CBM moves up but still lies below the reduction potential
+
+-4.2
+(a)
+-4.44 eV
+-4.9
+VBM
+PBEResult
+CBM
+Energy (eV)
+-5.6
+-5.67 eV
+-4.2
+(b)
+-4.44 eV
+-4.9
+SCAN Result
+-5.6
+-5.67 eV
+-5
+0
+5
+Strain (%)9
+value. The application of compressive strain up to -2% positions the VBM well below
+the oxidation potential of water but the CBM still lies below the reduction potential. On
+further increase in the compressive strain from -3 to -5%, both the CBM and VBM lie
+outside the range of water redox potentials hence enhancing the photocatalytic activities
+of the sample. When the heterostructure is subjected to vertical tensile strain from
++1% to +5%, both the CBM and VBM straddle the water redox potential range hence
+enhancing the photocatalytic response under visible light. There fore we conclude that
+the present heterostructure exhibits efficient charge separation for photocatalytic water
+splitting under tensile strain and larger compressive strain.
+The sample is further subjected to uniaxial and biaxial strain to analyse the
+photocatalytic performance. The uniaxial strain is applied by increasing or compressing
+the in-plane lattice constant along the X- direction keeping the Y value fixed. Similarly,
+the biaxial strain is applied by increasing or decreasing the X-Y lattice constant values
+simultaneously. It is to be noted that the interlayer separation is kept at its equilibrium
+value i.e 3.373 ˚A. The detailed graphical representation of band alignment under uniaxial
+strain is given in the Supporting information S3 a.
+It was observed that for PBE
+functionals the CBM and VBM both lie within the redox potential range for the entire
+range of uniaxial strain from -5 to +5%, thus indicating that uniaxial strain does not
+produce a significant improvement to be useful for photocatalytic water splitting.
+To get a better result, the strain calculation is repeated for the heterostructure
+using SCAN functional and is given in Supporting information S3b. It was observed
+that the application of larger uniaxial strain (-3 to -5%) puts the CBM and VBM outside
+the water redox potential range. But the tensile strain reduces the band edge separation
+even below the unstrained case. The VBM lies below the oxidation potential value but
+the CBM finds its position below the reduction potential. Therefore we found that the
+sample can be useful for photocatalytic water splitting for larger compressive uniaxial
+strain whereas the tensile strain does not produce an enhancement in the photocatalytic
+response.
+The present C2N/MoS2 heterostructure is also investigated under biaxial
+strain using both PBE and SCAN functionals (See Supporting information S4). The
+results are quite similar to that of the uniaxial strain.
+Like previous results, PBE
+functional does not produce a better result for the photocatalytic activity. A larger
+compressive strain puts the CBM above the reduction potential but the VBM lies above
+the oxidation potential of water, thus not providing a useful result for our purpose.
+Using SCAN functional, it was observed that from -2 to -5% compressive strain both
+the CBM and VBM straddle outside the water redox potential. On the other hand,
+tensile strain decreases the band edges position and put the CBM below the reduction
+potential of water. Therefore, the heterostructure can be used for photocatalytic studies
+for larger uniaxial and biaxial compressive strains.
+In order to illustrate the charge transfer process between the C2N and MoS2
+monolayers during the formation of the heterostructure, the charge density difference
+is executed using PBE functional.
+The charge density difference is calculated
+by subtracting the charge density of the individual C2N and MoS2 monolayers
+
+10
+Figure 5. Charge density difference of C2N/MoS2 heterostructure as a function of
+vertical (a) -5% compressive (b) 0% (c) +5% tensile strain
+from the C2N/MoS2 heterostructure.The charge density difference for the unstrained
+heterostructure is illustrated in Fig. 5b. In all our charge density figures, the cyan color
+indicates the charge depletion and the yellow color represents the charge accumulation
+process. From Fig. 5b, we observe that the charge accumulation mainly occurs in the
+interface region and partially in MoS2 monolayers, whereas most of the charge depletion
+occurs in the top C2N and bottom MoS2. The change in charge density under vertical
+strain is given in Fig. 5.
+From the Fig.5a, we observe that with an increase in compressive vertical strain by
+5%, there is an equal proportion of charge accumulation and depletion close to the MoS2
+layer whereas the charge depletion mainly occurs in the region close to C2N layer. The
+charge density given in cyan color near the C atom in the C2N monolayer indicates a
+charge depletion. Similarly, the charge density given in yellow color near the S atom in
+the MoS2 layer indicates a charge accumulation. Hence this observation indicates that
+there is a charge transfer from the S atom of MoS2 to the C atom of C2N monolayer. The
+intensification of the charge transfer process under large compressive strain indicates an
+enhanced interaction between the two monolayers. The strain-induced charge transfer
+in C2N based heterostructures is consistent with other literature reports [26, 27, 28].
+When the interlayer separation in increases to achieve a strain around +5%, the charge
+depletion from the C2N layer almost disappears and most of the charge accumulation
+occurs in the interface region close to MoS2. In other words, there is no appreciable
+charge redistribution close to the MoS2 layer. This may be the influence of the van
+der Waals gap between the two monolayers which varies as the structure moves from a
+compressive strain state to a tensile strain state.
+The effect of uniaxial compressive and tensile strain on the charge transfer process
+is illustrated in supplementary information Fig.
+S5.
+It indicates that there is no
+appreciable change in the charge density difference under uniaxial compressive and
+tensile strain (up to 5%). On the other hand application of biaxial strain indicates a
+significant influence on the charge density under compressive and tensile strains. From
+
+(a)
+E.= -5%
+(b)
+E,= 0%
+(c)
+E.= +5%11
+the Supplementary information Fig. S6, we observe that as the system is subjected to
+-5% compressive strain the charge depletion occurs in the C atom of the top layer. With
+system changes from large compression to large tensile strain state, the depletion near
+the C atom increases indicating more charge transfer from the S atom of MoS2 to the
+C atom of C2N layer. Therefore we observe that both vertical and biaxial strain affects
+the charge transfer process significantly as compared to the uniaxial strain state.
+3. Conclusion
+In
+summary,
+we
+have
+studied
+the
+photocatalytic
+performance
+of
+C2N/MoS2
+heterostructure as a function of uniaxial, biaxial, and vertical strain configuration
+through first-principles DFT calculations.
+The unstrained heterostructure possesses
+a direct band gap of 1.35 eV, where the VBM is populated by Mo ‘d’ states and the
+CBM is contributed by ‘C’ and ‘N’ ‘p’ states. The calculated position of CBM and
+VBM of individual monolayers indicates that the heterostructure possesses a type-II
+band alignment which is beneficial for charge separation across the two monolayers and
+preventing their recombination process. The SCAN functional provides better results
+for the band gap and band edge positions calculation as compared to that of the PBE
+result. Using the C2N/MoS2 heterostructure as a prototype, we have also found that the
+meta-GGA SCAN functional shows similar results as compared to the computationally
+expensive hybrid HSE functionals. The estimated CBM and CBM position as a function
+of vertical strain indicates that the water reduction and oxidation potential values lie
+within the band gap region with respect to the vacuum level for larger compressive
+and tensile strain. In contrast, the system exhibits good photocatalytic performance
+only for larger compression for uniaxial and biaxial strain states whereas the tensile
+strain reduces the separation between the VBM and CBM within the water redox
+potential value. The charge density difference indicates a significant charge transfer
+for vertical and biaxial configuration as compared to the uniaxial state. The present
+study can help researchers to reduce the computational cost by considering the meta-
+GGA functionals over HSE for electronic structure calculations of similar systems. Our
+calculation will also be extremely useful for designing artificial strained heterostructure
+for the experimental community for better device application for photocatalytic water
+splitting.
+4. Computational Methods
+The first principles electronic structure calculations were performed using density
+functional theory (DFT) with projector augmented-wave method [29] as implemented
+in the Vienna ab initio simulation package (VASP) [30]. The Perdew-Burke-Ernzerhof
+(PBE) [31] parametrization-based generalized gradient approximation (GGA) was
+chosen for the exchange-correlation functional. To accurately describe the interaction
+between the layered structures, we have included the van der Waals correction method
+
+REFERENCES
+12
+(DFT-D2) presented by Grimme [32].
+Initially, the gap between C2N and MoS2
+monolayers is kept at 3.5 ˚A which is further optimized before running the electronic
+structure calculation. As a benchmark, the system is further studied using strongly
+constrained and appropriately normed (SCAN) meta-GGA functionals to get more
+accurate results as compared to the PBE functional and also to be consistent with the
+reported experimental data. A vacuum layer of 20 ˚A was selected along the ‘Z’ direction
+to avoid interaction between the adjacent layers. The plane wave expansion cut-off was
+chosen to be 520 eV. The Brillouin zone integration was performed using a Γ-centered
+(9×9×1) k-mesh for the structural relaxation and electronic structure calculation of the
+isolated C2N and MoS2 monolayers. For the C2N/MoS2 heterostructure containing 354
+atoms, the geometry optimization and electronic structure calculation are performed
+using a Γ centered (2×2×1) k-point samplings.
+All the structures are allowed for
+relaxation to get the optimized atomic position until the total forces acting on each
+atom are less than 0.02 eV/˚A.
+Data availability statement
+The additional data that support the findings of this article are available in the
+Supplementary Information.
+Acknowledgments
+The authors would like to thank the National Institute of Science Education and
+Research (NISER), Department of Atomic Energy, Government of India, for funding
+the research work through project number RIN-4001. The authors acknowledge the
+high-performance computing facility at NISER.
+Data availability statement
+The additional data that support the findings of this article are available in the
+Supplementary Information.
+Conflict of interest
+The authors have no conflicts to disclose.
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+

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+page_content='Strain Induced Enhanced Photocatalytic Activities  in Layered Two Dimensional C2N/MoS2  Heterostructure: A Meta-GGA Study  Soumendra Kumar Das1, Lokanath Patra2, Prasanjit Samal1,  Pratap K Sahoo1  1School of Physical Sciences, National Institute of Science Education and Research  (NISER) Bhubaneswar, HBNI, Jatni, Khurda-752050, Odisha, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  2Department of Mechanical Engineering, University of California Santa Barbara, CA,  93106, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  E-mail: psamal@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='in, pratap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='sahoo@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
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+page_content='in  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The improved photocatalytic water splitting using 2D materials has  technological importance for economically viable renewable energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The present study  focuses on the effect of uniaxial, biaxial, and vertical strain on the energy gap and band  edge positions of C2N/MoS2 van der Waals heterostructures through first-principles  density functional theory using PBE and SCAN functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The calculations establish  that SCAN functional provides comparatively much better results as compared to the  PBE for the band gap and band alignment study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The heterostructure exhibits a type-  II band alignment which is beneficial for the efficient separation of charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' For a  good photocatalyst, the band edge positions should straddle the water redox potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It is observed that for both compressive and tensile vertical strain, the water redox  potential values lie within the valence band maximum (VBM) and conduction band  minimum (CBM) of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' On the other hand, for uniaxial and biaxial  strain, the system can be used as a useful photocatalyst only for larger compressive  strain, whereas for tensile strain, the energy gap between VBM and CBM keeps on  decreasing and lie within the water oxidation/reduction potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Our study also  establishes that the meta-GGA SCAN functional shows similar results as compared  to the computationally expensive hybrid HSE functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The present work can be  extremely useful for experimentalists to design artificial heterostructure devices for  better performance in photocatalytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Keywords: Photocatalytic water splitting, DFT, Van der Waals heterostructures, strain,  Type-II Band alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Submitted to: 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='03809v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='mtrl-sci]  10 Jan 2023  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Introduction  The emergence of photocatalytic water splitting has been a successful technology to  meet the demand for the energy crisis and environmental pollution created by our fast-  growing economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The development of high-performance photo-catalytic materials to  create hydrogen by using solar energy has been a serious focus of research for many  years [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The key factor for achieving highly efficient photocatalysts (PCs) is that  the band gap should be larger than the water redox potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' More specifically, the  conduction band minimum (CBM) of the PCs should be above the H+/H2 potential and  the valence band maximum (VBM) should be below H2O/O2 potential simultaneously,  thus requiring a minimum band gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='23 eV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In addition, literature reports have  established the importance of co-catalysts for boosting the electron-hole separation  and improving the reaction kinetics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Under such circumstances, two-dimensional  materials like graphene, hexagonal boron nitride (h-BN) mono layers, transition metal  dichalcogenides (TMDCs), C3N4, C2N, etc, have created a lot of interest, in meeting  the demand, because of their novel electronic, thermal and optoelectronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  In particular, MoS2 has a direct band gap (2 eV), and high carrier mobility in the  form of a single monolayer, which makes it an important candidate for photocatalytic  and photovoltaic applications [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Similarly, the porous C2N monolayer is found  to be a direct band gap semiconductor with a gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='96 eV [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' have  adopted a 2D/2D polymeric Z-scheme heterostructure by using a pair of ultrathin  g-C3N4 nanosheets in order to provide H2- and O2- evolving photocatalysts through  the strategy of electrostatic self-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Using Pt and Co(OH)2 as co-catalysts, the  heterostructure achieved a solar-to-hydrogen efficiency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='16 % which originates due to  the formation of direct Z-scheme charge transfer pathway through the interface between  H2- and O2- evolving components [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It has been suggested that the use of C2N and  MoS2 can be highly efficient for photocatalytic study and also can be complementary  to the use of graphene and h-BN [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Despite the extensive use of C2N and MoS2, there are some challenges as well for the  application of these materials for photocatalytic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The charge distribution of the  valence band maximum (VBM) and conduction band minimum (CBM) states for these  systems are not well separated in space resulting in reduced light absorbing efficiency  because of the recombination of the photoinduced electrons and holes [7, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore,  attempts have been made to use van der Waals heterostructures to fix the issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  electronic properties of C2N/InSe heterostructure are found to be greatly affected by  vertical strain and electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Without any electric field, the heterostructure possesses  a type-II band alignment with an indirect band gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='34 eV at an equilibrium  interlayer distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='325 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Application of an electric field or a change in interlayer  distance results in a transition from type-II to type-I band alignment and indirect to  direct band gap in this heterostructure [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Band gap and band offset engineering at C2N/MSe2 (M = Mo, W) interface have  established that the heterostructure possesses a narrow indirect band gap with type-II  3  band alignment which is favourable for the photogenerated electron-hole pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  application of vertical strain and electric field strongly modulate the magnitude of  band gap values and band offsets, but the type-II band alignment nature remains  preserved [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Strain engineering plays a significant role in tuning the electronic  properties and photocatalytic performance of 2D heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' have  studied the effect of strain on the electronic structure of C2N/MTe (M= Ga, In)  through DFT calculations which exhibit excellent optical properties with good structural  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It was observed that for C2N/GaTe heterostructure, the exciton-Bohr radius  remains unaffected by the application of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' On the other hand, compressive strain  reduces the exciton-Bohr radius in C2N/InTe system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The power conversion efficiency  shows an increase up to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='1% for C2N/GaTe with 4% strain and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='8% for C2N/GaTe  heterostructure with 6% strain [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' reported that the catalytic efficiency  of C2N/SiH heterojunction can be effectively adjusted by the application of -2% and  +4% biaxial strain for the hydrogen evolution reaction (HER) and oxygen evolution  reaction (OER), respectively [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The Cs3Bi2I9/C2N heterostructure exhibits a charge  distribution across the whole structure due to the difference in the work function between  the two monolayers and a charge transfer at the interface due to the formation of an  internal electric field [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The photocatalytic study in MoS2/ZnO heterostructure shows  an indirect band gap with type-II band alignment, a significant built-in potential of  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='42 eV, and a valence band offset of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='23 eV across the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The photogenerated  carriers are localized in different layers and can effectively generate hydrogen energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  In contrast, the MoSe2/ZnO heterostructure possesses a type-I band alignment with a  direct band gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='80 eV, built-in potential around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='64 eV, and a valence band offset  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='34 eV [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Despite the extensive studies on C2N based van der Waals heterostructures,  significant progress has not been achieved both theoretically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' As a  typical case, reports on the C2N/MoS2 heterostructures are very scarce and a systematic  investigation of the physical properties of this system is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In this article,  we report the effect of vertical, uniaxial, and biaxial strain on the photocatalytic  water splitting performance of C2N/MoS2 van der Waals heterostructures through first-  principles electronic structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Results and Discussion  The schematic of the crystal structure of C2N monolayer, MoS2 layer, and C2N/MoS2  heterostructure is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  In C2N monolayer, twelve C-N bonds are  alternately connected to the six C-C bonds in such a way that the periodic holes are  present without any atom in the middle of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The in-plane bond length is  estimated to be around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='43 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='47 ˚A for the C-C bond and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='34 ˚A for the C-N bond,  respectively which is also close to the previously reported values [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The optimized  lattice constant using PBE functional is estimated to be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='33 ˚A and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='19 ˚A for the C2N  and MoS2 monolayer, respectively which is in reasonable agreement with the reported  4  experimental and theoretical results [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The heterostructure is constructed by  making a supercell of (3×3×1) for C2N and (8×8×1) for MoS2 layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  corresponding lattice mismatch between the two monolayers is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='9% which is  within the acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The heterostructure after ionic relaxation results in a  corrugated structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 1d), very similar to silicene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The initial van der Waals gap  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='5 ˚A) between the layers reduces to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='373 ˚A with an optimized lattice constant of  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='47 ˚A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 1c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Schematic representation of the top view of isolated (a) C2N monolayer,  (b) MoS2 monolayer, (c) side view and (d) top view of C2N/MoS2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The schematic symbols of the C, N, Mo, and S atoms are indicated as blue, red, green,  and brown colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The calculated electronic structure of free-standing C2N monolayer, MoS2 using  the PBE functional are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 2a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band dispersion for both the material  indicates the presence of valence band maximum (VBM) and conduction band minimum  (CBM) at the same k value in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In addition, for both structures, the  VBM remains close to the Fermi level as compared to the CBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' This confirms that  both C2N and MoS2 monolayers are p-type direct band gap semiconductors and the  calculated energy gap values are estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='735 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='645 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It is interesting to note that the energy band dispersion in C2N monolayer is relatively  flat where as MoS2 shows highly dispersive bands both in VB and CB regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Hence,  it is expected that the MoS2 system will have a relatively low electron and hole effective  mass along the high symmetry path (Γ-M-K-Γ) as compared to that of C2N monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Therefore, the electron and hole mobilities of MoS2 would be larger than that of  the C2N monolayer, since the effective mass is inversely proportional to the carrier  5  mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Figure 2e illustrates the electronic structure of C2N/MoS2 heterostructure  which preserves the direct band gap behaviour of the isolated monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The VBM  is situated close to the Fermi level suggesting that the charge carriers are of p-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  However, the value of the energy gap is comparatively reduced and becomes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='353 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  We note that the ideal band gap for a semiconductor to use more visible light is around  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='5 eV [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Electronic band structure of (a) C2N, (b) MoS2 monolayer, (c) C2N/MoS2  heterostructure, (d-f) The total and projected density of states for the corresponding  system calculated using PBE functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The dashed line in each figure indicates the  Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It is well known that the PBE functional severely underestimates the energy gap  of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Again calculations involving the hybrid density functional like HSE-06,  B3LYP, etc are highly computationally expensive which is beyond the computational  resources available to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore we have further investigated the systems with SCAN  meta-GGA functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The obtained results are consistent with the experiment as well  as previously reported simulated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The contribution of different orbitals at the  band edges is evident from the total and orbital projected density of states analysis as  given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 2b-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 2b, it is clear that the valence band of C2N is dominated  by the C ‘2p’ states with a significant contribution from N ‘2p’ states as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The CBM  minimum is also populated by the hybridization of the ‘p’ states of C and N atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The partial DOS analysis of MoS2 establishes that the VBM is highly dominated by Mo  ‘4d’ states with a relatively less population than S ‘3p’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' On the other hand, the  CBM is mainly populated by Mo ‘4d’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The projected DOS for the heterostructure  3  (a)  3  1  (eV)  (e)  (c)  F  0  0  3-3  0  3  3  M  r  K  M  r  M  K  M  r  K  M  (b)  total dos  16  ((d):  total dos .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  DOS (StatesleV)  C_2P  Mo_4d  S_3p  N_2P  (f)  8  90  0  0  0  2  0  2  2  0  2  2  0  2  E-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' (eV)  E-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' (eV)  E-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' (eV)6  indicates that the VBM is populated by the states from MoS2 and the CBM is dominated  by the C2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It has been established that strain plays a significant role in efficiently tuning the  electronic, optical, and photocatalytic properties of van der Waals heterojunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In  this section, we studied the effect of vertical strain by changing the interlayer distance  between the C2N and MoS2 monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  However, we would like to mention that  experimentally, the interlayer distance can be effectively varied by changing pressure  with a scanning tunneling microscopy tip [21], through vacuum thermal annealing [22],  inserting a dielectric BN layer inside the van der Waals gap of the heterostructure [23],  using diamond anvil cells [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The vertical strain (∆D) can be defined as ∆D =  d−d0  d0  × 100, where d0 and d are the interlayer separation between C2N and MoS2 under  equilibrium and strained configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The effect of vertical strain on  the band gap of the heterostructure is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It is observed that with an  increase in tensile strain along the ‘z’ direction, the band gap value shows an increasing  trend and exhibits a linear variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Instead, when the compressive strain between  the layer is increased, the band gap decreases and follows a linear relationship with the  direction of applied strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band evolution is further analysed through the strain-  dependent projected density of states (PDOS) as shown in supporting information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It is clear that the compressive strain increases the population of Mo ‘4d’ states for the  VBM near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The CBM is mainly composed of a mixture of ‘p’ states  of C2N and MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' With an increase in tensile strain both Mo ‘4d’ and C2N, MoS2 ‘p’  states move away from the Fermi level resulting in a larger band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band gap  calculation is further analysed using the meta-GGA SCAN functional which indicates  an enhanced band gap value as compared to the PBE result for the entire range of  compressive and tensile strains considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band gap variation for the compressive  strain shows a nearly linear behaviour consistent with the PBE result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Similarly, for the  tensile configuration, the energy gap exhibits an increase in value up to 3% strain and  then remains constant with further expansion in interlayer separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Band gap tuning of C2N/MoS2 heterostructure under (a) vertical, (b)  uniaxial, and (c) biaxial strain using PBE and SCAN functional  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='5  (a)  (eV)  (q)  O-PBE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='2  O-SCAN  Vertical strain  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='6  Uniaxial strain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='6  Biaxialstrain  5  0  5  5  0  5  5  0  5  Strain (%)  Strain (%)  Strain (%)7  The present heterostructure is also investigated with the application of uniaxial  and biaxial strain in order to study the influence of in-plane orbital overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  PBE result for the band gap evolution as a function of uniaxial strain is represented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It is interesting to note that the band gap remains nearly linear for larger  compressive strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' As the compressive strain decreases from -5% to -1% the band gap  shows a mild increase in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The application of tensile strain along the uniaxial  direction shows a different trend as compared to the compressive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band gap  shows a systematic decrease in value with the increase in tensile strain from 0 to +5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The reduction in band gap may be due to the decrease in orbital overlapping between  the ‘p’ states of C and N and the ‘4d’ states of Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The corresponding partial density of  state (See supporting information Figure S2a, b) suggests that tensile strain increases  the DOS near the Fermi level for Mo ‘4d’ and C, N ‘p’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Similarly, the band  gap study was also executed by applying biaxial strain from -5% to +5% range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  system shows a completely different behaviour as compared to the result for vertical  strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The linear variation of the band gap for the compressive strain region remains  preserved as similar to the case of uniaxial strain with a linear variation with a slowly  decreasing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' However, if we apply tensile strain then the band gap decreases much  more rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The corresponding PDOS analysis indicates that for -5% compressive  strain, the VBM is mainly composed of C and N ‘2p’ states, and CBM is populated  by ‘p’ states of N atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' When the strain increases from -5% to +5%, Mo ‘4d’ states  become the highest occupied band near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The CBM comes closer to the  Fermi level and resulting in the reduced band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band gap calculation is further  analysed using SCAN functional for the uniaxial and biaxial strain configuration which  provides an enhanced band gap value as compared to the PBE functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' However, the  trend in variation of the band gap remains almost the same (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 3b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The essential requirement for any material for efficient photocatalytic water  splitting application is to identify appropriate band edge positions relative to the  hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) potentials of  the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The CBM of the material should lie above the reduction reaction potential  and the VBM should lie below the oxidation reaction potential of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' To estimate  the band edge positions relative to the oxidation and reduction potential, the absolute  energy position of the CBM and VBM are calculated with respect to the vacuum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The vacuum level for the material is obtained by calculating the local potential and  then taking the average of the constant potential region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In the present study, we have  investigated the photocatalytic activity of the C2N/MoS2 heterostructure by applying  strain in the uniaxial, biaxial, and vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  We note that the reduction  potential (EH+/H2) and oxidation potential (EO2/H2O) of water with respect to vacuum  level are -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='44 and -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='67 eV, respectively [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The band alignment is then calculated  by plotting the appropriate band edge position with respect to the vacuum level as a  function of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The calculated band edge position of the CBM and VBM under the influence of  vertical strain, using PBE functional is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It was observed that  8  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Band edge position of C2N/MoS2 heterostructure as a function of vertical  strain with respect to vacuum potential using (a) PBE and (b) SCAN functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  horizontal solid lines represent the water redox potentials  without applying any strain, the heterostructure exhibits band edge positions within  the water redox potentials of the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  This result indicates that the sample is  not useful to carry out photocatalytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The sample is subjected to  compressive (tensile) vertical strain by reducing (increasing) the interlayer separation  between the C2N and MoS2 monolayers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It was observed that by applying  compressive strain, the band edge positions got improved as compared to the zero  strain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' However, both the CBM and VBM lie within the water redox potential  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Applying tensile strain, the VBM and CBM move closer to the water redox  potential values but still lie below them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore the PBE result indicates that the  application of -5 to +5% vertical strain does not enhance the band edge positions to be  useful for photocatalytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' However, we would like to mention that PBE  functional severely underestimates the band gap and band edge positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore  the calculations are further executed using meta-GGA SCAN functional (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  result indicates a significant improvement as compared to that of the PBE functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It was observed that, without applying any strain, the VBM comes below the oxidation  reaction potential, whereas the CBM moves up but still lies below the reduction potential  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='2  (a)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='44 eV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='9  VBM  PBEResult  CBM  Energy (eV)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='67 eV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='2  (b)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='44 eV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='9  SCAN Result  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='67 eV  5  0  5  Strain (%)9  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The application of compressive strain up to -2% positions the VBM well below  the oxidation potential of water but the CBM still lies below the reduction potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' On  further increase in the compressive strain from -3 to -5%, both the CBM and VBM lie  outside the range of water redox potentials hence enhancing the photocatalytic activities  of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' When the heterostructure is subjected to vertical tensile strain from  +1% to +5%, both the CBM and VBM straddle the water redox potential range hence  enhancing the photocatalytic response under visible light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' There fore we conclude that  the present heterostructure exhibits efficient charge separation for photocatalytic water  splitting under tensile strain and larger compressive strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The sample is further subjected to uniaxial and biaxial strain to analyse the  photocatalytic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The uniaxial strain is applied by increasing or compressing  the in-plane lattice constant along the X- direction keeping the Y value fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Similarly,  the biaxial strain is applied by increasing or decreasing the X-Y lattice constant values  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It is to be noted that the interlayer separation is kept at its equilibrium  value i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='e 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='373 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The detailed graphical representation of band alignment under uniaxial  strain is given in the Supporting information S3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It was observed that for PBE  functionals the CBM and VBM both lie within the redox potential range for the entire  range of uniaxial strain from -5 to +5%, thus indicating that uniaxial strain does not  produce a significant improvement to be useful for photocatalytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  To get a better result, the strain calculation is repeated for the heterostructure  using SCAN functional and is given in Supporting information S3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' It was observed  that the application of larger uniaxial strain (-3 to -5%) puts the CBM and VBM outside  the water redox potential range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' But the tensile strain reduces the band edge separation  even below the unstrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The VBM lies below the oxidation potential value but  the CBM finds its position below the reduction potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore we found that the  sample can be useful for photocatalytic water splitting for larger compressive uniaxial  strain whereas the tensile strain does not produce an enhancement in the photocatalytic  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The present C2N/MoS2 heterostructure is also investigated under biaxial  strain using both PBE and SCAN functionals (See Supporting information S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  results are quite similar to that of the uniaxial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Like previous results, PBE  functional does not produce a better result for the photocatalytic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' A larger  compressive strain puts the CBM above the reduction potential but the VBM lies above  the oxidation potential of water, thus not providing a useful result for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Using SCAN functional, it was observed that from -2 to -5% compressive strain both  the CBM and VBM straddle outside the water redox potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' On the other hand,  tensile strain decreases the band edges position and put the CBM below the reduction  potential of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore, the heterostructure can be used for photocatalytic studies  for larger uniaxial and biaxial compressive strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  In order to illustrate the charge transfer process between the C2N and MoS2  monolayers during the formation of the heterostructure, the charge density difference  is executed using PBE functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The charge density difference is calculated  by subtracting the charge density of the individual C2N and MoS2 monolayers  10  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Charge density difference of C2N/MoS2 heterostructure as a function of  vertical (a) -5% compressive (b) 0% (c) +5% tensile strain  from the C2N/MoS2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='The charge density difference for the unstrained  heterostructure is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In all our charge density figures, the cyan color  indicates the charge depletion and the yellow color represents the charge accumulation  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 5b, we observe that the charge accumulation mainly occurs in the  interface region and partially in MoS2 monolayers, whereas most of the charge depletion  occurs in the top C2N and bottom MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The change in charge density under vertical  strain is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  From the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='5a, we observe that with an increase in compressive vertical strain by  5%, there is an equal proportion of charge accumulation and depletion close to the MoS2  layer whereas the charge depletion mainly occurs in the region close to C2N layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  charge density given in cyan color near the C atom in the C2N monolayer indicates a  charge depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Similarly, the charge density given in yellow color near the S atom in  the MoS2 layer indicates a charge accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Hence this observation indicates that  there is a charge transfer from the S atom of MoS2 to the C atom of C2N monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The  intensification of the charge transfer process under large compressive strain indicates an  enhanced interaction between the two monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The strain-induced charge transfer  in C2N based heterostructures is consistent with other literature reports [26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  When the interlayer separation in increases to achieve a strain around +5%, the charge  depletion from the C2N layer almost disappears and most of the charge accumulation  occurs in the interface region close to MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In other words, there is no appreciable  charge redistribution close to the MoS2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' This may be the influence of the van  der Waals gap between the two monolayers which varies as the structure moves from a  compressive strain state to a tensile strain state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The effect of uniaxial compressive and tensile strain on the charge transfer process  is illustrated in supplementary information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  It indicates that there is no  appreciable change in the charge density difference under uniaxial compressive and  tensile strain (up to 5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' On the other hand application of biaxial strain indicates a  significant influence on the charge density under compressive and tensile strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' From  (a)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='= -5%  (b)  E,= 0%  (c)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='= +5%11  the Supplementary information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' S6, we observe that as the system is subjected to  5% compressive strain the charge depletion occurs in the C atom of the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' With  system changes from large compression to large tensile strain state, the depletion near  the C atom increases indicating more charge transfer from the S atom of MoS2 to the  C atom of C2N layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Therefore we observe that both vertical and biaxial strain affects  the charge transfer process significantly as compared to the uniaxial strain state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Conclusion  In  summary,  we  have  studied  the  photocatalytic  performance  of  C2N/MoS2  heterostructure as a function of uniaxial, biaxial, and vertical strain configuration  through first-principles DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  The unstrained heterostructure possesses  a direct band gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='35 eV, where the VBM is populated by Mo ‘d’ states and the  CBM is contributed by ‘C’ and ‘N’ ‘p’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The calculated position of CBM and  VBM of individual monolayers indicates that the heterostructure possesses a type-II  band alignment which is beneficial for charge separation across the two monolayers and  preventing their recombination process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The SCAN functional provides better results  for the band gap and band edge positions calculation as compared to that of the PBE  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Using the C2N/MoS2 heterostructure as a prototype, we have also found that the  meta-GGA SCAN functional shows similar results as compared to the computationally  expensive hybrid HSE functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The estimated CBM and CBM position as a function  of vertical strain indicates that the water reduction and oxidation potential values lie  within the band gap region with respect to the vacuum level for larger compressive  and tensile strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' In contrast, the system exhibits good photocatalytic performance  only for larger compression for uniaxial and biaxial strain states whereas the tensile  strain reduces the separation between the VBM and CBM within the water redox  potential value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The charge density difference indicates a significant charge transfer  for vertical and biaxial configuration as compared to the uniaxial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The present  study can help researchers to reduce the computational cost by considering the meta-  GGA functionals over HSE for electronic structure calculations of similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Our  calculation will also be extremely useful for designing artificial strained heterostructure  for the experimental community for better device application for photocatalytic water  splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' Computational Methods  The first principles electronic structure calculations were performed using density  functional theory (DFT) with projector augmented-wave method [29] as implemented  in the Vienna ab initio simulation package (VASP) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The Perdew-Burke-Ernzerhof  (PBE) [31] parametrization-based generalized gradient approximation (GGA) was  chosen for the exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' To accurately describe the interaction  between the layered structures, we have included the van der Waals correction method  REFERENCES  12  (DFT-D2) presented by Grimme [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Initially, the gap between C2N and MoS2  monolayers is kept at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='5 ˚A which is further optimized before running the electronic  structure calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' As a benchmark, the system is further studied using strongly  constrained and appropriately normed (SCAN) meta-GGA functionals to get more  accurate results as compared to the PBE functional and also to be consistent with the  reported experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' A vacuum layer of 20 ˚A was selected along the ‘Z’ direction  to avoid interaction between the adjacent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The plane wave expansion cut-off was  chosen to be 520 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The Brillouin zone integration was performed using a Γ-centered  (9×9×1) k-mesh for the structural relaxation and electronic structure calculation of the  isolated C2N and MoS2 monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' For the C2N/MoS2 heterostructure containing 354  atoms, the geometry optimization and electronic structure calculation are performed  using a Γ centered (2×2×1) k-point samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  All the structures are allowed for  relaxation to get the optimized atomic position until the total forces acting on each  atom are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='02 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Data availability statement  The additional data that support the findings of this article are available in the  Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Acknowledgments  The authors would like to thank the National Institute of Science Education and  Research (NISER), Department of Atomic Energy, Government of India, for funding  the research work through project number RIN-4001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content=' The authors acknowledge the  high-performance computing facility at NISER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Data availability statement  The additional data that support the findings of this article are available in the  Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
+page_content='  Conflict of interest  The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE2T4oBgHgl3EQfUAe8/content/2301.03809v1.pdf'}
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+arXiv:2301.08671v1  [physics.flu-dyn]  20 Jan 2023
+Under consideration for publication in J. Fluid Mech.
+1
+Near-onset dynamics in natural doubly
+diffusive convection
+C´edric Beaume1, Alastair M. Rucklidge1 and Joanna Tumelty1
+1School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
+(Received xx; revised xx; accepted xx)
+Doubly diffusive convection is considered in a vertical slot where horizontal temperature
+and solutal variations provide competing effects to the fluid density while allowing the
+existence of a conduction state. In this configuration, the linear stability of the conductive
+state is known, but the convection patterns arising from the primary instability have
+only been studied for specific parameter values. We have extended this by determining
+the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers
+using a weakly nonlinear analysis. The resulting convection branches are extended using
+numerical continuation and we find large-amplitude steady convection states can coexist
+with the stable conduction state for sub- and supercritical primary bifurcations. The
+stability of the convection states is investigated and attracting travelling waves and
+periodic orbits are identified using time-stepping when these steady states are unstable.
+Key words: Authors should not enter keywords on the manuscript, as these must
+be chosen by the author during the online submission process and will then be added
+during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfm-
+keywords.pdf for the full list)
+1. Introduction
+Doubly diffusive convection can occur when a binary fluid is subject to external gra-
+dients of temperature and of concentration. It has primarily been studied in the context
+of oceanography as an important mechanism for heat and salt transport (Huppert &
+Turner 1981; Schmitt 1994), since approximately 44% of the world’s oceans are known to
+display this phenomenon (You 2002). Doubly diffusive convection can display a wealth of
+behaviour that depends on the respective orientations of the (thermal and solutal) driving
+gradients. At low latitude, oceans typically feature thermohaline staircases (Schmitt et al.
+1987, 2005), where the flow is characterised by well-mixed horizontal layers interspersed
+with interfaces displaying sharp upward pointing gradients of temperature and salinity.
+In configurations forced by upward gradients of salinity and temperature, fluids display
+strikingly complex dynamics characterised by an alternation of well-mixed convection
+zones and fingers transporting salt mostly vertically (Krishnamurti 2003, 2009). This
+salt fingering instability is a natural mechanism that enhances the local mixing of the
+oceans (Schmitt 1994). At high latitude, the forcing gradients typically point downwards
+and perturbations to the thermohaline staircase give rise to oscillatory dynamics in a
+behaviour called diffusive layering (Kelley et al. 2003; P´erez-Santos et al. 2014). Similar
+doubly diffusive phenomena are also found at the Earth’s core-mantle boundary (Lay
+et al. 2008), in astrophysical flows (Spiegel 1969, 1972; Bethe 1990) and in processes
+
+2
+C. Beaume, A. M. Rucklidge and J. Tumelty
+involving solidification (Wilcox 1993), such as in magma crystallisation (Huppert &
+Sparks 1984).
+Originally motivated by the above configurations, doubly diffusive convection has
+become a paradigm for the study of fluids as dynamical systems. A large variety of flow
+states comprised of convection rolls have been identified, including standing, travelling
+and modulated waves (Deane et al. 1988; Kolodner 1991; Predtechensky et al. 1994).
+Temporal complexity has also been found in various forms (Spina et al. 1998; Batiste
+et al. 2001) and is generated in a number of ways (Knobloch et al. 1986; Rucklidge
+1992; Beaume 2020). Work focusing on steady state dynamics also revealed intricate
+phenomena like spatial localisation in the presence (Mercader et al. 2009, 2011) and
+in absence (Beaume et al. 2011) of Soret effect. Localised convection states, called
+convectons, are found on solution branches exhibiting oscillatory trajectories in parameter
+space in a behaviour called snaking (Knobloch 2015). Travelling versions of convectons
+have also been found and produce an interesting hierarchy of interconnected instabilities
+(Watanabe et al. 2012, 2016).
+Practical considerations led to the study of inclined domains, where gravity and the
+driving gradients are no longer aligned (Paliwal & Chen 1980a,b; Bergeon et al. 1999),
+as well as cases in which the salinity and temperature gradients are not aligned with
+each other (Tsitverblit & Kit 1993; Tsitverblit 1995; Dijkstra & Kranenborg 1996).
+Motivated by solidification fronts (Wilcox 1993) and mixing currents in the vicinity
+of icebergs (Huppert & Turner 1981), this article focuses on a configuration, typically
+referred to as natural doubly diffusive convection, where the driving gradients are aligned
+but orthogonal to gravity.
+The bifurcation scenario for a range of small aspect-ratio domains was elucidated by
+Xin et al. (1998) and Bergeon & Knobloch (2002), and large aspect-ratio domains were
+found to support the existence of spatially localised states (Bergeon & Knobloch 2008b).
+More recent work focused on a full characterisation of these localised states and on the
+emergence of chaos in large aspect-ratio domains (Beaume et al. 2013a,b, 2018; Beaume
+2020). Most of the pattern formation introduced in the above references can be found
+close to onset but they have, mostly, been studied for a certain set of parameter values.
+Despite the fact that a comprehensive linear stability analysis in the special case of
+balancing thermal and solutal gradients has been available for more than two decades
+(Ghorayeb & Mojtabi 1997), the analysis for unbalanced gradients was only recently
+attempted by Shankar et al. (2021). Further, little is known about the dynamics near
+onset in the balanced case besides its linear regime, which makes it difficult to extrapolate
+the dynamics of the system away from the parameter values used in previous studies.
+Here, we perform a weakly nonlinear analysis and augment it by numerically continuing
+branches of spatially periodic states in a small-aspect ratio domain. Our present work
+extends the previous analyses to a wider range of parameter values. In the next section,
+we present the mathematical framework associated with our case of doubly diffusive
+convection. In Section 3, we detail the weakly nonlinear analysis of this system, followed
+by a characterisation of the nonlinear dynamics in Section 4. We conclude in Section 5
+with a short discussion.
+2. Mathematical formulation
+We consider the natural doubly diffusive convection of an incompressible fluid in a
+two-dimensional domain with periodic boundary conditions in the vertical direction.
+The side walls are rigid, impermeable and maintained at fixed temperatures and solutal
+concentration. The right wall is held at a higher temperature (T0 + ∆T ) and solutal
+
+Near-onset dynamics in natural doubly diffusive convection
+3
+x∗
+z∗
+u∗
+w∗
+g
+T ∗ = T0 + ∆T,
+C∗ = C0 + ∆C,
+u∗ = 0,
+w∗ = 0,
+T ∗ = T0,
+C∗ = C0,
+u∗ = 0,
+w∗ = 0,
+L
+Figure 1: Sketch of the two-dimensional domain of interest together with the dimensional
+form of the boundary conditions.
+concentration (C0 + ∆C) than the left wall, where the temperature is T0 and the solutal
+concentration is C0. This configuration is depicted in figure 1.
+The system is governed by the Navier–Stokes equation for fluid momentum, the in-
+compressibility condition and advection-diffusion equations for both the temperature and
+the concentration. Cross-diffusion due to the Soret and Dufour effects is not considered.
+The imposed temperature and solutal concentration differences are assumed to be small
+enough so that the Boussinesq approximation can be applied, whereby density variations
+are neglected except in buoyancy terms. The density of the fluid is assumed to have a
+linear dependence on its temperature and concentration:
+ρ∗ = ρ0 + ρT (T ∗ − T0) + ρC(C∗ − C0),
+(2.1)
+where ρ0 is the density of the fluid at temperature T0 and concentration C0 and ρT < 0
+(resp. ρC > 0) is the thermal (resp. solutal) expansion coefficient.
+We introduce the non-dimensional quantities:
+x = x∗
+L ,
+t =
+t∗
+L2/κ,
+u = u∗
+κ/L,
+T = T ∗ − T0
+∆T
+,
+C = C∗ − C0
+∆C
+,
+p =
+p∗
+ρ0κν/L2 ,
+(2.2)
+where L is the wall separation, κ is the rate of thermal diffusivity and ν is the kinematic
+viscosity. The non-dimensional governing equations for the fluid velocity u = uˆx + wˆz,
+the pressure p, the temperature T and the concentration C thus read:
+1
+Pr
+�∂u
+∂t + u · ∇u
+�
+= −∇p + ∇2u + Ra (T + NC)ˆz,
+(2.3)
+∇ · u = 0,
+(2.4)
+∂T
+∂t + u · ∇T = ∇2T,
+(2.5)
+∂C
+∂t + u · ∇C = 1
+Le∇2C,
+(2.6)
+where ˆz is the vertical ascending unit vector and where we have introduced the following
+
+4
+C. Beaume, A. M. Rucklidge and J. Tumelty
+dimensionless parameters:
+the Prandtl number Pr = ν
+κ,
+(2.7)
+the Rayleigh number Ra = gL3|ρT |∆T
+ρ0νκ
+,
+(2.8)
+the buoyancy ratio N = ρC∆C
+ρT ∆T ,
+(2.9)
+and the Lewis number Le = κ
+D,
+(2.10)
+where D is the rate of solutal diffusivity. The non-dimensional boundary conditions read:
+u = 0,
+w = 0,
+− ∂p
+∂x + ∂2u
+∂x2 = 0,
+T = 0,
+C = 0
+on
+x = 0,
+(2.11)
+u = 0,
+w = 0,
+− ∂p
+∂x + ∂2u
+∂x2 = 0,
+T = 1,
+C = 1
+on
+x = 1,
+(2.12)
+where the pressure boundary condition is the projection of the Navier–Stokes equation
+on the boundary. Each variable is periodic in the vertical direction.
+We restrict our attention to the case N = −1, where the full system (2.3–2.6, 2.11, 2.12)
+admits the steady conduction state with linear temperature and concentration profiles
+between the side walls:
+u = 0,
+T = x,
+C = x.
+(2.13)
+We further introduce convective variables as the departures of the temperature and
+concentration from the conduction state:
+Θ = T − x,
+(2.14)
+Φ = C − x.
+(2.15)
+Using these new variables, the conduction state takes the form
+u = 0,
+Θ = 0,
+Φ = 0,
+(2.16)
+and system (2.3)–(2.6) can be written as:
+1
+Pr
+�∂u
+∂t + u · ∇u
+�
+= −∇p + ∇2u + Ra (Θ − Φ)ˆz,
+(2.17)
+∇ · u = 0,
+(2.18)
+∂Θ
+∂t + u · ∇Θ = −u + ∇2Θ,
+(2.19)
+∂Φ
+∂t + u · ∇Φ = −u + 1
+Le∇2Φ.
+(2.20)
+This formulation involving the convective variables allows the identification of two
+symmetries of the system: the reflection S∆ and the continuous translation Tδ:
+S∆ :
+(x, z) �→ (1 − x, −z),
+(u, w, Θ, Φ) �→ −(u, w, Θ, Φ),
+(2.21)
+Tδ :
+(x, z) �→ (x, z + δ),
+(u, w, Θ, Φ) �→ (u, w, Θ, Φ).
+(2.22)
+With periodic boundary conditions, these generate the symmetry group O(2) and restrict
+the types of bifurcation that can occur from the conduction state.
+
+Near-onset dynamics in natural doubly diffusive convection
+5
+3. Weakly nonlinear predictions
+To predict the pattern formation present in our system, we start by performing the
+linear stability analysis of the conduction state (u, w, p, Θ, Φ) = (0, 0, 0, 0, 0), which was
+previously done by Ghorayeb & Mojtabi (1997) and by Xin et al. (1998). We briefly
+rederive their results in the following subsection so that they can be applied in the later
+weakly nonlinear analysis, where we derive Ginzburg–Landau equations to model the
+small-amplitude behaviour close to the primary bifurcation for all Lewis and Prandtl
+numbers.
+3.1. Linear stability analysis
+We first consider small-amplitude stationary normal mode perturbations to the con-
+duction state:
+(u, w, p, Θ, Φ)T = ǫ
+�
+(U1(x), W1(x), P1(x), Θ1(x), Φ1(x))T eikz + c.c.
+�
++ O(ǫ2),
+(3.1)
+where c.c. denotes the complex conjugate of the preceding term, ǫ ≪ 1, k is the vertical
+wavenumber and λ is the temporal growth rate of the perturbation. Inserting expansion
+(3.1) into system (2.17)–(2.20) and linearising the resulting system yields the eigenvalue
+problem:
+L(Ra)Ψ1 = 0,
+(3.2)
+for Ra and Ψ1 where
+Ψ1 = (U1, W1, P1, Θ1, Φ1)T eikz + c.c.,
+(3.3)
+and
+L(Ra) =
+
+
+
+
+
+
+∇2
+0
+−∂x
+0
+0
+0
+∇2
+−∂z
+Ra
+−Ra
+∂x
+∂z
+0
+0
+0
+−1
+0
+0
+∇2
+0
+−1
+0
+0
+0
+1
+Le∇2
+
+
+
+
+
+
+.
+(3.4)
+The complex functions U1, W1, P1, Θ1 and Φ1 satisfy Dirichlet boundary conditions on
+the side walls for the velocity, temperature and concentration perturbations:
+U1(x) = W1(x) = Θ1(x) = Φ1(x) = 0
+on x = 0, 1,
+(3.5)
+and the projection of the Navier–Stokes equation onto the boundary for the pressure
+perturbation:
+0 = −∂P1
+∂x + ∂2U1
+∂2x
+on x = 0, 1.
+(3.6)
+Solutions to (3.2–3.6) are independent of Pr and satisfy Φ1 = Le Θ1. Consequently,
+the only parameter dependence in the linear problem comes from the buoyancy term in
+the momentum equation, which takes the form Ra(1−Le)Θ1. The accordingly simplified
+version of equation (3.2) is then solved using a spectral eigenvalue solver based on a
+Chebyshev–Legendre collocation method for a range of k to determine the marginal
+stability curve in figure 2. This curve reveals a minimum at kc ≈ 2.5318 and Rac|Le−1| ≈
+6509, which corresponds to the primary instability of the conduction state. The absolute
+value here comes from the fact that the system resulting from left wall heating and that
+resulting from right wall heating are equivalent. Contour plots presenting the fields for
+the velocity components, streamfunction and convective variables of this eigenmode for
+Le = 11 are shown in figure 3. The conduction state is thus first unstable to a spatially
+
+6
+C. Beaume, A. M. Rucklidge and J. Tumelty
+Figure 2: Marginal stability curve for the onset of doubly diffusive convection. The
+conduction state is stable to modes with wavenumber k below the curve, and unstable
+to them above. The minimum of this curve is Rac|Le − 1| ≈ 6509 with wavenumber
+kc ≈ 2.5318 and corresponds to the location of the primary bifurcation.
+(a)
+(b)
+(c)
+(d)
+(e)
+Figure 3: Contour plots of a single wavelength of the real critical eigenvector Ψ1 for
+Le = 11 (kc ≈ 2.53). The profiles show the perturbations in (a) horizontal velocity, u,
+(b) vertical velocity, w, (c) velocity streamfunction, ψ, where u = −ψz and w = ψx, and
+in the convective variables (d) Θ and (e) Φ. Black (grey, dotted) lines indicate positive
+(negative, zero) values and are separated by 20% of the maximum absolute value.
+periodic state constituted of counter-rotating convection rolls that, when Le > 1, slant
+downwards from the hotter wall, filling the domain and extending to the cold wall.
+The conduction state can also undergo Hopf bifurcations, where the growth rate is
+purely imaginary. However, for the parameter values tested, these bifurcations occurred
+for Rayleigh numbers that are orders of magnitude larger than that of the primary
+stationary bifurcation and are therefore out of the scope of the present work.
+3.2. Weakly nonlinear analysis
+To investigate the weakly nonlinear regime around this primary bifurcation, we set
+Ra = Rac + ǫ2r with r = O(1) and ǫ ≪ 1 and assume that the system evolves on a
+slow temporal scale T1 = ǫ2t. We also introduce a long spatial scale, Z = ǫz, to allow
+small-amplitude states with long spatial modulations. The small-aspect ratio domains
+
+Near-onset dynamics in natural doubly diffusive convection
+7
+considered in the numerical computations in Section 4 do not permit these long-scale
+modulations, so terms involving derivatives with respect to Z may be ignored in the
+subsequent analysis with no effect on the main result of this section: the criticality of
+the primary bifurcation. However, this long spatial variable has been included here to
+broaden the scope of the analysis and will be considered in future work. We emphasise
+that each of the state variables of our system—u, w, p, Θ and Φ—depend upon the
+independent variables: x, z, Z and T1. Using this multiple-scales approach, the partial
+derivatives become
+∂
+∂t �→ ǫ2 ∂
+∂T1
+and
+∂
+∂z �→ ∂
+∂z + ǫ ∂
+∂Z .
+(3.7)
+Introducing the notation Ψ = (u, w, p, Θ, Φ)T , we can express each of the variables as a
+perturbation expansion in ǫ about the conduction state Ψ0 = (0, 0, 0, 0, 0)T:
+Ψ = Ψ0 + ǫΨ1 + ǫ2Ψ2 + . . . ,
+(3.8)
+where Ψj = (uj, wj, pj, θj, φj)T is the correction at O(ǫj) for j = 1, 2, . . . .
+The corrections are periodic in z and also satisfy homogeneous boundary conditions
+at each order in ǫ:
+uj = wj = θj = φj = 0
+on x = 0, 1,
+j = 1, 2, . . . ,
+(3.9)
+as well as the pressure boundary condition:
+∂2uj
+∂x2 − ∂pj
+∂x = 0
+on x = 0, 1,
+j = 1, 2, . . . .
+(3.10)
+The expansion (3.8) is substituted into the full system (2.17–2.20) and the perturbations
+are solved numerically order-by-order in ǫ using an extension of the aforementioned
+collocation method. By further extracting the parameter dependence of the perturbations
+at each order, we obtain a Ginzburg–Landau equation that can be applied for all
+parameter values and will indicate the criticality of the primary bifurcation.
+We proceed by detailing this formulation, which should be applied to the cases Le > 1
+and Le < 1 separately, owing to the parameter combination Ra(1 − Le) changing sign
+between them. However, the results for Le < 1 can be related to those for Le > 1 by
+using an alternative non-dimensionalisation to (2.2) involving the solutal diffusivity, D,
+instead of thermal diffusivity, κ, which results in a set of equations like (2.3–2.6), except
+with T and C exchanged and the Lewis, Prandtl and Rayleigh numbers replaced by
+the inverse Lewis number, Schmidt number, Sc = LePr, and solutal Rayleigh number,
+RaS = −RaNLe, respectively.
+The conduction state solves the system at leading order. At O(ǫ), the correction is
+given by the solution to linear system (3.2):
+Ψ1 = A1(Z, T1)
+�
+U1(x), W1(x), P1(x), Θ1(x), LeΘ1(x)
+�T
+eikcz + c.c.,
+(3.11)
+where kc is the critical wavenumber. No phase constraint is applied at this point, but the
+amplitude of the eigenfunction is fixed using:
+⟨U1 , U1⟩ + ⟨W1 , W1⟩ + ⟨P1 , P1⟩ + ⟨Θ1 , Θ1⟩ = 1,
+(3.12)
+with the inner product:
+⟨f , g⟩ = 1
+λc
+� λc
+0
+� 1
+0
+f
+T g dx dz,
+(3.13)
+
+8
+C. Beaume, A. M. Rucklidge and J. Tumelty
+fij
+j
+0
+1
+2
+1 U1 dU1
+dx + ikcW 1U1 + c.c.
+−2ikcU1
+U1 dU1
+dx + ikcW1U1
+2 U1 dW1
+dx + ikcW 1W1 + c.c. −2ikcW1 + P1 U1 dW1
+dx + ikcW 2
+1
+i 3 0
+−W1
+0
+4 U1 dΘ1
+dx + ikcW 1Θ1 + c.c.
+−2ikcΘ1
+U1 dΘ1
+dx + ikcW1Θ1
+Table 1: Functions fij (i = 1, 2, 3, 4, j = 0, 1, 2) in the nonlinear term N2 at O(ǫ2) in
+(3.14). The overbar denotes complex conjugation.
+where λc = 2π/kc is the wavelength of the critical eigenvector, the overbar denotes
+complex conjugation and the superscript T denotes the transposition operation when
+f is a vector. Due to the lack of available explicit expressions for the solutions to this
+perturbation problem, these inner products need to be computed numerically, which
+we achieved by using the Clenshaw–Curtis quadrature on the collocation nodes used
+in Section 3.1. The amplitude A1 evolves over both long spatial and temporal scales
+according to an amplitude equation that will be determined at higher order.
+At O(ǫ2), the linear operator L acts on the second-order terms and is forced by both
+the nonlinear terms between the O(ǫ) corrections and terms proportional to the slow
+spatial derivative of the O(ǫ) correction A1Z:
+L(Rac)Ψ2 =
+
+
+
+
+
+
+
+
+1
+Prf10|A1|2 +
+�
+A1Zf11eikcz + c.c.
+�
++ 1
+Pr
+�
+f12A2
+1e2ikcz + c.c.
+�
+1
+Prf20|A1|2 +
+�
+A1Zf21eikcz + c.c.
+�
++ 1
+Pr
+�
+f22A2
+1e2ikcz + c.c.
+�
+�
+A1Zf31eikcz + c.c.
+�
+f40|A1|2 +
+�
+A1Zf41eikcz + c.c.
+�
++
+�
+f42A2
+1e2ikcz + c.c.
+�
+Lef40|A1|2 +
+�
+A1Zf41eikcz + c.c.
+�
++ Le
+�
+f42A2
+1e2ikcz + c.c.
+�
+
+
+
+
+
+
+
+
+�
+����
+�
+N2
+,
+(3.14)
+where the functions fij(x) for i = 1, 2, 3, 4 and j = 0, 1, 2 are independent of Pr and Le
+and are given in table 1.
+To ensure the existence of a unique solution at this order, we derive a solvability
+condition using the Fredholm alternative theorem. This involves the adjoint operator to
+L, L†, defined through the relationship
+⟨f , Lg⟩ = ⟨L†f , g⟩,
+(3.15)
+which holds for all vector functions f and g. Integrating the left-hand side by parts, we
+find that the adjoint operator takes the form:
+L† =
+
+
+
+
+
+
+∇2
+0
+−∂x
+−1
+−1
+0
+∇2
+−∂z
+0
+0
+∂x
+∂z
+0
+0
+0
+0
+Rac
+0
+∇2
+0
+0
+−Rac
+0
+0
+1
+Le∇2
+
+
+
+
+
+
+,
+(3.16)
+together with the adjoint boundary conditions:
+u† = 0, w† = 0, θ† = 0, φ† = 0
+on x = 0, 1,
+(3.17)
+∂2u†
+∂x2 − ∂p†
+∂x = 0
+on x = 0, 1,
+(3.18)
+
+Near-onset dynamics in natural doubly diffusive convection
+9
+and periodicity in the vertical direction.
+The Fredholm alternative allows us to pose the adjoint problem:
+L†Ψ † = 0,
+(3.19)
+whose solution is unique up to a vertical translation and a multiplicative constant. This
+solution may be written in the form:
+Ψ † =
+�
+U †(x), W †(x), P †(x),
+1
+1 − LeΘ†(x), −
+Le
+1 − LeΘ†(x)
+�T
+eikcz + c.c.,
+(3.20)
+where the parameter dependence of the components has been extracted. The amplitude
+and phase are fixed by imposing the conditions:
+⟨U † , U †⟩ + ⟨W † , W †⟩ + ⟨P † , P †⟩ + ⟨Θ† , Θ†⟩ = 1,
+(3.21)
+and
+Im
+�
+⟨U † , U1⟩
+�
+= 0,
+(3.22)
+where Im represents the imaginary part, respectively.
+Using this adjoint solution, we then apply the O(ǫ2) solvability condition:
+⟨Ψ † , N2⟩ = 0.
+(3.23)
+Owing to the vertical wavenumber dependence of terms in N2, the only non-trivial
+contributions come from those proportional to A1Z and their complex conjugates and
+equation (3.23) then reduces to
+−2ikc⟨U †, U1⟩ − 2ikc⟨W †, W1⟩ + ⟨W †, P1⟩ − ⟨P †, W1⟩ − 2ikc⟨Θ†, Θ1⟩ = 0,
+(3.24)
+which may be further simplified to:
+�
+Ψ † , ∂LΨ
+∂kc
+�
+= 0.
+(3.25)
+Thus, this solvability condition is automatically satisfied as the primary bifurcation
+occurs at a quadratic minimum of the marginal stability curve (see figure 2).
+The O(ǫ2) system (3.14) can be solved to find that the second-order correction to the
+conduction state is
+Ψ2 = |A1|2Ψ 0
+2 + ((A2Ψ 1
+1 + A1ZΨ 1
+2 )eikcz + c.c.) + (A2
+1Ψ 2
+2 e2ikcz + c.c.),
+(3.26)
+where Ψ2 = (u2, w2, p2, θ2, φ2)T and the functions Ψ i
+2 for i = 0, 1, 2 have the following
+parameter dependence:
+Ψ 0
+2 =
+�
+0, 1
+Pr ˜w2 + (1 + Le) ˜w3, 1
+Pr ˜p2, ˜θ3, Le2˜θ3
+�T
+,
+(3.27)
+Ψ 1
+2 =
+�
+˜u7, ˜w7, ˜p7, ˜θ7, Le ˜θ7
+�T
+,
+(3.28)
+Ψ 2
+2 =
+� 1
+Pr
+�
+˜u4, ˜w4, ˜p4, ˜θ4, Le˜θ4
+�
++ (1 + Le)(˜u5, ˜w5, ˜p5, 0, 0)
++ (0, 0, 0, ˜θ5, Le2˜θ5) + Le(0, 0, 0, ˜θ6, ˜θ6)
+�T
+.
+(3.29)
+The newly introduced functions ˜ui, ˜wi, ˜pi and ˜θi for i = 2, ..., 7 are independent of Le
+and Pr and satisfy equations (A 1–A 5) in Section A.1 of the Appendix.
+
+10
+C. Beaume, A. M. Rucklidge and J. Tumelty
+Continuing to O(ǫ3), both the deviation away from the critical Rayleigh number and
+the slow time-dependence of the solution appear in the right-hand side of the resulting
+system, in addition to nonlinear terms between first and second-order corrections and
+terms with slow spatial derivatives. The system to solve at third-order is
+L(Rac)Ψ3 =
+
+
+
+
+
+
+
+
+
+
+1
+P r
+�
+∂u1
+∂T1 + J(u, u)
+�
+− 2 ∂2u2
+∂z∂Z − ∂2u1
+∂Z2
+1
+P r
+�
+∂w1
+∂T1 + J(u, w)
+�
+− r (θ1 − φ1) − 2 ∂2w2
+∂z∂Z − ∂2w1
+∂Z2 + ∂p2
+∂Z
+− ∂w2
+∂Z
+�
+∂θ1
+∂T1 + J(u, θ)
+�
+− 2 ∂2θ2
+∂z∂Z − ∂2θ1
+∂Z2
+�
+∂φ1
+∂T1 + J(u, φ)
+�
+−
+2
+Le
+∂2φ2
+∂z∂Z −
+1
+Le
+∂2φ1
+∂Z2
+
+
+
+
+
+
+
+
+
+
+�
+��
+�
+N3
+,
+(3.30)
+where the advective terms are
+J(u, f) = u1 · ∇f2 + u2 · ∇f1 + w1∂Zf1,
+(3.31)
+where f1 and f2, respectively, refer to the first- and second-order corrections of the
+variables f = u, w, θ and φ.
+The solvability condition at this order:
+⟨Ψ † , N3⟩ = 0,
+(3.32)
+is no longer trivially satisfied because some nonlinear terms contained in N3 have eikcz
+dependence arising from terms proportional to A1, |A1|2A1, A1ZZ, A2Z and their complex
+conjugates. However, the contributions to (3.32) from terms proportional to A2Z, cancel
+for the same reason that the solvability condition at O(ǫ2) was satisfied. Consequently,
+A2 remains arbitrary at this order. Collecting the remaining terms from equation (3.32),
+we obtain the Ginzburg–Landau equation (GLE), that holds for both Le > 1 and Le < 1:
+αA1T1 = γrA1 + β|A1|2A1 + δA1ZZ,
+(3.33)
+where table 2 indicates which terms of N3 contribute to each term above. This equation is
+equivariant under the O(2) symmetry so we may chose the phase of the O(ǫ) correction so
+that these coefficients are real. The coefficient δ is independent of the physical parameters
+Pr and Le, while α, β and γ satisfy the relations:
+α = 1
+Prα1 + (1 + Le)α2,
+(3.34)
+β =
+1
+Pr2 β1 + 1 + Le
+Pr
+β2 + (1 + Le2)β3 + Leβ4,
+(3.35)
+γ = (1 − Le)γ1,
+(3.36)
+where full expressions used to obtain αi, βi, γ1 and δ are provided in (A 6–A 9) and
+evaluated in table 4 in Section A.2. By dividing (3.33) through by α, equation (3.33) is
+more conveniently written as
+A1T1 = a1rA1 + a2|A1|2A1 + a3A1ZZ,
+(3.37)
+where a1 = γ/α, a2 = β/α and a3 = δ/α.
+The solutions to the Ginzburg–Landau equation (3.37) are good approximations of the
+small-amplitude solutions of the full doubly diffusive system (2.17–2.20). Of particular
+interest here are the two steady solutions that are invariant with respect to the long
+
+Near-onset dynamics in natural doubly diffusive convection
+11
+Term in the Ginzburg–Landau equation (3.33)
+αA1T1
+γrA1
+β|A1|2A1
+δA1ZZ
+Terms in N3 proportional to
+∂f1
+∂T1
+r(θ1 − φ1)
+u1 · ∇f2
+∂2f2
+∂z∂Z
+∂f1
+∂Z2
+u2 · ∇f1
+∂p2
+∂Z
+∂w2
+∂Z
+Table 2: Terms from N3 (see equation (3.30)) contributing to the Ginzburg–Landau
+equation (3.33). The column in which these terms are placed informs on the term to
+which they contribute. Here, f1 and f2, respectively refer to first- and second-order
+corrections of the variables f = u, w, θ and φ.
+spatial scale Z. The first of these solutions is the trivial solution:
+A1 = 0.
+(3.38)
+This solution is valid for all r and corresponds to the conduction state (2.16). The second
+important solution is:
+A1 =
+�
+−a1r
+a2
+�1/2
+eiχ,
+(3.39)
+where χ is an arbitrary phase. This solution relates to states of small-amplitude spatially-
+periodic convection that can be found near the primary bifurcation. These fluid states
+can then be approximated by
+(u, w, p, Θ, Φ)T ≈
+�
+−a1(Ra − Rac)
+a2
+�
+U1(x), W1(x), P1(x), Θ1(x), LeΘ1(x)
+�T
+eikcz + c.c,
+(3.40)
+where the phase χ has been absorbed into z via a vertical translation. These states only
+exist at small-amplitude for Rayleigh numbers that satisfy
+a1
+a2
+(Rac − Ra) > 0.
+(3.41)
+Consequently, the sign of the ratio a1/a2 determines the criticality of the primary
+bifurcation and the initial direction of branching.
+Using the numerical values in table 4, we computed ai over a range of parameter
+values. The coefficients a1 and a3 are positive for all Pr provided Le ̸= 1, whereas the
+sign of a2 changes as these parameters are varied. As a result, there exists a boundary
+in parameter space that separates regions where the primary bifurcation is subcritical
+(a2 > 0) from those where it is supercritical (a2 < 0). This boundary is shown in
+figure 4 and implies that, for any value of the Lewis number, there exists a critical
+value of the Prandtl number, Prc(Le), expressed in terms of the physical parameters
+in (A 15), above which the bifurcation is subcritical. This critical value tends to 0.376
+for small Lewis numbers while it approaches the asymptotic relation Prc ∼ 0.376/Le
+as the Lewis number tends to infinity. We further note that the parameter values for
+physical doubly diffusive systems from Schmitt (1983) all lie within the region where the
+primary bifurcation is subcritical. While we are unaware of further fluid systems lying
+within the supercritical region of parameter space, we expect that they exist since some of
+the physical systems identified in figure 4, including humidity/ heat and stellar interiors
+
+12
+C. Beaume, A. M. Rucklidge and J. Tumelty
+Figure 4: Boundary a2 = 0 in (Le, Pr) parameter space separating the region where the
+primary bifurcation from the conduction state is subcritical (above) from that where it is
+supercritical (below). The conduction state is linearly stable for all Ra at Le = 1 and this
+point is indicated by the open circle. The grey regions indicate parameter values from
+Schmitt (1983) for the physical doubly diffusive systems: (1) salt/sugar, (2) magmas,
+(3) oxide semiconductors, (4) heat/salt 0◦C, (5) heat/salt 30◦C, (6) humidity/heat, (7)
+liquid metals and (8) stellar interiors.
+(marked (6) and (8), respectively), have parameter values that are within an order of
+magnitude of the sub/supercritical boundary.
+We can gain physical insight into the criticality of the primary bifurcation by examining
+the contributions that each of the nonlinear terms from equations (2.3–2.6) make to a2,
+using a similar approach to the one Requil´e et al. (2020) applied to plane Poiseuille and
+plane Couette flows with viscous dissipation. The expression of the coefficient β (3.35)
+and the corresponding numerical values provided in table 4 in the Appendix, show that
+the inertial term u · ∇u (contributing to β1 and β2) provides a negative contribution
+to a2, whereas thermal u · ∇T and solutal u · ∇C advective terms (mostly contributing
+to β3 and ��4) provide a positive contribution to a2. The latter statement is further
+justified in Section A.3 in the Appendix. It is therefore solutal and thermal advection in
+the system that drives the subcriticality of the primary bifurcation, while inertial effects
+drive the supercriticality. Thus, reducing the Prandtl number reduces the subcriticality
+of the bifurcation since the effects of inertia are strengthened.
+The final term in the Ginzburg–Landau equation (3.37), a3A1ZZ, allows small-
+amplitude solutions of the doubly diffusive system to exhibit long scale amplitude
+modulation. These solutions include phase-winding states that describe patterns whose
+wavenumbers are close to the critical wavenumber kc (Cross et al. 1983), and spatially
+modulated states that can develop into localised states away from the primary bifurcation
+(Bergeon & Knobloch 2008a). These are out of the scope of the present work, but we
+will consider the effect of the term a3A1ZZ on spatially localised states in future work.
+4. Fully nonlinear behaviour
+Having established the region of (Le, Pr) parameter space in which the bifurcation is
+subcritical, we can now investigate the nonlinear behaviour of the system near the onset
+of convection. In particular, we focus on the structure and stability of the primary branch
+of spatially periodic convection states with wavenumber kc as it extends towards larger
+amplitudes. For this, we consider a single-wavelength domain with Lz = λc = 2π/kc,
+
+Near-onset dynamics in natural doubly diffusive convection
+13
+which precludes modulational instabilities arising in large domains that are captured by
+our weakly nonlinear analysis through the A1ZZ term in equation (3.37).
+We numerically continue the primary branch against the Rayleigh number across a
+range of Lewis (Le ∈ [5, 100]) and Prandtl (Pr ∈ [2 × 10−3, 10]) numbers. Cases for
+which Le < 1 can be extrapolated from our results by a suitable transformation. The
+solution branches will be identified on bifurcation diagrams showing either the total
+kinetic energy of steady states:
+E = 1
+2
+� λc
+0
+� 1
+0
+�
+u2 + w2�
+dx dz,
+(4.1)
+or the average velocity ∥u∥2 =
+�
+2E/λc, against the Rayleigh number Ra.
+Computations were carried out using a spectral element numerical method based on a
+Gauss–Lobatto–Legendre discretisation (Bergeon & Knobloch 2002) and supplemented
+by Stokes preconditioning with ∆t = 0.1, as detailed by Beaume (2017). Numerical
+results were validated against a discretisation of up to 4 spectral elements with 29 nodes
+in both the x and z directions. The stability of the steady states was computed using an
+Arnoldi method based on a time-stepping scheme (Mamun & Tuckerman 1995). Further
+direct numerical simulations used a stiffly stable second-order splitting scheme based on
+Karniadakis et al. (1991) with time-step ∆t = 10−3.
+4.1. Bifurcation structure
+The results can be summarised by dividing parameter space according to the qual-
+itative nature of the bifurcation diagram. Figure 5(a) indicates the four main regimes
+found. Region (1) describes the moderate and large Pr behaviour for all Le. In this
+region, the primary bifurcation is strongly subcritical and the primary branch has a
+single saddle-node, as shown in figure 5(b). Parameter values within this region have
+received the most attention in previous studies focusing on subcritical pattern formation
+(e.g. see (Xin et al. 1998; Bergeon & Knobloch 2008b)). Region (2) occupies a small
+region of parameter space above the boundary Pr = Prc, where the primary bifurcation
+is weakly subcritical, and separates the typical subcritical behaviour in region (1) from
+the supercritical behaviour in regions (3) or (4). The steady convection state branches
+typically have three saddle-nodes in region (2), as exemplified in figure 5(c). Regions (3)
+and (4) identify the two qualitatively different types of bifurcation diagrams observable
+when the primary bifurcation is supercritical. In both cases, the primary branch has two
+saddle-nodes, with the first lying in the supercritical region Ra > Rac. The difference
+between the regions is the location of the second saddle-node: in region (3), it is found
+for Ra < Rac (see figure 5(d)), whereas, in region (4), it is found in Ra > Rac (see
+figure 5(e)). Consequently, a large-amplitude convection state may coexist with the stable
+conduction state when the primary bifurcation is supercritical, but, for sufficiently small
+Pr, steady convection states are found entirely within the supercritical region, where the
+conduction state is unstable. There may exist a fifth region, where the primary branch
+increases monotonically in both Rayleigh number and in amplitude, but we have not
+identified it in this study.
+We now determine the structure of the primary branch as Pr decreases for a fixed value
+of Le. To achieve this, we follow the locations of its three saddle-nodes with respect to
+Ra and Pr. In doing so, we observed two different scenarios according to whether the
+pair of saddle-nodes are created on the lower or upper part of the primary branch.
+These are exemplified in figure 6 for Le = 11 (representative of 5 ⩽ Le ≲ 15) and
+Le = 20 (representative of 19 ≲ Le < 100). Since the transition between the two scenarios
+
+14
+C. Beaume, A. M. Rucklidge and J. Tumelty
+(a)
+(b) Region 1
+(c) Region 2
+(d) Region 3
+(e) Region 4
+Figure 5: (a) Enlargement of a subset of the parameter space shown in figure 4 showing
+four regions where the bifurcation diagrams exhibit qualitatively different behaviours.
+The thick line separates subcritical from supercritical branching, while the additional
+region boundaries are identified with either dotted or dashed lines. (b–e) Representative
+bifurcation diagrams for parameter values within each of the four regions. The stability
+of the branch segments are also indicated using thick lines for stable solutions, thin lines
+for solutions unstable to amplitude perturbations and dashed lines for solutions unstable
+to drift. The location of bifurcations depend upon the specific parameter values used, so
+those used for each sketch have been marked in panel (a).
+7(a)
+7(b)
+7(c)
+7(e)
+7(f)
+7(d)
+(a)
+7(g)
+7(h)
+7(k)
+7(l)
+7(i)
+7(j)
+(b)
+Figure 6: Location of the three saddle-node bifurcations of the primary branch in (Ra, Pr)
+parameter space for (a) Le = 11 and (b) Le = 20. The dashed line marks the critical
+Rayleigh number at which the primary bifurcation is found, Rac, and the cross marks the
+codimension two point (Rac, Prc) explained in the text. The insets provide enlargements
+of the area around Rac in each case. Arrows mark the bifurcation diagrams shown in
+figure 7.
+
+Near-onset dynamics in natural doubly diffusive convection
+15
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+Figure 7: Bifurcation diagram showing the primary branch of steady convection and the
+stability of the related states across the four regions, indicated in the top left corner.
+Thick lines indicate stable solutions, thin lines indicate solutions unstable to amplitude
+perturbations and dashed lines indicate solutions unstable to drift. Saddle-nodes are
+marked by symbols: SN1 (filled circle), SN2 (asterisk) and SN3 (triangle). The open circle
+corresponds to the destabilising drift bifurcation. The parameter values, also indicated
+by the arrows in figure 6, are: Le = 11, and (a) Pr = 1, (b) Pr = 0.1, (c) Pr = 0.042,
+(d) Pr = 0.032, (e) Pr = 0.01, (f) Pr = 0.005; as well as Le = 20, and (g) Pr = 1, (h)
+Pr = 0.1, (i) Pr = 0.023, (j) Pr = 0.02, (k) Pr = 0.01 and (l) Pr = 0.005. For Le = 11
+(resp. Le = 20), Prc ≈ 0.031, Prcusp ≈ 0.033 (resp. Prc ≈ 0.018, Prcusp ≈ 0.023).
+occupies a small region of parameter space within region (2) for 15 ≲ Le ≲ 19, we did
+not investigate it any further.
+To help interpret the plots in figure 6, figure 7 demonstrates the evolution of the
+bifurcation diagrams as Pr decreases for Le = 11 (panels (a–f)) and Le = 20 (panels
+(g–l)). The structure of the bifurcation diagrams in region (1), for high Pr, remain
+similar, as shown in figures 7(a) and 7(g). From the primary bifurcation, the primary
+branch extends towards lower Rayleigh numbers and proceeds to turn around at a saddle-
+node, hereafter referred to as SN1, before heading towards large amplitude convection
+states at large Ra. Figure 6 suggests that, as Pr → ∞, the location of SN1 tends to a
+constant Rayleigh number, dependent upon Le. This figure also shows that SN1 occurs
+at larger Ra as the Prandtl number is decreased and the primary bifurcation becomes
+less subcritical.
+Upon decreasing the Prandtl number, the primary branch undergoes a cusp bifurcation
+at Pr ≈ Prcusp(Le) > Prc(Le), while still subcritical, denoting the beginning of
+region (2). The cusp produces two additional saddle-nodes along the primary branch:
+SN2 and SN3. The exact process by which this is achieved depends on the Lewis number.
+For Le ≲ 15, the cusp bifurcation occurs at smaller amplitude than SN1 and the saddle-
+nodes are labelled SN3, SN2, SN1 as the branch is followed in the direction of increasing
+
+16
+C. Beaume, A. M. Rucklidge and J. Tumelty
+Figure 8: Saddle-node locations for Le = 5 (green), Le = 11 (purple), Le = 20 (blue)
+and Le = 50 (red). The black dashed line marks the location of the primary bifurcation
+and the red cross marks the codimension two point where the criticality of the primary
+bifurcation changes, at Pr = Prc. The black dashed line represents relationship (4.2).
+Saddle-nodes are marked by circles for SN1 asterisks for SN2 and triangles for SN3.
+energy (see, for example, figure 7(d) for Le = 11 and Pr = 0.032, near the cusp parameter
+value: Prcusp ≈ 0.033). In contrast, for Le ≳ 19, the cusp bifurcation occurs at higher
+amplitude than SN1 and saddle-nodes are labelled SN1, SN2, SN3, as shown in figure 7(i)
+for Le = 20, Pr = 0.023 ≈ Prcusp.
+Continuing to reduce Pr across region (2) (from Prcusp to Prc), the Rayleigh number
+associated with SN2 increases so that it reaches the supercritical region before Pr = Prc.
+During this transition, the saddle-node with smallest amplitude (SN3 for Le ≲ 15; SN1
+for Le ≳ 19) moves to larger Rayleigh numbers but with decreasing amplitude until it
+collides with the primary bifurcation at Pr = Prc and Ra = Rac, where the primary
+bifurcation changes from subcritical to supercritical. This process is highlighted in the
+insets of figure 6 and results in the primary branch possessing only two saddle-nodes in
+the supercritical regime (Pr < Prc).
+The locations of the remaining two saddle-nodes go toward larger Ra as Pr decreases
+and are found in the supercritical region (Ra > Rac) in region (4), as shown in
+figure 5. It is therefore clear that multiple steady convection states can exist for the
+same parameter values near the onset of convection, regardless of the criticality of the
+primary bifurcation. This result extends earlier observations on the number of saddle-
+node bifurcations occurring along the primary branch in related systems (Tsitverblit &
+Kit 1993).
+More insight into these results can be obtained by representing, as in figure 8, the
+location of the saddle-nodes for various Lewis numbers as a function of the reduced
+Prandtl number Pr/Prc and combined parameter Ra|Le−1|. These reduced parameters
+allows us to identify the location where the criticality of the primary bifurcation changes
+as the single coordinate point: Pr/Prc = 1, Ra|Le − 1| ≈ 6509.
+Figure 8 shows that, for Pr < Prc and the chosen values of the Lewis number, the
+location of the first supercritical saddle-node SN2 can be approximated by:
+RaSN2 ≈
+6460
+|Le − 1|
+� Pr
+Prc
+�−0.24
+.
+(4.2)
+
+Near-onset dynamics in natural doubly diffusive convection
+17
+For Pr < 10−2 (not shown), the location of saddle-node SN2 deviates from the relation
+above, indicating a potentially different asymptotic regime. These results also illustrate
+the large Pr behaviour of the subcritical saddle-node SN1: RaSN1|Le − 1| tends to a
+constant as the Prandtl number tends to infinity. This constant increases with Le and
+saturates for large values of the Lewis number. These results echo those obtained in
+doubly diffusive convection in a 2D vertical porous enclosure, where Mamou et al. (1998)
+used a parallel flow approximation to demonstrate that the Rayleigh number at which
+the subcritical saddle-node occurs is proportional to 1/(1 − Le) for large enough Lewis
+numbers.
+4.2. Solution profiles
+Despite the different scenarios obtained at different values of the Prandtl number (see
+figure 5), the steady convection states undergo similar structural changes along their
+branch, as evidenced in figure 9 for Le = 11 and Pr = 1, 0.032, 0.01 and 0.005. The
+streamfunction profiles are similar near the primary bifurcation regardless of the value of
+the Prandtl number (see second column of figure 9), which is in agreement with the linear
+stability results from figure 3(c). Moving along the branches in the direction of increasing
+energy, the first change that we observe is the strengthening of the anticlockwise roll,
+where fluid near the hotter wall moves upwards. This occurs in both the subcritical and
+the supercritical regimes, as can be seen in the third column of figure 9. Continuing the
+branches to the large-amplitude saddle-node and beyond, the amplitude of the weaker roll
+decreases, leaving room for the stronger roll to straighten. At large enough amplitude, an
+anticlockwise roll occupies the domain, irrespective of the value of Pr. Its amplitude grows
+as the upper branch is followed to larger values of Ra, where the Prandtl number starts
+to impact the flow: the roll occupies a smaller area at lower values of the Prandtl number,
+as seen within the final column of figure 9. This resembles the fly-wheel convection, with
+nearly circular streamlines, seen in low-Prandtl Rayleigh–B´enard convection as studied
+by Clever & Busse (1981).
+To characterise these observations in more detail, figure 10 reports the horizontal
+velocity profiles observed on the upper branch for Pr = 1, 0.1, 0.032 and 0.005. The
+decrease in roll size is apparent when Pr is decreased. This is particularly evident for
+Pr = 0.005, where the horizontal velocity remains small except within the range 0.6 ≲
+z ≲ 1.9, in such a way that the roll only occupies about half of the domain’s extent.
+Figure 10(a) additionally shows the transition to these states from the large rolls observed
+at O(1) Prandtl numbers. For Pr = 1, the maximum horizontal velocity is achieved far
+from the center of the roll, at z ≈ 0.44, 2.04, producing a region of strong shear between
+the rolls and gentle quasi-linear velocity variations inside the rolls. As Pr is lowered, these
+maxima move towards the centre of the roll by initially becoming less pronounced and
+creating flatter extrema (see figure 10(c)), followed by the emergence of peaks at z = 1
+and z ≈ 1.5. The maximum horizontal velocity does not change significantly within this
+range of Prandtl number values in such a way that the low Pr rolls represent narrow
+regions of strong shear surrounded by low amplitude flow.
+4.3. Stability of the nonlinear states
+The stability of states on the primary branch is controlled by two eigenmodes: an
+amplitude mode that preserves the S∆ symmetry of the system and a drift mode that
+breaks the S∆ symmetry. The translation mode, associated with vertical translations due
+to the symmetry Tδ, remains marginal along the branch and none of the other eigenmodes
+become destabilising over the range of parameters considered.
+
+18
+C. Beaume, A. M. Rucklidge and J. Tumelty
+Pr = 1
+Pr = 0.032
+Pr = 0.01
+Pr = 0.005
+Figure 9: Streamfunctions of the steady states on the primary branch when Le = 11
+for different values of the Prandtl number: top row Pr = 1, second row Pr = 0.032,
+third row Pr = 0.01 and bottom row Pr = 0.005. The left column shows the respective
+bifurcation diagrams and indicates with a cross the solutions that have been represented
+in the subsequent panels. Black (grey, dotted) contours indicate positive (negative, zero)
+values of the streamfunction. Contour intervals: first column, top two rows—10−4; first
+column, third row—10−5; first column, bottom row—2 × 10−5; second column, top two
+rows—0.02; second column, bottom two rows—0.01; third column—0.02; fourth and fifth
+columns—0.05.
+
+Near-onset dynamics in natural doubly diffusive convection
+19
+(a)
+(b)
+(c)
+(d)
+(e)
+Figure 10: Horizontal velocity and streamfunction of solutions from the upper segment of
+the primary branch at Ra = 700 for Pr = 1, 0.1, 0.032, 0.005 and Le = 11 represented
+via (a) the midline horizontal velocity (u(x = 0.5, z)) and streamfunction contours plots
+for (b) Pr = 1, (c) Pr = 0.1, (d) Pr = 0.032 and (e) Pr = 0.005 with contour intervals
+0.1.
+Close to the onset of convection, the amplitude mode is initially destabilising when
+the bifurcation is subcritical (Pr > Prc), whereas it is stabilising when the bifurcation is
+supercritical (Pr < Prc). This mode subsequently changes stability at successive saddle-
+nodes. In particular, it becomes stabilising at saddle-nodes SN1 and SN3, where the
+branch turns towards higher Ra, but becomes destabilising at SN2, where the branch
+turns towards lower Ra. As a result, the upper branches of steady convection states are
+always stable to amplitude perturbations for all Le and Pr.
+The drift mode is stabilising near the primary bifurcation at Ra = Rac for all Pr,
+but becomes destabilising at a drift-pitchfork bifurcation further along the branch at
+Ra = Rad, whose location depends upon both Le and Pr, as can be seen in figure 7.
+The marginal mode is identical to the translation mode at this bifurcation and its
+destabilisation leads to a pair of branches of travelling wave solutions, as shown in figure
+11(a) for Pr = 0.1 and Le = 11. Close to their onset, these states take the form of a
+single large-amplitude convection roll (see figure 11(c)) that slowly drifts either upwards
+or downwards. As these branches are followed beyond the drift bifurcation, an asymmetric
+streaming flow strengthens while the convection roll weakens and moves toward the wall
+where the streaming flow is the weakest. This transition is shown from figure 11(b) at
+Ra = 630 to figure 11(d) at Ra = 700. At the same time, the drift speed increases at
+a rate approximately proportional to √Ra − Rad, as shown in figure 11(e). This result
+extends the findings obtained for Le = 1.2, Pr = 1 by Xin et al. (1998) to a wider range
+of parameter values.
+The stability of the travelling waves is determined by the location of the drift bifurca-
+tion: these states are initially stable when the bifurcation occurs on the upper branch of
+steady convection states, whereas they are unstable when the bifurcation occurs along
+the lower branch. Both cases can be achieved for a given Le when Pr is varied, as figure 7
+illustrates for selected values of the Prandtl number with Le = 11 and Le = 20. For large
+values of the Prandtl number, the drift bifurcation occurs on the upper branch at large
+Rayleigh numbers. This location moves closer to the saddle-node with decreasing Prandtl
+numbers so that the two coincide at Pr = Pr∗ and Ra = Ra∗. For Le = 11, we found
+
+20
+C. Beaume, A. M. Rucklidge and J. Tumelty
+(a)
+(b)
+(c)
+(d)
+(e)
+Figure 11: Drift bifurcation and downward travelling waves for Pr = 0.1, Le = 11, for
+which Rad ≈ 638. (a) Bifurcation diagram showing the kinetic energy E as a function of
+the Rayleigh number Ra for steady states and travelling waves. Thick lines indicate stable
+solutions, thin lines indicate solutions unstable to amplitude perturbations and dashed
+lines indicate solutions unstable to drift. The drift bifurcation is shown by the open circle.
+(b) Stable convection state at Ra = 630 shown by contours of its streamfunction with
+intervals 0.1 (first red cross on panel (a)). Further panels show similar representations
+of stable travelling waves at: (c) Ra = 645 and (d) Ra = 700. (e) Squared drift speed
+along the stable branch as a function of the Rayleigh number. The dotted line shows the
+fitting law: vd ≈ 0.12
+√
+Ra − 640.
+that Pr∗ ≈ 0.042 and Ra∗ ≈ 614.9 (see figure 7(c) for a bifurcation diagram at similar
+values of the parameters). For smaller values of the Prandtl number, the drift bifurcation
+occurs along the lower branch of convection states and at a value of the Rayleigh number
+that increases as Pr is decreased. For all the parameter values tested, this bifurcation
+was found to occur at larger amplitude than saddle-node SN2 and, consequently, the
+small-amplitude steady convection states remain stable to drift.
+4.4. Dynamical attractors
+The temporal dynamics of the system change as the drift bifurcation passes below the
+subcritical saddle-node since all the steady convection states from the upper branch and
+the travelling wave states are destabilised in the process. Many initial conditions will
+consequently decay towards the conduction state at low Pr and Ra. This decay is not
+possible when the conduction state is unstable for Ra > Rac, where we find that the
+dynamics converge on time-dependent states.
+To understand how this behaviour arises, we unfold the saddle-node-pitchfork normal
+form near the codimension two point (Ra∗, Pr∗) where the drift bifurcation and saddle-
+node coincide. This unfolding takes the form (Guckenheimer & Holmes 1983):
+˙x = −µ1x + b1xz,
+(4.3)
+˙z = µ2 − x2 − z2 + b2z3,
+(4.4)
+where x represents the extent to which the state drifts, z represents the amplitude of the
+convection states, b1 > 0, b2 < 0, and µ1 and µ2 are two unfolding parameters that are
+introduced to respectively represent the deviations Pr − Pr∗ and Ra − Ra∗.
+When µ1 = µ2 = 0, the trivial state, (x, z) = (0, 0), undergoes a codimension two
+bifurcation. One of five phase portraits is observed in the vicinity of this bifurcation and
+these are shown in figure 12(a). In addition to the steady states previously discussed, the
+
+Near-onset dynamics in natural doubly diffusive convection
+21
+µ1
+µ2
+V
+IV
+III
+II
+I
+Pitchfork
+Heteroclinic
+connection
+Hopf
+Pitchfork
+Saddle-node
+(a)
+13(a)
+13(b)
+13(f)
+(b)
+13(c)
+13(d)
+13(e)
+(c)
+Figure 12: (a) Unfolding near the codimension two saddle-node-pitchfork bifurcation at
+µ1 = µ2 = 0 given by system (4.3, 4.4), after Guckenheimer & Holmes (1983). The
+different phase portraits are classified in five different regions labelled using Roman
+numerals and accompanied with a sketch of the corresponding phase space. In each
+of these sketches, the fixed points on the vertical line represent steady convection states.
+The vertical (resp. horizontal) direction is the eigendirection related to the amplitude
+(resp. drift) mode. (b) Analogy with the doubly diffusive convection problem is made by
+replacing µ1 by Pr−Pr∗ and µ2 by Ra−Ra∗ and regions of the (Ra, Pr) parameter space
+are shown as a function of the observed temporal behaviour for Le = 11. (c) Magnification
+of panel (b) near (Ra∗, Pr∗). Arrows indicate the values of Pr used to produce the
+bifurcation diagrams in figure 13. In the panels, the bifurcations are represented by:
+black and red solid lines (saddle-nodes), blue dotted lines (drift bifurcation), red dotted
+lines (Hopf bifurcation), red dot-dashed lines (heteroclinic connection) and in panels (b)
+and (c), the vertical dashed lines (primary stationary bifurcation of the conduction state).
+unfolding reveals the presence of periodic orbits. Relating the unfolding back to doubly
+diffusive convection, these correspond to relative periodic orbits consisting of drifting
+states that originate either from a travelling wave undergoing a Hopf bifurcation or from
+a global bifurcation where two steady convection states connect heteroclinically.
+Although the normal form (4.3, 4.4) only formally represents the dynamics of the
+full system close to the codimension two point, each of the regions shown in figure 12
+
+22
+C. Beaume, A. M. Rucklidge and J. Tumelty
+Stable in region?
+State Oa Ob Ia Ib IIa IIb IIIa IIIb IVa IVb IVc Va Vb
+O
+x
+x
+x
+x
+x
+x
+x
+SOCs
+x
+x
+x
+x
+SOCl
+x
+x
+T W
+x
+x
+PO
+x
+x
+Table 3: Stability of the observed doubly diffusive states within each region of the
+parameter space from figure 12. The naming convention used is as follows: O, conduction
+state; SOCs, small-amplitude stationary overturning convection; SOCl, large-amplitude
+stationary overturning convection; T W, travelling wave; and PO, relative periodic orbit.
+The regions Oa, ..., Vb refer to the regions introduced in figure 12.
+continues to be observed an appreciable distance away from this point. Figures 12(b)
+and (c) illustrate the extent of the corresponding regions in the doubly diffusive system
+when Le = 11 and we anticipate that similar results will hold for other values of
+the Lewis number. In this figure, the regions have been subdivided according to the
+types of stable attracting states that they display. The subdivisions occur owing to the
+instability of the conduction state at Rac and the creation of a pair of saddle-nodes at
+(Racusp, Prcusp), which enrich the previous unfolding. The resulting subregions, together
+with their associated attracting states, are summarised in table 3 and on the bifurcation
+diagrams in figure 13. As Pr varies, the system admits one of seven qualitatively distinct
+bifurcation diagrams. Six of these are presented in figure 13, which also indicate the
+range of kinetic energies over each relative periodic orbit attained via time-stepping.
+The seventh type of bifurcation diagram, where the primary branch lies entirely within
+the supercritical regime, is not shown but possesses similar features to that seen for
+Pr = 0.02 in figure 13(f) including stable small-amplitude steady convection states and
+relative periodic orbits.
+The three most relevant stable attracting states close to the primary bifurcation at high
+Pr (Pr > Pr∗ here) are: the conduction state (O), the large-amplitude steady convection
+states (SOCl) and the travelling wave states (T W). Below the onset of convection
+(region Oa), all initial conditions decay towards the first of these. In region Ia, above
+subcritical onset but before the drift instability, initial conditions converge towards SOCl,
+as evidenced by the energy-time and drift speed-time plot in figure 14(a). Increasing Ra
+beyond the drift instability into region IIa, SOCl is now unstable and the flow converges
+towards T W. Figure 14(b) shows that the former state may still be observed in the
+temporal dynamics as the initial condition first rapidly changes amplitude to approach
+SOCl before it builds vertical drift and converges to T W.
+The stable branch of travelling waves destabilises in a supercritical Hopf bifurcation
+that leads to a stable relative periodic orbit, as shown in figure 13 for Pr = 0.043.
+Figures 15(a)–(e) depict such an orbit shortly after the bifurcation at Ra = 650 and
+Pr = 0.043, where we see that the states exhibit small oscillations about a drifting state.
+The Hopf bifurcation moves towards lower Rayleigh numbers as Pr approaches Pr∗ from
+above, which reduces the extent over which stable T W are found. This continues until
+Pr = Pr∗, when stable T W cease to exist and the relative periodic orbit bifurcates
+directly from the codimension two bifurcation at the saddle-node.
+Upon further decreasing of the Prandtl number, so that the drift bifurcation occurs
+on the lower branch of steady convection, the system admits neither stable SOCl nor
+
+Near-onset dynamics in natural doubly diffusive convection
+23
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 13: Bifurcation diagrams showing the primary branch and other stable attracting
+states for Le = 11, and (a) Pr = 1, (b) Pr = 0.1, (c) Pr = 0.043, (d) Pr = 0.04,
+(e) Pr = 0.032 and (f) Pr = 0.02. The solid circles mark the saddle-nodes and open
+circles indicate where the drift bifurcation occurs. Thick (thin) lines represent states
+stable (unstable) to the amplitude mode whilst solid (dashed) lines show those stable
+(unstable) to the drift mode. Thick blue lines indicate the minimal and maximal energies
+achieved in the stable limit cycle, which starts in a Hopf bifurcation in (c) and in a
+heteroclinic bifurcation in (d,e,f). The unstable branches of travelling waves are not
+shown.
+stable T W. Instead, the bifurcation diagrams are similar to that shown for Pr = 0.04 in
+figure 13(d), where a branch of unstable T W extends from the drift bifurcation towards
+higher Rayleigh numbers and stable relative periodic orbits exist after a global bifur-
+cation, where the stable manifold of SOCl connects heteroclinically with the unstable
+manifold of the convection state on the lower branch and vice versa.
+The lack of stability of the nonlinear states before the heteroclinic connection lead
+all initial conditions to decay down to the conduction state in regions IVa and Va.
+Figures 14(d) and (f) illustrate this tendency for Pr = 0.032 when Ra = 630 and
+Ra = 620, respectively. In both cases, the amplitude of the initially imposed roll
+rapidly decreases to approach that of SOCl rolls. Afterwards, the drift speed of the
+state increases, as SOCl is unstable to drift, and reaches a maximum around t ≈ 50.
+The drift speed subsequently decays down to zero, due to the instability of T W, and
+
+24
+C. Beaume, A. M. Rucklidge and J. Tumelty
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 14: Energy-time (black) and drift speed-time (red) plots illustrating regions I–V in
+figure 12 with Le = 11. In each case, the initial state was the large-amplitude convection
+state at Ra = 700 for Pr = 0.1 that was perturbed in the direction of its unstable drift
+eigenmode. States approached during the trajectory are labelled as follows: (a) region
+Ia, convergence to SOCl when Pr = 0.1 and Ra = 630; (b) region IIb, convergence to
+T W when Pr = 0.1 and Ra = 660; (c) region IIIb, convergence to PO when Pr = 0.032
+and Ra = 660, (d) region IVa, convergence to O when Pr = 0.032 and Ra = 630; (e)
+region IVc, convergence to SOCs when Pr = 0.02 and Ra = 700; and (f) region Va,
+convergence to O when Pr = 0.032 and Ra = 620.
+the time-dependent state converges on the conduction state, which is the only stable
+attractor in these regions.
+Beyond the heteroclinic connection, initial conditions tend to converge towards the
+relative periodic orbit, as they invariably do in region IIIb, where the conduction state
+is unstable. Figure 14(c) illustrates this convergence starting from a large-amplitude roll
+with Pr = 0.032 and Ra = 660 perturbed in the direction of its unstable drift eigenmode.
+A single cycle of this orbit is shown in further detail in figures 15(f)–(j). This relative
+periodic orbit cycles between the three states: SOCl, T W and a steady small-amplitude
+convection state, in the following manner. The first stage of the orbit, from 15 ≲ t ≲ 40,
+resembles the temporal behaviour seen in region IIa (figure 14(b)), where the solution
+remains close to SOCl in profile (figure 15(h)) while the drift speed slowly increases in
+magnitude. Following this, between t ≈ 40 and t ≈ 54, the drift speed and kinetic energy
+rapidly increase as the profile of the state exhibits properties of the travelling wave (T W)
+solution (figure 15(i)). Between t ≈ 54 and t ≈ 68, both the drift speed and kinetic
+
+Near-onset dynamics in natural doubly diffusive convection
+25
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+Figure 15: Temporal evolution of downward-travelling states across one cycle of two
+relative periodic orbits at: (a)–(e) Ra = 650 with Pr = 0.043 and Le = 11 and (f)–(j)
+Ra = 660 with Pr = 0.032 and Le = 11 (as in figure 14(c)). (a) and (f) Anticlockwise
+trajectory of the periodic orbit in drift speed-energy phase space. Blue dots in (a) mark
+the conduction and steady convection states. (b) and (g) Energy-time (top) and drift
+speed-time (bottom) plots. (c)–(e) Streamfunctions of states along the orbit in (a) at (b)
+t = 0, (c) t = 8 and (d) t = 20 with contour intervals 0.1. (h)–(i) Streamfunctions of
+states along the orbit in (f) at (h) t = 26, (i) t = 52 and (j) t = 68, with contour intervals
+0.05. The streamfunctions have been translated vertically for better visual representation.
+energy decrease as the state approaches a small-amplitude, non-drifting convection state
+with inclined rolls (figure 15(j)). The final stage of this orbit is the transition from the
+small-amplitude back to large-amplitude steady convection, which is indicated by the
+monotonic increase in kinetic energy while maintaining vd ≈ 0 for t ≳ 70 and t ≲ 15 in
+figure 15(g).
+The heteroclinic connection leading to these orbits moves towards higher Rayleigh
+numbers as Pr decreases and coincides with SN2 for Pr ≲ 0.032 (see figures 13(e)
+and (f)). This suggests that a saddle-node infinite period (SNIPER) bifurcation explains
+the origin of the relative periodic orbits at low Prandtl and high Rayleigh numbers.
+However, by considering various properties of the relative periodic orbits for Pr = 0.032
+and Le = 11 as Ra approaches RaSN2 from above (figure 16), we additionally find that
+a gluing bifurcation occurs in the vicinity of the SNIPER bifurcation.
+At large Rayleigh numbers, a pair of relative periodic orbits with states drifting either
+upwards or downwards are related by the reflection symmetry. The maximal energy and
+drift speed attained along these orbits decrease with decreasing Rayleigh number, and the
+trajectories approach the stable and unstable manifolds of SOCl, as seen in figure 16(a).
+
+26
+C. Beaume, A. M. Rucklidge and J. Tumelty
+(a)
+(b)
+(c)
+Figure 16: Relative periodic orbits for Le = 11, Pr = 0.032 where RaSN2 ≈ 650.82. (a)
+Trajectories in (vd, E) phase space for Ra = 650.85 (blue), Ra = 660 (red) and Ra = 700
+(black). For Ra = 700 and Ra = 660, a pair of relative periodic orbits associated with
+either negative or positive drift velocity are shown, while for Ra = 650.85, a single
+periodic orbit with alternating negative and positive drift velocities is shown. (b) Period
+tP of orbits for selected Ra > RaSN2. The red dashed line shows that approximately
+tP ∝ (Ra − RaSN2)−0.56. (c) Energy-time plots for Ra = 700 (top), Ra = 660 (middle)
+and Ra = 650.85 (bottom).
+This leads to the two relative periodic orbits connecting in a gluing bifurcation around
+Ra ≈ 652 so that the trajectories become a single periodic orbits where states alternately
+drift in opposite directions. This is reminiscent of the pulsating waves seen in nonlinear
+magnetoconvection (Matthews et al. 1993).
+The resulting single periodic orbit persists until RaSN2, where it terminates in the
+SNIPER bifurcation. This is evidenced by the period of a single loop of the orbit scaling
+like tP ∝ (Ra − RaSN2)−0.56 as SN2 is approached, which is close to the expected
+tP ≈ |Ra − RaSN2|−0.5 scaling. The energy-time plots in figure 16(c) illustrate that
+the predominant increase in duration occurs near the small-amplitude steady convection
+state as the orbit approaches the steady state at SN2 in phase space. We also find that
+the time spent near SOCl increases, whilst the time where the state has large drift speed
+remains small, implying that the global bifurcation is due to the collision of the periodic
+orbit with the stable manifold of SOCl.
+The final attracting state that the flow may converge to is SOCs, as figure 14(e)
+illustrates for Pr = 0.02 and Ra = 700. This is possible for Pr < Prcusp in the
+supercritical regions Ob, IVc and Vb, where it is the only stable attracting state, and in
+the subcritical region IVb, where convergence towards the stable conduction state is also
+possible.
+5. Discussion
+This paper considers doubly diffusive convection driven by horizontal gradients of
+temperature and concentration, a configuration typically referred to as natural doubly
+diffusive convection. We have extended the linear stability analysis of Ghorayeb &
+Mojtabi (1997) by performing a thorough weakly nonlinear analysis of the system. This
+
+Near-onset dynamics in natural doubly diffusive convection
+27
+was complemented by a numerical exploration of the nonlinear regime, thereby also
+extending the analysis of Xin et al. (1998), who focused on Pr = 1 and Le = 1.2.
+From this analysis, we unravelled the relationships between saddle-nodes, drift and global
+bifurcations.
+We have identified regions where the resulting primary branch exhibits qualitatively
+different behaviour. For large values of the Prandtl number, the bifurcation is subcritical
+and hysteresis takes place between the conduction state and large-amplitude convection.
+Whereas, for Prandtl numbers below a critical value, the primary bifurcation is super-
+critical but this is preceded by the creation of two saddle-nodes without affecting the
+existence of large-amplitude convection. Despite this, we did not find any hysteresis in
+the supercritical regime owing to the presence of a destabilising drift bifurcation along
+the primary branch. The presence of multiple folds along a primary supercritical branch
+has already been observed in a non-homogeneous fluid system (Erenburg et al. 2003)
+but we believe that is the first time that it has been observed in homogeneously forced
+convection in such a small domain.
+By determining the stability of steady convection states along the primary branch,
+we identified a codimension two point between a large-amplitude saddle-node and a
+drift bifurcation. We analysed the dynamics around this codimension two point using
+its normal form and numerical simulations to investigate new Hopf and heteroclinic
+bifurcations giving rise to periodic orbits. Such time-dependent states are common
+features of low Prandtl number doubly diffusive convection (see also Umbr´ıa & Net
+(2019)). Finally, we provided a classification of the various regions in (Ra, Pr) parameter
+space according to the nature of their dynamical attractors, for a representative value of
+the Lewis number.
+We anticipate that the analysis provided in this paper may serve as a guide for future
+research in natural doubly diffusive convection by providing a comprehensive map of the
+near-onset dynamics as a function of the parameter values. Despite our attempt to be
+thorough, the characterisation of the nonlinear regime at very small Prandtl numbers,
+which is relevant in astrophysical contexts (see Garaud (2018)), remains to be explored.
+We observed that this regime behaves differently from extrapolated predictions from O(1)
+Prandtl numbers but have not pursued this any further.
+Lastly, the coexistence of steady overturning convection with the stable conduction
+state when the primary bifurcation is supercritical has important dynamical implications,
+which will be the subject of future exploration. In particular, it makes this system a
+candidate for spatially localised pattern formation in a supercritical fluid system, owing
+to the similarity of the primary branch structure with the Swift–Hohenberg equation
+considered by Knobloch et al. (2019).
+Acknowledgements
+This work was supported by the Leeds–York Natural Environment Research Council
+(NERC) Doctoral Training Partnership (DTP) SPHERES under grant NE/L002574/1
+and undertaken on ARC3, part of the High Performance Computing facilities at the
+University of Leeds, UK.
+Appendix A. Further expressions for the weakly nonlinear analysis
+A.1. Second-order corrections
+The solution to the system at O(ǫ2) (3.14) given in (3.26) involves parameter-free
+functions ˜ui, ˜wi, ˜pi and ˜θi for i = 2, ..., 7 within the expressions for Ψ 0
+2 , Ψ 1
+2 and Ψ 2
+2
+
+28
+C. Beaume, A. M. Rucklidge and J. Tumelty
+(3.27–3.29). These functions satisfy the forced linear equations:
+
+
+
+
+−D
+0
+0
+0
+0
+D2
+0
+0
+0
+0
+D2
+Rac(1 − Le)
+0
+0
+0
+D2
+
+
+
+
+
+
+
+
+˜p2
+˜w2
+˜w3
+˜θ3
+
+
+
+ =
+
+
+
+
+f10
+f20
+0
+f40
+
+
+
+ ,
+(A 1)
+
+
+
+
+D2 − 4k2
+c
+0
+−D
+0
+0
+D2 − 4k2
+c
+−2ikc
+Rac(1 − Le)
+D
+2ikc
+0
+0
+−1
+0
+0
+D2 − 4k2
+c
+
+
+
+
+
+
+
+
+˜u4
+˜w4
+˜p4
+˜θ4
+
+��
+
+ =
+
+
+
+
+f12
+f22
+0
+0
+
+
+
+ ,
+(A 2)
+
+
+
+
+
+
+D2 − 4k2
+c
+0
+−D
+0
+0
+0
+D2 − 4k2
+c
+−2ikc
+Rac(1 − Le)
+0
+D
+2ikc
+0
+0
+0
+−1
+0
+0
+D2 − 4k2
+c
+0
+−1
+0
+0
+0
+D2 − 4k2
+c
+
+
+
+
+
+
+
+
+
+
+
+
+˜u5
+˜w5
+˜p5
+˜θ5
+˜θ6
+
+
+
+
+
+
+=
+
+
+
+
+
+
+0
+0
+0
+f42
+0
+
+
+
+
+
+
+,
+(A 3)
+
+
+
+
+D2 − k2
+c
+0
+−D
+0
+0
+D2 − k2
+c
+−ikc
+Rac(1 − Le)
+D
+ikc
+0
+0
+−1
+0
+0
+D2 − k2
+c
+
+
+
+
+
+
+
+
+˜u7
+˜w7
+˜p7
+˜θ7
+
+
+
+ =
+
+
+
+
+f11
+f21
+f31
+f41
+
+
+
+ ,
+(A 4)
+where D =
+d
+dx, and ˜ui, ˜wi and ˜θi satisfy homogeneous boundary conditions and the
+pressure boundary conditions come from a projection of the Navier–Stokes equation
+onto the side walls:
+˜ui = 0,
+˜wi = 0,
+−∂˜pi
+∂x + ∂2˜ui
+∂x2 = 0
+˜θi = 0
+on
+x = 0, 1
+(A 5)
+A.2. Coefficients in the Ginzburg–Landau equation
+Expressions for the coefficients α, β, γ and δ in the Ginzburg–Landau equation (3.33)
+are obtained by evaluating:
+α = 1
+Pr
+�
+⟨U † , U1⟩ + ⟨W † , W1⟩
+�
++ (1 + Le)⟨Θ† , Θ1⟩
+= 1
+Prα1 + (1 + Le)α2,
+(A 6)
+β = −
+� 1
+Pr ⟨U † , N U
+3 ⟩ + 1
+Pr⟨W † , N W
+3 ⟩ +
+1
+1 − Le⟨Θ† , N Θ
+3 − LeN Φ
+3 ⟩
+�
+=
+1
+Pr2 β1 + 1 + Le
+Pr
+β2 + (1 + Le2)β3 + Leβ4,
+(A 7)
+γ = (1 − Le)⟨W †, Θ1⟩
+= (1 − Le)γ1,
+(A 8)
+δ = ⟨U † , U1⟩ + ⟨W † , W1⟩ + ⟨Θ† , Θ1⟩
++ 2ikc
+�
+⟨U † , ˜u7⟩ + ⟨W † , ˜w7⟩ + ⟨Θ† , ˜θ7⟩
+�
++ ⟨P † , ˜w7⟩ − ⟨W † , ˜p7⟩,
+(A 9)
+
+Near-onset dynamics in natural doubly diffusive convection
+29
+α1
+1.11 × 10−4
+β2 −1.63 × 10−8
+(Le > 1) γ1 −8.85 × 10−7
+α2
+2.27 × 10−4
+β3
+7.47 × 10−8
+(Le < 1) γ1
+8.85 × 10−7
+β1 −4.43 × 10−9
+β4
+1.58 × 10−7
+δ
+7.38 × 10−4
+Table 4: Numerical values of the coefficients α1, α2, β1, β2, β3, β4, γ1 and δ in (3.33).
+The sign of γ1 depends upon whether Le > 1 or Le < 1 as γ > 0 for all Le, while all
+other coefficients are independent of the parameters Le and Pr.
+where the nonlinear functions N F
+3 , for F = U, W, Θ, Φ, and coefficients βi for i = 1, 2, 3, 4
+are:
+N F
+3 = U1
+dF 0
+2
+dx + U1
+dF 2
+2
+dx + U 2
+2
+dF 1
+dx + 2ikcW 1F 2
+2 + ikcW 0
+2 F1 − ikcW 2
+2 F 1,
+(A 10)
+β1 = −
+�
+U †, U 1
+d˜u4
+dx + ˜u4
+dU 1
+dx + 2ikc˜u4W 1 + ikc ˜w2U1 − ikc ˜w4U 1
+�
+−
+�
+W †, U1
+d ˜w2
+dx + U1
+d ˜w4
+dx + ˜u4
+dW 1
+dx
++ ikcW 1 ˜w4 + ikcW1 ˜w2
+�
+,
+(A 11)
+β2 = −
+�
+U †, U 1
+d˜u5
+dx + ˜u5
+dU 1
+dx + 2ikc˜u5W 1 + ikc ˜w3U1 − ikc ˜w5U 1
+�
+−
+�
+W †, U1
+d ˜w3
+dx + U1
+d ˜w5
+dx + ˜u5
+dW 1
+dx
++ ikcW 1 ˜w5 + ikcW1 ˜w3
+�
+−
+�
+Θ†, U1
+d˜θ4
+dx + ˜u4
+dΘ1
+dx + 2ikcW 1˜θ4 + ikc ˜w2Θ1 − ikc ˜w4Θ1
+�
+,
+(A 12)
+β3 = −
+�
+Θ†, U1
+d˜θ3
+dx + U 1
+d˜θ5
+dx + ˜u5
+dΘ1
+dx + 2ikcW 1˜θ5 + ikc ˜w3Θ1 − ikc ˜w5Θ1
+�
+, (A 13)
+β4 = −
+�
+Θ†, U1
+d˜θ3
+dx + U 1
+�
+d˜θ5
+dx + d˜θ6
+dx
+�
++ 2˜u5
+dΘ1
+dx
++2ikc
+�
+W 1
+�
+˜θ5 + ˜θ6
+�
++ ˜w3Θ1 − ˜w5Θ1
+��
+.
+(A 14)
+These expressions for the parameter-free coefficients αi, βi, γ1 and δ are evaluated
+numerically and are given in table 4.
+Of particular interest is the boundary where the primary bifurcation changes from
+subcritical to supercritical. This occurs when β = 0, which we may find explicitly by
+taking the positive root of equation (A 7), to find:
+Prc = −(1 + Le)β2 +
+�
+(1 + Le)2β2
+2 − 4β1 [(1 + Le2)β3 + Leβ4]
+2[(1 + Le2)β3 + Leβ4]
+.
+(A 15)
+A.3. Effect of thermal and solution advective terms on a2
+To determine the contributions that each of the nonlinear terms make to a2, we
+introduce the factors ζ1 and ζ2 that multiply the thermal and solutal advective terms,
+respectively. We numerically perform the weakly nonlinear analysis for the modified
+
+30
+C. Beaume, A. M. Rucklidge and J. Tumelty
+0.0001
+0.01
+1
+100
+10000
+0
+(a)
+0.0001
+0.01
+1
+100
+10000
+0
+(b)
+Figure 17: Contours of the coefficient a2 as a function of ζ1 and ζ2, which respectively
+multiply thermal and solutal advective nonlinearities in (A 16–A19), for (a) Le = 11,
+Pr = 1 and (b) Le = 1/11, Pr = 1. The contour a2 = 0, which marks the boundary
+between subcriticality and supercriticality, is shown in bold.
+system:
+1
+Pr
+�∂u
+∂t + u · ∇u
+�
+= −∇p + ∇2u + Ra(T − C)ˆz,
+(A 16)
+∇ · u = 0,
+(A 17)
+∂T
+∂t + ζ1u · ∇T = ∇2T,
+(A 18)
+∂C
+∂t + ζ2u · ∇C = 1
+Le∇2C,
+(A 19)
+with ζ1, ζ2 ∈ [10−2, 104] and selected values of the Prandtl and Lewis numbers. The
+coefficient a2 tends to increase when one of ζ1 or ζ2 increases, while keeping the other
+fixed, as indicated by the contours in figure 17. Thus, temperature and solutal advection
+enhances the subcriticality of the primary bifurcation.
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+

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+arXiv:2301.01534v1  [astro-ph.SR]  4 Jan 2023
+Draft version January 5, 2023
+Typeset using LATEX default style in AASTeX631
+Origin of Quasi-Periodic Pulsation at the Base of Kink Unstable Jet
+Sudheer K. Mishra,1 Kartika Sangal,2 Pradeep Kayshap,3 Petr Jel´ınek,4 A.K. Srivastava,2 and S.P. Rajaguru1
+1Indian Institute of Astrophysics, Koramangala, Bangalore-560034, India.
+2Department of Physics, Indian Institute of Technology (BHU), Varanasi-221005, India
+3VIT Bhopal, Kothari Kalan, Sehore, Madhya-Pradesh 466114, India
+4University of South Bohemia, Faculty of Science, Department of Physics
+Braniˇsovsk´a 1760, CZ – 370 05 ˇCesk´e Budˇejovice, Czech Republic
+ABSTRACT
+We study a blowout jet that occurs at the west limb of the Sun on August 29th, 2014 using high-resolution
+1
+imaging/spectroscopic observations provided by SDO/AIA and IRIS. An inverse γ-shape flux-rope appears
+2
+before the jet– morphological indication of the onset of kink instability. The twisted field lines of kink-unstable
+3
+flux-rope reconnect at its bright knot and launch the blowout jet at ≈06:30:43 UT with an average speed of
+4
+234 km s−1. Just after the launch, the northern leg of the flux rope erupts completely. The time-distance
+5
+diagrams show multiple spikes or bright dots, which is the result of periodic fluctuations, i.e., quasi-periodic
+6
+fluctuations (QPPs). The wavelet analysis confirms that QPPs have a dominant period of ≈ 03 minutes. IRIS
+7
+spectra (Si iv, C ii, and Mg ii) may also indicate the occurrence of magnetic reconnection through existence
+8
+of broad & complex profiles and bi-directional flows in the jet. Further, we have found that line broadening
+9
+is periodic with a period of ≈ 03 minutes, and plasma upflow is always occurs when the line width is high,
+10
+i.e., multiple reconnection may produce periodic line broadening. The EM curves also show the same period
+11
+of ≈ 03 minutes in different temperature bins. The images and EM show that this jet’s spire is mainly cool
+12
+(chromospheric/transition region) rather than hot (coronal) material. Further, line broadening, intensity, and
+13
+EM curves have a period of ≈03 minutes, which strongly supports that multiple magnetic reconnection triggers
+14
+QPPs in the blowout jet.
+15
+Keywords: Blowout jet, Instability, Magnetic Reconnection, Magnetic; Magnetic fields, Corona
+1. INTRODUCTION
+Solar jets are an integral part of the solar atmosphere, and they are an important feature of the mass/energy cycle within the
+solar atmosphere. Solar jets occur everywhere in the solar atmosphere. The solar jets have been classified into different categories
+based on different criteria, namely, (1) based on morphology–standard jets & blowout jets (Moore et al. 2010, 2013), (2) based
+on the region of solar atmosphere where they occur– active-region jets, coronal hole jets, quiet-Sun jets, network jets, umbral
+jets, penumbral jets, and polar jets (Srivastava & Murawski 2011; Kayshap et al. 2013, 2018; Tian et al. 2014; Tiwari et al. 2016;
+Mulay et al. 2017; Srivastava et al. 2018), and (3) based on the filter in which they observed–e.g., H-α jet, extreme ultraviolet
+(EUV) jets, ultraviolet jet (UV) jets, X-ray jets, and radio jets (e.g., Shibata et al. 1992; Innes et al. 1997; Shibata et al. 2007;
+Sterling et al. 2015; Filippov et al. 2015; Shen et al. 2017; Zhang & Ji 2014; Ni et al. 2017; Zhang & Ni 2019). Solar surges are
+another category of solar jet-like features, and they are made of mainly cool plasma. Therefore, cool lines/filters (e.g., Hα, Ca ii
+H & K line, Mg ii h & k lines, Si iv, IRIS/SJI 1400 Å, AIA 304 Å, and many more) use to observe the solar surges (e.g., Sterling
+2000; Tziotziou et al. 2005; Kayshap et al. 2021). However, the surges can heat the plasma up to transition region/coronal tem-
+perature and then rapidly cools down to chromospheric temperatures (N´obrega-Siverio et al. 2016, 2017, 2018; De Pontieu et al.
+2021). In addition, Young (2015) have added one more category to the solar jets, i.e., dark jets– no intensity enhancement, but
+Corresponding author: Pradeep Kayshap
+[email protected]
+
+2
+Mishra et al.
+the signature exists in the Doppler velocity.
+The energy required to power the solar jets comes from the complex magnetic field through the most widely occurring
+process (i.e., magnetic reconnection) in the solar atmosphere. These solar jets are distributed all over the solar atmosphere
+as magnetic reconnection can occur anywhere within the solar atmosphere when suitable physical conditions are pronounced.
+Widely, the bipolar magnetic field reconnects with the pre-existing magnetic field, and produces the jets in the solar atmo-
+sphere (e.g., Yokoyama & Shibata 1995, 1996; Shimojo et al. 1996; Shibata et al. 2007; Nishizuka et al. 2008; Raouafi et al.
+2016). Moreno-Insertis & Galsgaard (2013) have performed a detailed numerical simulation, based on the magnetic reconnec-
+tion between an emerging bipolar magnetic field with the pre-existing open coronal magnetic field, to understand the triggering
+mechanisms and dynamics of solar jets. Further, in another remarkable work, Sterling et al. (2015) have shown that mini-filament
+can also trigger the solar jets, i.e., the eruption of the mini-filament trigger the magnetic reconnection which ultimately produces
+the solar jet. In this continuation, Wyper et al. (2017) has proposed the universal magnetic breakout model to trigger the jet,
+i.e., magnetic reconnection takes place due to the mini-filament eruption. However, it must be noted that this magnetic breakout
+model was already known & widely used in the triggering of the large-scale eruptions, and Wyper et al. (2018) have extended
+this model to the formation of solar jets. As per the magnetic breakout model, the magnetic reconnection takes place due to the
+mini-filament eruption, and it finally triggers the jet in the solar atmosphere. Hence, we can say that magnetic reconnection is an
+integral process of the formation of solar jets.
+There are at least three different theoretical explanations for plasma acceleration from various numerical simulations. In the first
+type of acceleration mechanisms, the plasma is accelerated from the magnetic reconnection site by the slingshot effect along the
+newly reconnected magnetic field lines (Yokoyama & Shibata 1996; Nishizuka et al. 2008; Moreno-Insertis et al. 2008). The re-
+leased energy from the magnetic reconnection can be deposited through various ways, e.g., adiabatic compression, Joule heating,
+accelerated particles, and shocks. Hence, this released energy from the magnetic reconnection can heat the plasma impulsively,
+and the strong pressure and temperature gradient will develop there that can induce the evaporation flows (Shimojo et al. 2001;
+Miyagoshi & Yokoyama 2003; Matsui et al. 2012). The evaporation flows is the second plasma acceleration mechanism for the
+solar jets, and it is induced by the magnetic reconnection. The speed in this mechanism (i.e., evaporation flow) of the jet plasma
+is much slower than the speed of plasma attained through the slingshot effect. When the twisted closed magnetic field lines
+reconnect with the untwisted open field lines then the twist will transfer to the newly reconnected magnetic field lines. And,
+the newly reconnected magnetic field lines will exhibit untwisting motions, which is the third type of acceleration mechanism
+induced by the magnetic reconnection. The models, which are based on this mechanism, are known as the untwisting models
+of the solar jets (e.g., Shibata & Uchida 1986; Schmieder et al. 1995; Canfield et al. 1996; Jibben & Canfield 2004; Kamio et al.
+2010; Archontis & Hood 2013; Moreno-Insertis & Galsgaard 2013; Fang et al. 2014). The helical/rotational motions (i.e., an
+important feature of the solar jets) can be explained through these untwisting models of the solar jets. In addition, it can be noted
+that the helical/rotational motion is a main feature of cool emissions of the solar jets (e.g., Canfield et al. 1996; Harrison et al.
+2001; Kamio et al. 2010; Hong et al. 2013). The blowout jet mainly emits at cooler temperature, and they always exhibit a strong
+rotation (e.g., Sterling et al. 2010; Moore et al. 2013).
+The physical process used in the untwisting models (i.e., reconnection between the twisted magnetic field and open magnetic
+field lines) is not the only way to trigger the solar jets and their helical or rotating motions. Various types of magnetohydrody-
+namic (MHD) instabilities (e.g., Rayleigh-Taylor (RT), Kelvin-Helmholtz (KH), Ballooning mode, convective-driven instability,
+radiatively-driven instability, heating-driven thermal instabilities, tearing mode, kink mode, sausage mode, helical/torsional
+mode, and current-sheet mode) are important physical process trigger the wide varieties of the solar phenomena, i.e., from the
+small-scale event (e.g., solar jets) to large-scale eruptions (Mishra et al. 2018; Mishra & Srivastava 2019; Mishra et al. 2021).
+Kink instability is one of the prominent mechanisms to trigger the solar jets (see review; Raouafi et al. 2016). The observational
+and theoretical studies suggest that the rotational/twisting motions of the solar jets are directly linked with the helical kink
+instability (Shibata & Uchida 1986; Pariat et al. 2009; Liu et al. 2019; Zaqarashvili et al. 2021).
+The embedded magnetic dipole is key feature of the numerical simulations based on the kink-instability. Generally, the MHD
+instabilities are associated with the embedded magnetic bipole in such numerical simulations, and the kink instability takes place
+when threshold in energy or helicity or twist exceeds some critical values (e.g., Pariat et al. 2009). The kink instability may
+evolve in a flux rope if the azimuthal component of the magnetic field exceeds some critical threshold (Aschwanden 2004).
+The theoretical and observational studies suggest different threshold values of twist/helicity for the kink instability, namely,
+
+QPP in Kink Unstable Jet
+3
+2π (Kruskal & Kulsrud 1958), 2.5π (Hood & Priest (1981)), 2.6π (e.g., Pariat et al. 2009, 2015),and 1.3 turns (Liu et al. 2019).
+Hence, we can say that the threshold value of twist to trigger kink-instability varies a lot, and it depends on the magnetic field
+conditions. Now, this kink instability forces a loss of the stability of the whole stable magnetic field configuration (i.e., the
+embedded magnetic flux in the uniform magnetic field; Pariat et al. 2009), and it leads to the magnetic reconnection that drives
+the helical solar jets (Pariat et al. 2015). However, such magnetic field configuration is not the only possibility for helical jets.
+The high-resolution observations have shown the existence of twisted flux rope, and the reconnection within the twisted magnetic
+structure can also produce such helical jets (Raouafi et al. 2010; Liu et al. 2011; Kayshap et al. 2013).
+The solar jets may also be the source of MHD waves (Cirtain et al. 2007; Zhelyazkov 2012) and quasi-periodic pulsations (e.g.,
+Morton et al. 2012; Zhang & Ji 2014). The quasi-periodic pulsations (QPPs) are a phenomenon frequently associated with solar
+flares, and these QPPs in solar flares occur over a vast range of periods, i.e, from a fraction of a second to several minutes (e.g.,
+Nakariakov et al. 2010, 2016; Kashapova et al. 2020). The different periods of QPPs may be related to the different physical
+processes in the solar atmosphere. Recently, Zimovets et al. (2021) have reviewed the observational and theoretical aspects of
+the QPPs in solar and stellar flares, and they suggest more than fifteen different mechanisms in support of the formation of QPPs
+in solar/stellar flares. All these physical mechanisms for QPPs are primarily associated with either MHD waves or quasi-periodic
+regimes of magnetic reconnection. As we know that QPPs are very common phenomena associated with solar/stellar flares
+while the observational detection of QPPs in solar jets is very rare. So far, there are only a few observational works that report
+the existence of the QPPs in the solar jets (e.g., Morton et al. 2012; Zhang & Ji 2014). Similar to the QPPs in solar flares,
+it is reported that multiple magnetic reconnection can trigger the QPPs in blowout jets also (Morton et al. 2012). Here, we
+mention that QPPs are not well studied in solar jets as there are only a few reports of QPPs in solar jets so far. And, on top of this
+fact, we further say that solar jet, kink-instability, and the formation of QPPs are not investigated as per the best of our knowledge.
+In the present scientific work, we provide a systematic observational study of solar jet triggered due to the kink instability, and
+subsequent evolution of quasi-periodic pulsations. The observations and data analysis are discussed in section 2. In section 3 we
+presents the observational results related to the eruption of kink unstable jets and quasi-periodic pulsation. Finally, the discussion
+and conclusion are presented in Section 4.
+2. OBSERVATIONS AND DATA ANALYSIS
+The Atmospheric Imaging Assembly (AIA) onboard Solar Dynamics Observatory (SDO) provides the full-disk images of
+the Sun in several filters (i.e., AIA 4500Å, AIA 1600 Å, AIA 1700 Å, AIA 304 Å, AIA 171 Å, AIA 211 Å, AIA 131 Å, and
+AIA 94 Å; Lemen et al. 2012). Some filters capture the emission from the extreme ultra-violet (EUV)/UV region, and one filter
+of AIA captures the emission from the visible waveband. Therefore, these AIA filters capture the emission of the Sun from the
+top of the photosphere to the lower corona. The spatial resolution of the EUV waveband is 1.5′′ with 0.6′′ per pixel width and
+a cadence of 12 seconds (see Lemen et al. 2012 for more details). We use multi-wavelength imaging observation obtained from
+AIA/SDO for this present work. The jet was triggered around ∼06:28 UT on August 29th, 2014 near the west limb of the Sun,
+and the jet event ended at ≈07:20 UT.
+The AIA images are used to understand the temporal evolution of this jet event. In addition to the temporal evolution, the AIA
+images can also be used to perform the differential emission measures (DEM). The Differential Emission Measure (DEM) is a
+very useful parameter to understand the thermal nature of this jet event. To estimate the DEM, we have used the method developed
+by Hannah & Kontar (2012). This method uses regularized inversion technique to perform the DEM using the warm/hot EUV
+channel of AIA/SDO (i.e., 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å). It is an automated method that uses the zeroth-order
+regularized inversion to return the DEM as a function of temperature. This regularized method provide a positive solution for
+the extracted DEM. We divide the temperature range between log T=5.0–7.5 K, with 25 temperature bins and an interval of log
+∆T=0.1 K.
+In addition to AIA imaging observations, we also use imaging (slit-jaw images(SJI)) and simultaneous spectroscopic observa-
+tions obtained from the Interface Region Imaging Spectrograph (IRIS). The IRIS telescope captures the emission from the lower
+solar atmosphere (chromosphere and transition region) using a slit-jaw imager (SJI), and IRIS takes images in far ultraviolet
+(FUV: 1331.56–1358.40 Å and 1390.00–1406.79 Å) and near ultraviolet (NUV: 2782.56–2833.89 Å) wavebands. IRIS does not
+provide full-disk images rather it provided small field-of-view (FOV) images of the Sun in some filters, namely, Mg ii 2796.0 Å
+filter, Si iv 1400 Å filter, C ii 1330 Å filter, and more filters (for more details see; De Pontieu et al. 2014). The IRIS has observed
+
+4
+Mishra et al.
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+Twist in jet
+Twist in jet
+Kink unstable jet
+Northern leg
+Southern leg
+AIA 304 06:29:07 UT
+N
+S
+W
+E
+Inverse γ
+�shape
+Inverse γ
+�shape
+Bright knot
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+Figure 1. The sequence of the images from SDO/AIA 304 Å with reverse color contrast shows the onset of the kink instability in a blowout jet.
+The morphological sign of kink instability (inverse γ-shape), bright knot, propagation of brightening along the twisted field lines, and rotation
+of the plasma thread (i.e., twist in the jet) have been identified, and these observational findings are indicated by the different arrows. The first
+panel shows the direction to understand the dynamics in the jet’s leg. The black and white arrow indicates the northern and southern legs of the
+eruptive jet (last panel).
+this jet event only in the C ii 1330 Å filter with the temporal cadence of 10 seconds, and the full field-of-view (FOV) of SJI is
+119′′×119′′.
+The IRIS also provides spectroscopic observations, and it observes some of the prominent spectral lines of interface-region, such
+as Mg ii k 2796Å, Mg ii h 2803 Å, C ii 1334.53 Å, C ii 1335.66 Å, Si iv 1393.76 Å, and Si iv 1402.77 Å. IRIS has observed the
+spectra of this particular jet event with an 8-step coarse raster observation of temporal cadence of 9.6 seconds. Hence, each raster
+file of this observation takes ∼77 seconds (i.e., 8.0×9.6 = 76.8 seconds).
+Lastly, we mention that we have also utilized wavelet analysis to diagnose the quasi-periodic behavior of various light curves
+extracted from this jet event. We adopt the method of Torrence & Compo (1998) for the wavelet analysis of the time series. We
+have applied the wavelet analysis on AIA 304 Å, AIA 171 Å, AIA 131 Å, and AIA 211 Å filter observations. We computed the
+wavelet power, global power, and 95% significance levels using the method developed by Torrence & Compo (1998).
+3. OBSERVATIONAL RESULTS
+We used high-resolution multi-wavelength imaging and spectroscopic data of AIA and IRIS to study an eruptive jet on August
+29th, 2014. This jet was situated near the west limb, and the jet was initiated around 06:28 UT. Here, we have described the whole
+jet event in upcoming subsections.
+3.1. Onset of the Kink-instability
+
+QPP in Kink Unstable Jet
+5
+We use a sequence of AIA 304 Å images to understand the spatio-temporal evolution of kink instability in a blowout jet (cf.,
+Figure 1). The images of Figure 1 are plotted in the reverse intensity. The direction system is displayed in the panel (a) of
+Figure 1. This blowout jet starts to lift at ≈06:28 UT from an active region NOAA 12146. At t = 06:29:07 UT, we see the inverse
+γ-shape structure outlined by a white-dashed line (panel a; Figure 1), which is a manifestation of the writhing motion near the
+base of the jet. This magnetized structure (i.e., inverse γ-shape structure) is developing with time, i.e., it is expanding and rising
+(panels (b) and (c); Figure 1; animation 1.mp4, animation 2.mp4). In the meantime, the inverse γ-shape structure performs an
+anticlockwise twist (see attached animation; animation 1.mp4), and as a result, it may become kink unstable. The characteristic
+inverse γ-shape evolves due to the conversion of the initial twist into the writhe (T¨or¨ok et al. 2010, 2014), and the inverse γ
+shape-like structure is a morphological signature for the onset of the kink instability.
+We see that the inverse γ-shape structure is lifting and expanding with time (panels (b) and (c); Figure 1). Now, at time t
+= 06:30:19 UT, the inverse γ-shape structure has fully developed, and again we have outlined the fully developed structure by
+the white dashed line (panel (d); Figure 1). We see little brightening around the knot (i.e., cross point) of the inverse γ-shape
+structure. In addition, we also see localized bright dots, and two bright dots are indicated by two black arrows (see panel d). The
+bright dots further propagate upward along the magnetic field lines as the jet develops with time. The brightening at the knot (as
+indicated by white arrows in the panel (e); Figure 1) is becoming stronger and wider with time (see, panels (e) and (f); Figure 1;
+animation 1.mp4, animation 2.mp4).
+Meanwhile, the plasma is moving up along the magnetic field that is forming the main body of this jet. This jet is not well-
+collimated as the width of the jet is increasing with time (see, panels (d), (e), (f), (g), (h), and (i); Figure 1; animation 1.mp4,
+animation 2.mp4). Hence, this jet is a typical blowout jet as per the morphological classification of the solar jets (e.g., Moore et al.
+2010, 2013). The jet is made of various plasma threads, and out of them, we have marked one plasma thread at 06:30:43 UT
+by a black arrow (panel (e); Figure 1). This particular thread is located at the top edge of the jet this time (i.e., 06:30:43 UT),
+and further, we have traced this particular plasma thread at various other times (please see the black arrows from panel (e) to (l);
+Figure 1). This particular thread follows anticlockwise motion with time, which suggests that the jet spire is rotating. In addition,
+we have also shown the evolution of the jet using IRIS 1330 Å filter (cf., figure A.1). This animated figure describes the main
+features of the jet as per the IRIS animations (animation 1.mp4, animation 2.mp4). Similar to AIA 304 Å, the IRIS/SJI 1330 Å
+filter observations also show the rotating motion of the jet. Hence, finally, we can say that the rotating motion of the jet plasma is
+also visible in the animated figure A.1.
+3.2. Multi-wavelength imaging observations of solar jet
+In the previous subsection (i.e., section 3.1), we have described the dynamics of the inverse γ-shape flux-rope and triggering
+of the blowout jet. In this subsection, we are investigating the dynamics of the blowout jet in cool (i.e., AIA 1600 Å, AIA 304 Å,
+and IRIS/SJI 1330 Å) and warm/hot temperature filters (i.e., AIA 171 Å, AIA 131 Å, and AIA 94 Å) to understand the multi-
+wavelength nature of this jet. The distinction between hot and cool temperature filters is relative, and the classification between
+hot and cool temperatures may changes from case to case. Here, in the present study, we say that AIA 1600 Å, AIA 304 Å,
+and IRIS/SJI 1330 Å are cool filters while AIA 171 Å, AIA 131 Å, and AIA 94 Å are warm/hot filters. We know that this
+inverse γ-shape magnetic structure lifts up with time (see; Figure 1). Here, in this subsection, we have discussed the multi-
+wavelength observations of the blowout jet. The Figure 2 shows the spatio-temporal evolution of the jet in AIA 1600 Å (top
+row), IRIS/SJI 1330 Å (second row), AIA 171 Å (third row), and AIA 131 Å wavebands. Similar to Figure 1, we have again
+displayed the directions in the panel Figure 2(a1). At t = 06:29:23 UT, the inverse γ-shape structure has significantly developed,
+and the top of this structure is well above the limb. This inverse γ-shape structure is clearly visible in cool temperature filters
+(panel (a1), (a2), (b1), and (b2);Figure 2), but it is not visible in the hot temperature filters (panels (c1), (c2), (d1), and (d2);
+Figure 2; animation 3.mp4). However, we see that the brightened base of inverse γ-shaped structure in the hot temperature filters.
+At the next time (t = 06:29:30 UT), we clearly see the jet in the cool temperature filter which is indicated by the black arrows
+(see, panels (a2) and (b2); Figure 2). At this time, we see a compact brightening in the vicinity of knot of inverse γ-shape structure
+(indicated by black arrows in panels (c2) and (d2); Figure 2) in the hot temperature filters (panels (c2) and (d2); Figure 2). We do
+not see the spire of jet in the hot temperature filters as we have seen in cool filters. At the next time (t = 06:31:18 UT), the jet is
+fully developed with two legs rooted at the solar surface (panels (a3) and (b3); Figure 2). The upper leg is termed as northern leg
+while the lower leg is termed as southern leg. And, these two legs are clearly indicated by the white and black arrows in the panels
+(a3) and (b3) of Figure 2, respectively. Interestingly, we do see the signature of bi-directional flows, i.e., plasma falls along both
+
+6
+Mishra et al.
+80
+100
+120
+140
+160
+Y (arcsec)
+80
+100
+120
+140
+160
+06:29:28 UT
+Inverse γ−shape
+(b1) IRIS/SJ I 1330
+06:30:16 UT
+Kink unstable jet
+(b2)IRIS/SJ I 1330
+06:32:25 UT
+(b3)
+a
+IRIS/SJ I 1330
+Northern leg
+Southern leg
+b
+IRIS/SJ I 1330
+06:35:56 UT
+(b4)IRIS/SJ I 1330
+80
+100
+120
+140
+160
+Y (arcsec)
+80
+100
+120
+140
+160
+06:29:43 UT
+(c1)
+AIA 171
+06:30:19 UT
+(c2)
+AIA 171
+06:31:19 UT
+AIA 171
+(c3)
+a
+b
+06:36:23 UT
+AIA 171
+(c4)
+80
+100
+120
+140
+160
+Y (arcsec)
+80
+100
+120
+140
+160
+06:29:23 UT
+Inverse γ−shape
+AIA 1600
+(a1)
+06:30:30 UT
+Kink unstable jet
+(a2)
+AIA 1600
+06:31:18 UT
+AIA 1600
+(a3)
+a
+b
+Northern leg
+Southern leg
+06:35:52 UT
+(a4)
+AIA 1600
+940
+960
+980
+1000
+1020
+X (arcsec)
+80
+100
+120
+140
+160
+Y (arcsec)
+940
+960
+980
+1000
+1020
+80
+100
+120
+140
+160
+06:29:32 UT
+AIA 131
+(d1)
+940
+960
+980
+1000
+1020
+X (arcsec)
+940
+960
+980
+1000
+1020
+06:31:08 UT
+AIA 131
+(d2)
+940
+960
+980
+1000
+1020
+X (arcsec)
+940
+960
+980
+1000
+1020
+06:32:32 UT
+AIA 131
+(d3)
+a
+b
+940
+960
+980
+1000
+1020
+X (arcsec)
+940
+960
+980
+1000
+1020
+06:36:08 UT
+AIA 131
+(d4)
+N
+S
+W
+E
+Figure 2. The figures show the multi-wavelength view of the kink unstable blowout jet observed on August 29th, 2014 by SDO/AIA and IRIS.
+It shows the morphological evolution of kink unstable blowout jet in the different layers of the solar atmosphere (i.e., AIA 1600 (top row),
+IRIS/SJI 1330 (second row), AIA 304 (third row), and AIA 131 (bottom row)). In this evolution, we have seen various important features of
+this blowout jet, namely, the development of inverse γ-shape in the cool filters (panels (a1) and (b1)), bright knot due to magnetic reconnection
+along with the triggering of blowout jet (panels (a2), (b2), (c2), and (d2)), bi-directional flows from bright know ((a3), (b3), (c3), and (d3)),
+and matured phase of jet with a cavity (panels (a4) and (b4)). Here, it should be noted that the jet is mainly visible in the cool filters, and does
+not emit much in hot filters. However, we see a compact bright structure in the hot filters (see (c2) and (d2)) which justifies the occurrence
+of internal magnetic reconnection in highly twisted magnetic field lines of inverse γ-shape near its apex. The northern and southern legs are
+indicated by white and black arrows in the panel (a3) and (b3). The two boxes ’a’ and ’b’ are shown in panels (a3), (b3), (c3), and (d3). We
+have used these boxes to investigate the total emission measure (EM) (see Figure 4(c) and(d)). The animation shows the (animation 3.mp4;
+AIA 304, AIA 171, AIA 131, and AIA 94 Å) triggering and the complete evolution of the kink unstable jet and their association with the
+multi-thermal plasma. The kink instability at the base of the jet, triggering of magnetic reconnection, the eruption of the jet, and associated
+dynamics are shown between 06:25 UT to 07:00 UT. The real-time duration of 12 s for this animation.
+legs below the bright knot (see red arrows in (a3), (b3), (c3), and (d3) panels; Figure 2; animation 3.mp4) while the plasma flows
+up above the bright knot (see blue arrows in (a3), (b3), (c3), and (d3) panels; Figure 2). This bi-directional flow is also clearly
+visible in the animation 1.mp4, animation 2.mp4, animation 3.mp4. This up-and-down flow of the plasma builds the main body
+of this blowout jet. Also, the bi-directional flows of the jet are visible in the cool filters. However, we only see some faint
+signatures of the blowout jet in the hot temperature filters (panels (c3) and (d3); Figure 2). At time t = 06:35:52 UT, the northern
+leg of inverse γ-shape flux-rope is completely disconnected from the solar limb while, the southern leg remains connected to
+the limb of the Sun (panels (a4), (b4), (c4), and (d4); Figure 2). Now, the jet is fully developed at this time along southern leg.
+Interestingly, we see big cavity in the jet plasma that is indicated by cyan color arrows (panels (a4) and (b4); Figure 2). It seems
+that some plasma from the main body of the jet has bifurcated, and the space between main body and bifurcated plasma of jet
+appears the black region (i.e., cavity).
+
+QPP in Kink Unstable Jet
+7
+3.3. Time-distance analysis of blowout jet
+We have performed the time-distance estimation along and across (perpendicular) the blowout jet to understand the kinematics
+of this blowout jet (cf., Figure 3). We displayed the late phase of the blowout jet along with one horizontal slit (S1) and four
+vertical slits (P1, P2, P3, and P4) in the panel (a) of Figure 3. All of the slits are shown by white dashed lines (panel (a)). As we
+have already indicated that jet is mainly visible in the cool temperature filters, therefore, the time-distance analysis is performed
+using the cool temperature filter, i.e., AIA 304 Å. The panel (b) of Figure 3 shows the time-distance diagram corresponding to
+the horizontal slit (i.e. slit S1). We have considered the width of 30 pixels around the horizontal slit (i.e., 15 pixels on each side
+of the slit) for the time-distance diagram shown in the panel (b) of Figure 3. We have drawn a path (dashed cyan line) on the
+ascending motion of jet plasma, and it is found that the jet plasma is moving up with the speed of 234.0 km s−1. Interestingly, on
+the close inspection, we noticed the multiple intensity enhancement at a regular intervals as indicated by the black arrows in the
+panel (b) of Figure 3. Here, we have identified at-lest three peaks.
+The right column of the Figure 3 shows four time-distance diagrams of the jet deduced from four different heights, i.e., along
+the slits P1 (panel (c)), P2 (panel (d)), P3 (panel (e)), and P4 (panel (f)). Again, we have used the same width of 30 pixels on both
+sides of all four slits (i.e., 15 pixels on both sides of the slit) in the production of time-distance diagrams. The panel (c) shows
+the time-distance diagram along the first vertical slit P1. We have already shown the presence of inverse γ-shape flux-rope, and
+the legs of this flux-rope show opposite motion before the jet eruption (see section 3.2). The slit P1 is located at the base of the
+jet, and covers both legs of the inverse γ-shape flux-rope. We observe two opposite motions of the plasma (white dotted curve on
+panel (c) of Figure 3). This particular pattern is visible due to the opposite motion of the legs of the flux rope.
+The time-distance diagram as per the second slit P2 is displayed in the panel (d) of Figure 3. In the first instance, we see two
+different intensity patches in the time-distance diagram, i.e., the first one is a long & broad patch (indicated by green arrow)
+while the second one is a narrow and slanted patch (indicated by cyan arrow). In the very initial phase, the blowout jet had a
+single body, while after some time the jet body was bifurcated into two parts with a cavity in between them as already explained
+in section 3.2. Even, in the reference image (panel (a); Figure 3), one can see the long straight main body of the jet, and some
+plasma fragments are distributed on a curved path below the main body of the jet. The base of slit P2 lies on the curved path, and
+then the slit crosses some part of the cavity before covering the main body of the jet. Hence, the bottom narrow slanted intensity
+patch (indicated by cyan color arrow) in the time-distance diagram forms due to the plasma fragments along a curved path. While
+the long & broad patch (indicated by green arrow) is due to the dynamics of the main jet body.
+The main body of the blowout jet contains small bright dots that are indicated by white arrows in the long straight intensity
+patch. The time-distance diagram along the slit S1 (i.e., panel (b); Figure 3) has already shown the multiple intensity peaks. We
+observe that same intensity enhancement is visible as multiple bright dots on regular time-interval in this time-distance diagram
+(panel (d) of Figure 3) drawn for the across the jet (i.e., slit P2). Here, at least we have identified three bright dots, and it should
+be noted that these bright dots are aligned on the slanted path in the broad & long patch.
+The time-distance diagram corresponding to slit P3 is displayed in the panel (e) of Figure 3. Similar to the previous time-
+distance diagram, at this height, we also see the bright dots in the main body of the jet. Here, again these bright dots are indicated
+by the white arrows in the time-distance diagram corresponding to slit P3 (panel (e); Figure 3), and these bright dots are on the
+slanted path in the long & broad patch (indicated by green arrow). The narrow slanted patch of intensity (indicated by cyan
+arrow) exists here as already seen in the panel (d) of Figure 3. The last vertical slit (i.e., slit P4) is located very far away from
+the base of the jet. And, the panel (f) of Figure 3 shows the time-distance diagram corresponding to this slit. Again, we see both
+patches of intensity as already seen in the panels (d) and (e). Although, intensity in the slanted patch is weak in comparison to
+the panels (d) and (e) of Figure 3. It is because that slit P4 is located near the top part of the jet. However, we see the multiple
+bright dots like the time-distance diagrams corresponding to the slits P2 (panel (d)) and P3 (panel (e) of Figure 3).
+3.4. Thermal structure of blowout jet
+We perform the differential emission measure (DEM) analysis to understand the thermal nature of this blowout jet. We use
+regularized inversion code developed by (Hannah & Kontar 2012) to extract the DEM using hot AIA/SDO filters. We use six hot
+optically thin EUV filters (e.g., 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å) of the AIA/SDO to estimate DEM coming from
+different temperatures bins. In the panels (a1), (a2), and (a3) of Figure 4, we have displayed three emission measure (EM) maps
+deduced from three different temperature ranges (i.e., log T/K = 5.7– 6.0, log T/K = 6.0– 6.3, and log T/K = 6.9– 7.2) during the
+onset of the kink-instability (i.e., t = ∼06:32:03 UT). Similarly, in the panels (b1), (b2), and (b3) of Figure 4, we have shown the
+
+8
+Mishra et al.
+Distance (Mm)
+0.0
+5.0
+10.0
+15.0
+Time in minutes (start from 06:25 UT)
+0
+20
+40
+60
+80
+100
+120
+20.0
+25.0
+234 km/s
+0
+20
+40
+60
+Distance (Mm)
+0
+20
+40
+60
+Distance (Mm)
+0.0
+5.0
+10.0
+15.0
+20.0
+25.0
+Time in minutes (start from 06:25 UT)
+0
+20
+40
+60
+Distance (Mm)
+0
+10
+20
+30
+Distance (Mm)
+Broad & long patch
+Slanted patch
+Broad & long patch
+Slanted patch
+950
+1000
+1050
+1100
+1150
+X (arcsecs)
+0
+50
+100
+150
+Y (arcsecs)
+950
+1000
+1050
+1100
+1150
+0
+50
+100
+150
+06:37:55 UT
+P1
+P2
+P3
+P4
+S1
+Slanted patch
+Broad & long patch
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 3. The panel (a) shows the intensity image from AIA 304 Å at t = 06:37:55 UT, i.e., from the decay phase of the jet. The over-plotted
+dashed lines are various slits along (i.e., S1) and across (i.e., P1, P2, P3, and P4) the jet, and they are used to produce the time-distance diagrams.
+The time-distance diagram along the slit S1 is shown in the panel (b), and we have drawn a line (i.e., dashed cyan line) along the ascending
+phase of the jet to estimate the speed of the blowout jet which is 234 km/s. We have also seen multiple spikes in this time-distance diagram
+which are indicated by the black arrows. In the right column, we have shown the time-distance diagrams along the slits P1 (panel (c)), P2 (panel
+(d)), P3 (panel (e)), and P4 (panel (f)). Here, we have seen the opposite motion of inverse γ-shape as indicated by the path drawn by the cyan
+dashed line in panel (c). Further, we have also seen the bright dots jet’s body which is indicated by white arrows (panels (d) and (e)). Here,
+we see a slanted intensity patch (panels (d), (e), and (f)) in the last three panels of the right column which is occurring due to the fragmented
+plasma on a curved path from the main body of the blowout jet.
+
+QPP in Kink Unstable Jet
+9
+same three EM maps during the developed phase of the blowout jet (i.e., t = ∼06:35:03 UT).
+In the initial phase, EM maps show a significant emission around the bright knot (as defined previously in sections 3.1 and 3.2)
+above the legs of inverse γ flux-rope as indicated by the black arrows in (a1), (a2), and (a3) panels of the Figure 4. The emission
+in the bright knot region exists over a very wide range of the temperature, i.e., log T = 5.7 to 7.2 K. Hence, it justifies the presence
+of multi-thermal plasma at the knot of inverse γ-shape flux-rope. After approximately 03 minutes, the emissions in the vicinity
+of the bright knot are significantly reduced (see all panels (b1), (b2), and (b3); Figure 4). Although we see little emission in the
+bottom region of the bright knot. And, a faint jet originating from this little emission area. The faint jet is indicated by the black
+arrows in the panels (b1), (b2), and (b3) of Figure 4.This EM study indicates that the spire of jet has very little hot emission. That
+is consistent with this jet showing up strongly in the cool filters, and being very faint in the hot filters.
+We have selected a box (a) within the bright knot region (see, black rectangular box in panel (a1); Figure 4) to know the
+temporal behavior of DEM. We classify the entire DEM into five different temperature ranges (bins), namely, 0.5–1.0, 1.0–2.0,
+2.0–4.0, 4.0–8.0, and 8.0–15 MK. Then, we estimated emission (EM =
+� Tmax
+Tmin DEM(T)dT) in all above specified temperature
+bins. Through this approach, we got five different EM curves, and they are displayed by five different colors in the panel (c) of
+Figure 4. It is visible that all five curves show a dominant peak during the formation of the bright knot. The total emission in all
+five temperature ranges is highest during the formation of the bright knot.
+We also extract the EM curves from another box (i.e., box (b)), which is situated inside the southern leg of the blowout jet
+(please see box (b) in panel (b1) of Figure 4). These EM curves are displayed in panel (d) of Figure 4. Interestingly, the EM
+curves from box (b) show periodic nature, unlike the nature of EM curves deduced from the box (a) (panel (c) of Figure 4). The
+box (a) is located in the northern leg of the blowout jet that erupts completely during the initial phase of the blowout jet. The
+plasma is completely swept away in the vicinity of the northern leg right after its eruption. Hence, we see only one dominant
+peak in EM curves extracted from the box (a). While, box (b) is situated near the base of the stable leg (i.e., southern leg) of this
+blowout jet. The periodic nature of EM curves extracted from the box (b) is present in all five temperature bins. We can easily
+locate at least three to four peaks in each EM curve, and it matches with the intensity light curve extracted from the different
+boxes from cool temperature filter AIA 304Å (see, section 3.6). The vertical black dotted line in the lower panels (i.e., panels (c)
+and (d); Figure 4) indicates the jet event start time (t = ∼06:28 UT).
+3.5. Spectroscopic diagnosis of blowout jet
+In addition to the slit jaw images, IRIS has also captured the near ultraviolet (NUV) as well as far ultraviolet (FUV) spectrum
+of this jet event. The panel (a) of Figure 5 shows the IRIS/SJI intensity map at a time of ∼06:32:06 UT. The IRIS has observed the
+event with a roll angle of 90◦. Therefore, this IRIS/SJI intensity map (i.e., panel (a)) of the Figure 5 is 90◦ rotated in comparison
+to other Figures (i.e., Figure 1, 2, and other Figures) of this manuscript. Here, we mention that direction system is added in the
+panel (a) of Figure 5. We have over-plotted all eight slit positions (i.e., eight vertical gray-dashed lines) on this IRIS/SJI intensity
+map (panel a). Then, we have chosen two locations to know the nature of Si iv, C ii and Mg ii profiles, namely, red plus sign
+(x = 993.12” and y = 95.19”) and blue plus sign (x = 997.12” and y = 93.52”). It should be noted that the selected locations
+(red and blue plus sign in panel a of Figure 5) are located in the vicinity of the bright knot region, i.e., most probably in the
+magnetic-reconnection region.
+In the imaging analysis, we have already shown the existence of the bi-directional flows in this jet event (see; section 3.2). And,
+as per the imaging analysis, the red plus sign lies in the down-flow region while the blue plus sign is in the up-flow region (see;
+Figure 2 and section 3.2). In Figure 5, we have displayed Si iv 1393.77 Å spectral profiles from red plus sign location by the red
+curve at three different times (i.e., 06:30:21 UT (panel b), 06:31:37 UT (panel c), and 06:32:54 UT (panel d)). Similarly, the Si iv
+1393.77 Å spectral profiles from blue plus locations are displayed by the black curve at three different times (i.e., t = 06:30:40 UT
+(panel (b)), 06:31:56 UT (panel (c)), and 06:33:13 UT (panel (d)). In the same way, we have shown the spectral profiles from red
+plus (red curve) and blue plus locations (black curve) of C ii 1335.66 Å spectral profiles (see panels (e), (f), and (g)) and Mg ii
+2796.35 Å (see panels (h), (i), and (j)).
+We have noticed that all black profiles (i.e., from Si iv, C ii, and Mg ii at all three times) are blue-shifted (upflows) while all the
+red profiles are red-shifted (down-flows). Hence, we can say that all three spectral lines (e.g., Si iv, C ii, and Mg ii) justify that
+red plus location is dominated by plasma downfall while the blue plus location is dominated by up flows, i.e., the spectra confirm
+the presence of bi-directional flows in the vicinity of the bright knot. Hence, finally, we can say that both (spectra and images)
+confirm the presence of bi-directional flows in the vicinity of the bright knot. In addition, we do see very broad profiles from
+Si iv, C ii, and Mg ii spectral lines during the initial/main phases of the blowout jet. Si iv 1393.77 Å spectral line is an optically
+
+10
+Mishra et al.
+80
+100
+120
+140
+160
+Y (arcsec)
+80
+100
+120
+140
+160
+log T=5.7-6.0
+a
+06:32:03 UT
+(a1)
+log T=6.0-6.3
+Bright knot
+(a2)
+log T=6.9-7.2
+(a3)
+940
+960
+980
+1000
+X (arcsec)
+80
+100
+120
+140
+160
+Y (arcsec)
+940
+960
+980
+1000
+80
+100
+120
+140
+160
+06:35:03 UT log T=5.7-6.0
+(b1)
+b
+Spire of jet
+940
+960
+980
+1000
+X (arcsec)
+940
+960
+980
+1000
+log T=6.0-6.3
+Spire of jet
+(b2)
+940
+960
+980
+1000
+X (arcsec)
+940
+960
+980
+1000
+log T=6.9-7.2
+Spire of jet
+(b3)
+19
+19
+20
+21
+22
+23
+24
+log10(EM)[cm
+-5 K
+-1]
+Bright knot
+Bright knot
+0
+500
+1000
+1500
+2000
+2500
+Time (seconds) start from 06:20 UT
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized EM
+(c)
+T[0.5--1.0]MK
+max[EM]=1.36*10
+23 cm
+-5 K
+-1
+T[1.0--2.0]MK
+max[EM]=6.24*10
+23 cm
+-5 K
+-1
+T[2.0--4.0]MK
+max[EM]=1.12*10
+24 cm
+-5 K
+-1
+T[4.0--8.0]MK
+max[EM]=3.92*10
+23 cm
+-5 K
+-1
+T[8.0--15.0]MK
+max[EM]=1.65*10
+23 cm
+-5 K
+-1
+0
+500
+1000
+1500
+2000
+2500
+Time (seconds) start from 06:20 UT
+0.4
+0.6
+0.8
+1.0
+Normalized EM
+(d)
+T[0.5--1.0]MK
+max[EM]=1.14*10
+23 cm
+-5 K
+-1
+T[1.0--2.0]MK
+max[EM]=5.54*10
+23 cm
+-5 K
+-1
+T[2.0--4.0]MK
+max[EM]=8.64*10
+23 cm
+-5 K
+-1
+T[4.0--8.0]MK
+max[EM]=3.56*10
+23 cm
+-5 K
+-1
+T[8.0--15.0]MK
+max[EM]=2.13*10
+23 cm
+-5 K
+-1
+Figure 4. The panels (a1), (a2), and (a3) shows EM maps in three different temperature ranges (i.e., log T/K = 5.7–6.0, 6.0–6.3, and 6.9–7.2)
+during the time t = 06:32:03 UT when the magnetic reconnection takes place around the bright knot of inverse-γ structure. The bright knot is
+indicated by the black arrows in all panels of the top row. Similarly, the middle row shows the same EM maps during the developed phase of
+this blowout jet (i.e., t = 06:35:03 UT). Here, we see very faint emission of the jet as indicated by black arrows in the middle row. Further, we
+have selected one box in the northern leg (i.e., box a) and another box in the southern leg (box b) to investigate the temporal variations of the
+EM curves in various temperature ranges. Various EM curves are shown in panels c (box a) and d (box b) of Figure 4 by various colors. The
+color scheme and temperature ranges are mentioned in both panels. All EM curves from the box a show only one peak while all EM curves
+from box b show three to four peaks. The box (a) is located in the northern leg which erupts in the initial phase of the jet.
+thin line, and normally, it is a single peak. However, in this blowout jet, the Si iv 1393.77 Å spectral line is either double peak or
+highly asymmetric profile (panels (b), (c), and (d) of Figure 5). On the other hand, C ii 1335.62 Å and Mg ii 2796.35 Å spectral
+lines are optically thick lines, and mostly, they appear as double peak profiles in the solar atmosphere. However, in this blowout
+jet, we see the very complex type of profiles from C ii 1335.62 Å (see panels (e), (f), and (g)) and Mg ii 2796.35 Å (see panels (h),
+
+QPP in Kink Unstable Jet
+11
+ IRIS/SJI- 06:32:06
+970
+980
+990 1000 1010
+X (arcsecs)
+70
+80
+90
+100
+110
+120
+Y (arcsecs)
+ (a)
+ N
+ S
+ W
+ E
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+10
+100
+1000
+ Intensity (DN)
+ Si IV: 1393.77 A
+ t = 06:30:40
+ t = 06:30:21
+ (b)
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+ 
+ t = 06:31:56
+ t = 06:31:37
+ (c)
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+ 
+ t = 06:33:13
+ t = 06:32:54
+ (d)
+-150
+-100
+-50
+0
+50
+100
+ Doppler shift (km/s)
+1
+10
+100
+ Intensity (DN)
+ C II: 1335.66 A
+ (e)
+-150
+-100
+-50
+0
+50
+100
+ Doppler shift (km/s)
+ 
+ (f)
+-150
+-100
+-50
+0
+50
+100
+ Doppler shift (km/s)
+ 
+ (g)
+-150
+-100
+-50
+0
+50
+100
+ Doppler shift (km/s)
+10
+100
+1000
+ Intensity (DN)
+ Mg II: 2796.35 A
+ (h)
+-150
+-100
+-50
+0
+50
+100
+ Doppler shift (km/s)
+ 
+ (i)
+-150
+-100
+-50
+0
+50
+100
+ Doppler shift (km/s)
+ 
+ (j)
+Figure 5. IRIS/SJI 1330 Å image depicts the blowout jet at time t = 06:32:06 UT along with 8-slit positions shown by white dashed lines in
+panel (a). IRIS has captured the spectra along these slits. Further, we have selected one location (i.e., red plus sign) in the northern leg while
+another location (blue plus sign) in the southern leg of the blowout jet. The panel (b), (c), and (d) show the spectral profiles of Si iv from both
+locations (i.e., the red curve from red plus location and black curve from blue plus location) at time t = 06:30 UT (panel b), 06:31 UT (panel
+(c)), and 06:32 UT (panel (d)). In the same fashion, we have shown C ii (panels (e), (f), and (g)) and Mg ii (panels h, i, and (j)) from both
+locations at the same three times. In general, all the spectral profiles from the red plus location are red-shifted (i.e., plasma downfall) while
+these profiles are blue-shifted (i.e., plasma upflows) from the blue plus location. In addition, all the spectral profiles are very complex profiles.
+(i), and (j)). Such type of complex profiles are reported in some small-scale energetic events (e.g., Peter et al. 2014; Young et al.
+2018).
+Further, we have taken five pixels around blue plus location (see; panel (a) Figure 5), and then all five profiles were averaged
+to get a single averaged Si iv profile at any particular instant of time. Through this approach, we have produced 33 averaged
+spectral profiles in a time range from 06:21:43 UT to 07:06:25 UT. This observation is an 8-step coarse raster observation with
+the cadence of 77 seconds. Therefore, the averaged spectral profile of any location is available after every 77 seconds (i.e.,
+8.0*cadence time).
+Some key averaged spectral profiles are displayed in the Figure 6. Panel (a) does not show the presence of the Si iv line as it
+is well before the jet event (t = 06:24:17 UT). While, t = 06:29:23 UT, we see the Si IV spectral line as the formation of the jet
+has already begun (see panel (b); Figure 6). We have fitted the line profile with the single Gaussian (blue curve) to estimate the
+peak intensity, Doppler velocity, and line width of the profile. The line width of this profile is high (i.e., 46.99 km/s) at time t =
+06:29:23 UT (panel (b)). At next time t = 06:31:56 UT, we found that the profile is very wide and asymmetric too. We have fitted
+this profile with the single Gaussian, and found a very high line width (i.e., 67.66 km/s). It should be noted that the line width
+has increased a lot (i.e., 67.66 km/s) in comparison to the previous time t = 06:29:29 UT (panel (b). After t = 06:31:56 UT, we
+notice a decrease pattern in the line width of Si iv 1393.77 Å profiles, i.e., the line width is 39.32 km/s at time t = 06:33:13 (panel
+
+12
+Mishra et al.
+ 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ Norm. Counts
+ Si IV: 1393.77 A
+ (a) t = 06:24:17
+ 
+ 
+ Width:46.99km/s
+ (b) t = 06:29:23
+ 
+ 
+ Width:67.66km/s
+ (c) t = 06:31:56
+ 
+ 
+ Width:39.32km/s
+ (d) t = 06:33:13
+ 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ Norm. Counts
+ Width:31.22km/s
+ (e) t = 06:34:30
+ 
+ 
+ Width:35.59km/s
+ (f) t = 06:35:46
+ 
+ 
+ Width:43.60km/s
+ (g) t = 06:37:03
+ 
+ 
+ Width:23.41km/s
+ (h) t = 06:38:20
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ Norm. Counts
+ Width:25.79km/s
+ (i) t = 06:39:36
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+ 
+ Width:31.64km/s
+ (j) t = 06:40:53
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+ 
+ Width:26.52km/s
+ (k) t = 06:42:09
+-200
+-100
+0
+100
+200
+ Doppler shift (km/s)
+ 
+ Width:22.09km/s
+ (l) t = 06:43:26
+Figure 6. The temporal evolution of normalized averaged Si-IV line profiles (i.e., averaged over the five pixels around the blue plus location
+shown in Figure 5) from the most probable reconnection region. It should be noted that all spectral profiles are normalized by their maximum
+counts. All the spectral profiles are fitted by single Gaussian (see blue curve in all panels). We do see periodic fluctuations in the line width of
+Si iv. The first panel does not show the line as it is before the triggering of the solar jets.
+(d) of Figure 6) and 31.22 km/s at time t = 06:34:30 UT (panel (e); Figure 6). Hence, for approximately 03 minutes (i.e., from
+06:31:56 UT to 06:34:40 UT), the line width shows a decreasing pattern as the line width falls from 67.66 km/s to 31.22 km/s.
+However, this decrease pattern in the line width breaks at time t = 06:35:46 UT as we see that the line width is now increasing
+with time (see; panels (f) and (g) of Figure 6). But again, the line width decreases with time t = 06:38:20 UT (panel (h); Figure 6).
+And one more time (i.e., third time), we see the same behavior of the line width, i.e., the line width increase (panels (i) and (j);
+Figure 6) and decrease further with time (panels (k) and (l) of Figure 6). This particular finding indicates that the line width of
+the Si iv spectral line has periodic behavior.
+To understand the periodic behavior of line width clearly, we have plotted the line width with time (see; panel (a) of Figure 7).
+As we know already, in the specified time range (i.e., t = 06:21:28 UT to 07:03:52 UT), there are 33 spectral lines, i.e., 33
+line widths values. The specified time range starts from 06:21:28 UT, however, the jet appears in the selected region around
+06:28:07 UT (see dashed green vertical line in panel (a); Figure 7). We know that there is no spectral line before 06:28:07 UT,
+therefore, all the points before the green dashed vertical line have zero line width. We see the increase and decrease in the line
+width on a regular interval of time (see; panel (a); Figure 7) . The oscillating behavior of the line width exists up to the time
+of t = 06:48 UT, i.e., up to the red-dashed vertical line. After ∼06:48 UT, the downfall phase of the blowout jet dominates, and
+
+QPP in Kink Unstable Jet
+13
+ 
+20
+40
+60
+80
+ Line Width (km/s)
+ 06:28:07 UT: Upflow
+ 06:48:33 UT: Downfall
+ (a)
+0
+500
+1000
+1500
+2000
+2500
+ Time (Seconds after 06:21:43 UT)
+-40
+-20
+0
+20
+40
+ Doppler Velocity (km/s)
+ (b)
+-60
+-40
+-20
+0
+20
+40
+ Doppler Velocity
+10
+20
+30
+40
+50
+60
+70
+ Line Width (km/s)
+R = -0.570
+y = -0.35x + 31.06
+ (c)
+Figure 7. The temporal evolution of line width and Doppler velocity from the most probable reconnection region (blue plus sign in Figure 5)
+is displayed in panels a and b, respectively. The triggering time of the jet is indicated by the green-dashed line, therefore, all the points before
+the green-dashed line are zero. The downfall phase of the blowout jet dominates after the red vertical dashed line. In the up flow phase of the
+jet, we do see the periodic behavior of the line width. Further, we also see the periodic behavior of Doppler velocity in the up-flow phase of the
+blowout jet (panel b). We have performed a correlation between line width and Doppler velocity which is shown in panel (c). It is found that
+line width is negatively correlated with the Doppler velocity, i.e., line width is during the up-flow, and the line width decreases as line profiles
+move towards the red-shifts.
+we don’t see much variations in the line width. So, finally, we can say that oscillations in the line width are present during the
+up-flow phase of the blowout jet.
+In addition to the line width, we have also shown the Doppler velocity with the time in panel (b) of Figure 7. The first few
+points are at zero Doppler velocity (up to the green dashed line) as the jet was not triggered by that time. And after that, we do
+see the fluctuation in the Doppler velocities (see points after the green dashed line). The careful inspection reveals that Doppler
+velocity is anti-correlated with line width during the up flow phase of the jet (i.e., points in between the green and red dashed
+lines). It means when the line width is high then Si iv line is blue-shifted and vice versa. As the line width decreases with time,
+then in response, the Si iv line moves towards the red shifts. We have already pointed out that there are a few cycles of periodic
+increase and decrease in the line width during the up-flow phase of the blowout jet. Similarly, we do see a kind of periodic
+behavior of Doppler velocity too. After 06:48:33 UT, all spectral profiles are red-shifted (see, points after the red dashed vertical
+line) as this is downfall dominant phase of the blowout jet. Further, we have checked the correlation between the line width and
+Doppler velocity for the up-flow dominated phase of the blowout jet (i.e., points between green and red dashed vertical lines)
+which is shown in the panel (c) of Figure 7. Now, it is well clear that line width and Doppler velocity are anti-correlated, i.e., the
+blue shifts (upflows) have high line width, and when line width moves towards the red shifts (downflows) the line width decrease.
+The Pearson coefficient is quite good (i.e., -0.57) for this correlation. Hence, we can say that Doppler velocity and line widths
+are anti-correlated during the up-flow phase of the jet.
+
+14
+Mishra et al.
+3.6. Quasi-periodic Pulsation
+We have found the presence of QPPs during the blowout jet. We have utilized AIA 304 Å, AIA 171 Å, AIA 131 Å, and
+AIA 211Å filters to study the QPPs in this blowout jet. We have selected four different boxes (i.e., B1, B2, B3, and B4) that are
+shown in the panel (a) of Figure 8 by different colors. Further, we estimated the averaged intensity curve (i.e., averaged over all
+pixels in the box) from all four boxes in all AIA filters (i.e, AIA 304 Å, AIA 171 Å, AIA 131 Å, and AIA 211Å). The panels (b)
+to (q) of Figure 8 show the wavelet analysis from four boxes. The wavelet analysis is performed on the averaged intensity light
+curves deduced from AIA 304 Å. The panel (b) of Figure 8 shows the averaged AIA 304 Å intensity curve (black curve) deduced
+from box B1 (black box in panel (a) of Figure 8). There is an over-plotted red curve that is a smoothed averaged intensity curve
+with a window of 15 points. Further, the panel (f) shows the detrended intensity curve, i.e., averaged AIA 304 Å light curve
+(black curve) - smoothed AIA 304 Å curve (red curve; panel (b); Figure 8). Then, we applied the wavelet analysis on this
+detrended light curve, and the deduced wavelet power map is shown in the panel (j) of Figure 8. Here, we do see a concentrated
+patch of power around the period of 03 minutes (i.e., 3.03 minutes) for a time range from 06 to 20 minutes. Further, we have
+estimated a 95% significance level that is important to check the reliability of any detected period in the wavelet analysis. And,
+the 95% significance level is shown by a white contour on the wavelet power map. Now, it is visible that a concentrated patch
+of the power lies within the 95% significance level contours. The cross-hatched gray area in the panel (j) of Figure 8 outlines
+the cone-of-influence (COI), and the powers inside this cross-hatched area are not reliable. But, here we can see that all the
+significant powers are outside of the COI. In the panel (n) of Figure 8, we have shown the global power (i.e., wavelet power
+averaged over time) against the period. The global power shows dominant peak at a period of ∼03 minutes, i.e., 3.03 minutes.
+We have applied the wavelet analysis in the same fashion to all other boxes (i.e., B2, B3, and B4) which are shown in panel (a)
+of Figure 8. The original & smoothed intensity curves, detrended intensity curve, wavelet power maps, and global power maps
+are shown in the same manner for box B2 (panels (c), (g), (k), and (o)), box B3 (panels (d), (h), (l), and (p)), and box B4 (panels
+(e), (i), (m), and (q)) in the Figure 8. Boxes B2 and B3 show very similar behavior as we found for box B1. The dominant period
+is also approximately 03 minutes (i.e., 2.34 minutes for box B2 (panel (o) of Figure 8) and box B3 (panel (p) of Figure 8)) for
+the almost same time range from 06 to 20 minutes.
+However, the intensity curve of the last box (i.e., B4) shows a sharp jump in the intensity for a short interval of time (i.e.,
+around 05 minutes only). It is unlike to the other boxes (B1, B2, and B3) as fluctuations sustain a bit longer therein (around 15
+minutes). We applied wavelet analysis in the same manner to box B4 also, and we found three concentrated patches of the power
+around the period of 06, 03, and 0.5 minutes (panel (q) of Figure 8). The wavelet power patches around the period of 06, 03, and
+0.5 minutes persist only for 10 minutes, 05 minutes, and less than one minute, respectively. Hence, these periods of box B4 (i.e.,
+06, 03, and 0.5 minutes; panel (q) of Figure 8) do not even complete two cycles, therefore, we are assuming them as non-reliable
+power. In addition, some part of the longer period (06 minutes) also lies within the COI (panel (m) of Figure 8). Hence, finally,
+we can say that none of the power patches in this wavelet power map of box B4 is reliable. And, we mention that the B4 does not
+has any periodicity unlike the other boxes (i.e., B1, B2, and B3). We have also shown the QPPs in the same fashion for AIA 171Å
+(cf., Figure B.1), AIA 131Å (cf., Figure B.2), and AIA 211Å (cf., Figure B.3) in the appendix B. The findings from these fitters
+are similar to what we have reported for AIA 304Å here.
+4. DISCUSSION AND CONCLUSIONS
+The present work provides an observation of the formation of blowout jet through kink instability. Initially, an inverse γ-shape
+flux-rope appears on the west limb on August 29th, 2014 that is a morphological indication for the onset of kink instability
+(T¨or¨ok & Kliem 2005; Pariat et al. 2009; Kayshap et al. 2013; Hassanin & Kliem 2016). The inverse γ-shape flux-rope activates
+around 06:28:00 UT, i.e., this structure rises, and expands with time. The twisted field lines are associated with inverse γ-shape
+flux-rope, and these magnetic field lines reconnect. The primary magnetic reconnection takes place around 06:31:00 UT near
+the apex of the inverse γ-shape flux-rope, i.e., in the vicinity of the bright knot. We have witnessed the bi-directional flows from
+the apex of the flux rope through the imaging analysis (section 3.2). Various spectral lines (i.e., Si iv, C ii, and Mg ii) clearly
+show red-shifted profiles (i.e., plasma downfall) below the apex, and all these profiles become blue-shifted (i.e., plasma up flow)
+above the apex of flux-rope (cf., Figure 5), i.e., bi-directional flows. Hence, both (images and spectra) confirm the presence
+of bidirectional flows which is a typical characteristic of the magnetic reconnection in the solar atmosphere (Innes et al. 1997;
+Huang et al. 2014; Innes et al. 2015; Yang et al. 2020; Chitta & Lazarian 2020; Bahauddin et al. 2021; De Pontieu et al. 2021;
+Antolin et al. 2021). Further, DEM analysis shows the presence of multi-thermal plasma around the knot of inverse γ-shape
+flux-rope in the wide temperature range (log T/K = 5.4–7.2; Figure 4). Hence, these observational findings (i.e., bi-directional
+
+QPP in Kink Unstable Jet
+15
+SDO AIA_4 304 29-Aug-2014 06:32:31.134 UT
+940
+960
+980
+1000
+1020
+X (arcsec)
+60
+80
+100
+120
+140
+160
+Y (arcsec)
+B1
+B2
+B3
+B4
+(a)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Norm Cnts
+-0.2
+0.0
+0.2
+0.4
+0.6
+Norm Cnts
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+Period (Minutes)
+1
+1
+2
+4
+6
+8
+10
+12
+Power 
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+Global Period (Minutes) 
+Global Period at max. power ( 3.03 min.)
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+Ł’
+Ł’
+Ł’
+2
+4
+6
+Power 
+Global Period at max. power ( 2.34 min.)
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+Ł<@DRRA
+Ł<@DRRA
+2
+4
+6
+8
+Power 
+Global Period at max. power ( 2.34 min.)
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+0.0
+3.7
+7.4
+11.1
+14.9
+1
+2
+3
+4
+5
+Power 
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+(m)
+(n)
+(o)
+(p)
+(q)
+Figure 8. The panel (a) displays SDO/AIA 304 Å image of the blowout jet at time t = 06:32:31 UT, and we select five boxes (B1, B2, B3, and
+B4) to deduce the emission curve from this filter. The temporal evolution of the intensity (i.e., black curve) in AIA 304 Å filter from box B1
+is displayed in the first panel of the first column. The over-plotted red dashed line is smoothed curve with a window of 15 points. The second
+panel of the first column shows the detrended curve, and the wavelet transform is applied to this detrended curve. The wavelet power map is
+displayed in the third panel with 95% significance (i.e., white contours). The power is mainly concentrated around ∼03 minutes. Finally, in
+the last panel of first column, the global wavelet power is displayed which again shows that global power peaks around ∼ 03 minutes (i.e., 2.78
+minutes). A similar analysis is shown for B2 (second column), B3 (third column), and B4 (fourth column), and the dominant period is ∼ 03
+minutes. While we did not find any significant period in B4.
+
+16
+Mishra et al.
+flows and multi-thermal plasma) indicate that the magnetic reconnection (primary) takes place around the knot of inverse γ-shape
+flux-rope which triggers the jet.
+Soon after the primary magnetic reconnection, the northern leg of the inverse γ-shape flux-rope completely erupts, and further,
+the jet has developed along only the southern leg. Interestingly, we have seen the multiple bright regions (with time) within the
+jet. Here, we have clearly seen the multiple bright spikes in the time-distance diagram estimated as per the slit (i.e., S1) along the
+blowout jet. Similarly, the time-distance diagrams estimated as per the slits (i.e., P2, P3, and P3) across the blowout jet show the
+multiple bright dots (cf.,(section 3.3)). It is trivial to understand that these multiple spikes (along the jet) or multiple bright dots
+(across the jet) are forming due to multiple enhancement in the intensity with time. Most probably, this multiple enhancements
+in the intensity supports the multiple magnetic reconnection scenario (Morton et al. 2012; Li et al. 2015; Kumar et al. 2017).
+Spectroscopic observations reveal a periodic enhancement in the line width of Si iv 1393.77 Å (cf., Figure 6 and 7). For the
+first time, the periodic enhancement of Si iv line width is being reported in this blowout jet event. On top of these crucial
+observational findings, we have seen that Si iv profiles are blue-shifted (upflows) when they are very broad (i.e., high line width).
+And, gradually, the profiles become narrower while they are moving toward the red-shifts (down flows). The periodic existence
+of such broadened blue-shifted Si iv profiles is most probably due to the occurrence of multiple magnetic reconnection in this
+blowout jet. Most importantly, our observations also reveal very complex and explosive type profiles of some prominent spectral
+lines (i.e., Si iv, C ii, and Mg ii) of the solar interface-region (cf., Figure 5). As we know that such complex & explosive type
+profiles are produced only due to the magnetic reconnection (Peter et al. 2014; Innes et al. 2015; Huang et al. 2017; Young et al.
+2018; Chitta & Lazarian 2020), all the observational findings indicate the occurrence of multiple magnetic reconnection.
+QPPs in the solar/stellar flares is an often phenomenon that occurs with few seconds to few minutes of oscillations pe-
+riod. Several reports discuss the triggering and related dynamics of QPPs in the solar atmosphere (e.g., Chen & Priest 2006;
+Inglis & Nakariakov 2009; Nakariakov & Melnikov 2009; Van Doorsselaere et al. 2016; Li et al. 2017; Nakariakov et al. 2018;
+McLaughlin et al. 2018; Kashapova et al. 2020; Shi et al. 2022; Zhou et al. 2022). The statistical studies of the intense solar
+flares suggest that the occurrence rate of QPPs reaches 30-80 % with intense flares lying above the M5 class (Sim˜oes et al. 2015).
+However, the QPPs occurrence rate reduces with the low intense solar flares (Zimovets et al. 2021). Using GOES X-ray data
+from 2011 to 2018, (Hayes et al. 2020) performed the statistical analysis for QPPs and their association with the different classes
+of solar flares. The authors claimed that the ≈46% of X-class and ≈29% of M-class flares show QPPs signature. However,
+only ≈9% of C-class flares exhibit QPPs signature. On the other hand, there are few reported observations of QPPs in jets
+(Morton et al. 2012; Zhang & Ji 2014; Shen et al. 2018). Interestingly, in the present work, the wavelet analysis of light curves
+from five different boxes in various AIA filters clearly demonstrates the existence of QPPs in the present blowout jet (section 3.6).
+The triggering mechanisms of QPPs are very crucial, and so far, more than 15 mechanisms have been proposed to under-
+stand the initiation mechanism of the QPPs (Nakariakov & Melnikov 2009; Van Doorsselaere et al. 2016; Zimovets et al. 2021).
+Broadly, these triggering mechanisms of QPP may be classified into two categories, namely, periodic spontaneous magnetic re-
+connection (Kliem et al. 2000; Karlick´y et al. 2005; Murray et al. 2009; Morton et al. 2012; Li et al. 2015; McLaughlin et al.
+2018) and the MHD waves that may induce the periodic magnetic reconnection (e.g., Ning et al. 2004; Foullon et al.
+2005; Nakariakov & Melnikov 2006; Nakariakov & Zimovets 2011; Tian et al. 2016; Zhang et al. 2016; Kumar et al. 2016;
+Zimovets et al. 2021). Ning et al. (2004) found recurring explosive events with a period of 3–5 minutes when the compressible
+waves push the oppositely directed field lines to reconnect. This process leads to multiple magnetic reconnection, and the QPPs
+were triggered by multiple magnetic reconnection (i.e., recurring explosive events). The other wave modes (e.g., fast mode MHD
+waves and global kink mode) may also trigger the periodic magnetic reconnection in a coronal loop that is situated near the
+flaring region. It initiates QPPs with a period of several minutes (e.g., Foullon et al. (2005); Nakariakov & Melnikov (2006)).
+In the present work, the detected QPPs from various intensity light curves in the blowout jet have a period of ∼ 03 minutes
+(section 3.6). The extracted EM from the blowout jet in the temperature range of 0.5–15 MK also shows the temporal variations
+on a time-scale of ∼ 03 minutes (section 3.4; Figure 4). Apart from the intensity and EM curves, the line-width of Si iv is also
+fluctuating on a time scale of approximately 03 minutes (section 3.5). Hence, consistently, we have found the fluctuations at a
+time scale of ∼ 03 minutes in various parameters (e.g., intensity, EM, and line width). Here, we would like to mention that this
+blowout jet triggers within an active region (AR). The umbra of the sunspot (i.e., photospheric and chromospheric atmosphere
+of AR) is filled with the 3-minute slow MHD waves (e.g.,Fleck & Schmitz 1991; Tian et al. 2014; Jess et al. 2012; Chae et al.
+2017; Felipe 2019, 2021; Botha et al. 2011; Farris & McAteer 2020; Kayshap et al. 2021). Hence, we conjecture that 03-minute
+oscillations are present within the triggering site of the blowout jet, and they may drive the periodic magnetic reconnection at
+time-scale of ∼ 03 minutes. Hence, most probably, we can say that the periodic magnetic reconnection produces the observed
+
+QPP in Kink Unstable Jet
+17
+periodic fluctuations (i.e., QPPs) in the intensity, EM, and line width.
+It should be noted that the magnetic reconnection between pre-existing and open coronal field and closed magnetic fields
+can produce a collimated jet without the rotation or twist, i.e., a kind of standard jet (e.g., Yokoyama & Shibata 1996;
+Miyagoshi & Yokoyama 2003; Moreno-Insertis et al. 2008; Moore et al. 2010; Liu et al. 2011; Moreno-Insertis & Galsgaard
+2013; Raouafi et al. 2016; Shen 2021). On the other hand, the magnetic reconnection between the pre-existing open coronal
+magnetic field and the twisted closed magnetic fields can produce newly twisted magnetic field lines which undergo the untwist-
+ing motions (e.g., Fang et al. 2014). The plasma flows along the newly twisted magnetic field lines that form the solar jet, and
+the rotational/helical motion of the solar jets is a result of untwisting motion of these newly twisted magnetic field lines. Hence,
+the magnetic reconnection is an important physical process to trigger the solar jets in the solar atmosphere (Shibata et al. 1992;
+Yokoyama & Shibata 1995; Shimojo et al. 1996, 1998; Shibata et al. 2007; Kayshap et al. 2013; Sterling et al. 2015; Tian et al.
+2014; Jel´ınek et al. 2015; Wyper et al. 2017; Kayshap et al. 2018; Srivastava et al. 2021). That is why this physical process
+(i.e., magnetic reconnection) is an integral feature of various 2.5D and 3D models of the solar jets (Yokoyama & Shibata 1996;
+Nishizuka et al. 2008; Moreno-Insertis et al. 2008; Gontikakis et al. 2009; Pariat et al. 2015, 2016). The present blowout jet
+shows a very strong rotation of the plasma column (see attached animation 1.mp4). The helical or rotational motions of the solar
+jets are an important indication of the kink instability (e.g., Shibata et al. 1996; Raouafi et al. 2016; Shen 2021). The numerical
+simulations have shown that destabilization of the system by global kink-instability (when helicity or twist exceeded the critical
+value) can trigger the magnetic reconnection through the separatrix surface, and the magnetic reconnection drives the helical
+solar jets (e.g., Pariat et al. 2009, 2010; Rachmeler et al. 2010; Pariat et al. 2015). Most of the magnetic field is either open or
+long curvature magnetic field in the vicinity of the blowout jet (see panels (c1), and (d1) of Figure 2). Here, we mention that
+kink instability destabilizes the pre-existing coronal magnetic field configuration through either magnetic reconnection between
+the kinked flux rope (i.e., kinked/erupting loops) and the pre-existing coronal fields or the internal magnetic reconnection within
+the kink unstable flux rope. Hence, due to the magnetic reconnection, the plasma flows along reconnected magnetic field lines
+which collectively form the spire of the blowout jet. This blowout jet is mainly visible in the cool-temperature filters, and the
+signature of this blowout jet is faint in the hot-temperature filters (see section 3.2). However, usually, the blowout jets have strong
+emissions at hot temperatures along with strong emissions at cool temperatures (e.g.,Moore et al. 2010, 2013). Here, it should
+be noted that surges emit mainly at the cool temperature. Therefore, in the present observational baseline, we don’t rule out the
+possibility of this jet being an chromospheric surge. However, the emission measure (EM) in high-temperature wavebands also
+shows some hot plasma emission (4-6 MK) from the spire of this jet. Therefore, this manuscript uses this feature as a blowout
+jet.
+Hence, kink-instability is an possibly an important driver of solar jets, and not only in the solar jets, but the kink-instability also
+plays crucial role in the triggering of the large-scale eruptions of the solar atmosphere (T¨or¨ok et al. 2004; T¨or¨ok & Kliem 2005;
+Srivastava et al. 2010, 2013; Kumar et al. 2012; Zhong et al. 2021). In case of observations related to the kinked solar flux-rope
+in the solar jets, Kayshap et al. (2013) have reported a kinked flux-tube that drives a solar jet, i.e., internal magnetic reconnection
+in the kinked flux-tube at the north polar triggers the polar jet. Further, Zhu et al. (2017) have also shown that kink instability
+triggers a blowout jet in the solar atmosphere. The present observation clearly shows the occurrence of the kinked flux rope at the
+west limb prior to the jet formation. Further, the imaging, as well as spectroscopic observations, confirms the multiple magnetic
+reconnection in support of the formation of this solar blowout jet.
+Hence, in conclusion, we say that the present observational baseline shows the inverse γ-shape, rotational or helical motion,
+and multiple magnetic reconnection in this blowout jet event. Most probably, these observational findings collectively indicate
+that the kink-instability triggers this blowout jet, and multiple magnetic reconnection leads to the formation of QPPs in this jet.
+5. ACKNOWLEDGEMENTS
+S. K. Mishra acknowledges the Indian Institute of Astrophysics (IIA, Bangalore) for providing the computational facilities
+and institute fellowship. K. Sangal would like to acknowledge the Council of Scientific & Industrial Research (CSIR), Govern-
+ment of India, for financial support through a Senior Research Fellowship (UGC-SRF). P. Jel´ınek acknowledges support from
+grant 21-16508J of the Grant Agency of the Czech Republic. A. K. Srivastava acknowledges the ISRO Project Grant (DS 2B-
+13012(2)/26/2022-Sec.2)for the support of his research. S.P. Rajaguru acknowledges support from SERB (Govt of India) research
+grant CRG/2019/003786. We acknowledge the use of (Hannah & Kontar 2012) for calculating the differential emission measure
+(DEM). Data courtesy of SDO/AIA and IRIS science team. We also acknowledge the use of (Torrence & Compo 1998) method
+to extract the wavelet power spectra and average period of QPPs.
+
+18
+Mishra et al.
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+
+QPP in Kink Unstable Jet
+21
+APPENDIX
+A. TEMPORAL EVOLUTION OF JET IN IRIS/SJI 1330 Å FILTER
+We have shown the evolution of the jet in IRIS/SJI 1330 Å filter observations (cf., figure A.1). At time t = 06:29 UT, we have
+seen the activation of the flux rope at the west limb of the Sun (panel a). Further, we have seen the formation of the bright knots
+that are most probably forming due to the magnetic reconnection (panel d), and this magnetic reconnection triggers the jet (panel
+d). The magnetic reconnection happens in the vicinity of bright knots, therefore, below the bright knots, the plasma falls back
+towards the limb (downflows) along both legs (indicated by red arrows in panel f). While above the bright knot the plasma flows
+upward as indicated by the blue arrow in panel (f). Around t = 06:33 UT, the northern leg of the jet erupts completely (panels
+h), and further, the jet evolves around the southern leg (panels h and i). In the later phase of the jet, we have seen that most
+of the plasma falls back towards the solar surface. The evolution and dynamics of the jet in this filter (i.e., IRIS/SJI 1330 Å)
+are very similar as we have already seen in the AIA 304 Å filter (cf., figure 1). The dynamics and evolution of this jet with
+the help of AIA 304 Å filter in the main text. Similar to AIA 304 Å filter, we have tracked a plasma thread that shows rotation
+with time (see curved white arrows from panels d to f). A similar evolution of this jet can be seen in the given animations (i.e.,
+animation 1.mp4–with annotation and animation 2.mp4–without annotation). The real-time duration of IRIS animations is 23
+seconds. Here, it should be noted that IRIS observations are rotated by a roll angle of 90◦, and we have provided the direction
+arrows in the first panel of figure A.1.
+B. QUASI PERIODIC PULSATIONS: AIA 171 Å, AIA 211 Å, AND AIA 131 Å
+In this appendix, we have shown the wavelet analysis for AIA 171 Å (Figure B.1), AIA 131 Å (Figure B.2), and AIA 211 Å
+(Figure B.3). We have shown wavelet power maps from all four boxes in the same fashion as we have described in section 3.6.
+Interestingly, the QPPs in these hot EUV wavebands of SDO/AIA also show a similar period as we found for AIA 304 Å
+(section 3.6).
+
+22
+Mishra et al.
+70
+80
+90
+100
+110
+120
+130
+Y (arcsec)
+70
+80
+90
+100
+110
+120
+130
+06:29:04 UT
+IRIS/SJI 1330
+(a)
+06:29:42 UT
+IRIS/SJI 1330
+(b)
+06:30:21 UT
+IRIS/SJI 1330
+(c)
+70
+80
+90
+100
+110
+120
+130
+Y (arcsec)
+70
+80
+90
+100
+110
+120
+130
+06:30:59 UT
+IRIS/SJI 1330
+(d)
+06:31:17 UT
+IRIS/SJI 1330
+(e)
+06:32:15 UT
+IRIS/SJI 1330
+(f)
+970
+980
+990
+1000 1010 1020 1030
+X (arcsec)
+70
+80
+90
+100
+110
+120
+130
+Y (arcsec)
+970
+980
+990
+1000 1010 1020 1030
+70
+80
+90
+100
+110
+120
+130
+06:32:54 UT
+IRIS/SJI 1330
+(g)
+970
+980
+990
+1000 1010 1020 1030
+X (arcsec)
+970
+980
+990
+1000 1010 1020 1030
+06:33:32 UT
+IRIS/SJI 1330
+(h)
+970
+980
+990
+1000 1010 1020 1030
+X (arcsec)
+970
+980
+990
+1000 1010 1020 1030
+06:34:10 UT
+IRIS/SJI 1330
+(i)
+Triggering of
+     fluxrope
+N
+S
+W
+E
+Triggering of
+   fluxrope
+Bright Knot
+Bright Knot
+Northern leg
+Southern leg
+Northern leg
+Southern leg
+Figure A.1. This figure shows the evolution of the jet in the IRIS/SJI 1330 Å filter. Firstly, we saw the activation of the kinked flux rope (panels
+b and c), and then the formation of bright knots due to the magnetic reconnection of twisted field lines (panel d). This magnetic reconnection
+leads downflows along both legs of the jet (see red arrows in panel f) and upflows along the spire of the jet (blue arrow in panel f). Further,
+after some time, one leg erupts completely (panels g and h), and the jet further develops along the southern leg (panels h and i). We also see the
+rotation of the plasma as indicated by white arrows from panel (d) to panel (f). At last, we mention that evolution of the jet in IRIS/SJI 1330 Å
+is similar as we have already seen in AIA 304 Å (cf., figure 1). IRIS animations (i.e., animation 1.mp4–with annotation and animation 2.mp4–
+without annotation) also show the same evolution of this jet. The IRIS animations start from 06:24:45 UT to 07:00:50 UT having a real time
+duration of 23 seconds.
+
+QPP in Kink Unstable Jet
+23
+ B1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ Norm Cnts
+0.0
+0.2
+0.4
+ Norm Cnts
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+Period (Minutes)
+1
+2
+4
+6
+8
+10 12 14
+ Power 
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+ Global Period (Minutes) 
+Global Period at max. power ( 2.78 min.)
+ B2
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+25
+25
+25
+1
+2
+3
+4
+5
+ Power 
+Global Period at max. power ( 2.34 min.)
+ B3
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+50
+50
+2
+4
+6
+ Power 
+Global Period at max. power ( 2.14 min.)
+ B4
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+50
+0.0
+3.7
+7.4
+11.1
+14.9
+1
+2
+3
+4
+ Power 
+Figure B.1. Same as figure 8 but for AIA 171 Å filter observations.
+
+24
+Mishra et al.
+ B1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ Norm Cnts
+-0.2
+0.0
+0.2
+0.4
+0.6
+ Norm Cnts
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+Period (Minutes)
+1
+2
+4
+6
+8
+10 12 14
+ Power 
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+ Global Period (Minutes) 
+Global Period at max. power ( 2.78 min.)
+ B2
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+Ł˘��UMGRAY
+Ł˘��UMGRAY
+1
+2
+3
+4
+5
+6
+ Power 
+Global Period at max. power ( 2.34 min.)
+ B3
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+‚‚W�GRAY
+‚‚W�GRAY
+‚‚W�GRAY
+1
+2
+3
+4
+5
+6
+7
+ Power 
+Global Period at max. power ( 2.14 min.)
+ B4
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+Ł˘��UMGRAY
+0.0
+3.7
+7.4
+11.1
+14.9
+1
+2
+3
+4
+ Power 
+Figure B.2. Same as figure 8 but for AIA 131 Å filter observations.
+
+QPP in Kink Unstable Jet
+25
+ B1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ Norm Cnts
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+ Norm Cnts
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+Period (Minutes)
+1
+1
+1
+2
+4
+6
+8
+10
+12
+ Power 
+0.5
+1.0
+2.0
+4.0
+8.0
+16.0
+ Global Period (Minutes) 
+Global Period at max. power ( 2.78 min.)
+ B2
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+����UMGRAY
+����UMGRAY
+����UMGRAY
+2
+4
+6
+8
+ Power 
+Global Period at max. power ( 2.55 min.)
+ B3
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+‚���UMGRAY
+‚���UMGRAY
+‚���UMGRAY
+1
+2
+3
+4
+5
+6
+ Power 
+Global Period at max. power ( 2.34 min.)
+ B4
+0
+5
+10
+15
+20
+25
+Time (Minutes)
+����EGRAY
+0.0
+3.7
+7.4
+11.1
+14.9
+1
+2
+3
+4
+ Power 
+Figure B.3. Same as Figure 8 but for AIA 211 Å filter observations.
+

NdE0T4oBgHgl3EQf0gIE/content/tmp_files/2301.02685v1.pdf.txt

@@ -0,0 +1,229 @@
+Prepared for submission to JHEP
+Gauss-Manin equations for propagators in the case
+of arbitrary masses
+S. Srednyak a
+aDuke University,
+Durham, USA
+Abstract: We derive the complete list of singularities of propagators in theories with ar-
+bitrary (complex) masses and for arbitrary diagram. We derive in a closed form differential
+equations for the propagator as a function of the momentum and masses.
+arXiv:2301.02685v1  [hep-th]  6 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Preliminaries
+1
+3
+Results
+2
+4
+Proofs
+2
+4.1
+Proof of Th1
+2
+4.2
+Proof of Th2
+3
+5
+Discussion
+3
+6
+Comparison with the known literature
+4
+7
+Conclusion
+4
+1
+Introduction
+In this paper we derive the Gauss-Manin connection for propagators for arbitrary mass case.
+Differential equations for perturbative amplitudes play an important role in their theory.
+They can be used for numerical evaluation of the amplitudes as well as for analytic study.
+There has been a large interest in this topic for the case of propagators. In particular, the
+papers [1–3] address the case of sunrise graphs. More recently, there is more work on the
+banana family [4, 5]. Beginning steps of all these analyses is the derivation of differential
+equations for the function represented by the diagram.
+We observe that singularities of propagators can be completely analysed in closed form.
+In particular, we derive explicit equations for all of the singularities of the diagram func-
+tion , relating them to the combinatorics of the diagram. Our derivation is based on the
+observation that in appropriate coordinates the Landau polynomials are linear functions of
+the coordinates.
+2
+Preliminaries
+We consider the standard propagators in massive theories [6] that can be written in the
+form
+Jm =
+�
+1
+�((qi + δip)2 + m2
+i )qmdLdq
+(2.1)
+where δi = 0, 1, in general depending on the choice of momentum flow.
+– 1 –
+
+3
+Results
+In this section we formulate our main results.
+Th1 (Characterization of singularity locus of the propagator). The singularity locus of
+the propagator is given by the set
+x = ri1mi1 + ri2mi2 + .... + rismis
+(3.1)
+where ris are rational numbers that depend on the diagram.
+There is finite set of such
+numbers, the cardinality of which grows exponentially with the number of loops.
+Prop. The singularities of the propagator in masses are located at the set
+�
+r′D
+i mi = 0
+(3.2)
+for some integer numbers r′D
+i
+that depend on the diagram .
+Th2 ( Differential equations for the propagator). The propagator satisfies the following
+system of equations
+∂fD
+∂zk
+= (
+�
+R={ri1...ris}
+AD
+k,R
+x + ri1mi1 + ri2mi2...rismis
+)fD
+(3.3)
+where the sum is extended over the set described in Th1. The matrices AD
+k,I depend only
+on coupling, dimension and diagram topology, and have no dependence on masses or the
+momentum.
+Analogous formula holds for the sum over all diagrams.
+4
+Proofs
+4.1
+Proof of Th1
+We will carry out the proof only for leading singularities. The analysis of subleading sin-
+gularities reduces to the case of sub diagrams. For leading singularity, there is a set of
+propagators Dis(p, qi) that develop a vanishing cycle in their intersection
+DI = ∩is∈IDis
+(4.1)
+We will consider the vanishing cycles at finite distance in q-space. The emergence of vanish-
+ing cycles corresponds to degeneracy of the system of normals to the varieties {q : Di(p, q) =
+0}. Note that these varieties are highly degenerate. Each of them has as singularity locus a
+(L−1)d-dimensional plane in q-space. Nonetheless, our criterion still works. It can be seen
+by applying the general theory of vanishing cycles as applied to singular schemes [7, 8].
+The criterion formulated above results in equations
+�
+bi(δs,ip + qs,i,1 + ... + qs,i,ri) = 0, s ∈ {0, ..., L}
+(4.2)
+for some bi that are not all zero. These equations can be solved for qi as
+qi = aip
+(4.3)
+– 2 –
+
+The singularities develop only when all vectors are collinear. Then the equations Ds(p, q) =
+0 can be rewritten in the form
+Ds = (δsp + qs,1 + ...qs,rs)2 − m2
+s = (δs + as,1 + ... + as,rs)2p2 − m2
+s = 0
+(4.4)
+or
+deltas + as,1 + ... + as,rs = ±ms/x
+(4.5)
+After elimination of the variables as,i we obtain our claim.
+The case of the vanishing cycle at infinity is simpler. In this case, we simply drop
+the terms m2
+i and consider qi as homogeneous coordinates on the projective space CPLd−1.
+Considerations similar to the above lead to the equation
+p2 = 0
+(4.6)
+which is the only leading Landau singularity at infinity.
+4.2
+Proof of Th2
+To prove Th2 we use the general fact that integrals of the type we consider satisfy a system
+of differential equations with respect to their parameters of the form
+∂f
+∂zk
+= (
+� Ak,I
+LI
+)f
+(4.7)
+where LI are the polynomials that give the singularity locus, and Ak,I are certain matrices (
+choice of which is not unique) that are polynomial in the parameters. For general reference
+see [griffiths, AGV]. The degree of the polynomials Ak,I must be strictly smaller than
+the degree of LI because otherwise solutions of this system of equations would have a
+singularity at infinity. In our case the polynomials are linear, therefore matices A must
+have no dependence on x. Likewise, they can have no dependence on masses because it
+would contradict regularity at infinity in mass space.
+5
+Discussion
+Our result fits propagators of quantum field theories into the family of equations
+∂f(x, l)
+∂xk
+= (
+� Ak,I
+lI
+)f
+(5.1)
+where Ak,I are constant matrices and lI are functions linear in the variables xi.
+This
+family of functions provides a natural generalization of the Grassmannian hypergeometric
+functions considered in [9, 10]. It is desirable to obtain combinatorial characterization of
+their solutions. To solve this problem, it is necessary to consider their dependence on the
+parameters lI,i. This dependence leads to the study of irregular singularities [11, 12]. Full
+solution of this problem would involve quantum groups [13, 14]
+– 3 –
+
+6
+Comparison with the known literature
+In papers [1, 3] deep properties of the sunrise family were studied.
+Sunrise graphs fall
+inside the class of the diagrams that we consider. While the equations derived in [1, 3] are
+seemingly more complicated, they in fact were derived before in [15]. The examination of
+formulas (7) and (12) of [15] shows that the equations of [15] can be transformed into the
+form stated in our theorem . The polynomial D in this paper can be decomposed into a
+product of linear forms, after which a partial fractioning procedure can be applied.
+Our method can be applied to the family of banana graphs [4, 5].
+7
+Conclusion
+In this paper we obtained Gauss-Manin connection for propagators that depend on an
+arbitrary set of masses. We hope our results can be useful for numerical computation of
+propagators and self energies.
+References
+[1] Luise Adams, Christian Bogner, and Stefan Weinzierl. The two-loop sunrise graph with
+arbitrary masses. J. Math. Phys., 54:052303, 2013.
+[2] Ettore Remiddi and Lorenzo Tancredi. Schouten identities for Feynman graph amplitudes;
+The Master Integrals for the two-loop massive sunrise graph. Nucl. Phys. B, 880:343–377,
+2014.
+[3] Spencer Bloch, Matt Kerr, and Pierre Vanhove. Local mirror symmetry and the sunset
+Feynman integral. Adv. Theor. Math. Phys., 21:1373–1453, 2017.
+[4] Albrecht Klemm, Christoph Nega, and Reza Safari. The l-loop Banana Amplitude from
+GKZ Systems and relative Calabi-Yau Periods. JHEP, 04:088, 2020.
+[5] Sebastian Pögel, Xing Wang, and Stefan Weinzierl. Bananas of equal mass: any loop, any
+order in the dimensional regularisation parameter. 12 2022.
+[6] Christian Bogner and Stefan Weinzierl. Feynman graph polynomials. International Journal
+of Modern Physics A, 25(13):2585–2618, 2010.
+[7] Masaki Kashiwara and Pierre Schapira. Microlocal study of sheaves. Number 51. Société
+mathématique de France, 1985.
+[8] Victor Ginsburg. Characteristic varieties and vanishing cycles. Inventiones mathematicae,
+84(2):327–402, 1986.
+[9] Kazuhiko Aomoto, Michitake Kita, Toshitake Kohno, and Kenji Iohara. Theory of
+hypergeometric functions. Springer, 2011.
+[10] IM Gelfand and RD MacPherson. Geometry in grassmannians and a generalization of the
+dilogarithm. Advances in mathematics, 44(3):279–312, 1982.
+[11] Michio Jimbo, Tetsuji Miwa, and Kimio Ueno. Monodromy preserving deformation of linear
+ordinary differential equations with rational coefficients: I. general theory and τ-function.
+Physica D: Nonlinear Phenomena, 2(2):306–352, 1981.
+[12] On the algebro-geometric integration¶.
+– 4 –
+
+[13] Alexander Varchenko. Multidimensional hypergeometric functions the representation theory
+of Lie Algebras and quantum groups, volume 21. World Scientific, 1995.
+[14] Xiaomeng Xu. Closure of stokes matrices i: caterpillar points and applications. arXiv
+preprint arXiv:1912.07196, 2019.
+[15] Michele Caffo, H. Czyz, S. Laporta, and E. Remiddi. The Master differential equations for
+the two loop sunrise selfmass amplitudes. Nuovo Cim. A, 111:365–389, 1998.
+– 5 –
+

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+page_content='  Differential equations for perturbative amplitudes play an important role in their theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  They can be used for numerical evaluation of the amplitudes as well as for analytic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  There has been a large interest in this topic for the case of propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' In particular, the  papers [1–3] address the case of sunrise graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' More recently, there is more work on the  banana family [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Beginning steps of all these analyses is the derivation of differential  equations for the function represented by the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  We observe that singularities of propagators can be completely analysed in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  In particular, we derive explicit equations for all of the singularities of the diagram func-  tion , relating them to the combinatorics of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Our derivation is based on the  observation that in appropriate coordinates the Landau polynomials are linear functions of  the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  2  Preliminaries  We consider the standard propagators in massive theories [6] that can be written in the  form  Jm =  �  1  �((qi + δip)2 + m2  i )qmdLdq  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='1)  where δi = 0, 1, in general depending on the choice of momentum flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  – 1 –  3  Results  In this section we formulate our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Th1 (Characterization of singularity locus of the propagator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The singularity locus of  the propagator is given by the set  x = ri1mi1 + ri2mi2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='. + rismis  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='1)  where ris are rational numbers that depend on the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  There is finite set of such  numbers, the cardinality of which grows exponentially with the number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The singularities of the propagator in masses are located at the set  �  r′D  i mi = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='2)  for some integer numbers r′D  i  that depend on the diagram .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Th2 ( Differential equations for the propagator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The propagator satisfies the following  system of equations  ∂fD  ∂zk  = (  �  R={ri1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='ris}  AD  k,R  x + ri1mi1 + ri2mi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='rismis  )fD  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='3)  where the sum is extended over the set described in Th1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The matrices AD  k,I depend only  on coupling, dimension and diagram topology, and have no dependence on masses or the  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Analogous formula holds for the sum over all diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  4  Proofs  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='1  Proof of Th1  We will carry out the proof only for leading singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The analysis of subleading sin-  gularities reduces to the case of sub diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' For leading singularity, there is a set of  propagators Dis(p, qi) that develop a vanishing cycle in their intersection  DI = ∩is∈IDis  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='1)  We will consider the vanishing cycles at finite distance in q-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The emergence of vanish-  ing cycles corresponds to degeneracy of the system of normals to the varieties {q : Di(p, q) =  0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Note that these varieties are highly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Each of them has as singularity locus a  (L−1)d-dimensional plane in q-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Nonetheless, our criterion still works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' It can be seen  by applying the general theory of vanishing cycles as applied to singular schemes [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  The criterion formulated above results in equations  �  bi(δs,ip + qs,i,1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' + qs,i,ri) = 0, s ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=', L}  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='2)  for some bi that are not all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' These equations can be solved for qi as  qi = aip  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='3)  – 2 –  The singularities develop only when all vectors are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Then the equations Ds(p, q) =  0 can be rewritten in the form  Ds = (δsp + qs,1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='qs,rs)2 − m2  s = (δs + as,1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' + as,rs)2p2 − m2  s = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='4)  or  deltas + as,1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' + as,rs = ±ms/x  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='5)  After elimination of the variables as,i we obtain our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  The case of the vanishing cycle at infinity is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' In this case, we simply drop  the terms m2  i and consider qi as homogeneous coordinates on the projective space CPLd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Considerations similar to the above lead to the equation  p2 = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='6)  which is the only leading Landau singularity at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='2  Proof of Th2  To prove Th2 we use the general fact that integrals of the type we consider satisfy a system  of differential equations with respect to their parameters of the form  ∂f  ∂zk  = (  � Ak,I  LI  )f  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='7)  where LI are the polynomials that give the singularity locus, and Ak,I are certain matrices (  choice of which is not unique) that are polynomial in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' For general reference  see [griffiths, AGV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The degree of the polynomials Ak,I must be strictly smaller than  the degree of LI because otherwise solutions of this system of equations would have a  singularity at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' In our case the polynomials are linear, therefore matices A must  have no dependence on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Likewise, they can have no dependence on masses because it  would contradict regularity at infinity in mass space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  5  Discussion  Our result fits propagators of quantum field theories into the family of equations  ∂f(x, l)  ∂xk  = (  � Ak,I  lI  )f  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='1)  where Ak,I are constant matrices and lI are functions linear in the variables xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  This  family of functions provides a natural generalization of the Grassmannian hypergeometric  functions considered in [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' It is desirable to obtain combinatorial characterization of  their solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' To solve this problem, it is necessary to consider their dependence on the  parameters lI,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' This dependence leads to the study of irregular singularities [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Full  solution of this problem would involve quantum groups [13, 14]  – 3 –  6  Comparison with the known literature  In papers [1, 3] deep properties of the sunrise family were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Sunrise graphs fall  inside the class of the diagrams that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' While the equations derived in [1, 3] are  seemingly more complicated, they in fact were derived before in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The examination of  formulas (7) and (12) of [15] shows that the equations of [15] can be transformed into the  form stated in our theorem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The polynomial D in this paper can be decomposed into a  product of linear forms, after which a partial fractioning procedure can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Our method can be applied to the family of banana graphs [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  7  Conclusion  In this paper we obtained Gauss-Manin connection for propagators that depend on an  arbitrary set of masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' We hope our results can be useful for numerical computation of  propagators and self energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  References  [1] Luise Adams, Christian Bogner, and Stefan Weinzierl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The two-loop sunrise graph with  arbitrary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content='  [2] Ettore Remiddi and Lorenzo Tancredi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Schouten identities for Feynman graph amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  The Master Integrals for the two-loop massive sunrise graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content='  [3] Spencer Bloch, Matt Kerr, and Pierre Vanhove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Local mirror symmetry and the sunset  Feynman integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content='  [4] Albrecht Klemm, Christoph Nega, and Reza Safari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The l-loop Banana Amplitude from  GKZ Systems and relative Calabi-Yau Periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' JHEP, 04:088, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  [5] Sebastian Pögel, Xing Wang, and Stefan Weinzierl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Bananas of equal mass: any loop, any  order in the dimensional regularisation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' 12 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  [6] Christian Bogner and Stefan Weinzierl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Feynman graph polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content=' Characteristic varieties and vanishing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content='  [9] Kazuhiko Aomoto, Michitake Kita, Toshitake Kohno, and Kenji Iohara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Theory of  hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  [10] IM Gelfand and RD MacPherson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Geometry in grassmannians and a generalization of the  dilogarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content=' Monodromy preserving deformation of linear  ordinary differential equations with rational coefficients: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' general theory and τ-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  Physica D: Nonlinear Phenomena, 2(2):306–352, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  [12] On the algebro-geometric integration¶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content='  – 4 –  [13] Alexander Varchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Multidimensional hypergeometric functions the representation theory  of Lie Algebras and quantum groups, volume 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' World Scientific, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content=' Closure of stokes matrices i: caterpillar points and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' arXiv  preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+page_content='  [15] Michele Caffo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Czyz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Laporta, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Remiddi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' The Master differential equations for  the two loop sunrise selfmass amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
+page_content=' Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE0T4oBgHgl3EQf0gIE/content/2301.02685v1.pdf'}
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+DRAFT VERSION JANUARY 11, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+An Analytical Theory for the Growth from Planetesimals to Planets by Polydisperse Pebble Accretion
+WLADIMIR LYRA,1 ANDERS JOHANSEN,2, 3 MANUEL H. CAÑAS,1 AND CHAO-CHIN YANG4
+1New Mexico State University, Department of Astronomy, PO Box 30001 MSC 4500, Las Cruces, NM 88001, USA
+2Center for Star and Planet Formation, GLOBE Institute, University of Copenhagen, Øster Voldgade 5-7, 1350 Copenhagen, Denmark
+3Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 221 00 Lund, Sweden
+4Department of Physics and Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL 35487-0324, USA
+Submitted to ApJ
+ABSTRACT
+Pebble accretion is recognized as a significant accelerator of planet formation. Yet, only formulae for single-
+sized (monodisperse) distribution have been derived in the literature. These can lead to significant underesti-
+mates for Bondi accretion, for which the best accreted pebble size may not be the one that dominates the mass
+distribution. We derive in this paper the polydisperse theory of pebble accretion. We consider a power-law
+distribution in pebble radius, and we find the resulting surface and volume number density distribution func-
+tions. We derive also the exact monodisperse analytical pebble accretion rate for which 3D and 2D accretion are
+limits. In addition, we find analytical solutions to the polydisperse 2D Hill and 3D Bondi limits. We integrate
+the polydisperse pebble accretion numerically for the MRN distribution, finding a slight decrease (by an exact
+factor 3/7) in the Hill regime compared to the monodisperse case. In contrast, in the Bondi regime, we find
+1-2 orders of magnitude higher accretion rates compared to monodisperse, also extending the onset of pebble
+accretion to 1-2 order of magnitude lower in mass. We find Myr-timescales, within the disk lifetime, for Bondi
+accretion on top of planetary seeds of masses 10−6 − 10−4M⊕, over a significant range of the parameter space.
+This mass range overlaps with the high mass end of the planetesimal initial mass function, and thus pebble
+accretion is possible directly following formation by streaming instability. This alleviates the need for mutual
+planetesimal collisions as a major contribution to planetary growth.
+Keywords: Pebble accretion, planet formation.
+1. INTRODUCTION
+Despite significant theoretical and observational advances
+in the past decade, a comprehensive theory of planet for-
+mation still remains elusive. Planet formation starts from
+the accumulation of sub-µm interstellar grains, growing by
+means of coagulation, in hit-and-stick low-velocity colli-
+sions (Safronov 1972; Nakagawa et al. 1981; Tominaga et al.
+2021). Laboratory experiments (Blum & Wurm 2008; Güt-
+tler et al. 2010) and numerical simulations (Güttler et al.
+2009; Geretshauser et al. 2010; Zsom et al. 2010) pro-
+vide evidence that this process is efficient in growing solid
+grains up to mm and cm radius (hereafter called “pebbles”)
+with growth beyond this size being unlikely, due to bounc-
+ing, fragmentation, and drift (Dullemond & Dominik 2005;
+Corresponding author: Wladimir Lyra
+[email protected]
+Brauer et al. 2008; Krijt et al. 2015), unless the possibility of
+very high porosities is introduced (Suyama et al. 2008, 2012).
+The streaming instability (Youdin & Goodman 2005;
+Youdin & Johansen 2007; Johansen & Youdin 2007; Kowa-
+lik et al. 2013; Lyra & Kuchner 2013; Krapp et al. 2019;
+Squire & Hopkins 2020; Schäfer et al. 2020; Paardekooper
+et al. 2020; Chen & Lin 2020; McNally et al. 2021; Lin
+2021; Flock & Mignone 2021; Zhu & Yang 2021; Yang &
+Zhu 2021) whereby the drift of grains through the gas is
+unstable, has been established as a mechanism to produce
+the first planetesimals (Johansen et al. 2007; Yang & Jo-
+hansen 2014; Carrera et al. 2015; Simon et al. 2016; Yang
+et al. 2017; Schaffer et al. 2018; Nesvorný et al. 2019; Li
+et al. 2019; Klahr & Schreiber 2021; Visser et al. 2021; Li &
+Youdin 2021), through concentration of pebbles into dense
+filaments that display a fractal structure with large overden-
+sities reached at the smallest scales of the simulations (Jo-
+hansen et al. 2015). Yet, growth by binary accretion of plan-
+arXiv:2301.03825v1  [astro-ph.EP]  10 Jan 2023
+
+2
+LYRA ET AL.
+etesimals into progressively larger objects, while able to ex-
+plain the growth of a giant planet’s core at 5 AU (if migration
+is ignored, Pollack et al. 1994), is not viable already at the or-
+bital position of Saturn, Uranus, or Neptune (Thommes et al.
+2003; Johansen & Bitsch 2019).
+This shortcoming of planetesimal accretion motivated the
+search for other avenues of planetary growth. Fast accre-
+tion rates of marginally coupled solids up to planetary masses
+were first seen in the simulations of Lyra et al. (2008). In that
+model, vortices trap pebbles and collapse them into Moon-
+mass objects via direct gravitational instability, which scoop
+up the remaining pebbles at a vertiginous rate, achieving
+Mars and Earth mass within a few hundred orbits. Whereas
+this growth was assisted by vortices, it illustrates that gas-
+assisted accretion of pebbles is potentially much faster than
+planetesimal accretion, due to the presence of gas drag as a
+dissipative mechanism. A similar result was found by Jo-
+hansen & Lacerda (2010), showing fast accretion rates onto
+a 100 km seed, highlighting the importance of pebble accre-
+tion for planetary growth, and suggesting for the first time
+that a significant fraction of the accretion of planetary bodies
+proceeds via pebbles (as opposed to planetesimals), before
+the dissipation of the gas disk.
+An analytical theory of pebble accretion was later devel-
+oped by Ormel & Klahr (2010) and Lambrechts & Johansen
+(2012), elucidating the existence of two regimes: one for
+small masses, where the seed mass accretes from a pebble
+headwind, a process reminiscent of Bondi-Hoyle-Lyttleton
+accretion (Bondi & Hoyle 1944; Hoyle & Lyttleton 1939);
+and another, for higher masses, where pebbles are accreted
+from the whole Hill sphere of the seed. These regimes were
+dubbed “drift-dominated” and “shear-dominated” by Ormel
+& Klahr (2010), respectively, whereas Lambrechts & Jo-
+hansen (2012) called them “Bondi” and “Hill”. As a rule of
+thumb, planetesimals accrete in the Bondi regime, protoplan-
+ets in the Hill regime (Ormel 2017; Johansen & Lambrechts
+2017), and both can yield orders-of-magnitude higher mass
+accretion rates than planetesimal accretion.
+Since its inception, the model has quickly risen to paradig-
+matic status, by virtue of a number of successes.
+Pebble
+accretion explains the formation of the gas giants (Lam-
+brechts & Johansen 2012), of the ice giants with low gas frac-
+tions (Lambrechts et al. 2014); the preponderance of super-
+Earths around other stars (Lambrechts et al. 2019; Bitsch
+et al. 2019b; Izidoro et al. 2021); it achieves a better planet
+population synthesis matching exoplanet populations than
+a planetesimal-based accretion model (Bitsch et al. 2019a;
+Drazkowska et al. 2022), and it is also compatible with the
+drift-dominated evolution of dust in T-Tauri disks (a flux of
+∼ 100 Earth masses over the disk lifetime, Appelgren et al.
+2020). Even the classical giant impact model for terrestrial
+planet formation (Raymond et al. 2004) is challenged now
+by a hybrid view where terrestrial planets accrete their mass
+from a combination of planetesimals and small pebbles (Jo-
+hansen et al. 2015, 2021).
+However, most previous works on pebble accretion con-
+sidered a monodisperse distribution of pebbles. In reality,
+the pebbles will have a distribution of sizes, ranging from
+sub-µm to mm or cm-size. A monodisperse distribution can
+be a reasonable assumption because, for the interstellar grain
+size distribution, following a power-law of -3.5 of the grain
+radius (Mathis et al. 1977; Hirashita & Kobayashi 2013,
+MRN henceforth) most of the mass resides in the largest peb-
+bles; a result that stands even after dust evolution away from
+MRN in the protoplanetary disk is considered (Birnstiel et al.
+2012). This makes the Hill regime of pebble accretion rela-
+tively insensitive to the dust spectrum, and either the dom-
+inant pebble size (Lambrechts & Johansen 2014) or a mass
+weighted representative pebble size (Guilera et al. 2020; Ven-
+turini et al. 2020) yield sensible results.
+Indeed, in a recent work, Andama et al. (2022), consider-
+ing polydisperse Hill accretion, find larger final core masses,
+not because of faster accretion rates, but because the smaller
+grains drift more slowly, lingering around for longer times
+than the largest pebbles, and thus extending the duration of
+accretion. Dr ˛a˙zkowska et al. (2021) also considering the Hill
+regime, focus on the beneficial aspects of fragmentation on
+keeping the pebbles sizes small, because too large pebbles
+accrete poorly.
+Both works consider a body already near
+the Bondi-Hill transition mass, a polydisperse size spectrum
+from the prescription of Birnstiel et al. (2012), and solve nu-
+merically for the mass accretion rates. Both works also high-
+light how the mass accretion rate is dependent on the embryo
+mass but not on pebble size.
+In stark constrast, in the Bondi regime the size distribution
+should matter significantly for the mass accretion rate itself.
+In the Bondi regime, the best accreted pebbles are those of
+friction time similar to the time the pebble takes to cross the
+Bondi radius, i.e., the Bondi time. For small enough seed
+mass, the larger, cm-sized, pebbles, drift so fast past the pro-
+toplanet that these pebbles essentially behave like planetesi-
+mals. In this case, the cross section for accretion is geometric
+(for high speeds), or gravitationally focused (for low speeds),
+and only slightly aided by gas drag. As a result, even though
+these pebbles dominate the mass budget, their mass accre-
+tion rate by the planetesimal can be lower than of the smaller
+pebbles for which Bondi accretion is more efficient. If that is
+the case, the pebble accretion rates in the Bondi regime may
+be underestimated by the current monodisperse prescriptions.
+Indeed, Lorek & Johansen (2022) recently find that planetes-
+imal accretion is insignificant beyond 5 AU, so the onset of
+pebble accretion has to overlap with the high-mass end of the
+planetesimal mass function if planet formation is to proceed.
+
+POLYDISPERSE PEBBLE ACCRETION
+3
+In this paper, we work out the polydisperse extension of
+pebble accretion. We find that indeed Bondi accretion is 1-2
+orders of magnitude more efficient in the polydisperse case.
+We also find that the onset of polydisperse Bondi accretion
+occurs at lower masses than monodisperse, by 1-2 orders of
+magnitude. Hill accretion is slightly less efficient, by a factor
+3/7, for the MRN distribution. We find the exact solution to
+the 2D-3D transition, as well as analytical expressions for the
+polydisperse 2D Hill and 3D Bondi accretion rates.
+This paper is structured as follows. In Sect. 2 we derive the
+grain size distribution functions; in Sect. 3 we apply them to
+pebble accretion, deriving the polydisperse model, and pro-
+ceeding with the analysis. In Sect. 4 we work out the an-
+alytical expressions for 2D Hill and 3D Bondi polydisperse
+accretion. A summary concludes the paper in Sect. 7. A ta-
+ble of mathematical symbols used in this work is shown in
+Table 1.
+2. DISTRIBUTION FUNCTIONS
+Consider the grain size distribution
+F(a,z) ≡ ∂n
+∂a
+(1)
+that defines the number density n; here, a is the grain radius
+and z the vertical coordinate. We integrate it to yield
+n(a,z) =
+� a
+0
+F(a′,z) da′,
+(2)
+and n(z) ≡ n(amax,z). The volume density is found by multi-
+plying F(a,z) by the mass m(a) of a single grain
+ρd(a,z) =
+� a
+0
+m(a′) F(a′,z) da′.
+(3)
+and again, ρd(z) ≡ ρd(amax,z).
+Due to sedimentation, we
+can write, for an equilibrium between diffusion and gravity
+(Dubrulle et al. 1995)
+F(a,z) ≡ f(a) e−z2/2H2
+d ,
+(4)
+defining the function f(a), which is the size distribution func-
+tion in the midplane. In Eq. (4), Hd is the grain scale height,
+a function of a (Klahr & Henning 1997; Lyra & Lin 2013)
+Hd = Hg
+�
+α
+St+α,
+(5)
+where Hg is the gas scale height, α is a dimensionless ver-
+tical diffusion parameter1, and St is the Stokes number, a
+1 This parameter is equivalent to the Shakura-Sunyaev parameter (Shakura
+& Sunyaev 1973) for isotropic turbulence of equal diffusion of mass and
+momentum (Youdin & Lithwick 2007; Yang et al. 2018).
+non-dimensionalization of the grain radius, normalized by
+the grain internal density ρ• and the gas column density Σg
+St ≡ π
+2
+a ρ•
+Σg
+.
+(6)
+2.1. The distribution function in the midplane
+To find f(a), consider spherical grains
+m(a) = 4π
+3 a3ρ•
+(7)
+and the column density
+Σd(a) ≡
+� ∞
+−∞
+ρd(a,z) dz.
+(8)
+Substituting Eq. (3), and integrating in z, we find
+Σd(a) = 25/2π3/2
+3
+� a
+0
+ρ• Hd a′ 3 f(a′) da′,
+(9)
+and the total column density Σd ≡ Σd(amax). We keep the
+internal density ρ• inside the integral because it is in gen-
+eral a function of radius, if grains have different composition.
+Given
+Σd(a) =
+� a
+0
+∂Σd(a′)
+∂a′
+da′,
+(10)
+we find, equating the integrands of Eq. (9) and Eq. (10), and
+solving for f(a)
+f(a) =
+3
+25/2π3/2Hgρ•
+�
+1+ St
+α a−3 ∂Σd(a)
+∂a
+,
+(11)
+where we also substituted Eq. (5) for Hd as a function of St.
+The distribution is determined if we find an expression for
+∂aΣd(a).
+2.1.1. Sedimented and unsedimented limits
+To find the general solution, we need to find the expression
+for ∂aΣd in Eq. (11). We do so by realizing that even though
+the midplane volume density is modified by sedimentation,
+the column density Σd is not. The two limits of f(a) are,
+first, the “sedimented” limit, for St ≫ α
+f(a)(sed) =
+3
+8πHgρ1/2
+• Σ1/2
+g
+α1/2 a−2.5 ∂Σd(a)
+∂a
+,
+(12)
+and, second, the unsedimented limit, for St ≪ α
+f(a)(unsed) =
+3
+25/2π3/2Hgρ•
+a−3 ∂Σd(a)
+∂a
+,
+(13)
+where we have substituted the Stokes number given by
+Eq. (6). Since the column density does not change with sed-
+imentation, we can find ∂aΣd by either limit.
+
+4
+LYRA ET AL.
+Table 1. Symbols used in this work.
+Symbol
+Definition
+Description
+Symbol
+Definition
+Description
+F
+Eq. (1)
+pebble size distribution
+ρd0
+Eq. (39)
+dust density at midplane
+a
+pebble radius
+δv
+Eq. (40)
+approach velocity
+z
+vertical coordinate
+S
+Eq. (34)
+stratification integral
+n
+Eq. (2)
+number density
+∆v
+Sub-Keplerian velocity reduction
+m
+Eq. (7)
+pebble mass
+Ω
+�
+GM⊙
+r3
+Keplerian frequency
+ρd
+Eq. (3)
+volume density
+ˆRacc
+Eq. (53)
+accretion radius
+Hd
+Eq. (5)
+pebble scale height
+χ
+Eq. (41)
+coefficient
+f
+Eq. (22)
+pebble distribution in midplane
+τ f
+St/Ω
+friction time
+Hg
+Ω/cs
+gas scale height
+tp
+Eq. (42)
+passing timescale
+α
+Shakura-Sunyaev viscosity
+γ
+Eq. (41)
+coefficient
+St
+Eq. (6)
+Stoker number
+G
+gravitational constant
+ρ•
+internal pebble density
+Mp
+planetesimal mass
+Σg
+Eq. (23)
+gas column density
+RH
+Eq. (43)
+Hill radius
+Σd
+ZΣg
+pebble column density
+tB
+Eq. (47)
+Bondi time
+k
+Eq. (14)
+power law of unsedimented distribution
+RB
+Eq. (46)
+Bondi radius
+p
+Eq. (16)
+power law of column density distribution
+Mt
+Eq. (49)
+transition mass
+q
+Eq. (17)
+power law of internal density
+MHB
+Eq. (48)
+Hill-Bondi transition mass
+D
+Eq. (20)
+Coefficient of column density distribution
+R
+3�
+3Mp
+4πρ•
+planetesimal radius
+Z
+Σd/Σg
+Dust-to-gas ratio
+vesc
+�
+2GMp
+R
+escape velocity
+ρ(0)
+•
+internal density of largest grain
+Stp
+Eq. (51)
+Stokes number past planetesimal
+r
+radial coordinate
+MBL
+Eq. (52)
+Bondi-geometric transition mass
+rc
+Eq. (23)
+cutoff radius
+tacc
+Eq. (54)
+accretion time
+W
+Eq. (27)
+column density distribution
+h
+Hg/r
+aspect ratio
+Racc
+Eq. (41)
+drag-modified accretion radius
+mp
+characteristic streaming instability mass
+ξ
+Eq. (41)
+coefficient
+ψ
+Eq. (64)
+shorthand
+˙M
+mass accretion rate
+T
+Eq. (24)
+gas temperature
+cs
+�
+Tcp(Γ−1)
+sound speed
+Γ
+adiabatic index
+µ
+mean molecular weight
+cp
+Rgas
+µ
+Γ
+(Γ−1)
+specific heat at constant pressure
+Rgas
+gas constant
+cv
+cp/Γ
+specific heat at constant volume
+We assume a power-law dependency for the unsedimented
+distribution in the midplane
+f(a)(unsed) ∝ a−k
+(14)
+where k is a constant (the MRN distribution corresponds to
+k = 3.5). Equating Eq. (13) and Eq. (14)
+∂Σd(a)
+∂a
+∝ ρ• a3 a−k.
+(15)
+We thus write
+∂Σd(a)
+∂a
+∝ a−p;
+(16)
+ρ• ∝ a−q;
+(17)
+p−q = k −3.
+(18)
+We can then write the column density distribution as a power
+law
+∂Σd(a)
+∂a
+= D a−p.
+(19)
+Integrating it in a, equating to Eq. (10), and solving for the
+constant D, we find
+D = (1− p)ZΣg
+a1−p
+max
+;
+(20)
+here we also substitute Σd = ZΣg, where Z is the metallicity.
+Considering now the variation of the internal density
+ρ•(a) = ρ(0)
+•
+� a
+amax
+�−q
+,
+(21)
+
+POLYDISPERSE PEBBLE ACCRETION
+5
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+15
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+f(a)
+Number density per radius
+=10
+5 general
+       sedimented
+=10
+4 general
+       sedimented
+=10
+3 general
+       sedimented
+unsedimented
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+13
+10
+12
+m(a)f(a)
+Mass density per radius
+Figure 1. Left: The grain distribution function f(a) in the midplane. Integrated over a, this function yields the number density n in the
+midplane. This model is calculated at 20 AU with density and temperature according to Eqs. (23) and (24), Z = 0.01, constant ρ•, and MRN
+(unsedimented, St ≪ α, black dashed line). Three values of α are shown (solid lines). The “sedimented” limits (St ≫ α, dotted lines) are shown
+for comparison. Right: mass density distribution, i.e., the left panel multiplied by the mass of a pebble. Integrated, this function yields the grain
+density ρd0 in the midplane. The distributions follow the unsedimented line for St ≲ α, and the sedimented line for St ≳ α, as expected. The
+mass function is constant with a in the sedimented limit because of the MRN choice: the a−3.5 power law is canceled by the combination of the
+mass of the particle a3 and the extra √a dependency from the sedimentation. Large dots mark the point where St = α.
+the full distribution is found at last
+f(a) =
+3(1− p)ZΣg
+25/2π3/2Hgρ(0)
+• a4−k
+max
+�
+1+aπ
+2
+ρ•(a)
+Σgα a−k.
+(22)
+Notice that to keep f(a) positive definite, the solution re-
+quires p < 1. For q = 0, Eq. (18) constrains k < 4.
+The gas density used is
+Σg = 103 gcm−2 � r
+AU
+�−1
+e−r/rc
+(23)
+i.e. the self-similar solution to the viscous evolution equa-
+tions (Lynden-Bell & Pringle 1974). Here r is the distance to
+the star, and rc a truncation radius. We choose rc =100 AU.
+For the temperature, we use the irradiated, radially optically
+thick, vertically optically thin model of Kusaka et al. (1970,
+see also Ida et al. 2016)
+T = 150K
+� r
+AU
+�−3/7
+(24)
+In addition, we assume metallicity Z = 0.01, adiabatic in-
+dex 1.4, and mean molecular weight 2.3.
+We plot the resulting distributions in Fig. 1, for 20 AU,
+maximum grain size amax = 1 cm, and k = −3.5. For internal
+density we use ρ(0)
+• = 3.5 g cm−3 and q = 0. The left panel
+shows f(a); the right panel m(a) f(a).
+The functions are
+shown for three values of α (solid lines). The unsedimented
+(St ≪ α, black dashed line) and “sedimented” (St ≫ α, dot-
+ted lines) limits are shown for comparison. We see that the
+sedimented distributions follow the unsedimented line for
+St ≲ α, and the sedimented line for St ≳ α, as expected. The
+flat profile for the sedimented cases is due to the MRN expo-
+nent, coupled with
+√
+St from the sedimentation.
+2.2. Column density
+For completeness, we define the vertically-integrated grain
+size distribution
+W(a) ≡
+� ∞
+−∞
+F(a,z) dz =
+√
+2π Hd f(a),
+(25)
+so that the pebble column density is
+Σd(a) =
+� a
+0
+m(a′) W(a′) da′.
+(26)
+Substituting Eq. (22) in Eq. (25), we find the column den-
+sity distribution function
+W(a) = 3(1− p)ZΣg
+4πρ(0)
+• a4−k
+max
+a−k,
+(27)
+which indeed yields Σd = ZΣg when integrated according to
+Eq. (26).
+3. PEBBLE ACCRETION
+Having found the size distribution function for the pebble
+density, we are in position to apply it to pebble accretion.
+Pebble accretion is usually split into three regimes of accre-
+tion (loosely coupled, Bondi, and Hill accretion), each with
+2D and 3D limits. We start by deriving the exact solution for
+the 2D-3D transition.
+
+6
+LYRA ET AL.
+3.1. Exact solution for the monodisperse 2D-3D transition
+The 3D and 2D limits of pebble accretion correspond to
+whether or not the accretion is embedded, i.e, if the accretion
+radius Racc exceeds the height of the pebble column. The
+quantity governing the transition is Racc/Hd, or rather
+ξ ≡
+�Racc
+2Hd
+�2
+,
+(28)
+which we will show a posteriori. The monodisperse mass
+accretion rates in these limits are (Lambrechts & Johansen
+2012)
+˙M3D = lim
+ξ→0
+˙M = πR2
+accρd0δv,
+(29)
+˙M2D = lim
+ξ→∞
+˙M = 2RaccΣdδv,
+(30)
+where δv is the velocity at which the pebble approaches the
+accretor, and ρd0 is the midplane density. In principle, we
+could apply Eq. (3) with Eq. (22) on Eq. (29); and Eq. (26)
+with Eq. (27) on Eq. (30), working with the two limits sepa-
+rately. Yet, given that ξ is a function of grain size, and there
+are other transitions to deal with (loose coupling/Bondi/Hill),
+it is preferable to work with a general expression for ˙M,
+which we derive in this section.
+Considering parallel horizontal chords of infinitesimal
+thickness in the vertical direction until the full accretion ra-
+dius is taken into account, the general expression for the mass
+accretion rate is
+˙M =
+� Racc
+−Racc
+2
+�
+R2acc −z2 ρd0 exp
+�
+− z2
+2H2
+d
+�
+δv dz.
+(31)
+Following Johansen et al. (2015) we define the stratification
+integral
+S ≡
+1
+πR2acc
+� Racc
+−Racc
+2
+�
+R2acc −z2 exp
+�
+− z2
+2H2
+d
+�
+dz,
+(32)
+so that the mass accretion rate is generalized into one expres-
+sion as
+˙M = πR2
+accρd0 S δv.
+(33)
+While Johansen et al. (2015) use a square approximation
+for the accretion radius, we find the exact solution of the
+stratification integral
+S = e−ξ [I0(ξ)+I1(ξ)],
+(34)
+where Iν(ξ) are the modified Bessel functions of the first
+kind, and ξ is given by Eq. (28). The exact monodisperse
+accretion rate is
+˙M = πR2
+accρd0 δve−ξ [I0(ξ)+I1(ξ)].
+(35)
+10
+1
+100
+101
+Racc/2Hp
+0.5
+1.0
+1.5
+2.0
+2.5
+M
+Pebble Accretion Rate
+Exact
+3D
+2D
+Square Approx.
+Square Approx. (0.79Racc)
+Figure 2. General expression for the monodisperse pebble accretion
+rate (Eq. 35). The 3D and 2D limits (eqs 29 and 30, respectively) are
+recovered. The square approximation is also shown for comparison.
+Indeed for ξ → 0, the Bessel functions tend to I0(0) = 1,
+I1(0) = 0, and we recover 3D accretion (Eq. 29). For ξ → ∞,
+both Bessel functions tend to eξ/√2πξ, and 2D accretion is
+recovered (Eq. 30). Fig. 2 shows the agreement graphically.
+The square approximation is shown for comparison.
+3.2. Polydisperse prescription
+To generalize Eq. (35) into a polydisperse description, we
+consider the integrated polydisperse accretion rate to be ˙M ≡
+˙M(amax), where
+˙M(a) =
+� a
+0
+∂ ˙M(a′)
+∂a′
+da′,
+(36)
+with
+∂ ˙M(a)
+∂a
+= πR2
+acc(a) δv(a) S(a) m(a) f(a).
+(37)
+Indeed, Eq. (36) with the integrand given by Eq. (37) is
+equivalent to Eq. (35) if
+� amax
+0
+R2
+acc(a) δv(a) S(a) m(a) f(a)da = ¯R2
+acc δv ¯Sρd0,
+(38)
+where the overline denotes that the quantity is an “effective”
+quantity, independent of pebble size. If the accretion radius
+Racc, the approach velocity δv, and the stratification integral
+S were independent of the grain radius a, Eq. (38) would be
+exactly equivalent to replacing the midplane dust density by
+the integrated grain size distribution
+ρd0 =
+� amax
+0
+m(a) f(a)da,
+(39)
+which is intuitive. We can now use much of the formalism
+of pebble accretion already derived in the literature. The ap-
+proach velocity δv is given by
+δv ≡ ∆v+ΩRacc,
+(40)
+
+POLYDISPERSE PEBBLE ACCRETION
+7
+where ∆v is the sub-Keplerian velocity reduction and Ω is
+the Keplerian frequency. The accretion radius is (Ormel &
+Klahr 2010)
+Racc ≡ ˆRaccexp
+�
+−χ(τ f /tp)γ�
+,
+(41)
+where τ f = St/Ω is the pebble friction time, χ = 0.4 and γ =
+0.65 are empirically-determined coefficients, and
+tp ≡
+GMp
+(∆v+ΩRH)3
+(42)
+is the characteristic passing time scale. Here G is the gravita-
+tional constant, Mp the mass of the planetesimal, and RH its
+Hill radius
+RH ≡
+�GMp
+3Ω2
+�1/3
+.
+(43)
+The variable ˆRacc depends on the accretion regime. For Hill
+accretion it is
+ˆR(Hill)
+acc
+=
+� St
+0.1
+�1/3
+RH,
+(44)
+and for Bondi accretion it is
+ˆR(Bondi)
+acc
+=
+�4τ f
+tB
+�1/2
+RB,
+(45)
+where
+RB ≡ GMp
+∆v2
+(46)
+is the Bondi radius and
+tB ≡ RB
+∆v
+(47)
+is the Bondi time. The transition mass between Bondi and
+Hill accretion is defined by (Ormel 2017)
+MHB = Mt
+8St,
+(48)
+where
+Mt ≡ ∆v3
+GΩ .
+(49)
+A third regime also exists, of accretion of loosely coupled
+pebbles, for which the accretion radius is the physical radius
+R augmented by the gravitational focusing cross-section
+R(geo)
+acc
+= R
+�
+1+ v2esc
+∆v2 ,
+(50)
+where vesc is the escape velocity of the planetary seed. In this
+regime the grains are so loosely coupled they behave almost
+like planetesimals, except for small enough grains, that re-
+main coupled to the gas and follow the gas streamlines. The
+quantity that defines this latter transition is (Ormel 2017)
+Stp = ∆v τ f
+R
+,
+(51)
+that is, the friction time normalized by the time to pass past
+the planetesimal; a planetesimal Stokes number (hence the
+“p” in Stp). For Stp < 1, we set R(geo)
+acc = 0. The transition mass
+MBL between Bondi and loosely coupled accretion happens at
+(Ormel 2017)
+MBL = Mt
+8 St.
+(52)
+3.3. Polydisperse vs Monodisperse
+We show in the left panel of Fig. 3 a reproduction of the
+monodisperse accretion rates from Johansen & Lambrechts
+(2017), for a = 10 cm, and at 5 AU. Even though the obser-
+vations do not support the existence of these large grains, we
+use it for benchmark purposes. The different lines show the
+pebble accretion rates in the Hill and Bondi regimes, as well
+as the loosely coupled regime for low masses.
+The Hill limit (blue dashed line) is recovered for Eq. (35)
+with ˆRacc given by Eq. (44), and δv = ΩR(Hill)
+acc . The Bondi
+limit (red dashed line) is recovered for Eq. (35) with ˆRacc
+given by Eq. (45), and δv = ∆v + ΩR(Bondi)
+acc
+. The actual solu-
+tion (black thick line) uses
+ˆRacc =
+�
+ˆR(Hill)
+acc
+if M ≥ MHB,
+ˆR(Bondi)
+acc
+if M < MHB,
+(53)
+and the general δv given by Eq. (40). The mass accretion
+rate is then the maximum between this and the loosely cou-
+pled accretion rates. The loosely coupled regime is given by
+Eq. (35) with δv = ∆v and Racc given by Eq. (50) if Stp ≥ 1,
+and zero otherwise.
+The right panel of Fig. 3 shows how the accretion rates dif-
+fer when we include a particle size distribution. In this panel
+we are showing the integrated accretion rate ˙M ≡ ˙M(amax)
+given by Eq. (36). The monodisperse line is shown for com-
+parison.
+3.3.1. Slightly lower efficiency in the Hill regime
+From comparing the plots in Fig. 3, we see that the poly-
+disperse accretion rate is slightly lower in the regime of Hill
+accretion; this occurs because, in the Hill regime, there is less
+mass at the biggest pebble size amax compared to monodis-
+perse (where all pebbles are of 10 cm).
+We work out in
+Sect. 4 this reduction factor to be exactly 3/7.
+3.3.2. Significantly higher efficiency in the Bondi regime
+In the Bondi regime, conversely, there are now pebbles
+to accrete of friction time similar to the Bondi time.
+In
+
+8
+LYRA ET AL.
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+M (M
+/yr)
+Accretion at 5AU - Monodisperse a =10 cm
+Actual
+Hill
+Bondi
+Loose Coupling
+Polydisperse
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/MEarth
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+M (M
+/yr)
+Accretion at 5AU - Polydisperse max(a )=10 cm
+Actual
+Hill
+Bondi
+Loose Coupling
+Monodisperse
+Figure 3. Comparison between monodisperse (left) and the integrated polydisperse (right) accretion rates (Eq. 36). The left panel uses the
+parameters of Fig. 4 of Johansen & Lambrechts (2017), except that we use the monodisperse general equation here derived (Eq. 35). A pebble
+size of 10 cm is not supported by observations but we keep this size for benchmarking purposes. The polydisperse accretion rate is reproduced
+in the left plot, and the monodisperse accretion rate in the right plot (grey lines), for comparison. The Hill accretion yields a lower accretion
+rate (3/7 lower than monodisperse) because other pebbles sizes are present, not only a = 10 cm. The main difference is the accretion rate for
+polydisperse Bondi accretion being up to two orders of magnitude more efficient than monodisperse, and the onset of pebble accretion happening
+over one order of magnitude lower in mass. This occurs because the best-accreted pebble is not present in the monodisperse distribution, and
+amax is too loosely coupled, accreting poorly. Notice the smooth transition from Bondi to Hill accretion with the exact 2D-3D transition.
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+102
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+Hill
+Bondi
+Loose Coupling
+ -17 
+ -16 
+ -15 
+ -14 
+ -13 
+ -12 
+ -11 
+ -10 
+ -9 
+ -8 
+ -7 
+ -6 
+ -5 
+ -4 
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+ln aM (M
+yr
+1) at 5AU
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+102
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+ -7 
+ -6 
+ -5 
+ -4 
+ -3 
+ -2 
+ -1 
+ 0 
+ -0.5 
+ 0.5 
+-8
+-6
+-4
+-2
+0
+log [ ln aM(a) /
+ln aM(amax)]  at 5AU
+Figure 4. Left: The polydisperse pebble accretion rate ∂ln a ˙M (Eq. 37), as a function of grain radius. In the Hill accretion regime the largest
+pebble present dominates the mass accretion rate. Conversely, for Bondi accretion, we see that at a given seed mass the differential accretion
+rate is non-monotonic with grain size. For low enough seed masses, the biggest grains, although dominating the mass distribution, accrete in
+the loosely coupled regime. Right: Same as the left plot, but normalized by the accretion rate for amax (proxy for monodispersive). The bright
+red contours are the regions were polydisperse accretion is enhanced over monodisperse. We see that it mostly corresponds to the region where
+monodisperse is in the loosely coupled regime, but polydisperse is already in Bondi. The best accreted pebbles are those for which the stopping
+time τ f equals the Bondi time tB. Absent in the monodispersive description, these pebbles may contribute less to the mass budget, but their
+enhanced accretion ends up dominating the mass accretion rate.
+the monodisperse regime there were only the 10 cm peb-
+bles that, for very low mass seed, behave like infinite St and
+do not accrete well. As a result, in the polydisperse case,
+Bondi accretion is more efficient than loosely coupled accre-
+tion over a wider range of low seed masses. At the mass
+where monodisperse experiences the onset of pebble accre-
+tion (about 10−4M⊕), the polydisperse distribution is well
+into the Bondi regime, which is about 100× more efficient.
+We also see that the onset of pebble accretion occurred be-
+tween 10−6 and 10−5M⊕, i.e., between 100-200 km. This is
+a significant early onset of pebble accretion, that may elim-
+inate the need for planetesimal accretion to bridge the gap
+between the largest masses formed by streaming instability
+and the onset of efficient pebble accretion (Johansen et al.
+2015; Schäfer et al. 2017; Li et al. 2019).
+We plot in Fig. 4 the differential mass accretion rate as a
+function of pebble size (horizontal axis) and seed mass (ver-
+tical axis). The left panel shows the polydisperse mass accre-
+tion rate ∂ln a ˙M, and the right panel shows the ratio between
+that and the same quantity for the largest grain size in the dis-
+
+POLYDISPERSE PEBBLE ACCRETION
+9
+tribution, which we take as a proxy for monodisperse. The
+three accretion regimes are labeled in the left plot; one sees
+the smooth transition between Hill and Bondi accretion, and
+the discontinuous transition from Bondi to loosely coupled.
+It is seen that, at a given mass, Hill accretion is monotonic
+with particle size, but Bondi accretion is not. A local maxi-
+mum of mass accretion rate occurs, corresponding to the size
+for which τ f = tBondi, which in turn leads to a linear depen-
+dency on the best accreted particle size for a given seed mass.
+The bright red parts of the right plot show where Bondi ac-
+cretion is more efficient than monodisperse. It is the more
+efficient accretion of these grains that boosts the Bondi ac-
+cretion rates in the polydisperse case. We see that it corre-
+sponds chiefly to the region of the parameter space for which
+monodisperse accretion was in the loosely coupled regime,
+but the polydisperse is well within Bondi. This confirms that
+indeed it is the accretion of the smaller, Bondi-optimal, peb-
+bles, that is increasing the accretion rate.
+3.4. Effect of distance
+We explore now the parameter space of stellocentric dis-
+tance; the results are shown in Fig. 5, showing the accretion
+rates at 10, 25, and 40 AU (notice also we decreased amax
+to 1 cm). The left plots show the integrated mass accretion
+rates ˙M, the middle plots the distribution ∂ln a ˙M, and the
+right plots the distribution normalized by the accretion rate
+for amax. The Hill accretion rate decreases only slightly with
+distance for this model, because the drop in Ω and Σd with
+distance is equally compensated by the increase in the Hill
+radius.
+As for the Bondi regime, we see that at the grain size where
+monodisperse would transition to loosely coupled, polydis-
+perse is still about two orders of magnitude more efficient,
+over all distances considered. The seed mass for onset of
+pebble accretion is also pushed down 1 order of magni-
+tude, from ∼ 5 × 10−5 to ∼ 5 × 10−6M⊕ at 10 AU. This is
+about 100-200 km radius (for internal densities 3.5 and 0.5
+g/cm3, respectively), reaching the range where pebble accre-
+tion onto the direct products of streaming instability is pos-
+sible. At 40 AU the onset of pebble accretion is pushed from
+∼ 10−3M⊕ in monodisperse to ∼ 10−4M⊕ in polydisperse. A
+significant reduction, but still in the mass range of planetary
+embryos, so planetesimals formed at that distance should re-
+main planetesimals. This is in accordance to the solar system
+constrain given by the existence of the cold classical Kuiper
+Belt objects at 40-50 AU, presumably undisturbed planetesi-
+mals.
+As distance increases, both the accretion rate and the size
+of the best accreted pebble decreases. While at 10 AU the
+best accreted size for a 10−5M⊕ seed (150-300 km radius)
+is 1 mm, at 40 AU it decreases to 10 µm. This has implica-
+tions for the densities of formed objects if the smaller pebbles
+have different composition, e.g. the smaller ones being sili-
+cate in nature and the larger ones being icy. Then a planetes-
+imal seed will preferentially accrete pebbles of rocky com-
+position until it grows enough in mass to start accreting ices
+efficiently.
+The left panel of Fig. 6 shows the integrated polydisperse
+pebble accretion rate as a function of distance, from 1 to
+100 AU. The mass accretion rate of a 10−4M⊕ seed at 20 AU
+is about 10−10M⊕yr−1. The thick black dashed line shows the
+typical mass of objects formed by streaming instability (Liu
+et al. 2020; Lorek & Johansen 2022). The thick grey dashed
+line shows 10 times that mass, proxy for the most massive
+objects formed directly by streaming instability.
+In the right panel we show the accretion time
+tacc ≡ Mp
+˙M ,
+(54)
+along with the same curves for objects formed by stream-
+ing instability. The plot shows that a 0.1 Pluto mass (2 ×
+10−4M⊕) seed has e-folding growth time of 1 Myr at 20 AU,
+and 10 Myr at 30 AU; that is, a Charon-mass planetary em-
+bryo can efficiently increase its mass by Bondi accretion dur-
+ing the lifetime of the disk. This implies that the formation of
+Pluto in the solar Nebula as far as 30 AU is possible by Bondi
+accretion of 10-100 µm grains onto a 0.1 Pluto mass seed.
+The plot also shows that up to 20 AU, the objects typ-
+ically formed by streaming instability (thick black dashed
+line) have growth times up to 3 Myr, within the lifetime of the
+nebula. Notice that, in the inner solar system, Bondi accre-
+tion on 10−6M⊕ seeds (≈ 100 km radius) at 3 Myr timescale
+is possible up to 3 AU. We conclude that Bondi accretion di-
+rectly on planetesimals is possible in the inner solar system,
+dismissing the need for mutual planetesimal collisions as a
+major contribution to planetary growth.
+3.4.1. Effect of maximum grain size
+In Fig. 7 we show the model for 3 different maximum grain
+sizes, from left to right: 3 mm, 1 mm, and 0.3 mm. The main
+feature is that, as the maximum grain size decreases, the mass
+accretion rate (accretion time) for given seed mass at a given
+distance decreases (increases).
+The 3 Myr contour reaches 10−6M⊕ at 3 AU for amax =
+3 mm, 10−6M⊕ at 2 AU for 1 mm, and 10−4M⊕ at 10 AU for
+0.3mm. The conclusion is similar: Myr-timescale Bondi ac-
+cretion on top of 100 km seeds (10−6 M⊕) is possible in the
+inner solar system. Except for the model with amax = 0.3 mm,
+the typical products of streaming instability can grow by peb-
+ble accretion in 3 Myr timescales.
+We calculate also a 10× more massive model. The higher
+dust mass also comes with a higher gas mass, and thus a re-
+duction in Stokes number for the same pebble size. It is un-
+clear a priori which effect dominates. In Fig. 8 we show the
+
+10
+LYRA ET AL.
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/MEarth
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+M (M
+/yr)
+Accretion at 10AU - Polydisperse max(a )=1 cm
+Actual
+Hill
+Bondi
+Loose Coupling
+Monodisperse
+10
+5
+10
+4
+10
+3
+10
+2
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+Hill
+Bondi
+Loose Coupling
+ -16 
+ -15 
+ -14 
+ -13 
+ -12 
+ -11 
+ -10 
+ -9 
+ -8 
+ -7 
+ -6 
+ -5 
+ -4 
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+ln aM (M
+yr
+1) at 10AU
+10
+5
+10
+4
+10
+3
+10
+2
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+ -6 
+ -5 
+ -4 
+ -3 
+ -2 
+ -1 
+ 0 
+ -0.5 
+ 0.5 
+-6
+-4
+-2
+0
+log [ ln aM(a) /
+ln aM(amax)]  at 10AU
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/MEarth
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+M (M
+/yr)
+Accretion at 25AU - Polydisperse max(a )=1 cm
+Actual
+Hill
+Bondi
+Loose Coupling
+Monodisperse
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+Hill
+Bondi
+Loose Coupling
+ -17 
+ -16 
+ -15 
+ -14 
+ -13 
+ -12 
+ -11 
+ -10 
+ -9 
+ -8 
+ -7 
+ -6 
+ -5 
+ -4 
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+ln aM (M
+yr
+1) at 25AU
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+ -6 
+ -5 
+ -4 
+ -3 
+ -2 
+ -1 
+ 0 
+ 1 
+ -0.5 
+ 0.5 
+-6
+-4
+-2
+0
+log [ ln aM(a) /
+ln aM(amax)]  at 25AU
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/MEarth
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+M (M
+/yr)
+Accretion at 40AU - Polydisperse max(a )=1 cm
+Actual
+Hill
+Bondi
+Loose Coupling
+Monodisperse
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+Hill
+Bondi
+Loose Coupling
+ -17 
+ -16 
+ -15 
+ -14 
+ -13 
+ -12 
+ -11 
+ -10 
+ -9 
+ -8 
+ -7 
+ -6 
+ -5 
+ -4 
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+ln aM (M
+yr
+1) at 40AU
+10
+4
+10
+3
+10
+2
+10
+1
+St
+10
+3
+10
+2
+10
+1
+100
+101
+a (mm)
+10
+6
+10
+4
+10
+2
+100
+Mp/MEarth
+ -6 
+ -5 
+ -4 
+ -3 
+ -2 
+ -1 
+ 0 
+ 1 
+ -0.5 
+ 0.5 
+-6
+-4
+-2
+0
+log [ ln aM(a) /
+ln aM(amax)]  at 40AU
+Figure 5. Left: Same as Fig. 3, right plot, but for the density and temperature of Eqs. (23) and (24), Z=0.01, and at different distances. Hill
+accretion is not much affected by distance, but Bondi accretion becomes increasingly less efficient as distance increases. Yet, the general trend
+remains, of polydisperse pebble accretion being 1-2 orders of magnitude more efficient than monodisperse at maximum, and showing an earlier
+onset in mass also by 1-2 orders of magnitude. Middle and Right: same as Fig. 4, at difference distances. The pebble size that maximizes
+Bondi accretion decreases as distance increases. This has interesting implications, because in the outer disk, the seeds, presumably icy, should
+accrete small grains, presumably silicates. This implies the possibility a two-mode formation of Kuiper belt objects: icy planetesimal produced
+by streaming instability of larger grains, followed by pebble accretion of smaller, silicate, grains.
+formation times for the model, using amax = 1cm. The for-
+mation times are overall shorter compared to the right panel
+of Fig. 6, pushing the 3 Myr e-folding contour to double the
+distance vis-à-vis the lower mass model (7 AU for 100 km,
+30 AU for 10−2 Pluto mass, and 60 AU for 10−1 Pluto mass).
+Even at this higher mass model, a 100 km seed has an e-
+folding growth time of over 100 Myr at 40 AU, and should
+remain planetesimals, as expected.
+3.4.2. Effect of sedimentation
+In Fig. 9 we show the e-folding growth times for the plan-
+etary seeds formed by streaming instability (typical objects
+and most massive objects), as a function of the turbulent vis-
+cosity parameter α.
+Its function in the model is only on
+how it influences sedimentation.
+The grey dotted line in
+the plot marks the threshold of 3 Myr. For moderately high
+turbulence (α = 10−3), the typical seeds have longer growth
+times than 3 Myr already beyond 6 AU. For lower turbulence,
+α = 10−5, as most pebbles are sedimented, the distance where
+growth occurs within 3 Myr increases to 40 AU. The most
+massive objects, well into the Bondi regime, all have fast
+growth times.
+4. ANALYTICAL SOLUTIONS
+In this section we derive the analytical solutions in the rel-
+evant limits of 2D Hill accretion and 3D Bondi accretion. In
+a polydisperse distribution, the pebble scale height is a func-
+tion of pebble radius, so the pebbles are not necessarily all in
+the 2D regime or all in the 3D regime. Also, because the tran-
+sitions between loosely coupled and Bondi, and from Bondi
+to Hill are St-dependent, the pebbles are not all in the same
+regime of accretion either.
+Yet, in practice these limits still yield reasonably accurate
+accretion rates. Because the distribution is top heavy, the
+2D Hill regime is applicable for large seed masses, that are
+accreting in this regime the biggest pebbles, which are re-
+sponsible for most of the mass accretion rate. The 3D Bondi
+
+POLYDISPERSE PEBBLE ACCRETION
+11
+100
+101
+102
+r/au
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+Polydisperse Accretion Rate - amax=1.0cm
+ -15 
+ -14 
+ -13 
+ -12 
+ -11 
+ -10 
+ -9 
+ -8 
+ -7 
+ -6 
+ -5 
+ -4 
+10 mp
+mp
+17
+15
+13
+11
+9
+7
+5
+3
+log10 [ M / (M
+yr
+1) ]
+100
+101
+102
+r/au
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+Accretion Timescale - amax=1.0cm
+ 4.0 
+ 4.5 
+ 5.0 
+ 5.5 
+ 6.0 
+ 6.5 
+ 7.0 
+ 7.5 
+ 8.0 
+ 8.5 
+ 9.0 
+10 mp
+mp
+3
+4
+5
+6
+7
+8
+9
+10
+log10 ( tacc/yr )
+Figure 6. Left: Integrated polydisperse pebble mass accretion rate, as a function of distance. The model uses the density and temperature of
+Eqs. (23) and (24), Z=0.01, and ρ• constant. The thick black dashed line shows the characteristic size of the planetesimals formed by streaming
+instability (Liu et al. 2020; Lorek & Johansen 2022); the grey line represents bodies of 10× the typical mass. Right: Accretion times M/ ˙M for
+the same model. The contour of 6.5 (3Myr) marks the boundary where accretion during the lifetime of the nebula is feasible by pebble accretion,
+without the need for planetesimal accretion. That contour corresponds to 3 AU, 10 AU, and 30 AU, for 10−6M⊕, 2×10−5M⊕, and 2×10−4M⊕,
+respectively. These masses correspond to 100 km radius, 10−2 and 10−1 Pluto masses, respectively. The typical products of streaming instability
+have <3 Myr growth times up to 30 AU.
+100
+101
+102
+r/au
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+Accretion Timescale - amax=3.0mm
+ 4.5 
+ 5.0 
+ 5.5 
+ 5.5 
+ 6.0 
+ 6.5 
+ 7.0 
+ 7.5 
+ 8.0 
+ 8.5 
+ 9.0 
+ 9.5 
+10 mp
+mp
+3
+4
+5
+6
+7
+8
+9
+10
+log10 ( tacc/yr )
+100
+101
+102
+r/au
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+Accretion Timescale - amax=1.0mm
+ 5.0 
+ 5.5 
+ 6.0 
+ 6.5 
+ 7.0 
+ 7.5 
+ 8.0 
+ 8.5 
+ 9.0 
+ 9.5 
+10 mp
+mp
+3
+4
+5
+6
+7
+8
+9
+10
+log10 ( tacc/yr )
+100
+101
+102
+r/au
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+Accretion Timescale - amax=0.3mm
+ 6.0 
+ 6.5 
+ 7.0 
+ 7.5 
+ 8.0 
+ 8.5 
+ 9.0 
+ 9.5 
+10 mp
+mp
+3
+4
+5
+6
+7
+8
+9
+10
+log10 ( tacc/yr )
+Figure 7. Same as Fig. 6, but exploring the parameter space of maximum grain radius amax, from left to right: 3 mm, 1 mm, and 0.3 mm. Upper
+plots show mass accretion rate, lower plots the accretion times. The trend seen is that Bondi accretion rates decrease with amax for the same
+seed mass and distance. The contour of 6.5 (3Myr) marks the boundary where accretion during the lifetime of the nebula is feasible by pebble
+accretion, without the need for planetesimal accretion. This translates into ≈ 3 AU for 100 km seeds (10−6M⊕), 10 AU for 0.01 Pluto mass
+(2×10−5M⊕), and up to 30 AU for 0.1 Pluto mass (2×10−4M⊕), for the first two models. The typical products of streaming instability grow in
+Myr timescales except for the last model, of maximum grain size 0.3 mm.
+regime is applicable as long as Racc < 2Hd (Eq. 28), which
+solving for mass yields
+Mp < ∆vΩH2
+gα
+GSt(St+α).
+(55)
+Normalizing by the transition mass Mt, we find
+Mp
+Mt
+≲
+α
+h2St(St+α),
+(56)
+where h ≡ Hg/r is the disk aspect ratio. For α ∼ 10−4 and
+h ∼ 10−2, 3D Bondi accretion should apply close to the tran-
+sition mass, except for big enough pebbles, as expected, be-
+cause these are too sedimented. Yet, as we have established,
+these pebbles contribute poorly to the mass accretion rate.
+For particles of τ f = tB, and assuming St ≫ α, we find
+Mp
+Mt
+≲
+� α
+h2
+�1/3
+,
+(57)
+i.e., within the expected ranges of α and h, the seed mass
+for which τ f = tB is within a factor of order unity from the
+transition mass. We conclude that a 3D approximation for
+the Bondi regime should lead to acceptable results.
+We work now the analytical expressions in these limits.
+4.1. Analytical Polydisperse 2D Hill accretion
+
+12
+LYRA ET AL.
+100
+101
+102
+r/au
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/M
+Accretion Timescale - amax=1.0cm - 10
+ 4.0 
+ 4.5 
+ 5.0 
+ 5.5 
+ 6.0 
+ 6.5 
+ 7.0 
+ 7.5 
+ 8.0 
+ 8.5 
+10 mp
+mp
+3
+4
+5
+6
+7
+8
+9
+10
+log10 ( tacc/yr )
+Figure 8. Same as the right panel of Fig. 6, but for 10 times the disk
+mass. Although the Stokes number decreases for the same particle
+radius, the increase in dust mass is the dominant effect, and accre-
+tion times decrease for the same seed mass and distance. Compared
+to the lower-mass model, the line of 3 Myr e-folding growth time
+is pushed to about twice the distance, allowing for pebble accre-
+tion on top of 100 km seeds (10−6M⊕) up to 7 AU. 200 km objects
+(10−5M⊕) can accrete pebbles efficiently up to 30 AU. At 40 AU ac-
+cretion on 100 km seeds takes over 100 Myr and they should remain
+planetesimals, consistent with evidence from the Solar System.
+100
+101
+102
+r/au
+105
+106
+107
+108
+tacc/yr
+Accretion times
+= 10
+3, mp
+= 10
+4, mp
+= 10
+5, mp
+= 10
+3, 10 mp
+= 10
+4, 10 mp
+= 10
+5, 10 mp
+Figure 9. Polydisperse pebble accretion timescales for different
+α values for the typical masses produced by streaming instability
+(solid lines), and ten times this mass (dashed lines), taken as proxy
+for the end of the streaming instability mass function. The grey dot-
+ted line marks 3 Myr. For α = 10−3, the typical seeds only grow
+within the lifetime of the nebula in the inner solar system, up to
+≈5-10 AU. For lower turbulence, α = 10−5, as most pebbles are sed-
+imented, the distance increases to 40 AU.
+We can integrate the polydisperse Hill regime analytically
+in the 2D limit by generalizing Eq. (30) with Σd given by
+Eq. (26)
+˙M2D,Hill = 2×102/3ΩR2
+H
+� amax
+0
+St(a)2/3 m(a)W(a)da. (58)
+Given the scalings St ∝ a1−q, m ∝ a3−q, and W ∝ a−k, the
+dependency of the integrand of Eq. (58) on a is
+∂ ˙M(a)
+∂a
+2D,Hill ∝ a(11−5q−3k)/3
+(59)
+Integrating it in a, we find the exact solution
+˙M2D,Hill =
+6(1− p)
+14−5q−3k
+�Stmax
+0.1
+�2/3
+Ω R2
+H Z Σg.
+(60)
+Eq. (60) differs from the monodisperse case (Eq. 30) by an
+efficiency factor
+� ˙Mpoly
+˙Mmono
+�
+2D,Hill
+=
+3(1− p)
+14−5q−3k
+�Stmax
+St
+�2/3
+.
+(61)
+For MRN (k = 3.5), q = 0, and St = Stmax, this yields
+� ˙Mpoly
+˙Mmono
+�k=3.5,q=0
+2D,Hill
+= 3
+7,
+(62)
+that is, about 43% of the monodisperse. Deviations from this
+number are due to not all pebbles being in the 2D Hill regime.
+For large enough seed mass, the deviations should be small,
+as indeed it is seen in the plots of Figs. 3 and 5.
+4.2. Analytical Polydisperse 3D Bondi accretion
+The Bondi accretion in the 3D regime limit is found by
+generalizing Eq. (29) with ρd0 given by Eq. (39)
+˙M3D,Bondi = 4πRB∆v2
+Ω
+×
+� amax
+0
+St e−2ψm(a) f(a)
+�
+1+2
+�
+StΩRB
+∆v
+�1/2
+e−ψ
+�
+da,
+(63)
+where we use the shorthand notation
+ψ ≡ χ[St/(Ωtp)]γ.
+(64)
+We will split Eq. (63) into two integrals
+˙M3D,Bondi = 4πRB∆v2
+Ω
+�� amax
+0
+e−2ψ St m(a) f(a) da
++ 2
+�ΩRB
+∆v
+�1/2 � amax
+0
+e−3ψ St3/2 m(a) f(a) da
+�
+.
+(65)
+The function f(a) has a dependency on
+�
+1+St/α, which
+makes these functions non-integrable sauf specific cases. We
+
+POLYDISPERSE PEBBLE ACCRETION
+13
+will thus use the following approximation, valid at x → 0 and
+x → ∞
+√
+1+x ≈ 1+√x.
+(66)
+While the error incurred with this approximation at x ≈ 1
+can be large, we are interested in the definite integral from
+0 to xmax. In this case, the error decreases if the range of
+integration is large enough, tending to zero for xmax → ∞, as
+shown in Fig. 10. Confident in the accuracy of Eq. (66), we
+write the approximate solution
+˙M3D,Bondi ≈ 3(1− p)ZΣgRB∆v2
+√
+2πHgΩρ(0)
+• a4−k
+max
+×
+�� amax
+0
+e−2ψ St m(a) a−kda
++ α−1/2
+� amax
+0
+e−2ψ St3/2 m(a) a−kda
++ 2
+�ΩRB
+∆v
+�1/2 � amax
+0
+e−3ψ St3/2 m(a) a−kda
++ 2
+�ΩRB
+α∆v
+�1/2 � amax
+0
+e−3ψ St2 m(a) a−kda
+�
+.(67)
+The four integrals are of the form below, for which there
+is an analytical solution in terms of lower incomplete gamma
+functions
+� amax
+0
+e− jasabda = γl
+� b+1
+s , jas
+max
+�
+sj(b+1)/s
+.
+(68)
+We thus write the solution of Eq. (67)
+˙M3D,Bondi ≈C1
+γl
+� b1+1
+s , j1as
+max
+�
+sj(b1+1)/s
+1
++C2
+γl
+� b2+1
+s , j2as
+max
+�
+sj(b2+1)/s
+2
++
+C3
+γl
+� b3+1
+s , j3as
+max
+�
+sj(b3+1)/s
+3
++C4
+γl
+� b4+1
+s , j4as
+max
+�
+sj(b4+1)/s
+4
+, (69)
+where the coefficients are
+10
+6
+10
+3
+100
+103
+106
+xmax
+1.0
+1.1
+1.2
+1.3
+1.4
+xmax
+0
+1 +
+x dx
+xmax
+0
+1 + x dx
+Figure 10. Approximating
+√
+1+x in by 1+√x (the asymptotic ex-
+pansion for x → 0 and x → ∞), to make the sedimented midplane
+distribution integrable analytically. As long as the function is inte-
+grated to a large value of xmax, the error incurred is small.
+s =γ(1−q)
+(70)
+b1 =4−2q−k
+(71)
+b2 = b3 =(9−5q−2k)/2
+(72)
+b4 =5−3q−k
+(73)
+St′ =
+π
+2Σg
+ρ(0)
+• aq
+max
+(74)
+j′ =χ
+� St′
+Ωtp
+�γ
+(75)
+j1 = j2 =2j′
+(76)
+j3 = j4 =3j′
+(77)
+m′ = 4π
+3 ρ(0)
+• aq
+max
+(78)
+K = 3(1− p)ZΣgRB∆v2
+√
+2πHgΩρ(0)
+• a4−k
+max
+(79)
+C1 =KSt′m′
+(80)
+C2 =KSt′3/2m′α−1/2
+(81)
+C3 =2KSt′3/2m′
+�ΩRB
+∆v
+�1/2
+(82)
+C4 =2KSt′2m′
+�ΩRB
+α∆v
+�1/2
+(83)
+Fig. 11 shows that the agreement between the numerical
+integration of Eq. (37) and the analytical solutions (Eq. (60)
+and Eq. (69)) is excellent in the range of validity. Having
+Eq. (60) and Eq. (69) as analytical expressions is of great
+interest for future studies including pebble accretion analyti-
+cally, instead of having to integrate the mass accretion rates
+numerically with the particle size distributions.
+
+14
+LYRA ET AL.
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/MEarth
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+M (M
+/yr)
+Polydisperse Numerical vs Analytical - 20AU - amax=1 cm
+Actual
+Hill (numerical)
+Hill 2D Analytical
+Bondi (numerical)
+Bondi 3D Analytical
+Figure 11. Agreement between the numerically calculated poly-
+disperse pebble accretion rate and the analytical solutions for 2D
+Hill accretion Eq. (60) and 3D Bondi accretion Eq. (69). While the
+Hill solution is exact, the Bondi solution is approximate. Yet, the
+agreement seen is excellent, because the best accreted pebbles in
+this regime are in the 3D range.
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Mp/MEarth
+10
+15
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+M (M
+/yr)
+Polydisperse - 20AU - amax=1.0 cm - Effect of k
+mono
+k=0.0
+k=2.0
+k=3.0
+k=3.5
+k=3.9
+Figure 12. Effect of varying the slope k of the grain size distribu-
+tion. The Hill regime is relatively insensitive to k, as this regime
+is dominated by the largest grains. For Bondi accretion, the mass
+accretion rates increase significantly as the slope steepens.
+5. EFFECT OF SLOPE K OF GRAIN SIZE
+DISTRIBUTION
+So far we have considered only the MRN value for the
+index k of the grain size distribution (Mathis et al. 1977),
+but this index should depend on the collisional evolution, ve-
+locities, and material strength of the pebbles (Kobayashi &
+Tanaka 2010; Kobayashi et al. 2016). Having found the ana-
+lytical solution for the accretion rates, we can more easily de-
+termine the impact of varying this parameter, which we show
+in Fig. 12. As the slope steepens, the mass accretion rate de-
+creases in the Hill regime. Compared to monodisperse, the
+effect is small, but accelerates as k approaches 4. This insen-
+sitivity is expected, as the Hill regime is dominated by the
+larger grains. The effect on mass accretion rate is more pro-
+nounced for the Bondi regime, as expected, as the amount
+of mass in different grain sizes more strongly affects this ac-
+cretion regime. As the slope steepens and more mass is made
+available in small grain sizes, the accretion rates onto smaller
+planetesimal seeds increases, although the effect is nonlinear.
+6. LIMITATIONS
+We are limited in this work by the vast expanses of the pa-
+rameter space and by the circular restricted 3-body problem
+solution that forms for the underlying assumption of the gas
+and pebble flow. While the former would be a valiant endeav-
+our, it is not the scope of this work to derive results applicable
+to all possible situations, but to derive the model in first place.
+As such, we kept our equations general in metallicity, inter-
+nal density, and grain size distribution, but apply it mostly for
+Z = 0.01, ρ(0)
+• = 3.5 g cm−3, q = 0. These parameters will vary
+with dust drift (lower the metallicity), composition (varying
+the internal density if ices or silicates), and porosity.
+As for going beyond the circular restricted 3-body prob-
+lem, recently the impact of the gravity of the planetary seed
+on the accretion flow has been calculated from hydrody-
+namical simulations (Okamura & Kobayashi 2021), for the
+Hill regime (Kuwahara & Kurokawa 2020a) and for the
+Bondi regime (Kuwahara & Kurokawa 2020b). In the Bondi
+regime, the trajectories are modified for St ≲ 10−3, with the
+gas flow reducing the accretion rate. Thus, Eq. (45) is over-
+estimated for small St. We find that the best accreted peb-
+bles, that give the bulk of the boost in Bondi accretion, are
+slightly above the St ∼ 10−3 transition found by (Kuwahara
+& Kurokawa 2020b); as such, this aspect of our results are
+not severely affected by the planet-induced flow.
+7. CONCLUSION
+In this paper, we worked out the theory of polydisperse
+pebble accretion, finding analytical solutions when possible.
+Our main findings are as follows:
+• We find that polydisperse Bondi accretion is 1-2 or-
+ders of magnitudes more efficient than in the monodis-
+perse case, This is because the best-accreted pebbles
+in the Bondi regime are those of friction time similar
+to Bondi time, not the largest pebbles present. The
+large pebbles, although dominating the mass budget,
+are weakly coupled across the Bondi radius and thus
+accrete poorly. The pebbles that are optimal for Bondi
+accretion may contribute less to the mass budget, but
+their enhanced accretion significantly impacts the mass
+accretion rate.
+• The onset of polydisperse pebble accretion is extended
+by 1-2 orders of magnitude lower in mass compared to
+monodisperse, for the same reason. The onset of peb-
+ble accretion with Myr-timescales reaches 100-350 km
+
+POLYDISPERSE PEBBLE ACCRETION
+15
+sized objects depending on stellocentric distances and
+disk model. For the model considered, Bondi accretion
+on Myr timescales, within the lifetime of the disk, is
+possible on top of 10−6M⊕ (100 km) seeds up to 4 AU,
+on top of 10−5M⊕ (200 km) seeds up to 10 AU, and
+on 10−4M⊕ (350 km) seeds up to 30 AU. A model 10
+times more massive doubles these distances.
+• In all models considered, at 40 AU a 100 km seed has
+growth time over 100 Myr, and should thus remain
+as planetesimals, in accordance with the existence of
+the cold classical Kuiper Belt population, presumably
+undisturbed planetesimals.
+• We find the analytical solution of the stratification in-
+tegral, and thus the exact solution for the 3D-2D tran-
+sition (Eq. 35),
+• We find analytical solutions for the polydisperse 2D
+Hill (Eq. 60) and 3D Bondi regime (Eq. 69). For the
+MRN distribution, the Hill accretion is a factor 3/7
+(about 42%) as efficient in polydisperse than monodis-
+perse.
+The fact that Myr-growth timescales, within the lifetime of
+the disk, is possible for polydisperse pebble accretion onto
+100-350 km seeds over a significant range of the parameter
+space, has significant implications. This mass range over-
+laps with the high mass end of the planetesimal initial mass
+function (Johansen et al. 2015; Schäfer et al. 2017; Li et al.
+2019), and thus pebble accretion is possible directly follow-
+ing formation by streaming instability, removing the need for
+planetesimal accretion. This conclusion is supported by the
+lack of of craters generated by 1-2 km on Pluto (Singer et al.
+2019), and recent findings by Lorek & Johansen (2022) that
+planetesimal accretion are not able to sustain accretion rates
+beyond 5 AU.
+While we do most of our numerical solutions with con-
+stant ρ•, we keep the analytical solutions general for varying
+this parameter, expecting that smaller pebbles should be of
+lower density, and the bigger pebbles of higher density, re-
+flecting different compositions (Morales et al. 2016). We no-
+tice that as the distance increases, the pebble size that maxi-
+mizes pebble accretion is increasingly smaller. This implies
+the possibility of a two-mode formation of Kuiper belt ob-
+jects: streaming instability of the largest pebbles forming icy
+objects of the order of ≳ 100 km in diameter, followed by
+pebble accretion leading to objects of the order of 1000 km,
+where silicates are incorporated mostly at the pebble accre-
+tion stage, due to their low Stokes number. This scenario
+would lead to a different composition for the smaller objects,
+mostly formed by ice streaming instability, and the larger ob-
+jects, grown by ice and silicate pebble accretion on top of
+the icy planetesimal seeds. A continuum of rock-to-ice frac-
+tion should be produced. Indeed a trend is clear in the Kuiper
+belt, of constant density around 0.5 g cm−3 for the smaller ob-
+jects (diameter less than 500 km), and increasing for larger
+objects (Brown 2012; Grundy et al. 2015; McKinnon et al.
+2017). We will explore how our findings in this paper can
+reproduce this result in a future work.
+ACKNOWLEDGMENTS
+WL acknowledges support from the NASA Theoretical
+and Computational Astrophysical Networks (TCAN) via
+grant 80NSSC21K0497, from the NASA Emerging Worlds
+program via grant 22-EW22-0005, and by NSF via grant
+AST-2007422.
+AJ is supported by the Swedish Research
+Council (Project Grant 2018-04867), the Danish National
+Research Foundation (DNRF Chair grant DNRF159), and
+the Knut and Alice Wallenberg Foundation (Wallenberg
+Academy Fellow Grant 2017.0287). A.J. further thanks the
+European Research Council (ERC Consolidator Grant 724
+687-PLANETESYS), the Göran Gustafsson Foundation for
+Research in Natural Sciences and Medicine, and the Wallen-
+berg Foundation (Wallenberg Scholar KAW 2019.0442) for
+research support. MHC is supported by grant 22-EW22-0005
+from the NASA Emerging Worlds program. We acknowl-
+edge conversations with Andrew Youdin, Jake Simon, Orkan
+Umurhan, Debanjan Sengupta, and Daniel Carrera.
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+Validating dark energy models using polarised Sunyaev-Zel’dovich effect with
+large-angle CMB temperature and E-mode polarization anisotropies
+Hiroto Kondo1,∗ Kiyotomo Ichiki1,2, Hiroyuki Tashiro1, and Kenji Hasegawa1
+1Graduate School of Science, Division of Particle and Astrophysical Science,
+Nagoya University, Chikusa-Ku, Nagoya, 464-8602, Japan
+2Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
+Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
+(Dated: February 1, 2023)
+The tomography of the polarized Sunyaev-Zeldvich effect due to free electrons of galaxy clusters
+can be used to constrain the nature of dark energy because CMB quadrupoles at different redshifts
+as the polarization source are sensitive to the integrated Sachs-Wolfe effect. Here we show that the
+low multipoles of the temperature and E-mode polarization anisotropies from the all-sky CMB can
+improve the constraint further through the correlation between them and the CMB quadrupoles
+viewed from the galaxy clusters. Using a Monte-Carlo simulation, we find that low multipoles of
+the temperature and E-mode polarization anisotropies potentially improve the constraint on the
+equation of state of dark energy parameter by ∼ 17 percent.
+I.
+INTRODUCTION
+Dark energy, which is causing the current accelerated
+expansion of the universe [1, 2], has two main effects on
+the temperature anisotropies of the cosmic microwave
+background (CMB). One is to change the angular dis-
+tance to the final scattering surface of the CMB, and the
+other is the Integrated Sachs-Wolfe (ISW) effect, which
+creates new temperature fluctuations due to the decay
+of the gravitational potential of the large-scale structure.
+The ISW effect is a characteristic effect that indicates
+that the universe is deviating from the matter-dominated
+one. However, because the temperature fluctuations pro-
+duced by this effect are smaller than those produced in
+the early universe in the standard cosmological model
+(so-called the SW effect), they are masked by the dis-
+persion of the fluctuations, making it difficult to obtain
+a statistically significant enough signal to approach the
+nature of dark energy. Therefore, the CMB constraint on
+dark energy-related parameters is weak because the ISW
+effect suffers from sizable cosmic variance errors in the
+CMB temperature anisotropy spectrum on large scales
+[3, 4].
+The Kamionkowski and Loeb method [5] is an effec-
+tive way to detect the ISW effect without this cosmic
+variance. This method uses the fact that the polariza-
+tion angle of CMB photons scattered by free electrons in
+a galaxy cluster is determined by the quadrupole tem-
+perature fluctuations of the CMB as seen from that clus-
+ter [6] and allow us to reconstruct the three-dimensional
+density fluctuations of the universe on large scales [7–10].
+While avoiding cosmic variance by fixing the realization
+of the initial density fluctuations, the direct detection
+of the ISW effect is possible by tomographic use of clus-
+ters of galaxies at various redshifts [11–13]. Our previous
+∗ [email protected]
+study using simple Monte Carlo simulations has shown
+that it is possible to constrain the dark energy equation of
+state parameters more accurately through the ISW effect
+than conventional methods based on the power spectra
+[14]. The method can also be useful for the studies of
+the power asymmetry of CMB polarization and density
+field [15], cosmic birefringence [16] and the reionization
+optical depth [17].
+In our previous study [14], we used the quadrupole
+anisotropies of the CMB as a diagnostic of the ISW effect.
+Specifically, we compared the quadrupole anisotropy of
+our CMB estimated from the three-dimensional density
+fluctuations on large scales reconstructed by the KL
+method, with the actual quadrupole anisotropy that can
+be directly observed by the all-sky CMB experiments
+such as Planck. In fact, it has been shown that the three-
+dimensionally reconstructed density fluctuations on large
+scales should be correlated not only with the quadrupoles
+but also with higher temperature multipoles and E-mode
+polarization fluctuations on large angular scales [18].
+Therefore, this paper aims to clarify to what extent the
+addition of these fluctuations as diagnostics improves the
+results obtained in previous studies.
+In the next section, we review our method developed
+in [14], and extend it by adding information on the tem-
+perature and E-mode polarization anisotropies on large
+angular scales. Section III presents our result of the fu-
+ture constraint on the dark energy equation of state pa-
+rameters based on Monte-Carlo simulations. In Section
+IV, we discuss and summarize this study.
+II.
+METHODOLOGY
+A.
+CMB polarization from galaxy clusters
+First, we consider the CMB polarization produced in
+galaxy clusters. The polarization is created by Thomson
+arXiv:2301.13676v1  [astro-ph.CO]  31 Jan 2023
+
+2
+scattering of the free electrons in galaxy clusters with the
+quadrupole component of the CMB anisotropy. There-
+fore, if a galaxy cluster is at the position ⃗x in the co-
+moving coordinate, we can observe the polarization from
+the galaxy cluster, which is produced by the Thomson
+scattering at the conformal time τx = τ0 − |⃗x| with the
+present conformal τ0.
+Accordingly, the observed polarization from galaxy
+clusters at ⃗x can be calculated with the Stokes parameter,
+Q(⃗x) and U(⃗x)
+Q(⃗x) ± iU(⃗x) = −
+√
+6
+10 τCTCMB(τx)
+×
+2
+�
+m=−2
+±2Y2m(ˆx)aT
+2m(⃗x, τx) ,
+(1)
+where τC
+is the optical depth of the galaxy clus-
+ter for Thomson scattering.
+In Eq. (1) aT
+2m(⃗x, τx) is
+the quadrupole component of the CMB temperature
+anisotropy observed at the position ⃗x and the conformal
+time τx.
+Now we consider the CMB temperature anisotropy on
+the position of the comoving coordinate ⃗x at the confor-
+mal time τx.
+The CMB temperature in the direction
+ˆn at ⃗x
+and τx can be decomposed into the isotropic part
+and anisotropic part,
+T(⃗x, ˆn, τx)
+=
+TCMB(⃗x, τx) +
+∆T(⃗x, ˆn, τx).
+Introducing the CMB anisotropy as
+∆(⃗x, ˆn, τx)
+≡
+∆T(⃗x, ˆn, τx)/TCMB
+and
+we
+expand
+the CMB anisotropy with sperical harmonic functions
+Ylm(ˆn),
+∆(⃗x, ˆn, τx) =
+∞
+�
+l=0
+l
+�
+m=−l
+aT
+lm(⃗x, τx)Ylm(ˆn) ,
+(2)
+where aT
+lm(⃗x, τ) is the coefficient of the spherical har-
+monic expansion and the coefficinet with ℓ = 2 is the
+quadrupole component aT
+2m(⃗x, τx).
+On the other hand, since the CMB anisotropy is the
+function of ⃗x and ˆn, we can decompose it by the plane
+wave function and the spherical harmonics,
+∆(⃗x, ˆn, τ) = 4π
+�
+d3kei⃗k·⃗x
+∞
+�
+l=0
+(−i)l∆T
+l (⃗k, τ)
+×
+l
+�
+m=−l
+Y ∗
+lm(ˆk)Ylm(ˆn) .
+(3)
+Therefore, the coefficient of the spherical harmonic ex-
+pansion in Eq. (2) can be written as
+aT
+lm(⃗x, τx) = (−i)l4π
+�
+d3kei⃗k·⃗x∆T
+l (⃗k, τx)Y ∗
+lm(ˆk) . (4)
+In our case, the cosmological linear perturbation the-
+ory is applicable to calculate ∆T
+l (⃗k, τ) in Eq. (3). Ac-
+cording to the cosmological linear perturbation theory,
+∆T
+l (⃗k, τ) can be obtained as
+∆T
+l (⃗k, τ) = ∆T
+l (k, τ)φini(⃗k) ,
+(5)
+where φini(⃗k) is the Fourier component of the initial cur-
+vature perturbations, and ∆T
+l (k, τ) is the liner transfer
+function which depends on the cosmological models and
+is obtained from the cosmological linear perturbation the-
+ory. We calculate ∆T
+l (k, τ) using a publicly available code
+CAMB [19].
+B.
+Monte-Carlo simulation
+Our aim of this paper is to study how much the KL
+methods with the future CMB temperature and polariza-
+tion measurement improve the constraint on the nature
+of dark energy. For this purpose, we demonstrate the KL
+methods by conducting the Monte Carlo simulation.
+In our simulation, to realize the CMB anisotropy at
+comoving position ⃗x we use transfer functions gener-
+ated by the publicly available code CAMB. Throughout
+this paper, we set Λ-CDM model with Ωbh2 = 0.0226,
+Ωch2 = 0.112, Ωνh2 = 0.00064, h = 0.7, as the reference
+cosmological models.
+The first step of the simulation is to generate the ini-
+tial fluctuation field φini(ki). Our initial fluctuation field
+is given as a Gaussian random field with the power spec-
+trum
+P(k) = k3
+2π2 P(k) = As
+� k
+k∗
+�ns−1
+,
+(6)
+where we set the parameters As = 2.1 × 10−9, ns = 0.96
+and k∗ = 0.05.
+In our methods, it is useful to employ the polar coordi-
+nate in Fourier k-space. To sample the Fourier mode, we
+divide the angular directions in Fourier space by Healpix
+[20] with Nside = 8. This means that the whole sky is di-
+vided into 768 sections. For the radial mode, we sample
+60 wave number modes uniformly in logathmical space
+with a range from k = 10−5 to 10−1. Thus, the over-
+all independent Fourier mode nk for this simulation is
+46080.
+Second, we simulate the polarization produced in clus-
+ters with the generated initial fluctuations φini(ki). In
+this process, we use the transfer function ∆(k, τx) with
+the fiducial equation of state of dark energy parame-
+ter w = −1. In our simulation, we set the number of
+galaxy clusters to Ncluster = 6000. We distribute them
+randomly in the angular direction and uniformly in red-
+shift ranging from z = 0 to 2. We calculate the polar-
+ization, Qfiducial(⃗xi) and Ufiducial(⃗xi), produced by each
+galaxy cluster at the position ⃗xi, following the procedure
+described in Sec. II A. To consider the observational un-
+certainties, Gaussian noise σobs/τ = 10−2 µK is added
+to each Qfiducial(⃗xi) and Ufiducial(⃗xi).
+Similarly, we simulate the CMB anisotropies directly
+observed at the origin with the generated initial fluctua-
+tions φini(ki). In both the temperature and the polariza-
+tion anisotropy (E-mode), we calculate the angular com-
+ponents, aT
+lmfiducial aE
+lmfiducial, in the range from l = 2
+
+3
+to 9.
+Here, as in the case for galaxy cluster polariza-
+tion, we add Gaussian noise σobs = 10−2 µK to alms as
+observational uncertainty.
+Fig.1 and 2 show one realization example of the Q maps
+for the polarization produced in galaxy clusters at z =
+0.01 and 0.3. The quadrupole of the CMB temperature
+observed by galaxy clusters at z = 0.01 is nearly identical
+to the CMB temperature quadrupole anisotropy at the
+origin. Therefore, according to Eq. (1), the pattern of
+the Q map on the sky is very similar to that of the CMB
+temperature quadrupole anisotropy at the origin. On the
+other hand, at z = 0.3, the quadrupole pattern observed
+at each galaxy cluster is slightly different.
+Therefore,
+the generated Q map has small-scale pattern due to the
+difference, although the large-scale pattern is similar to
+the Q map at z = 0.01.
+FIG. 1.
+Example Q polarization map observed at galaxy
+clusters at redshift z = 0.01.
+Because they are produced
+by quadrupoles that are nearly identical to the quadrupoles
+we observe today, they have a quadrupole pattern.
+FIG. 2. Same as Fig. 1, but at redshift z = 0.3. While fea-
+tures similar to the map at z = 0.01 remain, smaller patterns
+develop.
+The third step is the reconstruction of the initial fluc-
+tuations by the fitting of the polarization, Q and U, pro-
+duced by the galaxy clusters and the CMB temperature
+and polarization anisotropy, aT
+lm, aE
+lm, directly observed
+at the origin. We estimate the initial fluctuations to min-
+imize the function given by
+ftot = fpol + fT + fE + fprior
+(7)
+Each term in the right-hand side of the equation repre-
+sents the chi-square minimizations for fitting the polar-
+ization of the galaxy cluster Q(xi) and U(xi), the tem-
+perature anisotropy of the CMB aT
+lm and the polarization
+anisotropy of the CMB aE
+lm, and the prior, respectively.
+The chi-square minimizations for the polarization of
+the galaxy cluster Q(xi) and U(xi) can be written as
+fpol =
+Ncluster
+�
+i=1
+(Q(⃗xi) − Q(⃗xi)fiducial)2
+σ2
+pol
++(U(⃗xi) − U(⃗xi)fiducial)2
+σ2
+pol
+,
+(8)
+where Q(⃗xi) and U(⃗xi) is the polarization produced in
+galaxy clusters at ⃗xi with the estimated initial condition
+and Q(⃗xi)fiducial and U(⃗xi)fiducial is the polarization ob-
+tained in the simulation with adding the Gaussian noise
+with the variance σpol due to the uncertainty in the ob-
+servation of Q and U from galaxy clusters.
+We use CMB temperature anisotropy from l = 3 to 9
+for fitting
+fT =
+�
+l=3
+l
+�
+m=−l
+(aT
+lm − aT
+lmfiducial)2
+σ2
+T
+,
+(9)
+where aT
+lm is the temperature anisotropy evaluated from
+the estimated initial fluctuations, aT
+lmfiducial is the one
+obtained from the simulation, and σT is the uncertainty
+in observing CMB temperature anisotropy.
+For CMB polarization E-mode, l = 2 mode is also
+added to the fitting function
+fE =
+�
+l=2
+l
+�
+m=−l
+(aE
+lm − aE
+lmfiducial)2
+σ2
+E
+.
+(10)
+where aE
+lm is the E-mode polarization anisotropy evalu-
+ated from the estimated initial fluctuations, aE
+lmfiducial is
+the one obtained from the simulation, σE is the uncer-
+tainty in the observation of CMB E-mode polarization
+anisotropy.
+To improve the accuracy of the reconstruction, we also
+adopt a Gaussian prior based on power spectrum Pφ(k).
+fprior =
+nk
+�
+j
+R2
+ini(kj)
+2P(kj) .
+(11)
+where R2
+ini(k) is the Fourier component of the estimated
+initial fluctuations.
+Tuning the estimated initial fluctuations, R2
+ini(k), we
+search the set of R2
+ini(k) which can minimize the function
+f. The obtained set of R2
+ini(k) is the Fourier component
+
+Cluster Polarization Q redshift z=0.01
+-1.2863e-06
+1.6662e-06Cluster Polarization Q redshift z=0.3
+-1.1101e-06
+1.4673e-064
+of the estimated initial fluctuations which fit the polar-
+ization of the galaxy cluster and the CMB temperature
+and polarization anisotropy to the values in the fiucial
+mock simulation.
+In this process, the transfer functions are used to calcu-
+late the observable from the initial fluctuations φini(ki).
+Since the transfer function depends on the cosmological
+parameters, different cosmologies lead to different esti-
+mates of the initial fluctuations.
+In this work, we es-
+timate the initial fluctuations with several dark energy
+state parameters w = −1, −0.99, and −0.95 in order
+to verify the statistical power for the dark energy state
+parameter although the dark energy state parameter is
+fixed to w = −1 in the simulation.
+In the last step, we calculate the l = 2 mode temper-
+ature anisotropy aT
+2m
+est(0) observed at the origin using
+the estimated initial fluctuations and compare it to the
+true value aT
+2m
+true(0) ≡ aT
+2mfiducial(0) calculated from the
+mock simulation. Note that, in the fitting process, we
+do not use the l = 2 mode temperature anisotropy and
+reserve it for the comparison between one form the esti-
+mated initial fluctuations and the mock simulation data.
+Up to this point, the method has been applied to a sin-
+gle mock simulation. The sequence of steps is repeated
+one hundred times from the generation of the initial fluc-
+tuations and makes one hundred pairs of aT
+2m
+true(0) and
+aT
+2m
+est(0).
+The generated aT
+2m
+true(0) and aT
+2m
+est(0) pairs should
+agree within statistical error if they are generated us-
+ing the same transfer function. In application to actual
+observations, the cosmological parameters of the trans-
+fer function used in the estimation process should match
+those of the actual universe. Thus, the larger the dif-
+ference between pairs generated using different transfer
+functions, the more effective the method is able to con-
+strain the cosmological parameters.
+The accuracy of this method depends on errors in
+polarization measurements, the number of galaxy clus-
+ters, the optical depth of the clusters, and the redshift
+errors of the clusters.
+In this study, we assume the
+most ideal conditions, where the polarization measure-
+ment error and optical depth of the clusters are uniform
+σpol/τ = 10−2 µK, and the redshift error is negligible.
+The number of clusters used is assumed to be 6000 and
+randomly distributed.
+The error for the CMB all-sky
+observation is also used as σT = σE = 10−2 µK. The
+methodological, statistical uncertainty in this method is
+a complex mixture of these factors and can be calculated
+from the reconstruction error in the pair when the correct
+transfer function including w=-1 is used in the estima-
+tion.
+σ2
+method = 1
+N
+N
+�
+i=1
+1
+5
+�
+|∆aT
+20 i|2 + 2|∆aT
+21 i|2 + 2|∆aT
+22 i|2�
+(12)
+where N refers to the number of simulations used, and
+each ∆a2m are difference of pairs
+∆a2m = aT
+2m
+true(w = −1) − aT
+2m
+est(w = −1). (13)
+In Eq. (12), while the m = 0 component is a real number,
+the m = 1, 2 components are complex numbers, so the
+independent components are doubled, requiring a factor
+of 2 on the right side.
+In the setting of our simulation with Ncluster
+=
+6000, σpol/τ = 10−2 µK, Nside = 8 and nkmode = 60,
+the methodological statistical uncertainty is
+σmehthod ≃ 4.0 × 10−8.
+(14)
+We find out that, even when not including all-sky CMB
+observations of temperature fluctuations and polariza-
+tion, almost the same values were obtained as the
+methodological statistical uncertainty. Therefore, we can
+conclude that the dominant uncertainty of this recon-
+struction comes from the KL method.
+To examine statistical power, we define the chi-square
+statistic for the quadrupole as
+χ2(w) =
+1
+σ2
+method
+�
+|∆aT
+20|2 + 2|∆aT
+21|2 + 2|∆aT
+22|2�
+.(15)
+The chi-square is an indicator to show the goodness of
+fit between the cosmological model in the mock simula-
+tion and the one used for estimation. In our case, if the
+equation of state of dark energy, w, in the estimation is
+identical to the one in the simulation, it ideally follows
+the chi-square distribution with a degree of freedom of
+five. The chi-square values are larger when different w is
+used in the estimation process.
+In other words, the cosmological parameters can be
+varied and the cosmology can be restricted by compar-
+ing the differences in the chi-square values ∆χ2(w) =
+χ2(w) − χ2(w = −1).
+In other words, through the
+comparison of the difference in the chi-square values,
+∆χ2(w) = χ2(w) − χ2(w = −1), with changing w in the
+estimation, we can provide the observation constraint on
+w.
+III.
+RESULT
+In the previous study, only the polarization of the
+galaxy clusters was used in the fitting process to recon-
+struct the initial fluctuations. In this study, we investi-
+gate the improvement in statistical power for the dark
+energy equation of state parameter by adding tempera-
+ture anisotropy and polarization in the all-sky CMB ob-
+servations.
+We set the true equation-of-state parameters of dark
+energy w = −1. The difference in chi-square values for
+w = −0.99 is ⟨χ2(w = −0.99)⟩ = 1.14, 1.16, and 1.33
+respectively only galaxy clusters polarization case, the
+case with adding E-mode polarization, and the case with
+adding E-mode polarization and temperature anisotropy.
+We summarize the results in Table I. Fig.3 shows the
+histograms of ∆χ2 with 100 realizations in each case.
+Also, the difference in chi-square values for w = −0.95
+is ⟨χ2(w = −0.95)⟩ = 16.90, 17.85, and 19.93 for only
+
+5
+FIG. 3. Distribution of the difference of the chi-square statis-
+tic from the 100 simulations for w = 0.99.
+Different his-
+tograms show the cases obtained from fitting only to the po-
+larization of galaxy clusters, fitting with the E-mode, and fit-
+ting with the E-mode and temperature anisotropies of all-sky
+CMB observations, as indicated in the figure.
+galaxy clusters polarization case, the case with adding E-
+mode polarization, and the case with adding E-mode po-
+larization and temperature anisotropy, respectively. We
+summarize the results in Table II. Fig.4 shows the his-
+tograms of ∆χ2 with 100 realizations in each case.
+FIG. 4. Same as Fig. 3, but for w = 0.95.
+Observable
+σmehthod
+∆χ2
+Only cluster polarization
+4.060 × 10−8 1.137
+Cluster polarization + E-mode
+4.039 × 10−8 1.163
+Cluster polarization + E&T-mode 4.014 × 10−8 1.327
+TABLE I. ∆χ2 for parameters with w = −0.99
+Observable
+σmehthod
+∆χ2
+Only cluster polarization
+4.060 × 10−8 16.90
+Cluster polarization + E-mode
+4.039 × 10−8 17.85
+Cluster polarization + E&T-mode 4.014 × 10−8 19.93
+TABLE II. ∆χ2 for parameters with w = −0.95
+For both dark energy equation of state parameters, we
+obtained larger chi-square values when adding E-mode
+polarization and temperature anisotropy.
+This is due to the fact that E-mode polarization and
+temperature anisotropy in all-sky observations are asso-
+ciated with the polarization produced by galaxy clusters.
+Thus, combining all-sky CMB observations with the
+remote quadrupole technique using the polarization of
+galaxy clusters can more strongly constrain the cosmol-
+ogy.
+IV.
+SUMMARY AND DISCUSSION
+In this paper, we study how to constrain the nature
+of the dark energy using the ISW effect by combining
+information about the CMB quadrupole at high redshift
+obtained from the polarization of CMB photons pass-
+ing through a galaxy cluster based on the KL method
+with information about temperature and E-mode polar-
+ization fluctuations on large angular scales at z = 0. In
+conventional analyses based on power spectra, the SW
+contribution, which is unrelated to the dark energy ef-
+fect, acts like Gaussian noise and prevents the statistical
+detection of the ISW effect [12]. In contrast, our method
+can estimate and subtract the SW contribution by re-
+constructing the primordial density fluctuations in three
+dimensions. Thus, we can estimate the pure ISW effect
+due to dark energy.
+In our previous paper, to limit the equation of state
+for dark energy, we used only the z = 0 quadrupole,
+which is expected to correlate most with the polariza-
+tion of CMB photons scattered by clusters of galaxies.
+However, the polarization of CMB photons scattered by
+clusters of galaxies, especially at high redshifts, should
+correlate not only with the quadrupoles but also with
+higher multipoles at z = 0.
+Indeed, as shown in [18],
+CMB polarization generated due to a galaxy cluster at a
+higher redshift correlates not only with the quadrupoles
+but also with higher multipoles of the current CMB tem-
+perature fluctuations.
+Compared with the cluster polarization-only con-
+straint, our results showed that including E-mode po-
+larization (l > 2) and temperature anisotropies (l > 3)
+improves the constraining power for the dark energy
+parameter w by 18 percent if we compare w = −1
+and w = −0.95 dark energy models, assuming 6000
+clusters and polarization sensitivity of σpol/τ = 10−2.
+In our setup, this improvement comes almost equally
+from the E-mode polarization (l > 2) and temperature
+
+Ax2 = x2(w = - 0.99) - x2(w= - 1)
+cluster polarization only
++ E-mode
+8
++ T & E-mode
+6
+4 
+2
+0
+10-1
+100
+101
+102
+4x?Ax2 = x2(w= - 0.95) - x2(w= - 1)
+12
+cluster polarization only
++ E-mode
+10
++ T & E-mode
+8:
+6
+4
+2 
+0
+10-1
+100
+101
+102
+Ax?6
+anisotropies (l > 3). The improvement is due to the fact
+that the information on E-mode polarization and temper-
+ature anisotropy at z = 0 allowed us to solve part of the
+degeneracy between the 3D density fluctuation Fourier
+modes inferred from the polarization produced in galaxy
+clusters.
+ACKNOWLEDGMENTS
+This work is supported in part by the JSPS grant num-
+bers 18K03616,21H04467 and JST AIP Acceleration Re-
+search Grant JP20317829 and JST FOREST Program
+JPMJFR20352935 (K.I.), JP21K03533, and JP21H05459
+(H.T.), and JST SPRING, grant number JPMJSP2125
+(H.K.).
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@@ -0,0 +1,1688 @@
+A Cognitive Evaluation of Instruction Generation Agents
+tl;dr They Need Better Theory-of-Mind Capabilities
+♠Lingjun Zhao∗ and ♣Khanh Nguyen∗ and ♠♦Hal Daumé III
+♠University of Maryland–College Park
+♣Princeton University
+♦Microsoft Research
+[email protected]
+Abstract
+We mathematically characterize the cognitive
+capabilities that enable humans to effectively
+guide others through natural language.
+We
+show that neural-network-based instruction
+generation agents possess similar cognitive
+capabilities, and design an evaluation scheme
+for probing those capabilities.
+Our results
+indicate that these agents,
+while capable
+of effectively narrowing the search space,
+poorly predict the listener’s interpretations
+of their instructions and thus often fail to
+select the best instructions even from a small
+candidate set.
+We augment the agents with
+better theory-of-mind models of the listener
+and obtain significant performance boost in
+guiding real humans.
+Yet, there remains a
+considerable gap between our best agent and
+human guides.
+We discuss the challenges
+in closing this gap, emphasizing the need to
+construct better models of human behavior
+when interacting with AI-based agents.
+1
+Introduction
+Instruction generation refers to the problem of guid-
+ing humans to accomplish goals through natural
+language. While being able to hold fluent chit-
+chatting conversations with humans (Thoppilan
+et al., 2022; OpenAI, 2022), performances of AI-
+based agents in this problem are still far from per-
+fect (Zhao et al., 2021; Kojima et al., 2021; Wang
+et al., 2022). To build agents that communicate
+pragmatically like humans, we must equip them
+with cognitive capabilities similar to those of hu-
+mans. Accomplishing this goal requires (i) mathe-
+matically characterize the capabilities that are es-
+sential for human pragmatic communication and
+(ii) designing an evaluation scheme for assessing
+these capabilities of AI-based agents.
+In this paper, we present a framework for con-
+ducting fine-grained evaluation of the communica-
+tion capabilities of instruction generation agents.
+∗The first two authors contribute equally.
+Our evaluation focuses on cognitive capabilities
+that are known to be requisite for human-like prag-
+matic communication. The outcome of the evalua-
+tion indicates which cognitive capabilities require
+further development and thus can help developers
+direct their effort more deliberately and effectively.
+Figure 1 provides an overview of our approach.
+To identify the cognitive capabilities essential for
+pragmatic communication, we build on two lines
+of work from socio-cognitive science: Bayesian
+models of cooperative communication (Wang et al.,
+2020; Goodman and Frank, 2016; Shafto et al.,
+2014) and studies on how humans implement
+Bayesian reasoning (Sanborn and Chater, 2016;
+Sanborn et al., 2010; Vul et al., 2014; Mamassian
+et al., 2002). These models have been shown to be
+capable of predicting and explaining human behav-
+iors in various communication games. We propose
+a framework named bounded pragmatic agent that
+practically characterize the human cognitive pro-
+cess for instruction generation. We show that our
+framework can also describe the operation of a
+broad class of AI-based agents, including neural-
+network-based agents.
+Interpreting AI-based
+agents and humans under the same mathematical
+framework enables us to quantify their differences.
+We derive the optimality conditions that a bounded
+pragmatic agent must satisfy in order to generate
+optimally pragmatic instructions. These conditions
+correspond to well-known cognitive capabilities of
+humans: (i) the ability to efficiently generate rele-
+vant utterances (the search capability) (Bloom and
+Fischler, 1980; Gold et al., 2000; Trosborg, 2010)
+and (ii) the ability to accurately simulate the lis-
+tener’s interpretations of their utterances in the envi-
+ronment (the theory-of-mind capability) (Premack
+and Woodruff, 1978; Gopnik and Astington, 1988;
+Tomasello, 2019; Call and Tomasello, 2011). We
+then design an evaluation scheme for assessing
+these capabilities of an agent, measuring how close
+it is to satisfying our optimality conditions.
+arXiv:2301.05149v1  [cs.CL]  21 Dec 2022
+
+0.5
+How to bridge 
+this gap?
+Repeat K times
+   (i) Generate candidate (search capability)
+           ui ~ Sbase
+   (ii) Evaluate candidate (ToM capability)
+           score(ui) = LToM(e* | ui)
+Return argmaxu∈D score(u), D = { u1 ,..., uK }
+ToM capability
+Search capability
+𝚫search
+𝚫ToM
+Human
+Evaluated 
+agent
+Human-level 
+candidate 
+generation
+Human-level 
+candidate 
+evaluation
+Bounded pragmatic agent
+Recommendation:
+● Large 𝚫search, small 𝚫ToM  ⇒ improve inference algorithm
+● Large 𝚫ToM , small 𝚫search ⇒ enhance planning module
+(a)
+(b)
+(c)
+(d)
+Figure 1: An overview of our approach.
+We aim
+to build speaker agents that can guide humans to
+accomplish goals through natural language. Standard
+evaluation that computes task performance metrics
+is not helpful for directing the development of the
+evaluated agents (a).
+We propose a mathematical
+framework called “bounded pragmatic agent” that
+can characterize the operations of both AI-based
+and human speakers (b).
+Viewing AI-based agents
+and humans through this unifying lens enables us to
+compare them on more fine-grained capabilities (c),
+and better instruct future development of these agents
+towards leveling with human performance (d).
+We
+evaluate
+various
+neural-network-based
+agents1 on an instruction generation problem in
+photo-realistic 3D environments (Anderson et al.,
+2018b). To evaluate each capability of an agent,
+we compare it with the same agent but equipped
+with an optimal version of the evaluated capability,
+which is simulated by asking a human to perform
+that capability for the agent. Our evaluation reveals
+a crucial finding: most evaluated agents possess
+1We release our human-evaluation dataset and inter-
+face at https://lingjunzhao.github.io/coop_
+instruction.html.
+relatively efficient search capability but inadequate
+theory-of-mind capability. Specifically, on a major-
+ity of test cases, the agents can find an instruction
+that successfully guide humans by drawing a few
+samples. But they assign incorrect probabilities to
+the instructions and thus fail to select the best one
+as final outputs.
+We improve the theory-of-mind capability of the
+evaluated agents by equipping them with an ex-
+plicit pragmatic reasoning mechanism (Andreas
+and Klein, 2016; Fried et al., 2017), using state-
+of-the-art instruction-following agents (Magalhaes
+et al., 2019; Shen et al., 2022; Hong et al., 2021) as
+theory-of-mind models. We obtain significant im-
+provement over the original agents, shrinking the
+gap with human performance on test data by 36%.
+Towards eliminating the remaining gap, we illus-
+trate with empirical evidence a major challenge in
+developing better theory-of-mind models. Specif-
+ically, when employed, these models would be
+asked to evaluate AI-generated instructions, which
+may differ dramatically from human-generated in-
+structions. Hence, a standard supervised-learning
+training scheme that only exposes the model to
+human-generated instructions would be inadequate
+for learning reliable theory-of-mind models. We
+thus call for the construction of novel datasets, ap-
+proaches, and evaluation methods for developing
+these models.
+2
+Related Work
+Navigation Instruction Generation.
+Instruc-
+tion generation has been commonly studied in navi-
+gation settings (Anderson et al., 1991; Byron et al.,
+2010; Koller et al., 2010; Striegnitz et al., 2011;
+Goeddel and Olson, 2012; Fried et al., 2017, 2018).
+The Matterport3D simulator and the accompanying
+datasets (R2R (Anderson et al., 2018b), R4R (Jain
+et al., 2019), and RxR (Ku et al., 2020)) offer more
+challenging settings by combining photo-realistic
+scenes with long, verbally rich instructions. Recent
+work on evaluating instruction generation agents
+(Zhao et al., 2021) reveals the ineffectiveness of
+standard learning and modeling approaches to this
+problem. Wang et al. (2021) improve the accuracy
+and interpretability of instructions in the RxR set-
+ting. Kamath et al. (2022) leverage this model to
+synthesize additional data for training instruction-
+following agents. Our work offers useful principles
+for improving these models.
+
+o1o
+00Mathematical Models of Human Communica-
+tion.
+Human communication is a cooperative
+act (Grice, 1975; Scott-Phillips, 2014; Tomasello,
+2019). Pragmatic communication in humans may
+involve different cognitive capabilities like basic
+understanding of language and social rules (Tros-
+borg, 2010) and reasoning about the physical world
+(Bender and Koller, 2020) and human behavior
+(Ganaie and Mudasir, 2015; Enrici et al., 2019;
+Rubio-Fernandez, 2021). Our work describes simi-
+lar capabilities but provides a mathematical inter-
+pretation that allows for computational evaluation
+of those capabilities. Development of mathemat-
+ical models of human communication have been
+greatly useful for understanding human behaviors
+(Ho et al., 2016; Sumers et al., 2022) and building
+communication agents (Andreas and Klein, 2016;
+Fried et al., 2017, 2018; , FAIR; Lin et al., 2022).
+Numerous variants of these models have been pro-
+posed. Wang et al. (2020) present a comprehensive
+comparison of these variants and unify them under
+a framework inspired by optimal transport. Since
+we are interested more in characterizing general ca-
+pabilities than specific implementation, the model
+we propose in this work is a generalized version
+capturing the essence of these models.
+Evaluating Cognitive Capabilities of Neural
+Networks.
+A
+plethora
+of
+benchmarks
+for
+evaluating the cognitive capabilities of AI-based
+agents have been created, focusing on theory-of-
+mind capabilities (Le et al., 2019; Nematzadeh
+et al., 2018), grounding (Lachmy et al., 2021;
+Udagawa and Aizawa, 2019; Haber et al., 2019),
+commonsense reasoning (Talmor et al., 2018;
+Levesque et al., 2012; Zellers et al., 2019; Sap
+et al., 2019), etc. Recent work (Sap et al., 2022;
+Hu et al., 2022) examine performance of large
+language models on various cognitive tasks. They
+evaluate a capability by designing language tasks
+that are assumed to require the evaluated capability
+to solve. This approach is limited to large language
+models that can perform few-shot learning.
+A
+limitation of the approach is that it may not be
+possible to determine whether an agent solve the
+tasks in the intended way. Our evaluation scheme
+follows a different principle: we mathematically
+characterize exactly the capabilities we want to
+evaluate, and compare agents that possess different
+levels of these capabilities.
+3
+Problem Setting
+3.1
+Environment and Human Listener
+We consider a human listener h acting in a POMDP
+environment with state space S, action space Ah,
+transition function Eh(st+1 | st, at), start-state
+distribution Eh
+1 (s1), observation space Ω, and ob-
+servation function Oh(ot+1 | st+1). An instruction
+u ∈ U is an utterance consisting of words belong-
+ing to a vocabulary V. The human can follow in-
+structions to generate trajectories. For example, in
+an indoor navigation setting, upon hearing “go the
+kitchen and stop next to the oven”, a human can
+walk to the specified location. A T-step trajectory
+eh = (s1, oh
+1, ah
+1, · · · , sT , oh
+T , ah
+T ) is an execution
+of an instruction. The observable part of the tra-
+jectory ¯eh = (oh
+1, ah
+1, · · · , oh
+T , ah
+T ) is obtained by
+excluding the states from eh.
+To follow instructions, we imagine the human
+implements a policy πh(a | ¯e, u) that takes as
+input a partial observed trajectory ¯e and an in-
+struction u, and outputs a distribution over actions
+in Ah. Given an instruction u, a T-step trajec-
+tory is generated as follows. The human starts in
+s1 ∼ E1 and observes oh
+1 ∼ O(s1). At time step
+t, let ¯e1:t = (oh
+1, ah
+1, · · · , oh
+t ). The human chooses
+ah
+t ∼ πh(· | ¯e1:t, u), executes the action, and tran-
+sitions to st+1 ∼ Eh(st, ah
+t ). There, they perceive
+oh
+t+1 ∼ Oh(st+1). In the end, they issue a special
+stop action aT to terminate the trajectory. We de-
+fine Lh(e | u) as the probability of generating a
+trajectory e according to this process. We will refer
+to Lh as the real listener to distinguish it with the
+theory-of-mind listener, which is a mental model
+of the real listener that an agent constructs.
+3.2
+Pragmatic Instruction Generation
+In pragmatic instruction generation (PIGEN), the
+goal is to learn a speaker agent r that generates
+language instructions to guide a human listener
+h to reach states in the environment. The term
+“pragmatic” emphasizes that the agent generates
+language in a social context to achieve a commu-
+nication goal. In each PIGEN task, the speaker
+agent first imagines an intended trajectory e⋆ =
+(s1, or
+1, ar
+1, · · · , sT , or
+T , ar
+T ), which leads to the in-
+tended goal state sT from the state s1 that the hu-
+man is currently in. Because the human’s action
+space and observation function may differ from
+those of the agent, they may not be able to com-
+prehend e⋆ even if it is presented to them. Thus,
+the agent needs to translate the trajectory into an
+
+instruction ˆu that the human can understand and
+follow. To do so, it implements a speaker model
+Sr(u | e) that takes as input a trajectory and com-
+putes a distribution over instructions. The objective
+of the problem can be written formally as
+arg max
+Sr
+Ee⋆ [Lh(e⋆ | Gen(Sr, e⋆))]
+(1)
+where Gen(Sr, e⋆) is the process implemented by
+the agent for generating an instruction.
+The agent is evaluated using a dataset Deval
+of held-out trajectories.
+For each trajectory
+e⋆
+k ∈ Deval, We generate an instruction ˆuk =
+GEN(Sr, e⋆
+k) . The instruction is then presented
+to a human listener to follow, producing a trajec-
+tory eh
+k ∼ Lh(· | ˆuk). The performance of the
+agent, denoted by ρ(r), is the average similarity
+between the human-generated trajectory and the
+intended trajectory
+ρ(r) ≜
+1
+|Deval|
+�
+e⋆
+k∈Deval
+Ψ(eh
+k, e⋆
+k)
+(2)
+where Ψ is a similarity metric.
+4
+Building Agents that Communicate
+Pragmatically like Humans
+Faced with instances of the PIGEN problem
+daily, humans have evolved a highly efficient
+cognitive process for solving this problem. To
+build agents with a similar level of efficacy, we
+propose a mathematical model characterizing the
+human cognitive process for instruction generation
+(§ 4.1).
+We then derive the capabilities for an
+agent implementing that model to optimally solve
+PIGEN (§4.2). Finally, we present an evaluation
+scheme for collating these capabilities on a general
+class of speaker agents (§4.3).
+4.1
+A Mathematical Cognitive Model of
+Instruction Generation
+To formulate how humans generate instructions,
+we build on mathematical models of coopera-
+tive communication (Wang et al., 2020; Goodman
+and Frank, 2016; Shafto et al., 2014). We con-
+sider a general version where a speaker agent con-
+structs a pragmatic speaker model Sprag(u | e)
+based on two constituents: a base speaker model
+Sbase(u | e) and a theory-of-mind (ToM) listener
+model LToM(e | u). The base speaker represents
+general knowledge of the agent about the world and
+the language it speaks. The ToM listener reflects
+situated knowledge about the listener, simulating
+how they would behave in the environment given
+an instruction. The construction of Sprag is defined
+as a Bayesian belief update that alters the initial
+belief Sbase by re-weighting with LToM:
+Sprag(u | e) ∝ LToM(e | u)Sbase(u | e)
+(3)
+The pragmatic speaker utters an instruction of
+maximum probability under its model:
+ˆuprag ≜ arg max
+u∈U
+Sprag(u | e⋆)
+= arg max
+u∈U
+LToM(e⋆ | u)Sbase(u | e⋆) (4)
+This choice reflects that the speaker wants to maxi-
+mize the chance of the listener interpreting its in-
+struction correctly, but it is still influenced by prior
+knowledge.
+While this model accounts for human behaviors
+highly accurately on problems where U is a small
+discrete space (Frank and Goodman, 2012), in prob-
+lems where U is unbounded like PIGEN, it is un-
+likely that humans, which are known to be agents
+with bounded rationality (Simon, 1957), are able to
+implement the full Bayesian update in the model’s
+formulation. A hypothesis, which is supported by
+empirical evidence, is that humans approximate
+the update via Monte-Carlo sampling (Sanborn and
+Chater, 2016; Sanborn et al., 2010; Vul et al., 2014;
+Mamassian et al., 2002). Applying this hypothesis
+to our setting, we derive a more practical model
+of how human generate instructions, in which they
+perform the Bayesian update on a subspace Usub of
+U chosen by drawing samples from Sbase
+ˆubounded-prag ≜
+arg max
+u ∈ Usub ⊂ U
+LToM(e⋆ | u)
+(5)
+where Usub is a small set of candidate instructions
+generated by Sbase. We call an agent that gen-
+erates instructions according to Eq 5 a bounded
+pragmatic speaker (Figure 2). For such a speaker,
+instruction generation involves two cognitive tasks:
+the candidate generation task (performed by Sbase)
+and the candidate evaluation task (performed by
+LToM). The former task ensures that the generation
+of an instruction is efficient, while the latter
+guarantees the generated instruction conveys the
+intended meaning.
+4.2
+Essential Cognitive Capabilities of
+Pragmatic Instruction Generation Agents
+What cognitive capabilities enable humans to ef-
+fectively solve the PIGEN problem (section §3.2)?
+
+Theory-of-mind 
+Listener Model
+Base Speaker 
+Model
+ 
+Start
+Goal
+Human Listener
+Candidate set
+u1: Walk past the stairs and 
+out the door that leads 
+outside. Wait on the porch. 
+u2: Walk across the living 
+room and out the doors on the 
+other side. Stop just outside 
+the door
+…
+e1
+e2
+Figure 2: The cognitive process of a bounded pragmatic speaker. The speaker implements two models: a base
+speaker model and a theory-of-mind listener model. In every task, the speaker first imagines a trajectory it wants
+to convey to the human listener. To reduce the search space, it then uses the base speaker to generate a small set
+of relevant candidate instructions. After that, it employs the theory-of-mind model to simulate how the human
+listener would follow each instruction in the candidate set. The speaker finally elects the candidate instruction that
+causes the theory-of-mind listener to generate the trajectory most similar to the intended trajectory. The output
+instruction is finally sent to the human listener for a real execution in the environment.
+Viewing humans as bounded pragmatic agents, we
+can characterize those capabilities by identifying
+the requirements for a bounded pragmatic agent
+to optimize the PIGEN objective (Eq 1). A gen-
+eral condition is that the instruction ˆubounded-prag
+selected by the agent must satisfy
+ˆubounded-prag = u⋆ ≜ arg max
+u
+Lh(e⋆ | u)
+(6)
+where Lh is the real listener.
+We translate this condition into conditions for the
+constituent models, Sbase and LToM, of the agent.
+The condition for Sbase is that the candidate set Usub
+generated by it must contain the optimal instruction
+u⋆ (condition S ). Fulfilling this condition requires
+Sbase to be capable of quickly generating candidates
+and placing sufficiently high probability on u⋆ so
+that the instruction can be found by sampling a few
+candidates. We refer to this capability as the search
+capability of an agent.
+The condition for LToM is that it must rank u⋆
+first among the candidates (condition T ). Meeting
+this condition demands having the capability of
+mentally counterfactually simulating the behavior
+of the listener in an environment, and evaluating
+whether the communicated intention is actualized
+in the simulation. We refer to this capability as the
+ToM capability.
+The search and ToM capabilities are orthogonal
+and complementary. An agent with flawless ToM
+capability can evaluate the goodness of instructions
+given to it, but may not be able to efficiently gener-
+ate good instructions by itself. In contrast, an agent
+with effective search capability can quickly bring
+to attention highly relevant utterances but may not
+always select the best one for its communication
+purposes if it has a misleading ToM model.
+4.3
+Assessing the Cognitive Capabilities of an
+Instruction Generation Agent
+We consider a speaker agent r that learns a model
+Sr(u | e) and communicates a trajectory e⋆ by
+running an inference algorithm to compute an in-
+struction ˆuinfer ≈ arg maxu∈U Sr(u | e⋆). Gen-
+erative LSTM- or Transformer-based models that
+implement greedy or beam-search decoding are
+examples of such an agent.
+We notice that, like humans, r also possesses
+search and ToM capabilities. On one hand, it can
+generate candidate instructions like a base speaker
+by sampling from Sr or executing an inference al-
+gorithm. On the other hand, for a fixed e⋆, it can
+use Sr(u | e⋆) as a ToM model to rank instruc-
+tions. Improving these capabilities is crucial for r
+to better solve PIGEN. In fact, suppose Sr satis-
+fies conditions T and the following candidate set
+generated by Sr
+Ur
+sub ≜ {ˆuinfer} ∪ {ui ∼ Sr | 1 ≤ i ≤ N}
+(7)
+fulfills condition
+S . Then instead of running the
+
+do1o
+00o1oQ
+d456896934c56680.266
+efb1d36fd87d4287a115od9553e7o0df
+2inference algorithm, it can generate instructions as
+a bounded pragmatic agent as follows
+ˆu ≜ arg max
+u∈Ur
+sub
+Sr(u | e⋆)
+(8)
+and optimizes the PIGEN objective.
+To evaluate each capability of r, we measure
+the performance gap between the agent and a sky-
+line agent which is at human level in the evaluated
+capability but is equally good as r at the other ca-
+pability. Specifically, we define roracle-search to be
+an agent that employs Sr as the ToM model but
+is given a “gold” candidate set U⋆
+cand that always
+contains the ground-truth instruction u⋆. It outputs
+an instruction as follows
+ˆuoracle-search ≜ arg max
+u∈U⋆
+cand
+Sr(u | e⋆)
+(9)
+This agent has similar ToM capability as r but
+human-level search capability (in fact, its search
+capability satisfies condition
+S ). Next, we con-
+struct roracle-ToM which generates candidates using
+Sr but employs a real human to select the output
+instruction
+ˆuoracle-ToM ≜ arg max
+u∈Ur
+sub
+Lh(e⋆ | u)
+(10)
+where ˆuinfer is the instruction generated by the in-
+ference algorithm that r implements and Ur
+sub is de-
+fined as in Eq 7. The search capability of roracle-ToM
+is as good as r but its ToM capability is that of a
+human.
+We define the prospective performance gain
+(PPG) with respect to each capability as follows
+PPGsearch(r) ≜ ρ(roracle-search) − ρ(r)
+(11)
+PPGToM(r) ≜ ρ(roracle-ToM) − ρ(r)
+(12)
+where ρ computes the performance metric of an
+agent on evaluation data (Eq 2 of §3.2). The met-
+ric computes the potential improvement if one of
+the capability is enhanced. It implies which of the
+two capabilities of r is currently more deficient and
+thus informs future development direction for the
+agent. For example, if PPGsearch(r) is large and
+PPGToM(r) is small, it means that the evaluated
+agent is scoring the candidate instructions highly
+accurately but it is bad at finding high-score in-
+structions. In this case, developers may want to
+focus on devising a more effective inference al-
+gorithm for the agent. On the other hand, if the
+agent’s estimated scores are poorly calibrated, sig-
+nified by PPGToM(r) being large, building a bet-
+ter planning module that simulates the listener’s
+behavior more accurately would yield significant
+performance boost.
+5
+Improving ToM Capability with
+Ensemble Instruction-Following
+Agents
+We improve the ToM capability of an agent r by
+turning it into a bounded pragmatic agent that uses
+the original model Sr as the base speaker but is
+equipped with a better ToM model than Sr. A com-
+mon approach for building a ToM model is to learn
+an instruction-following policy ˆπ(a | u, ¯e) using
+the same dataset used for learning Sr (Andreas and
+Klein, 2016; Fried et al., 2017, 2018).
+We argue that this approach has a potential draw-
+back. A ToM model learned in this way is only
+exposed to human-generated input instructions. At
+deployment time, it would likely experience a co-
+variate shift because as a ToM model, the model
+is then asked to score instructions generated by
+a speaker model, not by humans. These instruc-
+tions may be incorrect, ungrammatical, or may sim-
+ply have a different style than human-generated
+instructions. This covariate shift would hamper the
+model’s judgement. Our preliminary experiments
+(Appendix § A.5) confirms that using a listener
+trained on only human-generated inputs as the ToM
+model hurts rather than improves the performance
+of various speakers.
+We show that this problem can be alleviated by
+employing ToM models that have calibrated uncer-
+tainty on unseen instructions. We obtain calibrated
+models through ensembling (Lakshminarayanan
+et al., 2017). Specifically, we randomly draw K
+90%-samples of the training dataset. We use each
+sample to train an instruction-following policy
+ˆπ(k)(a | u, ¯e); the policies are also initialized with
+different random seeds.
+When the agent has access to a simulation of
+the environment, it can leverage the simulation to
+construct better ToM models. Note that the proba-
+bility that a ToM model LToM assigns to an instruc-
+tion can be seen as an expectation of a 0-1 metric:
+LToM(e⋆ | u) = Ee∼LToM(·|u) [1{e = e⋆}], which
+does not award partial credit if e partially overlaps
+with e⋆. We make two changes: (i) replace the 0-1
+metric with a soft metric Ψ(e, e⋆) that can measure
+partial similarity between trajectories and (ii) ap-
+
+proximate the expectation by executing instruction-
+following policies ˆπ(k) in the environment to sam-
+ple trajectories. Our final ToM-augmented agent
+selects its instruction as follows
+ˆuaugment-ToM ≜ arg max
+u∈Ur
+sub
+LToM(u, e⋆)
+(13)
+LToM(u, e⋆) ≜
+1
+KM
+K
+�
+k=1
+M
+�
+j=1
+Ψ(ej(ˆπ(k), u), e⋆)
+Ur
+sub ≜ {ˆuinfer} ∪ {ui ∼ Sr | 1 ≤ i ≤ N}
+where e(π, u) denotes a trajectory obtained by con-
+tinuously sampling actions from a policy π condi-
+tioned on an instruction u.
+6
+Experimental Setup
+6.1
+Environment and Dataset
+We setup an instruction generation problem in 3D
+environments using the Matterport3D simulator
+(Anderson et al., 2018b). The simulator photo-
+realistically emulates the visual perception of a
+person walking in an indoor environment. Travel-
+ing in an environment is simulated as traversing in
+a graph where each node corresponds to a location.
+At any location, an agent is provided with RGB
+images capturing the 360-degree panoramic view
+when looking from that location.
+We train our speaker and listener models
+using the Room-to-Room (R2R) dataset which
+accompanies the simulator. The R2R dataset was
+originally created for training instruction-following
+agents. Each data point was collected by asking
+a crowd-worker to write a verbal description of a
+path in an environment. In the end, each path was
+annotated with three instructions. Each instruction
+contains 29 words on average. The dataset is split
+into a training set (61 environments, 4,675 paths),
+a seen validation set (340 paths) whose paths
+are sampled in the training environments, and an
+unseen validation set (11 environments unseen
+during training, 783 paths).
+We train the models using the training set and
+validate them on the unseen validation set for
+model selection. The final performance metrics
+are computed on the seen validation set.
+6.2
+Speaker Models
+We evaluate three speaker model architectures. The
+first is a GPT-2 model pre-trained on text (Radford
+et al., 2019) and fine-tuned on the R2R training
+set. The other two models are encoder-decoders:
+one implements an LSTM architecture similar to
+(Shen et al., 2022), and the other is based on a
+Transformer architecture (Vaswani et al., 2017).
+The parameters of these two models are randomly
+initialized.
+Training.
+We train the speakers with a standard
+maximum-likelihood objective using the AdamW
+optimizer (Loshchilov and Hutter, 2019) with
+a learning rate of 10−4.
+More detailed model
+implementation and hyperparameters are provided
+in § A.1.
+During training, we select the best
+model based on the unseen-validation BLEU score
+(Papineni et al., 2002) of the model-generated
+instructions with the respect to the ground-truth
+instructions.
+6.3
+Human Evaluation
+We evaluate each speaker model on 75 paths in
+the unseen validation data split. In the end, we
+have annotated 1,200 instructions generated by 16
+different systems (humans, 3 speaker models, and
+their ablated and augmented versions).
+To evaluate a speaker model, we present its gen-
+erated instructions to a human annotator and ask
+them to follow the instructions to navigate in Mat-
+terport3D environments. We adapt the PanGEA
+tool2 to setup a web navigation interface and cre-
+ate a task on Amazon Mechanical Turk (MTurk)
+to recruit human evaluators. We pay the evaluator
+$5.20 per task which takes about 25 minutes. For
+each evaluation task, we ask the human evaluator
+to follow six instruction-following sessions.
+Quality Assurance.
+One of the six sessions,
+which appears in all tasks, is a quality-control test
+featuring an easy-to-follow human-written instruc-
+tion. We only approve an evaluator if they navigate
+successfully to the goal destination in this test. Fol-
+lowing Zhao et al. (2021), we instruct the judges
+to not explore the environments unnecessarily and
+not wander back and forth unless they are lost. We
+record the trajectories created by the human and use
+them to compute the performance metrics. More
+details about the crowd-sourcing interface are given
+in Appendix §A.4.
+Performance Metrics.
+The quality of a speaker
+is determined by the similarity between the in-
+tended trajectory and the actual trajectories that the
+2https://github.com/google-research/
+pangea
+
+speaker’s instructions induce the human evaluators
+to generate. We compute these similarity metrics:
+• Success rate (SR) averages binary indicators
+of whether the final location of a human-
+generated trajectory is within 3 meters of the
+final location of the intended trajectory;
+• SPL (Anderson et al., 2018a) weights the suc-
+cess indicator with the ratio between the in-
+tended traveling distance and the actual one;
+• NDTW and SDTW are metrics based on dy-
+namic time-warping alignment (Magalhaes
+et al., 2019), capturing the similarity between
+two point sequences. NDTW computes only
+a sequence similarity score while SDTW
+weights the score with the success indicator.
+7
+Experiments
+We investigate the following questions:
+(a) How well do the speakers perform on our
+problem? We find that, while implementing
+advanced model architectures, these speakers
+perform poorly compared to human speakers.
+(b) What causes their performance deficiency?
+Using our evaluation scheme, we identify that
+the speakers possess decent search capability
+but inadequate ToM capability.
+(c) Can we improve the speakers by equip-
+ping them with better ToM models?
+We
+train ensembles of state-of-the-art instruction-
+following agents to serve as the ToM models
+for the speakers, and obtain significant im-
+provements.
+(d) What are the challenges in bridging the
+performance gap with human speakers?
+We show that state-of-the-art instruction-
+following agents are not optimally trained to
+serve as ToM models because they are mostly
+trained to predict how humans follow human-
+generated instructions, but as ToM models,
+they are required to accurately predict how
+humans follow model-generated instructions.
+How well do the speakers perform on our prob-
+lem?
+Figure 3 shows the performance of the three
+speaker models on variety of metrics. We also eval-
+uate the human-written instructions provided by
+the R2R dataset. Overall, there is a wide mar-
+gin between the models and the humans.
+The
+best model speaker (EncDec-Transformer) lags be-
+hind the humans by 21.6 NDTW points. We find
+that the encoder-decoder architecture with cross-
+attention of EncDec-Transformer outperforms the
+0
+10
+20
+30
+40
+50
+60
+70
+80
+NDTW
+SR
+SPL
+SDTW
+Metric Value
+Fine-tuned GPT-2
+EncDec-LSTM
+EncDec-Transformer
+Humans
+Figure 3: Performance of different speakers on held-
+out evaluation data. There is a considerable gap be-
+tween model and humans speakers.
++6.7
++1.2
++3.9
++35.2
++30.9
++25.8
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Fine-tuned GPT-2
+EncDec-LSTM
+EncDec-Transformer
+NDTW
+Original
+with Oracle Search
+with Oracle ToM
+Figure 4:
+Performance of the speakers and their
+human-augmented versions. Possessing human-level
+ToM capability improves performance of the speakers,
+showing that their original ToM capability is highly
+deficient compared to that of humans.
+decoder-only self-attention architecture of GPT-
+2 (+11.7 NDTW), indicating that fusing the vi-
+sion and language features too early in an archi-
+tecture may be detrimental. On the other hand,
+EncDec-Transformer leads over EncDec-LSTM by
+4.1 points, suggesting that the Transformer architec-
+ture is more effective than LSTM in this problem.
+What causes the speakers’ performance defi-
+ciency?
+Next, we investigate whether the lack of
+search or ToM capability is responsible for the per-
+formance deficiency of the speakers. Following our
+evaluation scheme, we compute the prospective per-
+formance gains when one of the capabilities were
+made optimal. The results presented in Figure 4
+show that it is an under-performed ToM capability
+that primarily causes the models to perform poorly.
+While equipping the models with optimal search
+capability only improves their performance by 30%
+on average, granting them optimal ToM capability
+nearly doubles their performance metrics. In fact,
+the search capability of the models is already as
+good as that of the humans we employ, because the
+models with optimal ToM capability achieve even
+
+Base speaker Sbase
+ToM listener LToM
+Fine-tuned GPT-2
+EncDec-LSTM
+EncDec-Transformer
+None
+37.7
+(▲ 0.0)
+45.3
+(▲ 0.0)
+49.4
+(▲ 0.0)
+Single VLN-BERT (Majumdar et al., 2020)
+38.9
+(▲ 1.2)
+39.8
+(▼ 5.5)
+46.2 (▼ 3.2)
+Ensemble of 10 EnvDrop-CLIP (Shen et al., 2022)
+37.8
+(▲ 0.1)
+53.1† (▲ 7.8)
+57.3† (▲ 7.9)
+Ensemble of 10 VLN
+⟳
+BERT (Hong et al., 2021)
+43.4
+(▲ 5.7)
+56.4‡ (▲ 11.1)
+54.2
+(▲ 4.8)
+Humans (skyline)
+72.9‡ (▲ 35.2)
+76.2‡ (▲ 30.9)
+75.2‡ (▲ 25.8)
+Table 1: Performance of the speakers when equipped with different ToM models. Employing ensemble instruction-
+following agents significantly improves their performance.
+‡ and † indicate results that are significantly higher
+than those of the corresponding “None” baseline (row 1) with p < 0.05 and p < 0.1, respectively (according to a
+two-related-sample t-test).
+Listener
+Instructions generated by
+VLN-BERT
+EnvDrop-CLIP
+VLN
+⟳
+BERT
+Humans (R2R dataset)
+65.4 (▼ 0.0)
+47.2 (▼ 0.0)
+65.0 (▼ 0.0)
+Fine-tuned GPT-2
+43.1‡ (▼ 22.3)
+31.6‡ (▼ 15.6)
+39.9‡ (▼ 25.1)
+EncDec-LSTM
+50.0‡ (▼ 15.4)
+43.7 (▼ 3.5)
+49.3‡ (▼ 15.7)
+EncDec-Transformer
+52.1‡ (▼ 13.3)
+41.5 (▼ 5.5)
+51.9‡ (▼ 13.1)
+Table 2: Agreement of human and model listeners on instructions generated by different speakers. The level
+of agreement decreases substantially when shifting from human-generated to model-generated instructions.
+‡
+indicate results that are significantly lower than the human skyline (row 1) with p < 0.05 (according to a
+two-related-sample t-test).
+slightly higher SDTW score than the human speak-
+ers (e.g., 75.2 of EncDec-Transformer compared
+to 71.0 of humans), though the differences are not
+statistically significant.
+Can we improve the speakers by equipping
+them with better ToM models?
+Following the
+procedure described in Section §5, we train var-
+ious state-of-the-art instruction-following agents
+to serve as ToM listener models for the speakers.
+These listeners are trained using maximum log-
+likelihood on the same data as the speakers. Perfor-
+mances of different combinations of speakers and
+listeners are given in Table 1. We attain the largest
+improvement of 7.9 NDTW points over the best
+base speaker (EncDec-Transformer) by augment-
+ing this speaker with an ensemble of 10 EnvDrop-
+CLIP listeners as the ToM model. We observe that
+ensemble models consistently outperform single
+models. More results about the detrimental effects
+of using single listeners on the speakers is given in
+Appendix §A.5. Despite the promising improve-
+ments, there remains a large gap of 17.9 NDTW
+points between our best speaker and the human
+speakers.
+What are the challenges in bridging the perfor-
+mance gap with human speakers?
+In the previ-
+ous set of experiments, a notable pattern emerges:
+the performance superiority of a listener on the
+R2R instruction-following problem, where it is
+asked to follow human-generated instruction, does
+not translate into a superiority in serving as a ToM
+model, where it is asked to rank model-generated
+instructions. To further illustrate this phenomenon,
+we measure the agreement between human listen-
+ers and model listeners on instructions generated
+by different speakers. We define the agreement
+score between a human Lh and a model ˆL as
+Agreement(Lh, ˆL)
+(14)
+= Averageu∈Deval (NDTW(eh(u), ˆe(u))) (15)
+where eh(u) and ˆe(u) are the trajectories gener-
+ated by Lh and ˆL given u, respectively, and Deval
+denotes the R2R seen validation set.
+As seen from Table 2, the listener agents agree
+more with the humans on human-generated instruc-
+tions than on model-generated ones. These re-
+sults can be explained through the lens of training-
+deployment covariate shift: during training, the
+model listeners are only trained to agree with hu-
+man listeners on human-generated instructions and
+
+thus does not know how to behave properly on
+other types of instructions.
+8
+Conclusion
+This work introduces a framework for analyzing of
+the cognitive capabilities of instruction generation
+agents. Our analysis highlights the necessity of
+constructing better ToM models for these agents.
+We argue that learning ToM models is faced with
+challenges that are distinct from those of learning
+instruction-following agents. We hope that our find-
+ings will motivate the community to create novel
+datasets, training methods, and evaluation proce-
+dures for tackling this problem.
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+Patrick Shafto. 2020. A mathematical theory of co-
+operative communication. Advances in Neural In-
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+ter Anderson. 2021.
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+
+Rowan Zellers, Yonatan Bisk, Ali Farhadi, and Yejin
+Choi. 2019.
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+sual commonsense reasoning. In Proceedings of the
+IEEE/CVF conference on computer vision and pat-
+tern recognition, pages 6720–6731.
+Ming Zhao, Peter Anderson, Vihan Jain, Su Wang,
+Alexander
+Ku,
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+Baldridge,
+and
+Eugene
+Ie. 2021.
+On the evaluation of vision-and-
+language navigation instructions.
+arXiv preprint
+arXiv:2101.10504.
+Hyperparam
+GPT-2
+Transformer
+Learning rate
+10−4
+10−4
+Batch size
+4
+32
+Optimizer
+AdamW
+AdamW
+Num. of training iterations
+2 × 105
+16 × 104
+Max. action steps
+15
+35
+Max. instruction length
+100
+80
+Image feature size
+2048
+512
+Orientation feature size
+128
+128
+Embedding dropout
+0.1
+0.3
+Hidden size
+768
+512
+Num. of hidden layers
+1
+1
+Hidden-layer dropout rate
+0.0
+0.6
+Num. of encoder layers
+-
+2
+Num. of decoder layers
+12
+2
+Transformer dropout rate
+0.1
+0.3
+Beam size
+5
+1
+Table 3:
+Hyperparameters for training the GPT-2
+EncDec-Transformer speakers.
+A
+Appendices
+A.1
+Implementation of Speaker Models
+The speaker models take a sequence of visual
+observations and actions from the trajectory
+e⋆ as input and output a text instruction u.
+The model is trained to estimate conditional
+probability Sθ(u|e⋆).
+The model and training
+hyperparameters are listed in Table Table 3.
+Input.
+The input trajectory e⋆ is a sequence of
+panoramic views and actions. Each panoramic
+view at time step t is represented by 36 vectors
+{ot,i}36
+i=1, each of which is a visual feature vec-
+tor extracted from a pre-trained vision model con-
+catenated with orientation features describing the
+agent’s current gaze direction. The image features
+of the GPT-2 model are extracted from a ResNet-
+152 model (He et al., 2016), whereas those of the
+encoder-decoder models are from a CLIP model
+(Radford et al., 2021). Each ground truth action
+a⋆
+t , which moves the agent to an adjacent location,
+is represented by image features from the gaze
+direction of the agent when looking towards that
+adjacent location, and orientation features captur-
+ing the direction of the adjacent location relative to
+the agent’s current gaze direction.
+Output.
+The output of a speaker model is a lan-
+guage instruction describing the input trajectory. At
+test time, the GPT-2 model employs beam search,
+
+Performance Metrics
+Speaker
+SR ↑
+SPL ↑
+NDTW ↑
+SDTW ↑
+Path Len ↓
+Interpretability ↑
+Finetuned GPT-2
+36.0
+27.8
+37.7
+24.5
+20.9
+2.9
+EncDec-LSTM
+49.3
+37.6
+45.3
+33.8
+17.4
+3.3
+EncDec-Transformer
+54.7
+43.8
+49.4
+40.4
+15.8
+3.4
+Humans (R2R dataset)
+76.0
+67.6
+71.0
+64.8
+14.2
+3.6
+Table 4: Humans evaluation results on instructions generated by the speaker models. The similarity metrics are
+defined in § 6.3. Path Len measures the average length of the generated trajectories. Interpretability indicates
+how easy or difficult to follow the instructions according to human evaluators (without knowing the ground-truth
+trajectory).
+and the encoder-decoder models generate instruc-
+tions via greedy decoding (Shen et al., 2022).
+Training Objective.
+We train the speakers with
+maximum-likelihood objective:
+max
+θ
+�
+(u⋆,e⋆)∈Dtrain
+|u⋆|
+�
+t=1
+log Sθ(u⋆
+t | e⋆, u⋆
+<t) (16)
+where θ is the speaker model parameters, u⋆
+t is t-th
+word of the ground-truth instruction, and u⋆
+<t is the
+first t − 1 words of the instruction.
+A.2
+Fine-tuning GPT-2 Speaker Model
+To represent the trajectory features as a sequence
+of feature vectors to feed into the GPT-2 model, we
+first average the view features ¯ot for each time step:
+¯ot = 1
+36
+36
+�
+i=1
+ot,i
+(17)
+We compute the input features e⋆
+t by concatenat-
+ing the panoramic view features and ground truth
+action features:
+e⋆
+t = [¯ot; a⋆
+t ]
+(18)
+The sequence of feature vectors e⋆ representing
+a trajectory is calculated as follows
+e⋆ = [tanh(e⋆
+1W); · · · ; tanh(e⋆
+T W)]
+(19)
+where W is parameters of a linear layer.
+For the instruction u⋆, we perform an embed-
+ding look-up of its words. Then, we first prompt
+the model with e⋆ and then train it to generate u⋆
+as a suffix.
+A.3
+Training Encoder-Decoder Models
+Our EncDec-LSTM model follows the implementa-
+tion of the speaker in Shen et al. (2022). We imple-
+ment the EncDec-Transformer model by replacing
+the LSTM layers of the speaker model described in
+Tan et al. (2019) with Transformer layers (Vaswani
+et al., 2017).
+A.4
+Human Evaluation Interface
+Figure 5 shows the interface for our human evalu-
+ation. After a human evaluator finishes following
+an instruction, we recorded the path they generate
+and compute similarity metrics with respect to the
+ground-truth path. After the instruction-following
+sessions, we ask each evaluator to assess the inter-
+pretability of the instructions by asking them how
+easy (or difficult) it was for them to follow the in-
+struction. We provide four rating levels ranging
+from “1: I couldn’t follow any part of the instruc-
+tion” to “4: very easy, the instructions gave accu-
+rate and sufficient information for me to follow”.
+The answer of the evaluators is converted to a score
+between one and four.
+Table 4 shows the human evaluation results of
+the three speaker models we evaluated.
+A.5
+Single vs. Ensemble Listeners
+As a preliminary experiment, we compare the ef-
+fectiveness of a single and an ensemble of 10
+VLN
+⟳
+BERT agents when serving as the ToM
+model of a speaker. Results in Figure 6 show that
+the ensemble listener is significantly better than the
+single listener for two different speakers.
+
+Figure 5: Human evaluation interface.
+SR
+SDTW
+NDTW
+SPL
+0
+15
+30
+45
+60
+Single ToM Listener
+Ensemble ToM Listeners
+Fine-tuned GPT-2 Speaker
+SR
+SDTW
+NDTW
+SPL
+0
+11.5
+23
+34.5
+46
+Single ToM Listener
+Ensemble ToM Listeners
+EncDec-LSTM Speaker
+Figure 6: Comparison of single and ensemble ToM lis-
+teners.
+
+TiPS: Hold and drag mouse to rotate current view. Double-clickto move. The YELLOW
+square indicates the next location you would be moving towards.
+You will be evaluating instruction #1992. If this number does not match the number after
+'?id=' in the page's link, please refresh the page after clearing your browser's caches and
+cookies.
+Instructions to be followed:
+Walk out of the living room towards the stairs, between the couch and the
+sitting area. Go up the three small stairs and stop at the top of the stairs.
+How easy was it to follow the instructions?
+O Very easy, the instructions gave accurate and sufficient information for me to follow
+O I could follow most of the instructions, but some minor parts were wrong or missing
+O I couldn't follow at least half of the instructions
+O I couldn't follow any part of the instruction
+Mechanical Turk Woker ID: Enter Worker ID
+Please close the tab ONLY after you see a green line indicating that your answer has been
+received.
+Submit

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+Astronomy & Astrophysics manuscript no. 44846corr
+©ESO 2023
+January 4, 2023
+Sub-Jovian desert of exoplanets at its boundaries
+Parameter dependence along the main sequence
+Gy. M. Szabó1, 2, 3 , Sz. Kálmán2, 4, 5, 6, 7 , L. Borsato
+8, V. Heged˝us
+3, 5, Sz. Mészáros1, 3, and R. Szabó4, 7, 9, 10
+1 ELTE Eötvös Loránd University, Gothard Astrophysical Observatory, Szombathely, Szent Imre h. u. 112., H-9700, Hungary
+2 MTA-ELTE Exoplanet Research Group, Szombathely, Szent Imre h. u. 112., H-9700, Hungary
+3 MTA-ELTE Lendület "Momentum" Milky Way Research Group, Hungary
+4 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, ELKH, Budapest, Konkoly-Thege Miklós út 15–17.,
+H-1121, Hungary
+5 ELTE Eötvös Loránd University, Doctoral School of Physics, Budapest, Pázmány Péter sétány 1/A, H-1117, Hungary
+6 Graduate School of Physics, University of Szeged, Szeged, Dóm tér 9., H-6720, Hungary
+7 CSFK, MTA Centre of Excellence, Budapest, Konkoly Thege Miklós út 15-17., H-1121, Hungary
+e-mail: [email protected]
+8 INAF-Osservatorio Astronomico di Padova Vicolo dell’Osservatorio 5, I-35122, Padova, Italy
+9 Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány 1/A, H-1171 Budapest, Hungary
+10 MTA CSFK Lendület Near-Field Cosmology Research Group
+Recieved ....; accepted ....
+ABSTRACT
+Context. The lack of sub-Jovian planets on orbits of Porb < 3 days is a puzzling aspect of galaxy formation with regard to the
+distribution of exoplanets whose origins are currently unresolved.
+Aims. The possible explanations behind the formation of the sub-Jovian or Neptunian desert include several scenarios that can lead to
+different shapes for the boundary, predicting various dependencies between the position of the boundary and the stellar parameters.
+Methods. We explored the exoplanet distribution in various 2D and 3D projections, revealing the stellar-dependent substructures in
+the Porb–MP and the Porb–RP parameter plane.
+Results. We demonstrate that the upper boundary includes a range of planets, namely, inflated hot Jupiters and normal hot Jupiters,
+in the two parameter planes, respectively. We confirm the dependence of the boundary on several stellar parameters and, based on a
+fuzzy clustering analysis, we provide quantitative formulae for the dependencies in groups of smaller and larger planets. The overall
+period-radius distribution shows chemical substructures as well, with the boundary being dependent on volatiles and alpha-elements,
+alongside marginal (to none) dependence found for refractory elements.
+Conclusions. These findings confirm multiple plausible causes for the formation of the desert, particularly preferring those scenarios
+related to the irradiation-driven loss of the atmospheres of moderately massive planets as the predominant process in shaping planetary
+distributions.
+Key words. Methods: statistical – Planets and satellites: formation – Astronomical databases: miscellaneous
+1. Introduction
+The observed absence of short-period planets (≲ 3 d) below the
+hot Jupiter clump and above the hot super-Earths is known as the
+“sub-Jovian desert” (Szabó & Kiss 2011; Benítez-Llambay et al.
+2011; Sanchis-Ojeda et al. 2014; Colón et al. 2015; Matsakos &
+Königl 2016; Eigmüller et al. 2017; Owen & Lai 2018; Szabó
+& Kálmán 2019) and “Neptunian desert” (Mazeh et al. 2016;
+Demangeon et al. 2018; Ionov et al. 2018; Mori et al. 2022). In
+recent years, several planets have been observed in these desert
+regions (Demangeon et al. 2018; Dragomir et al. 2020; Jenkins
+et al. 2020; Armstrong et al. 2020; Murgas et al. 2021; Mori
+et al. 2022), with some deserts possibly undergoing a conversion
+into a “savanna.” In this paper, we describe the desert-savanna
+boundaries in the parameter space.
+There is a known planet population both above and below
+its boundaries, therefore, the sub-Jovian or Neptunian desert or
+savanna itself is a puzzling structure of planetary distribution.
+The location of the desert is a meeting point for high-energy
+physics and atmospheric non-local thermodynamic equilibrium
+(NLTE) processes (representing the stellar activity and irradia-
+tion), planetary thermodynamics, and magneto-hydrodynamical
+interactions between the planet and the star, end-points of plane-
+tary migration, accretion processes in general, and planet forma-
+tion and evolution in extreme environments as well. Therefore,
+a test for planet formation theories is to check how they can ex-
+plain the presence of the desert and its boundaries.
+In Fig.1, we summarize the possible paths of how a young
+planet, once formed, can leave the desert. The indicative direc-
+tion of “leaving the desert” is denoted by numbers in the figure.
+The processes suggested for the different scenarios thus far in-
+clude: 1) the hyperinflation of low-mass gas giants at the bound-
+ary (Mordasini et al. 2015); 2) planet-growth-based processes,
+namely, it is has been well established that hot Jupiters have
+higher-than-expected (i.e., inflated) radii (Thorngren & Fortney
+2018), which can be attributed to irradiation from the host (e.g.,
+Sarkis et al. 2021). As the desert is observed in both the RP–P
+and MP – P planes (and at sub-Jovian planet sizes), these pro-
+cesses would necessarily have to include accretion as well as
+Article number, page 1 of 11
+arXiv:2301.01065v1  [astro-ph.EP]  3 Jan 2023
+
+IDIDIDIDIDA&A proofs: manuscript no. 44846corr
+1
+3
+4
+2
+5
+6
+-1
+0
+1
+2
+3
+-1.2
+-0.8
+-0.4
+0.0
+0.4
+log Rp/RJ
+log P [d]
+Fig. 1. Sketch of possible paths for already formed planets leaving the
+sub-Jovian desert (red arrows) and planets not allowed to reach the
+desert (blue arrows) overplotted to the distribution of known exoplan-
+ets. Theories and references behind the scenarios are summarized in the
+text.
+inflation. The theoretical calculations done so far (Emsenhuber
+et al. 2021a,b) were insufficient to explore this scenario in de-
+tail; 3) high-eccentricity migration (Giacalone et al. 2017; Owen
+& Lai 2018) suggests that the orbits of close-in planets with
+initially high eccentricities circularize with higher orbital peri-
+ods; 4) decreasing planet size can be the result of photoevapo-
+ration (Lopez & Fortney 2014; Lundkvist et al. 2016; Owen &
+Lai 2018), Roche-lobe-overflow (Kurokawa & Nakamoto 2014),
+“boil-off” (Owen & Wu 2016), and impact-based mass-loss
+(Schlichting et al. 2015; Schlichting & Mukhopadhyay 2018).
+Finally, (5) the tidal migration of hot Neptunes (Lin & Pa-
+paloizou 1986; Rozner et al. 2022), tidally trapped outward mi-
+gration (Masset et al. 2006), and disk migration (Mazeh et al.
+2016; Ataiee & Kley 2021) are other alternative scenarios.
+Besides scenarios involving migration and evaporation, Bai-
+ley & Batygin (2018) found that an accretion parameter corre-
+lated with planet mass and disk inner edge can explain the upper
+right side of the desert and explains its suggested slope of −2/7.
+Ataiee & Kley (2021) suggested a further possibility, where or-
+bital resonances change the orbital periods of the inner planets.
+The curious case of TOI-2196 b (Persson et al. 2022) would fit in
+well with migration-based scenarios, being a small and volatile-
+rich planet present in the savanna.
+Mazeh et al. (2016) derived relationships for the lower and
+upper boundaries of the desert in the RP–P plane finding that at
+the upper edge, RP ∝ P− 1
+3 , and at the lower edge, RP ∝ P
+2
+3 , thus
+suggesting different origins in the two size-regimes. Matsakos &
+Königl (2016) explained the two boundaries as the tidal disrup-
+tion barrier for gas giants following their high-eccentricity mi-
+grations and because of different mass-radius laws of low-mass
+and high-mass exoplanets, the two edges are formed differently.
+Owen & Lai (2018) suggested that the photoevaporation and the
+high-eccentricity migration to form the upper and lower bound-
+aries, respectively. Owen (2019) summarizes the processes that
+could have formed the desert boundaries from an evaporation
+approach, with a very detailed introduction to the astrophysical
+processes. A somewhat similar structure in the period-radius dis-
+tribution is related to the observed increasing sub-Saturn (≈ 4–
+8 R⊕) occurrence rate up to ∼300 days orbital period. Hallatt &
+Lee (2022) argued for a radiation-induced process (resulting in
+atmoshperic mass-loss), which is efficient at transforming sub-
+Saturns into sub-Neptunes (≲4 R⊕), with short orbital periods.
+The short-period end of this process is similar to the “process 3”
+in our Fig.1 and could explain the lower boundary of the desert
+as well.
+In our earlier paper Szabó & Kálmán (2019), we found that
+the boundaries of the desert depend more on fundamental stellar
+parameters and less on planetary mass, density, and size. In that
+work, the sample of exoplanets were smaller, which limited the
+further inference. In this paper, we re-apply the same analyses
+to the larger exoplanet data set at hand and we include a few
+new tests as well. We show that the boundaries of the desert in
+the period-mass (P – MP) and the period-radius (P – RP) planes
+are marked by a different planet population. The desert in the
+two main projections can be considered as a sign of multiple
+scenarios of planetary evolution acting differently in the various
+parameter spaces. Therefore, we revisited the P–MP and P–RP
+boundaries separately, which allowed for the findings of Szabó
+& Kálmán (2019) to be quantified. The dependence of the P–MP
+and P–RP boundaries on the stellar parameters are to be consid-
+ered as another piece of evidence to support this assertion. Fi-
+nally, we give expressions for both the planetary radius and mass
+in terms of stellar parameters and via fuzzy clustering. We con-
+clude that in all projections, planets form two main groups that
+behave differently, in a possible connection with the unresolved
+processes behind the formation of the sub-Jovian or Neptunian
+desert.
+2. Methods and data selection
+2.1. Data on confirmed planets from the NASA exoplanet
+archive
+The input to this analysis consists of two data sets. We followed
+Szabó & Kálmán (2019) in building up the sample from the
+NASA Exoplanet Archive (5009 planets in total1), following the
+same filtering as in Szabó & Kálmán (2019). This filtering has
+left out planets with less constrained parameters and kept the
+“golden sample” only. In building up the test sample, we defined
+the following selection criteria: (i) planetary mass MP < 13MJ
+and density ρP < 25ρJ and (ii) relative uncertainty for the plan-
+etary radius and mass of < 20% and < 60%, respectively, leav-
+ing us with a sample of 650 exoplanets. This is an increment
+with regard to the 406 planets in our previous analysis (Szabó
+& Kálmán 2019) and our current sample also represents a better
+quality because it also contains the filtering for the mass deter-
+mination errors. We accepted the stellar parameters provided in
+the original NASA Exoplanet Archive records.
+In Fig. 1, we show the distribution of our filtered sample
+(of 650 exoplanets) in the RP–P plane. Figures 2–8 show the
+same sample in both the radius-period and mass-period param-
+eter spaces. This design enables the visualization of the exact
+values of the continous variable, but has the limitation of being
+merely qualitative. To be able to quantitatively see the structures
+in the sample, we applied two-sampled Kolmogorov-Smirnov
+(KS) tests (see e.g., Feigelson & Babu 2012) to the projected
+parameters in different regions (univariate bands) of the data dis-
+tribution.
+1 The data were obtained from the Planetary Systems Compos-
+ite Data database https://exoplanetarchive.ipac.caltech.
+edu/cgi-bin/TblView/nph-tblView?app=ExoTbls&config=
+PSCompPars on 5 April 2022.
+Article number, page 2 of 11
+
+Gy. M. Szabó
+et al.: Sub-Jovian desert of exoplanets at its boundaries
+Still following Szabó & Kálmán (2019), we defined two sta-
+tistical regions in the RP – P plane as: the area between 0.28 and
+0.63 RJ is labeled A, and the one between 0.06 and 0.16RJ is
+labeled B. In the MP – P plane, we defined the C region between
+0.032 − 0.305 MJ and the D region between 0.001 − 0.010 MJ.
+To observe whether a stellar or planetary parameter is selec-
+tive at the position of the border (in other words, the position of
+the border depends on the examined parameter), we compared
+the period distribution of two subsamples in A to those in B; and
+two subsamples in C to those in D. The subsamples were sepa-
+rated at the median of the examined parameter. For example, we
+compared the period distribution of high-temperature and low-
+temperature stars in A and B, etc. We compared the resulting pe-
+riod distributions in a two-sampled KS test in all four regions and
+computed the p values. If the resulting p is small, we can con-
+clude that the two compared samples are drawn from different
+distributions; or, in other words, the third parameter can affect
+the distribution of exoplanets. Because regions A and C are in
+the desert and B and D are outside, this comparison helps iden-
+tify those parameters that affect the formation and the boundaries
+of the desert.
+Following the technique of colored scatter plots, we some-
+times identified several groups of exoplanets following a spe-
+cific pattern. These features can be proven with clustering meth-
+ods. Here, we followed the fuzzy clustering algorithm of Bezdek
+(1981), as implemented in the FKM function in the R software
+package2.
+2.2. APOGEE catalog of planet-host stars
+The second database we used for the analyses is based
+on Apache Point Observatory Galactic Evolution Experiment
+(APOGEE, Majewski et al. 2017), which provides homoge-
+neously derived atmospheric stellar parameters and eliminates
+uncertainties originating from the usage of different model atmo-
+spheres, spectral synthesis codes, line lists, and so on. We built
+an APOGEE exoplanet host stars catalog to test the findings on
+this homogeneous data set and we plot the data in Figs. 10-11.
+As part of the fourth iteration of Sloan Digital Sky Sur-
+vey (SDSS-IV,
+Blanton et al. 2017), in its final data release
+(DR17, Abdurro’uf et al. 2022), the APOGEE-2 survey derived
+and reported main atmospheric parameters and individual ele-
+ment abundances for 733,901 stars across the entire sky (Za-
+sowski et al. 2017) and focused on observing as many planet-
+host stars as possible, so it is an excellent choice to cross-match
+our sample with that of DR17. Observations were made with the
+identical SDSS spectrographs (Wilson et al. 2019) mounted on
+the 2.5-meter Sloan Foundation Telescope (Gunn et al. 2006)
+and the 2.5-meter Irénée du Pont Telescope of Las Campanas
+Observatory, Chile. The resolving power is R ∼ 22, 500 and the
+spectral coverage of the near-infrared H-band – from 15140 Å to
+16940 Å. This makes it possible to select planet-host stars near
+the Milky Way disk and dust obscured regions of the Milky Way.
+Besides observing stellar targets, APOGEE is sensitive to sub-
+stellar mass companions (exoplanet candidates) too, as multi-
+epoch visits along with high-precision RV measurements are
+available (Majewski et al. 2017).
+The APOGEE DR17 collaboration published both raw and
+calibrated values of effective temperature, surface gravity, and
+applied offset corrections to [M/H] based on solar neighbor-
+2 R Core Team (2017). R: A language and environment for statistical
+computing. R Foundation for Statistical Computing, Vienna, Austria.
+https://www.R-project.org/
+-1.0
+-0.5
+0.0
+log Rp/RJ
+-1
+0
+1
+2
+3
+4
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+log P [d]
+-1.2
+-0.8
+-0.4
+0.0
+0.4
+0.8
+1.2
+log ρP/ρJ
+Fig. 2. Sub-Jovian/Neptune desert of exoplanets in the RP – P and MP
+– P parameter spaces (top and bottom panel), colored by the planetary
+density. In the top panel, the upper boundary is marked by “normal”
+Jupiters with higher density (> 1ρJ), while in the bottom panel, the pop-
+ulation of inflated planets (red points) appear below the normal Jupiters
+and, thus, the inflated Jupiters mark the actual boundary.
+hood stars. Here, we take the calibrated effective temperatures
+and overall metallicities into account as Teff is in a better agree-
+ment with the infrared flux method (IRFM) scale than the raw
+spectroscopic values (Abdurro’uf et al. 2022). For this analy-
+sis, we carefully select the most accurate and precise APOGEE
+data. We globally filter out APOGEE stars labeled by the quality
+control flag STAR_BAD3 (Abdurro’uf et al. 2022) (but retained
+STAR_WARN), then we cross-matched the exoplanet catalog with
+the APOGEE DR17 data set in TOPCAT version 4.8-2 (Tool for
+OPerations on Catalogues And Tables, Taylor 2005) based on
+the equatorial coordinates of the host stars (∆ < 1 arcsec). The
+next step involved the implementation of several quality criteria:
+relative error lower than 20% for RP, VSCATTER < 1 km/s for
+RV variations, VERR < 1 km/s for RV error to filter out poten-
+tial variable stars, and M_H_ERR < 0.1 dex for the metallicity
+uncertainty.
+We note that the selected elements fall into the APOGEE
+data reduction categories of “most reliable” and “reliable”
+with regards to the overall quailty of their derivation (see the
+APOGEE DR17 website for more details).
+3. Results
+3.1. Different planets on the borders in projections
+In Fig. 2, we show the filtered NASA Exoplanet Archive planets
+in the RP–P and MP–P planes with the planet’s mean density as
+a coloring variable. The comparison of the two scatter plots re-
+veals an interesting feature. Planets with ≳ ρJ density, plotted as
+blue dots, follow a similar distribution in both planes, with the
+most important difference of the compressed size range of hot
+3 https://www.sdss.org/dr17/irspec/apogee-bitmasks/
+The web documentation include detailed flagging descriptions.
+Article number, page 3 of 11
+
+IDA&A proofs: manuscript no. 44846corr
+Jupiters (upper panel) that is related to well-understood plane-
+tary physics. However, the group of low-density planets (plot-
+ted in red) behave very differently between the two plots. These
+planets have ≲ MJ mass and they can be found at the middle of
+the mass distribution (lower panel), but they are among the most
+inflated planets, and can be observed on the top of hot Jupiters
+in the radius distribution (upper panel).
+This distinct clump of planets suffers from a very significant
+dislocation compared to the non-inflated or moderately inflated
+planets, which requires explanation. Also, the low-density plan-
+ets themselves form the upper boundary of the sub-Jovian (or
+Neptunian) desert in the MP–P projection, while they tend to
+be farther from the boundary than the hot Jupiters in the RP–
+P plane. Indeed, in the RP–P plane, the hot Jupiters themselves
+form the upper boundary of the desert.
+The anonymous referee has pointed out that similar struc-
+tures could be simple byproducts of the well-known increasing
+bulk density above a 1.0 MJ planetary mass, due to the increas-
+ing self-compression of the gas that builds up the planetary body.
+To test the robustness of this finding, we used an empirical model
+to describe the mass-density relations in general and we removed
+this contribution from the scatter plot.
+We calculated synthetic densities for each planet, following
+Traub (2011); Mordasini et al. (2012) and based on mean radii
+(Eq. (24) of (Mordasini et al. 2012). Since the applied model
+equation is designed to reproduce theoretical radii for planets,
+with semi-major axis a < 0.1 AU and 0.1 AU < a < 1 AU,
+we further restricted our sample to include planets with a < 1
+AU. Subtracting the logarithm of the resulting densities yielded
+the distribution shown in Fig. 3. We find that the upper edge of
+the desert in the MP – P plane (bottom panel of Fig. 3) is still
+generally dominated by low residual-density planets, which are
+then then observed to be further away from the sub-Jovian desert
+in the RP – P plane (top panel of Fig. 3).
+This suggests that the general density-mass relation does not
+explain the observed phenomenon. Indeed, the scatter plots col-
+ored with the residual log densities (Fig. 3) show a very similar
+distribution to what we see in Fig. 2 and we can have the sub-
+jective visual impression that Fig. 3 is even clearer. In any case,
+our conclusion is that the found double population of close-in
+giants is not a consequence of the density-mass relations, but it
+is rather superimposed onto the general pattern of density distri-
+butions. Taking the position of the planets with ρ/ρJ < 0 into
+account, it seems to be quite plausible that their presence is pri-
+mordially influenced by the desert boundaries. We also point out
+that the exoplanets with longer orbital periods (> 10 days) which
+have lower densities (red points on Fig. 2) remain roughly in the
+same position relative to the desert in both planes of Fig. 2.
+The analysis of the bulk densities (Fig. 2) thus suggests
+the upper boundary of the desert has a multiform nature: it is
+marked by ordinary hot Jupiters in one projection and inflated
+hot Jupiters in the other projection. Since the planetary physics
+behind these two distinct classes of hot Jupiters are different, this
+also brings up the possibility that the formation of the desert is a
+result of multiple processes (at least at the upper boundary): one
+process acting on ordinary jot Jupiters and one another acting on
+inflated hot Jupiters.
+The possible biases behind the observed distribution can be
+excluded by simple quantitative considerations. In particular,
+137 exoplanets falling into the ρP < 0.35 ρJ category were dis-
+covered primarily with 18 different photometric facilities includ-
+ing both ground-based and space telescopes, so their distribution
+in the parameter-space cannot be devoted to some instrument-
+specific selection distortion of some specific exoplanet discov-
+-1.0
+-0.5
+0.0
+log Rp/RJ
+-1
+0
+1
+2
+3
+4
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+log P [d]
+-1.2
+-0.8
+-0.4
+0.0
+0.4
+0.8
+1.2
+Residual log ρP/ρJ
+Fig. 3. Scatterplots of the distribution of exoplanets in the RP – P and
+MP – P planes as in Fig. 2 but with synthetic densities subtracted from
+the observed ones (see text for details).
+ery surveys. Also, from the entire sample of 650 exoplanets,
+only 23 of them were not discovered with the transit method
+(mostly radial velocity planets that had later been observed in
+transit as well) and none of them belong to the inflated group
+of planets (according to the definition given at the beginning of
+this paragraph). Therefore, the presence of radial velocity dis-
+coveries should not be considered as sources of biased data.
+Out of the 137 exoplanets with ρP < 0.35 ρJ, an overwhelm-
+ing majority (116) are members of planetary systems with a sin-
+gle known planetary mass object, with 19 systems containing 2
+planets and 2 containing 2 and 3 planets. The brightness distri-
+bution of host stars in the two samples is also very similar, with
+V = 10.56 ± 1.94 mag for the larger, 650-member sample and
+10.89 ± 1.59 mag for the low-density sample. We therefore see
+no evidence for observational biases. We can consider two possi-
+ble explanations for the multiple planet populations at the upper
+boundary: (i) either the inflated planets are in the early stages
+of planetary evolution and they are still in the process of initial
+contraction or losing their atmospheres; or (ii) they are more ma-
+ture planets, but have a different inner structure than the higher
+density hot Jupiters. In either case, the processes that form the
+desert must be selective for these formation scenarios, leading
+to the two distinct planet populations at the two different projec-
+tions of the upper boundary.
+To test these scenarios, a wider sample with more precisely
+determined exoplanet parameters is necessary. Our understand-
+ing of these questions will be greatly improved by the two up-
+coming ESA missions: Ariel (Tinetti et al. 2021) and PLATO
+(Rauer et al. 2014).
+3.2. Dependence of the boundary on stellar parameters
+In this subsection, we analyze the RP–P and MP–P scatter plots
+with regard to the distribution of stellar parameters (effective
+temperature, stellar radius, stellar mass, log g, and metallicity),
+Article number, page 4 of 11
+
+Gy. M. Szabó
+et al.: Sub-Jovian desert of exoplanets at its boundaries
+-1.0
+-0.5
+0.0
+log Rp/RJ
+A
+B
+A
+0.0
+0.5
+1.0
+Cum. distr.
+log Teff < 3.73
+log Teff > 3.73
+p =  0.000014
+B
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+log Teff < 3.67
+log Teff > 3.67
+p =  0.45
+log P [d]
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+C
+D
+C
+0.0
+0.5
+1.0
+Cum. distr.
+log Teff < 3.73
+log Teff > 3.73
+p =  0.00011
+D
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+log Teff < 3.53
+log Teff > 3.53
+p =  0.068
+log P [d]
+3.4
+3.5
+3.6
+3.7
+3.8
+3.9
+4.0
+log Teff [K]
+Fig. 4. Scatterplot showing the distribution of exoplanets in the RP –
+P (top left) and MP – P (top right) parameter spaces, colored by the
+temperature of the host star (see the color bar for the values). The middle
+and bottom panels show the cumulative distribution of the two samples
+divided at the median of Teff in the A, B, C, and D regions. The p-values
+of the KS-test are also shown.
+complementing and extending the results described in Szabó &
+Kálmán (2019).
+3.2.1. Effective temperature of the host
+Following the recipe described in Sect. 2, we split the sample
+of exoplanets at the median of the effective temperature of the
+host stars in different regions separately. The split values were
+5370 K, 4677 K, 5370 K, and 3388 K for the A, B, C, and D
+regions, respectively (see Fig. 4). We found that in both param-
+eter spaces, planets in A and C regions tend to have shorter or-
+bital periods around cooler hosts. The p-values of the KS tests
+were: 10−5 and 10−4 in A and C, respectively. This discrepancy
+in the cumulative distribution has not been observed in the B or
+D regions (p-values: 0.45 and 0.07), proving that the observed
+dependence is specifically located at the right boundary of the
+desert. Therefore, we confirm the findings of Szabó & Kálmán
+(2019) that the effective temperature of the host star plays a key
+role in the formation of the desert and it does so in both param-
+eter spaces.
+3.2.2. Stellar radius
+The boundary of the desert also shows dependence on RS in both
+the RP – P and MP – P planes, as visible in the scatter plots of
+Fig. 5. The stars below the median radii (0.94 R⊙ and 0.90 R⊙ in
+the A and C regions, respectively) are more likely to host plan-
+ets with shorter orbital periods. We note that the KS test yields
+p-values of 2 · 10−4 and 10−5 in A and C, respectively. There
+is no significant difference in the cumulative distribution plan-
+ets in the B and D regions. In B, the p-value is 0.43; while in
+D, the p = 0.013 still corresponds to 2.48σ, proving the statis-
+tical insignificance of the (otherwise apparent) differences seen
+in the bottom right panel of 5. As we find that stars with lower
+radii tend to host planets with shorter orbital periods and that this
+phenomenon is not present in the control regions, we conclude
+that the stellar radius also plays a key role in the formation of the
+desert.
+-1.0
+-0.5
+0.0
+log Rp/RJ
+A
+B
+A
+0.0
+0.5
+1.0
+Cum. distr.
+RS < 0.94 Rʘ
+RS > 0.94 Rʘ
+p =  0.00017
+B
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+RS > 0.72 Rʘ
+RS < 0.72 Rʘ
+p =  0.43
+log P [d]
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+C
+D
+C
+0.0
+0.5
+1.0
+Cum. distr.
+RS < 0.90 Rʘ
+RS > 0.90 Rʘ
+p =  0.00096
+D
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+RS < 0.34 Rʘ
+RS > 0.34 Rʘ
+p =  0.013
+log P [d]
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+log RS/Rʘ
+Fig. 5. Scatterplots showing the distribution of exoplanets in the RP – P
+and MP – P parameter spaces with RS used as the third parameter (top
+row) and the respective cumulative distributions (middle and bottom
+rows; same as Fig. 4).
+-1.0
+-0.5
+0.0
+log Rp/RJ
+A
+B
+A
+0.0
+0.5
+1.0
+Cum. distr.
+MS < 0.92 Mʘ
+MS > 0.92 Mʘ
+p =  0.000056
+B
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+MS < 0.76 Mʘ
+MS > 0.76 Mʘ
+p =  0.45
+log P [d]
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+C
+D
+C
+0.0
+0.5
+1.0
+Cum. distr.
+MS < 0.92 Mʘ
+MS > 0.92 Mʘ
+p =  0.055
+D
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+MS < 0.34 Mʘ
+MS > 0.34 Mʘ
+p =  0.013
+log P [d]
+-2.0
+-1.6
+-1.2
+-0.8
+-0.4
+0.0
+0.4
+log MS/Mʘ
+Fig. 6. Scatterplots showing the distribution of exoplanets in the RP – P
+and MP – P parameter spaces with MS used as the third parameter (top
+row) and the respective cumulative distributions (middle and bottom
+rows; same as Fig. 4).
+3.2.3. Stellar mass
+The scatter plot of exoplanets with the mass of the host star used
+as the third parameter is shown in Fig. 6. The split values of MS
+are 0.92 M⊙, 0.76 M⊙, 0.92 M⊙, and 0.34 M⊙ in A, B, C, and
+D, respectively. Dividing the samples in the four regions at these
+values leads to an ambiguous detection of MS dependence of the
+desert boundary (i.e., the distribution of planets at the size ranges
+of the desert). This is expressed in terms of the p-values of 6·10−5
+opposed to 0.45 in A and B regions, and then 0.055 opposed to
+0.013 in C and D regions. The dependence is present in the RP–
+P plane and is insignificant in the MP–P plane, consistently with
+the results of Szabó & Kálmán (2019). Therefore, we find that
+stellar mass has some influence most importantly on the radius
+distribution, but does not play an essential role in the formation
+of the desert on its own.
+Article number, page 5 of 11
+
+IDA&A proofs: manuscript no. 44846corr
+-1.0
+-0.5
+0.0
+log Rp/RJ
+A
+B
+A
+0.0
+0.5
+1.0
+Cum. distr.
+log g < 4.46
+log g > 4.46
+p =  0.040
+B
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+log g < 4.60
+log g > 4.60
+p =  0.22
+log P [d]
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+C
+D
+C
+0.0
+0.5
+1.0
+Cum. distr.
+log g < 4.48
+log g > 4.48
+p =  0.0055
+D
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+log g < 4.87
+log g > 4.87
+p =  0.37
+log P [d]
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+4.2
+4.4
+4.6
+4.8
+5.0
+5.2
+log g [cgs]
+Fig. 7. Same as Fig. 4 but with log g used as the third parameter instead
+of Teff.
+-1.0
+-0.5
+0.0
+log Rp/RJ
+A
+B
+A
+0.0
+0.5
+1.0
+Cum. distr.
+[M/H] < 0.16
+[M/H] > 0.16
+p =  0.088
+B
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+[M/H] < 0.01
+[M/H] > 0.01
+p =  0.27
+log P [d]
+-3
+-2
+-1
+0
+1
+log Mp/MJ
+C
+D
+C
+0.0
+0.5
+1.0
+Cum. distr.
+[M/H] < 0.11
+[M/H] > 0.11
+p =  0.0075
+D
+0.0
+0.5
+1.0
+Cum. distr.
+-1
+0
+1
+2
+3
+4
+[M/H] < -0.01
+[M/H] > -0.01
+p =  0.30
+log P [d]
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+[M/H]
+Fig. 8. Same as Fig. 4 but with [M/H] used as the third parameter instead
+of Teff.
+3.2.4. Stellar log g
+Exploring the effects of log g on the distribution of exoplanets
+in the two parameter spaces, we divided the samples at log g =
+4.46 in the A region, log g = 4.60 in B, log g = 4.48 in C, and
+log g = 4.87 in D. We find that there is no statistical difference
+in the cumulative distributions in either the A or B regions of the
+RP – P plane. This is expressed by the p-values of the KS-test:
+0.040 in A (corresponding to ∼ 2.1σ) and 0.22 in B. In the size
+ranges of the desert of the MP – P parameter space, there is a
+hint of a discrepancy between the cumulative distributions of the
+sample divided at the median at p = 0.0055 (corresponding to
+2.78σ). There is no such difference in the sample of the control
+region (p = 0.34).
+We therefore find no firm dependence of the desert on the
+surface gravity of the host star, contradicting to the earlier re-
+sults of Szabó & Kálmán (2019) (based on fewer planets and
+unconstrained for a reliable mass determination). The structure
+of Fig. 7 the most inflated hot Jupiters are hosted by stars with
+the lowest log g values (< 3.2) from the sample.
+3.2.5. Stellar metallicity
+In the filtered sample of 650 used in the exploration of the pre-
+vious parameters, there are 10 stars with unknown metallicities
+and we omitted those from the current analysis. The samples in
+A, B, C, and D are split at [M/H]= 0.16, 0.01, 0.11, and −0.01,
+respectively. There is no significant difference in the cumulative
+distribution of planets in the two examined areas of the RP – P
+plane (p-values are 0.088 and 0.27 in A and B). In the case of
+the MP – P plane, there is also only a hint that planets from the
+C region are more likely to have shorter orbital periods around
+younger, more metal-rich stars, expressed via p = 0.0075 (cor-
+responding to 2.67σ) in C compared to 0.30 in D.
+As in the case of log g, we find no firm dependence of the
+sub-Jovian desert of exoplanets on the metallicity of the host
+stars. This is somewhat contradictory to the results of Dong et al.
+(2018); Petigura et al. (2018) and Szabó & Kálmán (2019).
+3.3. Multimodal dependence of the planetary mass and
+radius on stellar parameters
+The experienced bimodality of the planet groups at the border
+of the desert Fig. 4 and the experienced differences behind the
+RP–P and MP–P relations reflect the presence of two different
+type of planets at the border: hot Jupiters at the upper bound-
+ary and super-Earths at the lower boundary. As the boundary
+of the desert in the mass and radius projections depend on stel-
+lar parameters, we expect to see various groups of planets with
+different parameter dependencies between the radius, mass, and
+stellar parameters.
+To explore distinct correlation laws between the planet and
+stellar parameters, we divided our sample into two subgroups
+in both the mass-metallicity and the radius-metallicity planes
+via a k-means fuzzy clustering algorithm (Ferraro et al. 2019).
+In the radius-metallicity plane, the resulting cluster consist-
+ing hot Jupiters had a median radius of 1.15 RJ (interquar-
+tile range of 1.01-1.33) and a cluster consisting Neptune- and
+Earth-sized planets had a median radius of 0.21 RJ (interquar-
+tile range of 0.14–0.27). When the clustering was done in the
+mass-metallicity distribution, the cluster of large planets had a
+median mass of 0.93 MJ (interquartile range of 0.57–1.80) and
+the subgroup of smaller planets has a mean mass of 0.026 MJ
+(interquartile range of 0.015–0.048).
+We fit linear regression models describing the dependence
+of planet mass and radius on Teff, MS , and [M/H]. Between the
+planet size and the effective temperature, we get:
+log
+� MP
+MJ
+�
+= 1.91(34) · log
+�Teff
+1 K
+�
+− 7.15(1.27),
+r = 0.48,
+(1)
+log
+�RP
+RJ
+�
+= 2.05(9) · log
+�Teff
+1 K
+�
+− 3.88(33),
+r = 0.50,
+(2)
+in the “large planet” cluster and
+log
+� MP
+MJ
+�
+= 2.04(25) · log
+�Teff
+1 K
+�
+− 9.09(91),
+r = 0.27,
+(3)
+log
+�RP
+RJ
+�
+= 0.85(12) · log
+�Teff
+1 K
+�
+− 3.83(46),
+r = 0.43,
+(4)
+in the “small planets” cluster. In the expressions, the ambiguity
+(standard deviation) of the last digits are shown in parentheses
+after the value of each coefficients.
+Article number, page 6 of 11
+
+Gy. M. Szabó
+et al.: Sub-Jovian desert of exoplanets at its boundaries
+3.4
+3.6
+3.8
+4.0
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+r =  0.50
+r =  0.43
+log Teff [K]
+log Rp/RJ
+-3
+-2
+-1
+0
+1
+r =  0.48
+r =  0.27
+log Mp/MJ
+-1.0
+-0.5
+0.0
+0.5
+r =  0.46
+r =  0.47
+log MS/Mʘ
+r =  0.55
+r =  0.22
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+r =  -0.18
+r =  0.17
+[M/H]
+r =  -0.03
+r =  0.29
+Fig. 9. Correlation between planet mass (top row), planetary radius (bottom row), and the main stellar parameters: Teff (left column), MS (middle
+column), and [M/H] (right column). The samples are divided into two groups via a fuzzy clustering algorithm applied to the distributions in
+the right panels. Blue points: Cluster of “larger planets.” Red points: Cluster of ‘smaller planets.” The fitted linear trends of Eqs. (2)–(11) are
+plotted with solid blue and red lines. Pearson’s r values, corresponding to the giant and smaller exoplanets are also displayed with blue and red,
+respectively.
+The expressions involving the stellar mass are:
+log
+� MP
+MJ
+�
+= 0.78(17) · log
+� MS
+M⊙
+�
+− 0.02(2),
+r = 0.55,
+(5)
+log
+�RP
+RJ
+�
+= 0.48(4) · log
+� MS
+M⊙
+�
+− 0.04(1),
+r = 0.46,
+(6)
+in the “large planet” cluster and
+log
+� MP
+MJ
+�
+= 0.94(9) · log
+� MS
+M⊙
+�
+− 1.46(3),
+r = 0.22,
+(7)
+log
+�RP
+RJ
+�
+= 0.37(5) · log
+� MS
+M⊙
+�
+− 0.64(1),
+r = 0.47,
+(8)
+in the “small planet” cluster. Such a correlation between planet
+size and stellar mass (but without clustering) was also noted by
+Fortney et al. (2007); Wu (2019), and Lozovsky et al. (2021). Wu
+(2019) found the linear coefficient between log MP and log MS to
+be unity, which is in good agreement with Eq. (7). These findings
+are also consistent with the qualitative observations from Fig. 6
+that the most massive stars from our sample (MS > 1.5 M⊙)
+host the largest and least dense hot Jupiters, those with radii of
+> 1.4 RJ.
+For the stellar metallicity dependencies, we get
+log
+� MP
+MJ
+�
+= −0.07(11) · [M/H] + 0.01(2),
+r = −0.03,
+(9)
+log
+�RP
+RJ
+�
+= −0.12(3) · [M/H] + 0.07(1),
+r = −0.18,
+(10)
+in the “large planet” cluster and
+log
+� MP
+MJ
+�
+= 0.94(9) · [M/H] − 1.45(3),
+r = 0.29,
+(11)
+log
+�RP
+RJ
+�
+= 0.17(7) · [M/H] − 0.70(7),
+r = 0.17
+(12)
+in the “small planet” cluster. Wu (2019) suggested that there is
+no correlation between stellar metallicity and planetary mass,
+which is compatible to the low values of correlation coefficients
+found in our analysis.
+The p value of these multiple correlations are in the range
+of 10−4 and 10−16 for all equations –except the ones involv-
+ing [M/H] (Eqs. 9-12), which have p values between 0.01–0.5,
+showing marginally significant or even, insignificant correla-
+tions. The bimodality of the correlations illustrate that there is
+no one-to-one relationship between planet sizes and stellar pa-
+rameters. It can be said, however, that in almost every case (with
+the exception of RP – MS ), a stronger correlation is found in
+the case of large exoplanets. In these linear models, no under-
+lying astrophysical connections are considered, they are based
+solely on statistics; however, in future works, they can be used
+to further consider the reasons and processes which shape these
+dependencies.
+3.3.1. Elemental abundances from APOGEE
+The APOGEE stellar parameters were taken from a homoge-
+neous, state-of-the-art quality catalog of stellar data, while the
+sources of stellar data in the NASA Exoplanet Archive are in-
+homogeneous, although the latter one includes many more plan-
+ets, which can increase the significance of the statistical analysis
+based upon it.
+From the APOGEE planet sample (Sect. 2.2), we repeated
+the fits described in Eqs. (2), (6), (3), and (7) leading to the fol-
+lowing regressions:
+log
+�RP
+RJ
+�
+= 1.21(43) · log
+�Teff
+1 K
+�
+− 4.65(1.60),
+r = 0.25 (13)
+log
+�RP
+RJ
+�
+= −0.03(5) · [M/H] − 0.11(3),
+r = −0.05 (14)
+Article number, page 7 of 11
+
+IDA&A proofs: manuscript no. 44846corr
+3.6
+3.7
+3.8
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+r =  0.25
+r =  0.18
+log Teff [K]
+log Rp/RJ
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+r =  -0.05
+r =  0.02
+[M/H]
+Fig. 10. Correlations between RP – Teff (left panel) and RP – [M/H] (same as the lower left and lower right panels of Fig. 9) showing the fuzzy
+clustering of APOGEE data and with the linear regressions of Eqs. (14)–(16) overplotted.
+-1
+0
+1
+2
+3
+4
+-1.0
+-0.5
+0.0
+log Rp/RJ
+log P [d]
+-0.5
+-0.3
+-0.1
+0.1
+0.3
+[M/H]
+-1
+0
+1
+2
+3
+4
+log P [d]
+-0.5
+-0.3
+-0.1
+0.1
+0.3
+Residual [M/H]
+Fig. 11. Distribution of exoplanets from the APOGEE planet host sample with [M/H] as the coloring parameter (left panel) and the residuals after
+a bilinear fit (right panel).
+in the “large planet” cluster and
+log
+�RP
+RJ
+�
+= 0.61(14) · log
+�Teff
+1 K
+�
+− 3.05(53),
+r = 0.18
+(15)
+log
+�RP
+RJ
+�
+= 0.02(5) · [M/H] − 0.79(1),
+r = 0.02
+(16)
+in the “small planet” cluster (these clusters and the linear models
+are shown in Fig. 10). Here, we can see that the APOGEE data
+reproduced the previously determined coefficients of Teff in both
+clusters, with larger ambiguities due to the significantly fewer
+planets in the currently availagble APOGEE sample. Also, the r
+regression coefficients are smaller. In the case of the R–Teff cor-
+relations, we see no significant dependence in the APOGEE data
+(the ambiguity of the coefficient of Teff is larger than the value,
+hence, 0 is within the range) and the r regression coefficients are
+also close to zero. Here, the APOGEE analysis reproduced the
+small correlation coefficients between RP and [M/H]. We note
+that as the uncertainties in the APOGEE data were also derived
+in a homogeneous way, we used these as weights for our regres-
+sion.
+The APOGEE data also contains the derived abundances4
+for a number of elements. We also repeated the linear modeling
+described above for individual elements inculding C, C I (union-
+ized C), N, O, Na, Mg, Al, Si, K, Ca, Mn, Co, Ni, Ce, and [C/O]
+to check for correlations between the relative abundance ratio of
+these elements and the planetary radius (Table 1). In both clus-
+ters of “larger” and “smaller” planets, the relationship between
+the planetary radius and any given X abundance is characterized
+by the slope of the line (k) and the intercept with the ordinate
+(n), such that:
+log RP
+RJ
+= k log X
+X⊙
++ n.
+(17)
+To show an example of the fitted distributions, the left panel
+of Fig. 11 shows the exoplanets in the RP – P plane with the
+4 https://www.sdss.org/dr17/irspec/abundances/
+Article number, page 8 of 11
+
+Gy. M. Szabó
+et al.: Sub-Jovian desert of exoplanets at its boundaries
+metallicity as the coloring parameter taken from the APOGEE
+planet host sample. The right panel shows the residuals after re-
+moving the bilinear trend according to Eq. (18), which is very
+similar to the uncorrected one. The comparison of the two panels
+show that the significant features which can be partly explained
+by the metallicity includes a negative metallicity gradient along
+the period (at the boundary, stars with a metallicity higher than
+0 are overabundant), mostly affecting the group of smaller plan-
+ets, and a gradient along the radius. The large end of “smaller
+planets” and hot Jupiters tend to have increased metallicity. The
+appropriate KS-test here did not reveal differences of the metal-
+licity distributions near and far from the edge of the desert. The
+plots for individual elements reproduced a very similar pattern,
+which we discuss below.
+Table 1 shows those elements which we have found to be
+significantly correlated with the planetary radius within either
+the “larger planet” or the “smaller planet” sample. As a conclu-
+sion, O, Na, and K can influence the size distribution within the
+smaller planet group, while marginal significance was found in
+the case of N and Si in the “smaller planet” group as well. We
+found that only Mg influences the radius of planets in the “large”
+groups.
+Correlations extending to the group of all planets, including
+both the "larger planet” and the “smaller planet” samples are fit-
+ted with a bilinear regression in the period–radius plane. This
+equation is expressed as follows:
+log X
+X⊙
+= a · log P + b · log RP
+RJ
++ c,
+(18)
+where a, b, and c are the coefficients of the linear regression. Ta-
+ble 2 shows those elements where a significant (p < 0.05) cor-
+relation was found. The most significant chemical parameter in
+the period–radius plane is the [M/H] itself, being very significant
+in both coordinates and we consider the overal metallicity as the
+most important forming parameter of the desert as well. The in-
+dividual abundances, on the other hand, give us more insights to
+how the overall metallicity determines the chemical dependence
+of the desert boundaries.
+The elements with significant effect of the period–radius dis-
+tribution, besides what is explained by the [Fe/H], include C, N,
+O, Mg, Al, Si, K, and Mn. In addition, we did not find significant
+dependence involving Co and Ce (although these elements were
+determined only for 14 exoplanet-host stars), as well as [Ca/Fe],
+[Ni/Fe], and [C/O].
+A comparison of Tables 1 and 2 shows that volatiles from
+the CNO process (especially N and O), moderately refractory
+alpha-process elements (Mg and Si) and K influence both the
+global structure of the planetary distribution – surrounding the
+desert as well – and they have positive correlations with the ra-
+dius of smaller planets. Stars with increased N, O, Na, and Si,
+or decreased K abundances tend to form larger planets is the
+“smaller planet” groups. In the “large planet” group, only Mg
+is inversely correlated to the planet sizes. This revealed a set of
+elements that have a significant internal influence to the sizes of
+smaller and larger planets are disjointed, which serves as strong
+evidence that the formation and evolution of smaller and larger
+planets follow a chemical bimodality.
+Table 2 shows that the difference between the smaller/larger
+groups of planets and the global period–radius maps is related to
+mostly those elements that also appeared in Table 1. Additional
+elements appear in Table 2 as C, Al, and Mn. This also corrob-
+orates the interpretation that elements forming in CNO cycle or
+alpha process in stellar nucleosynthesis have an influence on the
+period-radius distribution of the planetary system. Interestingly,
+all refractory elements showed no or insignificant correlations
+in both tests, which elements (having the highest condensation
+temperatures, Tcond > 1500 K) are often considered as the ini-
+tially condensed material in the protoplanetary disks (Scott &
+Sanders 2009), and the condensation center for less refractory
+and volatile materials in further steps. The distribution of plan-
+ets in the period-radius plane and around the desert seems to be
+related to more or less volatile elements in the atmosphere, rather
+than the refractory elements condensed in the cores. Therefore,
+the desert itself appears to be a predominantly atmospheric fea-
+ture, rather than a “core” feature, confirming the significance of
+atmospheric evaporation in its formation.
+4. Discussion
+Based on the data presented in Sects. 3.2.1–3.2.5, we could gen-
+erally confirm the results of Szabó & Kálmán (2019). Most im-
+portantly, the dependence of the desert boundary on the stellar
+effective temperature, which had been interpreted as an evidence
+for the photoevaporation, have been confirmed in the present
+study as well. An important new detail to this early interpreta-
+tion is the detection of a multiple and distinct population of hot
+Jupiters, having inflated and normal hot Jupiters at the bound-
+ary in the RP–P and the MP–P planes, respectively. Inflated hot
+Jupiters on the border can be a naturally seen as a new piece
+of evidence for the scenarios invoking photoevaporation (e.g.,
+Owen & Lai 2018), while the not inflated Jupiters at the bound-
+ary in the RP–P plane suggests a multiple track leading to the
+formation of the desert.
+We also found that the previously claimed metallicity depen-
+dence of the border are in general less significant than expected
+previously. The earlier interpretation of the claimed detection
+was also interpreted in connection to the evaporation scenarios,
+as the more effective cooling of a metal-rich atmosphere was
+claimed to form more irradiation-endurant exoplanets. Our new
+results have demonstrated that planetary occurrence indeed re-
+veals a dependence between the period, radius, and metallicity
+(in an agreement with the results of e.g. Demangeon et al. (2018)
+and Petigura et al. (2018)), even though it has not been confirmed
+as a specific characteristic of the planets near the desert bound-
+aries.
+The crowd of the exoplanets with spectroscopically detected
+atmospheric escape (dos Santos et al. 2021) can also be con-
+sidered as an evidence of irradiation effects forming the desert,
+because most of these planets are seen at the boundaries, or at
+least very close to it (Lecavelier des Etangs, personal commu-
+nication). It can also be pointed out from Fig. 4 that the hottest
+stars in our sample (with Teff ∼ 8000 – 10000 K) tend to host
+the largest and most massive hot Jupiters. A general trend that
+larger planets are more likely to be hosted by hotter stars can
+be observed from the median Teff values from A, B, C, and D
+regions. Similar statements can be made upon examination of
+Figs. 5 and 6; namely that stars with larger radius or mass are
+more likely to host larger planets. Lozovsky et al. (2021) sug-
+gested that larger radii of planets surrounding larger stars are a
+result of different compositions: larger stars tend to host planets
+with larger H-He mass fractions. Stars with higher metallicities
+are likely to host larger planets, extending to periods as long as
+≈20 days at least (Owen & Murray-Clay 2018). The explana-
+tion of the occurrence rate includes the cores of planets around
+more metal-rich stars to be more massive, hence, they can col-
+lect more initial atmospheric mass; and the photoevaporation of
+a more metal-rich atmosphere is more resistant to stellar irradia-
+Article number, page 9 of 11
+
+IDA&A proofs: manuscript no. 44846corr
+Table 1. Fitted linear models to the distribution of exoplanets in the planetary radius-abundance parameter spaces for the two clusters.
+Element
+N1
+k ± ∆k
+n ± ∆n
+r2
+p3
+N1
+k ± ∆k
+n ± ∆n
+r2
+p3
+Larger planets
+Smaller planets
+[N/Fe]
+106
+−0.04 ± 0.11
+−0.10 ± 0.03
+0.04
+0.72
+490
+0.11 ± 0.06
+−0.79 ± 0.01
+0.09
+0.06
+[O/Fe]
+106
+0.35 ± 0.23
+−0.13 ± 0.02
+0.15
+0.12
+567
+0.21 ± 0.10
+−0.83 ± 0.01
+0.09
+0.03
+[Na/Fe]
+85
+−0.08 ± 0.08
+−0.77 ± 0.09
+0.10
+0.34
+519
+0.20 ± 0.08
+−0.67 ± 0.02
+0.11
+0.01
+[Mg/Fe]
+106
+−0.51 ± 0.24
+−0.08 ± 0.03
+0.20
+0.04
+568
+−0.03 ± 0.09
+−0.79 ± 0.01
+0.01
+0.77
+[Si/Fe]
+106
+0.30 ± 0.32
+−0.10 ± 0.03
+0.09
+0.35
+568
+0.22 ± 0.12
+−0.80 ± 0.01
+0.07
+0.08
+[K/Fe]
+107
+−0.08 ± 0.17
+−0.11 ± 0.02
+0.04
+0.66
+566
+−0.16 ± 0.07
+−0.80 ± 0.01
+0.09
+0.03
+Notes. (1) Number of exoplanet hosts in the cluster. (2) Pearson’s r value. (3) p-values of the linear model.
+Table 2. Regression coefficients of Eq. (18).
+Element
+N1
+a ± ∆a
+b ± ∆b
+c ± ∆c
+p2
+[M/H]
+674
+−0.097 ± 0.012
+0.198 ± 0.022
+0.225 ± 0.021
+< 10−4
+[C/Fe]
+674
+0.020 ± 0.006
+−0.032 ± 0.010
+−0.062 ± 0.010
+0.0001
+[C I/Fe]
+673
+0.027 ± 0.006
+−0.040 ± 0.011
+−0.082 ± 0.011
+< 10−4
+[N/Fe]
+596
+0.010 ± 0.015
+0.111 ± 0.026
+0.116 ± 0.026
+0.0001
+[O/Fe]
+673
+0.023 ± 0.008
+0.001 ± 0.015
+0.040 ± 0.015
+0.0164
+[Mg/Fe]
+674
+0.030 ± 0.006
+−0.031 ± 0.011
+−0.030 ± 0.011
+< 10−4
+[Al/Fe]
+615
+0.021 ± 0.008
+0.001 ± 0.015
+0.095 ± 0.015
+0.0430
+[Si/Fe]
+674
+0.020 ± 0.005
+0.016 ± 0.009
+0.040 ± 0.009
+< 10−4
+[K/Fe]
+673
+0.026 ± 0.009
+−0.035 ± 0.016
+−0.022 ± 0.015
+0.0025
+[Mn/Fe]
+643
+−0.019 ± 0.006
+0.030 ± 0.011
+0.049 ± 0.011
+0.0005
+Notes. (1) Sample size. (2) p-value of the biliniear model.
+tion due to the increased cooling related to the photoevaporation
+of such an atmosphere.
+In Lecavelier Des Etangs (2007), additional important fac-
+tors have been introduced to the formalism, such as atmospheric
+loss due to tidal forces and the effect of he inclination. These two
+factors together have an influence on the lifetime of an evapo-
+rating atmosphere, leading to the conclusion that in the case of
+higher inclination values, increased density is required to retain
+the atmosphere during a specified lifetime. This result suggests
+that stars with higher temperatures (which tend to host more
+planets on high inclinations, Winn et al. (2010); Albrecht et al.
+(2012); Winn & Fabrycky (2015); Zhou et al. (2019)) can host
+only gaseous planets with a higher average atmospheric den-
+sity for a longer time. This conclusion would at least partly ex-
+plain the dependence of the desert boundary on the stellar effec-
+tive temperature, and the general dependence of planetary occur-
+rence on metallicity.
+The most important result of our analysis is the detection
+of multiple relations between MP and RP on one side and Teff
+and MS on the other side. These detections reflect the struc-
+tural difference between large, atmosphere-dominated planets
+and the smaller Neptunes and super-Earths. These multiple re-
+lations complicate the picture at the desert boundary because it
+is precisely these two kinds of planets that meet there. There are
+different kind of planets at the upper boundary and the lower
+boundary, which can naturally lead to different mass and radius
+dependencies on the stellar parameters. However, in the case of
+the metallicity dependencies, the sign of the slope is also differ-
+ent and we see a negative correlation between RP and [M/H] for
+large planets, along with a positive correlation for small planets
+(Eqs. 6 and 8). Therefore, we consider such multiple relations as
+an evidence for different processes forming the upper and lower
+boundaries of the desert (while there are at least two different
+processes at the upper boundary itself, as well; confirming the
+results from Ionov et al. (2018)).
+A widely claimed scenario here is the high-eccentricity mi-
+gration (Owen & Lai 2018), as well as the tidal disruption of
+giant planets (Matsakos & Königl 2016) or an XUV photoe-
+vaporation affecting sub-Saturns (Hallatt & Lee 2022). Deciding
+which one is the dominant at the lower boundary of the desert re-
+quires detailed studies of individual planets. A possible detection
+of the “sculpted Saturns” scenario would corroborate the predic-
+tion of the possible surviving super low-density sub-Saturns to
+the present day if they are born with even larger atmospheres
+than they currently harbor – in particular, Kepler 223 d has been
+claimed to be an example for such a planet on a 14.79 d period
+orbit (Hallatt & Lee 2022).
+There are several other questions that we were not able to
+answer directly from the present data – the most important being
+how the evaporation processes and the claimed high-eccentricity
+migration processes actually form the sub-Jovian/Neptunian
+desert. Also, because we found that the desert boundaries de-
+pend on stellar parameters, the simple linear and log-linear laws
+that mark the boundary of the desert (Mazeh et al. 2016) are in
+some way transited to a multilinear dependence. Here, we find
+a lack of predictions about the desert boundary in the R-P and
+M-P planes as a function of stellar temperature and metallicity,
+for instance, and a comparison is not readily possible. Instead,
+our aim in the present analysis is to show that the formation of
+the desert is indeed a complex astrophysical process itself. Thus,
+presenting planetary evolution processes with respect to funda-
+mental stellar parameters should indeed be the next step toward
+improving our understanding of this process.
+Our current conclusions were based on the currently avail-
+able NASA Exoplanet catalog, combined with the SDSS-IV
+APOGEE-2 catalog of exoplanet host stars with a limited num-
+ber of entries (< 700 planets in case of all elements). In the fu-
+ture, an extended stellar catalog from the SDSS-V Milky Way
+Mapper (MWM, Kollmeier et al. 2017) program will cover a
+much larger sample of planet-host stars, and a similar analysis
+involving these data will reveal deeper details of these correla-
+tions.
+5. Summary
+In this paper, we analyzed exoplanet occurrence with respect to
+stellar and planetary parameters. Our main results are as follows:
+– We demonstrated the multifaced nature of the upper bound-
+ary of the sub-Jovian or Neptunian desert: in the MP–P
+plane, the boundary is marked by inflated hot Jupiters; while
+in the RP–P plane, there are normal hot Jupiters at the bound-
+ary.
+– We confirmed the dependence of the period boundary on
+stellar parameters, such as effective temperature, stellar
+mass, and stellar radius.
+– The multiple populations and the parameter dependence of
+the boundary suggests multiple formation mechanisms.
+Article number, page 10 of 11
+
+Gy. M. Szabó
+et al.: Sub-Jovian desert of exoplanets at its boundaries
+– With fuzzy clustering, we also investigated double param-
+eter relations between planet mass and radius on one side,
+and effective temperature, stellar mass, and metallicity on the
+other.
+– The demarcation of these planet groups coincides with the
+position of the desert, suggesting that all the relationships
+connecting the planetary size and the fundamental stellar pa-
+rameters conjoin at the level of the size ranges of the desert
+and play a role in its formation.
+– In light of these results, we considered photoevaporation as
+the main process shaping the desert boundaries. We have
+discussed other possibilities such as high-eccentricity migra-
+tion, abrasion by irradiation, tidal loss of atmosphere, and
+internal structural differences.
+Acknowledgements. We acknowledge the support of the Hungarian National
+Research, Development and Innovation Office (NKFIH) grant K-125015, a
+PRODEX Experiment Agreement No. 4000137122, the Lendület LP2018-
+7/2022 grant of the Hungarian Academy of Science and the support of the city of
+Szombathely. LBo acknowledges the funding support from Italian Space Agency
+(ASI) regulated by “Accordo ASI-INAF n. 2013-016-R.0 del 9 luglio 2013 e
+integrazione del 9 luglio 2015”. Prepared with the professional support of the
+Doctoral Student Scholarship Program of the Co-operative Doctoral Program of
+the Ministry of Innovation and Technology financed from the National Research,
+Development and Innovation Fund.
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+ID

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+arXiv:2301.03177v1  [math.AC]  9 Jan 2023
+A FAMILY OF EXPLICIT WARING DECOMPOSITIONS OF A POLYNOMIAL
+KANGJIN HAN AND HYUNSUK MOON
+Abstract. In this paper we settle some polynomial identity which provides a family of explicit
+Waring decompositions of any monomial Xa0
+0 Xa1
+1 · · · Xan
+n
+over a field k. This gives an upper bound
+for the Waring rank of a given monomial and naturally leads to an explicit Waring decomposition of
+any homogeneous form and, eventually, of any polynomial via (de)homogenization. Note that such
+decomposition is very useful in many applications dealing with polynomial computations, symmetric
+tensor problems and so on. We discuss some computational aspect of our result as comparing with
+other known methods and also present a computer implementation for potential use in the end.
+1. Introduction
+Throughout the paper, let k be an infinite field and R = k[X0, . . . , Xn] = �
+d≥0 Rd be the ring of
+polynomials over k and Rd be the k-vector space of homogeneous polynomials (or forms) of degree
+d. For a given F ∈ Rd, a Waring decomposition of F over k is defined as a sum
+(1)
+F =
+r
+�
+i=1
+λiLd
+i ,
+where λi ∈ k and Li is a linear form over k. The smallest number r for which such a decomposition
+exists is called Waring rank of F over k and we denote it by rankk(F).
+Earlier studies of Waring decomposition and Waring rank, initiated by works of Sylvester and
+others, go back to the 19th century (see [9] for a historical background). But, despite their long
+history, the Waring ranks for general forms over the complex numbers, a long-standing conjecture
+in this field, were determined only in the 1990s by [1] and the complex Waring rank of monomials,
+a specific type of polynomial, has been found in recently [3, 5].
+Generally, it turns out that
+determination of the rank of a form is a very difficult task except some known cases (see e.g. [7]
+and references therein).
+Over the real numbers the Waring rank is even more difficult to compute in general. For some
+basic cases, the real Waring rank of a monomial Xa0
+0 Xa1
+1 · · · Xan
+n
+with ai > 0 is known; it is the
+degree when n = 1 [4] and 1
+2
+�n
+i=0(ai + 1) when min(ai) = 1 [6]. In [6], they also provide an upper
+bound for the real rank of any monomial
+1
+2aj
+�n
+i=0(ai + aj) where aj is min(ai), but it is not tight
+in general. The state of art result has been obtained recently by authors in [8] as follows:
+Theorem 1.1 ([8]). For a monomial Xa0
+0 Xa1
+1 . . . Xan
+n
+with each ai > 0, it holds that
+(2)
+rankR(Xa0
+0 Xa1
+1 . . . Xan
+n ) ≤ 1
+2
+� n
+�
+i=0
+(ai + 1) −
+n
+�
+i=0
+(ai − 1)
+�
+.
+Further, the same result is true for rankQ(Xa0
+0 Xa1
+1 . . . Xan
+n ).
+In this article we settle some identity which concerns an explicit Waring decompositions of any
+monomial. This can be used to recover the bound (2) in a direct way. In principle, via ‘apolarity’
+one could find an Waring decomposition of a given monomial M over R or Q using the sub-ideal of
+2010 Mathematics Subject Classification. 14P99, 12D05, 13P10, 14A25, 15A21, 14N15.
+Key words and phrases. Waring rank, Waring decomposition, Monomials, Symmetric tensor, Complexity.
+1
+
+the apolar ideal M⊥ which appeared in [8]. But, in general this involves such a huge amount of linear
+algebra computation of a large size system to determine whole coefficients of the decomposition
+that it is not easy to get the result actually in many cases.
+Here we prove some Waring-type identity which is parametrized by most numbers in the (infinite)
+base field k and has an interesting combinatorial nature (see Theorem 2.4 and Corollary 2.5). As
+a result, we can have a more direct formula for a Waring decomposition of M without relying on
+a massive linear algebra calculation.
+We discuss some consequence of the identities in Section 2 for finding an explicit Waring decom-
+position of not only a monomial, but also of any homogeneous form (to an arbitrary polynomial it
+can be easily applied via the process of (de)homogenization, too). Especially, in Section 3 we con-
+sider its computational aspect; it turns out that our method is asymptotically better in the number
+of summands than the method in [2] using a previously known decomposition (see Remark 3.3 for
+further discussion). We also present an example of the identity in Example 2.8 and a software
+implementation using a symbolic computer algebra system Macaulay2 [10] as well in Section 4.
+Finally, we’d like to mention that almost everything in the paper also does work over a finite
+field provided that the characteristic is relatively large. But, for brevity we here focus only on the
+case of an infinite field k.
+2. Main Result
+Definitions and Notations 2.1. First of all we define some notions and set notations on them.
+(1) Let a ∈ Zn+1
+≥0
+be a sequence of n + 1 nonnegative integers (note that such an a naturally
+corresponds to a monomial Xa0
+0 Xa1
+1 · · · Xan
+n
+∈ k[X0, . . . , Xn]). |a| means its sum �n
+0 ai and
+�|a|
+a
+�
+denotes the multinomial coefficient
+�
+|a|
+a0,a1,...,an
+�
+.
+(2) For a given a ∈ Zn+1
+≥0 , we set Za := {i | ai = 0}, the set of indices of zeros, Ea :=
+{i | ai is even}, the set of even indices and mi = ⌊ai−1
+2 ⌋ for i = 0, . . . , n.
+(3) Fix a ∈ Zn+1
+≥0 . For any set A with Za ⊂ A ⊂ Ea, we consider a set of ordered multiples
+KA := {(ki)i̸∈A | ki ∈ Z, 0 ≤ ki ≤ mi for i ̸∈ A} and SA := {(si)i̸∈A | si ∈ {0, 1}} which
+is the set of all ordered multiples for signs. As a reduction of each set, we also define two
+specific subsets
+KA := {(ki)i̸∈A | ki ∈ Z, 0 ≤ ki ≤ mi for i ̸∈ A and min{ki}i̸∈A = 0} ⊂ KA ,
+SA := {(si)i̸∈A | si ∈ {0, 1}, smin{i:i̸∈A} = 0} ⊂ SA .
+(4) For a given triple (A, k, s) where A is a set such that Za ⊂ A ⊂ Ea, k ∈ KA and s ∈ SA,
+we set ℓA,k,s :=
+�
+i̸∈A
+(−1)sitkiXi, a linear form in k[X0, X1, . . . , Xn].
+(5) Finally, we need to define the following combinatorial function depending on the value of
+ai;
+Fi(y) :=
+
+
+
+
+
+mi
+�
+j=1
+(y − tai−2j)
+, for mi > 0
+1
+, if mi = −1 or 0
+.
+Then Fi(y) is a polynomial function in k[y] and has degree mi unless mi = −1. For a
+monomial M and a polynomial f in k[y], we denotes the coefficient of M in the polynomial
+f by c(M, f).
+For the proof of Theorem 2.4, we also prove a lemma on the sum of signs.
+2
+
+Lemma 2.2. For a sequence of integers J ∈ Zk,
+�
+I∈{0,1}k
+(−1)
+�k
+i=1 Ji·Ii =
+k
+�
+i=1
+(1 + (−1)Ji) =
+�
+2k, if all the Ji are even
+0, if at least one of Ji is odd
+,
+where {0, 1}k denotes the set of all sequences of k numbers whose entry is 0 or 1.
+Proof.
+�
+I∈{0,1}k
+(−1)
+�k
+i=1 Ji·Ii =
+�
+I∈{0,1}k
+k
+�
+i=1
+(−1)Ji·Ii
+=
+�
+0≤I1≤1
+�
+0≤I2≤1
+· · ·
+�
+0≤Ik≤1
+(−1)J1·I1(−1)J2·I2 · · · (−1)Jk·Ik
+=
+�
+0≤I1≤1
+(−1)J1·I1 · · ·
+�
+0≤Ik≤1
+(−1)Jk·Ik
+=
+k
+�
+i=1
+�
+0≤Ik≤1
+(−1)Ji·Ii =
+k
+�
+i=1
+(1 + (−1)Ji)
+□
+Here is a small example of the equality of Lemma 2.2.
+Example 2.3. Let J = (3, 4, 5). Then k = 3 and the set {0, 1}3 is
+{(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.
+For each I ∈ {0, 1}3, �3
+i=1 Ii · Ji = I1J1 + I2J2 + I3J3 is given as
+0, 5, 4, 9, 3, 8, 7, and 12 .
+Hence, in one side we have
+�
+I∈{0,1}3
+(−1)
+�3
+i=1 Ii·Ji = (−1)0 + (−1)5 + (−1)4 + (−1)9 + (−1)3 + (−1)8 + (−1)7 + (−1)12
+= 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 = 0
+and in the other side
+k
+�
+i=1
+(1 + (−1)Ji) = (1 + (−1)3)(1 + (−1)4)(1 + (−1)5) = 0 .
+Now we prove our main theorem.
+Theorem 2.4. Let a ∈ Zn+1
+≥0
+be a sequence of n + 1 nonnegative integers with d = |a|. Then, it
+holds that
+(3)
+Da · Xa0
+0 Xa1
+1 · · · Xan
+n
+=
+�
+Za⊂A⊂Ea
+�
+(k,s)∈KA×SA
+CA,k,s(ℓA,k,s)d
+where
+Da = (−1)|Za| · 2n ·
+�d
+a
+�
+·
+�
+i̸∈Za
+Fi(tai)
+and
+CA,k,s = (−1)|A| · 2|A| · (−1)
+�
+i̸∈A ai·si �
+i∈A
+Fi(1)
+�
+i̸∈A
+c(yki, Fi)
+3
+
+Proof.
+(ℓA,k,s)d =
+� �
+i̸∈A
+(−1)sitkiXi
+�d =
+�
+|b|=d,A⊂Zb
+�d
+b
+� �
+i̸∈A
+(−1)si·bi(tki)biXbi
+i .
+Rewrite the right hand side of the equation as the coefficient and monomial with degree d as follows
+�
+Za⊂A⊂Ea
+�
+(k,s)∈KA×SA
+CA,k,s
+�
+�
+|b|=d,A⊂Zb
+�d
+b
+� �
+i̸∈A
+(−1)si·bi(tki)biXbi
+i
+�
+=
+�
+Za⊂A⊂Ea
+�
+|b|=d,A⊂Zb
+�d
+b
+� �
+i̸∈A
+Xbi
+i
+�
+(k,s)∈KA×SA
+CA,k,s
+� �
+i̸∈A
+(−1)si·bi(tki)bi
+�
+=
+�
+|b|=d
+�d
+b
+� n
+�
+i=0
+Xbi
+i
+�
+Za⊂A⊂Ea,A⊂Zb
+�
+(k,s)∈KA×SA
+CA,k,s
+� �
+i̸∈A
+(−1)si·bi(tki)bi
+�
+Let
+Tb =
+�
+Za⊂A⊂Ea,A⊂Zb
+�
+(k,s)∈KA×SA
+CA,k,s
+� �
+i̸∈A
+(−1)si·bi(tki)bi
+�
+then
+�
+Za⊂A⊂Ea
+�
+(k,s)∈KA×SA
+CA,k,s(ℓA,k,s)d =
+�
+|b|=d
+�d
+b
+�
+· Tb · Xb0
+0 Xb1
+1 · · · Xbn
+n
+We want to show that all Tb = 0 except b ̸= a and
+�d
+a
+�
+Ta = Da.
+Step 1
+Since all the linear forms ℓA,k,s do not have variables Xi with i ∈ Za, Tb = 0 for any b with
+Za ̸⊂ Zb. Hence we can assume that Za ⊂ Zb. So we can reduce
+Tb =
+�
+Za⊂A⊂Ea∩Zb
+�
+(k,s)∈KA×SA
+CA,k,s
+�
+(−1)
+�
+i̸∈A si·bi �
+i̸∈A
+(tki)bi
+�
+Step 2
+Tb =
+�
+Za⊂A⊂Ea∩Zb
+�
+k∈KA
+�
+s∈SA
+�
+(−1)|A| · 2|A| · (−1)
+�
+i̸∈A ai·si ·
+�
+i∈A
+Fi(1)
+�
+i̸∈A
+c(yki, Fi)
+· (−1)
+�
+i̸∈A bi·si �
+i̸∈A
+(tki)bi
+�
+=
+�
+Za⊂A⊂Ea∩Zb
+(−1)|A| · 2|A| ·
+�
+i∈A
+Fi(1)·
+� �
+k∈KA
+�
+i̸∈A
+c(yki, Fi)
+�
+i̸∈A
+(tki)bi
+·
+�
+s∈SA
+(−1)
+�
+i̸∈A ai·si(−1)
+�
+i̸∈A bi·si
+�
+By Lemma 2.2,
+�
+s∈SA
+(−1)
+�
+i̸∈A ai·si(−1)
+�
+i̸∈A bi·si =
+�
+s∈SA
+(−1)
+�
+i̸∈A(ai+bi)·si =
+
+
+
+
+
+2n−|A|,
+ai + bi ≡ 0(mod 2) for all
+i ̸∈ A, i ̸= min{i : i ̸∈ A}
+0,
+otherwise
+4
+
+since smin{i̸∈A} = 0. Let δ be the map from Z2 to {0, 1} such that δ(x, y) =
+�
+1, if x ≡ y(mod2)
+0, if x ̸≡ y(mod2)
+.
+Then
+�
+s∈SA
+(−1)
+�
+i̸∈A ai·si(−1)
+�
+i̸∈A bi·si = 2n−|A|
+�
+i̸∈A,i̸=min{i:i̸∈A}
+δ(ai, bi)
+For i ∈ A ⊂ Ea ∩ Za, ai and bi are both even. Hence d = �n
+i=0 ai = �n
+i=0 bi implies �
+i̸∈A ai ≡
+�
+i̸∈A bi. So if ai and bi have same parity for all i ̸∈ A, i ̸= min{i : i ̸∈ A}, then amin{i:i̸∈A} and
+bmin{i:i̸∈A} also have same parity. It means that
+�
+i̸∈A,i̸=min{i:i̸∈A}
+δ(ai, bi) =
+�
+i̸∈A
+δ(ai, bi)
+Since 2|A| · 2n−|A| = 2n do not relate to any summation and �
+i̸∈A δ(ai, bi) is only depend on the
+choice of A,
+Tb = 2n ·
+�
+Za⊂A⊂Ea∩Zb
+(−1)|A| �
+i∈A
+Fi(1)
+�
+i̸∈A
+δ(ai, bi)
+� �
+k∈KA
+�
+i̸∈A
+c(yki, Fi)
+�
+i̸∈A
+(tki)bi
+�
+Step 3 Let {0, . . . , n}\A = {i1, i2, . . . , ip} where p = n + 1 − |A|. Then
+�
+k∈KA
+�
+i̸∈A
+c(yki, Fi)
+�
+i̸∈A
+(tki)bi
+=
+�
+0≤ki1≤mi1
+�
+0≤ki2≤mi2
+· · ·
+�
+0≤kip≤mip
+p
+�
+j=1
+c(ykij , Fij)(tkij )bij
+=
+�
+0≤ki1≤mi1
+c(yki1, Fi1)(tki1)bi1 · · ·
+�
+0≤kip≤mip
+c(ykip, Fip)(tkip)bip
+=
+p
+�
+j=1
+�
+0≤kij ≤mij
+c(ykij , Fij)(tkij )bij =
+�
+i̸∈A
+�
+0≤ki≤mi
+c(yki, Fi)(tki)bi
+=
+�
+i̸∈A
+�
+0≤ki≤mi
+c(yki, Fi)(tbi)ki
+Since each Fi has degree mi for i ̸∈ A, �
+0≤ki≤mi c(yki, Fi)(tbi)ki = Fi(tbi). Hence
+Tb = 2n ·
+�
+Za⊂A⊂Ea∩Zb
+(−1)|A| �
+i∈A
+Fi(1)
+�
+i̸∈A
+Fi(tbi)
+�
+i̸∈A
+δ(ai, bi)
+Step 4
+Since Fi(tbi) = Fi(1) for i ∈ Ea ∩ Zb,
+�
+i∈A
+Fi(1)
+�
+i̸∈A
+Fi(tbi) =
+�
+i∈Ea∩Zb
+Fi(1)
+�
+i̸∈Ea∩Zb
+Fi(tbi).
+Also, since ai and bi are all even for i ∈ Ea ∩ Zb, �
+i̸∈A δ(ai, bi) = �
+i̸∈Ea∩Zb δ(ai, bi). So the only
+remaining term which is depended on the choice of A is (−1)|A|.
+Tb = 2n ·
+�
+i∈Ea∩Zb
+Fi(1)
+�
+i̸∈Ea∩Zb
+Fi(tbi)
+�
+i̸∈Ea∩Zb
+δ(ai, bi)
+�
+Za⊂A⊂Ea∩Zb
+(−1)|A|
+Step 5
+5
+
+If Za ̸= Ea ∩ Zb, then �
+Za⊂A⊂Ea∩Zb(−1)|A| = (−1)|Za|(1 − 1)|Ea∩Zb\Za| = 0. It means that
+Tb = 0 when Za ̸= Ea ∩ Zb.
+We can assume that Za = Ea ∩ Zb. Since �
+Za⊂A⊂Ea∩Zb(−1)|A| = (−1)|Za|,
+Tb = (−1)|Za| · 2n ·
+�
+i̸∈Za
+Fi(tbi)
+�
+i̸∈Za
+δ(ai, bi)
+For some i ̸∈ Za, if ai and bi do not have same parity, δ(ai, bi) = 0 and it implies that Tb = 0.
+Now we can assume that ai ≡ bi (mod 2) for all i ̸∈ Za. Suppose that there exist an index i such
+that ai ̸= 0 and bi = 0. Since ai and bi have the same parity, ai is even. It means that i ∈ Ea and
+i ∈ Zb. But i ∈ Ea ∩ Zb = Za contradicts to the supposition. It means that Za = Zb.
+If ai = 1, then bi ̸= 0 and so bi ≥ ai. If ai = 2, then bi ̸= 0 and bi ≡ 0 (mod 2). So bi ≥ 2 = ai.
+Suppose that ai > 2 and bi < ai for some i ̸∈ Za. Then bi = ai − 2j for some 1 ≤ j ≤ mi = ⌊ai−1
+2 ⌋
+since ai and bi have same parity and 0 < bi < ai. It implies that Fi(tbi) = �mi
+j=1(tbi − tai−2j) = 0
+for some i ̸∈ Za. Hence
+Tb = (−1)|Za| · 2n ·
+�
+i̸∈Za
+Fi(tbi) = 0
+So nonzero Tb occurs only when bi ≥ ai for all i ̸∈ Za. Since the total degree �n
+i=0 bi = �n
+i=0 ai =
+d are same, the only nonzero Tb occurs when bi = ai for all i.
+In conclusion,
+Da =
+�d
+a
+�
+· Ta =
+�d
+a
+�
+· (−1)|Za| · 2n ·
+�
+i̸∈Za
+Fi(tai)
+□
+How many do linear forms appear in the decomposition using (3)?
+If Ea ̸= {0, . . . , n}, the
+number of linear forms in Theorem 2.4 is given by
+�
+Za⊂A⊂Ea
+|KA||SA| =
+�
+Za⊂A⊂Ea
+2n−|A| �
+i̸∈A
+(mi + 1) = 1
+2
+�
+Za⊂A⊂Ea
+�
+i̸∈A
+2(mi + 1)
+= 1
+2
+�
+Za⊂A⊂Ea
+�
+i̸∈Ea
+(ai + 1)
+�
+i̸∈A,i∈Ea
+(ai) = 1
+2
+�
+i̸∈Ea
+(ai + 1)
+�
+Za⊂A⊂Ea
+�
+i̸∈A,i∈Ea
+(ai)
+= 1
+2
+�
+i̸∈Ea
+(ai + 1)
+�
+i∈Ea\Za
+(ai + 1) = 1
+2
+�
+i̸∈Za
+(ai + 1).
+In case of Ea = {0, . . . , n} (i.e. all the elements ai are even), the choice A = {0, . . . , n} should
+not be counted since there is no related linear form. Hence the toal number of linear forms can be
+computed as
+1
+2
+�
+Za⊂A̸⊂{0,...,n}
+�
+i̸∈A
+(ai) = 1
+2(
+�
+i̸∈Za
+(ai + 1) − 1) · · · (∗).
+Since, in the argument above we do not consider the parallel shift of the entries in the set KA due
+to non-zero scaling of corresponding linear forms, the number (∗) is larger than the upper bound
+given in the paper [8]. To remedy this, instead of KA we need to choose ki’s from
+KA = {(ki)i̸∈A | ki ∈ Z, 0 ≤ ki ≤ mi for i ̸∈ A, min{ki}i̸∈A = 0} ,
+which is a set of representatives of each class under the shift. Then, based on Theorem 2.4, we can
+obtain a more optimized (i.e number of linear forms reduced as much as possible) result as follows.
+6
+
+Corollary 2.5. Let a ∈ Zn+1
+≥0
+be a sequence of n + 1 nonnegative integers with |a| = d. Then, we
+have
+(4)
+Da · Xa0
+0 Xa1
+1 · · · Xan
+n
+=
+���
+Za⊂A⊂Ea
+�
+k∈KA
+�
+s∈SA
+CA,k,s(ℓA,k,s)d
+where
+Da = (−1)|Za| · 2n ·
+�d
+a
+�
+·
+�
+i̸∈Za
+Fi(tai)
+and
+CA,k,s =
+min{mi−ki}i̸∈A
+�
+j=0
+td·jCA,k+j·1,s .
+Proof. By Theorem 2.4,
+Da · Xa0
+0 Xa1
+1 · · · Xan
+n
+=
+�
+Za⊂A⊂Ea
+�
+g∈KA
+�
+s∈SA
+CA,g,s(ℓA,g,s)d.
+For a set (A, g, s) with Za ⊂ A ⊂ Ea, g ∈ KA, s ∈ SA, let m = min{gi}i̸∈A. Then
+ℓA,g,s =
+�
+i̸∈A
+(−1)sitgiXi = tm� �
+i̸∈A
+(−1)sitgi−mXi
+�
+= tmℓA,h,s
+where h = (gi − m)i̸∈A. Since min{hi}i̸∈A = min{gi} − min{gi} = 0, h ∈ KA. Hence
+�
+Za⊂A⊂Ea
+�
+g∈KA
+�
+s∈SA
+CA,g,s(ℓA,g,s)d
+=
+�
+Za⊂A⊂Ea
+�
+h∈KA
+�
+s∈SA
+�
+g∈KA,
+g−min{gi}·1=h
+CA,g,s(tmin{gi} · ℓA,h,s)d,
+and
+(5)
+�
+g∈KA,
+g−min{gi}·1=h
+CA,g,s · tmin{gi}·d =
+�
+h+m·1∈KA
+CA,h+m·1,s · tm·d .
+Since h + m · 1 ∈ KA if and only if 0 ≤ hi + m ≤ mi for i ̸∈ A,
+0 = max{−hi}i̸∈A ≤ m ≤ min{mi − hi}i̸∈A .
+Now the summation (5) becomes
+min{mi−hi}i̸∈A
+�
+m=0
+CA,h+m·1,s · tm·d =: CA,h,s .
+□
+Remark 2.6 (Waring rank and decomposition of a monomial over any infinite field k). We would
+like to remark that the Waring type identity in Corollary 2.5 gives an upper bound for the Waring
+rank as
+(6)
+rankk(M) ≤ 1
+2(
+n
+�
+i=0
+(ai + 1) −
+n
+�
+i=0
+(ai − 1)) .
+7
+
+for any monomial M = Xa0
+0 Xa1
+1 . . . Xan
+n
+with a0 ≥ a1 ≥ · · · ≥ an > 0 over an infinite field k in
+more direct way than in [8]; i.e. just by counting linear forms in the given explicit decomposition.
+It’s because the total number of linear forms appearing in (4) is counted by
+�
+Za⊂A⊂Ea
+|KA||SA| =
+�
+Za⊂A⊂Ea
+� �
+i̸∈A
+(mi + 1) −
+�
+i̸∈A
+(mi)
+�
+· 2n−|A|
+= 1
+2
+�
+Za⊂A⊂Ea
+� �
+i̸∈A
+2(mi + 1) −
+�
+i̸∈A
+2(mi)
+�
+= 1
+2
+�
+Za⊂A⊂Ea
+� �
+i̸∈Ea
+(ai + 1)
+�
+i∈Ea\A
+(ai) −
+�
+i̸∈Ea
+(ai − 1)
+�
+i∈Ea\A
+(ai − 2)
+�
+= 1
+2
+�
+i̸∈Ea
+(ai + 1)
+�
+Za⊂A⊂Ea
+�
+i∈Ea\A
+(ai) − 1
+2
+�
+i̸∈Ea
+(ai − 1)
+�
+Za⊂A⊂Ea
+�
+i∈Ea\A
+(ai − 2)
+= 1
+2
+�
+i̸∈Ea
+(ai + 1)
+�
+i∈Ea\Za
+(ai + 1) − 1
+2
+�
+i̸∈Ea
+(ai − 1)
+�
+i∈Ea\Za
+(ai − 1)
+= 1
+2
+� �
+i̸∈Za
+(ai + 1) −
+�
+i̸∈Za
+(ai − 1)
+�
+,
+which is the same number as in the upper bound (6).
+Remark 2.7 (Linear forms ℓA,k,s’s via Apolarity). The choice for our linear forms ℓA,k,s’s in this
+paper is originated from the apolarity using the ideal Ja(t) ⊂ (xa0+1
+0
+, . . . , xan+1
+n
+), which is first
+introduced in [8]. Each point of the zero set V (Ja(t)) in the projective n-space over k decides
+exactly a linear form ℓA,k,s in the present paper up to non-zero scaling.
+Example 2.8 (Case of X4
+0X3
+1X2
+2). Let a = {4, 3, 2}. Then Za = ∅, Ea = {0, 2} and m0 = 1, m1 =
+1, m2 = 0. There are four cases A ⊂ {0, 2} where A = ∅, {0}, {2}, {0, 2} and for each i we have
+F0 = (y − t2),
+F1 = (y − t),
+F2 = 1 .
+(1) A = ∅. Then, we get
+KA = KA = {(k0, k1, k2) | 0 ≤ k0 ≤ 1, 0 ≤ k1 ≤ 1, 0 ≤ k2 ≤ 0}
+= {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)}
+SA = {(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)}.
+The linear forms and coefficients indexed by (k, s) ∈ KA × SA is given by the following
+tables
+ℓ∅,k,s =
+s = 000
+001
+010
+011
+k = 000
+X0 + X1 + X2
+X0 + X1 − X2
+X0 − X1 + X2
+X0 − X1 − X2
+010
+X0 + tX1 + X2
+X0 + tX1 − X2
+X0 − tX1 + X2
+X0 − tX1 − X2
+001
+X0 + X1 + tX2
+X0 + X1 − tX2
+X0 − X1 + tX2
+X0 − X1 − tX2
+011
+X0 + tX1 + tX2
+X0 + tX1 − tX2
+X0 − tX1 + tX2
+X0 − tX1 − tX2
+C∅,k,s = (−1)0 · 20 · 1·
+s = 000
+001
+010
+011
+k = 000
+t3
+t3
+−t3
+−t3
+010
+−t2
+−t2
+t2
+t2
+001
+−t
+−t
+t
+t
+011
+1
+1
+−1
+−1
+8
+
+(2) A = {0} Then
+KA = KA = {(k1, k2) | 0 ≤ k1 ≤ 1, 0 ≤ k2 ≤ 0}
+= {(0, 0), (1, 0)}
+SA = {(0, 0), (0, 1)}.
+The linear forms and coefficients defined by (k, s) ∈ KA×SA is given by the following tables
+ℓ{0},k,s =
+-
+s = 00
+01
+k = 00
+X1 + X2
+X1 − X2
+10
+tX1 + X2
+tX1 − X2
+C{0},k,s = (−1)1 · 21 · (1 − t2)·
+-
+s = 00
+01
+k = 00
+−t
+−t
+10
+1
+1
+(3) A = {2} Then
+KA = {(k0, k1) | 0 ≤ k0 ≤ 1, 0 ≤ k1 ≤ 1}
+= {(0, 0), (0, 1), (1, 0), (1, 1)}
+KA = {(k0, k1) | 0 ≤ k0 ≤ 1, 0 ≤ k1 ≤ 1, min{k0, k1} = 0}
+= {(0, 0), (0, 1), (1, 0)}
+SA = {(0, 0), (0, 1)}.
+The linear forms and coefficients defined by (k, s) ∈ KA×SA is given by the following tables
+ℓ{2},k,s =
+-
+s = 00
+01
+k = 00
+X0 + X1
+X0 − X1
+01
+X0 + tX1
+X0 − tX1
+10
+tX0 + X1
+tX0 − X1
+11
+tX0 + tX1
+tX0 − tX1
+C{2},k,s = (−1)1 · 21 · 1·
+-
+s = 00
+01
+k = 00
+t3
+−t3
+01
+−t2
+t2
+10
+−t
+t
+11
+1
+−1
+Since tX0±tX1 and X0±X1 represent the same linear form, the proper coefficient C{2},(0,0),s
+of X0 ± X1 is ∓2(t3 + t9).
+(4) A = {0, 2} Then
+KA = {(k1) | 0 ≤ k1 ≤ 1}
+= {(0), (1)}
+KA = {(k1) | 0 ≤ k1 ≤ 1, min{k1} = 0}
+= {(0)}
+SA = {(0)}.
+The linear forms and coefficients defined by (k, s) ∈ KA×SA is given by the following tables
+ℓ{0,2},k,s =
+-
+s = 0
+k = 0
+X1
+1
+tX1
+9
+
+C{0,2},k,s = (−1)2 · 22 · (1 − t2) · 1·
+-
+s = 0
+k = 0
+−t
+1
+1
+Since the linear forms X1 and tX1 represent the same linear form, the proper coefficient
+C{0,2},(0),(0) of linear form X1 is 4(1 − t2)(−t + t9) = −4t11 + 4t9 + 4t3 − 4t.
+Finally, we compute
+D{4,3,2} = 22 ·
+�
+9
+4, 3, 2
+�
+· (t4 − t2)(t3 − t) = 5040t3(t2 − 1)2 .
+So, the identity (4) leads to
+5040t3(t2 − 1)2X4
+0X3
+1X2
+2 = t3(X0 + X1 + X2)9 + t3(X0 + X1 − X2)9 − t3(X0 − X1 + X2)9
+− t3(X0 − X1 − X2)9 − t2(X0 + tX1 + X2)9 − t2(X0 + tX1 − X2)9 + t2(X0 − tX1 + X2)9
++ t2(X0 − tX1 − X2)9 − t(X0 + X1 + tX2)9 − t(X0 + X1 − tX2)9 + t(X0 − X1 + tX2)9
++ t(X0 − X1 − tX2)9 + (X0 + tX1 + tX2)9 + (X0 + tX1 − tX2)9 − (X0 − tX1 + tX2)9
+− (X0 − tX1 − tX2)9 − 2(1 − t2)(−t)(X1 + X2)9 − 2(1 − t2)(−t)(X1 − X2)9
+− 2(1 − t2)(tX1 + X2)9 − 2(1 − t2)(tX1 − X2)9 − 2(t3 + t9)(X0 + X1)2 + 2(t3 + t9)(X0 − X1)9
+− 2(−t2)(X0 + tX1)9 − 2t2(X0 − tX1)9 − 2(−t)(tX0 + X1)9 − 2t(tX0 − X1)9
++ 4(1 − t2)(−t + t9)X9
+1
+and as dividing by D{4,3,2} = 5040t3(t2 − 1)2 we eventually have
+X4
+0X3
+1X2
+2 =
+1
+5040
+�
+1
+(t2 − 1)2 (X0 + X1 + X2)9 +
+1
+(t2 − 1)2 (X0 + X1 − X2)9 −
+1
+(t2 − 1)2 (X0 − X1 + X2)9
+−
+1
+(t2 − 1)2 (X0 − X1 − X2)9 −
+1
+t(t2 − 1)2 (X0 + tX1 + X2)9 −
+1
+t(t2 − 1)2 (X0 + tX1 − X2)9
++
+1
+t(t2 − 1)2 (X0 − tX1 + X2)9 +
+1
+t(t2 − 1)2 (X0 − tX1 − X2)9 −
+1
+t2(t2 − 1)2 (X0 + X1 + tX2)9
+−
+1
+t2(t2 − 1)2 (X0 + X1 − tX2)9 +
+1
+t2(t2 − 1)2 (X0 − X1 + tX2)9 +
+1
+t2(t2 − 1)2 (X0 − X1 − tX2)9
++
+1
+t3(t2 − 1)2 (X0 + tX1 + tX2)9 +
+1
+t3(t2 − 1)2 (X0 + tX1 − tX2)9 −
+1
+t3(t2 − 1)2 (X0 − tX1 + tX2)9
+−
+1
+t3(t2 − 1)2 (X0 − tX1 − tX2)9 −
+2
+t2(t2 − 1)(X1 + X2)9 −
+2
+t2(t2 − 1)(X1 − X2)9
++
+2
+t3(t2 − 1)(tX1 + X2)9 +
+2
+t3(t2 − 1)(tX1 − X2)9 − 2(t6 + 1)
+(t2 − 1)2 (X0 + X1)2 + 2(t6 + 1)
+(t2 − 1)2 (X0 − X1)9
++
+2
+t(t2 − 1)2 (X0 + tX1)9 −
+2
+t(t2 − 1)2 (X0 − tX1)9 +
+2
+t2(t2 − 1)2 (tX0 + X1)9 −
+2
+t2(t2 − 1)2 (tX0 − X1)9
+− 4(t2 + 1)(t4 + 1)
+t2
+X9
+1
+�
+,
+which gives us a 1-dimensional family of a Waring decomposition of X4
+0X3
+1X2
+2 (i.e. for all t ∈ k
+with 5040t3(t2 − 1)2 ̸= 0) of 1
+2(5 · 4 · 3 − 3 · 2 · 1) = 27 summands.
+3. Case of arbitrary homogeneous form
+In this section, we consider some consequence of the results in Section 2. Since a homogeneous
+form in k[X0, . . . , Xn] can be written as a k-linear combination of monomials of the same degree,
+10
+
+by our previous decomposition for a monomial, naturally we have a family of explicit Waring de-
+compositions of any homogeneous polynomial with k-coefficients. Finding such a sum of powers of
+linear forms representation of a given degree form is quite important in many areas of mathematics.
+For instance, when k = Q, it is closely related to the problem of integrating a polynomial function
+over a rational simplex, which is fundamental for applications such as discrete optimization, finite
+element methods in numerical analysis, and algebraic statistics computation (see e.g. [2, section 1]
+and references therein). For computational complexity of this integration problem, Waring decom-
+position can be used to obtain a polynomial time algorithm for evaluating integrals of polynomials
+of some fixed constraint (see [2, 3.3, 3.4]). Let’s briefly review some aspect of their result.
+Let ∆ be a k-dimensional rational simplex inside Rn and let f ∈ Q[X0, . . . , Xn] be a homogeneous
+polynomial with rational coefficients.
+To compute
+�
+∆ fdm, where dm is the integral Lebesgue
+measure on the affine hull ⟨∆⟩ of the simplex (see [2, 2.1] for the precise definition), we recall a
+useful formula due to M. Brion as follows.
+Proposition 3.1. (Brion) Let ∆ be the simplex that is the convex hull of (k+1) affinely independent
+vertices s1, s2, . . . , sk+1 in Rn. Let ℓ be a linear form which is regular w.r.t. ∆, i.e., ⟨ℓ, si⟩ ̸= ⟨ℓ, sj⟩
+for any pair i ̸= j. Then we have the following relation
+(7)
+�
+∆
+ℓDdm = k! vol(∆, dm)
+D!
+(D + k)!
+�k+1
+�
+i=1
+⟨ℓ, si⟩D+k
+�
+j̸=i⟨ℓ, si − sj⟩
+�
+.
+We note that, even when ℓ is not regular, there exists a similar expansion of the integral as a
+sum of residues (see e.g. [2, corollary 13]).
+Thus, once we represent a polynomial by a sum of power of rational linear forms, the integration
+immediately follows. And bounding the number of rational linear forms in the sum and finding
+all its Q-coefficients are among the main issues for the computational complexity of evaluating
+integrals of polynomials over a rational simplex.
+Now, let us consider the number of the summands in a given rational Waring decomposition of
+fgen, a general homogeneous polynomial of degree D in (n+1)-variables (here, we mean a ‘general’
+form by the one having all the monomials of total degree D).
+In [2], the authors used the following well-known identity, which is somewhat naive from the
+viewpoint of Waring rank, to consider their rational decomposition of fgen,
+(8)
+Xa0
+0 Xa1
+1 · · · Xan
+n
+= 1
+D!
+�
+0≤pi≤ai
+(−1)D−(p0+···+pn)�a0
+p0
+�
+· · ·
+�an
+pn
+�
+(p0X0 + · · · + pnXn)D ,
+where a = (a0, . . . , an) and D = |a| = a0 + · · · + an. To count the number of summands properly,
+one should group together proportional linear forms among the whole decomposition of fgen. The
+concept of primitive vectors (i.e. (p0, . . . , pn) ∈ Zn
+≥0 with gcd(p0, . . . , pn) = 1) precisely captures this
+number. Let F(n, D) be this number of minimal summands in the rational Waring decomposition
+of fgen using (8). Then, it is shown in [2, lemma 16] that F(n, D) is equal to
+(9)
+|{(p0, . . . , pn) ∈ Zn
+≥0, gcd(p0, . . . , pn) = 1, 1 ≤
+�
+i
+pi ≤ D}| =
+D
+�
+d=1
+µ(d) ·
+��n + 1 + ⌊D
+d ⌋
+n + 1
+�
+− 1
+�
+,
+where µ is the M¨obius function.
+Now, let us estimate this number as increasing the total degree in a fixed number of variables.
+As getting large D with a fixed n, by (9) the asymptotic behavior of F(n, D) can be calculated as
+(10)
+F(n, D) = δ(n, D)
+(n + 1)!Dn+1 + O(Dn) ,
+11
+
+where δ(n, D) =
+D
+�
+d=1
+µ(d)
+dn+1 . Since the Dirichlet series that generates the M¨obius function is the
+inverse of the Riemann zeta function ζ(s), which converges for Re(s) > 1, we see that δ(n, D) →
+1
+ζ(n+1) as D → ∞ and F(n, D) asymptotically has order of Dn+1 in this setting.
+On the other hand, if we regard the rational Waring decomposition of fgen by considering each
+monomial summand using our result, that is, Corollary 2.5, much less linear forms are needed to
+represent the polynomial asymptotically. Let K(n, D) be the number of minimal summands in the
+rational Waring decomposition of fgen via (4).
+Theorem 3.2. Let n ≥ 1, D ≥ 1 be positive integers and K(n, D) be as above. Then, we have the
+following formula
+(11)
+K(n, D) =
+n+1
+�
+r=1
+��⌊D−r
+2 ⌋ + r
+r
+�
+−
+�⌊D−r
+2 ⌋
+r
+��
+· 2r−1 ·
+�n + 1
+r
+�
+.
+In particular, when n is fixed and D → ∞, we have
+(12)
+K(n, D) = n + 1
+n!
+Dn + O(Dn−1) ,
+which has order of Dn asymptotically.
+Proof. Let L{i1,...,ir} be the set of all linear forms appeared in the Waring decomposition of fgen
+via (4) such that every member of L{i1,...,ir} is of the form λ1Xi1 + λ2Xi2 + · · · + λrXir for some
+nonzero λu’s. Then, by the proof of Corollary 2.5 we have
+L{i1,...,ir} :=
+�
+r
+�
+j=1
+(−1)sij tkij Xij |
+r
+�
+j=1
+aij = D, 0 ≤ kij ≤ mij = ⌊aij − 1
+2
+⌋
+, min{kij} = 0, sij ∈ {0, 1}, and si1 = 0
+�
+.
+In this set, each (ki1, ki2, . . . , kir) should satisfy the following inequality
+0 ≤
+r
+�
+j=1
+kij ≤
+r
+�
+j=1
+mij =
+r
+�
+j=1
+⌊aij − 1
+2
+⌋ · · · (∗∗)
+and note that both {ai1, ai2, . . . , air} and {ai1, . . . , aip −1, . . . , aiq +1, . . . , air} have the same r-tuple
+of kij’s satisfying (∗∗) whenever aip and aiq are all even. Thus, we can assume that at most one
+of aij is even. If D ≡ r (mod 2) and all the aij are odd, �r
+j=1 mij = �r
+j=1
+aij −1
+2
+= D−r
+2 . If
+D ̸≡ r(mod 2) and all the aij are odd except one, �r
+j=1 mij = D−r−1
+2
+. Hence, for a given D, we
+get
+L{i1,...,ir} :=
+�
+r
+�
+j=1
+(−1)sij tkij Xij | 0 ≤
+r
+�
+j=1
+kij ≤ ⌊D − r
+2
+⌋, min{kij} = 0, sij ∈ {0, 1}, and si1 = 0
+�
+so that the number of elements in L{i1,...,ir} as follows
+|L{i1,i2,...,ir}| =
+����
+�
+(kij) | 0 ≤
+r
+�
+j=1
+kij ≤ ⌊D − r
+2
+⌋, min{kij} = 0
+����� ·
+����
+�
+(sij) | sij ∈ {0, 1}, and si1 = 0
+�����
+=
+��⌊D−r
+2 ⌋ + r
+r
+�
+−
+�⌊D−r
+2 ⌋
+r
+��
+· 2r−1 .
+Finally, as multiplying
+�n+1
+r
+�
+for the possible choices for the indices {i1, . . . , ir} ⊂ {0, . . . , n}, the
+equation (11) is obtained.
+12
+
+As getting large D with fixed n, the highest power of D is obtained when r = n + 1. So, we
+estimate as
+K(n, D) ≈
+�� D−(n+1)
+2
++ n + 1
+n + 1
+�
+−
+� D−(n+1)
+2
+r
+��
+· 2n ·
+�n + 1
+n + 1
+�
+=
+�� D+(n+1)
+2
+n + 1
+�
+−
+� D−(n+1)
+2
+r
+��
+· 2n
+=
+�Dn+1 + (n + 1)2Dn + O(Dn−1)
+2n+1(n + 1)!
+− Dn+1 − (n + 1)2Dn + O(Dn−1)
+2n+1(n + 1)!
+�
+· 2n
+= n + 1
+n!
+Dn + O(Dn−1) ,
+where the asymptotic order is Dn which is better than Dn+1 in the case of F(n, D).
+□
+Remark 3.3. We make some remarks on the theorem above.
+(a) The formula (11) gives a new upper bound for Q-Waring rank of arbitrary rational homo-
+geneous polynomial. Unfortunately, this is not better than a naive bound
+�D+n
+n
+�
+, which
+comes from dim Q[X0, X1, . . . , Xn], asymptotically. But, the latter approach does not give
+an explicit linear forms of the power sum decomposition and its coefficients (remember that
+one should execute a massive computation to find them), whereas our method does provide
+a completely determined(!) power sum decomposition.
+(b) Note that F(n, D) and K(n, D) have different orders in the above asymptote by (10) and
+(12) (see also Table 1 for a significant difference between F(n, D) and K(n, D)).
+(c) In [3] the authors also provide a Waring decomposition of any monomial with determined
+coefficients over the complex number C. Since their rank is better than the bound (2), the
+approach using their decomposition would be surely better than the method via (4) for
+decomposing any homogeneous form. But, note that in their coefficient formula a complex
+number does occur in most cases (!), which puts a serious limitation for applying their
+method in application over real or rational numbers.
+(n, D)
+(2, 10)
+(2,50)
+(2,100)
+(3,10)
+(3,50)
+(3,100)
+(5,30)
+(5,50)
+(5,100)
+F(n, D)
+205
+18,970
+144,871
+831
+286,893
+4,207,287
+1,884,921
+31,651,125
+1,669,982,466
+K(n, D)
+133
+3,613
+14,713
+696
+83,416
+666,816
+1,305,092
+16,001,276
+502,701,736
+Table 1. Comparison of numbers of summands in the two Waring decompositions
+of fgen, a general homogeneous polynomial of degree D in n + 1 variables, based on
+a previously known method (8) in [2] and the method in this paper (4)
+4. Macaulay2 code for the decompositions
+Finally, we present a Macaulay2[10] code which computes the Waring-type polynomial identity
+concerning any given monomial over k in Section 2 and we execute it for X4
+0X3
+1X2
+2, the case in
+Example 2.8.
++ M2 −−no−r e ad l i n e −−print−width 79
+Macaulay2 ,
+version
+1.17
+with
+packages :
+ConwayPolynomials ,
+Elimination ,
+IntegralClosure ,
+InverseSystems , LLLBases ,
+MinimalPrimes ,
+PrimaryDecomposition ,
+ReesAlgebra ,
+Saturation ,
+TangentCone
+13
+
+i1
+: −−For a given
+sequence a ,
+find
+a l l
+the
+pairs
+(A, k , s )
+A k s l i s t e r=method ( ) ;
+i2
+:
+A k s l i s t e r ( List ):=a−>(
+n:=#a−1;
+A k s l i s t :={};
+E:= f or
+i
+from 0 to n
+l i s t
+i f ( even ( a i ))
+then
+i
+e l s e
+continue ;
+Z:= f or
+i
+from 0 to n
+l i s t
+i f ( a i ==0) then
+i
+e l s e
+continue ;
+Alist0 := subsets ( toList ( set (E)− set (Z ) ) ) ;
+A l i s t := f or
+i
+from 0 to #Alist0 −1 l i s t
+sort (Z | A l i s t 0 i ) ;
+f or
+i1
+from 0 to #Alist −1 do
+i f (#( A l i s t i 1 )!=n+1) then
+(
+A:= A l i s t i 1 ;
+notInA:= sort
+toList ( set ( 0 . . n)− set (A) ) ;
+KA:= toList (( f or
+i
+from 0 to #notInA−1 l i s t
+0 ) . .
+( f or
+i
+from 0 to #notInA−1 l i s t
+f l o o r (( a ( notInA i ) −1)/2)));
+KAbar:= f or
+i
+from 0 to #KA−1 l i s t
+i f (min( KA i)==0)
+then KA i
+e l s e
+continue ;
+SA:= toList (( f or
+i
+from 0 to #notInA−1 l i s t
+0 ) . .
+({0}| f or
+i
+from 1 to #notInA−1 l i s t
+1 ) ) ;
+f or
+i2
+from 0 to #KAbar−1 do (
+kk:=KAbar i2 ;
+f or
+i3
+from 0 to #SA−1 do(
+ss :=SA i3 ;
+A k s l i s t=append( Akslist ,{A, notInA , kk , ss }) ;
+)
+)
+)
+e l s e
+continue ;
+A k s l i s t
+) ;
+i3
+: −−For a given
+pair
+(A, k , s ) ,
+find a
+l i n e a r
+form
+l {A, k , s}
+linform=method ( ) ;
+i4
+:
+linform ( Ring ,
+List ):=(R, Aks)−>(
+(A, notInA , kk , I ):= toSequence (Aks ) ;
+bR:= baseRing (R) ;
+t0 :=sub (2 ,R) ;
+i f ( numgens bR!=0)
+then
+t0=bR 0 ;
+sum f or
+i
+from 0 to #notInA−1 l i s t
+( −1)ˆ( I i )∗( t0 )ˆ( kk i )∗R ( notInA i )
+) ;
+i5
+: −−For a given monomial ,
+find
+a l l
+the
+l i n e a r
+forms
+14
+
+linforms=method ( ) ;
+i6
+:
+linforms ( RingElement ):=(mon)−>(
+R:= ring mon;
+a:=( exponents mon) 0 ;
+A k s l i s t := A k s l i s t e r ( a ) ;
+f or
+i
+from 0 to #Akslist −1 l i s t
+linform (R, A k s l i s t i )
+) ;
+i7
+: −−For a given
+sequence a ,
+find
+F i
+F l i s t=method ( ) ;
+i8
+:
+F l i s t ( List , Ring , ZZ):=( a , S , ind)−>(
+mind:= f l o o r (( a ind −1)/2);
+bS=baseRing (S ) ;
+t0 :=sub (2 ,S ) ;
+i f ( numgens bS!=0)
+then
+t0=bS 0 ;
+i f (mind<=0) then sub (1 ,S)
+e l s e
+sub ( product
+f or
+i
+from 1 to mind
+l i s t
+S 0 −(t0 )ˆ( a ind −2∗ i ) , S)
+) ;
+i9
+: −−For a given
+pair
+(A, k , s ) ,
+find a
+c o e f f i c i e n t
+C {A, k , s}
+c f s=method ( ) ;
+i10
+:
+c f s ( List , Ring , List ):=( a ,R, Aks)−>(
+(A, notInA , kk , ss ):= toSequence (Aks ) ;
+bR:=baseRing (R) ;
+S:=bR[Y] ;
+C1:=(−2)ˆ(#A)∗ product
+f or
+i
+from 0 to #A−1 l i s t
+sub ( F l i s t (a , S , A i ) , sub ( matrix {{1}} ,bR ) ) ;
+C2:= product
+f or
+i
+from 0 to #notInA−1 l i s t
+sub ( c o e f f i c i e n t (( S 0 )ˆ( kk i ) , F l i s t (a , S , notInA i ) ) ,bR) ;
+C3:=sub (( −1)ˆ(sum f or
+i
+from 0 to #ss −1 l i s t
+a ( notInA i )∗ s s i ) ,bR) ;
+C1∗C2∗C3
+) ;
+i11
+: −−For a given
+pair
+(A, k , s ) ,
+find a
+p r i n c i p l e
+c o e f f i c i e n t
+C {A, k , s}
+pcfs=method ( ) ;
+i12
+:
+pcfs ( List , Ring , List ):=( a ,R, Aks)−>(
+(A, notInA , kk , ss ):= toSequence (Aks ) ;
+bR:=baseRing (R) ;
+t0 :=sub (2 ,R) ;
+15
+
+i f ( numgens bR!=0)
+then t0=bR 0 ;
+sum f or
+j
+from 0 to min( f or
+i
+from 0 to #notInA−1
+l i s t
+f l o o r (( a ( notInA i )−1)/2)− kk i )
+l i s t
+( t0 )ˆ( j ∗(sum a ))∗
+c f s (a ,R,{A, notInA , kk+( f or
+i
+from 0 to #notInA−1 l i s t
+j ) , ss })
+) ;
+i13
+: −−For a given monomial ,
+find
+the
+c o e f f i c i e n t
+D a
+D=method ( ) ;
+i14
+: D( RingElement ):=(mon)−>(
+a:=( exponents (mon)) 0 ;
+R:= ring mon;
+bR:=baseRing (R) ;
+S:=bR[Y] ;
+t0 :=sub (2 ,R) ;
+i f ( numgens bR!=0)
+then t0=bR 0 ;
+Z:= f or
+i
+from 0 to #a−1 l i s t
+i f ( a i ==0) then
+i
+e l s e
+continue ;
+sub((−1)ˆ(#Z)∗2ˆ(#a −1)∗((sum a ) ! / ( product
+f or
+i
+from 0 to #a−1 l i s t
+( a i ) ! ) ) ∗
+( product
+f or
+i
+from 0 to #a−1
+l i s t
+sub ( F l i s t (a , S , i ) , matrix {{ t0 ˆ( a i )}})) ,bR)
+) ;
+i15
+: −−For a given monomial ,
+find
+a l l
+the
+c o e f f i c i e n t s
+c o e f f s=method ( ) ;
+i16
+:
+c o e f f s ( RingElement ):=(mon)−>(
+R:= ring mon;
+a:=( exponents (mon)) 0 ;
+pl := A k s l i s t e r ( a ) ;
+f or
+i
+from 0 to #pl −1 l i s t
+pcfs (a ,R, p l i )
+) ;
+i17
+: −−Test
+f or m=X 0ˆ4X 1ˆ3X 2ˆ2
+−−Ring over a
+f r a c t i o n a l
+f i e l d
+of Q[ t ]
+T=QQ[ t ]
+o17 = T
+o17
+:
+PolynomialRing
+i18
+:
+fT=f r ac T
+o18 = fT
+16
+
+o18
+:
+FractionField
+i19
+: R=fT [ X 0 . . X 2 ]
+o19 = R
+o19
+:
+PolynomialRing
+i20
+: m=X 0ˆ4∗X 1ˆ3∗X 2ˆ2
+4 3 2
+o20 = X X X
+0 1 2
+o20
+: R
+i21
+: −−linearforms
+r e l at e d
+to m
+l s=linforms (m) ;
+i22
+: −−corresponding
+c o e f f i c i e n t s
+r e l at e d
+to m
+cs=c o e f f s (m) ;
+i23
+: −−check
+the
+equality
+of RHS and LHS of
+the
+Corollary
+rhs=sum f or
+i
+from 0 to #cs −1 l i s t
+c s i ∗( l s i )ˆ(( degree m) 0 )
+7
+5
+3
+4 3 2
+o23 = (5040 t
+− 10080 t
++ 5040 t
+)X X X
+0 1 2
+o23
+: R
+i24
+:
+lhs=D(m)∗m
+7
+5
+3
+4 3 2
+o24 = (5040 t
+− 10080 t
++ 5040 t
+)X X X
+0 1 2
+o24
+: R
+i25
+:
+lhs==rhs
+o25 = true
+17
+
+Acknowledgments
+The first author was supported by the National Research Foundation of Korea (NRF) grant
+funded by the Korea government (MSIT) (No. 2021R1F1A104818611) and the second author was
+supported by KIAS Individual Grant (MG083101) at Korea Institute of Advanced Study (KIAS).
+References
+[1] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom., 4 (1995), no. 2,
+201–222.
+[2] Baldoni, V., Berline, N., De Loera, J., K¨oppe, M., and Vergne, M., How to integrate a polynomial over a simplex,
+Mathematics of Computation, 80 (no. 273), 297–325 (2011).
+[3] W. Buczy´nska, J. Buczy´nski, Z. Teitler, Waring decompositions of monomials, J. Algebra 378, 45–57 (2013).
+[4] M. Boij, E. Carlini, A. V. Geramita, Monomials as sums of powers: the real binary case, Proc. Am. Math. Soc.
+139, 3039–3043 (2011).
+[5] E. Carlini, M. V. Catalisano, A.V. Geramita, The solution to Waring’s problem for monomials and the sum of
+coprime monomials, J. Algebra 370, 5–14 (2012).
+[6] E. Carlini, M. Kummer, A. Oneto and E. Ventura, On the real rank of monomials, Math. Z., 286 (2017), 571–577.
+[7] C. J. Hillar and L.-H. Lim, Most tensor problems are NP-hard, Journal of the ACM. 60 (6) (2013), 1–39.
+[8] K. Han and H. Moon, A New Bound for the Waring Rank of Monomials, SIAM J. Appl. Algebra Geom. 6 (3)
+(2022), 407–‌431.
+[9] A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lect. Notes in Math. vol.
+1721, Springer-Verlag, Berlin, Appendix C by Iarrobino and S.L. Kleiman, 1999.
+[10] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry,
+http://www.math.uiuc.edu/Macaulay2/.
+Kangjin Han, School of Undergraduate Studies, Daegu-Gyeongbuk Institute of Science & Tech-
+nology (DGIST), Daegu 42988, Republic of Korea
+Email address: [email protected]
+Hyunsuk Moon, School of Mathematics, Korea Institute for Advanced Study (KIAS), Seoul 02455,
+Republic of Korea
+Email address: [email protected], [email protected]
+18
+

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+Theoretical Analysis of Offline Imitation With Supplementary
+Dataset
+Ziniu Li *1,2, Tian Xu∗3, Yang Yu†3, and Zhi-Quan Luo†1,2
+1The Chinese University of Hong Kong, Shenzhen
+2Shenzhen Research Institute of Big Data
+3National Key Laboratory for Novel Software Technology, Nanjing University
+January 30, 2023
+Abstract
+Behavioral cloning (BC) can recover a good policy from abundant expert data, but may fail when expert
+data is insufficient. This paper considers a situation where, besides the small amount of expert data, a
+supplementary dataset is available, which can be collected cheaply from sub-optimal policies. Imitation
+learning with a supplementary dataset is an emergent practical framework, but its theoretical foundation
+remains under-developed. To advance understanding, we first investigate a direct extension of BC, called
+NBCU, that learns from the union of all available data. Our analysis shows that, although NBCU suffers
+an imitation gap that is larger than BC in the worst case, there exist special cases where NBCU performs
+better than or equally well as BC. This discovery implies that noisy data can also be helpful if utilized
+elaborately. Therefore, we further introduce a discriminator-based importance sampling technique to
+re-weight the supplementary data, proposing the WBCU method. With our newly developed landscape-based
+analysis, we prove that WBCU can outperform BC in mild conditions. Empirical studies show that WBCU
+simultaneously achieves the best performance on two challenging tasks where prior state-of-the-art methods
+fail.
+1
+Introduction
+Imitation learning (IL) methods train a good policy from expert demonstrations [Argall et al., 2009, Osa
+et al., 2018]. One popular approach is behavioral cloning (BC) [Pomerleau, 1991], which imitates the
+expert via supervised learning. Specifically, in IL, samples usually refer to state-action pairs/sequences
+from trajectories in the given dataset. Both the quality and quantity of trajectories are crucial to achieving
+satisfying performance. For instance, it is found that BC performs well when the dataset has a large amount
+of expert-level trajectories; see, e.g., [Spencer et al., 2021]. Nevertheless, due to the compounding errors
+issue [Ross and Bagnell, 2010], any offline IL algorithm, including BC, will fail when the number of expert
+trajectories is small [Rajaraman et al., 2020, Xu et al., 2021a]. To address the compounding errors issue, a
+naive solution is to ask the expert to collect more trajectories. However, querying the expert is expensive and
+intractable in applications such as healthcare and industrial control.
+To address the mentioned failure mode, we follow an alternative and emergent framework proposed in
+[Kim et al., 2022b, Xu et al., 2022a], which assumes, in addition to the expert dataset, a supplementary dataset
+is relatively cheap to obtain. In particular, this supplementary dataset could be previously collected by
+certain behavior policies; as such, it has lots of expert and sub-optimal trajectories. The key challenge here is
+*Equal contribution. Author ordering is determined by coin flip. Email: [email protected] and [email protected]
+†Corresponding author. Email: [email protected] and [email protected]
+1
+arXiv:2301.11687v1  [cs.LG]  27 Jan 2023
+
+to figure out which supplemental samples are helpful. We realize that a few advances have been achieved in
+this direction [Kim et al., 2022b,a, Xu et al., 2022a, Ma et al., 2022]. To leverage the supplementary dataset, a
+majority of algorithms train a discriminator to distinguish expert-style and sub-optimal samples, upon which
+a weighted BC objective is optimized to learn a good policy. For instance, Kim et al. [2022b] proposed the
+DemoDICE algorithm, which learns the discriminator via a regularized state-action distribution matching
+objective. As for the DWBC algorithm in [Xu et al., 2022a], the policy and discriminator are jointly trained
+under a cooperation framework.
+Though prior algorithms are empirically shown to perform well in some scenarios, the theoretical
+foundation of this new imitation problem remains under-developed. Specifically, researchers have a sketchy
+intuition that noisy demonstrations in the supplementary dataset may hurt the performance when we
+directly apply BC, and researchers hope to develop algorithms to overcome this challenge. However, the
+following questions have not been carefully answered yet: (Q1) precisely, what kind of noisy samples is
+harmful? (or what kind of supplementary samples is helpful?) (Q2) how to design algorithms in a principled
+way? Answers could deepen our understanding and provide insights for future advances.
+In this paper, we explore the above questions by investigating two (independent) variants of BC. To learn
+a policy, both algorithms apply a BC objective on the union dataset (the concatenation of the expert and
+supplementary dataset), but they differ in assigning weights of samples that appear in the loss function. To
+qualify the benefits of the supplementary dataset, we treat the BC only with the expert dataset as a baseline
+to compare.
+The first algorithm, called NBCU (naive BC with union dataset), assigns uniform weights for all samples.
+As a direct extension of BC, the theoretical study of NBCU provides answers to (Q1). In particular, we show
+that NBCU suffers an imitation gap that is larger than BC in the worst case (Theorem 2 and Proposition 1),
+implying its inferior performance in the general case. However, we discover special cases where NBCU
+performs better than or equally well as BC (Theorem 3 and Theorem 4). This discovery indicates that
+noisy data can also be helpful if utilized elaborately. To our best knowledge, such a message is new to the
+community. We provide the empirical evidence in Figure 1 and discuss these results in depth later.
+As NBCU (i.e., the direct extension of BC) may fail in the worst case, we develop another algorithm called
+WBCU (weighted BC with union dataset) to address this failure mode. In light of [Kim et al., 2022b, Xu et al.,
+2022a], WBCU trains a discriminator to weigh samples. We use the importance sampling technique [Shapiro,
+2003, Chapter 9] to design the weighting rule. Unlike prior works [Kim et al., 2022b, Xu et al., 2022a] that use
+a biased (or regularized) weighting rule, WBCU is theoretically sound and principled in the sense that the
+loss function is corrected as if samples from the supplementary dataset were collected by the expert policy;
+see Remark 1 for detailed discussion.
+Theoretically, we justify WBCU via a new landscaped-based analysis (Theorem 5). We characterize
+the landscape properties (e.g., the Lipschitz continuity and quadratic growth conditions) in Lemma 1 and
+Lemma 2. Interestingly, we find that the “smooth” function approximation for the discriminator plays a
+vital role in WBCU; without it, WBCU cannot be better than BC (Proposition 2). Our theory can provide
+actionable guidance for practitioners; see Section 5.2 for details. These results offer answers to (Q2).
+Finally, we corroborate our claims with experiments on the MuJoCo locomotion control. Given the same
+expert dataset (1 expert trajectory), we consider two tasks with different supplementary datasets. The first
+task (called full replay) follows the setting in [Kim et al., 2022b]: supplementary trajectories are from the
+replay buffer of an online SAC agent [Haarnoja et al., 2018]. The second task (called noisy expert) is a new
+test-bed: the supplementary dataset contains clean expert-style trajectories, and the noisy counterparts1.
+Please refer to Appendix D.1 for experiment details. Please see Figure 1 for the (averaged) normalized
+scores of learned policies over four representative MuJoCo environments2 (performance on individual
+environments is reported in Table 2 and Table 3 in Appendix). We find that WBCU simultaneously achieves
+the best performance on both two tasks, whereas prior state-of-the-art methods like DemoDICE [Kim et al.,
+2022b] and DWBC [Xu et al., 2022a] only perform well on one of two tasks. We will connect the experiment
+results with theoretical analysis in the main text.
+1Noisy counterparts cover expert states but are injected with action noise.
+2Ant-v2, HalfCheetah-v2, Hopper-v2, and Walker2d-v2.
+2
+
+Noisy Expert
+Full Replay
+Task
+0
+20
+40
+60
+80
+Normalized Score (%)
+38
+38
+49
+94
+62
+44
+35
+92
+64
+94
+Algorithm
+BC
+DemoDICE
+DWBC
+NBCU
+WBCU
+Figure 1: Averaged normalized scores of learned policies (over 4 MuJoCo environments) on two tasks with
+different supplementary datasets. A larger score means a better performance. Experiments show that 1)
+NBCU is worse than BC for the noisy expert task and better than BC for the full replay task; 2) only our
+method WBCU can perform well on both two tasks.
+2
+Related Work
+Adversarial Imitation Learning. Unlike BC, adversarial imitation learning (AIL) methods (e.g., GAIL [Ho
+and Ermon, 2016]) do not suffer the compounding errors issue when the expert trajectories are limited;
+see the empirical evidence in [Ho and Ermon, 2016, Ghasemipour et al., 2019, Kostrikov et al., 2019] and
+theoretical support in [Xu et al., 2020, 2022b]. In particular, AIL methods train a discriminator to perform the
+state-action distribution matching, which differs from BC. Nevertheless, AIL methods naturally work in
+the online setting (i.e., the interaction is allowed). If the transition function is not available, BC is minimax
+optimal [Rajaraman et al., 2020, Xu et al., 2021b] and offline counterparts of AIL are not better than BC [Li
+et al., 2022].
+Perhaps surprisingly, we find that the discriminator used in our WBCU algorithm plays a different role
+than AIL methods. We provide a detailed explanation in Section 5.
+Imitation Learning with Imperfect Demonstrations. The problem considered in this paper is closely
+related to imitation learning (IL) with imperfect demonstrations [Wu et al., 2019, Brown et al., 2019, Tangkaratt
+et al., 2020, Wang et al., 2021, Sasaki and Yamashina, 2021, Liu et al., 2022] in the sense that the supplementary
+dataset can also be viewed as imperfect demonstrations. However, our problem setting differs from IL with
+imperfect demonstrations in two key aspects. First, in IL with imperfect demonstrations, they either pose
+strong assumptions [Tangkaratt et al., 2020, Sasaki and Yamashina, 2021, Liu et al., 2022] or require auxiliary
+information (e.g., confidence scores on imperfect trajectories) on the imperfect dataset [Wu et al., 2019, Brown
+et al., 2019]. In contrast, we assume access to a small number of expert trajectories, which could be more
+practical. Second, most works [Wu et al., 2019, Brown et al., 2019, Tangkaratt et al., 2020, Wang et al., 2021]
+in IL with imperfect demonstrations require online environment interactions while we focus on the offline
+setting.
+Finally, we point out that the problem of offline imitation learning with a supplementary dataset is first
+considered in [Kim et al., 2022b, Xu et al., 2022a]. Subsequently, Ma et al. [2022], Kim et al. [2022a] study a
+related setting: learning from observation, where actions are missing and only states are observed. Existing
+works mainly focus on empirical studies, and the theoretical foundation remains under-developed.
+3
+
+3
+Preliminary
+Markov Decision Process. In this paper, we consider the episodic Markov decision process3 (MDP) M =
+(S, A, P, r, H, ρ) [Puterman, 2014]. The first two elements S and A are the state and action space, respectively.
+H is the maximum length of a trajectory and ρ is the initial state distribution. P = {P1, · · · , PH} specifies
+the non-stationary transition function of this MDP; concretely, Ph(sh+1|sh, ah) determines the probability of
+transiting to state sh+1 conditioned on state sh and action ah in time step h, for h ∈ [H], where the symbol
+[x] means the set of integers from 1 to x. Similarly, r = {r1, · · · , rH} is the reward function, and we assume
+rh : S × A → [0, 1], for h ∈ [H]. A time-dependent policy is πh : S → ∆(A), where ∆(A) is the probability
+simplex. πh(a|s) gives the probability of selecting action a on state s in time step h, for h ∈ [H]. When the
+context is clear, we simply use π to denote the collection of time-dependent policies {πh}H
+h=1.
+We measure the quality of a policy π by the policy value (i.e., environment-specific long-term return):
+V(π) = E
+�
+∑H
+h=1 r(sh, ah) | s1 ∼ ρ; ah ∼ πh(·|sh), sh+1 ∼ Ph(·|sh, ah), ∀h ∈ [H]
+�
+. To facilitate later analysis,
+we need to introduce the state-action distribution dπ
+h (s, a):
+dπ
+h (s, a) = P
+�
+sh = s, ah = a | s1 ∼ ρ; aℓ ∼ π(·|sℓ), sℓ+1 ∼ Pℓ(·|sℓ, aℓ), ∀ℓ ∈ [h]
+�
+.
+Here, we use the convention that dπ is the collection of all time-dependent state-action distributions.
+Sometimes, we need to consider the state distribution dπ(s), which shares the same definition with dπ(s, a)
+except that we only compute the probability of visiting a specific state s. It is easy to see that dπ
+h (s) =
+∑a dπ
+h (s, a), ∀h ∈ [H].
+Imitation Learning. The goal of imitation learning is to learn a high quality policy directly from expert
+demonstrations. To this end, we often assume there is a nearly optimal expert policy πE that could interact
+with the environment to generate a dataset (i.e., NE trajectories of length H):
+DE =
+�
+tr = (s1, a1, s2, a2, · · · , sH, aH) ; s1 ∼ ρ, ah ∼ πE
+h (·|sh), sh+1 ∼ Ph(·|sh, ah), ∀h ∈ [H]
+�
+.
+Then, the learner can use DE to mimic the expert. From a theoretical perspective, the quality of imitation
+is measured by the imitation gap: E
+�
+V(πE) − V(π)
+�
+, where π is the learned policy and the expectation is
+taken over the randomness of DE. We hope that a good learner can mimic the expert perfectly and thus the
+imitation gap is small. In this paper, we assume that the expert policy is deterministic, which is common in
+the literature [Rajaraman et al., 2020, 2021, Xu et al., 2020, 2021b]. Note that this assumption holds for tasks
+including MuJoCo locomotion control.
+Behavioral Cloning. Given an expert dataset DE, the behavioral cloning (BC) algorithm takes the
+maximum likelihood estimation:
+πBC ∈ max
+π
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+dE
+h(s, a) log πh(a|s),
+(1)
+where �
+dE
+h(s, a) is the empirical state-action distribution. By definition, �
+dE
+h(s, a) = nE
+h(s, a)/NE, where nE
+h(s, a)
+refers to the number of expert trajectories such that their state-action pairs are equal to (s, a) in time step h.
+In the tabular case, a closed-formed solution to Equation (1) is available:
+πBC
+h (a|s) =
+�
+�
+�
+nE
+h(s,a)
+nE
+h(s)
+if nE
+h(s) > 0
+1
+|A|
+otherwise
+(2)
+where nE
+h(s) ≜ ∑a′ nE
+h(s, a′). For visited states, BC can make good decisions by duplicating expert actions.
+However, BC has limited knowledge about the expert actions on non-visited states. As a result, it may suffer
+a large imitation gap when making a wrong decision on a non-visited state, leading to compounding errors
+[Ross and Bagnell, 2010].
+Imitation with Supplementary Dataset. BC will fail when the number of expert trajectories is small due
+3Our results can be translated to the discounted and infinite-horizon MDP setting. We consider the episodic MDP because it allows a
+simple way of dealing with the statistical estimation.
+4
+
+to the mentioned compounding errors issue. A naive solution is to collect more expert trajectories. In this
+paper, we follow an alternative and emergent framework in [Kim et al., 2022b, Xu et al., 2022a], where we
+assume that an offline supplementary dataset DS (i.e., NS trajectories of length H) is relatively cheap to obtain:
+DS =
+�
+tr = (s1, a1, s2, a2, · · · , sH, aH) ; s1 ∼ ρ, ah ∼ πβ
+h(·|sh), sh+1 ∼ Ph(·|sh, ah), ∀h ∈ [H]
+�
+,
+where πβ is a behavioral policy that could be a mixture of certain base policies, i.e., πβ = ∑K
+i=1 αiπi with
+∑K
+i=1 αi = 1 and αi ≥ 0 for i ∈ [K]. Algorithms can additionally leverage this supplementary dataset to
+mitigate the compounding errors issue.
+4
+Analysis of Naive Behavior Cloning with Union Dataset
+In this section, we consider a direct extension of BC for the problem of offline imitation with a supplementary
+dataset. Specifically, this algorithm called NBCU (naive BC with the union dataset) performs the maximum
+likelihood estimation on the union of the expert and supplementary datasets:
+πNBCU ∈ argmax
+π
+H
+∑
+h=1 ∑
+(s,a)
+�
+dU
+h (s, a) log πh(a|s),
+(3)
+where DU = DS ∪ DE and �
+dU
+h (s, a) is the empirical state-action distribution estimated from DU
+h (i.e., the
+subset of DU in step h). As an analogue to Equation (2), we have
+πNBCU
+h
+(a|s) =
+�
+�
+�
+nU
+h (s,a)
+nU
+h (s)
+if nU
+h (s) > 0
+1
+|A|
+otherwise
+(4)
+where, just like before, nU
+h (s, a) refers to the number of union trajectories such that their state-action pairs are
+equal to (s, a) in time step h. To analyze NBCU, we pose an assumption about the dataset collection.
+Algorithm 1 NBCU
+Input: Expert dataset DE and supplementary dataset DS.
+1: DU ← DE ∪ DS.
+2: Apply BC to learn a policy π by objective (3) with DU.
+Assumption 1. The supplementary dataset DS and expert dataset DE are collected in the following way: each time,
+we roll-out a behavior policy πβ with probability 1 − η and the expert policy with probability η, where η ∈ [0, 1]
+controls the fraction of expert trajectories. Such an experiment is independent and identically conducted by Ntot times.
+Under Assumption 1, we slightly overload our notations: let NE be the expected number of expert
+trajectories, i.e., NE = ηNtot, and NS be the expected number of supplementary trajectories, i.e., NS =
+(1 − η)Ntot.
+4.1
+Baseline: BC on the Expert Dataset
+To measure whether the supplementary dataset is helpful, we consider the BC only with the expert dataset as
+a baseline. This approach has been analyzed in [Rajaraman et al., 2020], and we transfer their results under
+our assumption4:
+4The proof of Theorem 1 builds on [Rajaraman et al., 2020] and the main difference is that the number of expert trajectories is a
+random variable in our set-up. Technically, we handle this difficulty by Lemma 3 in Appendix.
+5
+
+Theorem 1. Under Assumption 1. In the tabular case, if we apply BC only on the expert dataset, we have that5
+E
+�
+V(πE) − V(πBC)
+�
+≲ min
+�
+H, |S|H2
+NE
+�
+,
+where the expectation is taken over the randomness in the dataset collection.
+Proofs of all theoretical results are deferred to the Appendix.
+4.2
+Main Results of NBCU
+Now, we present our main claim about NBCU.
+Theorem 2. Under Assumption 1. In the tabular case, for any η ∈ (0, 1], we have
+E
+�
+V(πE) − V(πNBCU)
+�
+≲ min
+�
+H, (1 − η)
+�
+V(πE) − V(πβ)
+�
++ |S|H2 log(Ntot)
+Ntot
+�
+,
+where the expectation is taken over the randomness in the dataset collection.
+We often have V(πE) − V(πβ) > 0, as the behavior policy is usually inferior to the expert policy. In this
+case, even if Ntot is sufficiently large so that the second term is negligible, there still exists a positive gap
+between V(πE) and V(πNBCU). Fundamentally, this is because the behavior policy may collect non-expert
+actions, so the recovered policy may select a wrong action even on expert states, which results in bad
+performance. The following theorem shows that the gap V(πE) − V(πβ) is inevitable in the worst case.
+Proposition 1. Under Assumption 1. In the tabular case, there exists an MDP M, an expert policy πE and a behavior
+policy πβ, for any η ∈ (0, 1], when Ntot ≳ |S|, we have
+E
+�
+V(πE) − V(πNBCU)
+�
+≳ (1 − η)
+�
+V(πE) − V(πβ)
+�
+,
+where the expectation is taken over the randomness in the dataset collection.
+The construction of the hard instance in Proposition 1 is based on the following intuition: NBCU does
+not distinguish the action labels in the union dataset and treats them equally important; see Equation (4).
+Therefore, NBCU will learn bad decisions and suffer a non-vanishing gap when the dataset has a bad
+state-action coverage (i.e., primarily, action labels on the expert states are sub-optimal).
+4.3
+Positive Results of NBCU
+The previous results suggest that NBCU is not warranted to be better than BC. For some special cases
+(depending on the dataset coverage), however, we can bypass the hard instance in Proposition 1 and show
+that NBCU can perform well. We state two representative results as follows.
+Theorem 3. Under the same assumption with Theorem 2, additionally assume that πβ = πE. In the tabular case, for
+any η ∈ [0, 1], we have
+E
+�
+V(πE) − V(πNBCU)
+�
+≲ min
+�
+H, |S|H2
+Ntot
+�
+,
+where the expectation is taken over the randomness in the dataset collection.
+Theorem 3 is a direct extension of Theorem 1. The condition πβ = πE seems strong. Still, it may
+approximately hold in the following case: the supplementary dataset DS is collected by an online agent that
+improves its performance over iterations, in which πβ is close to πE asymptotically.
+5a(n) ≲ b(n) means that there exist C, n0 > 0 such that a(n) ≤ Cb(n) for all n ≥ n0. In our context, n usually refers to the number of
+trajectories.
+6
+
+Theorem 4. Under the same assumption with Theorem 2, additionally assume that πβ never visits expert states, i.e.,
+supp(dπβ
+h (·)) ∩ supp(dπE
+h (·)) = ∅6 for all h ∈ [H]. In the tabular case, for any η ∈ (0, 1], we have
+E
+�
+V(πE) − V(πNBCU)
+�
+≲ min
+�
+H, |S|H2
+NE
+�
+,
+(5)
+where the expectation is taken over the randomness in the dataset collection.
+It is commonly believed that noisy demonstrations are harmful to performance [Sasaki and Yamashina,
+2021]. Nevertheless, Theorem 4 provides a condition in which BC is robust to noisy samples. Technically,
+this robustness property stems from the theoretical analysis that only states along the expert trajectory are
+significant for the imitation gap, and noisy actions on the non-expert states do not contribute.
+Summary. Our results demonstrate that without a special design, the naive application of BC on the
+union dataset is not guaranteed to be better than BC. However, good results may happen if the dataset
+coverage is nice, suggesting that noisy demonstrations are not the monster in all cases. Please refer to Table 1
+for a summary.
+Table 1: Effects of state-action attributes of samples for NBCU. “” indicates a helpful sample, “” indicates
+a harmful sample, and “” means something in between.
+Expert state
+Non-expert state
+Expert action
+
+
+Non-expert action
+
+
+Connection with Experiments. From Figure 1, we already see two interesting phenomena: 1) NBCU
+is worse than BC for the noisy expert task; 2) NBCU is much better than BC for the full replay task. We
+use our theory to interpret these results. First, for the noisy expert task, our experiment setting (refer to
+Appendix D.1 for details) ensures that the supplementary dataset has noisy demonstrations on expert states,
+following the idea in Proposition 1. In this case, Theorem 2 predicts that NBCU is no better than BC. Second,
+for the full replay task, we remark that the replay buffer contains lots of expert-level trajectories (refer to
+Figure 5 in Appendix which shows that the online SAC converges to the expert-level performance quickly).
+Therefore, the dataset coverage is good in this scenario and Theorem 3 and Theorem 4 can explain the good
+performance of NBCU.
+5
+Analysis of Weighted Behavioral Cloning with Union Dataset
+In this section, we explore an alternative approach to leverage the expert and supplementary datasets.
+In light of [Kim et al., 2022b, Xu et al., 2022a], a discriminator is trained to score samples, upon which a
+weighted BC objective is used for policy optimization:
+πWBCU ∈ argmax
+π
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+�
+dU
+h (s, a) × [wh(s, a) log πh(a|s)] × I [wh(s, a) ≥ δ]
+�
+,
+(6)
+where �
+dU
+h (s, a) is the empirical state-action distribution estimated from DU
+h , wh(s, a) ∈ [0, ∞) is the weight
+decided by the discriminator, and δ ∈ [0, ∞) is a hyper-parameter. We point out that δ is introduced for
+theoretical analysis, and in practice we set δ = 0. We call this approach WBCU (weighted BC with the union
+dataset).
+Our key idea is the importance sampling technique [Shapiro, 2003, Chapter 9], which can transfer samples
+in the union dataset under the expert policy distribution. In this way, WBCU is expected to address the
+failure mode of NBCU. We elaborate on this point as follows. In the population level (i.e., there are infinitely
+samples), we would have �
+dU
+h = dU
+h , which is jointly determined by the expert policy and the behavioral
+6For a distribution p, supp(p) = {x : p(x) ̸= 0}.
+7
+
+policy. In this case, if wh(s, a) = dE
+h(s, a)/dU
+h (s, a), we would have �
+dU
+h (s, a)wh(s, a) = dE
+h(s, a). Accordingly,
+the objective (6) is to learn a policy as if samples were collected by the expert policy. In practice, dE
+h(s, a) and
+dU
+h (s, a) are unknown; instead, we only have finite samples from these two distributions. Therefore, we need
+to estimate the grounded importance sampling ratio dE
+h(s, a)/dU
+h (s, a).
+We emphasize that a simple two-step idea—first estimating dE
+h(s, a) and dU
+h (s, a) separately and then
+calculating their quotient—is intractable. This is because it is difficult to accurately estimate the probability
+density of high-dimensional distributions. Following the seminal idea in [Goodfellow et al., 2014], we take a
+one-step approach: we directly train a discriminator to estimate dE
+h(s, a)/dU
+h (s, a). Concretely, we consider
+time-dependent parameterized discriminators {ch : S × A → (0, 1)}H
+h=1, each of which has an objective
+max
+ch
+∑
+(s,a)∈S×A
+�
+dE
+h(s, a) [log (ch(s, a))] +
+∑
+(s,a)∈S×A
+�
+dU
+h (s, a) [log (1 − ch(s, a))] .
+(7)
+The above problem amounts to training a binary classifier (i.e., the logistic regression). In the population
+level, with the first-order optimality condition, we have
+c⋆
+h(s, a) =
+dE
+h(s, a)
+dE
+h(s, a) + dU
+h (s, a).
+(8)
+Then, we can obtain the importance sampling ratio dE
+h(s, a)/dU
+h (s, a) in the following way:
+wh(s, a) =
+c⋆
+h(s, a)
+1 − c⋆
+h(s, a).
+(9)
+Based on the above discussion, we outline the implementation of the proposed method WBCU in Algorithm 2.
+Algorithm 2 WBCU
+Input: Expert dataset DE and supplementary dataset DS.
+1: DU ← DE ∪ DS.
+2: Train a binary classifier c by objective (7) with DE and DU.
+3: Compute the importance sampling ratio w by Equation (9).
+4: Apply BC to learn a policy π by objective (6) with DU.
+Remark 1. The weighting rule of WBCU is unbiased in the sense that WBCU directly estimates the importance
+sampling ratio, while prior methods in [Kim et al., 2022b, Xu et al., 2022a] use biased/regularized weighting rules. As
+byproducts, WBCU has fewer hyper-parameters to tune.
+First, DemoDICE also implements the policy learning objective (6), but DemoDICE uses the weighting rule
+�w(s, a) ∝ d⋆(s, a)/dU(s, a) (refer to the formula between Equations (19)-(20) in [Kim et al., 2022b]), where d⋆ is
+computed by a regularized state-action distribution objective (refer to [Kim et al., 2022b, Equations (5)-(7)])7:
+d⋆ = argmin
+d
+DKL(d∥dE) + αDKL(d∥dU)
+s.t.
+d(s, a) ≥ 0
+∀s, a.
+∑
+a
+d(s, a) = (1 − γ)ρ(s) + γ ∑
+s′,a′
+P(s|s′, a′)d(s′, a′)
+∀s.
+where γ ∈ [0, 1) is the discount factor, α > 0 is a hyper-parameter. Due to the regularized term in the objective and the
+Bellman flow constraint, we have d⋆ ̸= dE.
+7For a moment, we use the notations in [Kim et al., 2022b] and present their results under the stationary and infinite-horizon MDPs.
+Same as the discussion of DWBC [Xu et al., 2022a].
+8
+
+9
+Discriminative Learning
+Expert Sample
+Supplementary Sample 
+ (Mode 1)
+Supplementary Sample 
+ (Mode 2)
+Decision Boundary  
+(by )¯θ
+Decision Boundary  
+(by 
+)
+θ⋆
+Figure 2: Illustration for the learning scheme of WBCU under Assumption 2.
+Second, DWBC considers a different policy learning objective (refer to [Xu et al., 2022a, Equation (17)]):
+min
+π
+α
+∑
+(s,a)∈DE
+[− log π(a|s)] −
+∑
+(s,a)∈DE
+�
+− log π(a|s) ·
+λ
+c(1 − c)
+�
++
+∑
+(s,a)∈DS
+�
+− log π(a|s) ·
+1
+1 − c
+�
+,
+(10)
+where α > 0, λ > 0 are hyper-parameters, and c is the output of the discriminator that is jointly trained with π (refer
+to [Xu et al., 2022a, Equation (8)]):
+min
+c
+λ
+∑
+(s,a)∈DE
+[− log c(s, a, log π(a|s))] +
+∑
+(s,a)∈DS
+[− log(1 − c(s, a, log π(a|s)))]
+− λ
+∑
+(s,a)∈DE
+[− log(1 − c(s, a, log π(a|s)))] .
+Since its input additionally incorporates log π, the discriminator is not guaranteed to estimate the state-action
+distribution. Thus, the weighting in Equation (10) loses a connection with the importance sampling ratio.
+Experiments in Figure 1 show that WBCU simultaneously works well on the noisy expert and full replay
+tasks, while prior methods like DemoDICE and DWBC perform well only on one of them. We believe the
+discussion in Remark 1 can partially explain the empirical observation. Next, we investigate the theoretical
+foundation of WBCU.
+5.1
+Negative Result of WBCU With Tabular Representation
+In this section, we consider parameterizing the discriminator c by a huge table (i.e., a vector with dimension
+|S||A|). We present a surprising counter-intuitive result.
+Proposition 2. In the tabular case (with δ = 0), we have πWBCU = πBC.
+Proposition 2 shows that even if we have a large number of supplementary samples and even if we use
+the importance sampling, WBCU is not guaranteed to outperform BC, based on the tabular representation.
+We highlight that this failure mode is because the discriminator has no extrapolation ability in this case.
+Specifically, for an expert-style sample (s, a) that only appears in the supplementary dataset, we have
+�
+dE
+h(s, a) = 0 and �
+dU
+h (s, a) > 0, so c⋆
+h(s, a) = �
+dE
+h(s, a)/(�
+dE
+h(s, a) + �
+dU
+h (s, a)) = 0. That is, such a good sample
+does not contribute to the learning objective (6). Intuitively, the tabular representation simply treats samples
+in a discrete way, which ignores the correlation between samples.
+9
+
+5.2
+Positive Result of WBCU With Function Approximation
+To bypass the hurdle in the previous section, we investigate WBCU with certain function approximation in
+this section. To avoid the tabular/discrete representation, we will consider “smooth” function approximators,
+which can model the internal correlation between samples. Specifically, we consider the discriminator to be
+parameterized by
+ch(s, a; θh) =
+1
+1 + exp(−⟨φh(s, a), θh⟩),
+(11)
+where φh(s, a) ∈ Rd is the feature vector (that can be learned by neural networks), and θh ∈ Rd is the
+parameter to train. Accordingly, the optimization problem becomes:
+min
+θh
+Lh(θh) ≜
+�
+∑
+(s,a)
+�
+dE
+h(s, a) [log (1 + exp (−⟨φh(s, a), θh⟩))] + ∑
+(s,a)
+�
+dU
+h (s, a) [log (1 + exp (⟨φh(s, a), θh⟩))]
+�
+.
+(12)
+Let θ⋆ = {θ⋆
+1, · · · , θ⋆
+H} be the optimal solution obtained from Equation (12). Due to the side information in
+the feature, samples are no longer treated independently, and the discriminator can perform a structured
+estimation. We clarify that to be consistent with the previous results, the policy is still based on the tabular
+representation. We discuss the general function approximation of policy in Appendix C.
+Since c⋆ is no longer analytic as in Equation (8), a natural question is: what can we say about it? Our
+intuition is stated as follows. Let DS
+h denote the set of samples in step h in DS. Since the behavior policy
+that collects DS
+h is diverse, we can imagine DS
+h contains two modes of samples: some of them actually are
+also collected by the expert policy while others are not. In the former case, we expect that wh(s, a) is large
+so that �
+dU
+h (s, a)wh(s, a) ≈ 1, indicating such a sample (s, a) is likely collected by the expert. In the latter
+case, we hope the discriminator can predict wh with a small value so that �
+dU
+h (s, a)wh(s, a) ≈ 0, indicating
+it is a non-expert sample. Notice that wh is monotone with respect to the inner product ⟨φh(s, a), θ⟩; refer
+to Equations (9)(11). Therefore, we conclude that a larger ⟨φh(s, a), θ⟩ means a significant contribution
+to the learning objective (6). Next, we demonstrate that the above intuition can be achieved under mild
+assumptions.
+Assumption 2 (Linear Separability). Let DS = DS,1 ∪ DS,2 and DS,1 ∩ DS,2 = ∅, where DS,1 is collected by the
+expert policy and DS,2 is collected by a sub-optimal policy (but the algorithm does not know this split). For each time
+step h ∈ [H], there exists a ground truth parameter ¯θh ∈ Rd, for any (s, a) ∈ DE
+h ∪ DS,1
+h
+and (s′, a′) ∈ DS,2
+h , it holds
+that
+⟨ ¯θh, φh(s, a)⟩ > 0, ⟨ ¯θh, φh(s′, a′)⟩ < 0.
+Readers may realize that Assumption 2 is closely related to the notion of “margin” in the classification
+problem. Define
+∆h(θ) ≜
+�
+min
+(s,a)∈DE
+h ∪DS,1
+h
+⟨θ, φh(s, a)⟩ −
+max
+(s′,a′)∈DS,2
+h
+⟨θ, φh(s′, a′)⟩
+�
+.
+From Assumption 2, we have ∆h( ¯θh) > 0. This means that there exists a classifier that recognizes samples
+from both DE
+h and DS,1
+h
+as “good” samples, which contributes to the objective (6). On the other hand, samples
+from DS,2
+h
+will be classified as “bad” samples, which do not matter for the learned policy. Note that such
+a nice classifier is assumed to exist, which is not identical to what is learned via Equation (12). Next, we
+theoretically control the (parameter) distance between two classifiers.
+Before further discussion, we note that ¯θh is not unique if it exists. Without loss of generality, we define
+¯θh as that can achieve the maximum margin (among all unit vectors8). To theoretically characterize the
+8Otherwise, the margin is unbounded by multiplying ¯θh with a positive scalar.
+10
+
+movement of θ⋆, we first characterize the landscape properties (e.g., Lipschitz continuity and quadratic
+growth conditions9) of ∆h and Lh(θ) in Lemma 1 and Lemma 2, respectively.
+Lemma 1 (Lipschitz Continuity). For any θ ∈ Rd, the margin function is Lh-Lipschitz continuous in the sense that
+∆h( ¯θh) − ∆h(θ) ≤ Lh
+�� ¯θh − θ
+�� ,
+where Lh =
+��φh(s1, a1) − φh(s2, a2)
+�� with (s1, a1) ∈ argmin(s,a)∈DE
+h ∪DS,1
+h ⟨θ, φh(s, a)⟩ and (s2, a2) ∈
+argmax(s,a)∈DS,2
+h ⟨θ, φh(s, a)⟩.
+Lemma 2 (Quadratic Growth). For any h, let Ah ∈ RNtot×d be the matrix that aggregates the feature vectors of
+samples in DU
+h . Consider the under-parameterization case that rank(Ah) = d. There exists τh > 0 such that
+Lh(θh) ≥ Lh(θ⋆
+h) + τh
+2
+��θh − θ⋆
+h
+��2 .
+Theorem 5. Under Assumption 2, for any h ∈ [H], if the following inequality holds
+�
+2
+�Lh( ¯θh) − Lh(θ⋆
+h)
+�
+τh
+< ∆h( ¯θh)
+Lh
+,
+(13)
+then we have ∆h(θ⋆
+h) > 0.
+To interpret Theorem 5, we note that ∆h(θ⋆
+h) > 0 means that the learned discriminator can perfectly
+distinguish the good samples (from DE
+h and DS,1
+h ) and bad samples (from DS,2
+h ). In other words, if the feature
+design is nice such that Inequality (13) holds, then the obtained classifier can still maintain the decision
+results by ¯θ; refer to Figure 2 for illustration. Consequently, all samples from DE
+h and DS,1
+h
+are assigned large
+weights. In this way, WBCU can leverage additional samples to outperform BC. Technically, ∆h(θ⋆
+h) > 0
+means that there exists a δ > 0 such that we have wh(s, a) > δ for (s, a) ∈ DE
+h ∪ DS,1
+h
+and wh(s, a) < δ for
+(s, a) ∈ DS,2
+h . As such, WBCU can utilize the good samples DS,1
+h
+and eliminate the bad samples DS,2
+h
+in theory.
+The detailed imitation gap bound depends on the number of trajectories in DE ∪ DS,1, and this result is
+straightforward, so we omit details here.
+Readers may ask whether inequality (13) can hold in practice. This question is hard to answer because
+the coefficient τh and Lh are data-dependent. Nevertheless, we can provide a toy example to illustrate that
+inequality (13) holds and give a sharp condition for d = 1; please refer to Appendix B for details. Further
+relaxation of the condition and assumption is left for future work.
+Summary. Our theoretical analysis (Proposition 2 and Theorem 5) discloses that the “smooth” function
+approximation is inevitable to achieve satisfying performance. For practitioners, our results would suggest
+that regularization techniques that control the smooth property of the function approximators may improve
+the performance, which we empirically verify below.
+Additional Experiments. We note that quite often, non-linear neural networks rather than linear func-
+tions are used in practice. For neural networks, the gradient penalty (GP) regularization10 is known to control
+the Lipschitz continuous property of the discriminator [Gulrajani et al., 2017]. In particular, a large gradient
+penalty loss can push the discriminator to prefer 1-Lipschitz continuous functions that are “smooth”. With
+the same set-up with experiments in Figure 1, we show that the gradient penalty is crucial for the practical
+performance of WBCU; see Figure 3. A similar phenomenon is also observed for the related algorithm
+DemoDICE; see Figure 6 in Appendix.
+6
+Conclusion
+We theoretically explore imitation learning with a supplementary dataset, and empirical results corroborate
+our findings. While our results have several desirable features, they also have shortcomings. One limitation
+9These terminologies are from the optimization literature (see, e.g., [Karimi et al., 2016, Drusvyatskiy and Lewis, 2018]).
+10This technique adds a squared loss of the gradient norm to the original loss function; see Appendix D.1 for details.
+11
+
+Noisy Expert
+Full Replay
+Task
+0
+20
+40
+60
+80
+Normalized Score (%)
+30
+6
+64
+94
+57
+93
+Algorithm
+WBCU(GP=0)
+WBCU(GP=1)
+WBCU(GP=10)
+Figure 3: Averaged normalized scores of trained policies of WBCU with gradient penalty (GP). Numbers in
+the legend indicate the scale of the GP regularization.
+is that we consider the tabular representation in policy learning. However, our theoretical implications may
+remain unchanged if function approximation is used. Please see Appendix C for discussion, which deserves
+further investigation.
+Exploring more applications of NBCU and WBCU is an interesting future work. For example, for large
+language models [Radford et al., 2018], we may have massive supplementary samples from the Web, while
+examples with human annotations are limited. Compared with the existing reinforcement-learning-based
+methods (see, e.g., [Stiennon et al., 2020]), training good policies may be easier and more efficient by the
+developed imitation learning approaches.
+Acknowledgements
+Ziniu Li would like to thank Yushun Zhang, Yingru Li, Jiancong Xiao, and Dmitry Rybin for reading the
+manuscript and providing valuable comments. Tian Xu would like to thank Fanming Luo, Zhilong Zhang,
+and Jingcheng Pang for reading the manuscript and providing helpful comments. Ziniu Li appreciates the
+helpful discussion with Congliang Chen about a technical lemma.
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+14
+
+A
+Proof of Results in Section 4
+Lemma 3. For any N ∈ N+ and p ∈ (0, 1), if the random variable X follows the binomial distribution, i.e.,
+X ∼ Bin(N, p), then we have that
+E
+�
+1
+X + 1
+�
+≤
+1
+Np.
+Proof.
+E
+�
+1
+X + 1
+�
+=
+N
+∑
+x=0
+�
+1
+x + 1
+�
+N!
+x!(N − x)! px(1 − p)N−x
+=
+1
+(N + 1)p
+N+1
+∑
+x=1
+�
+(N + 1)!
+x!(N + 1 − x)!
+�
+px(1 − p)N+1−x
+=
+1
+(N + 1)p
+�
+1 − (1 − p)N+1�
+≤
+1
+Np.
+A.1
+Proof of Theorem 1
+When |DE| ≥ 1, by [Rajaraman et al., 2020, Theorem 4.2], we have that
+V(πE) − EDE
+�
+V(πBC)
+�
+≤ 4|S|H2
+9|DE| .
+When |DE| = 0, we simply have that
+V(πE) − EDE
+�
+V(πBC)
+�
+≤ H.
+Therefore, we have the following unified bound.
+V(πE) − EDE
+�
+V(πBC)
+�
+≤
+|S|H2
+max{|DE|, 1} ≤ 2|S|H2
+|DE| + 1.
+The last inequality follows that max{x, 1} ≥ (x + 1)/2, ∀x ≥ 0. Finally, notice that |DE| ∼ Bin(Ntot, η). By
+Lemma 3, we have that
+V(πE) − E
+�
+V(πBC)
+�
+≤ E
+� 2|S|H2
+|DE| + 1
+�
+≤ 2|S|H2
+Ntotη
+= 2|S|H2
+NE
+,
+which completes the proof.
+A.2
+Proof of Theorem 2
+For analysis, we first define the mixture state-action distribution as follows.
+dmix
+h
+(s, a) ≜ ηdπE
+h (s, a) + (1 − η)dπβ
+h (s, a), dmix
+h
+(s) ≜ ∑
+a∈A
+dmix
+h
+(s, a), ∀(s, a) ∈ S × A, ∀h ∈ [H].
+Note that in the population level, the marginal state-action distribution of union dataset DU in time step h is
+exactly dmix
+h
+. That is, dU
+h (s, a) = dmix
+h
+(s, a), ∀(s, a, h) ∈ S × A × [H]. Then we define the mixture policy πmix
+induced by dmix as follows.
+πmix
+h
+(a|s) =
+�
+�
+�
+dmix
+h
+(s,a)
+dmix
+h
+(s)
+if dmix
+h
+(s) > 0,
+1
+|A|
+otherwise.
+∀(s, a) ∈ S × A , ∀h ∈ [H].
+15
+
+From the theory of Markov Decision Processes, we know that (see, e.g., [Puterman, 2014])
+∀h ∈ [H], ∀(s, a) ∈ S × A, dπmix
+h
+(s, a) = dmix
+h
+(s, a).
+Therefore, we can obtain that the marginal state-action distribution of union dataset DU in time step h is
+exactly dπmix
+h
+. Then we have the following decomposition.
+E
+�
+V(πE) − V(πNBCU)
+�
+= E
+�
+V(πE) − V(πmix) + V(πmix) − V(πNBCU)
+�
+= E
+�
+V(πE) − V(πmix)
+�
++ E
+�
+V(πmix) − V(πNBCU)
+�
+= V(πE) − V(πmix) + E
+�
+V(πmix) − V(πNBCU)
+�
+.
+For V(πE) − V(πmix), we have that
+V(πE) − V(πmix) =
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+dπE
+h (s, a) − dπmix
+h
+(s, a)
+�
+rh(s, a)
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+dπE
+h (s, a) − dmix
+h
+(s, a)
+�
+rh(s, a)
+= (1 − η)
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+dπE
+h (s, a) − dπβ
+h (s, a)
+�
+rh(s, a)
+= (1 − η)
+�
+V(πE) − V(πβ)
+�
+.
+The last equation follows the dual formulation of policy value [Puterman, 2014]. Besides, notice that
+E
+�
+V(πmix) − V(πNBCU)
+�
+is exactly the imitation gap of BC when regarding πmix and DU as the expert
+policy and expert demonstrations, respectively. By [Rajaraman et al., 2020, Theorem 4.4], we have that
+E
+�
+V(πmix) − V(πNBCU)
+�
+≲ |S|H2 log(Ntot)
+Ntot
+.
+Combining the above two equations yields that
+E
+�
+V(πE) − V(πNBCU)
+�
+≲ (1 − η)
+�
+V(πE) − V(πβ)
+�
++ |S|H2 log(Ntot)
+Ntot
+.
+A.3
+Proof of Proposition 1
+We consider the instance named Standard Imitation in [Xu et al., 2021a]; see Figure 4. In Standard Imitation,
+each state is an absorbing state. In each state, only by taking the action a1, the agent can obtain the reward
+of 1. Otherwise, the agent obtains zero rewards. The initial state distribution is a uniform distribution, i.e.,
+ρ(s) = 1/|S|, ∀s ∈ S.
+2
+Bandit
+1
+· · ·
+|S|�1
+|S|
+0
+1
+0
+1
+0
+1
+2
+�����������������
+�����������
+������������
+�����������������
+Figure 4: The Standard Imitation MDP in [Xu et al., 2021a].
+We consider that the expert policy πE always takes the action a1 while the behavioral policy πβ always
+takes another action a2. Formally, πE
+h (a1|s) = 1, ∀s ∈ S, h ∈ [H], πβ
+h(a2|s) = 1, ∀s ∈ S, h ∈ [H]. It is
+direct to calculate that V(πE) = H and V(πβ) = 0. The supplementary dataset DS and the expert dataset
+DE are collected according to Assumption 1. The mixture state-action distribution can be calculated as
+16
+
+∀s ∈ S, h ∈ [H],
+dmix
+h
+(s, a1) = ηdπE
+h (s, a1) + (1 − η)dπβ
+h (s, a1) = ηdπE
+h (s, a1) = ηρ(s),
+dmix
+h
+(s, a2) = ηdπE
+h (s, a2) + (1 − η)dπβ
+h (s, a2) = (1 − η)dπβ
+h (s, a2) = (1 − η)ρ(s).
+Note that in the population level, the marginal distribution of the union dataset DU in time step h is exactly
+dmix
+h
+. The mixture policy induced by dmix can be formulated as
+πmix
+h
+(a1|s) = η, πmix
+h
+(a2|s) = 1 − η, ∀s ∈ S, h ∈ [H].
+Just like before, we have dπmix
+h
+(s, a) = dmix
+h
+(s, a). The policy value of πmix can be calculated as
+V(πmix) =
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+dmix
+h
+(s, a)rh(s, a) =
+H
+∑
+h=1 ∑
+s∈S
+dmix
+h
+(s, a1) = ηH.
+The policy πNBCU can be formulated as
+∀h ∈ [H],
+πNBCU
+h
+(a|s) =
+�
+�
+�
+nU
+h (s,a)
+∑a′ nU
+h (s,a′)
+if ∑a′ nU
+h (s, a′) > 0
+1
+|A|
+otherwise
+(14)
+We can view that the BC’s policy learned on the union dataset mimics the mixture policy πmix. In the
+following part, we analyze the lower bound on the imitation gap of πNBCU.
+E
+�
+V(πE) − V(πNBCU)
+�
+= V(πE) − V(πmix) + E
+�
+V(πmix) − V(πNBCU)
+�
+= H − ηH + E
+�
+V(πmix) − V(πNBCU)
+�
+= (1 − η)(V(πE) − V(πβ)) + E
+�
+V(πmix) − V(πNBCU)
+�
+.
+Then we consider the term E
+�
+V(πmix) − V(πNBCU)
+�
+.
+V(πmix) − V(πNBCU) =
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+dπmix
+h
+(s, a) − dπNBCU
+h
+(s, a)
+�
+rh(s, a)
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) > 0}
++
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) = 0}.
+We take expectation over the randomness in DU on both sides and obtain that
+E
+�
+V(πmix) − V(πNBCU)
+�
+= E
+�
+�
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) > 0}
+�
+�
++ E
+�
+�
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) = 0}
+�
+� .
+(15)
+17
+
+For the first term in RHS, we have that
+E
+�
+�
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) > 0}
+�
+�
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)rh(s, a)E
+��
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+I{nU
+h (s) > 0}
+�
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)rh(s, a)P
+�
+nU
+h (s) > 0
+�
+E
+��
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+� ����nU
+h (s) > 0
+�
+= 0.
+The last equation follows the fact that when nU
+h (s) > 0, πNBCU
+h
+(a|s) is an unbiased maximum likelihood
+estimation of πmix
+h
+(a|s). For the remaining term, we have that
+E
+�
+�
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) = 0}
+�
+�
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)rh(s, a)E
+��
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+I{nU
+h (s) = 0}
+�
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)rh(s, a)P
+�
+nU
+h (s) = 0
+�
+E
+��
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+� ����nU
+h (s) = 0
+�
+=
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)rh(s, a)P
+�
+nU
+h (s) = 0
+� �
+πmix
+h
+(a|s) −
+1
+|A|
+�
+(a)
+=
+H
+∑
+h=1 ∑
+s∈S
+ρ(s)P
+�
+nU
+h (s) = 0
+� �
+η −
+1
+|A|
+�
+(b)
+= H
+�
+η −
+1
+|A|
+�
+∑
+s∈S
+ρ(s)P
+�
+nU
+1 (s) = 0
+�
+.
+In the equation (a), we use the fact that rh(s, a1) = 1, rh(s, a) = 0, ∀a ∈ A \ {a1}. In the equation (b), since
+each state is an absorbing state, we have that P
+�
+nU
+h (s) = 0
+� = P
+�
+nU
+1 (s) = 0
+�
+, ∀h ∈ [H].
+We consider two cases. In the first case of η ≥ 1/|A|, we directly have that
+E
+�
+�
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+ρ(s)
+�
+πmix
+h
+(a|s) − πNBCU
+h
+(a|s)
+�
+rh(s, a)I{nU
+h (s) = 0}
+�
+� ≥ 0.
+By Equation (15), we have that
+E
+�
+V(πmix) − V(πNBCU)
+�
+≥ 0,
+which implies that
+E
+�
+V(πE) − V(πNBCU)
+�
+≥ (1 − η)(V(πE) − V(πβ)).
+In the second case of η < 1/|A|, we have that
+H
+�
+η −
+1
+|A|
+�
+∑
+s∈S
+ρ(s)P
+�
+nU
+1 (s) = 0
+� (a)
+≥ −
+� 1
+|A| − η
+�
+H exp
+�
+− Ntot
+|S|
+�
+18
+
+≥ −(1 − η)H exp
+�
+− Ntot
+|S|
+�
+(b)
+≥ −(1 − η)H
+2
+.
+In the inequality (a), we use that
+∑
+s∈S
+ρ(s)P
+�
+nU
+1 (s) = 0
+�
+= ∑
+s∈S
+ρ(s)(1 − ρ(s))Ntot =
+�
+1 − 1
+|S|
+�Ntot
+≤ exp
+�
+− Ntot
+|S|
+�
+.
+The inequality (b) holds since we consider the range where Ntot ≥ |S| log(2). By Equation (15), we have that
+E
+�
+V(πmix) − V(πNBCU)
+�
+≥ −(1 − η)H
+2
+.
+This implies that
+E
+�
+V(πE) − V(πNBCU)
+�
+≥ (1 − η)(V(πE) − V(πβ)) − (1 − η)H
+2
+= (1 − η)
+2
+(V(πE) − V(πβ)).
+In both cases, we prove that E
+�
+V(πE) − V(πNBCU)
+� ≳ (1 − η)(V(πE) − V(πβ)) and thus complete the
+proof.
+A.4
+Proof of Theorem 3
+When πβ = πE, E
+�
+V(πE) − V(πNBCU)
+�
+is exactly the imitation gap of BC on DU with expert policy πE. By
+[Rajaraman et al., 2020, Theorem 4.2], we finish the proof.
+A.5
+Proof of Theorem 4
+According to the policy difference lemma in [Kakade and Langford, 2002], we have that
+V(πE) − V(πNBCU) =
+H
+∑
+h=1
+Esh∼dπE
+h (·)
+�
+QπNBCU
+h
+(sh, πE
+h (sh)) − ∑
+a∈A
+πNBCU
+h
+(·|sh)QπNBCU
+h
+(sh, a)
+�
+=
+H
+∑
+h=1
+Esh∼dπE
+h (·)
+�
+QπNBCU
+h
+(sh, πE
+h (sh))(1 − πNBCU
+h
+(πE
+h (sh)|sh))
+�
+≤ H
+H
+∑
+h=1
+Es∼dπE
+h (·)
+�
+Ea∼πNBCU
+h
+(·|s)
+�
+I
+�
+a ̸= πE
+h (s)
+���
+,
+where Qπ
+h (s, a) = E[∑H
+t=h r(st, at)|(sh, ah) = (s, a)] is the state-action value function for a policy π and πE
+h (sh)
+denotes the expert action. By assumption, we have ∀h ∈ [H], supp(dπE
+h (·)) ∩ supp(dπβ
+h (·)) = ∅. Therefore,
+we know that the union state-action pairs in DU does not affect πNBCU on expert states. As a result, πNBCU
+exactly takes expert actions on states visited in DE. Then we have that
+V(πE) − V(πNBCU) ≤ H
+H
+∑
+h=1
+Es∼dπE
+h (·)
+�
+I
+�
+s /∈ Sh(DE)
+��
+,
+where Sh(DE) is the set of states in time step h in DE. Taking expectation over the randomness within DE on
+both sides yields that
+V(πE) − EDE
+�
+V(πNBCU)
+�
+≤ HEDE
+�
+H
+∑
+h=1
+Es∼dπE
+h (·)
+�
+I
+�
+s /∈ Sh(DE)
+���
+.
+19
+
+By [Rajaraman et al., 2020, Lemma A.1], we further derive that
+V(πE) − EDE
+�
+V(πNBCU)
+�
+≤
+|S|H2
+max{|DE|, 1} ≤ 2|S|H2
+|DE| + 1.
+We take expectation over the binomial variable |DE| and have that
+V(πE) − E
+�
+V(πNBCU)
+�
+≤ E
+� 2|S|H2
+|DE| + 1
+�
+≤ 2|S|H2
+Ntotη
+= 2|S|H2
+NE
+,
+which completes the proof.
+B
+Proof of Results in Section 5
+B.1
+Proof of Proposition 2
+In the tabular case, with the first-order optimality condition, we have c⋆
+h(s, a) = �
+dE
+h(s, a)/(�
+dE
+h(s, a) + �
+dU
+h (s, a)).
+By Equation (9), we have
+�
+dU
+h (s, a)wh(s, a) = �
+dU
+h (s, a) ×
+�
+dE
+h(s, a)
+�
+dU
+h (s, a)
+= �
+dE
+h(s, a).
+Hence, the learning objective (6) reduces to (1).
+B.2
+Proof of Lemma 1
+Recall that
+∆h(θ) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+⟨θ, φh(s, a)⟩ −
+max
+(s′,a′)∈DS,2
+h
+⟨θ, φh(s′, a′)⟩.
+Then we have that
+∆h( ¯θh) − ∆h(θ) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+⟨ ¯θh, φh(s, a)⟩ −
+max
+(s′,a′)∈DS,2
+h
+⟨ ¯θh, φh(s′, a′)⟩
+−
+min
+(s,a)∈DE
+h ∪DS,1
+h
+⟨θ, φh(s, a)⟩ +
+max
+(s′,a′)∈DS,2
+h
+⟨θ, φh(s′, a′)⟩
+(a)
+≤ ⟨ ¯θh, φh(s1, a1)⟩ − ⟨ ¯θh, φh(s2, a2)⟩ − ⟨θ, φh(s1, a1)⟩ + ⟨θ, φh(s2, a2)⟩
+= ⟨ ¯θh − θ, φh(s1, a1) − φh(s2, a2)⟩
+(b)
+≤
+�� ¯θh − θ
+��
+���φh(s1, a1) − φh(s2, a2)
+��� .
+In inequality (a), we utilize the facts that (s1, a1) ∈ argmin(s,a)∈DE
+h ∪DS,1
+h ⟨θh, φh(s, a)⟩ and
+(s2, a2) ∈ argmax(s,a)∈DS,2
+h ⟨θh, φh(s, a)⟩. Inequality (b) follows the Cauchy–Schwarz inequality. Let Lh =
+��φh(s1, a1) − φh(s2, a2)
+�� and we finish the proof.
+B.3
+Proof of Lemma 2
+First, by Taylor’s Theorem, there exists θ′
+h ∈ {θ ∈ Rd : θt = θ⋆
+h + t(θh − θ⋆
+h), ∀t ∈ [0, 1]} such that
+Lh(θh) = Lh(θ⋆
+h) + ⟨∇Lh(θ⋆
+h), θh − θ⋆
+h⟩ + 1
+2
+�
+θh − θ⋆
+h
+�⊤ ∇2Lh(θ′
+h)
+�
+θh − θ⋆
+h
+�
+= Lh(θ⋆
+h) + 1
+2
+�
+θh − θ⋆
+h
+�⊤ ∇2Lh(θ′
+h)
+�
+θh − θ⋆
+h
+�
+.
+(16)
+20
+
+The last equality follows the optimality condition that ∇Lh(θ⋆
+h) = 0. Then, our strategy is to prove that the
+smallest eigenvalue of the Hessian matrix ∇2Lh(θ′
+h) is positive, i.e., λmin(∇2Lh(θ′
+h)) > 0. We first calculate
+the Hessian matrix ∇2Lh(θ′
+h). Given DE and DU, we define the function G : R(|DE|+|DU|) → R as
+G(v) ≜
+1
+|DE|
+|DE|
+∑
+i=1
+g(vi) +
+1
+|DU|
+|DU|
+∑
+j=1
+g(vj),
+where vi is the i-th element in the vector v ∈ R(|DE|+|DU|) and g(x) = log (1 + exp(x)) is a real-valued
+function. Besides, we use Bh ∈ R(|DE|+|DU|)×d to denote the matrix whose i-th row Bh,i = −yiφh(si, ai)⊤, and
+yi = 1 if (si, ai) ∈ DE
+h, yi = −1 if (si, ai) /∈ DE
+h. Then the objective function can be reformulated as
+Lh(θh) = ∑
+(s,a)
+�
+dE
+h(s, a) [log (1 + exp (−⟨φh(s, a), θh⟩))] + ∑
+(s,a)
+�
+dU
+h (s, a) [log (1 + exp (⟨φh(s, a), θh⟩))]
+=
+1
+|DE|
+∑
+(s,a)∈DE
+log (1 + exp (−⟨φh(s, a), θh⟩)) +
+1
+|DU|
+∑
+(s,a)∈DU
+log (1 + exp (⟨φh(s, a), θh⟩))
+= G(Bhθh).
+Then we have that ∇2Lh(θh) = B⊤
+h ∇2G(Bhθh)Bh, where
+∇2G(Bhθh) = diag
+�
+g′′((Bhθh)1)
+|DE|
+, . . . ,
+g′′((Bhθh)|DE|)
+|DE|
+,
+g′′((Bhθh)|DE|+1)
+|DE| + |DU|
+, . . . ,
+g′′((Bhθh)|DE|+|DU|)
+|DE| + |DU|
+�
+.
+Here g′′(x) = σ(x)(1 − σ(x)), where σ(x) = 1/(1 + exp(−x)) is the sigmoid function. The eigenvalues of
+∇2G(Bhθh) are
+�
+g′′((Bhθh)1)
+|DE|
+, . . . ,
+g′′((Bhθh)|DE|)
+|DE|
+,
+g′′((Bhθh)|DE|+1)
+|DE| + |DU|
+, . . . ,
+g′′((Bhθh)|DE|+|DU|)
+|DE| + |DU|
+�
+.
+Notice that θ′
+h ∈ {θ ∈ Rd : θt = θ⋆
+h + t(θh − θ⋆
+h), ∀t ∈ [0, 1]}. For a matrix A, we use λmin(A) to denote the
+minimal eigenvalue of A. Here we claim that the minimum of the minimal eigenvalues of ∇2G(Bhθt) over
+t ∈ [0, 1] is achieved at t = 0 or t = 1. That is,
+min{λmin(∇2G(Bhθt)) : ∀t ∈ [0, 1]} = min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))}.
+We prove this claim as follows. For any t ∈ [0, 1], we use {λ1(t), . . . , λ|DE|+|DU|(t)} to denote the eigenvalues
+of ∇2G(Bhθt). For each i ∈ [|DE| + |DU|], we consider λi(t) : [0, 1] → R as a function of t. Specifically,
+λi(t) =
+�
+�
+�
+g′′((Bhθ⋆
+h)i+t(Bh(θh−θ⋆
+h))i)
+|DE|
+,
+if i ∈ [|DE|]
+g′′((Bhθ⋆
+h)i+t(Bh(θh−θ⋆
+h))i)
+|DE|+|DU|
+,
+otherwise.
+We observe that g′′′(x) = σ(x)(1 − σ(x))(1 − 2σ(x)) which satisfies that ∀x ≤ 0, g′′′(x) ≥ 0, and ∀x ≥
+0, g′′′(x) ≤ 0. Therefore, we have that the minimum of λi(t) over t ∈ [0, 1] must be achieved at t = 0 or
+t = 1. That is,
+min
+t∈[0,1] λi(t) = min{λi(0), λi(1)}.
+(17)
+For any t ∈ [0, 1], we define it ∈ [|DE| + |DU|] as the index of the minimal eigenvalue of ∇2G(Bhθt), i.e.,
+λit(t) = λmin(∇2G(Bhθt)). Then we have that
+min{λmin(∇2G(Bhθt)) : ∀t ∈ [0, 1]} = min{λit(t) : ∀t ∈ [0, 1]}
+(a)
+= min{min{λit(0), λit(1)} : ∀t ∈ [0, 1]}
+= min{λi0(0), λi1(1)}
+21
+
+(b)
+= min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))}
+Equality (a) follows (17) and equality (b) follows that λi0(0) and λi1(1) are the minimal eigenvalues of
+∇2G(Bhθ0) and ∇2G(Bhθ1), respectively.
+In summary, we derive that
+min{λmin(∇2G(Bhθt)) : ∀t ∈ [0, 1]} = min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))},
+(18)
+which proves the previous claim.
+Further, we consider λmin
+�∇2Lh(θh)
+�
+.
+λmin
+�
+∇2Lh(θh)
+�
+=
+inf
+x∈Rd:∥x∥=1
+x⊤∇2Lh(θh)x
+=
+inf
+x∈Rd:∥x∥=1 (Bhx)⊤ ∇2G(Bhθh) (Bhx)
+=
+inf
+z∈Im(Bh) z⊤∇2G(Bhθh)z
+=
+�
+inf
+z∈Im(Bh) ∥z∥
+�2
+λmin(∇2G(Bhθh))
+≥
+�
+inf
+z∈Im(Bh) ∥z∥
+�2
+min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))}.
+Here Im(Bh) = {z ∈ Rd : z = Bhx, ∥x∥ = 1}. The last inequality follows Equation (18).
+In the under-parameterization case where rank(Ah) = d, we have that rank(Bh) = d. Thus, Im(Bh) is a
+set of vectors with positive norms, i.e., infz∈Im(Bh) ∥z∥ > 0. Besides, since g′′(x) = σ(x)(1 − σ(x)) > 0, we
+also have that
+min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))} > 0.
+In summary, we obtain that
+λmin
+�
+∇2Lh(θh)
+�
+≥
+�
+inf
+z∈Im(Bh) ∥z∥
+�2
+min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))} > 0.
+Then, with Equation (16), there exists τh =
+�
+infz∈Im(Bh) ∥z∥
+�2
+min{λmin(∇2G(Bhθ0)), λmin(∇2G(Bhθ1))} >
+0 such that
+Lh(θh) ≥ Lh(θ⋆
+h) + τh
+2
+��θh − θ⋆
+h
+��2 ,
+which completes the proof.
+B.4
+Proof of Theorem 5
+First, invoking Lemma 1 with θ = θ⋆
+h yields that
+∆h(θ⋆
+h) ≥ ∆h( ¯θh) − Lh
+�� ¯θh − θ⋆
+h
+�� .
+Here Lh = ∥φh(s, a) − φh(s′, a′)∥ with (s, a) ∈ argmin(s,a)∈DE
+h ∪DS,1
+h ⟨θ⋆
+h, φh(s, a)⟩ and
+(s′, a′) ∈ argmax(s,a)∈DS,2
+h ⟨θ⋆
+h, φh(s, a)⟩. Then, by Lemma 2, there exists τh > 0 such that
+Lh(θh) ≥ Lh(θ⋆
+h) + τh
+2
+��θh − θ⋆
+h
+��2 .
+22
+
+This directly implies an upper bound of the distance between θh and θ⋆
+h.
+��θh − θ⋆
+h
+�� ≤
+�
+2
+�Lh( ¯θh) − Lh(θ⋆
+h)
+�
+τh
+.
+When inequality (13) holds, we further have that
+��θh − θ⋆
+h
+�� < ∆h( ¯θh)/Lh. Then we get that
+∆h(θ⋆
+h) ≥ ∆h( ¯θh) − Lh
+�� ¯θh − θ⋆
+h
+�� > 0,
+which completes the proof.
+B.5
+An Example Corresponding to Theorem 5
+Example 1. To illustrate Theorem 5, we consider an example in the feature space R2. In particular, for time step
+h ∈ [H], we have the expert dataset and supplementary dataset as follows.
+DE
+h =
+��
+s(1), a(1)�
+,
+�
+s(4), a(4)��
+, DS
+h =
+��
+s(2), a(2)�
+,
+�
+s(3), a(3)��
+,
+DS,1
+h
+=
+��
+s(2), a(2)��
+, DS,2
+h
+=
+��
+s(3), a(3)��
+.
+The corresponding features are
+φh
+�
+s(1), a(1)�
+= (0, 1)⊤, φh
+�
+s(2), a(2)�
+=
+�
+−1
+2, 0
+�⊤
+,
+φh
+�
+s(3), a(3)�
+=
+�
+0, −1
+2
+�⊤
+, φh
+�
+s(4), a(4)�
+= (−1, 0)⊤.
+Notice that the set of expert-style samples is DE
+h ∪ DS,1
+h
+= {(s(1), a(1)), (s(2), a(2)), (s(4), a(4))} and the set of non-
+expert-style samples is DS,2
+h
+= {(s(3), a(3))}. It is direct to calculate that the ground-truth parameter that achieves
+the maximum margin among unit vectors is θh = (−
+√
+2/2,
+√
+2/2)⊤ and the maximum margin is ∆h(θh) =
+√
+2/2.
+According to Equation (12), for θh = (θh,1, θh,2)⊤, the optimization objective is
+Lh(θh) = ∑
+(s,a)
+�
+dE
+h(s, a) [log (1 + exp (−⟨φh(s, a), θh⟩))] + ∑
+(s,a)
+�
+dU
+h (s, a) [log (1 + exp (⟨φh(s, a), θh⟩))]
+= 1
+2 (log (1 + exp (−θh,2)) + log (1 + exp (θh,1)))
++ 1
+4
+�
+log (1 + exp (θh,2)) + log
+�
+1 + exp
+�
+−1
+2θh,1
+���
++ 1
+4
+�
+log
+�
+1 + exp
+�
+−1
+2θh,2
+��
++ log (1 + exp (−θh,1))
+�
+.
+We apply CVXPY [Diamond and Boyd, 2016] to calculate the optimal solution θ⋆
+h ≈ (−0.310, 0.993)⊤ and the
+objective values Lh(θ⋆
+h) ≈ 1.287, Lh(θh) ≈ 1.309. Furthermore, we calculate the Lipschitz coefficient Lh appears in
+Lemma 1.
+(s(2), a(2)) =
+argmin
+(s,a)∈DE
+h ∪DS,1
+h
+⟨θ⋆
+h, φh(s, a)⟩, (s(3), a(3)) ∈ argmax
+(s,a)∈DS,2
+h
+⟨θ⋆
+h, φh(s, a)⟩,
+Lh =
+���φh(s(2), a(2)) − φh(s(3), a(3))
+��� =
+√
+2
+2 .
+Then we calculate the parameter of strong convexity τh appears in Lemma 2. Based on the proof of Lemma 2, our
+strategy is to calculate the minimal eigenvalue of the Hessian matrix.
+23
+
+First, for θh = (θh,1, θh,2)⊤, the gradient of Lh(θh) is
+∇Lh(θh) = −
+∑
+(s,a)∈S×A
+�
+dE
+h(s, a)σ(−⟨φh(s, a), θh⟩) +
+∑
+(s,a)∈S×A
+�
+dU
+h (s, a)σ (⟨φh(s, a), θh⟩)
+=
+�1
+2σ(θh,1) − 1
+4σ(−θh,1) − 1
+8σ(−1
+2θh,1), 1
+4σ (θh,2) − 1
+2σ (−θh,2) − 1
+8σ(−1
+2θh,2)
+�⊤
+.
+Here σ(x) = 1/(1 + exp(−x)) for x ∈ R is the sigmoid function. Then the Hessian matrix at θh is
+∇2Lh(θh) =
+�
+�
+3
+4 f (θh,1) + 1
+16 f
+�
+1
+2θh,1
+�
+0
+0
+3
+4 f (θh,2) + 1
+16 f
+�
+1
+2θh,2
+�
+�
+� ,
+where f (x) = σ(x)(1 − σ(x)) and f (x) = f (−x). For any t ∈ [0, 1], the eigenvalues of the Hessian matrix at
+θt
+h = θh + t(θ⋆
+h − θh) are
+3
+4 f (θt
+h,1) + 1
+16 f
+�1
+2θt
+h,1
+�
+, 3
+4 f (θt
+h,2) + 1
+16 f
+�1
+2θt
+h,2
+�
+.
+Now, we calculate the minimal eigenvalues of ∇2Lh(θt
+h). We consider the function
+g(x) = 3
+4 f (x) + 1
+16 f
+�1
+2x
+�
+, ∀x ∈ [a, b].
+The gradient is
+g′(x) = 3
+4σ(x)(1 − σ(x))(1 − 2σ(x)) + 1
+32σ
+�1
+2x
+� �
+1 − σ
+�1
+2x
+�� �
+1 − 2σ
+�1
+2x
+��
+.
+We observe that ∀x ≤ 0, g′(x) ≥ 0, and ∀x ≥ 0, g′(x) ≤ 0. Thus, we have that the minimum of g(x) must be
+achieved at x = a or x = b. Besides, we have that g(x) = g(−x). With the above arguments, we know that the
+minimal eigenvalue is g(0.993) ≈ 0.163 and τh ≈ 0.163. Then we can calculate that
+�
+2
+�Lh( ¯θh) − Lh(θ⋆
+h)
+�
+τh
+≈ 0.520, ∆h( ¯θh)
+Lh
+= 1.
+The inequality (13) holds.
+B.6
+Additional Analysis of WBCU With Function Approximation
+Here we provide an additional theoretical result for WBCU with function approximation when d = 1.
+Theorem 6. Consider d = 1. For any h ∈ [H], if the following inequality holds
+1
+|DS|
+∑
+(s,a)∈DS
+h
+⟨φh(s, a), ¯θh⟩ <
+1
+|DE|
+∑
+(s′,a′)∈DE
+h
+⟨φh(s′, a′), ¯θh⟩.
+(19)
+then we have that ∆h(θ⋆
+h) > 0. Furthermore, the inequality (19) is also a necessary condition.
+Theorem 6 provides a clean and sharp condition to guarantee that the learned discriminator can perfectly
+classify the high-quality samples from DE
+h and DS,1
+h , and low-quality samples from DS,2
+h . In particular,
+inequality (19) means that by the ground truth parameter ¯θh, the average score of samples in supplementary
+dataset is lower than that of samples in expert dataset. This condition is easy to satisfy because the
+supplementary dataset contains bad samples from DS,2
+h
+whose scores are much lower. In contrast, DE only
+contains expert-type samples with high scores.
+24
+
+Proof. Recall the objective function
+min
+θh
+Lh(θh) = ∑
+(s,a)
+�
+dE
+h(s, a) log (1 + exp (−⟨φh(s, a), θh⟩)) + ∑
+(s,a)
+�
+dU
+h (s, a) log (1 + exp (⟨φh(s, a), θh⟩)) .
+Consider the function f (x) = log(1 + exp(x)). We have that f ′′(x) = σ(x)(1 − σ(x)) ≥ 0, where σ(x) =
+1/(1 + exp(−x)). As such, f ′′(x) is a convex function. Notice that the operations of composition with affine
+function and non-negative weighted sum preserve convexity [Boyd et al., 2004]. Therefore, Lh(θh) is a
+convex function with respect to θh. Then for any �θ ∈ R, it holds that
+Lh(θh) ≥ Lh(�θ) + ∇Lh(�θ)
+�
+θh − �θ
+�
+.
+Setting θh = θ⋆
+h and �θ = 0 yields that
+∇Lh(0)θ⋆
+h ≤ Lh(θ⋆
+h) − Lh(0).
+(20)
+The gradient of Lh(θh) is of the form
+∇Lh(θh) = −
+∑
+(s,a)∈S×A
+�
+dE
+h(s, a)σ(−⟨φh(s, a), θh⟩)φh(s, a) +
+∑
+(s,a)∈S×A
+�
+dU
+h (s, a)σ (⟨φh(s, a), θh⟩) φh(s, a).
+Here σ(x) = 1/(1 + exp(−x)) is the sigmoid function. Then we calculate the gradient at �θ = 0.
+∇Lh(0) = −
+∑
+(s,a)∈S×A
+�
+dE
+h(s, a)σ(0)φh(s, a) +
+∑
+(s,a)∈S×A
+�
+dU
+h (s, a)σ (0) φh(s, a)
+= 1
+2
+�∑(s,a)∈DU
+h φh(s, a)
+|DU|
+−
+∑(s,a)∈DE
+h φh(s, a)
+|DE|
+�
+=
+1
+2|DU|
+�
+� ∑
+(s,a)∈DU
+h
+φh(s, a) − |DU|
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+�
+=
+1
+2|DU|
+�
+� ∑
+(s,a)∈DS
+h
+φh(s, a) +
+∑
+(s,a)∈DE
+h
+φh(s, a) − |DU|
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+�
+=
+1
+2|DU|
+�
+� ∑
+(s,a)∈DS
+h
+φh(s, a) − |DS|
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+�
+= |DS|
+2|DU|
+�
+�
+1
+|DS|
+∑
+(s,a)∈DS
+h
+φh(s, a) −
+1
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+� .
+Furthermore, with inequality (19), we know that
+1
+|DS|
+∑
+(s,a)∈DS
+h
+φh(s, a) −
+1
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a) ̸= 0.
+Therefore, ∇Lh(0) ̸= 0. With the first-order optimality condition, we know that 0 is not the optimal solution
+and Lh(0) > Lh(θ⋆
+h). Combined with inequality (20), we have that
+∇Lh(0)θ⋆
+h ≤ Lh(θ⋆
+h) − Lh(0) < 0.
+This directly implies that
+�
+�
+1
+|DS|
+∑
+(s,a)∈DS
+h
+φh(s, a) −
+1
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+� θ⋆
+h < 0.
+25
+
+Besides, inequality (19) implies that
+�
+�
+1
+|DS|
+∑
+(s,a)∈DS
+h
+φh(s, a) −
+1
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+� ¯θh < 0.
+With the above two inequalities, we can derive that θ⋆
+h ¯θh > 0. Then we have that
+∆h(θ⋆
+h) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+θ⋆
+hφh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+θ⋆
+hφh(s′, a′)
+= θ⋆
+h
+¯θh
+�
+min
+(s,a)∈DE
+h ∪DS,1
+h
+¯θhφh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+¯θhφh(s′, a′)
+�
+= θ⋆
+h
+¯θh
+∆h( ¯θh)
+> 0.
+Thus, the sufficiency of condition (19) is proved. Next, we prove the necessity of condition (19). That is, if
+∆h(θ⋆
+h) > 0, then
+1
+|DS|
+∑
+(s,a)∈DS
+h
+⟨φh(s, a), ¯θh⟩ <
+1
+|DE|
+∑
+(s′,a′)∈DS
+h
+⟨φh(s′, a′), ¯θh⟩.
+Then we aim to prove that θ⋆
+h ¯θh > 0. It is easy to obtain that θ⋆
+h ̸= 0 and ¯θh ̸= 0 since ∆h(θ⋆
+h) > 0 and
+∆h( ¯θh) > 0. Then we consider two cases where ¯θh > 0 and ¯θh < 0.
+• Case I ( ¯θh > 0). In this case, we have that
+∆h( ¯θh) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+¯θhφh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+¯θhφh(s′, a′)
+= ¯θh
+�
+min
+(s,a)∈DE
+h ∪DS,1
+h
+φh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+φh(s′, a′)
+�
+.
+Then ∆h( ¯θh) > 0 implies that
+min
+(s,a)∈DE
+h ∪DS,1
+h
+φh(s, a) >
+max
+(s′,a′)∈DS,2
+h
+φh(s′, a′).
+(21)
+Then we claim that θ⋆
+h > 0. Otherwise, it holds that
+∆h(θ⋆
+h) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+θ⋆
+hφh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+θ⋆
+hφh(s′, a′)
+= θ⋆
+h
+�
+max
+(s,a)∈DE
+h ∪DS,1
+h
+φh(s, a) −
+min
+(s′,a′)∈DS,2
+h
+φh(s′, a′)
+�
+< 0.
+The last inequality follows θ⋆
+h < 0 and max(s,a)∈DE
+h ∪DS,1
+h φh(s, a) − min(s′,a′)∈DS,2
+h φh(s′, a′) > 0 due to
+inequality (21). Thus, the above inequality conflicts with ∆h(θ⋆
+h) > 0. The claim that θ⋆
+h > 0 is proved
+and ¯θhθ⋆
+h > 0.
+• Case II ( ¯θh < 0). Similarly, we get that
+∆h( ¯θh) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+¯θhφh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+¯θhφh(s′, a′)
+= ¯θh
+�
+max
+(s,a)∈DE
+h ∪DS,1
+h
+φh(s, a) −
+min
+(s′,a′)∈DS,2
+h
+φh(s′, a′)
+�
+.
+26
+
+Then ∆h( ¯θh) > 0 implies that
+max
+(s,a)∈DE
+h ∪DS,1
+h
+φh(s, a) <
+min
+(s′,a′)∈DS,2
+h
+φh(s′, a′).
+(22)
+Similar to the analysis in case I, we claim that θ⋆
+h < 0. Otherwise, it holds that
+∆h(θ⋆
+h) =
+min
+(s,a)∈DE
+h ∪DS,1
+h
+θ⋆
+hφh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+θ⋆
+hφh(s′, a′)
+= θ⋆
+h
+�
+min
+(s,a)∈DE
+h ∪DS,1
+h
+φh(s, a) −
+max
+(s′,a′)∈DS,2
+h
+φh(s′, a′)
+�
+< 0.
+The last inequality follows θ⋆
+h > 0 and min(s,a)∈DE
+h ∪DS,1
+h φh(s, a) − max(s′,a′)∈DS,2
+h φh(s′, a′) < 0 due to
+inequality (22). Thus, the above inequality conflicts with ∆h(θ⋆
+h) > 0. The claim that θ⋆
+h < 0 is proved
+and ¯θhθ⋆
+h > 0.
+To summarize, we have proved that ¯θhθ⋆
+h > 0. Similar to the proof of the sufficient condition, by the convexity
+of Lh(θh) and optimality of θ⋆
+h, we can obtain that
+�
+�
+1
+|DS|
+∑
+(s,a)∈DS
+h
+φh(s, a) −
+1
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+� θ⋆
+h < 0.
+This directly implies that
+�
+�
+1
+|DS|
+∑
+(s,a)∈DS
+h
+φh(s, a) −
+1
+|DE|
+∑
+(s,a)∈DE
+h
+φh(s, a)
+�
+� ¯θh < 0,
+which completes the proof of necessity.
+C
+Behavioral Cloning with General Function Approximation
+In the main text, the theoretical analysis for BC-based algorithms considers the tabular setting in policy
+learning where a table function represents the policy. Here we provide an analysis of BC with general function
+approximation in policy learning. Notice that the algorithms considered in this paper (i.e., BC, NBCU and
+WBCU) can be unified under the framework of maximum likelihood estimation (MLE)11. Therefore, the
+theoretical results in the main text can also be extended to the setting of general function approximation by a
+similar analysis.
+We consider BC with general function approximation, which is the foundation of analyzing NBCU and
+WBCU. In particular, we assume access to a policy class Π = {π = (π1, π2, . . . , πh) : πh ∈ Πh, ∀h ∈ [H]}
+and Πh = {πh : S → ∆(A)}, where πh could be any function (e.g., neural networks). For simplicity of
+analysis, we assume that Π is a finite policy class. The objective of BC is still
+πBC ∈ max
+π∈Π
+H
+∑
+h=1
+∑
+(s,a)∈S×A
+�
+dE
+h(s, a) log πh(a|s),
+but we do not have the analytic solution as in Equation (2).
+11Among these algorithms, the main difference is the weight function in the MLE objective; see Equations (1), (3) and (10).
+27
+
+Theorem 7. Under Assumption 1. In the general function approximation, additionally assume that πE ∈ Π, we have
+E
+�
+V(πE) − V(πBC)
+�
+≲ min
+�
+�
+�H, H2
+�
+log(|Π|HNE)
+NE
+�
+�
+�
+Compared with Theorem 1, we find that the only change in theoretical bound is that O(|S|/NE) is
+replaced with O(
+�
+log(|Π|)/NE). We note that such a change also holds for other algorithms (e.g., NBCU
+and WBCU). Therefore, our theoretical implications remains unchanged.
+Proof of Theorem 7. We apply the policy difference lemma in [Kakade and Langford, 2002] and obtain that
+V(πE) − V(πBC) =
+H
+∑
+h=1
+Esh∼dπE
+h (·)
+�
+QπBC
+h
+(sh, πE
+h (sh)) − ∑
+a∈A
+πBC
+h (·|sh)QπBC
+h
+(sh, a)
+�
+≤
+H
+∑
+h=1
+Esh∼dπE
+h (·)
+�
+QπBC
+h
+(sh, πE
+h (sh))(1 − πBC
+h (πE
+h (sh)|sh))
+�
+≤ H
+H
+∑
+h=1
+Es∼dπE
+h (·)
+�
+Ea∼πBC
+h (·|s)
+�
+I
+�
+a ̸= πE
+h (s)
+���
+= H
+H
+∑
+h=1
+Esh∼dE
+h(·)
+�
+TV
+�
+πE
+h (·|sh), πBC
+h (·|sh)
+��
+.
+With Theorem 21 in [Agarwal et al., 2020], when |DE| ≥ 1, for any δ ∈ (0, 1), with probability at least 1 − δ
+over the randomness within DE, we have that
+Esh∼dE
+h(·)
+�
+TV2 �
+πE
+h (·|sh), πBC
+h (·|sh)
+��
+≤ 2log(|Π|/δ)
+|DE|
+.
+With union bound, with probability at least 1 − δ, for all h ∈ [H], it holds that
+Esh∼dE
+h(·)
+�
+TV2 �
+πE
+h (·|sh), πBC
+h (·|sh)
+��
+≤ 2log(|Π|H/δ)
+|DE|
+,
+which implies that
+V(πE) − V(πBC) ≤ H
+H
+∑
+h=1
+Esh∼dE
+h(·)
+�
+TV
+�
+πE
+h (·|sh), πBC
+h (·|sh)
+��
+(a)
+≤ H
+H
+∑
+h=1
+�
+Esh∼dE
+h(·)
+�
+TV2 �
+πE
+h (·|sh), πBC
+h (·|sh)
+��
+≤
+√
+2H2
+�
+log(|Π|H/δ)
+|DE|
+.
+Inequality (a) follows Jensen’s inequality. Taking expectation over the randomness within DE yields that
+EDE
+�
+V(πE) − V(πBC)
+�
+≤ δH + (1 − δ)
+√
+2H2
+�
+log(|Π|H/δ)
+|DE|
+(a)
+=
+H
+2|DE| +
+�
+1 −
+1
+2|DE|
+� √
+2H2
+�
+log(2|Π|H|DE|)
+|DE|
+≤
+�√
+2 + 1
+�
+H2
+�
+log(2|Π|H|DE|)
+|DE|
+28
+
+≤ 4H2
+�
+log(4|Π|H|DE|)
+|DE|
+.
+Equation (a) holds due to the choice that δ = 1/(2|DE|). For |DE| = 0, we directly have that
+EDE
+�
+V(πE) − V(πBC)
+�
+≤ H.
+Therefore, for any |DE| ≥ 0, we have that
+EDE
+�
+V(πE) − V(πBC)
+�
+≤ 4H2
+�
+log(4|Π|H max{|DE|, 1})
+max{|DE|, 1}
+.
+We consider a real-valued function f (x) = log(cx)/x, ∀x ≥ 1, where c = 4|Π|H. Its gradient function is
+f ′(x) = (1 − log(cx))/x2 ≤ 0, ∀x ≥ 1. Then we know that f (x) is decreasing as x increases. Furthermore,
+we have that max{|DE|, 1} ≥ (|DE| + 1)/2, ∀|DE| ≥ 0. Then we obtain
+EDE
+�
+V(πE) − V(πBC)
+�
+≤ 4H2
+�
+log(4|Π|H max{|DE|, 1})
+max{|DE|, 1}
+≤ 4H2
+�
+2 log(4|Π|H(|DE| + 1))
+|DE| + 1
+.
+Taking expectation over the random variable |DE| ∼ Bin(Ntot, η) yields that
+E
+�
+V(πE) − V(πBC)
+�
+≤ 4H2E
+��
+2 log(4|Π|H(|DE| + 1))
+|DE| + 1
+�
+(a)
+≤ 4H2
+�
+E
+�2 log(4|Π|H(|DE| + 1))
+|DE| + 1
+�
+.
+Inequality (a) follows Jensen’s inequality. We consider the function g(x) = −x log(x/c), ∀x ∈ (0, 1], where
+c = 4|Π|H.
+g′(x) = −(log(x/c) + 1) ≥ 0, g′′(x) = − 1
+x ≤ 0, ∀x ∈ (0, 1].
+Thus, g(x) is a concave function. By Jensen’s inequality, we have that E[g(x)] ≤ g(E[x]). Then we can
+derive that
+E
+�
+V(πE) − V(πBC)
+�
+≤ 4H2
+�
+E
+�2 log(4|Π|H(|DE| + 1))
+|DE| + 1
+�
+= 4
+√
+2H2
+�
+E
+�
+g
+�
+1
+|DE| + 1
+��
+≤ 4
+√
+2H2
+�
+g
+�
+E
+�
+1
+|DE| + 1
+��
+(a)
+≤ 4
+√
+2H2
+�
+g
+� 1
+NE
+�
+≤ 4
+√
+2H2
+�
+log(4|Π|HNE)
+NE
+.
+In inequality (a), we use the facts that g′(x) ≥ 0 and E
+�
+1/(|DE| + 1)
+� ≤ 1/NE from Lemma 3. We complete
+the proof.
+29
+
+D
+Experiments
+D.1
+Experiment Details
+Dataset Collection. We train an online SAC agent [Haarnoja et al., 2018] with 1 million steps using the rlkit
+codebase12, which is the same as benchmark dataset D4RL13. We use the deterministic policy as the expert
+policy, which is common in the literature [Ho and Ermon, 2016] since the deterministic policy usually gives a
+better performance than the stochastic policy; see Figure 5 for the training curves of online SAC.
+0.00
+0.25
+0.50
+0.75
+1.00
+Interaction Step
+1e6
+0
+1000
+2000
+3000
+4000
+5000
+6000
+Evaluation Return
+Ant-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Interaction Step
+1e6
+2500
+0
+2500
+5000
+7500
+10000
+12500
+HalfCheetah-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Interaction Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Interaction Step
+1e6
+0
+1000
+2000
+3000
+4000
+5000
+Walker2d-v2
+Figure 5: Training curves of online SAC.
+In our experiments, the expert dataset has 1 expert trajectory that is collected by the trained SAC agent.
+Two tasks differ in the supplementary dataset.
+Noisy Expert Task. The supplementary dataset has 10 clean expert trajectories and 5 noisy expert
+trajectories. For noisy trajectories, the action labels are replaced with random actions (drawn from [−1, 1]).
+Full Replay Task. The supplementary dataset is from the replay buffer of the online SAC agent, which
+has 1 million samples (roughly 1000+ trajectories).
+Algorithm Implementation. The implementation of DemoDICE is based on the original authors’ code-
+base14. Same as DWBC15. We have fine-tuned the hyper-parameters of DemoDICE and DWBC in our
+experiments but find that the default parameters given by these authors work well. Following [Kim et al.,
+2022b], we normalize state observations in the dataset before training for all algorithms.
+We use gradient penalty (GP) regularization in training the discriminator of WBCU. Specifically, we add
+the following loss to the original loss (7):
+min
+θ
+(∥g(s, a; θ)∥ − 1)2 ,
+where g is the gradient of the discriminator c(s, a; θ).
+The implementation of NBCU and WBCU is adapted from DemoDICE’s. In particular, both the discrimi-
+nator and policy networks use the 2-layers MLP with 256 hidden units and ReLU activation. The batch size
+is 256 and learning rate (using the Adam optimizer) is 0.0003 for both the discriminator and policy. The
+number of training iterations is 1 million. We use δ = 0 and GP=1, unless mentioned. Please refer to our
+codebase16 for details.
+All experiments are run with 5 random seeds.
+D.2
+Additional Results
+In this section, we report the exact performance of trained policies for each MuJoCo locomotion control task.
+Training curves are displayed in Figure 8 and Figure 9. The evaluation performance of the last 10 iterations
+is reported in Table 2 and Table 3. The normalized score on a particular environment is calculated by the
+12https://github.com/rail-berkeley/rlkit
+13https://github.com/Farama-Foundation/D4RL
+14https://github.com/KAIST-AILab/imitation-dice
+15https://github.com/ryanxhr/DWBC
+16https://github.com/liziniu/ILwSD
+30
+
+following formula.
+Normalized Score =
+Algorithm Performance − Random Policy Performance
+Expert Policy Performance − Random Policy Performance.
+In Table 2 and Table 3, the normalized score is averaged over 4 environments.
+Training curves of WBCU and DemoDICE with gradient penalty are displayed in Figure 10, Figure 11,
+Figure 12, and Figure 13.
+Noisy Expert
+Full Replay
+Task
+0
+20
+40
+60
+80
+Normalized Score (%)
+31
+66
+40
+94
+49
+94
+Algorithm
+DemoDICE(GP=0)
+DemoDICE(GP=1)
+DemoDICE(GP=10)
+Figure 6: Averaged normalized scores of trained policies of DemoDICE with gradient penalty. Experiments
+show that the gradient penalty regularization also matters for DemoDICE.
+Table 2: Performance of the trained policies by BC, DemoDICE [Kim et al., 2022b], DWBC [Xu et al., 2022a],
+NBCU (Algorithm 1), and WBCU (Algorithm 2) on the noisy-expert task. Numbers correspond to the
+averaged evaluation return over 5 random seeds (± indicate the standard deviation); a lager return means
+better performance (same as other tables).
+Ant-v2
+HalfCheetah-v2
+Hopper-v2
+Walker2d-v2
+Normalized Score
+Random
+-325.6
+-280
+-20
+2
+0%
+Expert
+5229
+11115
+3589
+5082
+100%
+BC
+1759±287
+931±273
+2468±164
+1738±311
+38%
+DemoDICE(GP=0)
+1893±181
+2139±277
+1823±169
+563±34
+31%
+DemoDICE(GP=1)
+2492±235
+5553±501
+1320±319
+1153±220
+40%
+DemoDICE(GP=10)
+2523±244
+6020±346
+1990±90
+1685±160
+49%
+DWBC
+3270±238
+5688±557
+3317±59
+1985±175
+62%
+NBCU
+3259±159
+5561±539
+558±23
+518±56
+35%
+WBCU(GP=0)
+738±179
+3828±333
+2044±111
+477±81
+30%
+WBCU(GP=1)
+2580±224
+8274±488
+3217±203
+1932±329
+64%
+WBCU(GP=10)
+2130±259
+7813±472
+2930±229
+1510±476
+57%
+Remark 2. Readers may realize that NBCU actually is better than BC on the Ant-v2 and HalfCheetah-v2 environments
+for the noisy-expert task, which seems to contradict our theory. We clarify there is no contradiction. In fact, our theory
+implies that in the worst case, NBCU is worse than BC. As we have discussed in the main text, state coverage matters
+for NBCU’s performance. For the noisy expert task, we visualize the state coverage in Figure 7, where we use Kernel
+PCA17 to project states. In particular, we see that the state coverage is relatively nice on Ant-v2 and HalfCheetah-v2,
+17https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.KernelPCA.html. We use the “poly” ker-
+nel.
+31
+
+Table 3: Performance of the trained policies by BC, DemoDICE [Kim et al., 2022b], DWBC [Xu et al., 2022a],
+NBCU (Algorithm 1), and WBCU (Algorithm 2) on the full-replay task.
+Ant-v2
+HalfCheetah-v2
+Hopper-v2
+Walker2d-v2
+Normalized Score
+Random
+-325.6
+-280
+-20
+2
+0%
+Expert
+5229
+11115
+3589
+5082
+100%
+BC
+1759±287
+931±273
+2468±164
+1738±311
+38%
+DemoDICE(GP=0)
+4650±216
+9882±270
+39±0
+4149±288
+66%
+DemoDICE(GP=1)
+5009±98
+10701±70
+3427±104
+4560±146
+94%
+DemoDICE(GP=10)
+5000±124
+10781±67
+3394±93
+4537±125
+94%
+DWBC
+2951±155
+1485±377
+2567±88
+1572±225
+44%
+NBCU
+4932±148
+10566±86
+3241±276
+4462±105
+92%
+WBCU(GP=0)
+483±79
+−242±101
+346±93
+15±15
+6%
+WBCU(GP=1)
+4935±108
+10729±74
+3390±132
+4509±142
+94%
+WBCU(GP=10)
+4858±95
+10751±41
+3436±37
+4403±88
+93%
+and somewhat bad on Hopper-v2 and Walker2d-v2. This can help explain the performance difference among these
+environments in Table 2.
+32
+
+0
+50
+100
+150
+200
+250
+0
+10
+20
+30
+40
+DE
+DS, 1
+DS, 2
+(a) Ant-v2.
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2
+1
+0
+1
+2
+DE
+DS, 1
+DS, 2
+(b) Halfcheetah-v2.
+2
+0
+2
+4
+6
+4
+2
+0
+2
+4
+DE
+DS, 1
+DS, 2
+(c) Hopper-v2.
+3
+2
+1
+0
+1
+2
+3
+4
+2
+0
+2
+4
+6
+DE
+DS, 1
+DS, 2
+(d) Walker2d-v2.
+Figure 7: Visualization of the state coverage for the noisy expert task. According to the experiment set-up, red
+points correspond to good samples and blue points correspond to noisy and bad samples. Plots show that
+the state overlap between two modes of samples is large for Hopper-v2 and Walker2d-v2 and is limited for
+Ant-v2 and HalfCheetah-v2. Therefore, we can expect NBCU performs well on Ant-v2 and HalfCheetah-v2.
+33
+
+3000
+1500
+0
+1500
+3000
+4500
+6000
+Evaluation Return
+Ant-v2
+0
+2500
+5000
+7500
+10000
+HalfCheetah-v2
+BC
+DWBC
+DemoDICE
+Expert
+NBCU
+WBCU
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+1500
+3000
+4500
+6000
+Walker2d-v2
+Figure 8: Training curves of BC, DemoDICE [Kim et al., 2022b], DWBC [Xu et al., 2022a], NBCU (Algorithm 1),
+and WBCU (Algorithm 2) on the noisy expert task. Solid lines correspond to the mean performance and
+shaded regions correspond to the 95% confidence interval. Same as other figures.
+0
+1500
+3000
+4500
+6000
+Evaluation Return
+Ant-v2
+0
+2500
+5000
+7500
+10000
+HalfCheetah-v2
+BC
+DWBC
+DemoDICE
+Expert
+NBCU
+WBCU
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+1500
+3000
+4500
+6000
+Walker2d-v2
+Figure 9: Training curves of BC, DemoDICE [Kim et al., 2022b], DWBC [Xu et al., 2022a], NBCU (Algorithm 1),
+and WBCU (Algorithm 2) on the full replay task.
+34
+
+1500
+0
+1500
+3000
+4500
+6000
+Evaluation Return
+Ant-v2
+0
+2500
+5000
+7500
+10000
+HalfCheetah-v2
+Expert
+WBCU(GP=0.0)
+WBCU(GP=1.0)
+WBCU(GP=10.0)
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+1500
+3000
+4500
+6000
+Walker2d-v2
+Figure 10: Training curves of WBCU with gradient penalty on the noisy expert task.
+0
+1500
+3000
+4500
+6000
+Evaluation Return
+Ant-v2
+0
+2500
+5000
+7500
+10000
+HalfCheetah-v2
+Expert
+WBCU(GP=0.0)
+WBCU(GP=1.0)
+WBCU(GP=10.0)
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+1500
+3000
+4500
+6000
+Walker2d-v2
+Figure 11: Training curves of WBCU with gradient penalty on the noisy expert task.
+35
+
+3000
+1500
+0
+1500
+3000
+4500
+6000
+Evaluation Return
+Ant-v2
+0
+2500
+5000
+7500
+10000
+HalfCheetah-v2
+DemoDICE(GP=0.0)
+DemoDICE(GP=1.0)
+DemoDICE(GP=10.0)
+Expert
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+1500
+3000
+4500
+6000
+Walker2d-v2
+Figure 12: Training curves of DemoDICE with gradient penalty on the full replay task.
+0
+1500
+3000
+4500
+6000
+Evaluation Return
+Ant-v2
+0
+2500
+5000
+7500
+10000
+HalfCheetah-v2
+DemoDICE(GP=0.0)
+DemoDICE(GP=1.0)
+DemoDICE(GP=10.0)
+Expert
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+800
+1600
+2400
+3200
+4000
+Hopper-v2
+0.00
+0.25
+0.50
+0.75
+1.00
+Gradient Step
+1e6
+0
+1500
+3000
+4500
+6000
+Walker2d-v2
+Figure 13: Training curves of DemoDICE with gradient penalty on the full replay task.
+36
+

cdE2T4oBgHgl3EQfwgij/content/tmp_files/2301.04102v1.pdf.txt

@@ -0,0 +1,1715 @@
+Under consideration for publication in J. Fluid Mech.
+1
+Banner appropriate to article type will appear here in typeset article
+Effects of porous substrates on the structure of
+turbulent boundary layers
+P. Jaiswal1 and B. Ganapathisubramani1†
+1Aerodynamics & Flight Mechanics Group, University of Southampton, Southampton, SO16 7QF, UK
+(Received xx; revised xx; accepted xx)
+Three different porous substrates (with different pore sizes, 𝑠 and permeabilities, 𝐾) are
+used to examine their effect on the structure of boundary layer flow over them. The flow
+is characterised with single-point hot-wire measurements as well as planar Particle Image
+Velocimetry. In order to elucidate differences in shallow and deep flows past porous substrate,
+foams with two different thickness (ℎ) are used (for all three substrates). A wide range of
+Friction Reynolds number (2000 < 𝑅𝑒𝜏 < 13500) and Permeability based Reynolds number
+(1 < 𝑅𝑒𝐾 < 50) are attained. For substrates with 𝑅𝑒𝐾 ∼ 1, the flow behaviour remains
+similar to flow over impermeable smooth walls and as such Townsend’s hypothesis remains
+valid. In contrast, a substantial reduction in velocity disturbances and associated length
+scales is achieved for permeable (𝑅𝑒𝐾 > 1) and dense (relative to viscous scales, 𝑠+ < 60)
+foam at the thick substrate limit (ℎ/𝑠 > 10), which leads to the breakdown of outer-layer
+similarity. As porosity is increased, a thin substrate limit is reached (ℎ/𝑠), and the foam
+becomes sparse relative to viscous scales (𝑠+ > 100). For such foams, the flow conforms
+to outer-layer similarity and is more akin to flow over rough surfaces. Such substrates are
+unable to attenuate velocity disturbances and the dependence of substrate thickness (ℎ/𝑠) on
+spectral energy content of turbulent fluctuations ceases to exist. The present study shows that
+transition from thick to thin substrate flow behaviour depends not only on thickness-to-pore
+ratio (ℎ/𝑠) but also on substrate density relative to viscous scales of the flow (𝑠+).
+Key words: Authors should not enter keywords on the manuscript, as these must be chosen by
+the author during the online submission process and will then be added during the typesetting
+process (see Keyword PDF for the full list). Other classifications will be added at the same
+time.
+MSC Codes (Optional) Please enter your MSC Codes here
+1. Introduction
+Turbulent flow over and past porous surfaces is encountered in many engineering problems,
+ranging from flow over forest canopies (Finnigan 2000) to flows over and past river beds
+† Email address for correspondence: [email protected]
+Abstract must not spill onto p.2
+arXiv:2301.04102v1  [physics.flu-dyn]  10 Jan 2023
+
+2
+(Yovogan & Degan 2013). This makes the understanding of the flow behaviour over porous
+surfaces crucial. For a porous substrate, Rosti et al. (2015) showed that compared to porosity
+small changes in permeability can significantly alter the turbulence dynamics. The effects
+of wall-permeability for flows over and past porous foams was further studied in detail by
+Hahn et al. (2002); Breugem et al. (2006). Breugem et al. (2006) suggested that an isotropic
+porous substrate could be fully defined by three length scales, which are the square root
+of material permeability
+√
+𝐾, the substrate thickness ℎ, and the characteristic size of the
+‘roughness’ elements composing the substrate 𝑑𝑝. Breugem et al. (2006) stated that the
+effect of permeability on the flow is isolated if three conditions are meet: i) the wall thickness
+is larger than the flow penetration into the substrate, ii) the roughness Reynolds number
+𝑅𝑒𝑑 = 𝑑𝑝𝑈𝜏/𝜈 is small (𝑅𝑒𝑑 << 70, where 𝑈𝜏 is the skin-friction velocity and 𝜈 is the
+kinematic viscosity) and iii) the permeability Reynolds number 𝑅𝑒𝐾 =
+√
+𝐾𝑈𝜏/𝜈 is high
+(𝑅𝑒𝐾 >> 1).
+Studies by Breugem et al. (2006) and Manes et al. (2011) were able to meet the above
+mentioned criterion. Therefore, the effect of surface roughness can be neglected. It was shown
+that permeable wall can substantially alter eddy blocking, quasi-streamwise vortices and no
+slip at the wall (see Breugem et al. 2006, for instance). The modification of these properties
+by permeable wall, which are trademarks of the turbulent boundary layer, leads to a departure
+from outer-layer similarity in velocity statistics. Furthermore, the impact of permeable wall
+can also be felt by large-scale structures, and leads to non-existence of logarithmic mean
+velocity law (Breugem et al. 2006). Similarly, the wall roughness, by itself, can also alter
+turbulence dynamics by destroying non-linear self-sustaining cycles of turbulence (Jiménez
+2004). Although these mechanisms for permeable and rough surfaces are well reported in
+the literature, yet their relative contribution and interactions towards boundary-layer scales
+over a porous material, which is both rough and permeable, remains a matter for further
+investigation.
+One such aspect is the existence of Townsend’s outer-layer hypothesis (Townsend 1980)
+for a porous wall. According to Townsend’s hypothesis the outer layer flow is independent
+of the near-wall region; therefore, for a flow over a wall, the primary effect of the wall is
+impermeability and no-slip boundary condition. To this end several studies have demonstrated
+its validity for smooth walls (Chung et al. 2014). Townsend’s hypothesis has also been found
+to be valid for flow over rough walls, provided that the equivalent sand roughness height 𝑘𝑠
+is small compared to the boundary-layer thickness (Jiménez 2004). Therefore, in contrast to
+wall-permeability, the surface roughness with a reasonable scale separation (low 𝑘𝑠/𝛿) does
+not affect the logarithmic mean profiles and large-scale structures remain intact. In contrast
+for flows past porous surfaces, Breugem et al. (2006) and Suga et al. (2010) found that the
+outer-layer hypothesis holds for all but the wall-normal velocity component. Breugem et al.
+(2006) ascribed the absence of self-similarity in the wall-normal velocity profiles to the
+weakening of wall-blocking. The weakening of wall-blocking opens a path for inner-outer
+boundary layer communications through enhanced ejections and sweep (Breugem et al.
+2006) compared to a solid impermeable (smooth or rough) wall. Breugem et al. (2006) argue
+that the enhanced ejections and sweep are sufficient to nullify Townsend’s hypothesis, which
+requires the absence of inner layer scales influencing the outer wall flows. However, they
+(Suga et al. 2010; Breugem et al. 2006) were unable to decisively conclude if absence of
+outer-layer scaling is due to permeability or insufficient separation of scales because Breugem
+et al.’s (2006) numerical simulations were performed at a low Reynolds number.
+To overcome the limitation of the low Reynolds number that can be achieved with DNS,
+Manes et al. (2011) performed experimental measurements at a higher Reynolds number.
+Manes et al.’s (2011) data confirm the validity of Townsend’s outer-layer similarity hypothesis
+
+3
+for all the velocity components, for porous foams with negligible surface roughness. However,
+the thickness of their (Manes et al. 2011) porous substrate was much greater than the pore
+size. As such, their study is equivalent to flow past deep canopies (Sharma & García-Mayoral
+2020b), where the effect of foam thickness is no longer a relevant parameter.
+The analogy between flows past foams and canopies has been hypothesized by Efstathiou
+& Luhar (2018). Efstathiou & Luhar (2018) were able to show the effect of substrate thickness
+on the turbulent boundary layer, and near-wall flow physics by investigating porous materials
+with different ℎ/𝑠 ratio. Efstathiou & Luhar (2018) claim that the transition from thick to
+thin substrate behaviour occurs when the thickness and diameter of the porous substrate
+are of the same order of magnitude. One may argue that these porous materials, whose
+thickness and pore diameter are of the same order of magnitude, can be considered a
+“rough” wall. Consequently, for porous foams with thin substrate thickness, the relative
+contribution of roughness should be higher than permeability in setting the turbulence
+dynamics in the outer layer. In contrast, if the substrates thickness is an order of magnitude
+higher than the pore size, we would expect permeability to play an important role in setting
+the wall-boundary condition. Efstathiou & Luhar (2018) showed that for foams with finite
+thickness, the existence of Townsend’s outer-layer hypothesis remains valid. The foams tested
+by Efstathiou & Luhar (2018) had small values of permeability based Reynolds number
+(𝑅𝑒𝐾), especially those at the thick substrate limit. This suggests that permeability based
+scales were comparable to viscous scales in their study, as such it is unclear if the permeability
+played a role in setting wall-boundary conditions for the cases tested by Efstathiou & Luhar
+(2018). In addition to low values of 𝑅𝑒𝐾, Efstathiou & Luhar (2018) concluded the validity
+of Townsend’s hypothesis solely based on wall-parallel velocity statistics. The wall-normal
+statistics are especially sensitive to permeable surfaces as noted by Breugem et al. (2006);
+therefore the validity of Townsend’s outer-layer hypothesis for porous (rough and permeable)
+foams is an open question. Is the flow over such porous surfaces analogous to flows over
+rough surfaces away from the wall? If so, does the outer-layer similarity in velocity statistics
+holds for such porous foams? Thus the primary objective of the current paper is to test
+Townsend’s outer-layer hypothesis at a high Reynolds number for turbulent flows past porous
+foam with varying thicknesses, permeability and roughness.
+A potential similarity between flows past canopies (Sharma & García-Mayoral 2020b)
+and foam (Efstathiou & Luhar 2018) is that as the pore size is increased, a thin substrate
+limit is achieved where the velocity profile becomes fuller, ultimately resulting in loss of
+the inflection point. This ensures absence of any Kelvin-Helmholtz instability, which results
+in reduction in wall-parallel velocity spectra compared to cases where Kelvin-Helmholtz
+instability flow is present. Presence of KH instability was also reported by Kuwata & Suga’s
+(2017), they observed that the pressure fluctuations were correlated all along the span. On
+the one hand, numerical simulations (Motlagh & Taghizadeh 2016; Kuwata & Suga 2017)
+have been performed at much lower Reynolds numbers compared to experimental studies,
+on the other hand, experimental studies (Efstathiou & Luhar 2018) have reported this based
+only single-point statistics and for cases. Therefore, in the current study, PIV measurements
+were carried out for each wall topology. PIV inherently shows the flow structures, and one
+does not have to rely on the assumption of frozen turbulence to recover spatial information
+from temporal single-point velocity measurements.
+The dual-component planar PIV allows for the quantification of the wall-normal velocity
+disturbance field, which plays a decisive role in the generation of wall-pressure fluctuations
+(see chapter 8 of Blake 2017, for instance). The wall-pressure field is the primary driver of
+aerofoil self-noise, and changes in the two-point wall-normal velocity correlation have been
+used to assess the impact of porous materials on aerofoil self-noise (see Carpio et al. 2019, for
+instance). Carpio et al. (2019) showed that maximum noise reduction is achieved when the
+
+4
+two-point correlation of the wall-normal velocity disturbances is lower for a permeable wall
+compared to a solid wall. However, if the pore size is “too large” (hence high permeability), it
+can result in an increase in the two-point correlation of the wall-normal velocity disturbances
+(Carpio et al. 2019) and in the root mean square (RMS) values of wall-pressure (Breugem
+et al. 2006). This suggests that, for trailing-edge noise attenuation, there is a range of pore
+sizes and substrate thicknesses that can achieve noise reduction. Naturally, the question is
+what values of pore thickness and size can we expect to reduce the correlation of wall-normal
+velocity fluctuations?
+Furthermore, in such applications (trailing-edge noise reduction), flow past porous foams
+can transition from the thick foam limit (aerofoil mid-chord) to the thin foam limit (aerofoil
+trailing edge). At finite thickness limit, roughness layer can dictate the efficacy wall-
+permeability condition (White & Nepf 2007). Nevertheless, previous studies were, either
+done at low permeability based Reynolds number (Efstathiou & Luhar 2018), or have
+studied the flow past foams at deep/thick substrate limit (Manes et al. 2011), which is not
+representative of porous foams used in trailing-edge noise applications. Therefore, the second
+objective of this manuscript is to unravel flow structures present inflows over porous foam
+with varying thicknesses, which can provide direct experimental evidence on the existence
+or non-existence of KH type instability, and impact of porous substrates on the structure of
+turbulent boundary layer. To our knowledge, this is the first study of the spatial structure of
+turbulence over porous foams at high Reynolds number (𝑅𝑒𝜏 > 2000).
+As argued by Finnigan et al. (2009) and Manes et al. (2011) the imprint of Kelvin-
+helmoltz instability is best visible in streamwise velocity spectra. While Manes et al. (2011)
+performed measurements only at the thick foam limits, Efstathiou & Luhar (2018) were
+unable to report near-wall streamwise velocity spectra data due to noise. Additionally, both
+these measurements were performed at the dense foam limit. Therefore, this study fills the
+scientific gap by reporting streamwise energy spectra for flows over and past porous foams
+with varying thicknesses and pore densities at high Reynolds numbers. Thus, third and final
+objective of this study is to report wall-parallel turbulent kinetic energy spectra over a wide
+range of foam thickness and density, and delineate their impact on the associated time scales
+and turbulent energy.
+2. Methodology and outline
+In the present paper, both transitionally and fully rough flows above porous foams will be
+investigated at high Reynolds number. As the increase in porosity is achieved by increasing
+the pore size of the foams, the thickness and pore-size ratio range from ℎ/𝑠 = 0.7 to
+ℎ/𝑠 = 60. This allows investigation of differences between shallow and deep flows as the
+thickness is varied. At the same time, increasing or decreasing pores per inch should also
+permit one to cover the dense and sparse foam limits. If the nominal pore size (s) is taken as
+the characteristic length scale to compute the roughness Reynolds number 𝑠+ = 𝑠𝑈𝜏/𝜈 (see
+Efstathiou & Luhar 2018, for instance), then 𝑠+ for all but one case appears to be way beyond
+the condition (Reynolds number based on roughness<< 70) to decouple permeability from
+roughness (Breugem et al. 2006). Therefore, with the possible exception of thicker foam with
+highest number of cells at lowest free-stream velocity (𝑈∞) tested, all the test cases should
+experience both permeable and roughness effects. The present study aims to quantify the
+effects of porous wall and its overarching influence on the structure of turbulent boundary
+layer over the broadest range of Reynolds numbers. The Friction-based Reynolds number
+(𝑅𝑒𝜏 = 𝑈𝜏𝛿99/𝜈 where 𝑈𝜏 is the skin friction velocity and 𝛿99 is the boundary layer
+thickness) for the present study, was in the range 𝑅𝑒𝜏 ≈ 2000 − 13500. The permeability
+Reynolds number (𝑅𝑒𝐾 = 𝑈𝜏
+√
+𝐾/𝜈) was in the range 𝑅𝑒𝐾 ≈ 1 − 50. It is important to
+Focus on Fluids articles must not exceed this page length
+
+5
+note that the magnitude of 𝑈𝜏 used in Efstathiou & Luhar’s (2018) study was obtained from
+the smooth-wall region upstream of the porous substrate and does not include information
+about the effect of substrate permeability on the flow. In contrast, in the present study drag-
+balance measurements (see Ferreira et al. 2018, for details) are performed. The drag balance
+provides a direct measure of the local skin friction velocity, which will be applied as a
+reference velocity for normalisation in inner units.
+The paper is structured as follows: Details on porous materials, the test setup, experimental
+methods, and the associated measurement uncertainty can be found in section 3. Section 4
+reports the experimental findings in such a way that each of its three subsections aims to
+investigate the three objectives of the current manuscript. In order to evaluate as to whether
+the established scaling and similarity laws for (impermeable) rough wall flows can also be
+applied for permeable wall flow, the velocity statistics scaled with outer-layer variable are
+shown in section 4.1. Section 4.2 seeks to investigate the structure of turbulent flows over
+and past a porous wall, which is the second objective of this paper. Section 4.3 quantifies the
+wall-parallel turbulent kinetic energy spectra to elucidate the difference between shallow and
+deep flows past a porous foams. Section 5 provides an extended discussion on the findings.
+Finally, conclusions and perspectives are drawn in section 6.
+3. Experimental Set-up and Instrumentation:
+3.1. Porous substrate
+Three foams with varying number of cells per inch, each having two thicknesses, were
+used in the present study. One of these substrates was a high porosity open-cell reticulated
+polyurethane foam with porosity (empty volume over total volume) 𝜖 ≈ 0.97. The other
+two substrates were open-cell foams with moderate porosities 𝜖 ≈ 0.86 and 𝜖 ≈ 0.60.
+The pore size of the substrates were also obtained, and these are 𝑠 ≈ 3.84, 0.89, 0.25 mm
+ordered from the most porous to the least porous substrate. To measure these material
+properties all substrates were scanned (CT-scan) with a voxel resolution of 0.056 mm in
+all three dimensions. The data was later imported into the open source FijiJ software and
+the commercial Avizo software to estimate total porosity and pore size respectively. Total
+porosity was obtained by applying a Ozul threshold to the 3D stack of reconstructed images
+and pore sizes were obtained by applying iterative threshold and image segmentation. To
+put the total porosity values measured into context, spherical glass beads in a ‘uniformly
+random’ form have a porosity ranging from 𝜖 ≈ 0.64 to 𝜖 ≈ 0.36, cylindrical packings have
+a porosity range 𝜖 ≈ 0.59 to 𝜖 ≈ 0.32 and cylindrical fibres 𝜖 ≈ 0.919 to 𝜖 ≈ 0.682 as
+reviewed in Macdonald et al. (1979).
+3.2. Experimental test section
+All experimental investigations were conducted at the University of Southampton, in an
+open-circuit suction type wind tunnel. The wind tunnel has a working section of 4.5 [m]
+in length, with a 0.9 [m] height, and a 0.6 [m] cross-plane length. Over the bottom wall of
+the wind tunnel, the turbulent boundary-layer has a zero-pressure-gradient. The bottom wall
+of the test section is covered with the porous substrate. In the present paper, the coordinate
+system is defined such that the subscripts 1, 2 and 3 are used to define entities in wall-
+parallel, wall-normal and spanwise directions respectively (see figure 1). Furthermore, upper
+case letters are used to denote statistical mean, while lower case letters are used to denote
+the standard deviation. For instance, 𝑈1 and 𝑢2 denote the mean wall-parallel velocity and
+the standard deviation of wall-normal velocity respectively.
+
+6
+x1
+x3
+x2
+U∞
+Figure 1: Coordinate-system
+3.3. Hot-wire measurements
+Hot Wire Anemometry (HWA) measurements were performed with a single wire boundary-
+layer probe. This single wire, made with tungsten, has a 5 [𝜇m] diameter and a sensing length
+of 1 [mm]. The hot wire probe was connected to a DANTEC Streamline Pro anemometer,
+which was operating in Constant Temperature Anemometry (CTA) mode, at a fixed overheat
+ratio of 0.8. The signals were sampled at a rate of 20 kHz for a duration of at-least 3 minutes,
+which is equivalent to ∼ 20000 boundary-layer turnover time. The hot wire measurements
+allow a direct comparison of the velocity profiles measured using PIV. In the present
+manuscript single wire measurements were used to quantify temporal scales, and single-
+point velocity statistics.
+3.4. Particle Image Velocimetry
+In order to investigate flow structures in the mean flow direction, planar particle image
+velocimetry (PIV) measurements were performed. Images for the PIV measurements were
+taken with Lavision’s 16 Mpix CCD camera. The images were recorded in dual frame mode.
+For illumination, a dual pulse ND:YAG laser from Litron was used. A Magnum 1200 fog
+machine, equipped with a Glycerol-water based solution, was used to generate tracer particles
+for PIV measurements. The average size of the resulting tracer particles was approximately
+1 [𝜇𝑚]. A flat laser sheet of about 1 [mm] was generated by placing a cylindrical lens
+with a negative focal length after a set of spherical doublets. All the images were processed
+using Lavision’s commercial software Davis 8.2. In total about 2500 images were acquired
+at a sampling rate of 0.5 Hz, which ensures that the individual velocity field is statistically
+independent. The maximum free stream displacement was around 7 pixels, which implies a
+random error of 1.5% in PIV measurements. The velocity vector field were computed with a
+multi-grid cross-correlation scheme, which has a final window size of 24 × 24 [pixels2], and
+an overlap of 50% between the windows. Finally, PIV measurements have been performed
+at distance of approximately 3.0 [m] from the inlet, as shown in figure 2. Given the fact
+that the PIV measurement domain extends to almost twice the boundary-layer thickness, the
+streamwise averaged boundary-layer thickness is used to normalise flow quantities throughout
+the rest of the manuscript.
+3.5. Skin-friction measurements
+In the present work, a floating element balance presented by Ferreira et al. (2018) is used to
+quantify skin-friction coefficient (Cf ). On the bottom wall at approximately 3.3 [m] from
+the inlet a floating element balance (Ferreira et al. 2018) is flush mounted onto the wind
+tunnel floor. The measurement uncertainty in Cf, for all the cases reported in this study, using
+the floating element is about 5% (see Appendix of Gul & Ganapathisubramani 2021, for
+instance). The floating element balance is flush mounted with the wind tunnel floor and the
+gap surrounding the balance is taped over to prevent leaks. The porous foams are cut with
+
+7
+Figure 2: A schematic representation Planar PIV setup.
+Table 1: Uncertainty quantification for various measured quantities.
+Quantity Measured
+Uncertainty (95 % confidence)
+Tunnel Inlet velocity
+1 % 𝑈∞
+Random error mean velocity Planar PIV
+∼ 0.1 % 𝑈∞
+Averaging uncertainty 𝑅𝑖 𝑗 = 0.05 PIV
+4.0%
+Averaging uncertainty 𝑢2
+𝑖
+4.0% 𝑢𝑖
+Averaging uncertainty on autospectra
+1.96%
+Cf floating element
+5%
+0.1 [mm] precision to accommodate onto the surface of balance. Note that this precision
+is within the size of a pore for all surfaces and therefore its effect on the flow should be
+negligible. More information on the drag balance measurements can be found in Esteban
+et al. (2022).
+3.6. Measurement uncertainty
+The statistical quantities, such as mean and standard deviation, are estimated based on number
+of independent samples. The number of independent samples are 2500 (number of images)
+and 20000 (boundary-layer turnover time) for PIV and HWA respectively. Based on the
+number of independent samples, uncertainty in the averaging of statistical quantities can be
+estimated. Averaging uncertainty, for all the statistical quantities reported, are expressed with
+95% (20 : 1 odds) confidence (see Glegg & Devenport 2017, for instance).
+Finally, the statistical uncertainty in estimating two-point zero time delay correlation 𝑅𝑖 𝑗
+(see Benedict & Gould 1996, for instance), 𝜖𝑅𝑖 𝑗, can be estimated with 95% confidence by:
+
+Cameras
+Top view
+X3
+Laser
+X1
+Side view
+Laser sheet
+Laser
+X2
+th
+IX
+~3m8
+𝜖𝑅𝑖 𝑗 =
+2
+√
+𝑁
+× (1 − 𝑅2
+𝑖 𝑗)
+(3.1)
+where, 𝑁 is the number of independent samples. Finally, the measurement uncertainty for
+all the flow quantities have been summarised in table 1.
+4. Results
+The figure 3, shows instantaneous velocity field, with iso-contours of swirling strength (Λ𝑐𝑖)
+corresponding to 2−5% of the maximum value of Λ𝑐𝑖 as recommended by Zhou et al. (1999).
+As the measurements were performed only over the porous substrate, 𝑥2 = 0 corresponds to
+the surface of the foam. All the plots show similar contour levels indicating that the streamwise
+velocity fluctuations scaled with skin-friction velocity provides a reasonable “collapse” of the
+distribution. This suggests that streamwise velocity might conform to outer-layer similarity
+across all locations. However, the plots also show an overall attenuation in the extent of Λ𝑐𝑖
+for the 45 PPI foam suggesting some differences in the higher order statistics (and maybe in
+the vertical velocity component of some of these surfaces). This will be further explored in
+this section through more detailed statistical analysis.
+As mentioned earlier, the overall goal is to quantify the effect of increasing wall-porosity
+and relative foam thickness on the turbulent boundary-layer. Therefore, wall-porosity (for a
+given substrate thickness) was systematically increased at fixed inlet velocity. Although this
+ensures that the Reynolds number based on fetch (𝑅𝑒𝑥1) is the same for all the cases,
+the Reynolds number based on inner scales or the Kármán number is different. This
+is because for the cases tested, the permeability and roughness based Reynolds number
+increase simultaneously, and the inner velocity scales with the latter. Nevertheless, HWA
+measurements were performed at several flow speeds, which permits a broad coverage of
+parameter space and some iso-Kármán number data is also available. The 𝑘𝑠+ reported in
+the study were obtained using hot-wire measurements (see Esteban et al. 2022, for details).
+Finally, several non-dimensional parameters can be defined for given flow speed and porous
+substrate parameters, therefore, they are listed in table 2. As evidenced from table 2, cases
+over a wide range of Reynolds number have been investigated. Large values of 𝛿99/|𝑦𝑑| for
+the three porous cases reported confirms a large separation between inner and outer scales for
+permeable walls (Clifton et al. 2008). For references, |𝑦𝑑| corresponds to the absolute value
+of zero plane position (see Esteban et al. 2022). The lowest Reynolds number reported, in the
+present paper, is higher than most of the previous investigations (compared to Efstathiou &
+Luhar 2018, for instance), which permits a clear separation of scales and extending the study
+to both the transitionally and fully rough regimes. Finally, to cross validate PIV and HWA
+measurements, wall-normal profiles of wall-parallel mean velocity and its root mean squared
+values were compared. The mean velocity, obtained from PIV and HWA measurements,
+show a good agreement; therefore, to keep the manuscript succinct only a comparison of the
+variance of velocity fluctuations will be shown.
+4.1. Outer-layer scaling
+In order to validate Townsend’s (1980) outer-layer hypothesis, first and second order velocity
+statistics are plotted in outer-layer scaling, e.g. 𝛿99. The mean wall-parallel velocity in the
+defect form is shown in figure 4. The boundary-layer thickness 𝛿99 is used to scale the wall-
+normal distance while the inner-velocity 𝑈𝜏 is used to scale the wall-parallel velocity 𝑈1. The
+figure 4 shows good collapse beyond 𝑥2/𝛿99 = 0.3, as has been reported by earlier studies
+(Breugem et al. 2006; Manes et al. 2011; Efstathiou & Luhar 2018). In the present form
+
+9
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 3: Snapshot of instantaneous of streamwise velocity fluctuations field normalised
+by 𝑈𝜏. Green contours are iso-contour values for Λ𝑐𝑖. (a) 3 [mm] thick 10 PPI foam, (b)
+15 [mm] thick 10 PPI foam, (c) 3 [mm] thick 45 PPI foam, (d) 15 [mm] thick 45 PPI foam,
+(e) 3 [mm] thick 90 PPI foam, (f) 15 [mm] thick 90 PPI foam.
+(figure 4), the velocity deficit increases with increasing cells per inch in a porous substrate and
+the thickness of the substrate, and is consistent with the observations of Breugem et al. (2006).
+However, Efstathiou & Luhar (2018) had reported a slightly non-monotonic behaviour in
+velocity deficit. Efstathiou & Luhar (2018) had attributed this non-monotonic behaviour
+
+0.6
+660
+-
+C2
+0.40
+-10.84
+30.2
+0
+0.51.5
+C1/899-2
+-3
+4
+20.6
+660
+2
+0.40
+10.84
+3
+20.2
+0
+0.2
+0.40.6
+0.8
+C1/899-2
+-3
+4
+1.20.6
+66Q
+32
+0.400.84
+30.2
+0
+0.51
+1.5
+C1/099-2
+-3
+4
+20.6
+66Q
+C2
+0.40
+-10.84
+30.2
+0
+0.51
+1.5
+C1/ 099-2
+-30.6
+0.400.84
+3
+20.2
+0
+0.51
+1.5
+2
+C1/ 899-2
+-3
+4
+2.50.6
+660
+32
+0.40
+10.84
+30.2
+0
+0.51.5
+1
+C1/099-2
+-3
+.4
+210
+𝑠
+ℎ
+𝐾
+𝑈∞
+𝛿99
+𝑈𝜏
+𝑅𝑒𝜏
+𝑅𝑒𝐾
+𝑠+
+𝛿99/|𝑦𝑑|
+𝑘+𝑠
+ℎ/𝑠
+(mm)
+(mm) (10−9 m2) (m/s)
+(cm)
+(m/s)
+0.245 (90 PPI)
+3
+2.63
+10.4
+7.41
+0.494
+2349
+1.63
+7.76
+2470
+50.73 12.2
+0.245 (90 PPI)
+15
+2.63
+10.3
+9.09
+0.479
+2831
+1.60
+7.63
+313.45
+79.7
+61.2
+17.8
+10.67 0.926
+6756
+3.25
+15.51
+1333
+161.9
+0.89 (45 PPI)
+3
+36.2
+10.3
+8.44
+0.528
+2871
+6.47
+30.27
+401.9
+118.3
+3.3
+26.5
+9.45
+1.407
+8545
+17.20
+80.47
+525
+314.4
+0.89 (45 PPI)
+15
+36.2
+9.8
+9.66
+0.572
+3716
+7.32
+34.23
+112.3
+301.6 16.8
+17.8
+10.44 1.060
+7436
+13.55
+63.39
+99.4
+557.9
+3.84 (10 PPI)
+3
+245
+9.8
+9.21
+0.591
+3644
+19.58 151.93
+118
+367.7 0.78
+3.84 (10 PPI)
+15
+245
+10.3
+14.74 0.781
+7417
+24.90 193.22
+30.8
+2100
+3.9
+18.4
+14.71 1.410 13367 44.98 378.94
+29.5
+3791
+Table 2: Details of the porous-wall experimental data. The friction velocity (𝑈𝜏) is
+obtained from the direct measure of skin friction from the floating element drag balance.
+10-3
+10-2
+10-1
+100
+0
+5
+10
+15
+20
+Figure 4: Mean wall-parallel velocity deficit normalised by inner velocity. Legends: PIV
+data; 10 PPI: red circles, 45 PPI: purple squares, Red circles 10 PPI, purple squares 45
+PPI, 90 PPI: blue diamonds. Filled symbols correspond to the 15 [mm] thick substrate
+while hollow correspond to the 3 [mm] thick substrate.
+to the transition from deep to shallow flow over porous substrate. Furthermore, they had
+reported similar non-monotonic trend in higher velocity statistics.
+The variance of wall-parallel velocity disturbance (𝑢12 [m2/s2]) normalized by friction
+velocity (𝑈2
+𝜏), 𝑢+
+1, is shown in figure 5. A good collapse between PIV and HWA measurements
+are obtained except in the near-wall region. Near wall PIV measurements are compromised by
+Rapids articles must not exceed this page length
+
+11
+modulation error (Spencer & Hollis 2005) because the window of interrogation is larger than
+the near wall structures. The near wall data is also compromised due to laser light reflections,
+therefore the near-wall PIV data (∼ 2 [mm]) were omitted. Nevertheless, hot-wire data shows
+that the near-wall peaks are absent. In fact, for the 10 PPI 15 [mm] substrate, 𝑢+
+1 curve becomes
+essentially flat in the near-wall region. The inner peak in 𝑢+
+1 for smooth-wall is attributed
+to near-wall streaks (Hutchins & Marusic 2007) and splatting (Hunt & Morrison 2000).
+However, they both are subdued by wall permeability (Breugem et al. 2006) and surface
+roughness (Jiménez 2004) of the porous wall. For the 15 [mm] thick substrate, reduction
+in wall-parallel turbulence intensity scales with an increase in wall-permeability. This trend
+is consistent with the observations made by Manes et al. (2011). Furthermore, the peak in
+𝑢1+ for thin foam is considerably closer to the wall than the thick foam, which highlights
+the importance of permeability based Reynolds number 𝑅𝑒𝐾 and roughness based Reynolds
+number 𝑠+. For all the substrates tested over a broad range of Reynolds numbers (𝑅𝑒𝐾 and
+𝑠+), a good collapse of wall-parallel velocity fluctuations is obtained in the outer-layer region
+when plotted against outer-layer variable (𝛿99).
+The turbulent fluctuations for Reynolds stress and wall-normal velocity component are
+shown in figure 5 (b-c). The wall-normal velocity is especially susceptible to permeability
+(see Breugem et al. 2006, for instance). Slightly away from the wall 𝑥2/𝛿90 ∼ 0.1, the
+45 PPI foams shows largest differences in the wall-normal velocity disturbances possibly
+signaling increased permeability effects. The 10 PPI foam shows classic flat near wall-
+velocity fluctuations, as has been reported for fully-rough flows. The wall-normal component
+is associated with active motions, i.e. turbulent motion that contribute to Reynolds shear
+stress. The Reynolds stress, which is comprised of both active and inactive motions, also
+shows a significant spread in the outer layer for the 45 PPI foam. Therefore, impact of relative
+foam thickness (ℎ/𝑠) on velocity statistics is quiet substantial for this case. It is important to
+note that the spread in 𝑢1𝑢+
+2 profiles in the present study is similar to spread in wall-normal
+velocity variance reported by Manes et al. (2011). Therefore, the existence of outer-layer
+similarity for wall-normal velocity profiles is questionable even though the present study has
+been performed at very high Reynolds number. The 90 PPI foam has a very low permeability
+based Reynolds number 𝑅𝑒𝐾 ∼ 1 and a large separation between zero plane position 𝑦𝑑
+and boundary-layer thickness. Nevertheless, the absence of inner-peak should be noted,
+consequently it cannot be compared with smooth wall as reported by Esteban et al. (2022),
+who did an extensive study based on mean velocity profiles.
+Although Reynolds stress tensor component (−𝑢𝑖𝑢 𝑗) are statistical indicator of momentum
+transfer in the form of Reynolds shear stress, a more efficient dichotomy of outward-
+inward transport of momentum by turbulence can be obtained through quadrant analysis
+(Wallace 2016). Since the Reynolds stress tensor’s component −𝑢1𝑢2 is less than zero for
+well developed turbulent boundary-layer past a wall, only the ratio of negative quadrants Q2
+and Q4 is shown in figure 6. Notice that the outer-layer variables are now non-dimensionalised
+with 𝛿90 because PIV measurements for 10 PPI 15 [mm] substrate is unable to fully capture
+the boundary layer thickness 𝛿99. Nevertheless, it was verified that normalizing the plots
+with 𝛿90 or 𝛿99 had no impact on the outer-layer scaling. Therefore, subsequent plots will be
+normalised by 𝛿90. The quadrant Q2 is a summation of all the instants at which the wall-
+normal velocity is greater than its mean, while the wall-parallel velocity is lower than its
+time average. In contrast, quadrant Q4 is a summation of all the instants when 𝑢′
+1 is positive,
+while 𝑢′
+2 is negative. Q2, referred as ejections, marks the instances when a low speed fluid
+parcel is transported away from the wall. In contrast Q4, referred as sweep, is transport of
+high speed fluid parcel towards the wall. Figure 6 shows the relative contributions of 𝑄+
+2 and
+𝑄+
+4 quadrants as function of wall-normal distance. As a note of caution to the reader, higher
+
+12
+0.01
+0.1
+0.5
+1
+0
+1
+2
+3
+4
+(a)
+0.1
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+(b)
+0.1
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+(c)
+Figure 5: Outer-layer scaling of Reynolds shear stress profiles in the wall-normal direction.
+Velocity fluctuations are normalised by the inner velocity (𝑢𝜏2). 𝑥2/𝛿99 = 0 corresponds
+to the flow-substrate interface. Legends: PIV data; 10 PPI: red circles, 45 PPI: purple
+squares, Red circles 10 PPI, purple squares 45 PPI, 90 PPI: blue diamonds. Open symbols
+are for 3 [mm] thick substrate, while filled symbols correspond to 15 [mm] substrate.
+HWA data; 10 PPI:red lines, 45 PPI:purple lines, 90 PPI:blue lines. Solid lines correspond
+to 15 [mm] thick substrate, while the dotted lines correspond to 3 [mm] thick substrate.
+values of 𝑄+
+2 or 𝑄+
+4 does not imply higher overall levels of Reynolds stress −𝑢1𝑢2 as it is
+normalised by the later.
+While for the 3 [mm] substrates (6 (a and a)) a good collapse is achieved irrespective
+of foam permeability, for the 15 [mm] thick porous substrate weaker collapse among the
+various foams can be seen. These differences exist well into the outer layer for the case of 45
+PPI foam, which shows an increased 𝑄+
+2 and 𝑄+
+4 events. It is known that wall-permeability in
+absence of surface roughness, opens the path between near and outer wall regions (Breugem
+et al. 2006) and can invalidate the Townsend’s hypothesis. Similarly, Carpio et al. (2019)
+have reported an increase in Q2 and Q4 events with an increase in permeability. However,
+both these studies were performed at a low to moderate Reynolds numbers. While in our
+case, where both surface roughness and permeability are present, we see that increase in
+𝑄+
+2 and 𝑄+
+4 events are only seen by foam with intermediate permeability, where the flow
+remains transitionally rough, for which 𝑅𝑒𝜏 < 5000 (Esteban et al. 2022). This suggests
+that with an increase in roughness Reynolds number (𝑠+) and a accompanied transition to
+
+13
+0.2
+0.4
+0.6
+0.8
+1
+0
+20
+40
+60
+80
+100
+120
+(a)
+0.2
+0.4
+0.6
+0.8
+1
+0
+20
+40
+60
+80
+100
+120
+(b)
+0.2
+0.4
+0.6
+0.8
+1
+0
+20
+40
+60
+80
+100
+120
+(c)
+0.2
+0.4
+0.6
+0.8
+1
+0
+20
+40
+60
+80
+100
+120
+(d)
+Figure 6: (a) Relative contribution from 𝑄2/𝑄4 events for 3 [mm] thick substrate.
+(b) Relative contribution from 𝑄2/𝑄4 events for 15 [mm] thick substrate. Legends: 10
+PPI: red circles, 45 PPI: purple squares, Red circles 10 PPI, purple squares 45 PPI, 90
+PPI: blue diamonds. Open symbols are for 3 [mm] thick substrate, while filled symbols
+correspond to 15 [mm] substrate.
+a fully rough regime, the effect of permeability is subdued. The increase in 𝑄+
+4 events can
+be explained by relaxation in the wall-blocking condition by surface permeability. In order
+to fulfil the continuity condition, the 𝑄+
+2 events need to rise accordingly (Krogstad et al.
+1992). More importantly, it appears that 𝑄+
+2 and 𝑄+
+4 events do scale with wall permeability
+for transitionally rough flows, and that permeability opens a path of increased sweep events
+close to the wall. Therefore, the effects of porous walls extent to the outer-layer regions,
+this is sufficient to invalidate the Townsend’s outer-layer hypothesis. At shallow and deep
+substrate limits the permeability and pore size based Reynolds number are similar, the only
+noticeable difference are in the values of 𝑘+
+𝑠.
+To conclude, figures 4 and 5 show wall-parallel mean and variance collapse in the outer-
+layer when the velocity scales are normalised by 𝑈𝜏 and wall-normal distance by 𝛿99.
+However, as the substrate thickness and porosity is increased, the collapse for wall-normal
+component in the outer-layer becomes less evident. Furthermore, a good collapse in quadrants
+𝑄+
+2 and 𝑄+
+4 is observed in figure 6 for the thinner foam substrate. For thick substrates, collapse
+is achieved either when the substrate has permeability based Reynolds number comparable to
+viscous scales (90 PPI foam) or when the substrate is sparse and the flow transitions to fully
+rough regime (10 PPI foam). These results cast doubts on the validity of Townsend’s outer-
+layer hypothesis for turbulent flow past porous wall with varying thicknesses. Therefore, a
+detailed investigation on flow-structures will be performed in the following section.
+
+14
+4.2. The structure of turbulent boundary-layer over porous walls
+As mentioned in the introduction, for flow over and past porous foams no previous study
+at high Reynolds (𝑅𝑒𝜏 ∼ 2000) have reported multi-point correlation analysis, instead only
+single point statistics have been reported (Manes et al. 2011; Efstathiou & Luhar 2018).
+Therefore, in the current manuscript, two-point velocity correlation will be used to study the
+spatial structure of turbulence convecting over porous foams.
+In the present work, two-point correlation is denoted by:
+𝑅𝑖 𝑗(𝑥1, 𝑥1′, 𝑥2, 𝑥2′, 𝑥3, 𝑥3′) =
+𝑢𝑖(𝑥1, 𝑥2, 𝑥3)𝑢 𝑗(𝑥1′, 𝑥2′, 𝑥3′)
+𝑢𝑖(𝑥1, 𝑥2, 𝑥3) × 𝑢 𝑗(𝑥1′, 𝑥2′, 𝑥3′)
+(4.1)
+where 𝑢𝑖(𝑥1, 𝑥2, 𝑥3) is the𝑖-th component of the velocity fluctuation at the fixed or reference
+location while 𝑢 𝑗(𝑥1′, 𝑥2′, 𝑥3′) denotes the 𝑗-th component of the velocity fluctuations at the
+moving point. The terms 𝑢𝑖(𝑥1, 𝑥2, 𝑥3) and 𝑢 𝑗(𝑥1′, 𝑥2′, 𝑥3′) are mean turbulent fluctuations
+[m/s], at the fixed and moving point respectively. Equation (4.1), assumes that the flow is
+inhomogeneous in all three spatial directions. In the current study, we only treat the wall-
+normal location as the inhomogeneous direction.
+As explained in the previous sections, near wall PIV data (∼ 2 [mm]) could not be used due
+to modulation error and reflections close to the wall. Furthermore, it must be remembered
+that PIV measurements truncate both large and small scales. On the one hand, the size of the
+camera sensor sets the upper limit on the largest scale that can be imaged. While on the other
+hand, the smallest scale that can be captured is directly proportional to the final window size
+(Foucaut et al. 2004). Nevertheless, PIV measurements inherently show the spatial structure
+of turbulence without invoking Taylor’s hypothesis. The two-point correlation maps, obtained
+using PIV measurements, are plotted in figures 7, 8, and 9.
+Figure 7 shows the two-point zero time delay correlation for the wall-parallel velocity
+correlation in the wall-normal plane (𝑥1 − 𝑥2). Plots on left correspond to the 3 [mm] thick
+porous substrate and on the right correspond to 15 [mm] thick substrate. The correlation
+maps for near-wall fixed points (7 (b-d)) shows a poor collapse with the outer-layer variable
+𝛿90 in the outer-layer (Note that we are using 𝛿90 as the scaling variable instead of 𝛿99 as the
+full-extent of the boundary layer is not captured for one of the PIV cases). It is important
+to note that the entire extent of the wall-parallel velocity correlation could not be captured;
+therefore, only values of correlation above 0.5 are shown. As can be seen from figure 7,
+for any given point downstream of the fixed point, 𝑅11 appears to be inclined away from
+the wall. The characteristic inclination of 𝑅11 is linked to the statistical mean inclination
+of the hairpin structures with respect to the wall (Ganapathisubramani et al. 2005). In
+particular, a slight increase in angle could result in better access to higher momentum for
+hairpin structures. Sillero et al. (2014) reports the characteristic inclination of these hairpin
+structures are ∼ 10◦. Surface roughness is known to increase the inclination angle of 𝑅11.
+While Volino et al. (2007); Wu & Christensen (2010) have reported a slight increase (∼ 15◦)
+in the inclination angle of 𝑅11 compared to smooth walls, Krogstad & Antonia (1994) report
+almost a four-fold increase. In the present manuscript, average inclination were calculated
+following Volino et al.’s (2007) procedure of fitting a line in a least square sense that passes
+through the iso-contours of 𝑅11. The resulting angle close to the wall were found to be a
+function of the pore size (see table 3). In the present case, where both 𝑘+
+𝑠 and 𝑅𝑒𝐾 increase
+simultaneously, the 10 PPI has the highest inclination (∼ 20◦) compared to other surfaces.
+It is important to mention that other studies (Volino et al. 2007; Wu & Christensen 2010)
+had tested surfaces with varying roughness but the inclination was found to be independent
+of 𝑘+
+𝑠. The inclination appears to be independent of the thickness of foam, but scales with
+
+15
+-0.2
+0
+0.2
+0.4
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+(a)
+-0.2
+0
+0.2
+0.4
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+(b)
+-0.5
+0
+0.5
+0.2
+0.4
+0.6
+0.8
+1
+(c)
+-0.5
+0
+0.5
+0.2
+0.4
+0.6
+0.8
+1
+(d)
+Figure 7: Wall-parallel velocity two-point zero time delay correlation,
+R11(𝑥1, 𝑥′
+1, 𝑥2, 𝑥′
+2, 𝑥3, 𝑥3). Plots on the left are for 3 [mm] thick substrate while plots on
+the right are for 15 [mm] thick substrate. (a) Fixed point at (0.07 × 𝛿90); (b)Fixed point at
+(0.07 × 𝛿90); (c) Fixed point at (0.6 × 𝛿90); (d) Fixed point at (0.6 × 𝛿90). Legends: Red
+dotted lines for 10 PPI foam substrate, purple dashed lines 45 PPI foam substrate, and blue
+solid lines 90 PPI foam substrate. The iso-contour lines are from 0.5 to 0.9 with an
+increment of 0.1.
+pore density. Therefore, the increased inclination could due to deeper penetration (filling up)
+of flow past porous foams compared to skimming off for flow past foams with low porosity
+(e.g. 90 PPI). This suggests that with decreasing pore density, a transition to sparse canopy
+like behavior is obtained.
+In the case of 𝑅22, one can quantify overall correlation length as the field-of-view is
+large enough compared to overall extent of 𝑅22. The vertical velocity correlations, shown
+in figure 8, appears to be more sensitive to wall-permeability and thickness. Firstly, 𝑅22
+appears to be symmetric in the wall-parallel direction; however, a compression is observed
+in the wall-normal direction. Therefore, while permeable boundary condition with finite
+permeability does relax the wall blocking, it is not do enough to achieve symmetry in the
+wall-normal planes. The vertical velocity correlations for the 90 PPI foam remains invariant
+as the thickness of the substrate is increased. This clearly shows when 𝑅𝑒𝐾 ∼ 1, then foams
+behave like a smooth wall, and permeable effects are negligible. Interestingly, for the thicker
+foam substrate, as the porosity is increased, the overall extent of 𝑅22 first decreases (45 PPI)
+and then increases (10 PPI), as evidenced from figure 8 (b). In contrast, Carpio et al. (2019)
+
+16
+Foam
+ℎ
+Angle
+(PPI) (mm) (Degree)
+90
+3
+13
+15
+13.4
+45
+3
+15.8
+15
+16.2
+10
+3
+20.6
+15
+19.4
+Table 3: Angle of 𝑅11 at 0.07 × 𝛿90.
+had reported decrease in correlation in the extent of 𝑅22 with increasing permeability. It is
+noteworthy that while the flow over 45 PPI case 15 [mm] case is transitionally rough, the
+flow over 10 PPI case 15 [mm] case is fully rough (Esteban et al. 2022). Therefore, it appears
+that the permeable effects, which lead to the reduction in the extent of 𝑅22 is no longer
+dominant in sparse foams, where the flow transitions to a fully rough regime. Furthermore,
+when the overall extent of 𝑅22 is normalized by the boundary-layer thickness, a good collapse
+is obtained (8 (a-c)) for the thinner substrate. This indicates that reduction in the extent of
+wall-normal velocity correlation (𝑅22) with permeability is ineffective for the 3 [mm] thick
+substrate, yielding a better collapse for all the cases well into the outer layer. In contrast, for
+the thicker foam, the influence of permeability is present well into the outer-layer (8 (d)),
+provided permeability is greater than the viscous scales (𝑅𝑒𝐾 > 1) and the foam operates at
+a dense (𝑠+ < 100) and deep (ℎ/𝑠 > 10) limits. At sparse foam limit deeper flow penetration
+can be seen from figure (8 (b)), this is inline with the “filling up effect” remark made earlier
+in conjunction with mean inclination of the hairpin structures with respect to the wall.
+The two-point correlation of Reynolds stress component (the wall-parallel and wall-normal
+velocity fluctuations), 𝑅12, in 𝑥1 − 𝑥2 plane, is plotted in figure 9. The correlation 𝑅12 is
+representative of the extent to which the wall-parallel velocity are associated with a single
+ejection or sweep event, induced by the wall-normal velocity fluctuations. Recently, Gul
+& Ganapathisubramani (2021) showed for rough walls, 𝑅12 scales with the boundary-layer
+thickness and is independent of the 𝑘+
+𝑠 or the Kármán number. For the 15 [mm] substrate, as
+the porosity is increased, the 𝑅12 first decreases (45 PPI) and then increases (10 PPI). This
+behaviour is similar to 𝑅22 correlation map, as shown earlier. In particular, the extent of the
+correlation map 𝑅12 seems to be shortest for 45 PPI 15 [mm] thick substrate. In fact, the
+maximum iso-contour levels plotted, e.g. −0.4, is visibly absent for the 45 PPI 15 [mm] thick
+substrate case. Therefore, in response to the first objective of the present paper, the inability
+of boundary-layer thickness to collapse the overall extent of the 𝑅22 and 𝑅12 (well into the
+outer-layer) suggests that Townsend’s outer-layer similarity for these higher order quantities
+may not be valid for 15 [mm] thick porous substrates.
+Finally, Manes et al. (2011); Efstathiou & Luhar (2018) have linked improved mixing for
+the thickest and most permeable foams to the presence of KH instability. Furthermore, Manes
+et al. (2011) and Sharma & García-Mayoral (2020b) state such KH type instability occur at
+the interface of porous substrate, and at a distance 𝑥2/𝛿99 < 0.1. Indeed, KH instability are
+known to induce large quasi two-dimensional rollers, which leads to periodic organisation of
+wall-normal flow disturbances (see figure 18 of Jaiswal et al. 2022, for instance). Although
+
+17
+-0.1
+-0.05
+0
+0.05
+0.1
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+(a)
+-0.1
+-0.05
+0
+0.05
+0.1
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+(b)
+-0.4
+-0.2
+0
+0.2
+0.4
+0.2
+0.4
+0.6
+0.8
+1
+(c)
+-0.4
+-0.2
+0
+0.2
+0.4
+0.2
+0.4
+0.6
+0.8
+1
+(d)
+Figure 8: Wall-normal velocity two-point zero time delay correlation,
+R22(𝑥1, 𝑥′
+1, 𝑥2, 𝑥′
+2, 𝑥3, 𝑥3). Plots on the left are for 3 [mm] thick substrate while plots on
+the right are for 15 [mm] thick substrate. (a) Fixed point at (0.07 × 𝛿90); (b) Fixed point
+at (0.07 × 𝛿90); (c) Fixed point at (0.6 × 𝛿90); (d) Fixed point at (0.6 × 𝛿90). Legends:
+10 PPI:red dotted line, 45 PPI:purple dashed, 90 PPI:blue solid. The iso-contour lines are
+from 0.2 to 0.9 with an increment of 0.1.
+the figures 8 show limited streamwise extent, no periodic structures or modes associated
+with KH instability were observed for 𝑅22 or 𝑅12 (figures 8 (d)) within the measurement
+domain. This suggests that no KH type flow instability may be present in the cases that were
+investigated.
+As mentioned earlier, only the length scales associated with wall-normal can be quantified
+due to limited bandwidth (FOV) of our PIV measurements. In the present work, the turbulence
+correlation length is defined as:
+Λ|𝑘±
+𝑖 𝑗 (𝑥𝑖) =
+∫ ∞
+−∞
+𝑅𝑖 𝑗(𝑥𝑖, x𝑘±) d𝑥𝑘±
+(4.2)
+Here, x𝑘± is the separation vector in the direction 𝑘. The subscript ± denotes moving
+point direction. For instance, if the fixed point is located above the moving points for the
+wall-normal velocity calculations, then the integration of (4.2) will yield Λ|2−
+22 (𝑥2). This
+way of defining length scale is particularly appropriate for in-homogeneous turbulence (see
+Jaiswal et al. 2020, for instance). However, as noted by Sillero et al.’s (2014), a clear
+
+18
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0
+0.1
+0.2
+0.3
+0.4
+(a)
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0
+0.1
+0.2
+0.3
+0.4
+(b)
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+(c)
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+(d)
+Figure 9: The two-point zero time delay correlation of Reynolds stress component
+R12(𝑥1, 𝑥′
+1, 𝑥2, 𝑥′
+2, 𝑥3, 𝑥3). Plots on the left are for 3 [mm] thick substrate while plots on
+the right are for 15 [mm] thick substrate. (a) Fixed point at (0.07 × 𝛿90); (b) Fixed point
+at (0.07 × 𝛿90); (c) Fixed point at (0.6 × 𝛿90); (d) Fixed point at (0.6 × 𝛿90). Legends:
+10 PPI:red dotted line, 45 PPI:purple dashed, 90 PPI:blue solid. The three iso-contours
+levels correspond to the values of −0.4, −0.35 −0.30.
+physical interpretation of the length scale is difficult. The present study seeks to compare
+the effects of porous surfaces on the large scale turbulence structures, therefore a contextual
+interpretation of the length-scale can be used where it is a metric to quantify overall spatial
+correlation of velocity disturbances. 𝑅𝑖 𝑗 being a statistical quantity, equation (4.2) cannot
+be used directly without accumulating averaging 𝜖𝑅𝑖 𝑗 errors (see equation (3.1)). In order
+to avoid the accumulation of errors, length scales were estimated by fitting an exponential
+decay function (see Jaiswal et al. 2020, for implementation details).
+Figure 10 shows the longitudinal correlation length scales for wall-normal velocity
+component. Figure 10 (a-b), shows the wall-normal correlation lengths, Λ|2−
+22 (𝑥2) and
+Λ|2+
+22 (𝑥2), in the 𝑥1-𝑥2 plane. The length scale Λ|2−
+22 (𝑥2) should be particularly sensitive to the
+blocking effects induced by the wall. This is not surprising because the blocking effects are
+pre-dominant when approaching the wall (see Jaiswal et al. 2020, for instance). Nevertheless,
+difference in length scale for thicker 15 [mm] porous substrate is more visible. For instance,
+correlation length scales (Λ|2−
+22 (𝑥2)) for the 45 PPI substrate is smaller throughout the
+boundary layer. Surprisingly, the length scale Λ|2+
+22 (𝑥2) shows even more substantial reduction
+
+19
+0.1
+0.5
+1
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+(a)
+0.1
+0.5
+1
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+(b)
+Figure 10: Longitudinal length scales of wall-normal velocity component. (a) Integral
+length scale Λ|2+
+22 (𝑥2) (b) Integral length scale Λ|2−
+22 (𝑥2) Legends: Red circles 10 PPI,
+purple squares 45 PPI, and blue diamonds 90 PPI. Open symbols are for 3 [mm] thick
+substrate, while filled symbols correspond to 15 [mm] substrate.
+for the 45 PPI and 15 [mm] thick foam. Therefore, in response to the second objective of the
+present paper, no KH type flow instabilities were observed while substantial differences in
+length scales (Λ|2+
+22 (𝑥2)) are observed in the boundary-layer above the porous foam at a thick
+substrate limit (ℎ/𝑠 > 10).
+As mentioned previously, the length scale associated with wall-parallel velocity disturbance
+could not be quantified due to limited field of view. However, thanks to single-wire
+measurements, a high-fidelity estimation of time-scales associated with wall-parallel velocity
+fluctuations is possible. Similar to length scale, the time scale can be defined as:
+𝑇(xi) =
+∫ ∞
+−∞
+𝑅11
+�xi(𝑡), xi(𝑡 + d𝑡)�d𝑡
+(4.3)
+In order to reduce error in the estimation of𝑇(xi), the temporal correlations were fitted with
+an exponential decay function exp(−𝑎𝑥𝑘) to reduce the accumulation of errors in estimating
+𝑇(xi).
+The time scales thus calculated are plotted in figure 11. The time scales have been
+normalised with outer-layer variables 𝑈∞ and 𝛿90, as such the plots show time scale per
+unit boundary-layer turnover time. The figures 11 (a-b) show that the time-scales appear to
+collapse for the 3 [mm] substrate irrespective of the pore density (PPI). For the 15 [mm]
+substrate, the most porous foam (10 PPI) compares poorly in the near-wall region. More
+specifically, the 10 PPI 15 [mm] substrate has the shortest eddy turnover time. Since, the
+measurements were performed at different 𝑅𝑒𝜏, therefore, two additional cases at similar
+𝑅𝑒𝜏 and thickness are plotted in figures 11 (b). As can be seen from figure 11 (b), the 10 PPI
+15 [mm] substrate remains an outlier, as it has the shortest eddy turnover time. Nevertheless,
+at these Reynolds number 𝑅𝑒𝜏 ∼ 7000 the flow becomes fully rough (Esteban et al. 2022).
+Therefore, it is hypothesised that large energetic structures are pushed away from the wall, as
+fully rough regime develops. This is consistent with previous findings made by Squire et al.
+(2016) for flow past impermeable rough walls. This also explains a slight reduction in eddy
+turnover time for 45 and 90 PPI foams at 𝑅𝑒𝜏 ∼ 7000.
+The hot-wire data for all the cases can be further explored to examine the spectral content
+
+20
+0.2
+0.4
+0.6
+0.8
+1
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+(a)
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+(b)
+Figure 11: Integral scales of turbulence. (a) Integral time scale T for 3 [mm] thick
+substrates (b) Integral time scale T for 15 [mm] thick substrates. Legends: Red circles 10
+PPI, purple squares 45 PPI, and blue diamonds 90 PPI. Blue dashed and purple dotted
+lines correspond to 90 PPI and 45 PPI foams respectively, at similar 𝑅𝑒𝜏 ∼ 7000 and 15
+[mm] thick foam substrate.
+of the turbulent structures and the similarity (or lack thereof) between the different substrates
+can further elucidated.
+4.3. Spectral analysis of deep and shallow flows over porous foam.
+In order to obtain frequency related information of the turbulent kinetic energy associated
+with wall-parallel velocity disturbances, pre-multiplied wall parallel turbulent energy spectra
+(𝐸11) were computed and are depicted in this section at various wall-normal positions across
+the different porous substrates.
+Figure 12 shows contours pre-multiplied energy spectrogram (with normalised wall-
+normal position - 𝑥2/𝛿90- in the 𝑥-axis and normalised frequency - 𝐹+ = 𝐹𝜈/𝑈2
+𝜏 - in
+the 𝑥2-axis). Each sub-figure also shows an inset where the 𝑥2-axis is scaled in outer units
+(𝐹𝛿90/𝑈∞). Various cases have been ordered based on the thickness to pore ratio (ℎ/𝑠) at
+same fetch-based Reynolds number (𝑅𝑒𝑥1). The figures on the left column are for 3[mm]
+thick substrate while figures on the right correspond to 15[mm] thick substrate. The rows are
+arranged so that top row corresponds to porous substrates with highest permeability, while
+the lowest permeability substrates are at the bottom row. For the 3[mm] 90 PPI case, the
+peak is observed slightly away from the wall (figure 12 (e)). For the same substrate thickness,
+as permeability increases at first the peak moves closer to the wall (45 PPI case). However,
+with a further increase in permeability (10 PPI case) the peak in 𝐸11 (see figure 12 (a)) is
+smeared, and an energetic region slightly away from the wall is observed.
+For the case of 45 PPI substrate (figures 12c and 12d), the spectral peak is always located
+very close to the interface. A peak is possible even closer to the wall for 45 and 90 PPI,
+however it is unclear if this might extend in to the porous substrate. As the permeability and
+sparsity increases (from 90 PPI to 45 PPI), the substrate tends ot break-up near-wall structures
+and the energy is distributed across different scales. This results in the wall-normal shift of
+𝐸11 peak. Figures 12 (c-f) and (a-d) show energy spectra for cases at similar 𝑅𝑒𝜏. Figures 12
+(c-f) are at transitionally rough regime. Therefore, for transitionally rough regime (figures
+12 (c-f)) the spectral peak seems to be a function of ℎ/𝑠, as argued by (Efstathiou & Luhar
+2018). The preceding argument is valid only for 45 and 90 PPI at lowest flow speeds at which
+both are transitionally rough, and for which foam remains dense or 𝑠+ is small.
+
+21
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 12: Premultiplied 1-D Wall-parallel velocity energy spectra 𝐸11
+�
+F×E11
+𝑈2𝜏
+�
+as a
+function of frequency F+
+�
+F×𝜈
+𝑈2𝜏
+�
+, over the surface of foam, measured at a distance of 𝑥1
+= 3.3 [m] downstream of inlet. (a) 3 [mm] thick 10 PPI foam, (b) 15 [mm] thick 10 PPI
+foam, (c) 3 [mm] thick 45 PPI foam, (d) 15 [mm] thick 45 PPI foam, (e) 3 [mm] thick 90
+PPI foam, (f) 15 [mm] thick 90 PPI foam.
+
+2
+10-1
+1.5
+10-2
+1
+0.5
+10~3
+0
+10-2
+100
+C2/89010-1
+2
+1100
+1.5
+10-2
+10
+1
+10-3
+0.5
+10-2
+100
+C2/8902
+100
+10-1
+1.5
+10-2
+1
+0.5
+10-3
+n
+10-2
+100
+C2/8902
+10-1
+F
+1.5
+10
+10-2
+1
+0.5
+10-3
+10-2
+100
+C2/8902
+03100
+10-1
+1.5
+左 10-2
+1
+0.5
+10-3
+10-2
+100
+C2/0902
+100
+10-1
+1.5
+10
+10-2
+1
+0.5
+10-3
+0
+10-2
+100
+C2/89022
+(a)
+(b)
+(c)
+Figure 13: Premultiplied 1-D Wall-parallel velocity energy spectra 𝐸11
+�
+F×E11
+𝑈2𝜏
+�
+as a
+function of frequency F+
+�
+F×𝜈
+𝑈2𝜏
+�
+at similar 𝑅𝑒𝜏 ∼ 7000 and foam thickness 15 [mm].
+(a) 90 PPI foam. (b) 45 PPI foam. (c) 10 PPI foam.
+Finally, as shown in figures 12 (a) and (b) the peak in specral energy is closer 𝑥2/𝛿90 ∼ 0.1
+from the interface for the 10 PPI case. This “outer-peak” is typically associated with large-
+and very-large-scale motions (for flows over impermeable walls - both smooth and rough).
+This suggests that for for 10 PPI case, this outer region seems to be well developed for both
+thicknesses. This is concurrent with reduction of near-wall energy due to the “roughness”
+redistributing the energy across different scales and leading to eddies with shorter time scales
+close to the wall, as reported in figure 11 (b). Similar findings were made in previous studies
+over impermeable rough wall (see Schultz & Flack 2007; Squire et al. 2016, for instance),
+and presence of plateau in the turbulence intensity plots shown in figure 5. Therefore, with
+increasing values of 𝑠+ (hence 𝑘+
+𝑠), a reduction in near-wall energy is observed (compare plots
+12 b and e), and that in fully-rough regimes the peak in 𝐸11 is not a function of permeability
+based Reynolds number 𝑅𝑒𝐾. It may be argued that absence of near-wall peak in 𝐸11 can be
+a precursor or a result of fully-rough flow regime, which depends only on pressure drag. For
+flows above such sparse (𝑠+ ∼ 100) porous foam, roughness effects become dominant.
+To reinforce aforementioned claim that in the fully-rough regime the peak in spectra is no
+
+10-1
+2
+1100
+1.5
+10-2
+10
+1
+10-3
+0.5
+10-2
+100
+C2/8902
+10-1
+1.5
+10
+10-2
+左
+1
+10~3
+0.5
+10-2
+100
+C2/0902
+100
+10-1
+F
+1.5
+10-2
+1
+10~3
+0.5
+0
+10-2
+100
+C2/89023
+0.01
+0.1
+0.2
+0.5
+1
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+(a)
+0.01
+0.1
+0.2
+0.5
+1
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+(b)
+Figure 14: Shannon Entropy evaluated from streamwise velocity spectra. (a) 3 [mm] thick
+substrate Legends: Red circles 10 PPI 𝑅𝑒𝜏 = 3644,𝑅𝑒𝐾 = 19.5; Purple squares 45 PPI
+𝑅𝑒𝜏 = 2871,𝑅𝑒𝐾 = 6.47; Blue diamonds 90 PPI 𝑅𝑒𝜏 = 2349,𝑅𝑒𝐾 = 1.63; and Gray
+pentagon 45 PPI 𝑅𝑒𝜏 = 8545,𝑅𝑒𝐾 = 17.2.(b) 15 [mm] thick substrate Legends: Red
+circles 10 PPI 𝑅𝑒𝜏 = 7417,𝑅𝑒𝐾 = 24.9; Purple squares 45 PPI 𝑅𝑒𝜏 = 3716,𝑅𝑒𝐾 = 7.32;
+Blue diamonds 90 PPI 𝑅𝑒𝜏 = 2831,𝑅𝑒𝐾 = 1.6; Red dotted line 10 PPI
+𝑅𝑒𝜏 = 13367,𝑅𝑒𝐾 = 45; Purple dashed line 45 PPI 𝑅𝑒𝜏 = 7436,𝑅𝑒𝐾 = 13.55 and Blue
+solid line 90 PPI 𝑅𝑒𝜏 = 6756,𝑅𝑒𝐾 = 3.25.
+longer a function of ℎ/𝑠, figure 13 shows streamwise velocity spectra for the cases for where
+the flow is fully-rough (at 𝑅𝑒𝜏 ∼ 7000). As shown in figure 13, the peak in kinetic energy
+spectra associated to streamwise velocity appear to be centred 𝑥2/𝛿90 ∼ 0.1 confirming that
+for fully-rough regime position of peak in kinetic energy spectra no longer scales with ℎ/𝑠,
+as flow becomes increasingly sparse at higher Reynolds numbers (𝑠+). As a result, for fully
+rough flows over porous foams, the dependency of kinetic energy spectra peak on pore size
+saturates, and the near-wall region is dominated by the roughness sub layer while the effect
+of permeability is restricted.
+A quantitative comparison of the energy distribution across different cases can be obtained
+by computing the Shannon entropy of the spectral content of the streamwise velocity. Wesson
+et al. (2003) define Shannon entropy of the spectral content as:
+SH =
+N
+∑︁
+i
+−Si log Si
+log N
+(4.4)
+The Shannon entropy has also been used in the past by Manes et al. (2011); therefore
+a direct comparison with other studies if possible especially comparing surfaces with or
+without surface roughness over a permeable surface.
+As mentioned by Manes et al. (2011), Shannon entropy is a measure of scale heterogeneity
+and spectral shrinkage. In presence of coherent structures, the energy is concentrated
+around fewer scales that results in shrinkage of spectra, around the frequency (and hence
+wavenumber) of the corresponding coherent structure. As shown in figure 14, normalised
+Shannon entropy increases with a decrease in permeability. Furthermore, Shannon entropy is
+not a function of ℎ/𝑠, which is inline with the observations made from figures 12 and 13. For
+thinner 3 [mm] substrate, a good agreement is found beyond 0.1 × 𝛿90. Manes et al. (2011)
+had argued that the Shannon’s entropy should scale with permeability (𝑅𝑒𝐾). Manes et al.
+(2011) had associated this with the mixing-layer analogy. If 𝑅𝑒𝐾 determines the permeability
+
+24
+and the shear penetration depth is captured by the ratio 𝛿99/𝑦𝑑, then the 10 PPI, 15 [mm]
+thick substrate at 𝑅𝑒𝐾 ∼ 45 should have shown the lowest values of Shannon entropy (Figure
+14 dotted red line). Instead the same 10 PPI substrate at 𝑅𝑒𝐾 = 25) has lower values of
+Shannon entropy compared to 10 PPI substrate at 𝑅𝑒𝐾 = 45. Moreover, as shown in figure
+14, the Shannon entropy seems to be invariant to a single classical porous material parameters
+reported in the study. This is further reinforced by the fact that the 15 [mm] 45 PPI case at
+𝑅𝑒𝐾 of 19.5 and 𝑠+ of 91.2 shows much lower SH compared to 3 [mm] thick 10 PPI porous
+substrate at same 𝑅𝑒𝐾 but much higher 𝑠+.
+For similar 𝑅𝑒𝐾 ∼ 19, the Shannon entropy for the case of 45 PPI foam is vastly greater
+than the 10 PPI, for the 3 [mm] thick substrate. Therefore, wall-permeability alone does
+determine existence of large coherent structures in the case of porous foams with a finite
+thickness. For instance, at the deep foam limit the present 𝑅𝑒𝐾 determines scale heterogeneity
+only when 𝑅𝑒𝜏 is similar. The independence of SH from wall-permeability (𝑅𝑒𝐾) can be
+due to increase in sparsity (𝑠+) with increasing Reynolds number (𝑅𝑒𝜏), which may limit
+permeability effects of porous substrate.
+5. Discussion
+In the present study, relative foam thickness, pore density and size were varied to assess
+their impact on turbulent boundary layer above a foam. For thick foam substrates, a deep
+foam limit is achieved for foams with higher pore density (45 PPI and 90 PPI). Such
+deep foams remain at dense foam limit at low Reynolds number based on average pore
+size (𝑠+ < 50), and differences in outer-layer similarity are observed, provided that the
+permeability based Reynolds number is high enough (𝑅𝑒𝐾 > 1). In particular, velocity
+disturbances are substantially attenuated, and the extent of wall-normal velocity correlation,
+𝑅22, diminishes significantly. Therefore, 15 [mm] thick 45 PPI substrate has the lowest values
+of Λ|2+
+22 (𝑥2). The 15 [mm] thick 45 PPI foam also has the smallest extent of streamwise
+velocity streaks at a given ejection or sweep event (𝑅12) compared to the rest of the cases.
+More importantly, these differences persists well into the outer-layer. The 45 PPI foam has
+similar values of 𝑅𝑒𝐾 and 𝑠+ in deep and shallow substrate limits, and the only noticeable
+difference are measured in the values of 𝑘+
+𝑠. In other words, at thin substrate limit, the
+effect of solid wall below the foam substrate is non-negligible, as it attenuates the the zero
+displacement plane and hence the equivalent sand grain roughness. Therefore, for porous
+foams “thickness induced surface-roughness, 𝑘𝑠𝑟” (see Esteban et al. 2022, for details) can
+influence the outer-layer statistics. This is achieved by means of higher 𝑄+
+2 and 𝑄+
+4 events
+measured throughout the boundary layer, as such inner-layer is able to communicate with
+outer layer. Similar observations were made by Krogstad et al. (1992) for flows over and past
+impermeable rough wall. While it is tempting to draw an analogy between dense-deep foams
+and that of flow past dense canopy (Sharma & García-Mayoral 2020b), in present study no
+evidence of KH type flow instability is found for similar levels of sparsity and deep thickness
+limit. Nevertheless, it is hypothesized that the foam density limits required for the inception
+of KH instability could be lower in the case of porous substrate compared to the flow past
+dense canopy. This is backed by the findings of Manes et al. (2011); Efstathiou & Luhar
+(2018), who found the peak in wall-parallel velocity spectra associated with KH instability
+at lower Reynolds number (𝑠+) compared to the present study.
+As the wall-porosity is further increased for the same substrate thickness, the pore size
+becomes same order of magnitude as the substrate thickness (ℎ/𝑠 < 10) enabling access
+to shallow and sparse foam limits. At the sparse limit, it is hypothesized that as viscous
+scales shrink with increasing Reynolds number the reduction in velocity disturbances by
+porosity saturates. This is because the pore size becomes substantially larger than the near-
+
+25
+wall viscous scales and as such attenuation in wall-normal velocity correlation is no longer
+possible. This is also evidenced from the spectral heterogeneity, which no longer scales
+with wall-permeability (𝑅𝑒𝐾) at high sparsity limit (𝑠+ ∼ 100). Therefore, flow at high 𝑠+
+becomes analogous to flow past sparse canopies (Bailey & Stoll 2013; Sharma & García-
+Mayoral 2020a). The increase in velocity disturbances is also similar to ones observed in high
+sparsity limits for canopies (Sharma & García-Mayoral 2020a). This limits the attenuation of
+wall-normal velocity disturbances that drive wall-pressure fluctuations (Carpio et al. 2019).
+Similar to an impermeable rough wall, roughness sub-layer pushes the energetic flow away
+from the surface in the case of porous walls, as evidenced from streamwise kinetic energy
+spectra. Therefore, the wall-normal location of the velocity energy spectra peak depends
+on foam density (𝑠+) and flow regime, and spectral peak becomes independent of foam
+thickness (ℎ/𝑠) at sparse substrate limit. At the dense limit, the wall-normal location of the
+wall-parallel energy spectra scales with foam thickness (ℎ/𝑠), as observed by Efstathiou
+& Luhar (2018). An important distinction between flow past sparse porous substrates and
+roughness is that when the roughness is sparse, the wall becomes akin to smooth wall, while
+when the porosity is sparse, the flow becomes fully-rough. This is because permeability
+increases equivalent sand grain roughness (Esteban et al. 2022). Therefore, either when the
+substrate is very sparse (𝑠+ ⩾ 60) or when the substrate thickness becomes comparable to
+pore-size (ℎ ∼ 𝑠) the substrate acts as a rough wall. In contrast, when the permeability-based
+or the pore-based Reynolds numbers are comparable to that of viscous scales, then changes in
+outer-layer velocity statistics are negligible. Therefore, the outer-layer similarity is achieved
+for two extreme cases when either the substrate is akin to rough walls or similar to a smooth
+wall (𝑅𝑒𝐾 ∼ 1).
+6. Conclusions
+The present study quantifies the effect of wall porosity and substrate thickness on flows
+past porous foams. For a broad range of Reynolds numbers, the turbulent statistics, the
+spatio-temporal scales and energy spectra were quantified above the porous substrate within
+the boundary layer. In particular, the present manuscript extends current state-of-the-art
+(Manes et al. 2011; Efstathiou & Luhar 2018) to include the effects of foam density (𝑠+) and
+relative foam thickness (ℎ/𝑠) on turbulent boundary layer over porous walls over a range of
+𝑅𝑒𝐾. We cover both transitionally- and fully-rough regimes and quantify the turbulent flow
+structures through the use of two-point correlations. The foam thickness-to-pore size range
+from ℎ/𝑠 ≈ 0.7 − 60, while various Reynolds numbers range from 𝑅𝑒𝜏 ≈ 2000 − 13500,
+𝑅𝑒𝐾 ≈ 1 − 50 and 𝑠+ ≈ 75 − 400.
+Two research questions has driven the present study: 1) Is the flow over such porous
+surfaces analogous to flows over rough surfaces away from the wall? If so, does the outer-
+layer similarity in velocity statistics holds for such porous foams? 2) For what values of
+pore thickness and size can we expect to reduce the correlation of wall-normal velocity
+fluctuations?
+As it turns out these questions are interlinked for the case of flow past a permeable foam. In
+particular, the present study shows a substantial reduction in the correlations of the velocity
+fluctuations (𝑅12 and 𝑅22), at deep-dense substrate limits with high permeability based on
+Reynolds numbers (𝑅𝑒𝐾 > 1), which appears to break the Townsend’s outer-layer similarity
+and provide an avenue for using porous walls in aerofoil self-noise reduction applications.
+In particular, this is achieved by an increased relative vertical momentum exchange by an
+increase in ejection 𝑄+
+2 and sweep 𝑄+
+4 events across the boundary layer. Therefore, wall-
+permeability boundary condition is felt across the boundary-layer, resulting in substantial
+reduction in velocity disturbance field above the porous wall. As such, the present study
+
+26
+shows that the success of outer-layer similarity depends on the flow regime (transitionally
+rough or fully rough), pore density (𝑠+), permeability (𝑅𝑒𝐾) and relative foam thickness
+(ℎ/𝑠). Therefore, in the outer-layer, the flow over porous surfaces is analogous to flows over
+rough surfaces only at the shallow or sparse foam limits at high Reynolds number (𝑅𝑒𝜏). This
+is also evident from the fact that at sparse foam limits, spectral heterogeneity and the peak
+in spectral energy content become independent of permeability and relative foam thickness,
+respectively.
+The influence of permeability, surface roughness, and substrate structure, as well as their
+non-linear interactions, needs to be explored further. Future work should include systematic
+variations of surface roughness for a given permeability (and vice versa). This can potentially
+be achieved by adding a high permeability surface (that is rough) on top of a surface with a
+given permeability, which will enable a better understanding of the effects of roughness and
+permeability on turbulent flow over porous surfaces.
+Acknowledgements
+We acknowledge the support from E. Rodríguez-López and M.A Ferreira in the data
+acquistion phase as well as Luis Esteban-Blay and Tim Schoelle for their help in the initial
+data curation.
+Funding
+We acknowledge the financial support from EPSRC (Grant Ref no: EP/S013296/1) and
+European Office for Airforce Research and Development (Grant No: FA9550-19-1-7022,
+Programme Manager: Dr. Doug Smith).
+Declaration of interests
+The authors report no conflict of interest.
+Data availability statement
+All data supporting this study will be made openly available from the University of
+Southampton repository upon publication.
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+

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+Anomalous behaviors in (non-)relativistic Brownian motions
+Weiguo Chen,1 Carsten Greiner,2 and Zhe Xu ∗1
+1Department of Physics, Tsinghua University and Collaborative Innovation
+Center of Quantum Matter, Beijing 100084, China
+2Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at Frankfurt,
+Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
+Anomalous behaviors such as the memory of the initial state, the ballistic diffusion, and the
+break of the equipartition theorem and the ergodicity in Brownian motions are investigated by
+solving analytically the generalized Langevin equation of non-relativistic Brownian particles with
+colored noise. These behaviors can also be observed in the Brownian motion of relativistic particles
+by solving numerically the generalized Langevin equation for specially chosen memory kernels. Our
+analyses give rise to think about the possible anomalous motion of heavy quarks in relativistic
+heavy-ion collisions.
+I.
+INTRODUCTION
+The Brownian motion is a famous stochastic process
+exhibiting the relationship between the fluctuation and
+the dissipation in statistical physics. Mathematically it is
+described by the Langevin equation. The common knowl-
+edge from text books by solving the Langevin equation
+is that the Brownian motion is a random motion of the
+Brownian particle in a fluid or gas.
+In the long time
+limit and with the ensemble average, the Brownian par-
+ticle will reach the thermal equilibrium with the mat-
+ter where it is suspended and diffuse linearly with time.
+On the other hand, for specific interactions between the
+Brownian particle and the particles which the matter is
+made of, the fluctuation as well as the dissipation corre-
+late with their former values in the motion. The current
+motion is determined by the motion in the past by means
+of the memory kernel in the generalized Langevin equa-
+tion. This memory property can give rise to the different
+diffusion from the linear one, called as the anomalous
+diffusion [1–8], and break the equipartition theorem and
+the ergodicity [9, 10]. One goal of this paper is to find
+out the general behavior of the thermal equilibrium, dif-
+fusion, and ergodicity of the Brownian particle for any
+kind of memory kernels.
+The motion of heavy quarks in the quark-gluon plasma
+produced in relativistic heavy-ion collisions has been con-
+sidered as a Brownian motion in the QCD matter and de-
+scribed by the relativistic form of the Langevin equation
+[11–13]. However, this treatment cannot simultaneously
+explain the experimental data of the energy loss and the
+collective flow of hadronic particles stemming from heavy
+quarks [11, 14–24]. Could an anomalous motion of heavy
+quarks explain the data? To answer this question one has
+to verify at first that the anomalous behavior that can
+occur in the motion of non-relativistic Brownian particles
+can also occur in the motion of relativistic Brownian par-
+ticles. This is another goal of this paper. The verification
+∗[email protected]
+is not trivial, since the relativistic Langevin equation is
+not a linear equation and cannot be solved analytically.
+The paper is organized as follows. In Sec. II we solve
+the generalized Langevin equation of non-relativistic
+Brownian particles analytically by employing the Laplace
+technique. The memory kernels are classified into three
+categories. All the behaviors are normal in the first cat-
+egory. The diffusion is normal. The equipartition the-
+orem and the ergodicity hold. In the second category,
+the memory effect occurs, the diffusion is ballistic, and
+the equipartition theorem and the ergodicity are bro-
+ken.
+In the third category, besides the memory effect
+and the break of the ergodicity, an oscillating behavior
+appears, which brings the Brownian particle out of and
+back to the equilibrium periodically. We give examples
+for each category of the memory kernels and present the
+analytical results of the averaged kinetic energy, displace-
+ment squared, and the velocity correlation function of the
+Brownian particle. In Sec. III the Langevin equation of
+relativistic Brownian particles is solved numerically for
+the memory kernels given in the previous section. Ex-
+cept for the oscillation, other anomalous behaviors are
+also seen in relativistic Brownian motions.
+We give a
+summary in Sec.
+IV and show the details in Laplace
+transformations in Appendix.
+II.
+THE LANGEVIN EQUATION OF
+NON-RELATIVISTIC BROWNIAN PARTICLES
+A.
+White vs. colored noise
+The motion of non-relativistic Brownian particles is
+described historically by the Langevin equation
+m ˙v = −γv + ξ .
+(1)
+The change of the momentum of the Brownian particle
+stems from interactions of the Brownian particle with
+the molecules of the matter, where the Brownian particle
+is suspended. The interactions are summed up into the
+mean force, which is the friction or dissipation term −γv,
+and the random force, which is the fluctuation or noise
+arXiv:2301.12450v1  [hep-ph]  29 Jan 2023
+
+2
+term ξ. The ensemble average of the noise, < ξ >, is
+zero, while the time correlation of the noise is
+< ξi(t)ξj(t′) >= δijαδ(t − t′) .
+(2)
+This noise is called white noise, since the Fourier trans-
+form of the time correlation to the frequency space is a
+constant.
+The strength of the noise, α, and the coefficient of the
+friction term, γ, should be related, since both the fluctu-
+ation and the dissipation have the same origin, namely
+interactions of the Brownian particle with the molecules
+of the matter. Their relation can be derived as the fol-
+lowing. Suppose the Brownian motion starts at t0, the
+solution of Eq. (1) is
+v(t) = v(t0)e− γ
+m (t−t0) + 1
+m
+� t
+t0
+dt′ e− γ
+m (t−t′)ξ(t′) .
+(3)
+We have then
+< v(t) · ξ(t) >= 3α
+2m
+(4)
+by using Eq. (2). Making the scalar product of Eq. (1)
+and v and taking the ensemble average, we obtain
+< mdv
+dt · v >= −γ < v · v > + < v(t) · ξ(t) > .
+(5)
+The left hand side is equal to m
+2
+d<v2>
+dt
+and is zero in the
+long time limit, when the Brownian particle relaxes to a
+stable state. Thus,
+3α
+2m = γ < v2 > .
+(6)
+Assuming that the state, to which the Brownian particle
+relaxes, is a thermal state, we have m < v2 > /2 =
+3kBT/2 according to the equipartition theorem.
+T is
+the temperature of the matter surrounding the Brownian
+particle. We obtain
+α = 2kBTγ .
+(7)
+This is the basic form of the fluctuation-dissipation theo-
+rem. In turn, putting Eq. (7) into Eq. (6) we can obtain
+the equipartition of the kinetic energy. Therefore, in the
+case of the Brownian motion with white noise, it is not
+necessary to question which is more fundamental between
+the fluctuation-dissipation theorem and the assumption
+of the equilibrium of the Brownian particle.
+From Eq. (3) we see that in the long time limit, v(t)
+does not depend on v(t0). This shows that in the long
+time limit the Brownian particle looses the memory of its
+initial velocity. We denote it as no memory effect.
+The integral of v in Eq. (3) gives the solution of the
+displacement of the Brownian particle ∆x = x(t)−x(t0).
+In the long time limit one can easily obtain
+< (∆x)2 >= 6Dt ,
+(8)
+where D = kBT/γ is the diffusion constant. We call such
+diffusion the normal diffusion.
+We now turn to the generalized form of the Langevin
+equation (1), which reads
+m ˙v(t) = −
+� t
+0
+dt′ Γ(t − t′)v(t′) + ξ(t)
+(9)
+with
+< ξi(t)ξj(t′) >= δijA(t − t′) .
+(10)
+Here we have shifted t0 to 0. The generalized Langevin
+equation describes a non-Markov process. The interac-
+tions of the Brownian particle with the molecules of the
+matter at earlier times also affect the current change of
+momentum. This is, on the one hand, revealed by the
+memory kernel Γ(t−t′). On the other hand, the noise at
+the current time correlates with those at earlier times as
+A(t − t′). In this case, the Fourier transform of A(t − t′)
+to the frequency space has a structure other than a con-
+stant. Such noise is denoted as the colored noise. The
+fluctuation-dissipation theorem is now generalized to
+A(t − t′) = kBT Γ(t − t′)
+(11)
+according to the second kind of Kubo’s law [25]. Particu-
+larly, for A(t−t′) = αδ(t−t′) and Γ(t−t′) = 2γδ(t−t′),
+the generalized Langevin equation (9) returns to Eq. (1)
+with white noise (2), and the generalized fluctuation-
+dissipation theorem (11) returns to its basic form (7). We
+note that the correlation A(t − t′) is such kind of func-
+tions, which Fourier transforms are non-negative [26, 27].
+This applies also for Γ(t − t′) due to the fluctuation-
+dissipation theorem.
+The question now is whether the Brownian particle
+under the colored noise will relax to the thermal equilib-
+rium state. In the next subsection we will show that the
+answer to this question depends on the actual functional
+form of the memory kernel Γ(t − t′). For some classes
+of Γ(t − t′), the Brownian particle will not relax to the
+thermal equilibrium with the surrounding matter. More-
+over, even in the long time limit the Brownian particle
+still keeps the memory on its initial state, its diffusion
+shows an anomalous behavior, and the ergodicity is bro-
+ken. These results are based on the hypothesis that the
+fluctuation-dissipation theorem Eq. (11) is a fundamen-
+tal assumption.
+B.
+Analytical solutions
+In this subsection we solve the generalized Langevin
+equation (9) and give the analytical results of < v2 >,
+< (∆x)2 >, and the correlation function of the velocity
+for some chosen memory kernels.
+Performing the Laplace transformation of Eq. (9) we
+obtain
+msv(s) − mv(0) = −Γ(s)v(s) + ξ(s) ,
+(12)
+
+3
+where s is defined in the complex space. We have then
+v(s) = v(0) + 1
+mξ(s)
+s + 1
+mΓ(s)
+.
+(13)
+Defining the response function in the Laplace space as
+G(s) =
+1
+s + 1
+mΓ(s) ,
+(14)
+Eq. (13) is rewritten to
+v(s) = v(0)G(s) + 1
+mG(s)ξ(s) .
+(15)
+We then perform the inverse Laplace transformation of
+Eq. (15) and obtain
+v(t) = v(0) G(t) + 1
+m
+� t
+0
+dt′G(t − t′)ξ(t′) .
+(16)
+At t = 0 we find G(t = 0) = 1. With Eq. (16) we have
+< v2 > (t) = v2(t)G2(t) + 1
+m2
+� t
+0
+dt′G(t − t′) ×
+×
+� t
+0
+dt′′G(t − t′′) < ξ(t′′) · ξ(t′) >
+= v2(0)G2(t) + 3kBT
+m2
+� t
+0
+dt′G(t − t′) ×
+×
+� t
+0
+dt′′G(t − t′′)Γ(t′′ − t′) .
+(17)
+After some steps, which can be found in Appendix, we
+get the final result
+< v2 > (t) = v2(0)G2(t) + 3kBT
+m
+�
+1 − G2(t)
+�
+.
+(18)
+In particular, for Γ(t − t′) = 2γδ(t − t′), we get Γ(s) =
+γ, G(s) = 1/(s + γ/m). By taking the inverse Laplace
+transformation we get G(t) = e−γt/m. The solution (16)
+is identical to Eq. (3). In the long time limit, G(t →
+∞) = 0, we get 1
+2m < v2 >= 3
+2kBT, which agrees with
+the equipartition theorem at the thermal state.
+Before we present the results of < v2 > (t) for some
+chosen memory kernels, we now discuss generally the long
+time behavior of < v2 > and give the answer whether the
+Brownian particle will relax to the thermal state. From
+Eq. (18) it is obvious that if the Brownian particles are
+initially in thermal state, i.m., < v2 > (0) = 3kBT/m,
+they stay always in thermal state, < v2 > (t) = 3kBT/m,
+regardless of the actual form of the memory kernel. For
+the case that the Brownian particles are initially out of
+thermal equilibrium, they will approach to the thermal
+state, only if G(t → ∞) = 0. In the following we exam-
+ine G(t → ∞) for non-thermal initial state of Brownian
+particles.
+Performing the inverse Laplace transformation we have
+G(t) = L−1[G(s)] =
+1
+2πi
+� β+i∞
+β−i∞
+ds G(s)est
+=
+1
+2πi
+� β+i∞
+β−i∞
+ds
+1
+s + 1
+mΓ(s)est .
+(19)
+The integral area should be chosen to ensure that the
+integral is convergent. So we choose the left half of the
+complex plane with respect to the imaginary axis (the
+real part is negative). Along the semicircle with the in-
+finite radius the integral vanishes. Suppose G(s) have n
+single poles sj = σj + iωj, j = 1, 2, · · · , n. G(s) can be
+written to
+G(s) =
+n
+�
+j=1
+aj
+s − sj
+,
+(20)
+where
+aj = lim
+s→sj(s − sj)G(s) .
+(21)
+Thus we obtain
+G(t) =
+n
+�
+j=1
+aj esjt
+(22)
+according to the residue theorem.
+We classify memory kernels according to the position
+of poles of G(s).
+At first, if no poles are located on
+the imaginary axis, the real part of esjt is eσjt with a
+negative σj.
+Therefore, we have G(t → ∞) = 0.
+In
+this case, the Brownian particle will relax to the thermal
+equilibrium with the surrounding matter. For example,
+we choose [28, 29]
+Γ1(t − t′) = γ
+2τ e− |t−t′|
+τ
+.
+(23)
+Its Laplace transform is
+Γ1(s) =
+γ
+2(1 + sτ) .
+(24)
+G1(s) has two poles located in the left half of the complex
+plane and not on the imaginary axis. Therefore, G1(t →
+∞) = 0.
+Secondly, we consider the case that only one of the
+poles is located on the imaginary axis and specially at
+s = 0. In this case it should be Γ(s = 0) = 0 [see Eq.
+(14)]. We have G(t → ∞) = a1, where
+a1 = lim
+s→0 sG(s) = lim
+s→0
+s
+s + 1
+mΓ(s)
+=
+1
+1 + 1
+m
+dΓ(s)
+ds
+���
+s=0
+≡
+1
+1 + Q
+(25)
+
+4
+with
+Q =
+1
+m
+d
+ds
+� ∞
+0
+dt Γ(t)e−st
+����
+s=0
+= − 1
+m
+� ∞
+0
+dt Γ(t)t .
+(26)
+We see that G(t → ∞) is nonzero.
+The equipartition
+theorem is broken and the Brownian particle will relax
+to a certain state, but not to the thermal equilibrium with
+the surrounding matter. In addition, from Eq. (18) we
+see that the term v2(0)G2(t) contributes to < v2 > (t),
+which indicates that in the long time limit the Brownian
+particle still keeps the memory of its initial state. This
+is the memory effect.
+From Eq. (18) we also see that G(t) is smaller than 1,
+because < v2 > is always positive, also for v(0) = 0. If
+v(0) = 0, the kinetic energy m < v2 > /2 will reach a
+smaller value than that from the equipartition theorem.
+G(t) < 1 also leads to a1 < 1 and thus Q > 0.
+We
+realize that in this case negative correlations, Γ(t) < 0
+at some time interval, will occur.
+This indicates that
+the mean force does not always decelerate the Brownian
+particle.
+It will also accelerate the Brownian particle.
+This might be the physical reason, why the Brownian
+particle cannot approach to the thermal equilibrium with
+the surrounding matter.
+We choose for example [28]
+Γ2(t − t′) = γ
+4τ
+�
+1 − |t − t′|
+τ
+�
+e− |t−t′|
+τ
+,
+(27)
+which falls down to be negative at t − t′ = τ and ap-
+proaches to 0 at large t − t′ from the negative side. This
+correlation resembles that for studying the non-Markov
+dissipative evolution of the chiral fields [27]. The Laplace
+transform is
+Γ2(s) =
+γsτ
+4(1 + sτ)2 .
+(28)
+G2(s) has one pole at s = 0 and other two poles in the
+left half of the complex plane and not on the imaginary
+axis.
+Thirdly, some poles are located on the imaginary axis,
+but not at s = 0. These poles appear in pairs symmetric
+to s = 0, s1,2 = ��iω1 for instance. We can write G(s) to
+the form
+G(s) =
+k
+�
+j=1
+�
+aj
+s − iωj
++
+a∗
+j
+s + iωj
+�
++
+n
+�
+j=2k+1
+aj
+s − sj
+. (29)
+We obtain G(∞) = limt→∞
+�k
+j=1 2[Re(aj) cos(ωjt) −
+Im(aj) sin(ωjt)]. The oscillations of G(t) are not damped
+in the long time limit due to the absence of the real part
+of the poles. In this case the Brownian particle will not
+even approach to a steady state. On the other hand, G(t)
+oscillates across zero. When G(t) is zero, the Brownian
+particle is at the thermal equilibrium. So, in the long
+time limit, the Brownian particle goes out of and returns
+to the thermal equilibrium circularly.
+We present now the analytical results of G(t) for some
+chosen memory kernels Γ(t − t′). For Γ1(t − t′) [see Eqs.
+(23)] we obtain results for two cases. If B ≡
+�
+1 − 2γτ/m
+is real, we have
+G11(t) =
+�
+cosh
+�
+B t
+2τ
+�
++ 1
+B sinh
+�
+B t
+2τ
+��
+e− t
+2τ . (30)
+If B is pure imaginary, we define B′
+=
+−iB
+=
+�
+2γτ/m − 1 and have
+G12(t) =
+�
+cos
+�
+B′ t
+2τ
+�
++ 1
+B′ sin
+�
+B′ t
+2τ
+��
+e− t
+2τ .
+(31)
+For Γ2(t − t′) [see Eq. (27)] we obtain
+G2(t) =
+1
+1 + C +
+�
+C
+1 + C cos
+�√
+C t
+τ
+�
++
+√
+C
+1 + C sin
+�√
+C t
+τ
+��
+e− t
+τ ,
+(32)
+where C = γτ/(4m).
+For the last case we choose
+Γ3(t−t′) = γ
+1 + ω2τ 2
+2τ(ω2τ 2 + 1 − ϵ)[1−ϵ cos(ω|t−t′|)]e− |t−t′|
+τ
+,
+(33)
+which has a oscillating disturbance.
+The amplitude ϵ
+should be
+ϵ ≤ 4
+W
+��
+(W + 1)(W + 4) − W − 2
+�
+(34)
+with W = ω2τ 2, so that the Fourier transform of Γ3(t−t′)
+is non-negative. The Laplace transform of Γ3(t − t′) is
+Γ3(s) = γ′
+�
+1
+1 + sτ − ϵ
+1 + sτ
+W + (1 + sτ)2
+�
+,
+(35)
+where
+γ′ = γ
+W + 1
+2(W + 1 − ϵ) .
+(36)
+G3(s) has four poles. Two of them are located in the
+left half of the complex plane, while another two poles
+are on the imaginary axis, when γ′ is the solution of the
+equation
+2(1 − ϵ)2τ 2γ′2 + mτ[(1 − ϵ)(8 + 5W) − 9W]γ′
++2m2(W + 1)(W + 4) = 0 .
+(37)
+The condition that the above equation has a solution is
+ϵ ≥ 4
+W
+��
+(W + 1)(W + 4) − W − 2
+�
+.
+(38)
+Together with the condition (34) we have
+ϵ = 4
+W
+��
+(W + 1)(W + 4) − W − 2
+�
+.
+(39)
+
+5
+We see that ϵ is fixed for given ωτ. With this we obtain
+γτ
+m = 2(W + 1 − ϵ)
+W + 1
+�
+(W + 1)(W + 4)
+1 − ϵ
+.
+(40)
+The strength of the memory kernel, γ, is not a free pa-
+rameter. It relates to the frequency of the oscillation ω,
+the decay time scale τ, and the mass of the Brownian
+particle m. This indicates that the case that poles are
+located on the imaginary axis (not at the origin) hardly
+occurs. This is different from Γ1(t − t′) and Γ2(t − t′),
+in which γ and τ are free parameters. After a lengthy
+derivation we obtain
+G3(t) =
+1
+1 + 1
+2
+�
+W +1
+W +4
+cos
+��
+Z − 6W − 3
+6
+t
+τ
+�
++
+1
+1 + 2
+�
+W +4
+W +1
+�
+3
+√
+3
+√
+Z
+sin
+� √
+Z
+2
+√
+3
+t
+τ
+�
++ cos
+� √
+Z
+2
+√
+3
+t
+τ
+��
+e− 3t
+2τ ,
+(41)
+where Z = 8W + 4
+�
+(W + 1)(W + 4) + 5.
+From the dependence of G11 on B, G12 on B′, G2 on
+C, and G3 on W we realize that all G’s, as functions of
+t/τ, are fixed by the only parameter γτ/m. The same is
+FIG. 1: Examples of the memory kernels (upper panel) and
+the response functions (lower panel) as functions of t/τ. The
+memory kernels are multiplied by τ 2/m.
+FIG. 2: The time evolution of the average kinetic energy of
+the Brownian particle under the white noise and the colored
+noise with the chosen memory kernels. The initial momentum
+of the Brownian particle is set to be zero.
+also for the memory kernels Γ1, Γ2, and Γ3 multiplied by
+τ 2/m. Figure 1 shows the chosen memory kernels (times
+τ 2/m) and the respective G’s as functions of t/τ. The
+curves of Γ1 with B′ = 3 and Γ3 with W = 2 are divided
+by 10.
+In Fig. 2 we show the time evolution of the average
+kinetic energy of the Brownian particle, < Ek >= m <
+v2 > /2, scaled by kBT, according to Eq.
+(18).
+The
+initial momentum of the Brownian particle, p0 = mv(0),
+is set to be zero. The solid curve depicts the result under
+the white noise, where G(t) = e−t/τ as derived before.
+Here τ denotes m/γ and thus, γτ/m = 1. Other curves
+depict the results under the colored noise with the mem-
+ory kernels given before and shown in Fig. 1. We see
+that the averaged kinetic energy of the Brownian parti-
+cle under the white noise and the colored noise with Γ1
+approaches to the value according to the equipartition
+theorem.
+This corresponds to the case that the poles
+of the response function G(s), the Laplace transform of
+G(t), locate in the left half of the complex plane and not
+on the imaginary axis. For the case with Γ2, where one
+of the poles of G(s) is at the origin, s = 0, the aver-
+aged kinetic energy is less than the value according to
+the equipartition theorem. Finally, for the case with Γ3,
+where two of the poles of G(s) are on the imaginary axis,
+the averaged kinetic energy oscillates. At the peaks it
+reaches the value according to the equipartition theorem.
+The difference between the peak and the trough equals
+to the square of the amplitude of the first term of G3(t)
+times 3/2 [see Eqs. (41) and (18)] and varies between
+24/25 for W → 0 and 2/3 for W → ∞.
+By comparing with Fig. 2, the memory effect can be
+observed in Fig. 3, where the initial momentum is set
+to be p0 = √mkBT. Again, in the long time limit, the
+
+-1, B=0.5
+0.4
+- I1, B'=3, divided by 10
+.I2, C=0.25
+-I3, W=2, divided by 10
+0.0
+(a)
+ - G11, B=0.5
+- G12, B'=3
+... G2, C=0.25
+_G3, W=2
+G(t/ T)
+0
+(b)
+10
+15
+20
+t/ T3
+white noise
+- 1. B=0.5
+- - I1, B'=3
+2
+-I3, W=2
+/ kB
+po/VmkB T =0
+5
+10
+t /T6
+FIG. 3: Same as Fig. 2. The initial momentum is set to be
+p0 = √mkBT.
+averaged kinetic energy of the Brownian particle under
+the white noise and the colored noise with Γ1 approaches
+to the value according to the equipartition theorem. In
+these cases no memory effect is expected. From the re-
+sults with Γ2 and Γ3 we do see the memory effect. The
+larger the initial momentum, the larger is the averaged
+kinetic energy of the Brownian particle with Γ2 in the
+long time limit. In the case of Γ3, for p0 < √3mkBT, the
+averaged kinetic energy at long times reaches the value
+according to the equipartition at the peaks of the oscil-
+lation. The difference between the peak and the trough
+is decreasing to zero, when the initial momentum is in-
+creasing from zero to √3mkBT. For p0 > √3mkBT, the
+averaged kinetic energy at long times reaches the value
+according to the equipartition at the troughs of the oscil-
+lation. The difference between the peak and the trough
+is increasing for the increasing initial momentum.
+We now study the diffusion behavior of the Brown-
+ian particle with the colored noise.
+Since v = ˙x, the
+Langevin equation (9) can be rewritten to
+m¨x = −
+� t
+0
+dt′Γ(t − t′) ˙x(t′) + ξ(t) .
+(42)
+Performing the Laplace transformation to Eq. (42) and
+proceeding the same steps such as solving Eq. (9), we
+obtain the analytical results of < (∆x)2 >,
+< (∆x)2 > (t) = v2(0)H2(t) + 3kBT
+m
+�
+2
+� t
+0
+dt′H(t − t′)
+−H2(t)
+�
+,
+(43)
+where
+H(t) = L−1[H(s)] and H(s) = 1
+sG(s) .
+(44)
+Since H(t = 0) = 0, we have
+˙H(t) = G(t).
+For the
+known G(t) the analytical results of < (∆x)2 > (t) can
+be easily calculated. In the following we discuss its long
+time behavior.
+(i) For the case that the poles of G(s) are located in
+the left half of the complex plane and not on the
+imaginary axis such as for the white noise and for
+the colored noise with Γ1, H(s) has an additional
+pole at s = 0 due to Eq. (44). In the long time limit
+we have H(t) → G(s = 0) and thus < (∆x)2 >
+(t) → 6kBTG(s = 0)t/m, which is proportional
+to t. This is the normal diffusion. For the white
+noise, G(s = 0) = m/γ, we get < (∆x)2 > (t) →
+6kBTt/γ = 6Dt ∼ t, which agrees with Eq. (8)
+in the previous subsection. For Γ1, G1(s = 0) =
+2m/γ, we get < (∆x)2 > (t) → 12kBTt/γ ∼ t.
+Therefore,
+m < (∆x)2 > |wn
+kBTτ 2
+→ 6 t
+τ ,
+(45)
+m < (∆x)2 > |Γ1
+kBTτ 2
+→ 12 m
+γτ
+t
+τ .
+(46)
+(ii) For the case that one pole of G(s) is located at
+s = 0 and the other poles are located in the left
+half of the complex plane and not on the imaginary
+axis such as for the colored noise with Γ2, H(s) has
+a two-fold pole at s = 0. In the long time limit
+H(t) goes to the residue of H(s)est at s = 0, which
+is d[s2H(s)est]/ds at s = 0. For Γ2 it equals to
+2τC/(1 + C)2 + t/(1 + C) with C = γτ/(4m). We
+obtain < (∆x)2 > (t) → [v2(0)+3kBTC/m]t2/(1+
+C)2 ∼ t2 and
+m < (∆x)2 > |Γ2
+kBTτ 2
+→ mv2(0)/(kBT) + 3C
+(1 + C)2
+� t
+τ
+�2
+.
+(47)
+The diffusion with a parabolic dependence of <
+(∆x)2 > on t is an anomalous diffusion and called
+the ballistic diffusion.
+(iii) For the case that some pols of G(s) are located on
+the imaginary axis but not at s = 0 and the other
+poles are located in the left half of the complex
+plane such as for the colored noise with Γ3, the
+poles on the imaginary axis lead to the oscillation
+of H(t) with constant amplitudes. The long time
+behavior is dominated by the pole of H(s) at s = 0
+like in case (i). For Γ3 we get G3(s = 0) = 2m/γ,
+< (∆x)2 > (t) → 12kBTt/γ ∼ t and
+m < (∆x)2 > |Γ3
+kBTτ 2
+→ 12 m
+γτ
+t
+τ .
+(48)
+This is again the normal diffusion.
+Figure 4 shows the time evolution of the average dis-
+placement squared of the Brownian particle scaled by
+kBTτ 2/m. The results with the white noise, Γ1 (B =
+0.5), and Γ2 are divided by 10. We see that the average
+
+3.
+white noise
+- I1, B=0.5
+ - 1,B'=3
+.. 2, C=0.25
+2
+-I3, W=2
+/ KB
+po/VmkB T =1
+V
+5
+10
+0
+t /T7
+FIG. 4:
+The time evolution of the average displacement
+squared of the Brownian particle under the white noise and
+the colored noise with various memory kernels.
+The ini-
+tial momentum of the Brownian particle is set to be p0 =
+√mkBT. The results with the white noise, Γ1 (B = 0.5), and
+Γ2 are divided by 10.
+displacement squared of the Brownian particle under the
+white noise and the colored noise with Γ1 is linear in time
+in the long time limit, whereas it is parabolic in time for
+the colored noise with Γ2 and is oscillating along a linear
+line in time for the colored noise with Γ3.
+Finally we discuss briefly the ergodicity in the Brown-
+ian motion with the colored noise. The ergodicity states
+that the ensemble average of a variable equals its time
+average in the infinite-time limit.
+The necessary con-
+dition of the ergodicity is given by Khinchin’s theorem
+[8], which declares that the auto-correlation CO(t, t′) =<
+O(t)O(t′) > − < O(t) >< O(t′) > of a stochastic vari-
+able O(t) should vanish in the limit t → ∞ at fixed
+t′ ≫ t0. For the sufficient condition of the ergodicity,
+the newly defined quantity WO =
+� ∞
+t′ CO(t, t′)dt should
+be finite and non-zero [30]. In the following we examine
+Cv(t, t′) and Wv for various given memory kernels.
+The derivation of < v(t) · v(t′) > from Eq.
+(16) is
+similar to that of < v2 > (t). The result is
+Cv(t, t′) = 3kBT
+m
+[G(t − t′) − G(t)G(t′)] .
+(49)
+For the white noise and the colored noise with the first
+class of the memory kernel such as Γ1(t − t′), G(t →
+∞) = 0 and we have limt→∞ Cv(t, t′) = 0 at fixed t′. To
+prove the validity of the ergodicity, we calculate Wv for
+these two cases. We obtain
+Wwn
+v
+= 3kBT
+γ
+�
+1 − e−2γt′/m�
+≈ 3kBT
+γ
+(50)
+for large t′ in the case of the white noise, and
+W 1
+v ≈ 6kBT
+γ
+(51)
+for large t′ in the case of the colored noise with Γ1. In
+these two cases the ergodicity holds.
+For the colored noise with other memory kernels such
+as Γ2 and Γ3, the limits of G2(t) and G3(t) [see Eqs.
+(32) and (41)] do not go to zero for t → ∞.
+Thus,
+limt→∞ Cv(t, t′) does not go to zero either. In these two
+cases the ergodicity is broken. In general, the ergodic-
+ity is broken when G(s) has poles on the imaginal axis.
+This result does not depend on the initial state, which
+is different from the equilibation. As we have noticed,
+the equipartition theorem is broken for the cases with
+Γ2 and Γ3, if the initial Brownian particles are out of
+thermal equilibrium.
+C.
+Numerical calculations
+In this subsection, we solve the Langevin equation (9)
+numerically. Comparing to the conventional numerical
+method for solving ordinary differential equations, here
+we have to generate series of noise from its time correla-
+tion [see Eqs. (10) and (11)]. The details of the genera-
+tion of white and colored noise and the numerical method
+can be found in Ref. [27].
+We have two purposes for performing the numerical
+calculations. At first we test our numerical computations
+by comparing the numerical results with the analytical
+ones, in order to prepare a well-tested numerical code for
+solving the relativistic Langevin equation, which is in-
+troduced and investigated in the next section. Secondly,
+we want to calculate the momentum distribution of the
+Brownian particle in the long time limit in the cases of
+the colored noise with Γ2 and Γ3, since obviously these
+distributions cannot be obtained analytically.
+From the previous subsection we notice that the
+Langevin equation (9) can be nondimensionalized. Defin-
+ing the dimensionless quantities ˜p = mv/√mkBT and
+˜t = t/τ we obtain
+d˜p
+d˜t = −
+� ˜t
+0
+d˜t′ ˜Γ(˜t − ˜t′)˜p(˜t′) + ˜ξ(˜t) ,
+(52)
+where
+˜Γ(˜t − ˜t′) = τ 2
+m Γ(t − t′) ,
+(53)
+which is dimensionless.
+Examples are plotted in the
+upper panel of Fig.
+1.
+The dimensionless noise is
+˜ξ(˜t) = ξ(t)τ/√mkBT with
+< ˜ξi(˜t)˜ξj(˜t′) >= δij ˜Γ(˜t − ˜t′) .
+(54)
+Remind that τ is the decay time scale of the memory
+kernels. For the given memory kernels [see Eqs. (23),
+(27), and (33)] the only free parameter in Eq. (52) is
+γτ/m.
+Figure 5 shows one colored noise sequence in the case
+of Γ3 with W = 2. Its time correlation function is shown
+
+140
+white noise
+-I1,B=0.5
+120
+ - 1,B'=3
+ I2, C=0.25
+100
+dy)
+ I3, W=2
+po
+80
+=1
+Λ
+VmkB T
+xV
+60
+V
+m
+40
+20
+5
+10
+15
+20
+25
+30
+35
+40
+t /T8
+FIG. 5: Colored noise sequence.
+FIG. 6: Time correlation function of the noise with Γ3.
+in Fig. 6, where the numerical result agrees well with
+the given correlation function. 50000 series of noise have
+been generated.
+We have checked the time evolution of the average ki-
+netic energy and displacement squared. The numerical
+results (not shown) agree well with the analytical ones
+presented in the previous subsection, see Figs. 2, 3, and
+4.
+The momentum distributions of the Brownian particle
+at long times are shown in Fig. 7 and compared with the
+Maxwell-Boltzmann distributions. For the case with Γ3,
+the distributions are calculated at the times when the av-
+erage kinetic energy reaches its maximum (peak) as well
+as its minimum (trough), see Fig. 2. From Fig. 7 we
+see that the distribution of the Brownian particle with
+Γ3 at the peak agrees well with the Maxwell-Boltzmann
+FIG. 7: The momentum distribution of the Brownian particle
+in the long time limit.
+distribution with the temperature T of the surrounding
+matter. This is expected from the study in the previ-
+ous subsection that the Brownian particle is in thermal
+equilibrium at peaks of the average kinetic energy in the
+long time limit.
+It is also seen, as expected, that the
+momentum distributions of the Brownian particle with
+Γ2 and with Γ3 at the trough are not in thermal equilib-
+rium with the surrounding matter. However, they look
+like the Maxwell-Boltzmann distributions with different
+“temperatures”.
+If so, then these “temperatures” can
+be obtained by the values of the average kinetic energy
+in the long time limit according to Eq. (18). For the
+Brownian motion with Γ2 we have
+3
+2kBT ′ = 1
+2m < v2 > (t → ∞)
+= 3
+2kBT
+�
+1 − G2
+2(t → ∞)
+�
+= 3
+2kBT
+�
+1 −
+1
+(1 + C)2
+�
+,
+(55)
+while for the Brownian motion with Γ3 at the trough we
+have
+3
+2kBT ′′ = 3
+2kBT
+�
+1 − G2
+3(t → ∞, at the trough)
+�
+= 3
+2kBT
+�
+��1 −
+1
+�
+1 + 1
+2
+�
+W +1
+W +4
+�2
+�
+�� .
+(56)
+We plot the Maxwell-Boltzmann distributions with T ′
+and T ′′ in Fig.
+7 and see agreements of these distri-
+butions with the numerical results. This indicates that
+in the long time limit the Brownian particle always gets
+thermalized. It may be the result of the use of the Gaus-
+sian noise. The “temperature” that the Brownian parti-
+
+noise sequence (dimensionless)
+2
+4
+6
+8
+10
+t /T0.4
+numerical simulations
+expected
+normalized correlation
+0.2
+0
+0
+2
+4
+6
+8
+t / T1.5
+- I2(t), C=0.25
+--- I3(t), W=2, trough
+- I3(t), W=2, peak
+- MB distribution with T
+MB distribution with T'
+- MB distribution with T
+d pN/ Np
+po=0
+0.5
+0
+2
+4
+6
+p9
+cle feels is in some cases not the temperature of the mat-
+ter, where the Brownian particle is suspended. Therefore,
+the attempt to use the Brownian particle to probe an un-
+known matter may become difficult because of possible
+occurrence of anomalous behaviors.
+III.
+THE LANGEVIN EQUATION OF
+RELATIVISTIC BROWNIAN PARTICLES
+Assume that the matter, where the Brownian particle
+is suspended, remains at rest. Then, the Langevin equa-
+tion (9) can be extended to a form applied to relativistic
+particles [31–37],
+˙p(t) = −
+� t
+0
+dt′ Γ(t − t′)p(t′)c2
+E(t′) + ξ(t) ,
+(57)
+where p is the momentum and E =
+�
+p2c2 + m2
+0c4 is the
+energy of the Brownian particle with the rest mass m0.
+Its reduced form with the white noise is
+˙p = −γ pc2
+E + ξ .
+(58)
+The time correlation functions of the noise are same as
+those in Eqs. (10) and (2), respectively.
+Similar to Eq.
+(3), the formal solution of Eq.
+(58)
+reads
+p(t) = p(t0)e−γ
+� t
+t0
+dt′c2
+E
++
+� t
+t0
+ds e−γ
+� t
+s
+ds′c2
+E ξ(s) .
+(59)
+We have then < p · ξ >= 3α/2, which is same as Eq.
+(4) in the case for non-relativistic Brownian particles.
+Taking the ensemble average of the scalar product of Eq.
+(58) with p, we obtain
+1
+2
+d
+dt < p2 >= −γ < p2c2
+E
+> +3α
+2 .
+(60)
+If the Brownian particle reaches the thermal equilibrium
+in the long time limit, the left hand side of the above
+equation vanishes and < p2c2/E >= 3kBT by using the
+relativistic Boltzmann distribution f = exp[−E/(kBT)].
+The same fluctuation-dissipation theorem as Eq. (7) is
+derived in the relativistic case.
+Therefore, we assume
+that the general fluctuation-dissipation theorem (11) is
+also valid for relativistic Brownian particles, because a
+memory kernel having the δ-function form reduces the
+Langevin equation from Eq. (57) to Eq. (58) and the
+fluctuation-dissipation theorem from Eq. (11) to Eq. (7).
+We notice that from the assumption of the fluctuation-
+dissipation theorem we can also derive that the relativis-
+tic Brownian particle under the white noise will reach the
+thermal equilibrium in the long time limit.
+It is obvious that both Eqs.
+(57) and (58) are not
+linear differential-integral equations and thus, cannot be
+solved analytically by using the Laplace transformation
+as performed in the non-relativistic case. Even the solu-
+tion (59) can only be calculated numerically, since E on
+the right hand side of Eq. (59) is a function of p at times
+before t. Even though we can derive that the Brownian
+particle under the white noise can reach the thermal equi-
+librium in the long tine limit, but we cannot determine
+the behavior of the diffusion and ergodicity analytically.
+In the following we solve the Langevin equations (57)
+and (58) by using the numerical code mentioned and well
+tested in the previous section. We will study the equili-
+bration, memory effect, diffusion and ergodicity of Brow-
+nian particles under the white noise and the colored noise
+with the memory kernels Γ1, Γ2, and Γ3 given in the pre-
+vious section.
+Like Eq.
+(52), the relativistic Langevin equation
+(57) can also be nondimensionalized, when we define
+˜p = pc/(kBT),
+˜m0 = m0c2/(kBT), ˜t = t/τ, and
+˜ξ = ξτc/(kBT). We have then
+d˜p
+d˜t = −
+� ˜t
+0
+d˜t′ ˜Γ(˜t − ˜t′)
+˜p(˜t′)
+�
+˜p2 + ˜m2
+0
++ ˜ξ(˜t) ,
+(61)
+where
+˜Γ(˜t − ˜t′) = τ 2c2
+kBT Γ(t − t′)
+(62)
+and
+< ˜ξi(˜t)˜ξj(˜t′) >= δij ˜Γ(˜t − ˜t′) .
+(63)
+τ is the decay time scale in the given memory kernels.
+Looking at the memory kernels Γ1, Γ2, and Γ3 in Eqs.
+(23), (27), and (33), we find that ˜m0 and γτc2/(kBT) are
+the free parameters in Eq. (61). Note that ωτ in Γ3 is
+not an additional parameter, since ωτ (or W) is fixed by
+γτ/m0 in Eq. (40), where we replace m by m0. We also
+realize that τ does not exist in the Langevin equation
+with the white noise, Eq. (58). In this case we define
+τ = kBT/(γc2).
+Figure 8 shows the time evolution of the average ki-
+netic energy of the relativistic Brownian particle, Ek =
+E − m0c2 scaled by kBT, under the colored noise with
+Γ1. The initial momentum of the Brownian particle is
+zero.
+The parameter γτc2/(kBT) is set to be 6.
+For
+τ = 1 fm/c and T = 300 MeV, γ/2 is consistent with the
+drag coefficient in the Brownian motion of charm quarks
+in the quark-gluon plasma created in relativistic heavy-
+ion collisions [14, 19, 24, 38–45]. We vary the rest mass
+of the Brownian particle to see the relativistic effect and
+to check the numerical calculations.
+We see that in the long time limit, the average kinetic
+energy increases from the non-relativistic limit, 3kBT/2,
+to the ultra-relativistic limit, 3kBT. All the final values
+with different masses agree with those according to the
+equipartition theorem, which indicates that the Brown-
+ian particle under the colored noise with Γ1 reaches the
+thermal equilibrium with its surrounding matter. The
+numerical result of the relativistic Langevin equation
+
+10
+FIG. 8: The time evolution of the average kinetic energy of
+the relativistic Brownian particle under the colored noise with
+Γ1 and with various rest masses. The initial momentum of
+the particle is set to be zero.
+with a large mass ˜m0 = 40 is compared with the an-
+alytical one of non-relativistic Langevin equation. The
+perfect agreement demonstrates the solid numerical cal-
+culations.
+The time evolution of the average kinetic energy of
+the relativistic Brownian particle with the mass ˜m0 =
+1.4 under the white noise and the colored noise with the
+memory kernels Γ1, Γ2, and Γ3 are depicted in Fig. 9.
+The initial momentum is zero. The curve of Γ1 is the
+same as that in Fig. 8. For Γ3 with W = 2 we obtain
+γτ/m0 = 13.8 according to Eq. (40) by replacing m by
+m0. Thus, we have γτc2/(kBT) = 19.3 for ˜m0 = 1.4.
+FIG. 9: Same as Fig. 8 but under the white noise and the
+colored noise with various memory kernels.
+From Fig. 9 we see that the average kinetic energy of
+the Brownian particle under the white noise and the col-
+ored noise with Γ1 and Γ3 approaches to the same value
+in the long time limit, which is the value according to
+the equipartition theorem. The oscillating behavior in
+the non-relativistic limit with Γ3 does not occur here.
+Remind that Eq. (40) is a strict condition for the oscil-
+lating behavior in the non-relativistic case, which is not
+necessarily the same one met in the relativistic case.
+The long time limit of the average kinetic energy of
+the Brownian particle with Γ2 is smaller than that with
+other memory kernels. The anomalous behavior already
+seen in the non-relativistic case appears in the relativistic
+case too. We want to know whether we can make use
+of the formulas derived in the non-relativistic limit in
+the previous section to analyze the results achieved in
+the relativistic case. From Eq. (18) we get the average
+kinetic energy in the long time limit,
+< Ek > |t→∞
+kBT
+= < Ek > |t=0
+kBT
+1
+(1 + C)2
++3
+2
+�
+1 −
+1
+(1 + C)2
+�
+,
+(64)
+where Ek = p2/(2m) and C = γτ/(4m). We use this
+formula and replace m by m0. For p0 = 0, γτc2/(kBT) =
+6, and ˜m0 = 1.4 we obtain < Ek > |t→∞/(kBT) = 1.15,
+which agrees with the numerical result seen in Fig. 9.
+The memory effect of the initial momentum is shown
+in Fig.
+10.
+In these calculations the initial momen-
+tum is set to be p0c/(kBT) = 2.
+By comparing with
+Fig. 9 we see that while the long time limit of the av-
+erage kinetic energy of the relativistic Brownian parti-
+cle under the white noise and the colored noise with Γ1
+and Γ3 do not show any dependence of the initial mo-
+mentum, it does appear for the particle with Γ2. Using
+FIG. 10: Same as Fig. 9, but the initial momentum of the
+particle is set to be p0c/(kBT) = 2.
+
+white noise
+YTc2
+3
+kB T
+YTc2
+YTc2
+19.3
+=6
+kB T
+kB T
+T
+/kB
+Λ
+po c
+mo
+.4
+kB
+0
+5
+10
+15
+20
+25
+30
+t/Tmo
+mo C
+:0
+:40
+kB T
+mo
+1.4
+non-relativistic limit
+kB T
+3
+T
+kB
+2
+V
+po c
+0
+0
+5
+10
+15
+20
+25
+30
+t/Twhite noise
+12.
+YTC2
+3
+kB T
+YTc2
+YTc2
+19.3
+:6
+kB T
+kB T
+T
+kB
+V
+po c
+mo
+0
+1.4
+kB
+0
+5
+10
+15
+20
+25
+30
+t/T11
+FIG. 11:
+The time evolution of the average displacement
+squared of the relativistic Brownian particle under the white
+noise and the colored noise with various memory kernels. The
+initial momentum of the particle is set to be p0 = 0.
+the formula in the non-relativistic limit (64), we obtain
+< Ek > |t→∞/(kBT) = 1.48 for p0c/(kBT) = 2, which
+also agrees with the numerical result seen in Fig. 10.
+We depict the time evolution of the average displace-
+ment squared, scaled by τ 2c2, in Fig. 11. The behaviors
+of the diffusion are the same as in the non-relativistic
+limit. The particle diffusion under the white noise and
+the colored noise with Γ1 and Γ3 are normal, while it
+is ballistic for the particle with Γ2.
+Using the formu-
+las in the non-relativistic limit Eqs.
+(45) - (48) times
+kBT/(m0c2),
+< (∆x)2 > |wn
+τ 2c2
+→ 6 kBT
+m0c2
+t
+τ ,
+(65)
+< (∆x)2 > |Γ1,Γ3
+τ 2c2
+→ 12 kBT
+γτc2
+t
+τ ,
+(66)
+< (∆x)2 > |Γ2
+τ 2c2
+→ kBT
+m0c2
+p2
+0/(m0kBT) + 3C
+(1 + C)2
+� t
+τ
+�2
+,
+(67)
+we compare these non-relativistic results with the rela-
+tivistic ones. We find perfect agreements for the colored
+noise with Γ1 and Γ3, an approximate agreement for the
+white noise, and a difference of almost a factor of 2 for
+the colored noise with Γ2.
+Finally we calculate the auto-correlation function of
+momentum at different times, in order to study the va-
+lidity of the ergodicity in the motion of relativistic Brow-
+nian particles. Figure 12 shows the numerical results at
+a fixed large t′/τ = 20.
+The momentum is scaled by
+kBT/c. We clearly see that Cp with Γ2 is non-zero in the
+long time limit, while the other two correlations with Γ1
+and Γ3 approach to zero. Therefore, the motion with Γ2
+breaks the validity of the ergodicity. The motions with
+Γ1 and Γ3 hold the ergodicity, since Wp, i.e., the integral
+FIG. 12: The auto-correlation function of momentum of the
+relativistic Brownian particle under the colored noise with Γ1,
+Γ2, and Γ3.
+of Cp, in the two cases are obviously finite and non-zero.
+IV.
+SUMMARY
+In this paper we solved analytically the generalized
+Langevin equation of non-relativistic Brownian particles
+with the memory kernel and colored noise by employing
+the Laplace transformation technique. According to the
+position of the poles of the response function G(s), the
+memory kernels are classified to three categories, which
+result in different behaviors of the thermal equilibrium,
+the memory effect, the particle diffusion, and the ergod-
+icity.
+Specifically, the first category includes the cases
+that all the poles of G(s) are located in the left half of
+the complex plane but not on the imaginary axis. In the
+long time limit, the Brownian particle approaches to the
+thermal equilibrium with the surrounding matter, has no
+memory of the initial state, and diffuses normally. The
+ergodicity holds. The second category includes the cases
+that one pole is located on the imaginal axis and specially
+at the origin s = 0 and the other poles are located in the
+left half of the complex plane.
+Usually, the Brownian
+particle cannot reach the equilibrium with the surround-
+ing matter, but will reach an equilibrium with a different
+“temperature” rather than that of the surrounding mat-
+ter. This “temperature” as well as the average kinetic en-
+ergy of the Brownian particle depend on its initial state,
+which shows the memory effect. The particle diffusion is
+ballistic and the ergodicity is broken. The third and last
+category includes the cases that some poles in pairs are
+located on the imaginal axis but not at the origin and
+the other poles are located in the left half of the complex
+plane. The Brownian particle approaches to and departs
+from the thermal equilibrium with the surrounding mat-
+
+500
+white noise
+YTc2
+:6
+400
+kB 
+YTc2
+=6
+kB T
+YTc2
+300
+19.3
+kB
+ od
+mo (
+V
+0
+1.4
+KR
+△x
+200
+V
+100
+10
+20
+30
+40
+0
+50
+60
+t/T15
+YTc2
+:6
+kB T
+:6
+10
+kB T
+YTC
+19.3
+kB T
+Cp(t/T, t'/t)
+po c
+mo 
+:1.4, t/t=20
+5
+kB T
+kB T
+20
+30
+40
+50
+60
+t / T12
+ter periodically. The amplitude of the oscillation in the
+average kinetic energy depends on the initial state, which
+is again the memory effect. The diffusion of the Brownian
+particle is normal, but the ergodicity is broken.
+For relativistic Brownian particles, the Langevin equa-
+tion cannot be solved analytically because of the non-
+linearity.
+Therefore, we do not have the general con-
+clusion about the behavior of the thermal equilibrium,
+memory effects, the diffusion and the ergodicity for any
+given memory kernel. On the other hand, we solved the
+relativistic Langevin equation numerically for three typ-
+ical memory kernels chosen as the examples in the non-
+relativistic case. Similar results as obtained in the non-
+relativistic case for the first and second category of mem-
+ory kernels are also seen in the relativistic case. In other
+words, there are indeed memory kernels, with which the
+relativistic Brownian particle cannot reach the thermal
+equilibrium, has memory effects of the initial state, and
+diffuses anomalously.
+Its motion breaks the ergodicty.
+Moreover, by regarding the relativistic particle as the
+non-relativistic one (by replacing the mass by the rest
+mass), the average kinetic energy and the average dis-
+placement squared (except for one case with Γ2) can
+be well described by the formulas derived in the non-
+relativistic case.
+Anomalous behaviors in Brownian motions may chal-
+lenge the probe of an unknown matter by using a Brow-
+nian particle. On the other hand, our present investiga-
+tion may give rise to think about the memory effect and
+anomalous diffusion of heavy quarks in the quark-gluon
+plasma created in relativistic heavy-ion collisions.
+Acknowledgments
+This work was financially supported by the National
+Natural Science Foundation of China under Grants No.
+11890710, No. 11890712, and No. 12035006. C.G. ac-
+knowledges support by the Deutsche Forschungsgemein-
+schaft (DFG) through the grant CRC-TR 211 “Strong-
+interaction matter under extreme conditions.”
+Appendix A: Laplace transformation
+The Laplace transform of a function f(t) is defined as
+L[f(t)] =
+� t
+0
+dtf(t)e−st ≡ f(s) ,
+(A1)
+where s is a complex variable. The inverse transform is
+L−1[f(s)] =
+� β+i∞
+β−i∞
+dsf(s)est .
+(A2)
+Some useful properties of the Laplace transformation are
+listed below:
+L
+�df(t)
+dt
+�
+= sf(s) − f(t = 0) ,
+(A3)
+L
+�� t
+0
+dt′f(t − t′)g(t′)
+�
+= f(s)g(s) ,
+(A4)
+L
+�d2f(t)
+dt2
+�
+= s2f(s) − sf(t = 0) − df(t)
+dt (t = 0) .
+(A5)
+We now derive Eq. (18) from Eq. (17)
+< v2 > (t) = v2(0)G2(t) + 3kBT
+m2
+� t
+0
+dt′G(t − t′) ×
+×
+� t
+0
+dt′′G(t − t′′)Γ(t′′ − t′) .
+Because the integrals for t′′ ≥ t′ and t′′ ≤ t′ are same, we
+rewrite the above equation to
+< v2 > (t) = v2(0)G2(t) + 3kBT
+m2 2
+� t
+0
+dt′G(t − t′) ×
+×
+� t
+0
+dt′′G(t − t′′)Γ(t′′ − t′)θ(t′′ − t′) .
+(A6)
+By τ = t′′ − t′, we have
+� t
+0
+dt′′G(t − t′′)Γ(t′′ − t′)θ(t′′ − t′)
+=
+� t−t′
+−t′
+dτG(t − t′ − τ)Γ(τ)θ(τ)
+=
+� t−t′
+0
+dτG(t − t′ − τ)Γ(τ)
+= L−1
+�
+L
+�� t−t′
+0
+dτG(t − t′ − τ)Γ(τ)
+��
+= L−1 [G(s)Γ(s)] .
+Since G(s) = 1/[s + Γ(s)/m] and G(t = 0) = 1, we get
+G(s)Γ(s) = m[sG(s) − G(t = 0)] = mL
+�dG(t − t′)
+d(t − t′)
+�
+and
+L−1 [G(s)Γ(s)] = mdG(t − t′)
+d(t − t′) = −mdG(t − t′)
+dt′
+.
+Putting this in Eq. (A6) we obtain finally
+< v2 > (t) = v2(0)G2(t) − 3kBT
+m
+2
+� t
+0
+dt′G(t − t′)dG(t − t′)
+dt′
+= v2(0)G2(t) + 3kBT
+m
+�
+1 − G2(t)
+�
+.
+
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