Add files using upload-large-folder tool

-9FJT4oBgHgl3EQfqCwO/content/tmp_files/2301.11602v1.pdf.txt

@@ -0,0 +1,2070 @@
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+ABSTRACT. Given a (finite) simplicial complex, we define its i-th Laplacian polytope as the
+convex hull of the columns of its i-th Laplacian matrix. This extends Laplacian simplices of
+finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of
+these polytopes, we focus on the d-th Laplacian polytope of the boundary of a pd ` 1q-simplex
+Bpσd`1q. If d is odd, then as for graphs, the d-th Laplacian polytope turns out to be a pd ` 1q-
+simplex in this case. If d is even, we show that the d-th Laplacian polytope of Bpσd`1q is
+combinatorially equivalent to a d-dimensional cyclic polytope on d ` 2 vertices. Moreover, we
+provide an explicit regular unimodular triangulation for the d-th Laplacian polytope of Bpσd`1q.
+This enables us to to compute the normalized volume and to show that the h˚-polynomial is
+real-rooted and unimodal, if d is odd and even, respectively.
+1. INTRODUCTION
+Over decades, several lattice polytopes arising from graphs have been studied, extensively.
+Prominent examples include matching polytopes, cut polytopes, edge polytopes, adjacency
+polytopes of several types, among which are symmetric edge polytopes (see e.g., [23, 4, 19,
+21, 28, 10]). Following this line of research, in 2017, Braun and Meyer [6] initiated the study
+of Laplacian simplices that are defined as the convex hull of the columns of the classical
+Laplacian matrix of a simple graph (see also [24, 3]). Since each simple graph can be seen
+as a 1-dimensional simplicial complex and since to each simplicial complex, we can associate
+Laplacian matrices, defined via their boundary maps in simplicial homology, it is natural to
+extend the definition of Laplacian simplices to arbitrary simplicial complexes and their Lapla-
+cians. More precisely, given a simplicial complex ∆ (with a fixed ordering of the vertex set) and
+its i-th Laplacian matrix Lip∆q :“ Bi`1B⊺
+i`1 ` B⊺
+i Bi, we define the i-th Laplacian polytope Ppiq
+∆
+of ∆ as the convex hull of the columns of Lip∆q. Here, Bi and Bi`1 denote boundary maps in
+simplicial homology.
+We initiate the study of Laplacian polytopes by establishing first some general combinatorial
+and geometric properties and then by focusing on a particular case. More precisely, we consider
+the situation that the underlying simplicial complex ∆ is the boundary of the pd ` 1q-simplex,
+denoted by Bpσd`1q, and that we take its highest Laplacian LdpBpσd`1qq. For simplicity, we
+set PBpσd`1q :“ Ppdq
+Bpσd`1q. If d is even, it is easily seen, that, as for graphs, PBpσd`1q is a pd ` 1q-
+simplex. If d is odd, the situation is more complicated. By deriving a complete facet description
+of PBpσd`1q in this case, we are able to show that PBpσd`1q is combinatorially equivalent to a d-
+dimensional cyclic polytope on d `2 vertices.
+It was shown in [6] that Laplacian simplices have unimodal h˚-vectors for certain classes of
+graphs, including trees, odd cycles and complete graphs. Inspired by these results, we study
+properties of the h˚-vectors of general Laplacian polytopes. This is further motivated by the
+general question under which conditions a lattice polytope has a unimodal h˚-vector. It was
+conjectured by Hibi and Ohsugi that this is true for reflexive lattice polytopes that have the
+integer decomposition property (IDP) [27], and, recently, Adiprasito, Papadakis, Petrotou and
+Steinmeyer could confirm this conjecture in the positive [1]. However, it is still mysterious what
+happens if the polytope is not reflexive. We consider this question for the Laplacian polytope
+1
+arXiv:2301.11602v1  [math.CO]  27 Jan 2023
+
+2
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+PBpσd`1q of the boundary of the pd `1q-simplex. Even in this seemingly most simple situation,
+Ppdq
+∆
+turns out to be not reflexive and hence the mentioned results towards unimodality do not
+apply. However, the following result shows that Ppdq
+∆
+has at least the integer decomposition
+property.
+Theorem A. PBpσd`1q has a regular unimodular triangulation for every integer d ě 0.
+We note that, combined with [2, Theorem 1.3], this result implies that the h˚-vector of
+PBpσd`1q is decreasing in its second half which is obviously implied by but weaker than uni-
+modality. The main ingredient for Theorem A is the so-called interior polytope of PBpσd`1q,
+that is defined as the convex hull of the interior lattice points of PBpσd`1q. Indeed, this polytope
+turns out to be reflexive (after translation to the origin) and miraculously, PBpσd`1q happens to
+be the second dilation of it (after translating both polytopes to the origin). Using edgewise
+subdivisions, we provide an explicit construction of a regular unimodular triangulation for the
+interior polytope which then extends to such a triangulation of PBpσd`1q by [18, Theorem 4.8].
+As a byproduct, we can also compute the normalized volume of PBpσd`1q (see Corollary 6.7).
+Theorem A combined with the results on the interior polytope enables us to show the following
+statement:
+Theorem B.
+(a) h˚ ´
+PBpσd`1q;t
+¯
+has only real roots if d P N is odd.
+(b) h˚ ´
+PBpσd`1q
+¯
+is unimodal with peak in the middle for every d P N.
+We note that if d is odd, then the statement in pbq is just an easy consequence of the one in
+paq. We conjecture paq to be true also if d is even.
+The paper is organized as follows. Section 2 provides necessary background on simplicial
+complexes, their Laplacian matrices and lattice polytopes. Section 3 collects basic properties of
+the Laplacian matrix LdpBpσd`1qq of the boundary of a simplex. In Section 4, we introduce the
+i-th Laplacian polytope Ppiq
+∆ of a simplicial complex ∆. Among others, we compute its number
+of vertices (Proposition 4.4), the dimension of PBpσd`1q (Lemma 4.6) and show that PBpσd`1q is
+always simplicial (Theorem 4.8). The goal of Section 5 is to derive a complete facet description
+of PBpσd`1q and to show that it is combinatorially equivalent to a d-dimensional cyclic polytope
+on d `2 vertices if d is even (Theorem 5.3 and Theorem 5.4). Section 6 is devoted to the proofs
+of Theorems A and B, including the construction and study of the interior polytope of PBpσd`1q.
+Finally, in Section 7 we state some open problems and possible future directions.
+2. PRELIMINARIES
+In this section, we provide the necessary background on simplicial complexes, Laplacian
+matrices and polytopes. For more information on these topics we refer to [30, 17, 26, 11, 18, 15].
+Moreover, we assume the reader to have basic knowledge about graphs (see e.g., [12]).
+2.1. Simplicial complexes and Laplacian matrices. Given a finite set V, a simplicial com-
+plex ∆ on vertex set V is a collection of subsets of V that is closed under inclusion. Elements
+of ∆ are called faces and inclusion-wise maximal faces are called facets. The dimension of a
+face F is dimpFq :“ |F| ´ 1 and we use Fip∆q to denote the set of i-dimensional faces of ∆.
+The dimension of ∆ is defined as dimp∆q :“ maxpi : Fip∆q ‰ Hq. If all facets have the same
+dimension, ∆ is called pure. 0-dimensional and 1-dimensional faces of ∆ are called vertices and
+edges, respectively. The sets of vertices and edges of ∆ induce a graph in a natural way, which
+we call the 1-skeleton or graph of ∆. Given a pd ´ 1q-dimensional simplicial complex ∆, its
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+3
+f-vector fp∆q “ pf´1p∆q, f0p∆q,..., fd´1p∆qq is defined by fip∆q :“ |t f P ∆ : dimpFq “ iu| for
+´1 ď i ď d ´1 and its h-vector hp∆q “ ph0p∆q,h1p∆q,...,hdp∆qq by the polynomial identity
+(2.1)
+dÿ
+k“0
+hkp∆qtd´k “
+dÿ
+k“0
+fk´1p∆qpt ´1qd´k.
+The polynomials fp∆;tq :“ řd´1
+i“´1 fip∆qti and hp∆;tq :“ řd
+i“0 hip∆qti are called the f- and h-
+polynomial of ∆, respectively.
+In order to introduce general Laplacian matrices of a simplicial complex ∆, we need to recall
+basic notions from simplicial homology. For this purpose, let ∆ be a pd ´ 1q-dimensional
+simplicial complex on vertex set V and assume that the vertices are ordered. Without loss
+of generality, assume V “ rns “ t1,...,nu endowed with the natural ordering induced by N. We
+denote by Cip∆q the Q-vector space with basis teσ : σ P Fip∆qu and set Cip∆q “ t0u for i ď ´1
+and i ą d ´1. The i-th boundary map is the linear map Bi : Cip∆q Ñ Ci´1p∆q defined by
+(2.2)
+Bipeσq :“
+i`1
+ÿ
+k“1
+p´1qk´1eσztjku,
+where σ “ t j1 ă ¨¨¨ ă ji`1u P Fip∆q. By abuse of notation, we will use Bi to denote both, the map
+and its corresponding matrix. The i-th Laplacian matrix of ∆ is defined as Lip∆q :“ Bi`1B⊺
+i`1 `
+B⊺
+i Bi.
+Note that Lip∆q provides an endomorphism of Cip∆q which depends on the chosen
+ordering of the vertices. We recall that Hip∆;Qq :“ kerpBiq{ImpBi`1q is the i-th (simplicial)
+homology group of ∆.
+To provide an explicit description of Lip∆q, we need some further notation. Faces F,G P Fip∆q
+are called lower adjacent if F XG P Fi´1p∆q. If, additionally, eFXG appears with the same sign
+in BipeFq and BipeGq, we call F XG the similar common lower simplex of F and G. Otherwise,
+F XG is referred to as the dissimilar common lower simplex of F and G. The upper degree of
+F P Fip∆q, denoted degUpFq, is the number of pi`1q-faces of ∆ containing F. We will use the
+following description of Lip∆q from [15, Theorem 3.3.4]:
+Theorem 2.1. Let ∆ be a simplicial complex on vertex set rns, ordered 1 ă ¨¨¨ ă n, and let i P N
+with 0 ď i ď dimp∆q. For F,G P Fip∆q, let ℓF,G denote the entry of Lip∆q in row and column
+corresponding to F and G, respectively. Then, Lip∆q is symmetric. Moreover:
+(i) If i “ 0, then ℓF,G “ degUpFq if F “ G, ℓF,G “ ´1 if F Y G P Fi`1p∆q, and ℓF,G “ 0,
+otherwise.
+(ii) If i ą 0, then
+ℓF,G “
+$
+’
+’
+’
+&
+’
+’
+’
+%
+degUpFq`i`1,
+if F “ G,
+1,
+if F ‰ G, F YG R Fi`1p∆q, F XG P Fi´1p∆q similar
+´1,
+if F ‰ G, F YG R Fi`1p∆q, F XG P Fi´1p∆q dissimilar
+0,
+otherwise.
+Note that if i “ 0 in the previous theorem, then L0p∆q coincides with the classical Laplacian
+matrix of the graph of ∆ (from graph theory).
+2.2. (Lattice) polytopes. A polytope P is the convex hull of finitely many points in Rd. If
+dimP “ k, we call P a k-polytope. A linear inequality a⊺x ď b for a P Rd and b P R is called a
+valid inequality for P if a⊺y ď b for all y P P. A (proper) face of P is a (non-empty) set of the
+form PXtx P Rd : a⊺x “ bu for some valid inequality a⊺x ď b with a ‰ 0. Faces of dimension
+0, dimP´2 and dimP´1 are called vertices, ridges and facets, respectively. We use V pPq and
+FpPq to denote the set of vertices and facets of P, respectively. A valid inequality a⊺x ď b is
+
+4
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+facet-defining if F “ PXtx P Rd : a⊺x “ bu for some F P FpPq. The facet-ridge graph GpPq
+of P is the graph on vertex set FpPq where tF,Gu is an edge if and only if F and G intersect
+in a ridge. If V pPq Ď Zd, P is called a lattice polytope. Two lattice polytopes P, Q Ď Rd
+are unimodular equivalent, denoted as P – Q, if there exist a unimodular matrix U P Rdˆd
+and a vector b P Zd such that U ¨ P ` b “ Q. We use ∆d to denote the standard d-simplex,
+i.e., ∆d “ convtt0u Y tei P Rd : i P rdsuu, where e1,...,ed denote the standard unit vectors.
+A polytope P is simplicial if all of its facets are simplices. The normalized volume of a d-
+dimensional lattice polytope P Ď Rd is given by nvolpPq “ d!¨volpPq, where volpPq denotes the
+usual Euclidean volume. A lattice d-simplex ∆ with normalized volume 1 is called unimodular.
+In this case, ∆ – ∆d. A polytope P is reflexive if P “ tx P Rd : Ax ď 1u for an integral matrix
+A, where 1 denotes the all ones vector. In this case, 0 is the unique interior lattice point of P.
+A triangulation T of a lattice d-polytope P is a subdivision into lattice simplices of dimension
+at most d. We denote the set of vertices in T by V pT q. A triangulation is unimodular if all its
+simplices are. T is called regular if there exists a height function ωP : V pT q Ñ R such that T
+is the projection of the lower envelope of the convex hull of tpv,ωPpvqq : v P V pT qu Ď Rd`1
+to the first d coordinates. We note that every triangulation is in particular a simplicial complex.
+Let P Ď Rd be a lattice d-polytope. Ehrhart [14] proved that the number of lattice points in the
+n-th dilation of P, i.e., |nPXZd| is given by a polynomial EPpnq of degree d in n for all integers
+n ě 0. The Ehrhart series of P is
+ÿ
+ně0
+EPpnqtn “
+h˚pP;tq
+p1´tqd`1 “ h˚
+0pPq`h˚
+1pPqt `¨¨¨`h˚
+s pPqts
+p1´tqd`1
+,
+where h˚pP;tq P Zrts is a polynomial of degree at most d, called h˚-polynomial of P. The vector
+h˚pPq “ ph˚
+0pPq,...,h˚
+s pPqq is called h˚-vector of P. We will often omit P from the notation and
+just write h˚ “ ph˚
+0,...,h˚
+s q if P is clear from the context. By [29, Theorem 2.1], it is well-
+known that h˚
+i pPq is non-negative for all i. If P admits a unimodular triangulation T , then
+h˚pPq “ hpT q [29, Corollary 2.5]. Moreover, if T is a regular unimodular triangulation of P,
+then
+h˚
+tpd`1q{2upPq ě ¨¨¨ ě h˚
+d´1pPq ě h˚
+dpPq
+[2, Theorem 1.3]. It was shown by Hibi in [20] that a lattice d-polytope P Ď Rd is reflexive (up
+to unimodular equivalence) if and only if P contains a unique interior lattice point, and h˚pPq is
+palindromic, i.e., h˚
+i pPq “ h˚
+d´ipPq for all 0 ď i ď td{2u.
+3. LAPLACIAN MATRICES OF BOUNDARIES OF SIMPLICES
+In this section we investigate basic properties of the Laplacian matrix of the boundary of a
+simplex that will be useful for deriving properties of the corresponding Laplacian polytopes in
+Section 4.
+We start with an easy general statement.
+Lemma 3.1. Let ∆ be a d-dimensional simplicial complex. Then
+rankLdp∆q “ fdp∆q´dimQ Hdp∆;Qq.
+Proof. We have the following chain of equalities:
+rankLdp∆q “ rankpB⊺
+dBdq “ fdp∆q´dimQ kerpB⊺
+dBdq “ fdp∆q´dimQ kerpBdq,
+where the last equality follows from the fact that kerpBdq “ kerpB⊺
+dBdq. Since dim∆ “ d, we also
+have Hdp∆;Qq “ kerpBdq, which shows the claim.
+□
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+5
+In the following, we let σd`1 “ 2rd`2s be the pd `1q-simplex and we use Bpσd`1q to denote
+its boundary, i.e., Bpσd`1q “ σd`1ztrd ` 2su. Let Fi “ rd ` 2sztd ` 3 ´ iu for 1 ď i ď d ` 2
+and order the columns and rows of LdpBpσd`1qq according to F1,...,Fd`2. We first provide an
+explicit description of the d-th Laplacian matrix in this case.
+Theorem 3.2. Let ∆ “ Bpσd`1q. Then, Ldp∆q P Zpd`2qˆpd`2q, L0p∆q “
+ˆ
+0
+0
+0
+0
+˙
+and, for
+d ě 1, 1 ď i, j ď d `2, we have
+Ldp∆qi j “
+#
+d `1,
+if i “ j,
+p´1qi` j´1,
+otherwise.
+Proof. Since fdpBpσd`1qq “ d `2, we have Ldp∆q P Zpd`2qˆpd`2q.
+Assume d “ 0. As B0 is the zero map, the statement is immediate.
+Now let d ě 1. Since dim∆ “ d, it follows that degUpFq “ 0 for any d-face F of ∆. Using
+Theorem 2.1 this implies that Ldp∆qii “ d `1 for all 1 ď i ď d `2.
+Now, let i ‰ j.
+Since Ldp∆q is symmetric, we can assume that i ă j.
+Fi and Fj have
+the common lower simplex Fi X Fj “ rd ` 2sztd ` 3 ´ i,d ` 3 ´ ju ‰ H. By Equation (2.2),
+eFiXFj appears with sign p´1qd`3´ j in Bdperd`2sztd`3´iuq and it appears with sign p´1qd`2´i
+in Bdperd`2sztd`3´ juq. These signs coincide, meaning that Fi X Fj is a similar common lower
+simplex of Fi and Fj, if and only if i` j is odd. The claim follows from Theorem 2.1.
+□
+The next lemma will be crucial for determining the dimension of the Laplacian polytope of
+Bpσd`1q in Lemma 4.6.
+Lemma 3.3. Let ∆ “ Bpσd`1q. Then, Ldp∆q has rank d ` 1 and every pd ` 1q-element subset
+of the columns (resp. rows) of Ldp∆q is linearly independent.
+Proof. The first statement follows from Lemma 3.1 and the fact that Hdp∆;Qq “ Q. Let 1 ď i ď
+d `2. Let Ai be the pd `1qˆpd `1q-matrix obtained from Ldp∆q by removing the i-th row and
+column. By definition, Ai “ Ldp∆ztFiuq. Since Hdp∆ztFiuq,Qq “ 0, this matrix has full rank.
+As adding any extra row or column to Ai does not change the rank, the claim follows.
+□
+Lemma 3.4. Let ∆ “ Bpσd`1q. Then
+rank
+ˆ
+Ldp∆q
+1¨¨¨1
+˙
+“
+#
+d `1,
+if d is even,
+d `2,
+if d is odd.
+Proof. First assume that d is even. We define λ “ pλ1,...,λd`2q⊺ P Rd`2 by
+λj “
+#
+0,
+if j is odd,
+2
+d`2,
+if j is even.
+Using Theorem 3.2 it is straight-forward to verify that Ldp∆q ¨ λ “ 1 which, combined with
+Lemma 3.3 shows the claim.
+Now let d be odd and assume by contradiction that rank
+ˆ
+Ldp∆q
+1¨¨¨1
+˙
+ă d `2. Lemma 3.1 and
+Lemma 3.3 imply that rank
+ˆ
+Ldp∆q
+1¨¨¨1
+˙
+“ rankLdp∆q. Hence, there exists λ “ pλ1,...,λd`2q⊺ P
+Rd`2, such that Ldp∆q¨λ “ 1. Let Ldp∆qrd`1s be the matrix obtained from Ldp∆q by deleting
+the last row. Then, we also have Ldp∆qrd`1s ¨ λ “ 1 and it follows from Lemma 3.3 that, up
+
+6
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+to the choice of the last coordinate λd`2, the vector λ is unique. Indeed, a direct computation
+shows that, if λd`2 “ µ for some µ P R, then we must have
+(3.1)
+λ j “
+$
+’
+&
+’
+%
+pd`2q¨µ`1
+d`2
+,
+if j is odd,
+´pd`2q¨µ´1
+d`2
+,
+if j is even.
+However, denoting by rd`2 the last row of Ldp∆q, it holds that rd`2 ¨λ “ 0 ‰ 1, which yields a
+contradiction.
+□
+4. GENERAL PROPERTIES OF LAPLACIAN POLYTOPES
+The goal of this section is to generalize Laplacian simplices – as introduced and studied in
+[6, 24] – that are associated to a graph to arbitrary simplicial complexes and their Laplacian
+matrices. After stating some basic general properties of what we call Laplacian polytopes, we
+focus on boundaries of simplices and their highest Laplacians.
+In the following, given a matrix M, we use convpMq to denote the polytope given by the
+convex hull of the columns of M.
+Definition 4.1. Let ∆ be a d-dimensional simplicial complex on rns, ordered 1 ă ¨¨¨ ă n, and
+let 0 ď k ď d. The k-th Laplacian polytope of ∆ is defined as the convex hull of the columns of
+Lkp∆q, i.e.,
+Ppkq
+∆
+– convpLkp∆qq Ď R fkp∆q.
+We want to remark that the 0-th Laplacian polytope of a simplicial complex coincides with
+the Laplacian simplex of its 1-skeleton, as defined in [6]. The next example shows that different
+orderings of the vertex set of ∆ may result in polytopes of different dimensions.
+Example 4.2. Let G be the 4-cycle on r4s with EpGq “ t12,23,34,14u. If the vertices of G are
+ordered 1 ă 2 ă 3 ă 4, then Pp1q
+G
+is a 3-simplex. If the vertices of G are ordered 1 ă 2 ă 4 ă 3,
+then Pp1q
+G
+is a 2-dimensional rectangle.
+Example 4.3. Pp2q
+Bpσ3q is given by the convex hull of the columns of the following matrix:
+L2pBpσ3qq
+“
+¨
+˚
+˚
+˝
+3
+1
+´1
+1
+1
+3
+1
+´1
+´1
+1
+3
+1
+1
+´1
+1
+3
+˛
+‹‹‚.
+It will follow from Lemma 4.10 that Pp2q
+Bpσ3q is unimodular equivalent to the square in R2 with
+vertices p1,´1q,p´1,1q,p3,1q and p1,3q.
+We start by showing that every column of Lkp∆q yields a vertex of Ppkq
+∆ .
+Proposition 4.4. Let ∆ be a d-dimensional simplical complex and 0 ď k ď d an integer. Then,
+Ppkq
+∆
+has fkp∆q many vertices.
+Proof. Set m :“ fkp∆q and let vpiq denote the i-th column of Lkp∆q. We assume by contradiction
+that there exists 1 ď i ď m, a set S Ď rmsztiu and λj P R with λj ą 0 and ř
+jPS λ j “ 1 such that
+vpiq “ ř
+jPS λ jvpjq. Setting λ j “ 0 if j R S Y tiu and λi “ ´1, we see that λ :“ pλ1,...,λmq⊺ P
+kerpLkp∆qq and hence λ P kerpBkq by [25, Corollary 1.3.1]. Let wpℓq denote the ℓ-th column of
+Bk. If wpiq
+ℓ “ 1, then, since wpjq
+ℓ
+P t´1,0,1u, λ j ą 0 and ř
+jPS λj “ 1, we must have wpjq
+ℓ
+“ 1
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+7
+for all j P S. By the same reasoning, it follows that wpjq
+ℓ
+“ ´1 for all j P S if wpiq
+ℓ “ ´1. As all
+columns of Bk have the same number of non-zero entries, we conclude wpiq “ wpℓq for all ℓ P S,
+which is a contradiction.
+□
+The next proposition gives a sufficient criterion for Ppdim∆q
+∆
+being a simplex.
+Proposition 4.5. Let ∆ be a d-dimensional simplicial complex. If Hdp∆;Qq “ 0, then Ppdim∆q
+∆
+is
+an pfdp∆q´1q-simplex.
+Proof. Lemma 3.1 implies that rankLdp∆q “ fdp∆q. Consequently, the columns of Ldp∆q are
+linearly independent which shows the claim.
+□
+In the following, we focus on the d-th Laplacian polytope of Bpσd`1q. To simplify notation,
+we set PBpσd`1q “ Ppdq
+Bpσd`1q. We use spiq to denote the i-th column of LdpBpσd`1qq. Moreover,
+given a subset S Ď rd ` 2s, we denote by LdpSq the matrix obtained from LdpBpσd`1qq by
+deleting the rows with indices in S.
+Combining Lemma 3.4 and [17, p. 4] the following formula for the dimension of PBpσd`1q is
+immediate.
+Lemma 4.6. Let ∆ “ Bpσd`1q. Then,
+dimP∆ “
+#
+d,
+if d is even,
+d `1,
+if d is odd.
+The previous statement together with Proposition 4.4 allows us to conclude:
+Corollary 4.7. Let d P N with d ě 1 and ∆ “ Bpσd`1q. Then, P∆ has d`2 vertices. In particular,
+P∆ is a pd `1q-simplex, if d is odd.
+Corollary 4.7 trivially implies that PBpσd`1q is a simplicial polytope if d is odd. The same
+statement turns out to be true for d even.
+Theorem 4.8. PBpσd`1q is simplicial for every d P N.
+Proof. Let ∆ “ Bpσd`1q. If d is odd, then the claim is trivially true by Corollary 4.7.
+Now, let d be even. If d “ 0, then P∆ is just the origin and as such simplicial. Let d ě 2 and
+let F be the vertices of a facet of P∆. Combining Lemma 4.6 and Corollary 4.7 it follows that
+d ď |F| ď d `1. If, by contradiction, |F| “ d `1, then Lemma 3.3 implies that the convex hull
+of F is d-dimensional, i.e., F cannot be a facet. Consequently, F is a simplex, which finishes
+the proof.
+□
+As, by Lemma 4.6, the Laplacian polytope of Bpσd`1q is never full-dimensional, our next
+goal is to construct a polytope that is unimodular equivalent to PBpσd`1q and full-dimensional
+with respect to its ambient space. We first need to introduce some further notation.
+We let 1even and 1odd denote the 0´1-vectors in Rd`2 whose even and odd entries are equal
+to 1, respectively. Given these definitions, we can easily compute the affine hull of PBpσd`1q.
+Lemma 4.9. Let d P N with d ě 1 and ∆ “ Bpσd`1q.
+affpP∆q “
+#␣
+x P Rd`2 : p1odd ´1evenq⊺ ¨x “ 0
+(
+,
+if d is odd,
+␣
+x P Rd`2 : 1⊺
+odd ¨x “ 1⊺
+even ¨x “ d`2
+2
+(
+,
+if d is even.
+Proof. By Lemma 4.6, it is enough to show that all vertices of P∆ lie in the specified subspaces
+of dimension d `1 and d, respectively. This can be seen by a direct computation.
+□
+
+8
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+The next lemma gives the desired unimodular equivalent polytopes.
+Lemma 4.10. Let d P N. The polytope PBpσd`1q is unimodular equivalent to convpLdpt1uqq and
+convpLdpt1,2uqq if d is odd and even, respectively.
+Proof. Define matrices A,B P Zpd`2qˆpd`2q as follows
+A “
+¨
+˚
+˚
+˝
+1⊺
+odd ´1⊺
+even
+0
+...
+Ed`1
+0
+˛
+‹‹‚
+and
+B “
+¨
+˚
+˚
+˚
+˚
+˝
+1⊺
+odd
+1⊺
+even
+0
+0
+...
+...
+Ed
+0
+0
+˛
+‹‹‹‹‚
+,
+where Ed and Ed`1 denote identity matrices. Note that A and B are unimodular. By Lemma 4.9,
+we conclude that
+A¨PBpσd`1q “ t0uˆconvpLdpt1uqq,
+if d is odd and
+B¨PBpσd`1q “ tppd `2q{2,pd `2q{2quˆconvpLdpt1,2uqq,
+if d is even. This finishes the proof.
+□
+In the following, we use rPBpσd`1q to denote the unimodular equivalent polytope to PBpσd`1q
+as constructed in Lemma 4.10. By abuse of notation, we will also refer to rPBpσd`1q as d-th
+Laplacian polytope of Bpσd`1q. We also want to remark that, if d is odd, we have the following,
+easy-to-show containment relation: rPBpσd`1q Ď rPBpσd`2q.
+5. THE FACET DESCRIPTION AND THE COMBINATORIAL TYPE OF PBpσd`1q
+While, for odd d, we have already seen that rPBpσd`1q is a simplex, the goal of this section is
+to determine the combinatorial type of PBpσd`1q if d is even. To reach this goal, we will first
+provide a complete irredundant facet description of PBpσd`1q.
+We fix some notation. Let bpℓq denote the vertex of rPBpσd`1q, that is given by the ℓ-th column of
+Ldpt1,2uq. By Theorem 3.2, we have bpℓq
+k
+“ d `1 if k “ ℓ´2 and bpℓq
+k
+“ p´1qk`ℓ´1, otherwise.
+Proposition 5.1. Let d ě 2 be even. Then the following inequalities are facet-defining and
+irredundant for rPBpσd`1q
+(i) 1⊺ ¨x ď d `2,
+(ii) 1⊺
+odd ¨x´xi ď d`2
+2 , where i P rds is even,
+(iii) 1⊺
+even ¨x´xj ď d`2
+2 , where j P rds is odd,
+(iv) xi `xj ě 0, where 1 ď i ă j ď d such that i` j is odd.
+Moreover, the vertices, that attain equality in (i)–(iv), are given by the sets tbpℓq : 3 ď ℓ ď d`2u,
+tbpℓq : ℓ P rd`2szt1,i`2uu, tbpℓq : ℓ P rd`2szt2, j`2uu and tbpℓq : ℓ P rd`2szti`2, j`2uu,
+respectively.
+Proof. We first consider the inequality in (i). If �� P t1,2u, then bpℓq P t´1,1ud with alternating
+entries and hence 1⊺ ¨ bpℓq ă d ` 2. Let 3 ď ℓ ď d ` 2. As d is even, it follows from above
+that bpℓq has one entry equal to d ` 1, d
+2 entries equal to 1 and d
+2 ´ 1 entries equal to ´1. This
+implies 1⊺ ¨bpℓq “ d `2. Hence, the inequality in (i) defines a facet, whose vertices are given by
+tbpℓq : 3 ď ℓ ď d `2u, where we use that the affine hull of the latter set is pd ´1q-dimensional
+by Lemma 3.3.
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+9
+Similarly, it is straightforward to verify that the inequalities in (ii)–(iv) are valid for rPBpσd`1q
+and that the given sets of vertices are the ones attaining equality. As those all differ and their
+affine hulls all have dimension d ´1, it follows that the inequalities are irredundant.
+□
+For the sake of completeness we add the description of the facets of rPBpσd`1q for d odd.
+Remark 5.2. If d ě 3 is odd, using Theorem 3.2 it is not hard to see, that the following
+inequalities are facet-defining for rPBpσd`1q
+(i) 1⊺ ¨x ď d `2,
+(ii) 2¨1⊺
+odd ¨x´xi ď d`2
+2 , where i P rd `1s is even,
+(iii) 2¨1⊺
+odd ¨x`xj ď d `2, where j P rds is odd.
+It is easy to verify that these inequalities are irredundant and as, by Lemma 4.6, rPBpσd`1q is a
+simplex, they provide the complete facet description of rPBpσd`1q. We omit an explicit proof since
+this description will not be needed.
+We state the first main result of this section.
+Theorem 5.3. For d even, rPBpσd`1q is completely described by the inequalities in Proposition 5.1.
+Moreover, this description is irredundant. In particular, rPBpσd`1q has pd`2q2
+4
+many facets.
+Proof. We let Ă
+F denote the set of facets of rPBpσd`1q, provided by Proposition 5.1 and we write
+G Ă
+F for the subgraph of the facet-ridge graph of rPBpσd`1q that is induced on vertex set Ă
+F. It
+follows from Theorem 4.8, that the facet-ridge graph of rPBpσd`1q is d-regular and connected.
+Since any d-regular subgraph does not have a proper d-regular subgraph, for the first statement,
+it suffices to show that G Ă
+F is d-regular.
+Since G Ă
+F is a subgraph of GprPBpσd`1qq, its maximal degree is at most d. Hence, to show the
+claim, it suffices to show that |EpG Ă
+Fq| “
+d¨|VpG Ă
+F q|
+2
+.
+We first count the vertices of G. Using Proposition 5.1, we get that
+(5.1)
+|VpG Ă
+Fq| “ 1` d
+2 ` d
+2 `
+ˆd
+2
+˙2
+“ pd `2q2
+4
+,
+Here, the last term in the middle comes from the fact that the inequalities in (iv) are indexed by
+sets ti, ju where i P t2ℓ : ℓ P rd
+2su and j P t2ℓ´1 : ℓ P rd
+2su.
+It remains to count the number of edges of G Ă
+F. In the following, we identify a facet in Ă
+F
+with its set of vertices. Given this, we use the following short hand notation for the different
+types of facets in Ă
+F.
+(i) F “ tbpℓq : 3 ď ℓ ď d `2u;
+(ii) Ei “ tbpℓq : ℓ P rd `2szt1,i`2uu, where i P rds is even;
+(iii) Oj “ tbpℓq : ℓ P rd `2szt2, j `2uu, where j P rds is odd;
+(iv) Fk,m “ tbpℓq : ℓ P rd `2sztk `2,m`2uu for 1 ď k ă m ď d such that k `m is odd.
+We immediately get that
+(a) |F XEi| “ |F XOj| “ d ´1 for all even i P rds and all odd j P rds;
+(b) |F XFk,m| “ d ´2 for all 1 ď k ă m ď d;
+(c) |Ei XE j| “ d ´1 for all odd i, j P rds with i ‰ j;
+(d) |Ei XO j| “ d ´2 for all even i P rds and all odd j P rds;
+(e) |Ei XFk,m| “ d ´1 iff i P tk,mu, i even, k `m odd, and |Ei XFk,m| “ d ´2, otherwise;
+(f) |Oi XO j| “ d ´1 for all even i, j P rds with i ‰ j;
+
+10
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+(g) |O j XFk,m| “ d ´1 iff j P tk,mu, j odd, k `m odd, and |Oj XFk,m| “ d ´2, otherwise.
+(h) |Fi, j XFk,m| “ d ´1 iff |ti, j,k,mu| “ 3 and |Fi,j XFk,m| “ d ´2, otherwise.
+Since edges of G Ă
+F are given by tuples of facets intersecting in d ´1 vertices, we get d edges in
+(a), 0 edges in (b) and (d),
+`d{2
+2
+˘
+edges in each of (c) and (f),
+`d
+2
+˘2 edges in each of (e) and (g)
+and 2¨ d
+2 ¨
+`d{2
+2
+˘
+edges in (h). This yields
+|EpG Ă
+Fq| “ d `2¨
+ˆd{2
+2
+˙
+`2¨
+ˆd
+2
+˙2
+`d ¨
+ˆd{2
+2
+˙
+“ dpd `2q2
+8
+“
+d ¨|VpG Ă
+Fq|
+2
+.
+It follows that G Ă
+F is d-regular. The In particular-statement follows from (5.1).
+□
+The previous theorem allows us to determine the combinatorial type of rPBpσd`1q if d is even. It
+is well-known (see e.g., [17, Section 6.1]) that there are only finitely many combinatorial types
+of simplicial d-polytopes with d ` 2 vertices. More precisely, any simplicial d-polytope with
+d ` 2 vertices is obtained as the convex hull of a d-simplex T d and a vertex v that is beyond
+k facets of T d, where 1 ď k ď d ´ 1. It is easily seen that the combinatorial type of such a
+polytope only depends on k. Following Gr¨unbaum, we use T d
+k to denote the corresponding
+combinatorial type. Given that we know the number of facets of rPBpσd`1q (see Theorem 5.3), we
+can immediately determine its combinatorial type.
+Theorem 5.4. Let d be even. Then rPBpσd`1q is of combinatorial type T d
+d
+2 . In particular, rPBpσd`1q
+is combinatorially equivalent to a d-dimensional cyclic polytope on d `2 vertices.
+Proof. By Theorem 5.3, rPBpσd`1q has pd`2q2
+4
+facets. Using [17, Section 6.1, Theorem 2], it
+follows that this number has to be equal to
+ˆd `2
+2
+˙
+´
+ˆk `1
+2
+˙
+´
+ˆd `1´k
+2
+˙
+,
+where rPBpσd`1q is of combinatorial type T d
+k . Solving for k yields k “ d
+2. The second statement
+follows from [17, Section 6.1, Theorem 1].
+□
+We remark that from the previous theorem, we also get a precise formula for the f- and
+h-vector of rPBpσd`1q (see, e.g., [17]).
+Remark 5.5. Given the precise description of the facets from the proof of Theorem 5.3, it is not
+hard to write down a shelling order for rPBpσd`1q (d even). Namely, one particular shelling is
+given by
+F,E2,E4,...,Ed,O1,O3,...,Od´1,F1,2,F1,4,...,F1,d,F2,3,F2,5,...,F2,d´1,...,Fd´1,d.
+6. REGULAR UNIMODULAR TRIANGULATIONS AND h˚-VECTORS
+This section is divided into two parts. The goal of the first is to prove Theorem A, namely,
+that rPBpσd`1q admits a regular unimodular triangulation .
+As a byproduct we will also be
+able to compute the normalized volume rPBpσd`1q. In the second part, we provide the proof
+of Theorem B.
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+11
+FIGURE 1. rPBpσ3q and its interior polytope QBpσ3q translated to the origin.
+6.1. Triangulations through interior polytopes. If d is even, one of our main tools towards
+the formulated goal is the so-called interior polytope QBpσd`1q of rPBpσd`1q, defined as follows:
+QBpσd`1q “ conv
+´
+rPBpσd`1qzB
+´
+rPBpσd`1q
+¯
+XZd¯
+.
+Figure 1 depicts rPBpσ3q and its interior polytope QBpσ3q, both translated to the origin.
+Surprisingly, it turns out that PBpσd`1q and its interior polytope are combinatorially equivalent.
+More precisely, the following stronger statement is true:
+Theorem 6.1. Let d P N be even. Then the following statements hold:
+(a) The complete and irredundant facet description of QBpσd`1q is given by:
+(i) 1⊺ ¨x ď d `1,
+(ii) 1⊺
+odd ¨x´xi ď d
+2 for even i P rds,
+(iii) 1⊺
+even ¨x´xj ď d
+2 for odd j P rds,
+(iv) xi `xj ě 1 for 1 ď i ă j ď d such that i` j is odd.
+(b) QBpσd`1q ´1 is reflexive. In particular, 1 is the unique interior lattice point of QBpσd`1q.
+(c) 2¨
+´
+QBpσd`1q ´1
+¯
+“ rPBpσd`1q ´1.
+Proof. We let Q “ rPBpσd`1q ´1. The vertices of Q are given by upℓq :“ bpℓq `1 for 1 ď ℓ ď d `2.
+It is immediate that all coordinates of upℓq are divisible by 2. Hence, 1
+2Q is a lattice polytope.
+Using Theorem 5.3, it follows that the facets of 1
+2Q are given by
+‚ 1⊺ ¨x ď 1,
+‚ 1⊺
+odd ¨x´xi ď 1 for even i P rds,
+‚ 1⊺
+even ¨x´xj ď 1 for odd j P rds,
+‚ xi `xj ě ´1 for 1 ď i ă j ď d such that i` j is odd,
+which shows that 1
+2Q is reflexive. It remains to show that 1
+2Q`1 “ QBpσd`1q. Since 1
+2Q`1 is
+a lattice polytope, it follows that 1
+2Q ` 1 Ď QBpσd`1q. For the other inclusion it suffices to note
+that the facets of 1
+2Q and rPBpσd`1q are parallel and that they have distance
+1
+?
+d,
+?
+2
+?
+d`2,
+?
+2
+?
+d`2 and
+1
+?
+2 to each other for facets of the form in (i), (ii), (iii) and (iv), respectively. This implies that
+there is no lattice point in rPBpσd`1qzpp1
+2Q`1qYB rPBpσd`1qq and hence QBpσd`1q Ď 1
+2Q`1.
+□
+
+m
+312
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+We define vectors cp1q,...,cpd`2q P Rd
+by cpℓq
+k
+“ d`2
+2
+if k “ ℓ ´ 2 and cpℓq
+k
+“
+maxp0,p´1qk`ℓ´1q, otherwise. Combining Proposition 5.1 and Theorem 6.1 (c), we get the
+following description of the vertices of QBpσd`1q and its facets.
+Corollary 6.2. The vertices of QBpσd`1q are the vectors cp1q,...,cpd`2q. Moreover, the vertices,
+that attain equality in Theorem 6.1 (i)–(iv), are given by the sets tcpℓq : 3 ď ℓ ď d ` 2u,
+tcpℓq : ℓ P rd `2szt1,i`2uu, tcpℓq : ℓ P rd `2szt2, j`2uu and tcpℓq : ℓ P rd `2szti`2, j`2uu,
+respectively.
+We now recall several definitions and facts concerning regular unimodular triangulations (see
+[18, Subsection 2.3.2.] for more on these topics).
+Given full-dimensional polytopes P Ď Rd and P1 Ď Rd1 of positive dimension, their join P˚P1
+is the pd `d1 `1q-dimensional polytope defined by
+Pˆt0d1uˆt0u Y t0duˆP1 ˆt1u.
+The next statement, which is well-known, will be crucial for the construction of a regular
+unimodular triangulation of rPBpσd`1q.
+Theorem 6.3. Let P Ď Rd and P1 Ď Rd1 be polytopes of dimension d and d1, respectively. Let
+S “ tSi : i P rnsu and S1 “ tS1
+j : j P rmsu be triangulations of P and P1, respectively, where Si
+and S1
+j denote the full-dimensional cells. If both S and S1 are regular and unimodular, then
+T “
+␣
+Si ˚S1
+j : i P rns, j P rms
+(
+is a regular unimodular triangulation of P˚P1.
+We will also make use of the following statement, see [18, Theorem 4.8].
+Theorem 6.4. If P has a (regular) unimodular triangulation T , then so has any dilation cP,
+where c is a positive integer.
+A well-studied subdivision, which is related to the Veronese construction in algebra but also
+appears in topology [7, 13, 16, 8], is the so-called rth edgewise subdivision of a simplicial
+complex. In the following, we review this definition for the special case that ∆ is the pn ´
+1q-dimensional simplex on vertex set V “ te1,e2,...,enu Ď Rn. For a positive integer r, let
+Ωr “ tpi1,...,inq P Nn : i1 ` i2 ` ¨¨¨ ` in “ ru denote the set of lattice points r∆ X Zn. For
+x “ px1,...,xnq P Zn, we define
+ϕpxq :“ px1,x1 `x2,...,x1 `¨¨¨`xnq P Rn.
+The rth edgewise subdivision of ∆ is the simplicial complex esdrp∆q on vertex set Ωr, for which
+F Ď Ωr is a face if for all x,y P F
+ϕpxq´ϕpyq P t0,1un
+or
+ϕpyq´ϕpxq P t0,1un.
+By definition, the geometric realization of the rth edgewise subdivision of ∆ gives a lattice
+triangulation of r∆. It is known that this triangulation is regular [8, Proposition 6.4.], which
+is also unimodular since all maximal simplices have normalized volume 1. In the following,
+we will use esdrp∆q to denote both, the triangulation as a simplicial complex and its geometric
+realization. Given any pn ´ 1q-dimensional unimodular simplex Γ Ď Rn, esdrp∆q, naturally
+induces a regular unimodular triangulation of rΓ (by applying the corresponding unimodular
+transformation).
+Slightly abusing notation, we will refer to this triangulation as edgewise
+subdivision of Γ or even of rΓ, denoted esdrpΓq. Moreover, the restriction of esdrpΓq to any
+face F P Γ equals esdrpFq as a simplicial complex and as geometric realization.
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+13
+Example 6.5. Figure 2 depicts the 3rd edgewise subdivision of the 2-dimensional simplex ∆2 :“
+convpp0,0q,p1,0q,p0,1qq as triangulation of 3∆2. The vertex labels correspond to the vertex
+labels from the original definition and not the lattice points.
+FIGURE 2. The triangulation of 3¨∆2 given by esd3p∆2q.
+We now outline our strategy to show that rPBpσd`1q has a regular unimodular triangulation.
+We first prove that rPBpσd`1q and the facets of QBpσd`1q, if d is odd and even, respectively, are
+unimodular equivalent to joins of dilated standard simplices. If d is odd and even, we can hence
+triangulate rPBpσd`1q and facets of QBpσd`1q as join of edgewise subdivisions. If d is odd, the claim
+follows by Theorem 6.3. If d is even, we next show that these triangulations are consistent on
+intersections of facets. By coning with 1, we get a unimodular triangulation of QBpσd`1q (see
+Theorem 6.1 (d)) and hence of rPBpσd`1q by Theorem 6.4. The regularity follows by using that
+the triangulation is regular on single facets and that each facet is triangulated in the same way.
+The next statement yields the first step in the outlined strategy.
+Proposition 6.6.
+(a) Let d ě 2 be even and F P F
+´
+QBpσd`1q
+¯
+. Then
+F –
+ˆd `2
+2
+∆ d´2
+2 ´1 d´2
+2
+˙
+˚
+ˆd `2
+2
+∆ d´2
+2 ´1 d´2
+2
+˙
+.
+(b) Let d ě 1 be an odd integer. Then
+rPBpσd`1q –
+`
+pd `2q∆ d`1
+2 ´2¨1
+˘
+˚
+`
+pd `2q∆ d´1
+2 ´2¨1
+˘
+.
+Proof. The proof of (a) is divided into four cases, according to the four classes of facets from
+Theorem 6.1 (a).
+Let F “ tx P Rd : 1⊺ ¨ x ď d ` 1u. By Corollary 6.2, the vertices of F are cp3q,...,cpd`2q.
+We now consider the matrix A, whose ℓ-th column equals cp2ℓ`1q if 1 ď ℓ ď d
+2 and cp2ℓ`2´dq if
+d
+2 `1 ď ℓ ď d. If we reorder the rows of A, by taking first the rows with odd index and then the
+ones with even index, increasingly, we obtain a matrix S, which looks as follows:
+S “
+˜ d`2
+2 ¨E d
+2
+1 d
+2 ˆ d
+2
+1 d
+2 ˆ d
+2
+d`2
+2 ¨E d
+2
+¸
+,
+
+4
+(0, 0, 3)
+(1, 0,2)
+(0,1,2)
+(2, 0, 1)
+(1,1,1
+(0, 2, 1)
+(3, 0, 0)
+(2,1,0)
+(1, 2, 0)
+(0,3, 0)
+-114
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+where 1kˆk denotes the pk ˆ kq-matrix with all entries equal to 1. Clearly, F – convpSq. Let
+E1
+k P Zpk´1qˆk be the pk ˆkq-identity matrix with its first row removed and let
+U “
+¨
+˚
+˚
+˚
+˝
+E1
+d
+2
+0 d´2
+2 ˆ d
+2
+0 d´2
+2 ˆ d
+2
+E1
+d
+2
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+˛
+‹‹‹‚P Zdˆd.
+It is easily seen that U is unimodular and a direct computation shows that
+U ¨pS´1dˆdq “
+¨
+˚
+˚
+˚
+˚
+˝
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0 d´2
+2 ˆ d
+2
+0 d´2
+2 ˆ d
+2
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+˛
+‹‹‹‹‚
+,
+where 0kˆk denotes the pk ˆkq-matrix with all entries equal to 0 and M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+denotes the matrix whose columns are the vertices of d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2 in the obvious order.
+Since F – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates
+and by the definition of the join. We also note that the vertices of F corresponding to the vertices
+of the dilated simplices are tcp2ℓ`1q : 1 ď ℓ ď d
+2u and tcp2ℓq : 2 ď ℓ ď d
+2 `1u.
+Similarly, one can show that for the facets defined by
+‚ 1⊺
+odd ¨x´xi ď d
+2, where i P rds is even,
+‚ 1⊺
+even ¨x´xj ď d
+2, where j P rds is odd,
+‚ xi `xj ě 1 for 1 ď i ă j ď d such that i` j is odd,
+respectively, the vertices
+‚ tcp2ℓ`1q : 1 ď ℓ ď d
+2u and tcp2ℓq : 1 ď ℓ ď d
+2 `1,ℓ ‰ i`2
+2 u,
+‚ tcp2ℓ`1q : 0 ď ℓ ď d
+2,ℓ ‰ j`1
+2 u and tcp2ℓq : 2 ď ℓ ď d
+2 `1,ℓ ‰ i`2
+2 u,
+‚ tcp2ℓ`1q : 0 ď ℓ ď d
+2,ℓ ‰ j`1
+2 u and tcp2ℓq : 1 ď ℓ ď d
+2 `1,ℓ ‰ i`2
+2 u,
+respectively, correspond to the vertices of the dilated simplices. The rather technical proofs can
+be found in the appendix. Similarly, (b) will be shown in the appendix.
+□
+We recall and prove Theorem A.
+Theorem A. PBpσd`1q has a regular unimodular triangulation for every integer d ě 0.
+Proof. Since rPBpσd`1q and PBpσd`1q are unimodular equivalent, it suffices to show the statement
+for rPBpσd`1q. First assume that d is odd. By Proposition 6.6 (b), we know that
+rPBpσd`1q –
+`
+pd `2q∆ d`1
+2 ´2¨1
+˘
+˚
+`
+pd `2q∆ d´1
+2 ´2¨1
+˘
+.
+Since the pd ` 2qnd edgewise subdivision is a regular unimodular triangulation of the pd `
+2qnd dilation of any unimodular simplex (as well as of any translation), we conclude with
+Theorem 6.3 that rPBpσd`1q has a regular unimodular triangulation.
+Next assume that d is even. If d “ 0, rPBpσd`1q is just a point and there is nothing to show.
+Let d ě 2. We construct a regular unimodular triangulation of the interior polytope QBpσd`1q.
+By Proposition 6.6 every facet F of QBpσd`1q is unimodular equivalent to
+(6.1)
+ˆd `2
+2
+∆ d´2
+2 ´1 d´2
+2
+˙
+˚
+ˆd `2
+2
+∆ d´2
+2 ´1 d´2
+2
+˙
+.
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+15
+By the same reasoning as for d odd, we can triangulate F as join of edgewise subdivisions of
+unimodular simplices. In this way, we obtain regular unimodular triangulations of each facet
+of QBpσd`1q. We now show that the union of these triangulations, yields a triangulation of the
+boundary of QBpσd`1q. For this aim, let F and G be facets of QBpσd`1q and let T pFq and T pGq
+be the considered triangulations. Let us further denote by Fi and Gi, where i P r2s, the vertex
+sets corresponding to the vertex sets of the dilated (and translated) simplices in (6.1). It follows
+from the end of the proof of Proposition 6.6 that (after possible renumbering)
+pF1 YF2qXpG1 YG2q “ pF1 XF2qYpG1 XG2q.
+This directly yields that the restrictions of T pFq and T pGq to F X G coincide: Indeed, they
+are given as the join of the edgewise subdivisions of the dilated (and translated) simplices on
+vertex sets F1 XF2 and G1 XG2. This shows that the union of the triangulations of the facets is
+indeed a triangulation of the boundary of QBpσd`1q, which is, in particular, unimodular. Since,
+by Theorem 6.1 (b), QBpσd`1q ´1 is reflexive, we can extend this triangulation to a unimodular
+triangulation of QBpσd`1q by coning over the unique interior lattice point 1. In the following, we
+call this triangulation T .
+It remains to show that T is a regular triangulation. The previous paragraph implies that
+the induced triangulations on facets QBpσd`1q are all regular and unimodular equivalent to each
+other. In particular, there exists a simultaneous lifting function h yielding the triangulation of
+an arbitrary facet. Fix a facet F and let T pFq be the induced triangulation on F. Since F is a
+simplex, we can assume that hpvq “ 1 for any vertex v P F. Moreover, for any lattice point u in F,
+that is not a vertex, we have hpuq ă 1, since otherwise u would not be a vertex of T pFq. Hence,
+there exists a non-negative function g, whose values are bounded by 1, that vanishes on the
+vertices of F such that h “ 1´g. Moreover, for any ε ą 0, hε “ 1´εg is also a lifting function
+for F yielding T pFq. Finally, ignoring 1 and lifting all other lattice points in QBpσd`1q according
+to the simultaneous lifting function hε, gives a lifting function such that the projection of the
+lower envelope yields T on the boundary of QBpσd`1q and potentially additional faces in the
+interior. Lifting 1 at height 0, gives a lifting of all lattice points of QBpσd`1q. If ε is sufficiently
+small, one can guarantee that the triangulation obtained as the lower envelope is the cone with
+1 over the boundary of the previous triangulation (ignoring 1) since potential interior faces that
+we had seen before, do no longer lie in the lower envelope.
+The claim follows by Theorem 6.1 (c) and Theorem 6.4.
+□
+Analyzing the proof of Theorem A, we can compute the normalized volume of PBpσd`1q:
+Corollary 6.7. The normalized volume of PBpσd`1q is pd `2qd.
+Proof. We compute the normalized volume of rPBpσd`1q, which equals the one of PBpσd`1q, by
+counting the number of maximal simplices in the unimodular triangulation T constructed in
+the proof of Theorem A.
+First assume that d is odd. We have seen that T is unimodular equivalent to
+esdd`2
+´
+∆ d`1
+2
+¯
+˚esdd`2
+´
+∆ d´1
+2
+¯
+.
+Since the rth edgewise subdivision of an m-simplex, has rm maximal simplices, it follows that
+the number of maximal simplices in the constructed unimodular triangulation of rPBpσd`1q equals
+pd `2q
+d´1
+2 ¨pd `2q
+d`1
+2 “ pd `2qd.
+Let d be even. We first compute the normalized volume of QBpσd`1q. Combining Theorem 5.3
+and Theorem 6.1, it follows that QBpσd`1q has exactly pd`2q2
+4
+facets. By the proof of Theorem A
+
+16
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+each of these has a unimodular triangulation that is unimodular equivalent to
+esd d`2
+2
+´
+∆ d´2
+2
+¯
+˚esd d`2
+2
+´
+∆ d´2
+2
+¯
+.
+As in the case that d is odd, we conclude that each facet is triangulated into
+`d`2
+2
+˘ d´2
+2 ¨
+`d`2
+2
+˘ d´2
+2
+“
+`d`2
+2
+˘d´2 many maximal simplices and hence QBpσd`1q has normalized volume pd`2qd
+2d
+. Since,
+by Theorem 6.1 (c), rPBpσd`1q `1 “ 2¨QBpσd`1q, the claim follows.
+□
+6.2. Unimodality and real-rootedness. The goal of this subsection is to prove Theorem B.
+If d is even, then by the proof of Theorem A, QBpσd`1q has a regular unimodular triangulation.
+Since it is also reflexive (after translation) by Theorem 6.1 (b), the next statement is immediate
+from [9, Theorem 1] (see also [2, Theorem 1.3]):
+Lemma 6.8. Let d be an even positive integer. Then h˚pQBpσd`1qq is symmetric and unimodal.
+To show unimodality of h˚prPBpσd`1qq, if d is even, we need to analyze the change of the h˚-
+vector under the second dilation of a polytope (cf., Theorem 6.1 (c)). Given a d-dimensional
+lattice polytope P, it follows, e.g., from [7, Theorem 1.1] (see also [5, 22]) that
+(6.2)
+h˚
+i p2Pq “
+dÿ
+j“0
+ˆd `1
+2i´ j
+˙
+h˚
+jpPq.
+We need the following technical but crucial lemma.
+Lemma 6.9. Let i P N and rj :“
+` d`1
+2i`2´ j
+˘
+´
+`d`1
+2i´ j
+˘
+. Then for k P N, we have
+´rr2i`2´ d`3
+2 s´k “ rt2i`2´ d`3
+2 u`k.
+Proof. We set aj “
+` d`1
+2i`2´ j
+˘
+and bj “
+`d`1
+2i´ j
+˘
+. The claim follows if both
+ar2i`2´ d`3
+2 s´k “ bt2i`2´ d`3
+2 u`k
+and
+br2i`2´ d`3
+2 s´k “ at2i`2´ d`3
+2 u`k
+hold. Due to the symmetry of the binomial coefficient it suffices to show that
+(i) p2i`2´r2i`2´ d`3
+2 s`kq`p2i´t2i`2´ d`3
+2 u´kq “ d `1
+(ii) p2i´r2i`2´ d`3
+2 s`kq`p2i`2´t2i`2´ d`3
+2 u´kq “ d `1.
+It is obvious that (i) and (ii) are equivalent. The claim follows from direct computations.
+□
+The next statement will be the key ingredient to show that h˚prPBpσd`1qq is unimodal.
+Proposition 6.10. Let b “ pb0,...,bdq be a symmetric and unimodal sequence of non-negative
+reals. Let c “ pc0,...,cdq be defined by
+ci “
+dÿ
+j“0
+ˆd `1
+2i´ j
+˙
+b j.
+Then,
+c0 ď c1 ď ¨¨¨ ď ct d`1
+2 u.
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+17
+Proof. We define r j as in Lemma 6.9. Note that rj ě 0 if and only if j ě 2i ` 2 ´ d`3
+2 . For
+0 ď i ă d`1
+2 , we have
+ci`1 ´ci “
+dÿ
+j“0
+„ˆ
+d `1
+2i`2´ j
+˙
+´
+ˆd `1
+2i´ j
+˙ȷ
+b j “
+dÿ
+j“0
+r jbj
+“
+2p2i`2´ d`3
+2 q
+ÿ
+j“0
+r jbj `
+dÿ
+j“2p2i`2´ d`3
+2 q`1
+r jbj
+“
+r2i`2´ d`3
+2 s
+ÿ
+j“1
+rt2i`2´ d`3
+2 u` j
+´
+bt2i`2´ d`3
+2 u` j ´br2i`2´ d`3
+2 s´ j
+¯
+`r2i`2´ d`3
+2 b2i`2´ d`3
+2 `
+dÿ
+j“2p2i`2´ d`3
+2 q`1
+r jbj,
+where for the last equality, we use Lemma 6.9 and we set `r2i`2´ d`3
+2 b2i`2´ d`3
+2
+“ 0 if d is
+even. Since b j ě 0 and rj ě 0 for j ě 2i` 2´ d`3
+2 , it follows that the single summand and the
+sum in the last line of the above computation are both non-negative. Concerning the first sum,
+the coefficients r2i`2´ d`3
+2 ` j are non-negative and therefore, in order to show non-negativity of
+ci`1 ´ci, it suffices to show that for 1 ď j ď r2i`2´ d`3
+2 s, we have
+bt2i`2´ d`3
+2 u` j ě br2i`2´ d`3
+2 s´ j.
+This directly follows from the unimodality and symmetry of the sequence b if 2i`2´ d`3
+2 ` j ď
+d`1
+2 . Assume 2i`2´ d`3
+2 ` j ą d`1
+2 . Since i ď d
+2, we have
+d `1
+2
+ă 2i`2´ d `3
+2
+` j ď d `2´ d `3
+2
+` j “ d `1
+2
+` j ď
+Zd `1
+2
+^
+` j.
+Using that b is symmetric and unimodal it follows that
+bt2i`2´ d`3
+2 u` j ě bt d`1
+2 u` j “ bd´t d
+2u´ j ě br2i`2´ d`3
+2 s´ j.
+This shows the claim.
+□
+We now recall and prove Theorem B:
+Theorem B.
+(a) h˚ ´
+PBpσd`1q;t
+¯
+has only real roots if d P N is odd.
+(b) h˚ ´
+PBpσd`1q
+¯
+is unimodal with peak in the middle for every d P N.
+Proof. Since PBpσd`1q has a regular unimodular triangulation T by Theorem A, we have
+h˚pPBpσd`1qq “ hpT q. If d is odd, such a triangulation is given by
+esdd`2
+´
+∆ d`1
+2
+¯
+˚esdd`2
+´
+∆ d´1
+2
+¯
+and its h-polynomial equals h
+´
+esdd`2
+´
+∆ d`1
+2
+¯
+;t
+¯
+¨ h
+´
+esdd`2
+´
+∆ d´1
+2
+¯
+;t
+¯
+. Since both factors
+are real-rooted by [22, Corollary 4.4], so is h˚ ´
+PBpσd`1q;t
+¯
+.
+
+18
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+Suppose that d is even. Combining Lemma 6.8, (6.2) and Proposition 6.10, we get that
+h˚pPBpσd`1qq is increasing up to the middle, i.e.,
+h˚
+0pPBpσd`1qq ď h˚
+1pPBpσd`1qq ď ¨¨¨ ď h˚
+d
+2 pPBpσd`1qq.
+Since, by Theorem A, PBpσd`1q has a regular unimodular triangulation it follows by [2, Theorem
+1.3] that h˚pPBpσd`1qq is decreasing beyond the middle, i.e.,
+h˚
+d
+2 pPBpσd`1qq ě ¨¨¨ ě h˚
+dpPBpσd`1qq.
+The claim follows.
+□
+We would like to remark that even though the interior polytope QBpσd`1q has a symmteric
+h˚-vector, this is not true for PBpσd`1q.
+7. OPEN PROBLEMS
+We end this article with some obvious directions for future research.
+We have initiated the study of Laplacian polytopes Ppiq
+∆ by studying the special case that ∆ is
+the boundary of a pd ` 1)-simplex and i “ d. It is therefore natural to consider the following
+very general problem.
+Problem 7.1. Study geometric and combinatorial properties of Ppiq
+∆
+for (classes of) simplicial
+complexes and general 0 ď i ď dim∆. In particular: What is the normalized volume? When
+do these polytopes have a regular unimodular triangulation? What properties do the h˚-vector
+and the h˚-polynomial have?
+In view of Proposition 4.5, a good starting point might be to study Ppdq
+∆
+for simplicial d-balls,
+since in this case we already know that Ppdq
+∆
+is a simplex. As part of this problem, it might be
+useful to consider how Laplacian polytopes change under certain operations on the simplicial
+complex, e.g., deletion/contraction of vertices, taking links, connected sums, joins. We want to
+remark that for i “ 1 we get Laplacian simplices as studied in [6] and [24].
+We have shown that PBpσd`1q has a regular unimodular triangulation by explicitly constructing
+one. However, for more general classes of simplicial complexes, a better approach might be to
+compute a Gr¨obner basis of the toric ideal. This gives rise to the following problem whose
+solution would also contribute to Problem 7.1:
+Problem 7.2. Describe a Gr¨obner basis of the toric ideal of Ppiq
+∆ in terms of the combinatorics
+of ∆. When does there exist a squarefree Gr¨obner basis (giving rise to a regular unimodular
+triangulation)?
+We want to emphasize that the Laplacian polytope depends on the ordering of the vertices of
+∆ (see Example 4.2). It is therefore natural to ask the following question:
+Question 7.3. Which orderings yield (up to unimodular or combinatorial equivalence) the same
+Laplacian polytope? How many equivalence classes are there?
+Apart from these more general problems, there are several open questions that are directly
+related to our results. In Corollary 6.7, we have computed the normalized volume of PBpσd`1q
+explicitly and thereby have obtained a precise formula for the sum of the h˚-vector entries.
+Using the explicit regular unimodular triangulation from Theorem A and inclusion-exclusion
+we can also express the h˚-polynomial as alternating sum, where all summands are products
+of h˚-polynomials of edgewise subdivisions of dilated simplices of varying dimension. Note
+that for d odd, we only have one summand. However, this does not yield a direct combinatorial
+interpretation of the entries of the h˚-vector. We therefore propose the following problem:
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+19
+Problem 7.4. Find a combinatorial interpretation of the entries of the h˚-vector of PBpσd`1q (see
+Table 1 for the h˚-vectors if 1 ď d ď 8).
+d
+h˚ `
+PBσd`1
+˘
+1
+p1,2,0q
+2
+p1,10,5q
+3
+p1,22,78,24,0q
+4
+p1,131,726,419,19q
+5
+p1,149,4049,8558,3750,300,0q
+6
+p1,1478,38179,126372,85623,10422,69q
+7
+p1,926,157566,1135846,2188310,1150800,145600,3920,0q
+8
+p1,17617,1581403,6864069,43252570,31729319,6314903,239867,251q
+TABLE 1. The h˚-vectors of PBσd`1 for d “ 1,...,8.
+Finally, in view of Theorem B (a), we have the following conjecture:
+Conjecture 7.5. Let d be even. Then, h˚ ´
+PBpσd`1q;x
+¯
+is real-rooted.
+We have verified this conjecture computationally up to d “ 10. For this problem, we suspect
+that an approach via interlacing sequences might be helpful, but we have not been able to carry
+it out so far.
+8. APPENDIX
+We provide the missing parts of the proof of Proposition 6.6. We recall some notation. We
+denote by E1
+k P Zpk´1qˆk the pk ˆkq-identity matrix with its first row removed and by 1mˆn and
+0mˆn the pm ˆ nq-matrices whose entries are all equal to 1 and 0, respectively. Moreover, we
+denote by M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+the matrix whose columns are the vertices of d`2
+2 ∆ d´2
+2 ´
+1 d´2
+2 ˆ d
+2 in the obvious order.
+Proof of Proposition 6.6 (a). Let d ě 2 and for a fixed even integer i P rds, consider the facet
+F “ tx P Rd : 1⊺
+odd ¨x´xi ď d
+2u of QBpσd`1q. By Corollary 6.2, the vertices of F are tcpℓq : ℓ P
+rd ` 2szt1,i ` 2uu. We now consider the matrix B P Zdˆd whose ℓ-th column equals cp2ℓ`1q if
+1 ď ℓ ď d
+2 and cp2ℓ´dq if d
+2 ` 1 ď ℓ ď i`d
+2
+and cp2ℓ`2´dq if i`d
+2 ` 1 ď ℓ ď d. If we reorder the
+rows of B, by taking first the rows with odd index, increasingly, followed by the row with index
+i and then the remaining rows with even index, increasingly, we obtain a matrix S, which looks
+as follows:
+S “
+¨
+˚
+˝
+E d
+2 ¨ d`2
+2
+1 d
+2 ˆ d
+2
+1
+¨¨¨
+1
+0
+¨¨¨
+0
+1 d´2
+2 ˆ d
+2
+E1
+d
+2 ¨ d`2
+2
+˛
+‹‚.
+Clearly, F – convpTq. Let
+U “
+¨
+˚
+˚
+˚
+˝
+E1
+d
+2
+0 d´2
+2 ˆ d
+2
+0 d´2
+2 ˆ d
+2
+E1
+d
+2
+0
+¨¨¨
+0
+´1 0
+¨¨¨
+0
+1
+¨¨¨
+1
+´1 0
+¨¨¨
+0
+˛
+‹‹‹‚P Zdˆd.
+
+20
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+It is easy to see that U is unimodular and a direct computation shows that
+U ¨pS´1dˆdq “
+¨
+˚
+˚
+˚
+˚
+˝
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0 d´2
+2 ˆ d
+2
+0 d´2
+2 ˆ d
+2
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+˛
+‹‹‹‹‚
+.
+Since F – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates
+and by the definition of the join. We also note that the vertices of F corresponding to the vertices
+of the dilated simplices are tcp2ℓ`1q : 1 ď ℓ ď d
+2u and tcp2ℓq : 2 ď ℓ ď d
+2 `1, ℓ ‰ i`2
+2 u.
+For a fixed odd integer j P rds, consider the facet G “ tx P Rd : 1⊺
+even ¨ x ´ xj ď d
+2u of
+QBpσd`1q. By Corollary 6.2, the vertices of F are tcpℓq : ℓ P rd ` 2szt2, j ` 2uu. We now
+consider the matrix C P Zdˆd whose ℓ-th column equals cp2ℓ´1q if 1 ď ℓ ď j`1
+2
+and cp2ℓ`1q if
+j`3
+2 ď ℓ ď d
+2 and cp2ℓ`2´dq if d
+2 `1 ď ℓ ď d. If we reorder the rows of C by taking first the rows
+with odd index k P rdszt ju, increasingly, followed by row j and then the rows with even index,
+increasingly, we obtain a matrix S, which looks as follows:
+S “
+¨
+˚
+˝
+E1
+d
+2 ¨ d`2
+2
+1 d´2
+2 ˆ d
+2
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+1 d
+2 ˆ d
+2
+E d
+2 ¨ d`2
+2
+˛
+‹‚.
+Clearly, G – convpSq. Let
+U “
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+E d
+2 ´1
+ˇˇˇˇˇ
+0 d´2
+2 ˆ d`2
+2
+0 d´2
+2 ˆ d
+2
+ˇˇˇˇˇ
+E1
+d
+2
+0
+¨¨¨
+0
+ˇˇ
+1
+¨¨¨
+1
+0¨¨¨
+0
+´1
+ˇˇˇ
+1
+¨¨¨
+1
+˛
+‹‹‹‹‹‹‹‹‹‚
+P Zdˆd.
+It is easy to see that U is unimodular and a direct computation shows that
+U ¨pS´1dˆdq “
+¨
+˚
+˚
+˚
+˚
+˝
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0 d´2
+2 ˆ d
+2
+0 d´2
+2 ˆ d
+2
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+˛
+‹‹‹‹‚
+.
+Since G – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates
+and by the definition of the join. We also note that the vertices of G corresponding to the vertices
+of the dilated simplices are tcp2ℓ`1q : 0 ď ℓ ď d
+2,ℓ ‰ j`1
+2 u and tcp2ℓq : 2 ď ℓ ď d
+2 `1,ℓ ‰ i`2
+2 u.
+For fixed integers 1 ď i ă j ď d of different parity consider the facet H “ tx P Rd : xi`xj ě 1u
+of QBpσd`1q. Without loss of generality assume that i is odd j is even. By Corollary 6.2, the
+vertices of F are tcpℓq : ℓ P rd `2szti`2, j`2uu. We now consider the matrix D P Zdˆd whose
+ℓ-th column equals cp2ℓ´1q if 1 ď ℓ ď i`1
+2 , cp2ℓ`1q if i`3
+2 ď ℓ ď d
+2, cp2ℓ´dq if d
+2 `1 ď ℓ ď j`d
+2
+and
+cp2ℓ`2´dq if j`d
+2 `1 ď ℓ ď d. If we reorder the rows of D by taking first the rows with odd index
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+21
+k P rdsztiu, increasingly, followed by row i, followed by the rows with even index ℓ P rdszt ju,
+increasingly, followed by row j as the last row, we obtain a matrix S, which looks as follows:
+S “
+¨
+˚
+˚
+˚
+˝
+d`2
+2 ¨E1
+d
+2
+11
+d
+2
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+11
+d
+2
+d`2
+2 ¨E1
+d
+2
+1
+¨¨¨
+1
+0
+¨¨¨
+0
+˛
+‹‹‹‚.
+Clearly, H – convpSq. Let
+U “
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+E d
+2 ´1
+ˇˇˇˇˇ
+0p d
+2 ´1qˆp d
+2 `1q
+01
+d
+2
+ˇˇˇˇˇE d
+2 ´1
+ˇˇˇˇˇ0p d
+2 ´1qˆ1
+´e⊺
+d
+´pe d
+2 `edq⊺
+˛
+‹‹‹‹‹‹‹‹‚
+P Zdˆd.
+It is easy to see that U is unimodular and a direct computation shows that
+U ¨pS´1dˆdq “
+¨
+˚
+˚
+˚
+˚
+˝
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0 d´2
+2 ˆ d
+2
+0 d´2
+2 ˆ d
+2
+M
+´
+d`2
+2 ∆ d´2
+2 ´1 d´2
+2 ˆ d
+2
+¯
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+1
+¨¨¨
+1
+˛
+‹‹‹‹‚
+.
+Since H – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates
+and by the definition of the join. We also note that the vertices of H corresponding to the vertices
+of the dilated simplices are tcp2ℓ`1q : 0 ď ℓ ď d
+2,ℓ ‰ i`1
+2 u and tcp2ℓq : 1 ď ℓ ď d
+2 ` 1,ℓ ‰
+j`2
+2 u.
+□
+Proof of Proposition 6.6 (ii). Let d ě 1 be an odd integer.
+We define vectors up1q,...,
+ud`2 P Rd`1 by upℓq
+k
+“ d ` 1 if k “ ℓ ´ 1 and upℓq
+k
+“ p´1qk`ℓ´1q, otherwise. By Lemma 4.10,
+up1q,...,upd`2q are the vertices of rPBpσd`1q. We now consider the matrix E P Zpd`1qˆpd`2q whose
+ℓ-th column equals bp2ℓ´1q if 1 ď ℓ ď d`3
+2
+and up2ℓ´pd`3qq if d`3
+2 `1 ď ℓ ď d `2. If we reorder
+the rows of E, by taking first the rows with even index and then the ones with odd index,
+increasingly, we obtain a matrix Q “ pqk,ℓq P Zpd`1qˆpd`2q with
+‚ qk,k`1 “ d `1 for k P rd `1s,
+‚ qk,ℓ “ 1 if k ď d`1
+2
+and ℓ ą d`3
+2 , or k ą d`1
+2
+and ℓ ď d`3
+2
+‚ qk,ℓ “ ´1, otherwise.
+Clearly, rPBpσd`1q – convpQq. Let
+U “
+¨
+˚
+˚
+˚
+˚
+˚
+˝
+E d`1
+2
+0 d`1
+2 ˆ d`1
+2
+0
+0 d´1
+2 ˆ d`1
+2
+...
+E d´1
+2
+0
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+˛
+‹‹‹‹‹‚
+P Zpd`1qˆpd`1q.
+
+22
+MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE
+It is easy to see that U is unimodular and a direct computation shows that
+U ¨pQ´1pd`1qˆpd`2qq “
+¨
+˚
+˚
+˝
+M
+´
+pd `2q∆ d`1
+2 ´2¨1
+¯
+0
+0
+M
+´
+pd `2q∆ d´1
+2 ´2¨1
+¯
+0
+¨¨¨
+0
+1
+¨¨¨
+1
+˛
+‹‹‚.
+Since rPBpσd`1q – convpU ¨pQ´1pd`1qˆpd`2qqq, the claim follows by definition of the join.
+□
+REFERENCES
+[1] K. Adiprasito, Stavros A. Papadakis, V. Petrotou, and J. Steinmeyer. Beyond positivity in ehrhart theory,
+2022.
+[2] C.A. Athanasiadis. h˚-vectors, eulerian polynomials and stable polytopes of graphs. Electron. J. Combin.,
+11(2):Research Paper 6, 13 pp. (electronic), 2004/06.
+[3] G. Balletti, T. Hibi, M. Meyer, and A. Tsuchiya. Laplacian simplices associated to digraphs. Arkiv f¨or
+matematik, 56, 12 2018.
+[4] F. Barahona and A. Mahjoub. On the cut polytope. Mathematical Programming, 36:157–173, 06 1986.
+[5] M. Beck and A. Stapledon. On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series.
+Math. Z., 264(1):195–207, 2010.
+[6] B. Braun and M. Meyer. Laplacian simplices. Advances in Applied Mathematics, 114, 2017.
+[7] F. Brenti and V. Welker. The veronese construction for formal power series and graded algebras. Advances in
+Applied Mathematics, 42(4):545–556, 2009.
+[8] M. Brun and T. R¨omer. Subdivisions of toric complexes. Journal of Algebraic Combinatorics, 21, 2004.
+[9] W. Bruns and T. R¨omer. h-vectors of gorenstein polytopes. Journal of Combinatorial Theory, Series A,
+114:65–76, 01 2007.
+[10] Alessio D’Al`ı, Martina Juhnke-Kubitzke, Daniel K¨ohne, and Lorenzo Venturello. On the gamma-vector of
+symmetric edge polytopes. Preprint arXiv: https://arxiv.org/abs/2201.09835, 2022.
+[11] J.A. De Loera, J. Rambau, and F. Santos. Triangulations. Structures for algorithms and applications,
+volume 25. 2010.
+[12] R. Diestel. Graph Theory, volume 173. 2017.
+[13] H. Edelsbrunner and D. R. Grayson. Edgewise subdivision of a simplex. In Proceedings of the Fifteenth
+Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pages 24–30. ACM, New York,
+1999.
+[14] E. Ehrhart. Sur les poly`edres rationnels homoth´etiques `a n dimensions. C. R. Acad. Sci. Paris, 254:616–618,
+1962.
+[15] T.E. Goldberg. Combinatorial laplacians of simplicial complexes. A Senior Project submitted to The Division
+of Natural Science and Mathematics of Bard College, 5 2002.
+[16] D. R. Grayson. Exterior power operations on higher K-theory. K-Theory, 3(3):247–260, 1989.
+[17] B. Gr¨unbaum. Convex Polytopes. Graduate Texts in Mathematics, 2003.
+[18] C. Haase, A. Paffenholz, L. Piechnik, and F. Santos. Existence of unimodular triangulations - positive results.
+Memoirs of the American Mathematical Society, 270, 05 2014.
+[19] J. Herzog, T. Hibi, and H. Ohsugi. Edge Polytopes and Edge Rings, pages 117–140. 09 2018.
+[20] T. Hibi. Dual polytopes of rational convex polytopes. Combinatorica, 2(2):237–240, 1992.
+[21] A. Higashitani, K. Jochemko, and M. Michałek. Arithmetic aspects of symmetric edge polytopes.
+Mathematika, 65:763–784, 05 2019.
+[22] K. Jochemko. On the real-rootedness of the veronese construction for rational formal power series.
+International Mathematics Research Notices, 2018:4780–4798, 2018.
+[23] L. Lov´asz and M. D. Plummer. Matching theory. Annals of Discrete Mathematics, 29, 1986.
+[24] M. Meyer and Pllaha T. Laplacian simplices ii: A coding theoretic approach, 2018.
+[25] R. Mulas, D. Horak, and J. Jost. Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and
+Topology, pages 1–58. Springer International Publishing, Cham, 2022.
+[26] J.R. Munkres. Elements of Algebraic Topology. Addison Wesley Publishing Company, 1984.
+[27] H. Ohsugi and T. Hibi. Special simplices and Gorenstein toric rings. J. Combin. Theory Ser. A, 113(4):718–
+725, 2006.
+[28] H. Ohsugi and A. Tsuchiya. Pq-type adjacency polytopes of join graphs. 03 2021.
+[29] R.P. Stanley. Decompositions of rational convex polytopes. Annals of Discrete Math., 6:333–342, 1980.
+
+LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES
+23
+[30] G. Ziegler. Elements of Algebraic Topology. Graduate Texts in Mathematics, 1995.
+UNIVERSIT ¨AT OSNABR ¨UCK, INSTITUT F ¨UR MATHEMATIK, 49069 OSNABR ¨UCK, GERMANY
+Email address: [email protected]
+UNIVERSIT ¨AT OSNABR ¨UCK, INSTITUT F ¨UR MATHEMATIK, 49069 OSNABR ¨UCK, GERMANY
+Email address: [email protected]
+

-dAzT4oBgHgl3EQfFfqG/content/tmp_files/2301.01012v1.pdf.txt

@@ -0,0 +1,2638 @@
+arXiv:2301.01012v1  [math.AP]  3 Jan 2023
+NOVEL SPATIAL PROFILES OF POPULATION DISTRIBUTION OF
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION
+INFECTION MECHANISM AND SMALL MOVEMENT RATE FOR
+THE INFECTED INDIVIDUALS
+RUI PENG, ZHI-AN WANG, GUANGHUI ZHANG AND MAOLIN ZHOU
+Abstract. In this paper, we are concerned with two SIS epidemic reaction-diffusion
+models with mass action infection mechanism of the form SI, and study the spatial profile
+of population distribution as the movement rate of the infected individuals is restricted to
+be small. For the model with a constant total population number, our results show that
+the susceptible population always converges to a positive constant which is indeed the
+minimum of the associated risk function, and the infected population either concentrates
+at the isolated highest-risk points or aggregates only on the highest-risk intervals once the
+highest-risk locations contain at least one interval. In sharp contrast, for the model with
+a varying total population number which is caused by the recruitment of the susceptible
+individuals and death of the infected individuals, our results reveal that the susceptible
+population converges to a positive function which is non-constant unless the associated risk
+function is constant, and the infected population may concentrate only at some isolated
+highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or
+occupy the whole habitat, depending on the behavior of the associated risk function and
+even its smoothness at the highest-risk locations. Numerical simulations are performed to
+support and complement our theoretical findings.
+1. Introduction and existing results
+The outbreak of the novel coronavirus disease 2019 (COVID-19) continues to spread
+rapidly around the world, and it has caused tremendous impacts on public health and
+the global economy. As it is commonly recognized, population movement is a significant
+factor in the spread of many reported infectious diseases including COVID-19 [5, 9, 25],
+Date: January 4, 2023.
+2010 Mathematics Subject Classification. 35J57, 35B40, 35Q92, 92D30.
+Key words and phrases. Reaction-diffusion SIS epidemic model; mass action infection mechanism; spa-
+tial profile; small movement rate; heterogeneous environment.
+R. Peng: Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China.
+Email: pengrui [email protected].
+Z.-A. Wang: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung
+Hom, Kowloon, Hong Kong. Email: [email protected].
+G. Zhang: School of Mathematics and Statistics, Huazhong University of Science and Technology,
+Wuhan, 430074, China. Email: [email protected].
+M. Zhou: Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China.
+Email: [email protected].
+R. Peng was partially supported by NSF of China (Nos. 12271486, 12171176), Z.-A. Wang was partially
+supported by the Hong Kong Scholars Program (Project ID P0031250) and an internal grant from the
+Hong Kong Polytechnic University (Project ID P0031013), G. Zhang was partially supported by NSF of
+China (No. 12171176, 11971187) and the Fundamental Research Funds for the Central Universities (No.
+5003011008), and M. Zhou was partially supported by the Nankai Zhide Foundation and NSF of China
+(No. 11971498).
+1
+
+2
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+and the lockdown and quarantine has turned out to be one of the most effective measures
+to reduce or even eliminate the infection [30, 60]. On the other hand, the importance of the
+population heterogeneity has also been observed in the complicated dynamical behaviour
+of the transmission of COVID-19 [7, 8, 17].
+To gain a deeper understanding of the impact of population movement and heterogeneity
+on the transmission of epidemic diseases from a mathematically theoretical viewpoint, in
+the present work we are concerned with two SIS reaction-diffusion systems with mass action
+infection mechanism in a heterogeneous environment. We aim to study the spatial profile of
+population distribution as the movement rate of the infected individuals is controlled to be
+sufficiently small. Such kind of information may be useful for decision-makers to predict the
+pattern of disease occurrence and henceforth to conduct more effective strategies of disease
+eradication. The mass action infection mechanism was first proposed in the seminal work
+of Kermack and McKendrick [26], in which the disease transmission was assumed to be
+governed by a bilinear incidence function SI (one may also refer to [27–29] or [54]). The
+systems under consideration in this paper are possibly the simplest yet basic SIS epidemic
+models.
+The first model we will deal with in this work is the following coupled reaction-diffusion
+equations in one-dimensional space:
+
+
+
+
+
+
+
+
+
+
+
+St − dSSxx = −β(x)SI + γ(x)I,
+0 < x < L,
+t > 0,
+It − dIIxx = β(x)SI − γ(x)I,
+0 < x < L,
+t > 0,
+Sx = Ix = 0,
+x = 0, L,
+t > 0,
+S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥, ̸≡ 0,
+0 < x < L.
+(1.1)
+Here, S(x, t) and I(x, t) are respectively the population density of the susceptible and in-
+fected individuals at position x ∈ [0, L] and time t; the homogeneous Neumann boundary
+condition means that no population flux crosses the boundary x = 0, L; dS and dI are pos-
+itive constants measuring the motility of susceptible and infected individuals, respectively;
+and the functions β and γ are H¨older continuous positive functions in [0, L] representing
+the disease transmission rate and the disease recovery rate, respectively.
+Integrating the sum of the equations of (1.1), combined with the homogeneous Neumann
+boundary value conditions, we observe that
+� L
+0
+(S(x, t) + I(x, t)) dx =
+� L
+0
+(S0(x) + I0(x)) dx =: N,
+∀t ≥ 0.
+Thus, the total population number in (1.1) is conserved all the time.
+The system (1.1) was investigated in the recent works [16, 65, 68]; in particular, when
+the movement of either the susceptible or infected population is restricted to be slow, the
+authors explored the profile of the spatial distribution of the disease modelled by (1.1). The
+understanding of such a profile amounts to determine the behavior of the so-called endemic
+equilibrium with respect to the small diffusion rate dS or dI. The endemic equilibrium of
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+3
+(1.1) is a positive steady state solution, which satisfies the following elliptic system:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−dSSxx = −β(x)SI + γ(x)I,
+0 < x < L,
+−dIIxx = β(x)SI − γ(x)I,
+0 < x < L,
+Sx = Ix = 0,
+x = 0, L,
+� L
+0
+(S(x) + I(x)) dx = N.
+(1.2)
+According to [16, 65, 68], if minx∈[0,L]
+γ(x)
+β(x) < N
+L , for any small dI > 0, (1.2) admits at least
+one positive solution (S, I), which is called an endemic equilibrium (EE for abbreviation)
+in terms of epidemiology; moreover, (S, I) satisfies S, I ∈ C2([0, L]) and S, I > 0 on [0, L].
+As remarked in [68], it is a challenging problem to study the spatial profile of EE of
+(1.2) with respect to the small movement rate dI of the infected population; in [65], the
+authors provided a first result in this research direction. Indeed, they proved the following
+conclusion.
+Theorem 1.1. [65, Theorem B] Assume that minx∈[0,L]
+γ(x)
+β(x) < N
+L . Then as dI → 0, the
+EE (S, I) of (1.2) satisfies (up to a sequence of dI) that S → ˆS uniformly on [0, L], where
+ˆS ∈ C([0, L]) with min[0,L]
+γ(x)
+β(x) ≤ ˆS(x) ≤ max[0,L]
+γ(x)
+β(x), and I → µ weakly for some Radon
+measure µ with nonempty support in the sense of
+� L
+0
+I(x)ζ(x)dx −→
+�
+[0,L]
+ζ(x)µ(dx),
+∀ζ ∈ C([0, L]).
+(1.3)
+Obviously, Theorem 1.1 does not give a precise description for ˆS and µ and hence the
+spatial profile of the susceptible and infected populations remains obscure. From the aspect
+of disease control, it becomes imperative to know an informative behavior of µ. In this
+paper, we manage to give a satisfactory result on the profile of ˆS and µ.
+In (1.1), some important factors such as the death and recruitment rates of population
+are ignored so that the total population number is a constant. In order to take into account
+the death and recruitment rates of population, the following reaction-diffusion epidemic
+system was proposed in [40]:
+
+
+
+
+
+
+
+St − dSSxx = Λ(x) − S − β(x)SI + γ(x)I,
+0 < x < L, t > 0,
+It − dIIxx = β(x)SI − [γ(x) + η(x)] I,
+0 < x < L, t > 0,
+Sx = Ix = 0,
+x = 0, L, t > 0,
+S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥, ̸≡ 0,
+0 < x < L.
+(1.4)
+The recruitment term of the susceptible population is represented by the function Λ(x)−S
+so that the susceptible is subject to the linear growth/death ([4, 24]); η(x) accounts for
+the death rate of the infected. Here, Λ, η are assumed to be positive H¨older continuous
+functions on [0, L]. All other parameters have the same interpretation as in (1.1).
+It is easily seen that the following elliptic problem
+−dSSxx = Λ(x) − S,
+0 < x < L;
+Sx(0) = Sx(L) = 0
+(1.5)
+
+4
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+admits a unique positive solution ˜S. Then ( ˜S, 0) is a unique disease-free equilibrium of
+(1.4). An EE of (1.4) satisfies the following ODE system:
+
+
+
+
+
+
+
+−dSSxx = Λ(x) − S − β(x)SI + γ(x)I,
+0 < x < L,
+−dIIxx = β(x)SI − [γ(x) + η(x)] I,
+0 < x < L,
+Sx = Ix = 0,
+x = 0, L.
+(1.6)
+As one of the main results of [40], the following conclusion on the profile of EE of (1.6)
+with respect to small dI was established.
+Theorem 1.2. [40, Theorem 3.2] Assume that the set {x ∈ [0, L] : β(x) ˜S(x) > γ(x) +
+η(x)} is non-empty. As dI → 0, then any EE (S, I) of (1.6) satisfies (up to a subsequence
+of dI) that S → ˆS
+uniformly on [0, L], where ˆS ∈ C([0, L]) and ˆS > 0 on [0, L], and
+� L
+0 Idx → ˆI for some positive constant ˆI.
+As in Theorem 1.1, Theorem 1.2 does not characterize the precise distribution of the
+susceptible and infected populations. In this paper, we will also provide a clear picture of
+the population distributions for (1.6) as the movement rate dI tends to zero. It turns out
+that the spatial profiles of the disease distribution modelled by (1.2) and (1.6) are rather
+different.
+The rest of paper is organized as follows. In section 2, we state the main theoretical
+results, and section 3 is devoted to their proofs. In section 4, we carry out the numerical
+simulations and discuss the implications of our results in terms of disease control. In the
+appendix, we recall some known facts which will be used in the paper.
+2. Statement of main results
+In this section, we state the main findings of this paper on models (1.2) and (1.6).
+To proceed, we underline some terminologies frequently used throughout the paper. For
+model (1.2), we call γ(x)
+β(x) the risk function, and call each element of the set
+�
+x ∈ [0, L] :
+γ(x)
+β(x) = minx∈[0,L]
+γ(x)
+β(x)
+�
+the highest-risk point (or location). Similarly, for model (1.6), we
+call γ(x)+η(x)
+β(x)
+the risk function, and call each element of the set
+�
+x ∈ [0, L] :
+γ(x)+η(x)
+β(x)
+=
+minx∈[0,L]
+γ(x)+η(x)
+β(x)
+�
+the highest-risk point (or location).
+2.1. Results for model (1.2). For the sake of convenience, we set
+k(x) = γ(x)
+β(x),
+kmin = min
+x∈[0,L] k(x),
+and
+Θk =
+�
+x ∈ [0, L] : k(x) = kmin
+�
+.
+We note that when the risk function k(x) = k is a positive constant, it follows from [65]
+that S(x) ≡ k is a constant, and in turn by the equation of I, we immediately see that
+I = N
+L − k is also a positive constant provided that k < N
+L . In what follows, we do not
+consider such a trivial case and assume that k(x) is non-constant on [0, L].
+We now state our main result on the asymptotic behavior of any EE (S, I) of (1.2) as
+dI → 0 as follows.
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+5
+Theorem 2.1. Assume that k(x) is non-constant and kmin < N
+L . Then as dI → 0, the EE
+(S, I) of (1.2) satisfies
+S(x) → kmin
+uniformly for x ∈ [0, L].
+(2.7)
+The following assertions hold for the asymptotic behavior of I.
+(i) If Θk = {x0}, then we have
+I(x) → (N − Lkmin)δ(x0) weakly in the sense of (1.3),
+where δ(x0) is the Dirac measure centered at x0. Moreover, I(x) → 0 locally uni-
+formly in [0, L] \ {x0}.
+(ii) If Θk = [̺1, ̺2] for some 0 < ̺1 < ̺2 < L, then we have
+I(x) → 0
+uniformly on [0, ̺1] ∪ [̺2, L],
+and
+I(x) → ˆI(x)
+uniformly for x ∈ [̺1, ̺2],
+where ˆI ∈ C2([̺1, ̺2]), ˆI > 0 in (̺1, ̺2), and ˆI is the unique positive solution of
+
+
+
+
+
+
+
+
+
+−ˆIxx = β(x)
+dS (ˆa − ˆI)ˆI,
+̺1 < x < ̺2,
+ˆI = 0,
+x = ̺1, ̺2,
+� ̺2
+̺1
+ˆI dx = N − Lkmin,
+(2.8)
+where the positive constant ˆa is uniquely determined by the integral constraint in
+(2.8).
+Regarding Theorem 2.1, we would like to make some comments in order as follows.
+Remark 2.1. In addition to the two cases treated in Theorem 2.1, we can handle some
+more general cases. In particular, we would like to make the following comments.
+(i) If the set Θk contains only finitely many isolated points, say {xi}j
+i=1 for some j ≥ 2,
+then one can slightly modify the proof of Theorem 2.1(i) to show that S → kmin
+uniformly on [0, L], and I → 0 locally uniformly in [0, L] \ ({xi}j
+i=1), and
+I(x) →
+j
+�
+i=1
+ciδ(xi) weakly in the sense of (1.3),
+where δ(xi) is the Dirac measure centered at xi and the nonnegative constants ci
+fulfill �j
+i=1 ci = N − Lkmin. Nevertheless, we can not determine the exact values
+of ci; in other words, as dI → 0, it is unclear to us whether I concentrates at all
+xi (1 ≤ i ≤ j) or only some of them. The numerical results suggest that the former
+alternative holds; see Figure 1 in section 4.
+(ii) If the set Θk contains at least one proper interval of [0, L], by adapting the argument
+of Theorem 2.1(ii), we can show that S → kmin uniformly on [0, L], and I → ˆI
+uniformly on [0, L] with
+ˆI = 0
+on [0, L] \ Θk,
+�
+Θk
+ˆI dx = N − Lkmin.
+
+6
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+In particular, if Θk =
+� �j∗
+i=1[̺i, ̺i]
+� � � �{xi}j∗
+i=0
+�
+for some j∗ ≥ 1, j∗ ≥ 0, then
+we can prove that
+ˆI = 0
+on [0, L] \ (
+j∗
+�
+i=1
+(̺i, ̺i)),
+and in (̺i, ̺i) (1 ≤ i ≤ j∗), either ˆI = 0 or ˆI > 0. Without loss of generality,
+assuming that ˆI(x) > 0 for x ∈ � ˆj∗
+i=1(̺i, ̺i) for some 1 ≤ ˆj∗ ≤ j∗, then in each
+such (̺i, ̺i), we can conclude that ˆI solves
+�
+−ˆIxx = β(x)
+dS (ˆa − ˆI)ˆI,
+̺i < x < ̺i,
+ˆI = 0,
+x = ̺i, ̺i,
+where the positive constant ˆa is uniquely determined by
+ˆj∗
+�
+i=1
+� ̺i
+̺i
+ˆI dx = N − Lkmin.
+However, it seems rather challenging to prove whether ˆI is positive on all intervals
+(̺i, ̺i) (1 ≤ i ≤ j∗) or only on some of them. Our numerical results suggest that
+the former alternative holds; see Figure 2 in section 4.
+(iii) The assertion in (ii) above suggests that if the highest-risk locations contain at
+least one interval, then the disease can not stay on any possible isolated highest-risk
+points once the infected individuals move slowly.
+Remark 2.2. In the case (ii) of Theorem 2.1, if ̺1 = 0 (or ̺2 = L), the results of Theorem
+2.1 still hold true if we replace the Dirichlet boundary condition of ˆI in (2.8) at ̺1 = 0 (or
+̺2 = L) by the Neumann boundary condition ˆIx(0) = 0 (or ˆIx(L) = 0). A similar remark
+applies to the case discussed in Remark 2.1(ii) above.
+Remark 2.3. After this paper was finished, we noticed the work [10] in which the authors
+derived (2.7) and the convergence of the I-component in the case (i) of Theorem 2.1 in
+any spatial dimension in a more general setting; see Theorem 2.5(i) there. However, their
+result does not establish the convergence of the I-component within Θk in the case (ii) of
+Theorem 2.1 nor in the more general case mentioned by Remark 2.1; on the other hand,
+our proof of (2.7) and the convergence of the I-component outside of Θk is rather different
+from that of [10].
+2.2. Results for model (1.6). We now turn to system (1.6). For the sake of simplicity,
+we assume that Λ in (1.6) is a positive constant, and also denote
+h(x) = γ(x) + η(x)
+β(x)
+,
+hmin = min
+x∈[0,L] h(x),
+and
+Θh =
+�
+x ∈ [0, L] : h(x) = hmin
+�
+.
+Clearly, ˜S(x) = Λ. We also enhance the existence condition of EE of (1.6) in Theorem 1.2
+by imposing the following condition:
+Λ > h(x)
+for all x ∈ [0, L].
+(2.9)
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+7
+Now we can state our main findings on the asymptotic behavior of any EE (S, I) of (1.6)
+as dI → 0. The first result reads as follows.
+Theorem 2.2. Assume that (2.9) holds. As dI → 0, then any EE (S, I) of (1.6) satisfies
+(up to a subsequence of dI) that S → ˆS
+uniformly on [0, L], and I → µ weakly in the
+sense of (1.3), where µ is some Radon measure and ˆS solves weakly in W 1,2(0, L) the free
+boundary problem:
+−dS ˆSxx = Λ − ˆS − η(x)µ({x})
+��
+{x∈[0, L]: ˆS(x)=h(x)},
+x ∈ (0, L).
+(2.10)
+Here, µ({x})
+��
+{x∈[0, L]: ˆS(x)=h(x)} is the restriction of µ on the set {x ∈ [0, L] : ˆS(x) = h(x)};
+otherwise, µ({x}) = 0. Moreover we have the following properties for µ and ˆS.
+(i) The Radon measure µ satisfies
+µ({x ∈ [0, L] : ˆS(x) ̸= h(x)}) = 0,
+µ({x ∈ [0, L] : ˆS(x) = h(x)}) > 0.
+(2.11)
+(ii) The function ˆS ∈ C([0, L]) satisfies
+hmin ≤ ˆS(x) ≤ h(x),
+∀x ∈ [0, L],
+(2.12)
+Θh ⊂
+�
+x ∈ [0, L] :
+ˆS(x) = h(x)
+�
+;
+(2.13)
+If x1, x2 ∈ Θh with x1 < x2 and (x1, x2) ∩ Θh = ∅, then
+hmin < ˆS(x),
+∀x ∈ (x1, x2).
+(2.14)
+Theorem 2.2 asserts that ˆS touches h at all highest-risk points. In what follows, our goal
+is to examine the properties ˆS for some specific risk function h, which in turn provides us
+with a more precise description of the profile of µ. Indeed, we can obtain the following
+result for (1.6).
+Theorem 2.3. Let ˆS and µ be given as in Theorem 2.2. Assume that h ∈ C2([0, L]) and
+(2.9) holds. The following assertions hold.
+(i) If −dShxx ≤ Λ − h in (0, L), hx(0) ≥ 0 and hx(L) ≤ 0, then we have
+ˆS(x) = h(x),
+∀x ∈ [0, L],
+(2.15)
+µ({x}) = Λ − h(x) + dShxx(x)
+η(x)
+,
+a.e. for x ∈ (0, L).
+(2.16)
+(ii) If hx is non-decreasing on [0, L] and Θh = {τ0} for some 0 ≤ τ0 ≤ L, then the
+following assertions hold.
+(a) When 0 < τ0 < L, we have
+ˆS(x) = h(x),
+∀x ∈ [τ1, τ2],
+(2.17)
+and in [0, τ1) ∪ (τ2, L], ˆS < h satisfies
+
+
+
+
+
+
+
+−dS ˆSxx(x) = Λ − ˆS,
+x ∈ (0, τ1) ∪ (τ2, L),
+ˆSx(0) = 0,
+ˆSx(L) = 0,
+ˆS(τ1) = h(τ1),
+ˆS(τ2) = h(τ2),
+(2.18)
+and µ satisfies
+µ({x}) = Λ − h(x) + dShxx(x)
+η(x)
+,
+a.e. for x ∈ (τ1, τ2),
+(2.19)
+
+8
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+µ({x}) = 0,
+∀x ∈ [0, τ1) ∪ (τ2, L],
+(2.20)
+where the numbers τ1, τ2 with 0 < τ1 < τ0 < τ2 < L are uniquely determined
+by
+e2d−1/2
+S
+τ1 − 1
+e2d−1/2
+S
+τ1 + 1
+= −d1/2
+S hx(τ1)
+Λ − h(τ1) ,
+e2d−1/2
+S
+(τ2−L) − 1
+e2d−1/2
+S
+(τ2−L) + 1
+= −d1/2
+S hx(τ2)
+Λ − h(τ2) .
+(2.21)
+(b) When τ0 = L, then we have the following assertions.
+(b-1) If
+e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+> −
+d1/2
+S
+hx(L)
+Λ−h(L) , then (2.17) and (2.19) hold with [τ1, τ2] re-
+placed by [τ1, L], µ([0, τ1)) = 0, and on [0, τ1], ˆS satisfies
+�
+−dS ˆSxx(x) = Λ − ˆS,
+x ∈ (0, τ1),
+ˆSx(0) = 0,
+ˆS(τ1) = h(τ1),
+(2.22)
+where 0 < τ1 < L is uniquely determined by the first equation in (2.21).
+(b-2) If e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+≤ −
+d1/2
+S
+hx(L)
+Λ−h(L) , then ˆS is the unique positive solution of
+�
+−dS ˆSxx(x) = Λ − ˆS,
+x ∈ (0, L),
+ˆSx(0) = 0,
+ˆS(L) = h(L),
+(2.23)
+and µ satisfies
+µ([0, L)) = 0,
+µ({L}) = ΛL −
+� L
+0 ˆS(x)dx
+η(L)
+.
+(2.24)
+(c) When τ0 = 0, then we have the following assertions.
+(c-1) If e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+>
+d1/2
+S
+hx(0)
+Λ−h(0) , then (2.17) and (2.19) hold with [τ1, τ2] replaced
+by [0, τ2], µ((τ2, L]) = 0, and on [τ2, L], ˆS satisfies
+�
+−dS ˆSxx(x) = Λ − ˆS,
+x ∈ (τ2, L),
+ˆSx(L) = 0,
+ˆS(τ2) = h(τ2),
+(2.25)
+where 0 < τ2 < L is uniquely determined by the second equation in (2.21).
+(c-2) If e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+≤
+d1/2
+S
+hx(0)
+Λ−h(0) , then ˆS is the unique positive solution of
+�
+−dS ˆSxx(x) = Λ − ˆS,
+x ∈ (0, L),
+ˆSx(L) = 0,
+ˆS(0) = h(0),
+(2.26)
+and µ satisfies
+µ((0, L]) = 0,
+µ({0}) = ΛL −
+� L
+0 ˆS(x)dx
+η(0)
+.
+(2.27)
+(iii) If hx is non-decreasing on [0, ̺1]∪[̺2, L] and Θh = [̺1, ̺2] for some 0 < ̺1 < ̺2 < L,
+then all the assertions in (ii)-(a) above hold, where the numbers τ1, τ2 satisfying
+0 < τ1 < ̺1 < ̺2 < τ2 < L are uniquely determined by (2.21).
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+9
+For model (1.2), our result shows that the infected population concentrates or aggregates
+only at the highest-risk locations. In sharp contrast, for model (1.6), our result suggests
+that the disease will occupy a neighborhood of the interior highest-risk locations or even
+occupy the whole habitat [0, L], or concentrates only at the boundary highest-risk location,
+depending on the risk function h. More detailed discussions on the implications of our
+theoretical results, along with numerical simulations, will be given in section 4.
+We would like to make some remarks on Theorem 2.3 as follows.
+Remark 2.4. It is worth mentioning that all the statements in Theorem 2.3 except the
+expression (2.19) for the Radon measure µ remain true provided that the risk function
+h ∈ C1([0, L]). Such a comment also applies to Lemmas 3.1-3.4 in the forthcoming section.
+Remark 2.5.
+(i) It is clear that Theorem 2.3(i) holds if h < Λ is a constant or more
+generally h is a unique solution to the following problem:
+�
+−dShxx = Λ − h,
+x ∈ (0, L),
+h(0) = σ1,
+h(L) = σ2,
+where 0 < σ1, σ2 < Λ.
+When hx(0) > 0, the change of the derivatives from Sx(0) = 0 to ˆSx(0) = hx(0) >
+0 would suggest that I should experience the concentration phenomenon at x = 0
+(that is, I(0) → ∞) as dI → 0. The same remark applies to the case of hx(L) < 0.
+(ii) In contrast to Theorem 2.3(i), it is easily seen that ˆS ̸≡ h on [0, L] provided that
+−dShxx(x∗) > Λ − h(x∗) for some x∗ ∈ (0, L).
+(iii) Clearly, the assertions of Theorem 2.3(ii)-(b1) hold if hx(L) = 0 and the assertions
+of Theorem 2.3(ii)-(c1) hold if hx(0) = 0.
+(iv) In a general case that Θh contains an interior isolated point and hx is non-decreasing
+in a neighbourhood of such a point, we can conclude that (2.15) and (2.16) hold in
+some neighbourhood of this point; if Θh contains an interval, a similar conclusion
+also holds. See Lemma 3.1 and Lemma 3.3 below.
+3. Proof of main results: Theorems 2.1, 2.2 and 2.3
+This section is devoted to the proof of Theorems 2.1, 2.2 and 2.3.
+3.1. Proof of Theorem 2.1. In this subsection, we present the proof of Theorem 2.1.
+Proof of Theorem 2.1. First of all, we recall that for any EE (S, I) of (1.2), from [65] (see
+(3.3) there), the following holds:
+kmin ≤ S(x) ≤ max
+[0,L] k(x),
+∀x ∈ [0, L].
+(3.1)
+By the positivity of I and the uniqueness of the principal eigenvalue, it is clear from the
+equation of I that
+λ1(dI, γ − βS) = 0,
+∀dI > 0,
+where λ1(dI, γ − βS) is defined as in the appendix. Using Theorem 1.1, as dI → 0 (up to a
+subsequence), we see that S → ˆS uniformly on [0, L] for some positive function ˆS. Hence,
+by Lemma 5.1 in the appendix and the continuous dependence of the principal eigenvalue
+on the weight function γ − βS, we have
+0 = lim
+dI→0 λ1(dI, γ − βS) = min
+x∈[0,L][γ(x) − β(x) ˆS(x)].
+
+10
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+This obviously implies that
+ˆS(x) ≤ k(x),
+∀x ∈ [0, L] and
+ˆS(y0) = k(y0)
+(3.2)
+for some y0 ∈ [0, L].
+From Theorem 1.1, we recall that I → µ weakly for some Radon measure µ with
+µ([0, L]) > 0 in the following sense
+� L
+0
+I(x)ζ(x)dx →
+� L
+0
+ζ(x)µ(dx),
+∀ζ ∈ C([0, L]),
+as dI → 0.
+(3.3)
+We now integrate the first equation in (1.2) by parts over [0, L] and use the boundary
+conditions to deduce that
+� L
+0
+[β(x)S(x) − γ(x)]I(x)dx = 0,
+∀dI > 0.
+(3.4)
+Letting dI → 0 in (3.4), combined with (3.3) and the fact that S → ˆS uniformly on [0, L]
+as dI → 0, we infer that
+�
+[0,L]
+[β(x) ˆS(x) − γ(x)]µ(dx) = 0,
+(3.5)
+which, together with (3.2), gives
+�
+{x∈[0,L]: ˆS(x)<k(x)}
+β(x)[ ˆS(x) − k(x)]µ(dx) =
+�
+[0,L]
+β(x)[ ˆS(x) − k(x)]µ(dx) = 0.
+As a result, we find that
+µ({x ∈ [0, L] : ˆS(x) < k(x)}) = 0
+(3.6)
+and
+µ({x ∈ [0, L] : ˆS(x) = k(x)}) = µ([0, L]) > 0.
+(3.7)
+In view of (3.4) and
+� L
+0 (S(x) + I(x)) dx = N, for any dI > 0 we have
+� L
+0
+S(x)I(x)dx ≤
+1
+min[0,L] β(x)
+� L
+0
+γ(x)I(x)dx ≤ max[0,L] γ(x)
+min[0,L] β(x) N,
+∀dI > 0.
+(3.8)
+Then, applying the L1-theory for elliptic equation (see Lemma 5.2 in the appendix) to the
+S-equation, one sees that for any 1 ≤ r < ∞,
+∥S∥W 1,r(0,L) ≤ C,
+∀dI > 0.
+(3.9)
+Hereafter, C or C(ǫ) is a positive constant independent of dI > 0 but may be different
+from place to place. Taking r = 2 in (3.9), we note that W 1,2(0, L) is a Hilbert space and
+W 1,2(0, L) is compactly embedded to C([0, L]). Thus, we may assume that S → ˆS weakly
+in W 1,2(0, L) and S → ˆS uniformly on [0, L] as dI → 0. Now, for any ζ ∈ W 1,2(0, L) (and
+so ζ ∈ C([0, L])), we get from the S-equation that
+dS
+� L
+0
+Sx(x)ζx(x)dx =
+� L
+0
+[−β(x)S(x) + γ(x)]I(x)ζ(x)dx,
+∀dI > 0.
+(3.10)
+By virtue of (3.3), (3.6) and (3.7), we can send dI → 0 in (3.10) to obtain
+dS
+� L
+0
+ˆSx(x)ζx(x)dx = 0,
+∀ζ ∈ W 1,2(0, L).
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+11
+This means that ˆS is a weak (and then a classical) solution of
+−uxx(x) = 0,
+x ∈ (0, L);
+ux(0) = ux(L).
+Consequently, ˆS must be a positive constant. It then follows from (3.2) that ˆS = kmin,
+and so S(x) → kmin uniformly on [0, L].
+In the sequel, we are going to determine the limit of I.
+We first consider case (i):
+Θk = {x0} is a singleton. By what was proved above, it is easily seen that
+I(x) → (N − Lkmin)δ(x0)
+weakly in the sense of (1.3),
+where δ(x0) is the Dirac measure centered at x0.
+It remains to show I(x) → 0 locally uniformly in [0, L] \ {x0}. We only consider the
+case of x0 ∈ (0, L), and the case x0 = 0 or L can be handled similarly. Since S(x) → kmin
+uniformly on [0, L], by the definition of kmin, we know from the I-equation that, given small
+ǫ > 0, Ixx > 0 on [0, x0 −ǫ]∪[x0 +ǫ, L] as long as dI is small enough. As Ix(0) = Ix(L) = 0,
+I is increasing in [0, x0 − ǫ] while is decreasing in [x0 + ǫ, L]. Thus, due to the arbitrariness
+of ǫ, it readily follows from (3.6) that I(x) → 0 locally uniformly in [0, x0) ∪ (x0, L], as
+claimed.
+We next consider case (ii): Θk = [̺1, ̺2] ⊂ (0, L).
+First of all, we can assert that
+I(x) → 0 locally uniformly in [0, L] \ [̺1, ̺2] by a similar argument as in case (i). In what
+follows, we will analyze the limiting behavior of I in the interval [̺1, ̺2]. To this end, let
+us introduce the following function
+w(x) = S(x) − kmin
+dI
+,
+x ∈ [0, L].
+Due to (3.1), w ≥ 0 on [0, L]. In addition, by our assumption, one notices that w solves
+−dSwxx(x) = −β(x)Iw,
+x ∈ [̺1, ̺2],
+(3.11)
+and I satisfies
+−Ixx(x) = β(x)wI,
+x ∈ [̺1, ̺2].
+(3.12)
+Since
+� L
+0 I(x)dx ≤ N, for any small ǫ > 0, Lemma 5.3(b) in the appendix can be applied
+to (3.11) to assert that
+max
+x∈[̺1+ǫ,̺2−ǫ] w(x) ≤ C(ǫ)
+min
+x∈[̺1+ǫ,̺2−ǫ] w(x).
+(3.13)
+We now claim that w is uniformly bounded on [̺1 +ǫ, ̺2 −ǫ] for all small dI > 0. Other-
+wise, there is a sequence of dI, labelled by itself for simplicity, such that the corresponding
+solution sequence {(w, I)} satisfies
+max
+x∈[̺1+ǫ,̺2−ǫ] w(x) → ∞,
+as dI → 0.
+(3.14)
+By (3.13), w → ∞ uniformly on [̺1 + ǫ, ̺2 − ǫ] as dI → 0. To produce a contradiction,
+let us denote λD
+1 to be the principal eigenvalue of the following eigenvalue problem with
+Dirichlet boundary conditions:
+�
+−ϕxx = λϕ,
+x ∈ (̺1 + ǫ, ̺2 − ǫ)
+ϕ(̺1 + ǫ) = ϕ(̺2 − ǫ) = 0.
+(3.15)
+Apparently, λD
+1 > 0. For all small dI > 0, by (3.14) we may assume that
+β(x)w(x) > 2λD
+1
+on [̺1 + ǫ, ̺2 − ǫ].
+
+12
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+Thus, it follows from (3.12) that I ∈ C2([0, L]) is a positive and strict supersolution of the
+following operator in the sense of [57, Definition 2.1]:
+�
+Lu := −uxx − 2λD
+1 u,
+x ∈ (̺1 + ǫ, ̺2 − ǫ),
+∀u ∈ C2([0, L]),
+u(̺1 + ǫ) = u(̺2 − ǫ) = 0.
+By means of [57, Proposition 2.1], the principal eigenvalue, denoted by ˜λD
+1 , of the eigenvalue
+problem
+�
+Lϕxx = λϕ,
+x ∈ (̺1 + ǫ, ̺2 − ǫ),
+ϕ(̺1 + ǫ) = ϕ(̺2 − ǫ) = 0
+satisfies ˜λD
+1 > 0.
+On the other hand, the uniqueness of the principal eigenvalue of problem (3.15) implies
+˜λD
+1 + 2λD
+1 = λD
+1 , and so ˜λD
+1 = −λD
+1 < 0, leading to a contradiction. The previous claim
+is thus verified. Due to the arbitrariness of ǫ, we have shown that w is locally uniformly
+bounded in (̺1, ̺2) with respect to all small dI > 0.
+Furthermore, by Lemma 5.2 in the appendix, it is easy to see from (3.12) that I is locally
+uniformly bounded in (̺1, ̺2) independent of all small dI > 0. The standard regularity
+theory for elliptic equations can be applied to (3.11) and (3.12), respectively to deduce that
+w and I are locally bounded (independent of small dI) in (̺1, ̺2) in the usual C2+α-norm
+for some α ∈ (0, 1). Then, by a diagonal argument, we may assume that
+(w, I) → ( ˆw, ˆI)
+in C2
+loc(̺1, ̺2),
+as dI → 0.
+Clearly, by (3.12), ( ˆw, ˆI) satisfies
+−ˆIxx(x) = β(x) ˆw ˆI,
+x ∈ (̺1, ̺2).
+(3.16)
+Furthermore, by adding (3.11) and (3.12), one easily sees that ( ˆw, ˆI) solves
+−(dS ˆw + ˆI)xx = 0
+in (̺1, ̺2).
+This indicates that
+dS ˆw(x) + ˆI(x) = ˆa + ˆbx,
+x ∈ (̺1, ̺2)
+(3.17)
+for some constants ˆa, ˆb.
+In what follows, we aim to determine ˆa and ˆb. By a simple observation, (w, I) satisfies
+�
+−(dSw + I)xx = 0,
+x ∈ (0, L),
+(dSw + I)x = 0,
+x = 0, L.
+Thus, dSw + I = cdI is a positive constant on [0, L] for any dI > 0. Recall that w, I are
+locally uniformly bounded in (̺1, ̺2). Hence, as dI → 0, we may assume that
+dSw + I = cdI → ˆc ∈ [0, ∞)
+uniformly on [0, L].
+From (3.17) it follows that ˆc = ˆa and ˆb = 0. In addition, our analysis indicates that w and
+I are uniformly bounded on [0, L]. Precisely, it holds that
+w(x),
+I(x) ≤ C,
+∀x ∈ [0, L].
+(3.18)
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+13
+We now use the equation of I, together with the fact of w, I ≥ 0 and the definition of
+k, to find that
+−Ixx = β(x) [S − k(x)] I
+dI
+= β(x)
+�S − kmin
+dI
++ kmin − k(x)
+dI
+�
+I
+(3.19)
+≤ β(x)wI,
+x ∈ (0, L).
+Multiplying both sides in (3.19) by I and integrating over (0, L), we obtain
+� L
+0
+(Ix)2dx ≤
+� L
+0
+βwI2dx ≤ C
+due to (3.18). This and (3.18) imply that ∥I∥W 1,2(0,L) ≤ C. Since W 1,2(0, L) is compactly
+embedded to C([0, L]), we can assume that I → ˆI uniformly on [0, L]. By what was proved
+before, ˆI = 0 on [0, ̺1] ∪ [̺2, L], and by (3.16) and (3.17), on [̺1, ̺2], ˆI solves
+�
+−ˆIxx = β(x)
+dS (ˆa − ˆI)ˆI,
+̺1 < x < ̺2,
+ˆI = 0,
+x = ̺1, ̺2.
+(3.20)
+Because of
+� L
+0 (S(x) + I(x)) dx = N and S → kmin uniformly on [0, L] as dI → 0, it is
+easily seen that
+� ̺2
+̺1
+ˆI dx = N − Lkmin > 0.
+(3.21)
+Thanks to the Harnack inequality (see Lemma 5.3(b)) and (3.21), we have from (3.20) that
+ˆI > 0 in (̺1, ̺2). By (3.17) and the fact of ˆb = 0, clearly ˆa > 0.
+It is well known that given ˆa > 0, the positive solution of problem (3.20), if it exists,
+must be unique, denoted by ˆIˆa; moreover, if 0 < ˆa1 < ˆa2, then ˆIˆa1(x) < ˆIˆa2(x) for all
+x ∈ (̺1, ̺2). With these facts, one can check that the positive constant ˆa is uniquely
+determined by (3.21) in an implicit manner. Therefore, all the assertions in case (ii) have
+been verified. The proof is thus complete.
+□
+3.2. Proof of Theorem 2.2. We are now in a position to give the proof of Theorem 2.2.
+Proof of Theorem 2.2. First of all, one can follow the analysis of Theorem 2.1, combined
+with the result of Theorem 1.2 and its proof (see [40, Theorem 3.2]), to show that as
+dI → 0, any EE (S, I) of (1.6) satisfies (up to a subsequence of dI) that S → ˆS weakly in
+W 1,2(0, L) and uniformly on [0, L], and I → µ weakly in the sense of (1.3) for some Radon
+measure µ and positive function ˆS ∈ W 1,2(0, L), and
+0 < ˆS(x) ≤ h(x),
+∀x ∈ [0, L],
+(3.22)
+and (2.11) hold.
+For any ζ ∈ W 1,2(0, L) (and so ζ ∈ C([0, L])), we use the S-equation to obtain
+dS
+� L
+0
+Sxζxdx =
+� L
+0
+[Λ − S − β(x)SI + γ(x)I]ζdx
+=
+� L
+0
+[Λ − S − η(x)I]ζdx −
+� L
+0
+[β(x)S − (γ(x) + η(x))]Iζdx
+(3.23)
+
+14
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+for all dI > 0. In view of (3.22) and (2.11), we send dI → 0 to infer that
+� L
+0
+[β(x)S − (γ(x) + η(x)]Iζdx →
+�
+[0,L]
+β(x)[ ˆS − h(x)]ζµ(dx) = 0.
+Thus, by letting dI → 0, it follows from (3.23) that
+dS
+� L
+0
+ˆSxζxdx =
+� L
+0
+(Λ − ˆS)ζdx −
+� L
+0
+η(x)ζµ(dx),
+∀ζ ∈ W 1,2(0, L).
+(3.24)
+Together with (2.11), this means that ˆS ∈ W 1,2(0, L) is a weak solution of (2.10).
+In what follows, for a general positive H¨older continuous function h, we will prove three
+claims:
+Claim 1. If the minimum of h is attained at x = 0 (resp. at x = L), then ˆS must touch
+h at this point; that is, ˆS(0) = h(0) = hmin (resp. ˆS(L) = h(L) = hmin).
+We only handle the case that hmin is attained at x = 0, and the other case can be treated
+similarly. Since ˆS ≤ h on [0, L], we suppose that ˆS(0) < h(0) and so ˆS(x) < h(x) on [0, ǫ0]
+for some small ǫ0 > 0. Thus, from (2.10), we have −dS ˆSxx = Λ − ˆS, ∀x ∈ (0, ǫ0]. A simple
+analysis shows that
+ˆS(x) = c1ed−1/2
+S
+x + c2e−d−1/2
+S
+x + Λ, x ∈ (0, ǫ0]
+(3.25)
+for some constants c1, c2. On the other hand, using the S-equation, we integrate on [0, x]
+to deduce
+−Sx(x) = 1
+dS
+� x
+0
+[Λ − S(y) − β(y)S(y)I(y) + γ(y)I(y)]dy, x ∈ [0, ǫ0].
+(3.26)
+From the proof of [40, Theorem 3.2], we know that
+� L
+0
+S(x)I(x)dx ≤ C,
+� L
+0
+I(x)dx ≤ C, and S(x) ≤ C,
+∀x ∈ [0, L],
+(3.27)
+for some positive constant C, independent of dI > 0.
+In the sequel, the constant C allows to vary from line to line but does not depend
+on dI > 0. It immediately follows from (3.26) that Sx is uniformly bounded on [0, ǫ0],
+independent of dI > 0. Note that µ([0, ǫ0]) = 0 due to (2.11), and I → µ weakly in the
+sense of (1.3). Given any small ǫ > 0, we can find a small ρ > 0 so that for all 0 < dI ≤ ρ,
+� ǫ0
+0
+I(x)dx ≤ ǫ +
+�
+[0,ǫ0]
+µ(dx) = ǫ.
+Now, for any x1, x2 ∈ [0, ǫ0] satisfying |x1 − x2| < ǫ, we have
+��Sx(x1) − Sx(x2)
+�� = 1
+dS
+���
+� x2
+x1
+[Λ − S(y) − β(y)S(y)I(y) + γ(y)I(y)]dy
+���
+≤ C|x1 − x2| + C
+� x2
+x1
+I(y)dy
+≤ C|x1 − x2| + C
+� ǫ0
+0
+I(y)dy ≤ Cǫ
+provided that 0 < dI ≤ ρ. This shows that Sx is equi-continuous on [0, ǫ0] once 0 < dI ≤ ρ.
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+15
+Hence, we can apply the well-known Ascoli-Arzel`a theorem, up to a further subsequence
+of dI, to conclude that Sx is uniformly convergent on [0, ǫ0] as dI → 0. As
+S(x) − S(0) =
+� x
+0
+Sx(y)dy,
+S → ˆS uniformly on [0, ǫ0],
+it is easily seen that S → ˆS in C1([0, ǫ0]). Thus, ˆSx(0) = 0, and in turn we get from (3.25)
+that c1 = c2. Because of ˆS ≤ h on [0, L] and the condition (2.9), we have c1 = c2 < 0, and
+so
+ˆSx(x) = c1[ed−1/2
+S
+x − e−d−1/2
+S
+x] < 0,
+∀x ∈ (0, ǫ0].
+This means that ˆS is decreasing on [0, ǫ0].
+By virtue of h(0) ≤ h(x) for all x ∈ [0, L] and (2.11), one can extend the above analysis to
+assert that ˆS is decreasing on [0, L] and so ˆS < h on [0, L]. This clearly gives µ([0, L]) = 0,
+a contradiction with µ([0, L]) > 0 due to (2.11) again. Hence, we must have ˆS(0) = h(0) =
+hmin.
+Claim 2. If ˆS attains its local minimum at some x0 ∈ (0, L), then ˆS must touch h at
+this point; that is, ˆS(x0) = h(x0).
+Suppose that ˆS(x0) < h(x0) due to ˆS ≤ h. Thus, there is a small ǫ0 > 0 such that
+ˆS(x) < h(x) for all x ∈ [x0 −ǫ0, x0 + ǫ0] ⊂ (0, L). By (2.11), µ([x0 −ǫ0, x0 + ǫ0]) = 0 and so
+−dS ˆSxx = Λ − ˆS
+on [x0 − ǫ0, x0 + ǫ0].
+As before, ˆS takes the form of (3.25) on [x0−ǫ0, x0+ǫ0] for some constants c1, c2. Obviously,
+ˆSx(x0) = 0, which leads to c2 = c1e2d−1/2
+S
+x0, and so c1 < 0. Thus, it holds that
+ˆS(x) = c1[ed−1/2
+S
+x + ed−1/2
+S
+(2x0−x)] + Λ,
+x ∈ [x0 − ǫ0, x0 + ǫ0]
+(3.28)
+for some constant c1 < 0. In view of (3.28), basic computation gives that ˆS is increasing
+on [x0 −ǫ0, x0] while is decreasing on [x0, x0 + ǫ0]. This implies that x0 is a local maximum
+of ˆS, a contradiction with our assumption. As a result, ˆS must touch h at x = x0.
+Claim 3. If the minimum of h is attained at some point y0 ∈ (0, L), then ˆS must touch
+h at this point; that is, ˆS(y0) = h(y0) = hmin.
+Suppose that ˆS(y0) < h(y0) = hmin. There are two possible cases to happen in the
+interval [0, y0): Case 1. ˆS never touches h in [0, y0), that is, ˆS < h in [0, y0); Case 2. ˆS
+touches h somewhere in [0, y0).
+When Case 1 occurs, by (2.11), we know that ˆS must touch h in (y0, L]. Let y1 be the
+first point (from the left side) at which ˆS touches h. That is, y1 ∈ (y0, L], and
+ˆS(x) < h(x),
+∀x ∈ (y0, y1),
+ˆS(y1) = h(y1) ≥ hmin.
+On the other hand, since ˆS < h in [0, y0), we can follow the analysis used in Claim 1 to
+show that ˆS is decreasing on [0, y1]. This is an obvious contradiction with ˆS(y0) < hmin ≤
+ˆS(y1).
+When Case 2 occurs, we denote by y2 ∈ [0, y0) the first point from the right side such
+that ˆS touches h in [0, y0). That is,
+ˆS(x) < h(x),
+∀x ∈ (y2, y0),
+ˆS(y2) = h(y2) ≥ hmin.
+If ˆS does not touch h in (y0, L]. By a similar argument to the proof of Claim 1 and
+appealing to the fact of Sx(L) = 0, one sees that ˆS is increasing in (y2, L], leading to
+
+16
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+ˆS(y2) < ˆS(y0), which contradicts with ˆS(y2) ≥ hmin > ˆS(y0).
+Hence, it is necessary
+that ˆS touches h in (y0, L]. Let y3 be the first point where ˆS touches h in (y0, L]. Thus,
+ˆS(x) < h(x) for all x ∈ (y0, y3) and ˆS(y3) = h(y3) ≥ hmin. Therefore, ˆS(x) < h(x) in
+the interval (y2, y3), ˆS(y0) < h(y0) = hmin and ˆS(y2), ˆS(y3) ≥ hmin. This implies that on
+[y2, y3], ˆS must attain its minimum at some y4 ∈ (y2, y3). By Claim 2, we can conclude
+that ˆS(y4) = h(y4), a contradiction again. So far, we have verified Claim 3.
+A similar reasoning as that of proving Claim 3 yields ˆS ≥ hmin on [0, L]. Thus (2.12)
+holds. Thanks to Claim 1 and Claim 3, (2.13) is true. It is also apparent that Claim 2
+implies (2.14). The proof is now complete.
+□
+3.3. Proof of Theorem 2.3. This subsection is devoted to the proof of Theorem 2.3. We
+begin with some lemmas as follows.
+Lemma 3.1. Assume that h ∈ C2([0, L]) and hx is non-decreasing in some neighborhood
+of ̺0 ∈ Θh. Let ˆS and µ be given as in Theorem 2.2. Then there exists a small ǫ0 > 0 such
+that
+ˆS(x) = h(x),
+∀x ∈ (̺0 − ǫ0, ̺0 + ǫ0) ∩ (0, L)
+and
+µ({x}) = Λ − h + dShxx
+η(x)
+,
+a.e. for x ∈ (̺0 − ǫ0, ̺0 + ǫ0) ∩ (0, L).
+Proof. By Theorem 2.2, we know that ̺0 ∈ {x ∈ [0, L] :
+ˆS(x) = h(x)}. In the sequel, we
+only consider the case of ̺0 ∈ (0, L), and the case of ̺0 = 0 or L can be treated similarly.
+There are three possibilities we have to distinguish:
+(1) ̺0 is an isolated point in the set {x ∈ [0, L] : ˆS(x) = h(x)};
+(2) ̺0 is an accumulation point in {x ∈ [0, L] : ˆS(x) = h(x)};
+(3) there is a small ǫ0 > 0 such that (̺0 − ǫ0, ̺0 + ǫ0) ⊂ {x ∈ [0, L] : ˆS(x) = h(x)}.
+In what follows, we will exclude (1) and (2). If (1) happens, then
+ˆS(̺0) = h(̺0) = hmin and
+ˆS < h
+in (̺0 − ǫ1, ̺0 + ǫ1) \ {̺0}
+for some small ǫ1 > 0.
+Note that µ([0, L]) < ∞. In view of this fact, one can apply the interior regularity theory
+for elliptic equations to (2.10) and assert that ˆS ∈ C1(0, L). Clearly, hx(̺0) = 0. Since
+ˆS(̺0) = h(̺0) = hmin and ˆS ≥ hmin due to (2.12), we infer that ˆSx(̺0) = 0.
+On the other hand, by (2.10), ˆS satisfies
+−dS ˆSxx = Λ − ˆS
+in (̺0 − ǫ1, ̺0 + ǫ1) \ {̺0}.
+(3.29)
+By using ˆSx(̺0) = 0 and (3.29), one can easily see that ˆS is increasing in (̺0 −ǫ1, ̺0) while
+ˆS is decreasing in (̺0, ̺0 + ǫ1). This implies that ˆS < hmin in (̺0 − ǫ1, ̺0 + ǫ1) \ {̺0},
+contradicting against (2.12). Thus, (1) is impossible.
+If (2) happens, without loss of generality, we can find two points, say z1, z2 with ̺0 <
+z1 < z2 < ̺0 + ǫ2 for some small ǫ2 > 0 such that
+ˆS(z1) = h(z1),
+ˆS(z2) = h(z2) and
+ˆS < h in (z1, z2).
+(3.30)
+By taking ǫ2 to be smaller if necessary, we may assume that hx(z1) ≤ hx(z2) due to the
+monotonicity of hx. Then, ˆS solves (3.29) in (z1, z2). By means of (3.30), we have
+ˆSx(z1) ≤ hx(z1),
+ˆSx(z2) ≥ hx(z2),
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+17
+leading to ˆSx(z1) ≤ ˆSx(z2). However, it follows from (3.29) that ˆSxx < 0 in (z1, z2), which
+gives ˆSx(z1) > ˆSx(z2), a contradiction. Hence, the possibility (2) has been ruled out.
+The above argument shows that (3) must hold. Now, since ˆS = h on [̺0 − ǫ0, ̺0 + ǫ0],
+we can multiply both sides of (2.11) by any function ζ ∈ C2([0, L]) with compact support
+on [̺0 − ǫ0, ̺0 + ǫ0] and integrate to conclude that
+dShxx + Λ − h − η(x)µ({x}) = 0,
+a.e. for x ∈ (̺0 − ǫ0, ̺0 + ǫ0),
+which yields the expression of µ({x}).
+□
+Lemma 3.2. Assume that h ∈ C2([0, L]), hx is non-decreasing on [0, L], and Θh = {τ0}
+for some τ0 ∈ (0, L). Then there exist two numbers τ1, τ2 with 0 < τ1 < τ0 < τ2 < L such
+that
+ˆS(x) = h(x),
+∀x ∈ [τ1, τ2],
+(3.31)
+and on [0, τ1) ∪ (τ2, L], ˆS satisfies
+
+
+
+
+
+
+
+−dS ˆSxx(x) = Λ − ˆS,
+x ∈ (0, τ1) ∪ (τ2, L),
+ˆSx(0) = ˆSx(L) = 0,
+ˆS(τ1) = h(τ1),
+ˆS(τ2) = h(τ2),
+(3.32)
+and µ satisfies
+µ({x}) = Λ − h + dShxx
+η(x)
+,
+a.e. for x ∈ (τ1, τ2),
+(3.33)
+µ({x}) = 0,
+∀x ∈ [0, τ1) ∪ (τ2, L].
+(3.34)
+Proof. Let us denote
+τ1 = inf{τ ∈ [0, τ0) :
+ˆS(x) = h(x), ∀x ∈ [τ, τ0]},
+τ2 = sup{τ ∈ (τ0, L] :
+ˆS(x) = h(x), ∀x ∈ [τ0, τ]}.
+Lemma 3.1 implies that τ1 and τ2 are well defined, and 0 ≤ τ1 < τ0 and τ0 < τ2 ≤ L. In
+addition, (3.31) and (3.33) hold.
+In light of the monotonicity of hx on [0, L], it is easily seen from the proof of Lemma
+3.1 that if τ1 > 0, then ˆS can not touch h in (0, τ1) and in turn µ([0, τ1)) = 0; similarly, if
+τ2 < L, ˆS can not touch h in (τ2, L) and so µ((τ2, L]) = 0.
+If τ1 > 0 and τ2 < L, we can use the analysis as in the proof of Claim 1 of Theorem 2.2 to
+conclude that ˆSx(0) = ˆSx(L) = 0. As µ([0, τ1) ∪ (τ2, L]) = 0, by (2.10) and the continuity
+of ˆS, a standard compactness argument of elliptic equations yields that ˆS solves (3.32) in
+the classical sense. Clearly, the solution of (3.32) is unique.
+It remains to prove τ1 > 0 and τ2 < L. Note that the monotonicity of hx, Θh = {τ0}
+and hx(τ0) = 0 ensure hx(0) < 0 and hx(L) > 0. Suppose that τ1 = 0, and so (3.31) holds
+on [0, τ2]. Now, given τ ∈ (0, τ0], integrating the S-equation over [0, τ] and using (3.31),
+
+18
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+we infer that
+−dSSx(τ −) =
+� τ
+0
+[Λ − S(y) − β(y)S(y)I(y) + γ(y)I(y)]dy
+=
+� τ
+0
+[Λ − S(y) − η(y)I(y)]dy +
+� τ
+0
+[γ(y) + η(y) − β(y)S(y)]I(y)dy
+→
+�
+[0,τ]
+[Λ − h(y) − η(y)µ](dy) =
+� τ
+0
+[−dShxx(y)]dy
+= −dShx(τ) + dShx(0),
+as dI → 0.
+That is, for any τ ∈ (0, τ0], it holds that
+Sx(τ −) → hx(τ) − hx(0),
+as dI → 0.
+Since hx is non-decreasing on [0, τ0] and hx(0) < 0, there exists a small ǫ0 > 0 such that
+for all x ∈ [τ0 − ǫ0, τ0],
+Sx(x−) ≥ 1
+2[hx(τ0) − hx(0)] = −1
+2hx(0) > 0
+for all small dI > 0. This implies that S is increasing on [τ0 − ǫ0, τ0] for all such small
+dI > 0. In view of S → h uniformly on [τ0 − ǫ0, τ0] as dI → 0, h must be non-decreasing
+on [τ0 − ǫ0, τ0], which is a contradiction against our assumption. Hence, τ1 > 0. Similarly,
+we have τ2 < L by using hx(L) > 0. As a consequence, we deduce (3.34). The proof is
+complete.
+□
+Similar to the argument of Lemma 3.1, we can conclude the following result.
+Lemma 3.3. Assume that h ∈ C2([0, L]), [̺1, ̺2] ⊂ Θh and hx is non-decreasing in some
+neighborhood of ̺1, ̺2. Let ˆS and µ be given as in Theorem 2.2. Then there exists a small
+ǫ0 > 0 such that
+ˆS(x) = h(x),
+∀x ∈ (̺1 − ǫ0, ̺2 + ǫ0) ∩ (0, L)
+and
+µ({x}) = Λ − h + dShxx
+η(x)
+,
+a.e. for x ∈ (̺1 − ǫ0, ̺2 + ǫ0) ∩ (0, L).
+Based upon Lemma 3.3, we can deduce the following result.
+Lemma 3.4. Assume that h ∈ C2([0, L]), Θh = [̺1, ̺2] and hx is non-decreasing on
+[0, ̺1] ∪ [̺2, L]. Let ˆS and µ be given as in Theorem 2.2. Then there exist two numbers
+τ1, τ2 with 0 < τ1 < ̺1 < ̺2 < τ2 < L such that all the assertions in Lemma 3.2 hold.
+With the aid of Lemmas 3.1-3.4, we are now in a position to prove Theorem 2.3.
+Proof of Theorem 2.3. We first prove (i). We proceed indirectly and suppose that ˆS ̸≡ h
+on [0, L]. Since ˆS touches h at least at the highest-risk point due to Theorem 2.2, we can
+find an interval, denoted by [ℓ1, ℓ2] ⊂ [0, L], such that ˆS < h in (ℓ1, ℓ2) and at the boundary
+point x = ℓi for i = 1, 2, either ˆS touches h (and so ˆS(ℓi) = h(ℓi)) or ˆS(ℓi) < h(ℓi). In
+the latter case, it is necessary that ℓi = 0 or L, and the analysis to deduce Claim 1 in the
+proof of Theorem 2.2 shows that ˆSx(ℓi) = 0. In any case, clearly ˆS satisfies
+�
+−dS ˆSxx = Λ − ˆS,
+x ∈ (ℓ1, ℓ2),
+ˆS(ℓi) = h(ℓi) or
+ˆSx(ℓi) = 0,
+i = 1, 2.
+(3.35)
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+19
+Thus, by our assumption, h is a sub-solution to problem (3.35), and max{Λ, maxx∈[0,L] h(x)}
+is a super-solution to (3.35). The well-known technique of sub-supersolution iteration, com-
+bined with the uniqueness of solutions to problem (3.35), allows us to conclude that ˆS ≥ h
+on [ℓ1, ℓ2], which leads to a contradiction. Hence, (2.15) holds, and (2.16) follows from
+(2.10) by using a test-function argument similarly as before. Therefore, (i) is proved.
+We next prove (ii). First of all, let us consider the case of τ0 ∈ (0, L). In this case,
+the assertions (2.17)-(2.20) follow from Lemma 3.2, and it remains to show that τ1, τ2 are
+uniquely determined by (2.21). As ˆS < h in [0, τ1), we have
+ˆS(x) = c1[ed−1/2
+S
+x + e−d−1/2
+S
+x] + Λ,
+∀x ∈ [0, τ1]
+for some c1 < 0. It then follows from ˆS(τ1) = h(τ1) that
+ˆS(x) = −
+Λ − h(τ1)
+ed−1/2
+S
+τ1 + e−d−1/2
+S
+τ1 (ed−1/2
+S
+x + e−d−1/2
+S
+x) + Λ,
+∀x ∈ [0, τ1].
+Note that ˆS is convex while h is concave in the interval [0, τ1), and moreover, ˆS ∈ C1([0, L])
+as shown before. Hence, ˆS must be tangent to h at x = τ1, which in turn implies that τ1 is
+the unique solution to ˆSx(τ1) = hx(τ1). Thus, τ1 is uniquely determined by the following
+equation:
+ed−1/2
+S
+τ1 − e−d−1/2
+S
+τ1
+ed−1/2
+S
+τ1 + e−d−1/2
+S
+τ1 = −d1/2
+S hx(τ1)
+Λ − h(τ1) .
+Similarly, τ2 is uniquely determined by the second equation of (2.21). The assertions in
+(ii)-(a) have been verified.
+We now consider the case of τ0 = L. In view of our assumption, clearly hx(0) < 0,
+hx(L) ≤ 0, and ˆS(L) = h(L).
+Assume that e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+> −
+d1/2
+S
+hx(L)
+Λ−h(L) . In order to deduce the desired conclusion in (ii)-
+(b1), one can follow the analysis of Lemmas 3.1 and 3.2. By checking the analysis there,
+one just needs to show that τ1 defined in the assertion (ii)-(a) satisfies τ1 > 0. It turns
+out that this amounts to rule out the situation that ˆS < h in [0, L). Suppose that ˆS < h
+in [0, L). Then, arguing as before, we see that ˆS satisfies −dS ˆSxx = Λ − ˆS in (0, L) and
+ˆSx(0) = 0. Solving this problem, we get ˆS(x) = c1[ed−1/2
+S
+x + e−d−1/2
+S
+x] + Λ for some c1 < 0.
+It then follows from ˆS(L) = h(L) that
+c1 = −
+Λ − h(L)
+ed−1/2
+S
+L + e−d−1/2
+S
+L.
+Thus, we get
+ˆSx(L) = −d−1/2
+S
+(Λ − h(L))e2Ld−1/2
+S
+− 1
+e2Ld−1/2
+S
++ 1
+.
+By means of ˆS < h in [0, L) and ˆS(L) = h(L), it is necessary that ˆSx(L) ≥ hx(L), which
+leads to
+e2Ld−1/2
+S
+− 1
+e2Ld−1/2
+S
++ 1
+≤ −d1/2
+S hx(L)
+Λ − h(L) ,
+contradicting with our assumption. Therefore, τ1 > 0 must hold, and (ii)-(b1) is proved.
+
+20
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+Assume that e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+≤ −
+d1/2
+S
+hx(L)
+Λ−h(L) . We first show that τ1 > 0 is impossible. On the
+contrary, we suppose that τ1 > 0, and by the above analysis, τ1 must solve the first equation
+of (2.21). Let us consider the following auxiliary problem:
+f(τ) = e2τd−1/2
+S
+− 1
+e2τd−1/2
+S
++ 1
++ d1/2
+S hx(τ)
+Λ − h(τ) ,
+τ ∈ [0, L].
+Since hx(τ) is non-decreasing, hx(τ) ≤ 0 on [0, L], h(τ) is non-increasing and h(τ) > Λ
+on [0, L], it is easy to check that
+d1/2
+S
+hx(τ)
+Λ−h(τ) is non-decreasing on [0, L]. Clearly, e2τd−1/2
+S
+−1
+e2τd−1/2
+S
++1
+is
+increasing on [0, L]. Therefore, f(τ) is increasing on [0, L]. Observe that f(L) = e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
++
+d1/2
+S
+hx(L)
+Λ−h(L) ≤ 0 due to our assumption. This implies that the first equation of (2.21) has no
+solution with respect to τ1 in [0, L), arriving at a contradiction. Hence, ˆS < h in [0, L) and
+µ([0, L)) = 0, and so ˆS solves (2.23). It remains to prove (2.24). Indeed, by integrating
+the sum of (1.6), we obtain
+ΛL −
+� L
+0
+S(x)dx =
+� L
+0
+η(x)I(x)dx,
+∀dI > 0.
+Letting dI → 0 yields
+ΛL −
+� L
+0
+ˆS(x)dx =
+�
+[0,L]
+η(x)µ(dx) = η(L)µ({L}).
+Here we used the fact of µ([0, L)) = 0. This gives (2.24), and thus the assertions in (ii)-(b2)
+hold true.
+The case of τ0 = 0 can be treated similarly as above. In view of Lemma 3.4 and the
+analysis above, the assertions in (iii) follow immediately. The proof is completed.
+□
+4. Discussions and numerical simulations
+In recent years, many reaction-diffusion models have been proposed to investigate the
+transmission dynamics of infectious diseases in a heterogeneous environment. For example,
+models associated with (1.1) have been studied in [2, 16, 18, 19, 35, 36, 39, 40, 49–52, 55,
+56, 59, 61, 68]. When the random diffusion is not present, such kind of models have been
+explored in [1, 3, 20, 21, 38, 42, 62, 66, 67] and the references therein. One may also refer
+to [14, 22, 23, 32, 33, 37, 41, 58, 63, 64, 70, 71] for relevant studies on the effect of random
+diffusion on the dynamics of infectious diseases.
+In this paper, we have investigated the steady state solution (namely, EE) of the SIS
+epidemic reaction-diffusion models (1.2) and (1.6), in which the disease transmission is
+governed by the well-known mass action infection mechanism, due to Kermack and McK-
+endrick [26]. In model (1.2), the total population number of the susceptible and infected
+populations is a constant, while in model (1.6), the total population number is varying,
+which results from the inclusion of the recruitment for the susceptible population and the
+death of the infected population. Our purpose is to determine the spatial profile of EE
+as the movement rate dI of the infected individuals tends to zero. Such kind of informa-
+tion may be useful for decision-makers to predict the pattern of disease occurrence and
+henceforth to develop effective disease control strategies.
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+21
+The previous works [39, 65] derived partial results regarding the spatial profile of EE
+for (1.2) and (1.6) as dI → 0; however, a precise characterization for the distribution of
+susceptible and infected populations is lacking. In the present work, we have provided a
+comprehensive understanding on this issue. Below we shall summarize the main theoretical
+findings of this paper, which will also be supported or complemented by our numerical
+simulation results.
+4.1. Profile of EE of model (1.2) as dI → 0. As pointed out before, when the risk
+function k(x) = γ(x)
+β(x) is a constant on the entire habitat [0, L], then (k, N
+L −k) is the unique
+EE of (1.2) provided that k < N
+L , while ( N
+L , 0) is the unique disease-free equilibrium of
+(1.2) provided that k ≥ N
+L . Indeed, in such a trivial case, one can follow the same analysis
+as in [16, Theorem 4.1] to conclude that (k, N
+L − k) is a global attractor of (1.1) if k < N
+L
+and ( N
+L , 0) is a global attractor of (1.4) if k ≥ N
+L . Thus, unless otherwise specified, we
+always assume below that the risk function k(x) = γ(x)
+β(x) is non-constant on [0, L].
+According to Theorem 2.1, for model (1.2), one finds that the susceptible population S
+converges to the positive constant kmin as dI → 0, which means that the susceptible will
+always distribute homogeneously on the entire habitat once the movement of the infected
+individuals is restricted to be sufficiently small. Nevertheless, the profile of the infected
+population I as dI → 0 crucially depends on the distribution behavior of the highest-risk
+set Θk of the risk function k(x). More precisely, concerning the profile of I for model (1.2),
+we have the following findings.
+(i) If Θk consists of a single point, then I must concentrate only at such a highest-risk
+point.
+(ii) If Θk contains only multiple isolated points, it follows from Remark 2.1 that I will
+also concentrate at least at one of those highest-risk points, and the disease will vanish
+elsewhere. As shown in Figure 1(a)-(b)-(c) for three typical cases, our simulation results
+suggest that I should concentrate at all such highest-risk points, though the population
+number of I at each such highest-risk point may vary, depending on the functions β, γ.
+(iii) If Θk contains at least one proper interval, then no concentration phenomenon
+occurs for the disease distribution, and the infected population will aggregate only on such
+intervals consisting of highest-risk points, regardless of whether there are isolated highest-
+risk points or not (see Figure 2(a)-(b)). Indeed, our numerical results indicate that the
+infected population will aggregate on all such intervals consisting of highest-risk points (see
+Figure 2(c)); however the population number of I at each such interval may be different,
+depending on the functions β, γ.
+4.2. Profile of EE of model (1.6) as dI → 0. For model (1.6), for the general H¨older
+continuous risk function h, under the condition (2.9), as dI → 0, we know from Theorem
+2.2 that the susceptible population S converges to a positive function ˆS, which is non-
+constant unless h is constant. The infected population I converges to a positive Radon
+measure µ, whose support is contained in the region in which ˆS touches h; in other words,
+the disease stays only within the place where the susceptible population distributes along
+the risk function. If the risk function h is of C2, we see from Lemma 3.1 and Lemma
+3.3 that the infected population aggregates at least in a neighborhood of the highest-risk
+locations.
+Furthermore, when h ∈ C2([0, L]), in light of Theorem 2.3, one can draw the following
+conclusions concerning the asymptotic profile of I.
+
+22
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+0
+0.2
+0.4
+0.6
+0.8
+1
+−5
+0
+5
+10
+15
+20
+25
+30
+35
+40
+x
+ 
+ 
+k(x)
+S(x)
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+−10
+0
+10
+20
+30
+40
+50
+60
+70
+80
+x
+ 
+ 
+S(x)
+I(x)
+0
+0.5
+1
+0.4
+0.6
+0.8
+1
+x
+ 
+ 
+k(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+−10
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+x
+ 
+ 
+S(x)
+I(x)
+0
+0.5
+1
+0.4
+0.6
+0.8
+1
+x
+ 
+ 
+k(x)
+(a) Θk = {1
+2}
+(b) Θk = {1
+8, 1
+2}
+(c) Θk = {1
+8, 3
+8, 1}
+Figure 1. Numerical simulations of the solution profile of model (1.2),
+where L = 1, N = 2, dS = 1, dI = 10−7, β(x) = 1 + 1
+2 sin(2πx), γ(x) =
+k(x)β(x), kmin
+=
+1
+2 and k(x) is chosen as follows.
+In (a), k(x) =
+1 + 1
+2 cos(2πx).
+In (b), k(x) = 1 − 4x, 0 ≤ x <
+1
+8; k(x) = 4x,
+1
+8 ≤
+x <
+1
+4; k(x) =
+3
+2 − 2x,
+1
+4 ≤ x <
+1
+2; k(x) = x,
+1
+2 ≤ x ≤ 1.
+In (c),
+k(x) = 1 − 4x, 0 ≤ x < 1
+8; k(x) = 4x,
+1
+8 ≤ x < 1
+4; k(x) = 2 − 4x,
+1
+4 ≤ x <
+3
+8; k(x) = 4x − 1,
+3
+8 ≤ x < 1
+2; k(x) = 3
+2 − x,
+1
+2 ≤ x ≤ 1.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+x
+ 
+ 
+k(x)
+S(x)
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+x
+ 
+ 
+k(x)
+S(x)
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+x
+ 
+ 
+k(x)
+S(x)
+I(x)
+(a) Θk = [ 1
+4, 3
+4]
+(b) Θk = [ 1
+4, 1
+2] ∪ { 7
+8}
+(c) Θk = [0, 1
+16] ∪ { 3
+8} ∪ [ 5
+8, 3
+4]
+Figure 2. Numerical simulations of the solution profile of model (1.2),
+where L = 1, N = 2, dS = 1, dI = 10−5, β(x) = 1, γ(x) = k(x)β(x), kmin = 1
+2
+and k(x) is chosen as follows. In (a), Θk = [ 1
+4, 3
+4], k(x) = 1
+2 + 5(x − 1
+4)2, 0 ≤
+x < 1
+4; k(x) = 1
+2,
+1
+4 ≤ x < 3
+4; k(x) = 1
+2 + 5(x − 3
+4)2,
+3
+4 ≤ x ≤ 1.
+In (b),
+Θk = [ 1
+4, 1
+2] ∪ { 7
+8}, k(x) = 1
+2 + 4(x − 1
+4)2, 0 ≤ x < 1
+4; k(x) = 1
+2,
+1
+4 ≤ x <
+1
+2; k(x) = 1
+2 + 4(x − 1
+2)2,
+1
+2 ≤ x < 3
+4; k(x) = 1
+2 + 16(x − 7
+8)2,
+3
+4 ≤ x ≤ 1. In
+(c), Θk = [0, 1
+16] ∪ { 3
+8} ∪ [ 5
+8, 3
+4], k(x) = 1
+2, 0 ≤ x <
+1
+16; k(x) = 8x,
+1
+16 ≤ x <
+1
+8; k(x) = 1,
+1
+8 ≤ x < 1
+4; k(x) = 2 − 4x,
+1
+4 ≤ x < 3
+8; k(x) = 8x − 5
+2,
+3
+8 ≤ x <
+1
+2; k(x) = 11
+2 − 8x,
+1
+2 ≤ x < 5
+8; k(x) = 1
+2,
+5
+8 ≤ x < 3
+4; k(x) = 2
+3x,
+3
+4 ≤ x ≤ 1.
+(i) For any risk function h satisfying −dShxx ≤ Λ − h in (0, L), hx(0) ≥ 0, hx(L) ≤ 0,
+and condition (2.9) (for instance, h < Λ is a positive constant), the infected population
+must occupy the entire habitat, and it also forms the concentration phenomenon at the
+boundary point x = 0 (or x = 1) if hx(0) > 0 (or hx(1) < 0), which is also the highest-risk
+location; see Theorem 2.3(i) and the numerical illustrations in Figure 3(a)-(b)-(c).
+(ii) For any convex risk function h (i.e., hxx ≥, ̸≡ 0 on [0, L]) fulfilling (2.9), the infected
+population usually stays only in part of the habitat. In particular, by Theorem 2.3(ii)(iii),
+we can observe the following behaviors.
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+23
+0
+0.5
+1
+0.95
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+1.35
+1.4
+x
+ 
+ 
+h(x)
+0
+0.5
+1
+0.95
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+1.35
+1.4
+x
+ 
+ 
+S(x)
+0
+0.5
+1
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+x
+ 
+ 
+I(x)
+0
+0.5
+1
+0.95
+1
+1.05
+1.1
+1.15
+x
+ 
+ 
+h(x)
+0
+0.5
+1
+0.95
+1
+1.05
+1.1
+1.15
+x
+ 
+ 
+S(x)
+0
+0.5
+1
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+x
+ 
+ 
+I(x)
+0
+0.5
+1
+0.95
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+x
+ 
+ 
+h(x)
+0
+0.5
+1
+0.95
+1
+1.05
+1.1
+1.15
+1.2
+1.25
+1.3
+x
+ 
+ 
+S(x)
+0
+0.5
+1
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+x
+I(x)
+ 
+ 
+I(x)
+(a) h(x) = 1 + 5x2(1 − x)2
+(b) h(x) = 1 + x2(1 − x)
+(c) h(x) = 1 + x(1 − x)
+Figure 3. Numerical simulations of the solution profile of model (1.6),
+where β(x) = 1 + 1
+2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − η(x), dS = 1, dI =
+10−8, Λ = 10. In (a), h(x) = 1 + 5x2(1 − x)2, in (b), h(x) = 1 + x2(1 − x),
+and in (c), h(x) = 1 + x(1 − x).
+(ii-a) If the highest-risk set Θh contains only one point, denoted by τ0, then the distri-
+bution behavior of the infected population is affected by whether τ0 is a boundary point or
+an interior point. More precisely, when τ0 is an interior point, then the infected population
+resides in a certain left neighborhood of τ0, staying away from the boundary points x = 0
+and x = 1. In fact, such a neighborhood can be calculated through the formula (2.21).
+One may further refer to Figure 4(a).
+However, if τ0 is a boundary point, say τ0 = L, then the infected population stays in a
+certain neighborhood of L provided e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+> −
+d1/2
+S
+hx(L)
+Λ−h(L) , while the infected population
+concentrates only at L provided e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
+≤ −
+d1/2
+S
+hx(L)
+Λ−h(L) . Since hx(L) ≤ 0 in this situation,
+the infected population stays in a certain neighborhood of L provided for all dS > 0 if
+hx(L) = 0. If hx(L) < 0, it should be noted that the function q(dS) = d−1/2
+S
+e2Ld−1/2
+S
+−1
+e2Ld−1/2
+S
++1
++
+hx(L)
+Λ−h(L) deceases in dS ∈ (0, ∞), limdS→0 q(dS) = ∞ and limdS→∞ q(dS) =
+hx(L)
+Λ−h(L) < 0. As a
+result, there is a unique d∗
+S > 0 such that q(d∗
+S) = 0, and in turn the infected population
+stays in a left neighborhood of L for 0 < dS < d∗
+S , and the infected population concentrates
+only at L for all dS ≥ d∗
+S.
+(ii-b) If the highest-risk set Θh contains only an interval, then the infected population
+resides in a certain neighborhood of such an interval. Again, such a neighborhood can be
+calculated through the formula (2.21). See the numerical simulation in Figure 4(b).
+(ii-c) For a general H¨older continuous risk function h, we can conclude that the disease
+must exist in all isolated highest-risk point(s) and a neighborhood of each highest-risk
+interval if exists; nevertheless, it is challenging to give a precise characterization for the
+distribution behavior of the susceptible and infected populations, due to the mathematical
+difficulties on the analysis of the free boundary problem (2.10). We have performed the
+numerical simulations in Figure 5(a)-(b) as an illustration.
+In what follows, we would like to make some more discussions on (ii-a) above in the case
+that τ0 is a boundary point. For example, we take τ0 = L, and also assume that hx(L) < 0.
+On the one hand, by fixing hx(L), we have known from (ii-b) that large diffusion rate dS
+can result in the disease concentration only at the location L and small diffusion rate dS
+
+24
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+2
+4
+6
+8
+10
+12
+x
+ 
+ 
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+1.1
+1.2
+1.3
+x
+ 
+ 
+τ1
+ τ2
+h(x)
+S(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+20
+25
+30
+x
+ 
+ 
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.5
+0.6
+0.7
+0.8
+x
+ 
+ 
+τ1
+ τ2
+h(x)
+S(x)
+(a) Θh = {1
+2}
+(b) Θh = [1
+4, 3
+4]
+Figure 4. Numerical simulations of the solution profile of model (1.6),
+where β(x) = 1 + 1
+2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − η(x), dS =
+1, dI = 10−10, Λ = 10, and h(x) = 1 + (x − 1
+2)2 in (a), while in (b), h(x) =
+1
+2 + 5(x − 1
+4)2, 0 ≤ x < 1
+4; h(x) = 1
+2,
+1
+4 ≤ x < 3
+4; h(x) = 1
+2 + 5(x − 3
+4)2,
+3
+4
+≤ x < 1.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+20
+25
+30
+35
+x
+ 
+ 
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.5
+0.6
+0.7
+0.8
+x
+ 
+ 
+h(x)
+S(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+x
+ 
+ 
+I(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+x
+ 
+ 
+h(x)
+S(x)
+(a) Θh = [1
+4, 1
+2] ∪ {7
+8}
+(b) Θh = [0, 1
+16] ∪ {3
+8} ∪ [5
+8, 3
+4]
+Figure 5. Numerical simulations of the solution profile of model (1.6),
+where dS = 1, dI = 10−5, β(x) = 1 + 1
+2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) −
+η(x), Λ = 10. In (a) and (b), h(x) is chosen to be the same as k(x) in Figure
+2(b) and Figure 2(c), respectively.
+will cause the disease to distribute in a left neighborhood of L. On the other hand, once
+dS is fixed, the concentration phenomenon happens only if −hx(L) is properly large. This
+motivates us to see whether a similar concentration phenomenon could occur at an interior
+isolated highest-risk point if the risk function h is merely H¨older continuous. To illustrate
+this phenomenon, let us consider the following risk function whose curve is the connection
+of two segments:
+h(x) =
+�
+a1
+�
+x − L
+2
+�
++ Λ
+4 ,
+x ∈
+�
+0, L
+2
+�
+,
+a2
+�
+x − L
+2
+�
++ Λ
+4 ,
+x ∈
+� L
+2 , L
+�
+,
+(4.1)
+with a1 < 0, a2 > 0. Obviously, h is merely Lipschitz continuous at x = L
+2 . Our numer-
+ical simulation results demonstrate that if the slopes |a1|, a2 are properly large, then the
+infected population will concentrate at x = L
+2 (Figure 6(a)); if |a1|, a2 are small, then the
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+25
+infected population will aggregate in a neighborhood of x = L
+2 (Figure 6(b)); and if |a1| is
+small while a2 is large, then the infected population will aggregate in a left-neighborhood
+of x =
+L
+2 (Figure 6(b)). These profiles behave rather differently from that in Theorem
+2.3(ii) for h ∈ C2([0, L]), as shown by Figure 4(a). Therefore, the numerical results reveal
+that the smoothness of h may have a substantial effect on the spatial distribution of the
+disease.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+50
+100
+150
+200
+250
+x
+ 
+ 
+I(x)
+0
+0.5
+1
+2
+3
+4
+5
+6
+7
+8
+x
+ 
+ 
+h(x)
+S(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+20
+25
+30
+35
+40
+x
+ 
+ 
+I(x)
+ 
+ 
+0
+0.5
+1
+2
+3
+4
+5
+6
+7
+8
+x
+ 
+ 
+h(x)
+S(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+20
+40
+60
+80
+100
+120
+x
+ 
+ 
+I(x)
+0
+0.5
+1
+2
+3
+4
+5
+6
+7
+8
+x
+ 
+ 
+h(x)
+S(x)
+(a) a1 = −10, a2 = 10
+(b) a1 = −1, a2 = 1
+(c) a1 = −1, a2 = 10
+Figure 6. Numerical simulations of the solution profile of model (1.6),
+where β(x) = 1 + 1
+2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − η(x), L = 1, dS =
+1, dI = 10−5, Λ = 10 and h(x) is given by (4.1).
+4.3. Conclusion. The discussions in the above two subsections, together with the numer-
+ical simulations, show that the spatial profile of the susceptible and infected populations
+of (1.2) and (1.6) with respect to small movement rate of the infected individuals are
+rather different. This is caused by the presence of the recruitment term for the suscep-
+tible population and the death rate for the infected population. On the other hand, we
+would like to mention that the recent works [11–14, 31, 34, 69] studied various kinds of
+reaction-diffusion-advection SIS epidemic models, in which the advection term represents
+some passive movement in a certain direction, e.g., due to external environmental forces
+such as water flow [46–48, 57], wind [15] and so on. In particular, if an advection is present
+in (1.2) and stands for, for instance, the water flow, it was proved in [13, Theorem 1.4]
+that, as dI → 0, the susceptible population converges to a positive function while the in-
+fected population concentrates only at the downstream of the water flow; a similar result
+can be shown to hold for the corresponding system (1.6). Such a distribution behavior is
+essentially different from that of (1.2) and (1.6) with small dI.
+In summary, our results here, combined with those of [13, 31, 40], suggest that the re-
+cruitment term for the susceptible population, the death rate for the infected population
+(even the smoothness of the associated risk function) as well as the advection can lead
+to significant impacts on the disease transmission and thus decision-makers should attach
+great importance to these factors when taking measures such as the lockdown and quaran-
+tine to control the movement or immigration of the infected individuals so as to eliminate
+the disease infection.
+
+26
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+5. Appendix
+In this appendix, we always let Ω be a smooth and bounded domain in Rn (n ≥ 1). Given
+f ∈ C(Ω), consider the following eigenvalue problem with Neumann boundary condition:
+�
+−D∆φ + f(x)φ = λφ
+in Ω,
+∂φ
+∂ν = 0
+on ∂Ω,
+(5.2)
+where ν(x) is the unit exterior normal vector of ∂Ω at x, and the coefficient D is a positive
+constant.
+We start with a well-known fact concerning the asymptotic behavior of the principal
+eigenvalue of (5.2) with respect to small diffusion; one may refer to, for example, [45,
+Lemma 3.1].
+Lemma 5.1. Let λ1(D, f) be the principal eigenvalue of (5.2). Then it holds that
+lim
+D→0 λ1(D, f) = min
+x∈Ω f(x).
+We next recall the L1-estimate for the weak solution (due to [6]) of the following linear
+elliptic problem:
+−∆w + c(x)w = g
+in Ω,
+∂w
+∂ν = 0 on ∂Ω.
+(5.3)
+Lemma 5.2. (a) (Global estimates)
+Assume that c ∈ L∞(Ω), g ∈ L1(Ω) and let w ∈
+W 1,1(Ω) be a weak solution of (5.3). Then, for any r ∈ [1, n/(n − 1)), we have w ∈ W 1,r(Ω)
+and the following estimate
+∥w∥W 1,r(Ω) ≤ C∥g∥L1(Ω),
+where the positive constant C is independent of w.
+(b) (Interior estimates) Assume that Ω′ ⊂⊂ Ω is a smooth domain, c ∈ L∞(Ω), g ∈
+L1(Ω), and let w ∈ W 1,1(Ω) be a weak solution to the equation −∆w + c(x)w = g. Then,
+for any r ∈ [1, n/(n − 1)), we have w ∈ W 1,r(Ω′) and the following estimate
+∥w∥W 1,r(Ω′) ≤ C∥g∥L1(Ω),
+where the positive constant C is independent of w.
+At last, we state a Harnack-type inequality for weak solutions (see, e.g., [43] or [53]),
+whose strong form was obtained in [44].
+Lemma 5.3. (a) (Global Harnack inequality)
+Let c ∈ Lr(Ω) for some r > n/2.
+If
+w ∈ W 1,2(Ω) is a non-negative weak solution of the boundary value problem
+−∆w + c(x)w = 0 in Ω,
+∂w
+∂ν = 0 on ∂Ω,
+then there is a constant C, determined only by ∥c∥r, r and Ω such that
+sup
+Ω
+w ≤ C inf
+Ω w.
+(b) (Local Harnack inequality) Let Ω′ ⊂⊂ Ω be a smooth domain and c ∈ Lr(Ω) for some
+r > n/2. If w ∈ W 1,2(Ω) is a non-negative weak solution of the equation −∆w+c(x)w = 0,
+then there is a constant C, determined only by ∥c∥r, r, Ω and Ω′, such that
+sup
+Ω′ w ≤ C inf
+Ω′ w.
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+27
+References
+[1] L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS
+epidemic patch model, SIAM J. Appl. Math., 67(2007), 1283-1309.
+[2] L.J.S. Allen, B.M. Bolker, Y.Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS
+epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21(2008), 1-20.
+[3] L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Spatial patterns in a discrete-time SIS patch model,
+J. Math. Biol., 58(2009), 339-375.
+[4] R.M. Anderson, R.M. May, Population biology of infectious diseases, Nature, 280(1979), 361-367.
+[5] D. Balcan, et al., Multiscale mobility networks and the spatial spreading of infectious diseases, Proc.
+Natl Acad. Sci. USA, 106(2009), 21484-21489.
+[6] H. Brezis, W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Jpn.,
+25(1973), 565-590.
+[7] T. Britton, F. Ball, P. Trapman, A mathematical model reveals the influence of population hetero-
+geneity on herd immunity to SARS-CoV-2, Science, 369(2020), 846-849.
+[8] T. Britton, F. Ball, P. Trapman, The disease-induced herd immunity level for Covid-19 is substantially
+lower than the classical herd immunity level, preprint, arXiv:2005.03085.
+[9] D. Brockmann, D. Helbing, The hidden geometry of complex, network-driven contagion phenomena,
+Science, 342(2013), 1337-1342.
+[10] K. Castellano, R.B. Salako, On the effect of lowering population’s movement to control the spread of
+an infectious disease, J. Differential Equations, 316(2022), 1-27.
+[11] R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic
+model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 26(2021), 2997-3022.
+[12] R. Cui, K.-Y. Lam, Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model
+in advective environments, J. Differential Equations, 263(2017), 2343-2373.
+[13] R. Cui, H. Li, R. Peng, M. Zhou, Concentration behavior of endemic equilibrium for a reaction-
+diffusion-advection SIS epidemic model with mass action infection mechanism, Calc. Var. Partial
+Differential Equations, 60(2021), paper no. 184, 38 pp.
+[14] R. Cui, Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equa-
+tions, 261(2016), 3305-3343.
+[15] K.A. Dahmen, D.R. Nelson, N.M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41(2000),
+1-23.
+[16] K. Deng, Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh
+Sect. A, 146(2016), 929-946.
+[17] F. Di Lauro, et al., The impact of network properties and mixing on control measures and
+disease-induced herd immunity in epidemic models:
+a mean-field model perspective, preprint,
+arXiv:2007.06975.
+[18] Z. Du, R. Peng, A priori L∞-estimate for solutions of a class of reaction-diffusion systems, J. Math.
+Biol., 72(2016), 429-1439.
+[19] D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79(2019), 1581-1606.
+[20] D. Gao, C-P. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc., 148(2020),
+1709-1722.
+[21] D. Gao, S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci.,
+232(2011), 110-115.
+[22] J. Ge, K. Kim, Z. Lin, H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk
+domain, J. Differential Equations, 259(2015), 5486-5509.
+[23] S. Han, C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear
+incidence, Appl. Math. Lett., 98(2019), 114-120.
+[24] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42(2000), 599-653.
+[25] J.S. Jia, et al., Population flow drives spatio-temporal distribution of COVID-19 in China, Nature,
+582(2020), 389-394.
+[26] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–I, Proc.
+Roy. Soc. London Ser. A, 115(1927), 700-721.
+[27] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–I, Bull.
+Math. Biol., 53(1991), 33-55.
+
+28
+R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU
+[28] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–II. The
+problem of endemicity, Bull. Math. Biol., 53(1991), 57-87.
+[29] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–III. Fur-
+ther studies of the problem of endemicity, Bull. Math. Biol., 53(1991), 89-118.
+[30] M.U.G. Kraemer, et al., The effect of human mobility and control measures on the COVID-19 epi-
+demic in China, Science, 368(2020), 493-497.
+[31] K. Kuto, H. Matsuzawa, R. Peng, Concentration profile of the endemic equilibria of a reaction-
+diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56(2017), paper
+no. 112, 28 pp.
+[32] C. Lei, F. Li, J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence
+in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23(2018), 4499-4517.
+[33] C. Lei, J. Xiong, X. Zhou, Qualitative analysis on an SIS epidemic reaction-diffusion model with
+mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete
+Contin. Dyn. Syst. Ser. B, 25(2020), 81-98.
+[34] C. Lei, X. Zhou, Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-
+advection SIS epidemic model with spontaneous infection, Discrete Contin. Dyn. Syst. Ser. B, to
+appear.
+[35] B. Li, Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal.
+Appl., 475(2019), 1910-1926.
+[36] B. Li, H. Li, Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew.
+Math. Phys., 68(2017), no. 4, Art. 96, 25 pp.
+[37] B. Li, J. Zhou, X. Zhou, Asymptotic profiles of endemic equilibrium of a diffusive SIS epidemic
+system with nonlinear incidence function in a heterogeneous environment, Proc. Amer. Math. Soc.,
+148(2020), 4445-4453.
+[38] H. Li, R. Peng, Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch
+models, J. Math. Biol., 79(2019), 1279-1317.
+[39] H. Li, R. Peng, F.-B. Wang, Vary total population enhances disease persistence: qualitative analysis
+on a diffusive SIS epidemic model, J. Differential Equations, 262(2017), 885-913.
+[40] H. Li, R. Peng, Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with
+mass action mechanism and birth-death effect: analysis, simulations, and comparison with other
+mechanisms, SIAM J. Appl. Math., 78(2018), 2129-2153.
+[41] H. Li, R. Peng, T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-
+dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31(2018), 26-56.
+[42] M.Y. Li, Z. Shuai, Global stability of an epidemic model in a patchy environment, Can. Appl. Math.
+Q., 17(2009), 175-187.
+[43] G.M. Lieberman, Bounds for the steady-state Sel’kov model for arbitrary p in any number of dimen-
+sions, SIAM J. Math. Anal., 36(2005), 1400-1406.
+[44] C.S. Lin, W.M. Ni, I. Takagi,
+Large amplitude stationary solutions to a chemotaxis system, J.
+Differential Equations, 72(1988), 1-27.
+[45] Y. Lou, T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without
+dominance, J. Differential Equations, 225(2006), 624-665.
+[46] F. Lutscher, M.A. Lewis, E. McCauley, Effects of heterogeneity on spread and persistence in rivers,
+Bull. Math. Biol., 68(2006), 2129-2160.
+[47] F. Lutscher, E. McCauley, M.A. Lewis, Spatial patterns and coexistence mechanisms in systems with
+unidirectional flow, Theor. Popul. Biol., 71(2007), 267-277.
+[48] F. Lutscher, E. Pachepsky, M.A. Lewis, The effect of dispersal patterns on stream populations, SIAM
+Rev., 47(2005), 749-772.
+[49] P. Magal, G. Webb, Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity,
+31(2018), 5589-5614.
+[50] P. Magal, G. Webb, Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models,
+SIAM J. Appl. Math., 79(2019), 284-304.
+[51] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model.
+Part I, J. Differential Equations, 247(2009), 1096-1119.
+[52] R. Peng, S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model,
+Nonlinear Anal., 71(2009), 239-247.
+
+TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM
+29
+[53] R. Peng, J. Shi, M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis
+and saturation law, Nonlinearity, 21(2008), 1471-1488.
+[54] R. Peng, Y. Wu, Global L∞-bounds and long-time behavior of a diffusive epidemic system in a
+heterogeneous environment, SIAM J. Math. Anal., 53(2021), 2776-2810.
+[55] R. Peng, F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion
+model: Effects of epidemic risk and population movement, Phys. D. 259(2013), 8-25.
+[56] R. Peng, X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Non-
+linearity, 25(2012), 1451-1471.
+[57] R. Peng, X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic-
+parabolic problem with applications, Calc. Var. Partial Differential Equations, 54(2015), 1611-1642.
+[58] P. Song, Y. Lou, Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment,
+J. Differential Equations, 267(2019), 5084-5114.
+[59] J. Suo, B. Li, Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent
+incidence function in a heterogeneous environment, Math. Biosci. Eng., 17(2019), 418-441.
+[60] H. Tian, et al., An investigation of transmission control measures during the first 50 days of the
+COVID-19 epidemic in China, Science, 368(2020), 638-642.
+[61] Y. Tong, C. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially
+heterogeneous environment, Nonlinear Anal. Real World Appl., 41(2018), 443-460.
+[62] C. Vargas-De-Leon, A. Korobeinikov, Global stability of a population dynamics model with inhibition
+and negative feedback, J. Math. Medicine and Biol., 30(2013), 65-72.
+[63] J. Wang, X. Wu, Dynamics and profiles of a diffusive Cholera model with bacterial hyperinfectivity
+and distinct dispersal rates, J. Dyn. Diff. Equat., 2021, https://doi.org/10.1007/s10884-021-09975-3
+[64] J. Wang, J. Wang, Analysis of a reaction-diffusion Cholera model with distinct dispersal rates in the
+human population, J. Dyn. Diff. Equat., 33(2021), 549-575.
+[65] X. Wen, J. Ji, B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model
+with mass action infection mechanism, J. Math. Anal. Appl., 458(2018), 715-729.
+[66] D. Wodarz, J.P. Christensen, A.R. Thomsen, The importance of lytic and nonlytic immune responses
+in viral infections, Trends Immunol., 23(2002), 194-200.
+[67] D. Wodarz, M.A. Nowak, Immune response and viral phenotype:
+do replication rate and cy-
+topathogenicity influence virus load? J. Theor. Med., 2(2000), 113-127.
+[68] Y. Wu, X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass
+action infection mechanism, J. Differential Equations, 261(2016), 4424-4447.
+[69] J. Zhang, R. Cui, Asymptotic behavior of an SIS reaction-diffusion-advection model with saturation
+and spontaneous infection mechanism, Z. Angew. Math. Phys., 71(2020), paper no. 150, 21 pp.
+[70] S. Zhu, J. Wang, Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear
+source in spatially heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 25(2020), 1999-
+2019.
+[71] S. Zhu, J. Wang, Asymptotic profiles of steady states for a diffusive SIS epidemic model with spon-
+taneous infection and a logistic source, Commun. Pure Appl. Anal., 19(2020), 3323-3340.
+

.gitattributes

@@ -191,3 +191,8 @@ dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf filter=lfs diff=lfs merge=lfs -tex
 MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf filter=lfs diff=lfs merge=lfs -text
 BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf filter=lfs diff=lfs merge=lfs -text
 2tFKT4oBgHgl3EQf7y43/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf filter=lfs diff=lfs merge=lfs -text
+6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf filter=lfs diff=lfs merge=lfs -text
+wdE3T4oBgHgl3EQfOgk_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+YNFQT4oBgHgl3EQfdzaO/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf filter=lfs diff=lfs merge=lfs -text

69AyT4oBgHgl3EQfcvd4/content/tmp_files/2301.00289v1.pdf.txt

@@ -0,0 +1,604 @@
+Stability Analysis of Picard Iteration for Coupled Neutronics/Thermal-Hydraulics Simulations 
+Dean Wang 
+Nuclear Engineering Program, The Ohio State University, Columbus, OH 43210 
+[email protected] 
+David P. Griesheimer 
+Bettis Atomic Power Laboratory, Bechtel Marine Propulsion Corporation, West Mifflin, PA 15122 
+[email protected] 
+INTRODUCTION 
+Reactor core analysis often needs to solve a multiphysics 
+nonlinear coupled system, including neutron transport, 
+thermal-hydraulics, and other important physics phenomena. 
+One straightforward method for solving such a coupled 
+system is Picard fixed-point iteration [1], which alternates 
+between solving individual physics problems separately. 
+However, many numerical studies show that Picard iteration 
+can be unstable, and a user-defined relaxation is usually 
+required to achieve convergence [2-4]. 
+In this paper, we present a formal Fourier analysis (FA) 
+of Picard iteration for the coupled neutronics/thermal 
+hydraulics (N/TH) problem and derive theoretical predictions 
+for the spectral radius of Picard iteration for such coupled 
+calculations as a function of the temperature difference 
+between the fuel and coolant, temperature coefficients of 
+cross sections (i.e., Doppler feedback), scattering ratio, and 
+core height. An optimal underrelaxation factor is also derived 
+based on the Fourier analysis.    
+ 
+FORMULATION AND ALGORITHM 
+We consider the following simple one-group, planar-
+geometry k-eigenvalue problem on the domain 0	 ≤ 	𝑥	 ≤ 	𝐿 
+with reflective boundary conditions: 
+𝜇
+!"($,&)
+!$
++ Σ((𝑇)𝜓(𝑥, 𝜇)  
+=
+)
+* Σ+(𝑇)𝜙(𝑥) +
+)
+*,!"" 𝜈Σ-(𝑇)𝜙(𝑥) ,               (1) 
+and the simplified heat transfer equation for a single typical 
+pressurized water reactor (PWR) fuel pin:  
+𝑇 = 𝑇. + 𝐴Σ-(𝑇)𝜙(𝑥) ,  
+        (2) 
+with 
+𝐴 = 𝜋𝑟-/
+* 𝜅𝑅( ,                             (3a) 
+and  
+𝑅( = 6
+)
+01," +
+)
+*12#3# +
+)
+*1,$ ln 9
+2$%
+2$&: +
+)
+*12$%3; ,        (3b) 
+where 
+𝜓 = neutron angular flux 
+𝜙 = ∫
+𝜓(𝑥, 𝜇)𝑑𝜇
+)
+4)
+, neutron scalar flux 
+Σ( = macroscopic total cross section  
+Σ+ = macroscopic scattering cross section 
+Σ- = macroscopic fission cross section 
+ν = average neutron yields per fission 
+𝑘5-- = effective multiplication factor 
+𝜅 = average energy released per fission 
+𝑇 = volume averaged fuel temperature 
+𝑇. = bulk coolant temperature 
+𝑟-/ = fuel radius 
+𝑟67 = cladding inner radius 
+𝑟6/ = cladding outer radius 
+𝑟8 =
+2$&92$%
+*
+, mean radius in the gap 
+𝑘- = fuel thermal conductivity 
+𝑘6 = cladding thermal conductivity 
+ℎ8 = effective gap conductance 
+ℎ = coolant convection heat transfer coefficient 
+Note that the linear heat generation rate (or linear power) 
+of the fuel rod, 𝑞′, can be calculated by 
+𝑞:(𝑥) = 𝜋𝑟-/
+* 𝜅Σ-(𝑇)𝜙(𝑥) . 
+             (4) 
+Picard iteration is used to solve the above coupled N/TH 
+system as follows. The transport equation is solved first, then 
+the fuel temperature is calculated using the newly obtained 
+thermal power (neutron flux). An underrelaxation factor is 
+introduced in the temperature update. Note that the transport 
+iteration is fully converged during each TH update. 
+𝜇
+!"(()*)($,&)
+!$
++ Σ(C𝑇(,)D𝜓(,9))  
+=
+)
+* Σ+C𝑇(,)D𝜙(,9))(𝑥) +
+)
+*,!"" 𝜈Σ-C𝑇(,)D𝜙(,9)) ,      (5) 
+𝑇∗ = 𝑇. + 𝐴Σ-C𝑇(,)D𝜙(,9))(𝑥) ,                (6a) 
+𝑇(,9)) = 𝜔𝑇∗ + (1 − 𝜔)𝑇(,) ,                  (6b) 
+where 𝜔 is the underrelaxation factor and the superscript 𝑘 
+denotes the iteration number.  
+ 
+LINEARIZATION 
+To perform Fourier analysis of the coupled N/TH 
+problem, we need to first linearize the system of equations. 
+We define the following linearized variables: 
+𝜓(𝑥, 𝜇) = 𝜓<(𝑥, 𝜇) + 𝜀𝜓)(𝑥, 𝜇) ,            (7a) 
+
+𝜙(𝑥) = 𝜙<(𝑥) + 𝜀𝜙)(𝑥) ,                   (7b) 
+𝑘5-- = 𝑘5--,/ , 
+ 
+           (7c) 
+𝑇(𝑥) = 𝑇< + 𝜀𝑇)(𝑥) , 
+ 
+  (7d) 
+Σ7(𝑇) = Σ7< + Σ7)(𝑇 − 𝑇<)  
+= Σ7< + 𝜀Σ7)𝑇)(𝑥) ,  𝑖 = 𝑡, 𝑠, 𝑓, 𝑎               (7e) 
+Note that 𝑘5-- = 𝑘5--,/ due to the flux normalization. 
+The cross sections are assumed to be linearly dependent on 
+the fuel temperature. However, other feedback mechanisms 
+such as thermal expansion [5] and moderator temperature 
+feedbacks can be treated as well. 
+Substituting Eqs. (7a) - (7e) into (5), after some algebra 
+we obtain by neglecting the 𝑂(𝜀*) terms 
+𝜇
+!"*
+(()*)($,&)
+!$
++ Σ(<𝜓)
+(,9))(𝑥, 𝜇) + Σ()𝑇)
+(,)(𝑥)𝜓<(𝑥, 𝜇)  
+=
+)
+* Σ+<𝜙)
+(,9))(𝑥) +
+)
+* Σ+)𝑇)
+(,)(𝑥)𝜙<  
++
+)
+*,!"",- 𝜈Σ-<𝜙)
+(,9))(𝑥) +
+)
+*,!"",- 𝜈Σ-)𝑇)
+(,)(𝑥)𝜙< .  (8) 
+For reflective BC, 𝜓< =
+=-
+* , and Σ>< =
+)
+,!"",- 𝜈Σ-<, then 
+we rewrite Eq. (8) as  
+𝜇
+!"*
+(()*)($,&)
+!$
++ Σ(<𝜓)
+(,9))(𝑥, 𝜇)  
+=
+)
+* Σ(<𝜙)
+(,9))(𝑥) −
+)
+* Σ(<𝛾𝑇)
+(,)(𝑥) ,                (9) 
+where 
+𝛾 = (1 − 𝑐<) Q
+?.*
+?.- −
+?"*
+?"-R 𝜙< ,              (10a) 
+with 
+Σ>) = Σ() − Σ+) ,                         (10b) 
+ 
+𝑐< =
+?/-
+?0- .                                 (10c) 
+Substituting Eqs. (7b), (7d), and (7e) into (6a) and (6b) 
+respectively, we obtain  
+𝑇)
+∗(𝑥) = 𝐴Σ-<𝜙)
+(,9))(𝑥) + 𝐴Σ-)𝜙<𝑇)
+(,)(𝑥) ,     (11) 
+𝑇)
+(,9))(𝑥) = 𝜔𝑇)
+∗(𝑥) + (1 − 𝜔)𝑇)
+(,)(𝑥) .       (12) 
+Then we substitute Eq. (11) into (12) to give   
+𝑇)
+(,9))(𝑥)  
+= 𝜔𝐴Σ-<𝜙)
+(,9))(𝑥) + C1 − 𝜔 + 𝜔𝐴Σ-)𝜙<D𝑇)
+(,)(𝑥) . (13) 
+For brevity we drop the subscript “1” in the flux and 
+temperature variables without confusion 
+𝜇
+!"(()*)($,&)
+!$
++ Σ(<𝜓(,9))(𝑥, 𝜇)  
+=
+)
+* Σ(<𝜙(,9))(𝑥) −
+)
+* Σ(<𝛾𝑇(,)(𝑥) ,             (14) 
+𝑇(,9))(𝑥) 
+= 𝜔𝐴Σ-<𝜙(,9))(𝑥) + C1 − 𝜔 + 𝜔𝐴Σ-)𝜙<D𝑇(,)(𝑥). (15) 
+ 
+FOURIER ANALYSIS 
+We introduce the inverse Fourier transforms:  
+𝜙(,)(𝑥) = ∫
+𝑎(,)(𝜉)𝑒7?0-@$𝑑𝜉
+9A
+4A
+ ,            (16a) 
+𝜓(,)(𝑥, 𝜇) = ∫
+𝑏(,)(𝜉, 𝜇)𝑒7?0-@$𝑑𝜉
+9A
+4A
+ ,         (16b) 
+𝑇(,)(𝑥) = ∫
+𝑐(,)(𝜉)𝑒7?0-@$𝑑𝜉
+9A
+4A
+.              (16c) 
+The same Fourier ansatz is used for the temperature as 
+for the neutron flux because the fuel temperature is roughly 
+proportional to the neutron flux as shown in Eq. (6a). The 
+solutions are required to satisfy the boundary conditions. The 
+discrete Fourier error mode 𝜉 for the reflective boundary 
+conditions are given below in Eq. (17). If the periodic 
+boundary conditions are used, then they are simply multiplied 
+by a factor of 2. 
+𝜉 =
+1
+?0-B 𝑗 ,    𝑗 = ±1, ±2, …                  (17) 
+where 𝐿 is the reactor core height (or the fuel rod length), i.e., 
+the slab thickness in our model problem.  
+By substituting Eqs. (16a) - (16c) into (14) and noting 
+that each of the Fourier modes is independent, we obtain 
+Σ(<(𝑖𝜉𝜇 + 1)𝑏(,9))(𝜉, 𝜇)  
+=
+)
+* Σ(<𝑎(,9))(𝜉) −
+)
+* Σ(<𝛾𝑐(,)(𝜉) .             (18) 
+We rewrite Eq. (18) as 
+𝑏(,9))(𝜉, 𝜇) =
+)
+*
+)
+(7@&9)) 𝑎(,9))(𝜉) −
+)
+*
+C
+(7@&9)) 𝑐(,)(𝜉). (19) 
+By noting that ∫
+)
+*
+)
+(7@&9)) 𝑑𝜇
+)
+4)
+= tan4)(𝜉)/𝜉 and 
+𝑎(,9))(𝜉) = ∫
+𝑏(,9))(𝜉, 𝜇)𝑑𝜇
+)
+4)
+, we integrate the above 
+equation with respect to 𝜇 to obtain  
+𝑎(,9))(𝜉) = −
+CD12(@)
+)4D12(@) 𝑐(,)(𝜉) , 
+         (20) 
+where 
+𝜌EF(𝜉) =
+GHI3*(@)
+@
+ . 
+ 
+(21) 
+Note that 𝜌EF is the spectral radius function for the 
+standard power iteration (PI) algorithm. 
+Substituting Eqs. (16a) and (16c) into (15), we obtain 
+𝑐(,9))(𝜉) = 𝜔𝐴Σ-<𝑎(,9))(𝜉) 
++C1 − 𝜔 + 𝜔𝐴Σ-)𝜙<D𝑐(,)(𝜉) .                (22) 
+Substituting Eq. (20) into (22), we have 
+𝑐(,9))(𝜉)  
+= ^1 − 𝜔 + 𝜔𝐴Σ-)𝜙< − 𝜔𝐴Σ-<
+CD12(@)
+)4D12(@)_ 𝑐(,)(𝜉) . (23) 
+
+Thus, we obtain the spectral radius function of Picard 
+iteration for the coupled N/TH problem as 
+𝜚(𝜉) = 1 − 𝜔 + 𝜔𝐴Σ-)𝜙< − 𝜔𝐴Σ-<
+CD12(@)
+)4D12(@) .     (24) 
+Substituting Eqs. (4) and (10a) into (24), we obtain   
+𝜚(𝜉) = 1 − 𝜔 a
+1 − 𝜋𝑟-/
+* 𝜅𝑅(Σ-)𝜙< +
+𝜋𝑟-/
+* 𝜅𝑅(Σ-<𝜙<
+()46-)J4.*
+4.-4
+4"*
+4"-
+K
+)4D12(@)
+𝜌EF(𝜉)
+b. (25) 
+Substituting Eq. (4) into (25), we have   
+𝜚(𝜉) = 1 − 𝜔 )1 − 𝑞!𝑅" ,
+#!"
+#!# − -
+#$"
+#$# −
+#!"
+#!#.
+$%&#
+$%'%&()) 𝜌+,(𝜉)01. (26) 
+By noting that 𝑞:𝑅( = 𝑇 − 𝑇., Eq. (26) can be rewritten 
+as 
+𝜚(𝜉) = 1 − 𝜔 21 − (𝑇 − 𝑇-) 4
+#!"
+#!# − -
+#$"
+#$# −
+#!"
+#!#.
+$%&#
+$%'%&()) 𝜌+,(𝜉)56 . 
+       (27) 
+Finally, the spectral radius of Picard iteration for the 
+coupled N/TH nonlinear system is given as 
+𝜌 = max
+@ |𝜚(𝜉)| .                              (28) 
+ 
+RESULTS 
+The spectral radius of the Picard iteration method for the 
+coupled N/TH system is a function of the temperature 
+difference between the fuel and coolant, temperature 
+coefficients of fission and absorption cross sections, 
+scattering ratio, and spectral radius of the standard PI 
+algorithm (or essentially the error mode).  
+The spectral radius function of the PI algorithm, 
+𝜌EF(𝜉) = tan4)(𝜉)/𝜉, attains the largest value at the error 
+mode  𝜉 = 𝜋/(Σ(<𝐿), which is the most slowly converging 
+mode. It is well known that the PI becomes increasingly slow 
+(𝜌 → 1) as the problem domain becomes large, though the 
+method is unconditionally stable because its spectral radius 
+always remains below 1. On the other hand, 𝜌EF(𝜉) tends to 
+zero as 𝜉 limits to infinity. In addition, it is interesting to point 
+out that the term  𝜌EF(𝜉)/(1 − 𝜌EF(𝜉)) in Eq. (26) or (27) can 
+be approximated by 3/𝜉* for 𝜉 small (e.g., the relative 
+difference is less than 1% when 𝜉 < 0.2). With such 
+approximation, we have actually obtained the spectral radius 
+for the diffusion solution coupled with TH. 
+For light water reactors, the cross-section temperature 
+coefficients are typically very small. Table I summarizes the 
+one-group cross section data for a typical PWR.  
+TABLE I. Typical PWR Data 
+Σ(< 
+(cm4)) 
+𝜈Σ-/ 
+(cm4)) 
+𝑐/ 
+Σ-)/Σ-< 
+(K4)) 
+Σ>)/Σ>< 
+(K4)) 
+0.718 
+0.0297 0.96 −1.99 × 104L 8.67 × 104M 
+Note that the temperature coefficient of the absorption 
+cross section is negative, whereas that of the fission cross 
+section is positive. For such problems, the convergence of the 
+unrelaxed Picard iteration is determined by the smallest error 
+mode  𝜉 = 𝜋/(Σ(<𝐿), and the spectral radius is given as 
+𝜌 = (𝑇 − 𝑇.) 6Q
+?.*
+?.- −
+?"*
+?"-R
+)46-
+)4D12(@) 𝜌EF(𝜉) −
+?"*
+?"-; .   (29) 
+Fig. 1 shows that the spectral radius increases with the 
+increasing reactor core height and eventually Picard iteration 
+fails to converge when the reactor core height is larger than a 
+critical value (for the given total cross section). If the 
+temperature difference between the fuel and coolant increases 
+(e.g., the increasing thermal resistance or linear heat 
+generation rate 𝑞:), then the coupling becomes less stable as 
+the spectral radius becomes larger.   
+ 
+Fig. 1. Spectral radius vs. core height.  
+To verify the FA results, we compute numerical 
+convergence rates based on a 1-D model problem, which is 
+the homogeneous slab with the reflective boundary on both 
+sides. The Gauss-Legendre S12 quadrature set is used for 
+angular discretization and the Diamond Difference (DD) 
+method is employed for spatial discretization. Note that the 
+angular quadrature and the mesh size used are sufficiently 
+fine to minimize the numerical errors. A simple heat balance 
+model is used to calculate the fuel and coolant temperatures 
+at each axial cell. Fig. 1 shows that the numerical results for 
+the problem are in excellent agreement with the FA results.  
+For the relaxed case, i.e., 0 < 𝜔 < 1, underrelaxation 
+can not only help to stabilize Picard iteration but improve the 
+convergence rate as shown in Fig. 2.  
+ 
+Fig. 2. Spectral radius vs. underrelaxation. 
+For this case, the core height 𝐿	 = 	150 cm, and the 
+typical PWR data in Table I is used. Again, the FA 
+predictions are consistent with the numerical results. 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1
+10
+100
+Spectral Radius
+L (cm)
+FA (T-Tm = 275K)
+FA (T-Tm = 500K)
+Numerical
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+Spectral Radius
+Underrelaxation, 𝜔
+FA
+Numerical
+
+To derive the optimal underrelaxation factor 𝜔/N(, it is 
+noted that when 𝜔 < 𝜔/N(, max
+@ |𝜚(𝜉)| is found at 𝜉 = ∞, 
+where 𝜌EF(𝜉) = 0, and  
+𝜌 = 1 − 𝜔 61 − (𝑇 − 𝑇.)
+?"*
+?"-; ,   
+  (30) 
+while for 𝜔 > 𝜔/N(, max
+@ |𝜚(𝜉)| is found at 𝜉 = 𝜋/(Σ(<𝐿), 
+and  
+𝜌 = −1 + 𝜔 '1 − (𝑇 − 𝑇!) +
+"!"
+"!# − ,
+"$"
+"$# −
+"!"
+"!#-
+#$%#
+#$&%&'
+'
+()#*( 𝜌)* .
++
+")#,/01. (31) 
+Then the optimal 𝜔/N( can be obtained by equating Eqs. 
+(30) and (31):  
+𝜔/N( =
+*
+*4(O4O5)P*
+4"*
+4"-4J4.*
+4.-4
+4"*
+4"-
+K
+*3$-
+*36127
+8
+40-9:
+D12Q
+8
+40-9RS
+ .  (32) 
+For the case shown in Fig 2, the FA predicted optimal 
+underrelaxation factor is the same as the numerical result, 
+𝜔/N( = 0.66. Note that this case is unstable (𝜌 = 1.042) 
+unless underrelaxation is applied. It indicates that the 
+theoretical estimate of the optimal underrelaxation factor is 
+quite accurate. The optimal underrelaxation depends on 
+various parameters as indicated by Eq. (32). For example, it 
+varies with the core height (for this case, Σ(< = 0.718 cm4)) 
+as depicted in Fig. 3. The more underrelaxation is needed for 
+higher cores (or longer fuel rods). It also indicates that the 
+higher fuel/coolant temperature difference (i.e., larger linear 
+power or thermal resistance), the more underrelaxation is 
+necessitated for stabilizing Picard iteration.  
+ 
+Fig. 3. Optimal underrelaxation vs. core height. 
+ 
+CONCLUSIONS 
+ We have presented a formal Fourier analysis to 
+theoretically predict the convergence properties of Picard 
+fixed-point 
+iteration 
+for 
+coupled 
+neutronics/thermal-
+hydraulics calculations. The work provides a more rigorous 
+theoretical basis for applications of Picard iteration for such 
+calculations. The derived closed form estimate for the 
+spectral radius of the Picard coupling method is a function of 
+various reactor parameters such as the fuel and coolant 
+temperature difference (which instead depends on the rod 
+linear power and thermal resistance), fuel temperature 
+feedback (Doppler effect), scattering ratio, and reactor core 
+height. It implies that Picard iteration is more stable for 
+smaller reactors and lower rod linear power (or thermal 
+resistance). In addition, it is worth noting that for LWRs the 
+Doppler feedback plays a more dominant role in Picard 
+iteration than the moderator temperature (density) feedback. 
+This finding is consistent with numerical experiments 
+reported in Ref. 4. We will report the analysis in the future.  
+A long-standing issue with Picard iteration is that it 
+oftentimes relies on underrelaxation to stabilize the coupled 
+calculation. However, a priori optimal underrelaxation was 
+previously not available for a specific problem. We hope that 
+our new theoretical result can provide a valuable estimate of 
+underrelaxation for stabilizing coupled neutronics/thermal-
+hydraulics calculations. It is now possible to determine local 
+optimal underrelaxation factors and apply them to different 
+fuel rods or assemblies in the reactor.  
+The relaxed Picard iteration is similar to the undamped 
+Anderson acceleration with depth 𝑚 = 1 (AA-1), in which 
+only one previous iterate is used [6,2,3]. However, it is 
+expected that the Picard with optimal underrelaxation will 
+outperform the AA-1 algorithm since the linear coefficients 
+of AA-1 are determined by minimizing the norm of an affine 
+combination of residual vectors and they are generally 
+different from the optimal underrelaxation factor. 
+Although we have only focused on the Fourier analysis 
+for the simple PWR model problem, the methodology 
+presented should be applicable for other types of reactors and 
+more realistic problems such as multigroup problems. In 
+addition, it can be also applied to other coupling methods.  
+ 
+REFERENCES  
+1. C. T. KELLY, Iterative Methods for Linear and 
+Nonlinear Equations, SIAM, Philadelphia (1995). 
+2. A. TOTH et al., “Analysis of Anderson Acceleration on 
+a Simplified Neutronics/Thermal Hydraulics System,” 
+Proceedings of Joint International Conference on 
+Mathematics and Computation (M&C), Supercomputing 
+in Nuclear Applications (SNA) and the Monte Carlo 
+(MC) Method 2015, Nashville, TN, April 19-23, 2015. 
+3. S. HAMILTON, et al., “An assessment of coupling 
+algorithms for nuclear reactor core physics simulations,” 
+J. Comput. Phys., 311, 194 (2017). 
+4. D. F. GILL, et al., “Numerical Methods in Coupled 
+Monte Carlo and Thermal-Hydraulic Calculations,” 
+Nucl. Sci. Eng., 185, 722 (2019). 
+5. D. WANG and F. ABDULLATIF, “Neutron Transport 
+Problems with Nonlinear Temperature Feedback,” 
+Proceedings 
+of 
+International 
+Conference 
+on 
+Mathematics and Computational Methods Applied to 
+Nuclear Science and Engineering 2021 (M&C 2021), 
+Virtual Meeting, October 3-7, 2021, pp. 1326-1335 
+(2021). 
+6. D. G. ANDERSON, “Iterative Procedures for Nonlinear 
+Integral Equations,” J. Assoc. Comput. Mach., 12, 547 
+(1965). 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1
+10
+100
+𝜔opt
+L (cm)
+T-Tm = 275K
+T-Tm = 500K
+

69AyT4oBgHgl3EQfcvd4/content/tmp_files/load_file.txt

@@ -0,0 +1,261 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf,len=260
+page_content='Stability Analysis of Picard Iteration for Coupled Neutronics/Thermal-Hydraulics Simulations  Dean Wang  Nuclear Engineering Program, The Ohio State University, Columbus, OH 43210  wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='12239@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='edu  David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Griesheimer  Bettis Atomic Power Laboratory, Bechtel Marine Propulsion Corporation, West Mifflin, PA 15122  david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='griesheimer@unnpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='gov  INTRODUCTION  Reactor core analysis often needs to solve a multiphysics  nonlinear coupled system, including neutron transport,  thermal-hydraulics, and other important physics phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  One straightforward method for solving such a coupled  system is Picard fixed-point iteration [1], which alternates  between solving individual physics problems separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  However, many numerical studies show that Picard iteration  can be unstable, and a user-defined relaxation is usually  required to achieve convergence [2-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  In this paper, we present a formal Fourier analysis (FA)  of Picard iteration for the coupled neutronics/thermal  hydraulics (N/TH) problem and derive theoretical predictions  for the spectral radius of Picard iteration for such coupled  calculations as a function of the temperature difference  between the fuel and coolant, temperature coefficients of  cross sections (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', Doppler feedback), scattering ratio, and  core height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' An optimal underrelaxation factor is also derived  based on the Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  FORMULATION AND ALGORITHM  We consider the following simple one-group, planar-  geometry k-eigenvalue problem on the domain 0  ≤  𝑥  ≤  𝐿  with reflective boundary conditions:  𝜇  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "($,&)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='$  + Σ((𝑇)𝜓(𝑥, 𝜇)  =  )  Σ+(𝑇)𝜙(𝑥) +  )  ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"" 𝜈Σ-(𝑇)𝜙(𝑥) ,               (1)  and the simplified heat transfer equation for a single typical  pressurized water reactor (PWR) fuel pin:  𝑇 = 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' + 𝐴Σ-(𝑇)𝜙(𝑥) ,  (2)  with  𝐴 = 𝜋𝑟-/  𝜅𝑅( ,                             (3a)  and  𝑅( = 6  )  01," +  )  12#3# +  )  1,$ ln 9  2$%  2$&: +  )  12$%3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' ,        (3b)  where  𝜓 = neutron angular flux  𝜙 = ∫  𝜓(𝑥, 𝜇)𝑑𝜇  )  4)  , neutron scalar flux  Σ( = macroscopic total cross section  Σ+ = macroscopic scattering cross section  Σ- = macroscopic fission cross section  ν = average neutron yields per fission  𝑘5-- = effective multiplication factor  𝜅 = average energy released per fission  𝑇 = volume averaged fuel temperature  𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' = bulk coolant temperature  𝑟-/ = fuel radius  𝑟67 = cladding inner radius  𝑟6/ = cladding outer radius  𝑟8 =  2$&92$% , mean radius in the gap  𝑘- = fuel thermal conductivity  𝑘6 = cladding thermal conductivity  ℎ8 = effective gap conductance  ℎ = coolant convection heat transfer coefficient  Note that the linear heat generation rate (or linear power)  of the fuel rod, 𝑞′, can be calculated by  𝑞:(𝑥) = 𝜋𝑟-/  𝜅Σ-(𝑇)𝜙(𝑥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  (4)  Picard iteration is used to solve the above coupled N/TH  system as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The transport equation is solved first, then  the fuel temperature is calculated using the newly obtained  thermal power (neutron flux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' An underrelaxation factor is  introduced in the temperature update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Note that the transport  iteration is fully converged during each TH update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  𝜇  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "(()*)($,&)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='$  + Σ(C𝑇(,)D𝜓(,9))  =  )  Σ+C𝑇(,)D𝜙(,9))(𝑥) +  )  ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"" 𝜈Σ-C𝑇(,)D𝜙(,9)) ,      (5)  𝑇∗ = 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' + 𝐴Σ-C𝑇(,)D𝜙(,9))(𝑥) ,                (6a)  𝑇(,9)) = 𝜔𝑇∗ + (1 − 𝜔)𝑇(,) ,                  (6b)  where 𝜔 is the underrelaxation factor and the superscript 𝑘  denotes the iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  LINEARIZATION  To perform Fourier analysis of the coupled N/TH  problem, we need to first linearize the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  We define the following linearized variables:  𝜓(𝑥, 𝜇) = 𝜓<(𝑥, 𝜇) + 𝜀𝜓)(𝑥, 𝜇) ,            (7a)  𝜙(𝑥) = 𝜙<(𝑥) + 𝜀𝜙)(𝑥) ,                   (7b)  𝑘5-- = 𝑘5--,/ ,  (7c)  𝑇(𝑥) = 𝑇< + 𝜀𝑇)(𝑥) ,  (7d)  Σ7(𝑇) = Σ7< + Σ7)(𝑇 − 𝑇<)  = Σ7< + 𝜀Σ7)𝑇)(𝑥) ,  𝑖 = 𝑡, 𝑠, 𝑓, 𝑎               (7e)  Note that 𝑘5-- = 𝑘5--,/ due to the flux normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  The cross sections are assumed to be linearly dependent on  the fuel temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' However, other feedback mechanisms  such as thermal expansion [5] and moderator temperature  feedbacks can be treated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (7a) - (7e) into (5), after some algebra  we obtain by neglecting the 𝑂(𝜀*) terms  𝜇  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "*  (()*)($,&)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='$  + Σ(<𝜓)  (,9))(𝑥, 𝜇) + Σ()𝑇)  (,)(𝑥)𝜓<(𝑥, 𝜇)  =  )  Σ+<𝜙)  (,9))(𝑥) +  )  Σ+)𝑇)  (,)(𝑥)𝜙<  +  )  ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "",- 𝜈Σ-<𝜙)  (,9))(𝑥) +  )  ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "",- 𝜈Σ-)𝑇)  (,)(𝑥)𝜙< .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (8)  For reflective BC, 𝜓< =  =-  , and Σ>< =  )  ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "",- 𝜈Σ-<, then  we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (8) as  𝜇  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "*  (()*)($,&)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='$  + Σ(<𝜓)  (,9))(𝑥, 𝜇)  =  )  Σ(<𝜙)  (,9))(𝑥) −  )  Σ(<𝛾𝑇)  (,)(𝑥) ,                (9)  where  𝛾 = (1 − 𝑐<) Q  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='. *  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='.- −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "*  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "-R 𝜙< ,              (10a)  with  Σ>) = Σ() − Σ+) ,                         (10b)  𝑐< =  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='/-  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='0- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (10c)  Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (7b), (7d), and (7e) into (6a) and (6b)  respectively, we obtain  𝑇)  ∗(𝑥) = 𝐴Σ-<𝜙)  (,9))(𝑥) + 𝐴Σ-)𝜙<𝑇)  (,)(𝑥) ,     (11)  𝑇)  (,9))(𝑥) = 𝜔𝑇)  ∗(𝑥) + (1 − 𝜔)𝑇)  (,)(𝑥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (12)  Then we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (11) into (12) to give  𝑇)  (,9))(𝑥)  = 𝜔𝐴Σ-<𝜙)  (,9))(𝑥) + C1 − 𝜔 + 𝜔𝐴Σ-)𝜙<D𝑇)  (,)(𝑥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (13)  For brevity we drop the subscript “1” in the flux and  temperature variables without confusion  𝜇  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "(()*)($,&)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='$  + Σ(<𝜓(,9))(𝑥, 𝜇)  =  )  Σ(<𝜙(,9))(𝑥) −  )  Σ(<𝛾𝑇(,)(𝑥) ,             (14)  𝑇(,9))(𝑥)  = 𝜔𝐴Σ-<𝜙(,9))(𝑥) + C1 − 𝜔 + 𝜔𝐴Σ-)𝜙<D𝑇(,)(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (15)  FOURIER ANALYSIS  We introduce the inverse Fourier transforms:  𝜙(,)(𝑥) = ∫  𝑎(,)(𝜉)𝑒7?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='0-@$𝑑𝜉  9A  4A  ,            (16a)  𝜓(,)(𝑥, 𝜇) = ∫  𝑏(,)(𝜉, 𝜇)𝑒7?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='0-@$𝑑𝜉  9A  4A  ,         (16b)  𝑇(,)(𝑥) = ∫  𝑐(,)(𝜉)𝑒7?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='0-@$𝑑𝜉  9A  4A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (16c)  The same Fourier ansatz is used for the temperature as  for the neutron flux because the fuel temperature is roughly  proportional to the neutron flux as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The  solutions are required to satisfy the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The  discrete Fourier error mode 𝜉 for the reflective boundary  conditions are given below in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' If the periodic  boundary conditions are used, then they are simply multiplied  by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  𝜉 =  1  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='0-B 𝑗 ,    𝑗 = ±1, ±2, …                  (17)  where 𝐿 is the reactor core height (or the fuel rod length), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=',  the slab thickness in our model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  By substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (16a) - (16c) into (14) and noting  that each of the Fourier modes is independent, we obtain  Σ(<(𝑖𝜉𝜇 + 1)𝑏(,9))(𝜉, 𝜇)  =  )  Σ(<𝑎(,9))(𝜉) −  )  Σ(<𝛾𝑐(,)(𝜉) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (18)  We rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (18) as  𝑏(,9))(𝜉, 𝜇) =  ) )  (7@&9)) 𝑎(,9))(𝜉) −  ) C  (7@&9)) 𝑐(,)(𝜉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (19)  By noting that ∫  ) )  (7@&9)) 𝑑𝜇  )  4)  = tan4)(𝜉)/𝜉 and  𝑎(,9))(𝜉) = ∫  𝑏(,9))(𝜉, 𝜇)𝑑𝜇  )  4)  , we integrate the above  equation with respect to 𝜇 to obtain  𝑎(,9))(𝜉) = −  CD12(@)  )4D12(@) 𝑐(,)(𝜉) ,  (20)  where  𝜌EF(𝜉) =  GHI3*(@)  @  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  (21)  Note that 𝜌EF is the spectral radius function for the  standard power iteration (PI) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (16a) and (16c) into (15), we obtain  𝑐(,9))(𝜉) = 𝜔𝐴Σ-<𝑎(,9))(𝜉)  +C1 − 𝜔 + 𝜔𝐴Σ-)𝜙<D𝑐(,)(𝜉) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (22)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (20) into (22), we have  𝑐(,9))(𝜉)  = ^1 − 𝜔 + 𝜔𝐴Σ-)𝜙< − 𝜔𝐴Σ-<  CD12(@)  )4D12(@)_ 𝑐(,)(𝜉) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (23)  Thus, we obtain the spectral radius function of Picard  iteration for the coupled N/TH problem as  𝜚(𝜉) = 1 − 𝜔 + 𝜔𝐴Σ-)𝜙< − 𝜔𝐴Σ-<  CD12(@)  )4D12(@) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (24)  Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (4) and (10a) into (24), we obtain  𝜚(𝜉) = 1 − 𝜔 a  1 − 𝜋𝑟-/  𝜅𝑅(Σ-)𝜙< +  𝜋𝑟-/  𝜅𝑅(Σ-<𝜙<  ()46-)J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' *  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='-4  4"*  4"-  K  )4D12(@)  𝜌EF(𝜉)  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (25)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (4) into (25), we have  𝜚(𝜉) = 1 − 𝜔 )1 − 𝑞!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='𝑅" ,  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='# − -  #$"  #$# −  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content="  $%&#  $%'%&()) 𝜌+,(𝜉)01." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (26)  By noting that 𝑞:𝑅( = 𝑇 − 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (26) can be rewritten  as  𝜚(𝜉) = 1 − 𝜔 21 − (𝑇 − 𝑇-) 4  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='# − -  #$"  #$# −  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"  #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content="  $%&#  $%'%&()) 𝜌+,(𝜉)56 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  (27)  Finally, the spectral radius of Picard iteration for the  coupled N/TH nonlinear system is given as  𝜌 = max  @ |𝜚(𝜉)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (28)  RESULTS  The spectral radius of the Picard iteration method for the  coupled N/TH system is a function of the temperature  difference between the fuel and coolant, temperature  coefficients of fission and absorption cross sections,  scattering ratio, and spectral radius of the standard PI  algorithm (or essentially the error mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  The spectral radius function of the PI algorithm,  𝜌EF(𝜉) = tan4)(𝜉)/𝜉, attains the largest value at the error  mode  𝜉 = 𝜋/(Σ(<𝐿), which is the most slowly converging  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' It is well known that the PI becomes increasingly slow  (𝜌 → 1) as the problem domain becomes large, though the  method is unconditionally stable because its spectral radius  always remains below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' On the other hand, 𝜌EF(𝜉) tends to  zero as 𝜉 limits to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' In addition, it is interesting to point  out that the term  𝜌EF(𝜉)/(1 − 𝜌EF(𝜉)) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (26) or (27) can  be approximated by 3/𝜉* for 𝜉 small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', the relative  difference is less than 1% when 𝜉 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' With such  approximation, we have actually obtained the spectral radius  for the diffusion solution coupled with TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  For light water reactors, the cross-section temperature  coefficients are typically very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Table I summarizes the  one-group cross section data for a typical PWR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Typical PWR Data  Σ(<  (cm4))  𝜈Σ-/  (cm4))  𝑐/  Σ-)/Σ-<  (K4))  Σ>)/Σ><  (K4))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='718  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='0297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='96 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='99 × 104L 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='67 × 104M  Note that the temperature coefficient of the absorption  cross section is negative, whereas that of the fission cross  section is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' For such problems, the convergence of the  unrelaxed Picard iteration is determined by the smallest error  mode  𝜉 = 𝜋/(Σ(<𝐿), and the spectral radius is given as  𝜌 = (𝑇 − 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=') 6Q  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='. *  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='.- −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "*  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "-R  )46-  )4D12(@) 𝜌EF(𝜉) −  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "*  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "-;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (29)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 1 shows that the spectral radius increases with the  increasing reactor core height and eventually Picard iteration  fails to converge when the reactor core height is larger than a  critical value (for the given total cross section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' If the  temperature difference between the fuel and coolant increases  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', the increasing thermal resistance or linear heat  generation rate 𝑞:), then the coupling becomes less stable as  the spectral radius becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Spectral radius vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' core height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  To verify the FA results, we compute numerical  convergence rates based on a 1-D model problem, which is  the homogeneous slab with the reflective boundary on both  sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The Gauss-Legendre S12 quadrature set is used for  angular discretization and the Diamond Difference (DD)  method is employed for spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Note that the  angular quadrature and the mesh size used are sufficiently  fine to minimize the numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' A simple heat balance  model is used to calculate the fuel and coolant temperatures  at each axial cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 1 shows that the numerical results for  the problem are in excellent agreement with the FA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  For the relaxed case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', 0 < 𝜔 < 1, underrelaxation  can not only help to stabilize Picard iteration but improve the  convergence rate as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Spectral radius vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' underrelaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  For this case, the core height 𝐿  =  150 cm, and the  typical PWR data in Table I is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Again, the FA  predictions are consistent with the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  1  10  100  Spectral Radius  L (cm)  FA (T-Tm = 275K)  FA (T-Tm = 500K)  Numerical  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='8  1  Spectral Radius  Underrelaxation, 𝜔  FA  Numerical  To derive the optimal underrelaxation factor 𝜔/N(, it is  noted that when 𝜔 < 𝜔/N(, max  @ |𝜚(𝜉)| is found at 𝜉 = ∞,  where 𝜌EF(𝜉) = 0, and  𝜌 = 1 − 𝜔 61 − (𝑇 − 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=')  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "*  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' "-;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=" ,  (30)  while for 𝜔 > 𝜔/N(, max  @ |𝜚(𝜉)| is found at 𝜉 = 𝜋/(Σ(<𝐿),  and  𝜌 = −1 + 𝜔 '1 − (𝑇 − 𝑇!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=') +  "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"  "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='# − ,  "$"  "$# −  "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='"  "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content="#-  #$%#  #$&%&'  '  ()#*( 𝜌)* ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  +  ")#,/01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (31)  Then the optimal 𝜔/N( can be obtained by equating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  (30) and (31):  𝜔/N( = 4(O4O5)P*  4"*  4"-4J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' *  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='-4  4"*  4"-  K  3$-  36127  8  40-9:  D12Q  8  40-9RS  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (32)  For the case shown in Fig 2, the FA predicted optimal  underrelaxation factor is the same as the numerical result,  𝜔/N( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Note that this case is unstable (𝜌 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='042)  unless underrelaxation is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' It indicates that the  theoretical estimate of the optimal underrelaxation factor is  quite accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The optimal underrelaxation depends on  various parameters as indicated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' For example, it  varies with the core height (for this case, Σ(< = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='718 cm4))  as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The more underrelaxation is needed for  higher cores (or longer fuel rods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' It also indicates that the  higher fuel/coolant temperature difference (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', larger linear  power or thermal resistance), the more underrelaxation is  necessitated for stabilizing Picard iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Optimal underrelaxation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' core height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  CONCLUSIONS  We have presented a formal Fourier analysis to  theoretically predict the convergence properties of Picard  fixed-point  iteration  for  coupled  neutronics/thermal-  hydraulics calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The work provides a more rigorous  theoretical basis for applications of Picard iteration for such  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' The derived closed form estimate for the  spectral radius of the Picard coupling method is a function of  various reactor parameters such as the fuel and coolant  temperature difference (which instead depends on the rod  linear power and thermal resistance), fuel temperature  feedback (Doppler effect), scattering ratio, and reactor core  height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' It implies that Picard iteration is more stable for  smaller reactors and lower rod linear power (or thermal  resistance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' In addition, it is worth noting that for LWRs the  Doppler feedback plays a more dominant role in Picard  iteration than the moderator temperature (density) feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  This finding is consistent with numerical experiments  reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' We will report the analysis in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  A long-standing issue with Picard iteration is that it  oftentimes relies on underrelaxation to stabilize the coupled  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' However, a priori optimal underrelaxation was  previously not available for a specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' We hope that  our new theoretical result can provide a valuable estimate of  underrelaxation for stabilizing coupled neutronics/thermal-  hydraulics calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' It is now possible to determine local  optimal underrelaxation factors and apply them to different  fuel rods or assemblies in the reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  The relaxed Picard iteration is similar to the undamped  Anderson acceleration with depth 𝑚 = 1 (AA-1), in which  only one previous iterate is used [6,2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' However, it is  expected that the Picard with optimal underrelaxation will  outperform the AA-1 algorithm since the linear coefficients  of AA-1 are determined by minimizing the norm of an affine  combination of residual vectors and they are generally  different from the optimal underrelaxation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  Although we have only focused on the Fourier analysis  for the simple PWR model problem, the methodology  presented should be applicable for other types of reactors and  more realistic problems such as multigroup problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' In  addition, it can be also applied to other coupling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  REFERENCES  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' KELLY, Iterative Methods for Linear and  Nonlinear Equations, SIAM, Philadelphia (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' TOTH et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', “Analysis of Anderson Acceleration on  a Simplified Neutronics/Thermal Hydraulics System,”  Proceedings of Joint International Conference on  Mathematics and Computation (M&C), Supercomputing  in Nuclear Applications (SNA) and the Monte Carlo  (MC) Method 2015, Nashville, TN, April 19-23, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' HAMILTON, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', “An assessment of coupling  algorithms for nuclear reactor core physics simulations,”  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', 311, 194 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' GILL, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', “Numerical Methods in Coupled  Monte Carlo and Thermal-Hydraulic Calculations,”  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', 185, 722 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' WANG and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' ABDULLATIF, “Neutron Transport  Problems with Nonlinear Temperature Feedback,”  Proceedings  of  International  Conference  on  Mathematics and Computational Methods Applied to  Nuclear Science and Engineering 2021 (M&C 2021),  Virtual Meeting, October 3-7, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' 1326-1335  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' ANDERSON, “Iterative Procedures for Nonlinear  Integral Equations,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content=', 12, 547  (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}
+page_content='2  1  10  100  𝜔opt  L (cm)  T-Tm = 275K  T-Tm = 500K' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'}

6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d26dd90780495a7b4eea87a3e37d73b9849276a0f8b4254310b51daa9689aa57
+size 2487273

6tAzT4oBgHgl3EQfEvoZ/content/tmp_files/load_file.txt

@@ -0,0 +1,393 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf,len=392
+page_content='Detector and physics simulation using heavy ion collisions at  NICA-SPD  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Denisenko1,a) and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Pandey2,b)  1Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna-141980,  Moscow Region, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  2Graduate Engineer Trainee, Larsen & Toubro Limited, Faridabad,  Haryana, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  a)iden@jinr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='ru  b)rishav160999@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='com  Abstract  The space-time picture of hadron formation in high-energy collisions with nuclear  targets is still poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  The tests of hadron formation was suggested for the  first stage of SPD running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  They will require measuring charged pion and proton  spectra with the precision better than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' A research has been carried out to check  feasibility of such studies at SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' In this work, 12C − 12C and 40Ca − 40Ca heavy ion  collisions at center of mass energy of 11 GeV/nucleon were simulated using the SMASH  event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Firstly, the generator-level events were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The distribution of  track multiplicities and momentum distributions of different types of charged particles  were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Secondly, the generated events passed through the full reconstruction  using the SpdRoot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  At this stage particles were identified using dE/dx  measurement and time-of-flight information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It allowed us to estimate charge track  multiplicities in the tracking system and purities of charge particles spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The results  on multiplicity are important to estimate occupancies in the tracking system, while the  results on the pion and proton momentum spectra show that particle identification  should be acceptable for validation of hadron formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' This is the first study  of moderate ion collisions for the SPD Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Keywords:  Hadron formation effects, Heavy ion collision, SMASH, NICA-SPD, Rapidity,  Charged track multiplicity, Particle physics event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='00997v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='ins-det]  3 Jan 2023  1  INTRODUCTION  The SPD detector is primarily optimized to study spin dependent gluon structure of proton and  deuteron using open charm production, charmonia production and prompt photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' At the same  time, its physics program includes studies of various aspects of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The work is devoted to studies  of hadron formation in nuclear collisions proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Hadrons produced in hadron collisions emerge in the form of prehadrons, which interact with  nucleons with reduced strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' This suppression is poorly known and is described in model de-  pendent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' This suppression results in different spectra of final particles as is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1  for rapidity distributions (in a similar way it affects the pT spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Naturally, these spectra can  be used to study hadron formation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The required precision of such measurements is 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  The aim of this work is to evaluate feasibility of such measurements with MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Here,  ion collisions of 12C − 12C and 40Ca − 40Ca at √s = 11AGeV were generated using the SMASH  (Simulating Many Accelerated Strongly-interacting Hadrons) event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Afterwards, the the  full simulation and reconstruction was performed using the SpdRoot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 1: Rapidity spectra of protons and charged pions in 12C − 12C and 40Ca − 40Ca collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  2  12C + 12C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' s= 11 GeV  40Ca + 40Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' sN = 11 GeV  105  106  Protons  Protons  w/oformation  w/oformation  default  default  QDM  QDM  do/dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' mb  104  peut = 2 GeV/c -  Peut = 2 GeV/c -  Peut = 1 GeV/c  Peut = 1 GeV/c  103  104  102  103  4 -3 -2 -1  0  1  2  3  4  4  3 -2 -1  0  1  2  3  4  y  y  12C + 12C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' sN= 11 GeV  40Ca + 40Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' sNR = 11 GeV  104  105  do/dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' mb  do/dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' mb  103  104  w/oformation  w/oformation  default  default  QDM  QDM  Peut = 2 GeV/c  Peut = 2 GeV/c  Pcut = 1 GeV/c  pcut = 1 GeV/c  102  103  4 -3 -2  1  0  1  2  3  4  4 -3 -2 -1  0  1  2  3  4  y  y2  NICA FACILITY  The NICA (Nuclotron based Ion Collider fAcility) collider at Joint Institute for Nuclear Research  in Dubna is is being built to provide beams for two experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The first experiment, MPD (Multi  Purpose Detector), will study properties of dense baryonic matter (matter present at extreme high  density in QCD phase diagram) like Quark Gluon Plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The second experiment, SPD (Spin  Physics Detector), is devoted to study of spin related phonomena and QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Once the NICA collider  will be operational, scientists will be able to create a special state of matter in laboratory which  existed for very short interval of time (˜20µ sec) just after the big bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' This special state is called  as QGP (Quark Gluon Plasma) and it filled the entire universe shortly after the big bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  The main parts of NICA facility consists of two independent injector complex (injector for light  ions, and injector for heavy ions-KRION 6T), Light Ion Linear Accelerator (LU20) for accelerating  light ions like protons (H+), deutrons, and α-particles upto 5 MeV of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='E, then Heavy Ion Linear  Accelerator (HILAC) to accelerate heavy ions upto Au to a maximum K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='E of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2 MeV/n, then a  Super Conducting (SC) Booster Synchrotron to create ultra high vacuum and to provide complete  stripping of heavy ions, then a SC Heavy Ion Synchrotron Nuclotron to accelerate both light and  heavy ions to required beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The accelerated beams will collide at two different locations  where MPD detector and SPD detector are being built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The schematic view of NICA complex is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 2: Schematic view of NICA complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  3  SPD DETECTOR  The Spin Physics Detector [2,3] is a 4π universal detector optimized to study spin-related phenomena  via open charm, charmonia and promopt photons in the collisions of polarized p-p or d-d beams  with √sNN up to 27 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' However, at first stage of NICA-SPD, the expected collision energy  will be from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4 up to 10 GeV, and later on after first upgrade, it is expected to reach upto 27  GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The general layout depicting isometric projection of SPD setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The main  parts involved in advanced tracking and particle identification capabilities have been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' (i)  The beam pipe passes through the center of the detector, carries the accelerated beams of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' (ii)  The MicroMegas detector is to improves the momentum resolution and tracking efficiency of the  tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' (iii) The Straw Tracker (ST) detector is for the reconstruction of the primary and  3  BM@N Detector  SPD  Transport Channel  HILAC  Collider  LU20  Booster  MPD  Nuclotronsecondary particle tracks and for determination of their momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' (iv) The Time Of Flight (TOF)  detector, is a part of Particle Identification (PID) system, and is used for identification of particles  like π, k, and p with long trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' (v) The magnet system shown by red color provides 1T  of magnetic field along the beam axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' This setup is limited to first stage of SPD operation, and  will be considered only for the identification of stable charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Neutral particles, like n0,  photons will be detected at later stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The main parts of SPD first stage have been explained in  detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' There is a possibility to have TOF system for the first stage studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 3: Layout of the SPD setup proposed for first stage at NICA-SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1  CENTRAL TRACKER  The innermost detector of SPD consists of a MicroMegas-based Central Tracker (MCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Its purpose  is to identify the primary vertex coordinate and to improve momentum resolution and tracking  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  It is based on MicroMegas (Micro Mesh Gaseous Structure) technology and detects  charged particle by amplifying the charges produced due to ionization of the gas molecules present  in detector volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' When an ionizing particle track passes through detector volume, it ionizes the  gas molecules and creates few hundreds of e−-ion pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Electrons are accelerated opposite to the  direction of applied electric field of 600 V/cm in ionization gap, while ions are attracted towards  cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' When the e− crosses micromesh, it faces intense electric field (> 30 KV/cm) and gains  enough energy to ionize other gas molecules in its path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' During this process an avalanche of e−-ion  pair is produced (1e− produces 104 e−-ion pairs) which is significant to create an electronic signal  which is read out by readout electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  STRAW TRACKER  ST is mainly for the reconstruction of primary and secondary particle tracks and measuring their  momenta, but also participates in identification of π, K, and p on via energy deposit (dE/dx)  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It consists of two major parts - barrel (covers radius from 270 to 850 mm) and two  end-caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The barrel is divided into 8 modules enclosed in a carbon fiber capsule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Each module has  30 double layers of straw tubes (dia 1cm) which runs parallel (long straw tubes) and perpendicular  4  Straw tracker  Magnet  Range system  MicroMegas Endcap  RangesystemEndcap  MicroMegas  Beam-beamcounter  Beam pipe  Strawtracker Endcap  zoomx4  Zero degree calorimeter(short straw tubes) to the beam axis and contains 1500 and 6000 parallel and perpendicular straw  tubes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Straw tubes are made of polyethylene terephthalate and outer surface is coated  with very thin layer of Cu and Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Carbon capsule is meant to protect the outer surface of these  tubes from humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' One side and two opposite ends of capsule are provided with small holes  where end plugs are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' FEE are connected to these end plugs to read the detector signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Any  particle which passes through the long straws will send detector signal to both opposite ends while a  particle passing through short straw will send detector signal to any one side of capsule where FEE  is attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Thus, long straws will be read from two opposite ends while short straws will be read  from one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The end-caps of ST are divided into 3 modules and each module has 4 hexadecimal  cameras (U, V, X, Y) to record the four coordinates of any physical quantity like four-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  The FEE to be used can be similar to the one used at NA64 experiment (for the search of dark  matter), or DUNE experiment (to detect and study properties of neutrino).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='3  TIME OF FLIGHT DETECTOR  TOF detector is the part of PID system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Similar to ST, the TOF provides identification of π, k, and  p by measuring their flight time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The energy loss data registered by ST can be used together with the  data from TOF for correct identification of particle tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The TOF distinguishes charged particles  (mainly π and k) in the momentum range up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The major parts of TOF comprises of a  barrel and two end-caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' For the first stage of NICA-SPD, two different designs of TOF has been  suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' First one is TOF based on multigap timing Resistive Plate Chambers (mRPC), which  will consist 220 rectangular plate chambers (160 for the barrel and 30 each for end-caps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Second  one is based on Plastic Scintillator Tiles and will comprise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1K small scintillator tiles (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4K for  barrel and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4K for each end-caps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Scintillator has a property of emitting light in visible region  when an ionizing radiation passes through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' So, in this design when a particle passes through  TOF, scintillated photons are produced which are detected by four Si Photo Multipliers (SiPMs)  present at each sensor board attached at two extreme ends of scintillator tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  4  EVENT GENERATION  12C −12C and 40Ca−40Ca heavy ion collisions at √s = 11 AGeV with maximum impact parameter  set to 8 fm for C-C and 11 fm for Ca-Ca were simulated using SMASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The fermi motion was  assumed to be “frozen” and 100K events were generated for each heavy ion collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The SMASH  input file for C-C collision is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  *********** SMASH INPUT ************  config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='yaml file for C-C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Logging:  default: INFO  General:  Modus:  Collider  Time_Step_Mode: Fixed  Delta_Time:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1  End_Time:  200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='0  Randomseed:  1  Nevents:  100000  5  Output:  Output_Interval: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='0  Particles:  Format:  ["Oscar2013"]  Modi:  Collider:  Projectile:  Particles: {2212: 6, 2112: 6} #C-12  Target:  Particles: {2212: 6, 2112: 6} #C-12  Sqrtsnn: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='0  Impact:  Sample: "quadratic"  Range: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='0, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='0]  Fermi_Motion: "frozen"  ************************************  Multiplicity of generated charged particles for C − C and Ca − Ca collisions are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The peaks at 12 for 12C+12C collisions and at 40 for 40Ca+40Ca collisions correspond to  events where no interaction occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The rapidity distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The spectra  obtained from SMASH output show qualitative agreement with the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Peaks for protons  correspond to particles moving close to the initial beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Moreover, fractions of different  particle types can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It can be seen that for |y| < 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' within the acceptance of the  detector) charge particles are dominated by pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Apart from p±, π±, & K±, marginal numbers  of sigmas, cascades, and omegas were also generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The PID efficiency depends on the particle  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  The momentum spectra for protons, pions and kaons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 6 in the  midrapidity region (|y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5 for which theoretical predictions has been given) Most of the pions  have momentum below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8 GeV and protons - below 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It means that types of these particles  should be well resolved by dE/dx measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' When studying pion or proton spectra, there is  high probability of kaon/pion misidentification, but fraction of such events is strongly suppressed  by small initial kaon numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  6  (a)  (b)  Figure 4: Generator-level multiplicity of charged particles for 12C − 12C collision (a) and 40Ca − 40Ca collisions (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (a)  (b)  Figure 5: Rapidity distribution of charged particles in 12C − 12C (a) and 40Ca − 40Ca (b) collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  7  Total Multiplicity of Charged Particles, C-12 + C-12  104  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' of events  103  102  10  0  10  20  30  40  50  60  70  80  90  100  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' of charged particlesTotal Multiplicity of Charged Particles, Ca-40 + Ca-40  104  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' of events  103  0  10  20  30  40  50  60  70  80  90  100  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' of charged particlesRapidity distribution of charged particles, C-12 + C-12  protons  105  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='. pions  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' kaons  104  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' of charged particles  103  102  10  3  2  3  5  Rapidity of charaed particles (yRapidity distribution of charged particles, Ca-40 + Ca-40  106  protons  pions  105  kaons  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' of charged particles  104  103  102  10  5  3  2  Y  Rapidity of charged particles (y)(a) p distribution of p± in 12C − 12C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (b) p distribution of p± in 40Ca − 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (c) p distribution of π± in 12C − 12C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (d) p distribution of π± in 40Ca − 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (e) p distribution of k± in 12C − 12C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (f) p distribution of k± in 40Ca − 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 6: Total momentum distribution of protons, pions, and kaons at generator level in 12C − 12C and 40Ca− 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  8  Total momentum distribution of pions, C-12 + C-12  16000  14000  12000  pions  10000  8000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  6000  4000  2000  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  Total momentum of pions (p)Total momentum distribution of pions, Ca-40 + Ca-40  60000  50000  40000  ON  30000  20000  10000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  Total momentum of pions (p)Total momentum distribution of kaons, C-12 + C-12  1000  800  of kaons  600  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  400  200  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  Total momentum of kaons (p)Total momentum distribution of kaons, Ca-40 + Ca-40  4000  3500  3000  kaons  2500  2000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  1500  1000  500  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  Total momentum of kaons (p)Total momentum distribution of protons, C-12 + C-12  1600  1400  1200  protons  1000  800  ON  600  400  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  Total momentum of protons (p)Total momentum distribution of protons, Ca-40 + Ca-40  6000  5000  of protons  4000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  3000  2000  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  Total momentum of protons (p)5  DETECTOR SIMULATION AND EVENT RECONSTRUCTION  The detector simulation and reconstruction was performed with the SpdRoot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' To read  SMASH generated events the SpdRoot code was modified and additional C++ class was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  During the simulation stage the particles were transported through the detector geometrical model  using Geant4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  At the reconstruction stage, Geant4 tracks and vertices were reconstructed and  particle identification with dE/dx and time of flight measurements was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' For the PID  three hypotheses were considered: pion, kaon and proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The reconstructed ionization energy  losses and “measured” time of flight were used to construct conditional probabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' p(t|pid),  where t is the measured time and pid is a particle type hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Out of 100K events generated  by SMASH, first 1K events were considered for detector simulation due to slow data processing in  SpdRoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  6  ANALYSIS  A physical analysis was performed using C++ codes and ROOT library based on SpdRoot output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  All tracks reconstructed in the detector with measured momentum were accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' For the particle  type the one that gives the largest conditional probability is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Multiplicity, as well as  kinematic distributions for pions, kaons and protons are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' For particle momentum spectra  there are no notable differences between C − C and for Ca − Ca collisions, so only the first ones  will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1  CHARGED TRACK MULTIPLICITY  The SPD detector set-up is optimized for p − p and d − d collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Thus knowing charged track  multiplicities for ion collisions is important to estimate CT and ST occupancies and feasibility of  such studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 7 shows the total multiplicity of charged particles reconstructed by the tracking  system in 12C − 12C and 40Ca − 40Ca collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The numbers of reconstructed tracks are much  lower compared to generator-level studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It is because the geometry of the tracking system is such  that, tracks with polar angle, θ < 10◦ or > 170◦ do not hit the tracker and passes along the beam  pipe itself, so such tracks are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Also, there were events without nuclei interactions which  resulted in no track reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' So, to avoid a large peak at zero due to mentioned reasons, the  X-axis count starts from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  PION MOMENTUM SPECTRUM (12C − 12C)  The spectra of particles identified as pions separately by ionization losses and by TOF are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 8 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The spectra show resemblance with the generator plot of pion momentum distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Based pn MC-truth information backround from misidentification other charged particles  (K±, p±, e±, & µ±) is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The obtained distribution for “pions identified as pions” only slightly  deviates from distribution of all selected pion candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The estimated relative contamination of  the pion spectra is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It can seen that purity above 90% can be obtained up to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2 GeV using either dE/dx or TOF measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  9  Figure 7: Charged track multiplicity reconstructed by in 12C − 12C (left), 40Ca − 40Ca (right) collisions (shown by red) and  number of particles for which TOF information is available (shown by blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (a) Total momentum distribution of reconstructed charged particles  identified as π± by ionization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (b) Total momentum distribution of reconstructed charged particles  identified as π± by TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 8: Total momentum distribution of reconstructed π± candidates in 12C − 12C collision (Detector level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 9: Purity of the selected pion candidates as a function of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  10  Total multiplicity of charged particles passing through tracking system, C12-C12  60  TOF  50  ST  40  events  30  NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  20  10  一  10  20  30  40  50  60  70  80  90  10090  TOF  80  ST  70  events  60  50  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  40  30  20  10  0  10  20  30  40  50  60  70  80  90  100Charged Particles ldentified as Pions by ST, C12-C12  350  Pions identified as pions  Kaons identified as pions  Protons identified as pions  300  Electrons identified as pions  Muons identified as pions  250  Chargedparticlesidentifiedaspions  200  150  100  50  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  p(GeV/c)Charged Particles ldentified as Pions by TOF, C12-C12  Pions identified as pions  300  Kaons identified as pions  Protons identified as pions  250  Electrons identified as pions  Muons identified as pions  Charged particles identified as pions  200  Counts  150  100  50  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  p(GeV/c)Pion spectra precision, C12-C12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  Counts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  Precision recordedbyTOF  Precision recorded by ST  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  p(GeV/c)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='3  KAON MOMENTUM SPECTRUM (12C − 12C)  The kaon momentum spectrum was explicitly mentioned among observables to study hadron for-  mation effects in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Nevertheless, kaon production may be interesting for the reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The  obtained spectra of kaon candidates is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 10 separately for ionization losses and TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  First of all, the shown data lack statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Secondly, it can bee seen that there is a huge contamina-  tion from misidentified pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' This is explained by very small fraction of generated kaons and the  fact that probability to select misidentified particle is proportional to their number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The relative  fraction of correctly identified kaons in shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (a) Total momentum distribution of reconstructed charged particles  identified as K± by ionization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (b) Total momentum distribution of reconstructed charged particles  identified as K± by TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 10: Total momentum distribution of reconstructed K± candidates in 12C − 12C collision (Detector level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 11: Purity of the selected kaon candidates as a function of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  11  Charged Particles ldentified as Kaons by ST, C12-C12  Kaons identified as kaons  60  Pions identified as kaons  Protons identified as kaons  Electrons identified as kaons  50  Muons identified as kaons  Charged particles identified askaons  40  Counts  30  20  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  p(GeV/c)Charged Particles ldentified as Kaons by TOF, C12-C12  25  Kaons identified as kaons  Pions identified as kaons  Protons identifiedaskaons  20  Electrons identified as kaons  Muons identified as kaons  Charged particles identified as kaons  15  Counts  10  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  0  2  p(GeV/c)Kaon spectra precision, C12-C12  Precision recorded by TOF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='9  Precision recorded by ST  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  unts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  Col  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  2  p(GeV/c)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  PROTON MOMENTUM SPECTRUM (12C − 12C)  Finally, proton momentum spectra have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' In this study protons and antiprotons were  considered together, but the fraction of produced antiprotons is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The proton candidate  distributions and the contributions from misidentification are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The purity of the  selected samples is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' It can be seen dE/dx measurements alone will not allow  precise determination of proton spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The reasonably good results can be expected only in  case of combined identification by ionization losses and TOF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (a) Total momentum distribution of reconstructed charged particles  identified as p± by ionization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  (b) Total momentum distribution of reconstructed charged particles  identified as p± by TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 12: Total momentum distribution of reconstructed p± candidates in 12C − 12C collision (Detector level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Figure 13: Purity of the selected proton candidates as a function of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  12  Charged Particles ldentified as Protons by ST, C12-C12  90  Protons identified as protons  Kaons identified as protons  80  Pions identified as protons  Electrons identified as protons  Muons identified as protons  70  Charged particles identified as protons  60  Counts  50  40  30  20  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  0  5  p(GeV/c)Charged Particles ldentified as Protons by TOF, C12-C12  90  Protons identified as protons  Kaons identified as protons  80  Pions identified as protons  Electrons identified asprotons  70  Muons identified as protons  Charged particles identified as protons  60  Counts  50  40  30  20  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  0  5  p(GeV/c)Proton spectra precision, C12-C12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='8  Counts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='2  Precision recorded by TOF  Precision recorded by ST  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='5  5  p(GeV/c)7  SUMMARY  The goal of this work was to check the feasibility of hadron formation effects studies at the first  stage of SPD operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' For this purpose an analysis of 12C − 12C and 40Ca − 40Ca collisions were  performed at the generator level and then the full event reconstruction was done at detector level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  The multiplicity distributions indicate that occupancies of tracking detectors should be checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  Part of the events with high number of charged tracks may not be fully reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Particle  identification with ionization losses and TOF was considered separately (for future dE/dx only or  their combination can be expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' The purity of the measured charged pion distribution for both  types of ion collisions using dE/dx only is rather good and meets mentioned before requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' In  case of combination of information from ionization losses and time of flight system purity of proton  distribution may be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  References  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Abramov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Aleshko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Baskov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Boos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Bunichev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Dalkarov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' El-Kholy,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Galoyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Guskov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Kim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' 52 (2021) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='6, 1044-1119  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='1134/S1063779621060022 [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='08477 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' Abazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content=' [SPD proto], [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='00442 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  [3] SPD TDR [unpublished].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'}

CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/2301.02413v1.pdf.txt

@@ -0,0 +1,3062 @@
+Benchmarking Gaussian Basis Sets in
+Quantum-Chemical Calculations of
+Photoabsorption Spectra of Light Atomic
+Clusters
+Vikram Mahamiya,∗,† Pritam Bhattacharyya,∗,†,‡ and Alok Shukla∗,†
+†Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076,
+India
+‡Present Address: Institute for Theoretical Solid State Physics, Leibniz IFW Dresden,
+Helmholtzstr. 20, 01069 Dresden, Germany
+E-mail: [email protected]; [email protected]; [email protected]
+Abstract
+The choice of Gaussian basis functions for computing the ground-state properties of
+molecules, and clusters, employing wave-function-based electron-correlated approaches,
+is a well-studied subject.
+However, the same cannot be said when it comes to the
+excited-state properties of such systems, in general, and optical properties, in particular.
+The aim of the present study is to understand how the choice of basis functions affects
+the calculations of linear optical absorption in clusters, qualitatively, and quantitatively.
+For this purpose, we have calculated linear optical absorption spectra of several small
+charged and neutral clusters, namely, Li2, Li3, Li4, B+
+2 , B+
+3 , Be+
+2 , and Be+
+3 , using a
+variety of Gaussian basis sets. The calculations were performed within the frozen-core
+approximation, and a rigorous account of electron correlation effects in the valence
+1
+arXiv:2301.02413v1  [physics.chem-ph]  6 Jan 2023
+
+sector was taken by employing various levels of configuration interaction (CI) approach
+both for the ground and excited states.
+Our results on the peak locations in the
+absorption spectra of Li3 and Li4 are in very good agreement with the experiments.
+Our general recommendation is that for excited-state calculations, it is very important
+to utilize those basis sets which contain augmented functions. Relatively smaller aug-
+cc-pVDZ basis sets also yield high-quality results for photoabsorption spectra, and are
+recommended for such calculations if the computational resources are limited.
+Introduction
+Gaussian basis functions (GBFs) were initially proposed by Boys for use in computational
+atomic and molecular quantum mechanics,1 and over the years have become the preferred
+basis functions in quantum chemistry.1 The reason behind the popularity of GBFs is the
+so-called Gaussian product theorem1,2 which allows for analytical results for the expressions
+of multi-center integrals involving various physical quantities. Nevertheless, one has to be
+always careful about various convergence related issues when using GBFs, because, unlike
+Slater basis functions, they do not exhibit correct asymptotic behavior far away from the
+nuclei.
+This generally leads to the requirement that a large number of GBFs should be
+used to achieve convergence, leading to huge memory and CPU-time requirements because
+the required number of integrals scale as ≈ N 4, where N is the total number of basis
+functions. Keeping this in mind, several groups have studied the convergence properties
+of GBFs over the years, and have come up with schemes to balance accuracy with the
+computational effort (see, e.g., Refs.3,4 for comprehensive reviews). Huzinaga was one of
+the earliest researchers to optimize GBFs for Hartree-Fock (HF) calculations on atoms.5
+Ruedenberg and coworkers devised the so-called even-tempered basis set,6,7 while Huzinaga
+and coworkers developed well-tempered basis functions.8 Huzinaga and coworkers further
+developed several contracted basis sets,9,10 discussed in detail in Ref.4 Pople and coworkers
+developed a large number of basis sets3,4 which enjoy continued popularity even in present
+2
+
+times. One of the most popular minimal basis sets introduced by Pople and coworkers is STO-
+3G contracted basis set,11 whose purpose was to emulate Slater-type orbitals, using GTFs.
+Split-valence basis sets are among the most popular extended basis sets introduced by Pople
+et al.,12–15 in which for inner shells contracted minimal basis functions are used, but for the
+valence shells a split set of basis functions is employed, which consist of both contracted and
+primitive GTFs. Depending upon the contraction schemes, these basis sets were given names
+such as 3-21G,13 4-31G,12 6-21G,14 and 6-311G15, etc. Pople and coworkers also proposed
+further enlarged basis sets containing polarization and diffuse functions of higher angular
+momenta, which have since become popular choices in quantum chemistry.15–20 Dunning
+and coworkers introduced a series of extended basis sets, called “correlation-consistent” (CC)
+basis sets, which are of varying sizes, containing both polarization and diffuse functions.21–23
+The basic idea behind these CC basis sets is that they recover a significant amount of electron
+correlation energy in post-Hartree-Fock treatments of corresponding atoms. In addition to
+the basis sets mentioned here, numerous other sets of basis functions have been developed
+over the years, for which we refer the reader to review articles by Davidson and Feller,3 and
+Huzinaga.4
+Even though so many basis sets have been developed by numerous groups, in most of the
+reports the criteria for their selection appears to be driven by a good description of the ground
+state energies of the atoms involved either at the Hartree-Fock level, or in electron-correlated
+calculations.3,4 Previously, Balakina et al.24 have explored the basis set dependence of the
+linear and non-linear optical properties of conjugated organic molecule p-nitroaniline. They
+reported that the [4s3p2d/3s] basis set also provides similar results as aug-cc-pVDZ basis
+set for the calculations of (hyper)polarizability. Parsons et al.25 have explored the basis
+set dependence of optical rotation calculations of various types of gauges. They found that
+the origin-invariant length gauge (LG-OI) gauge with aug-cc-pVTZ basis set provides a
+balance of cost and accuracy for DFT method.
+Reis and Papadopoulos26 reported that
+the inclusion of f-functions in the Dunning’s basis sets does not have a large effect on the
+3
+
+electric properties of B4 cluster. Lauderdale and Coolidge27 have explored the effect of basis
+sets on the non-linear optical properties (hyperpolarizabilities) of linear diacetylenes using
+time-dependent Hartree-Fock theory. They found that the inclusion of a diffuse ‘d’ function
+to a standard double-zeta plus polarization basis can significantly improve the frequency-
+dependent hyperpolarizability. Jabłonski and Palusiak28 have explored the influence of basis
+sets in Hartree-Fock (HF) and DFT/B3LYP calculations for the values atoms in molecules
+(AIM) parameters. They found that smaller Dunning’s basis sets, including cc-pVDZ and
+aug-cc-pVDZ provide poor results as compared to medium-sized Pople-type basis sets. We
+are not aware of a systematic study in which the basis sets have been examined from the
+perspective of their performance in excited state calculations. Furthermore, we have also
+not come across a study which examines the basis sets from the point of view their ability to
+compute optical properties of atoms and molecules, which involves calculations of transition
+dipole moments, in addition to excited state energies, and wave functions.
+In order to
+fill this void, we decided to undertake a systematic investigation of the influence of basis
+sets on the qualitative and quantitative description of optical absorption spectra of atomic
+clusters. In this paper, we have performed calculations of linear optical absorption spectra
+of several small neutral and cationic clusters, e.g., Li2, Li3, Li4, B+
+2 , B+
+3 , Be+
+2 , and Be+
+3 ,
+using the configuration-interaction (CI) approach. For this purpose, a number of basis sets,
+namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-
+cc-pVTZ, were employed, and their influence on the convergence of excited state energies,
+wave functions, and transition dipole moments has been systematically examined. In this
+study, the reason behind our choice of smaller sized atomic clusters and their ions, as against
+larger ones, is that it is possible to perform highly accurate CI calculations on smaller systems
+so that the difference between results obtained with different basis sets will be due the nature
+of basis sets, and not due to the CI approach employed. Based upon our calculations, the
+main conclusion is that it is very important to include diffuse basis functions in the basis set
+in order to obtain a good description of the photoabsorption spectra.
+4
+
+Theoretical Approach and Computational Details
+General Methodology
+All the calculations were performed using the first-principles wave-function-based electron-
+correlated approaches, using the standard Hamiltonian within the Born-Oppenheimer ap-
+proximation. The molecular orbitals are expressed in terms of the linear combination of
+Cartesian-Gaussian type basis functions, also called atomic orbitals (AOs). Although, for
+such calculations, a number of program packages are available, we employed GAUSSIAN1629
+and MELD30 for our calculations. The geometries of all the clusters considered in this work
+were optimized using GAUSSIAN16 package29 at the coupled-clusters singles-double (CCSD)
+level of theory, employing a large augmented correlation-consistent polarized valence triple-
+zeta (aug-cc-pVTZ) basis set.
+We perform excited-states calculations for various clusters employing their ground-state
+optimized geometries, using the configuration-interaction (CI) methodology at various levels
+of approximation, as implemented in the program package MELD30. The CI calculations
+yield the vertical excitation energies, the ground and excited state wave functions, and the
+transition dipole matrix elements connecting the ground and the excited states, which, in
+turn, are used to compute the optical absorption spectra of various clusters.
+The level
+of CI employed in the calculations depends on the size of cluster, the number of valence
+electrons in cluster, and the number of active orbitals. The linear optical absorption spectra
+of Li2, Li3, Li4, Be+
+2 , B+
+2 clusters were computed at the full CI (FCI) level, while for Be+
+3
+and B+
+3 calculations were performed at the quadruple CI (QCI), and the multi-reference
+singles-doubles CI (MRSDCI) levels, respectively.
+We start the calculations on a given cluster by first performing restricted Hartree-Fock
+(RHF) calculations on it, and obtain the molecular orbitals (MOs), expressed as linear com-
+binations of the chosen AOs. In order to perform CI calculations, the one- and two-electron
+Hamiltonian matrix elements are transformed from the AO representation to the MO rep-
+5
+
+resentation. For the FCI calculations, all possible configurations obtained by placing all the
+valence electrons of the cluster in the given set of MOs, in all possible ways, consistent with
+the Pauli exclusion principle. In the QCI approach, we first choose a reference configuration,
+and then generate configurations which are singly-, doubly-, triply-, and quadruply-excited
+with respect to it. For the ground-state calculations, the reference configuration is normally
+taken to be the RHF configuration, while for the excited-state calculations one chooses an ex-
+cited configuration which is closest to the excited state one is trying to calculate. However,
+both the FCI and the QCI approaches can lead to a very large number of configurations
+if the number of electrons and the MO basis is large, thus, making the calculations in-
+tractable. Therefore, for the larger clusters, we employed the multi-reference singles-doubles
+configuration-interaction (MRSDCI) approach, as implemented in the MELD package. In
+this approach, the singly- and doubly-excited configurations are generated from a list of
+configurations called the reference configurations, chosen by the user. We performed the
+MRSDCI calculations in an incremental manner, by starting out with a small set of refer-
+ence configurations that are close to the states (ground or excited) we are targeting. Then
+we analyze the optical absorption spectra of the cluster calculated from that MRSDCI cal-
+culation, and identify a new set of configurations which need to be included in the list of
+reference configurations based upon their contributions in the wave functions of the targeted
+states. The procedure is iterated until the calculated optical absorption spectrum converges
+to within a user-defined threshold. In all the CI calculations, the configurations are actu-
+ally configuration-state functions (CSFs) which are eigenstates of the point-group symmetry
+operators, and the total spin operators S2 and Sz.31–42
+The linear optical absorption spectrum of a given cluster is calculated under the electric-
+dipole approximation, using the formula
+σ(ω) = 4πα
+�
+i
+ωio|⟨i|ˆe.r|0⟩|2γ2
+(ωi0 − ω)2 + γ2
+(1)
+6
+
+Above: (i) σ(ω) represents the optical absorption cross section, (ii) ω is the frequency of
+incident light, (iii)ˆe denotes the polarization direction of the incident light, (iv) r is the posi-
+tion operator, (v) α is the fine structure constant, (vi) ℏωi0 is the energy difference between
+ground state (0) and the ith excited state (i), and (vii) γ is the uniform line width associated
+with each excited state. The line width γ is taken to be 0.1 eV in all our calculations. The
+sum over index i denotes the sum over all possible excited states. We have restricted this
+sum in our calculations up to the states corresponding to excitation energies of 10 eV, or
+less. Additionally, the oscillator strength fn corresponding to an optical transition from the
+ground state to the n-th excited state is computed using the standard formula
+fn = 2me
+3ℏ2 ∆En
+�
+j=x,y,z
+�
+α
+|⟨nα|Oj|0⟩|2
+(2)
+above me is the electron mass, |0⟩ and |nα⟩ are, respectively the CI wave functions of the
+ground state and the excited state in question, with α being a degeneracy label, Oj denotes
+j-th Cartesian component of the electric-dipole operator, while ∆En = En − E0 is the
+excitation energy of the excited state.
+Computational Parameters
+In this section, we will discuss the convergence of the results with respect two parameters,
+related to the basis-set-size: (a) number of active orbitals in the CI calculations, and (b)
+number of CSFs included in the calculations.
+Active molecular orbitals
+It is well-known that the computational cost at configuration interaction (CI) level of theory
+increases as N 6
+act, where Nact is the total number of active molecular orbitals used in the
+CI calculations.
+Therefore, the time needed to perform a CI calculation will proliferate
+rapidly with the increasing values of Nact. We have adopted two approaches to reduce the
+7
+
+size of the active MO set: (a) we adopt the frozen-core approximation to eliminate the core
+orbitals of each atom of the cluster, and (b) for certain cases involving large CI matrices,
+we delete all those virtual (unoccupied) orbitals from our calculations whose single-particle
+energies are larger than 1 Hartree. The frozen-core approximation is a standard approach
+which also has the added advantage of considerably reducing the number of active electrons
+(nelec) in the calculation. The “1 Hartree cutoff” also doesn’t reduce the accuracy of the
+calculations because we are interested in low-lying optical excitations below 10 eV, while
+our cutoff eliminates only those orbitals from the calculations whose energy is larger than
+27.21 eV. Both these approximations have been investigated rigorously in our group in earlier
+calculations.36,41–43.
+To be specific, in the present set of calculations, we have considered all the virtual orbitals
+for Li2 and Be+
+2 clusters, while for Li3, Li4, Be+
+3 , B+
+2 and B+
+3 clusters we have imposed the 1
+Hartree cutoff.
+Size of CI expansion
+Another important parameter that controls the quality of calculations is the total number
+of CSFs, Ntotal, included in the CI expansion of the many-particle wave functions of the
+clusters concerned, both for their ground and the excited states. As mentioned earlier, for
+a given set of active electrons and MOs, the best possible CI expansion corresponds to the
+FCI expansion, which becomes intractable for systems with large values of nelec and Nact.
+However, whenever FCI is not possible, we employ one of the restricted CI approaches such
+as the QCI or the MRSDCI methods. Of the two, it is crucial to examine the convergence of
+the MRSDCI approach which is based upon singles and doubles excitations from a number
+of reference configurations (Nref) leading to the final CI expansion with Ntotal CSFs. We
+examined the convergence of the optical absorption spectrum for the B+
+3 cluster calculated
+using the MRSDCI method, with respect to Nref, and Ntotal, as presented in Fig. 1.
+8
+
+Figure 1: Convergence of the optical absorption spectrum of the B+
+3 computed using the
+MRSDCI method, with the increasing numbers of reference configurations (Nref). For cal-
+culations labeled MRSDCI1, MRSDCI2, and MRSDCI3, values of Nref were 58, 101, and
+144, respectively.
+In the figure, we plot the absorption spectra of B+
+3 obtained from three MRSDCI calcula-
+tions of increasing sizes labeled as MRSDCI1, MRSDCI2, and MRSDCI3. In these calcula-
+tions, the values of parameters Nref and Ntotal were Nref= 58, Ntotal= 4007873, Nref= 101,
+Ntotal= 5781436, and Nref= 144, Ntotal= 8422193, respectively. From Fig. 1., it is obvious
+that the spectra obtained using MRSDCI2 and MRSDCI3 calculations are very close to each
+other, signaling convergence with respect to the size of the MRSDCI expansion.
+Results and Discussion
+Before discussing the results of our calculations of the optical absorption spectra of various
+clusters, we first summarize their ground state geometries in Table 1, optimized at the CCSD
+9
+
+300
+MRSDCI1
+250
+MRSDCI2
+MRSDCI3
+Intensity(arb. units)
+200
+150
+100
+50
+0
+0
+8
+10
+Energy(eV)level of theory, employing GAUSSIAN16 suite of programs29, and large aug-cc-pVTZ basis
+sets.
+Table 1: The nature of the structure, along with the point group symmetry utilized, during
+the coupled-cluster singles-doubles (CCSD) geometry optimization calculations are presented
+below. Additionally, for each cluster, the symmetry of the ground-state wave function, total
+Hartree-Fock (HF) energy in Hartree (Ha), total CCSD energy (Ha), and the correlation
+energy (eV) are also presented. During the calculations, aug-cc-pVTZ basis set was employed
+for each cluster.
+Cluster
+Structure
+Point group
+Symmetry of the
+HF energy
+CCSD energy
+Correlation energy
+GS wave function
+(Ha)
+(Ha)
+(eV)
+Li2
+Linear
+D2h
+1Ag
+-14.8715509
+-14.9033549
+0.87
+Li3
+Linear
+D2h
+2Ag
+-22.3088776
+-22.3454346
+0.99
+Li3
+Isosceles triangle
+C2v
+2A1
+-22.3170594
+-22.3557287
+1.05
+Li4
+Rhombus
+D2h
+1Ag
+-29.7619144
+-29.8354840
+2.00
+Be+
+2
+Linear
+D2h
+2Ag
+-28.9205835
+-28.9672583
+1.27
+Be+
+3
+Linear
+D2h
+2Ag
+-43.5410215
+-43.6332325
+2.51
+B+
+2
+Linear
+D2h
+2Ag
+-48.8344277
+-48.9626672
+3.49
+B+
+3
+Equilateral triangle
+D3h
+1A
+′
+1
+-73.4445744
+-73.7127527
+7.27
+For each cluster, the table lists the nature of its ground-state structure, point group
+employed in the calculations, symmetry of the ground state wave function, total energy of the
+ground state, and the correlation energy. In Table 2, the details related to our CI calculations
+performed for computing the optical absorption spectra of various clusters are provided. For
+various clusters, the table lists: (a) the type of CI calculation, (b) the point-group symmetry
+employed in the calculations, (c) irreducible representations considered for each point group,
+and (d) for each irreducible representation, the size of the CI expansion (Ntotal) for each
+cluster are depicted. From Table 2 it is obvious that most of the CI calculations were of the
+FCI type, which are exact for the chosen set of active MOs. Furthermore, in the calculations
+in which approaches such as QCI or MRSDCI were used, the size of the CI expansion is
+quite large. This means that the CI calculations performed in this work are fairly large
+scale, indicating that the computed optical absorption spectra are numerically accurate.
+Next, for these clusters, we discuss in detail the calculated ground state geometries, followed
+by their optical absorption spectra.
+10
+
+Table 2: For each cluster, the type of CI approach used for the calculations of the optical
+properties, point group symmetry employed during the CI calculations, and the total number
+of configurations (Ntotal) in the calculation are listed below. The value of Ntotal corresponds
+to aug-cc-pVTZ basis set based CI calculations.
+Cluster
+Structure
+Method
+Point group
+Symmetry
+Ntotal
+used
+Li2
+Linear
+FCI
+C1
+1A
+5886
+Li3
+Linear
+FCI
+C1
+2A
+575960
+Li3
+Isosceles triangle
+FCI
+C2v
+2A1
+137956
+2B1
+129520
+2B2
+137396
+Li4
+Rhombus
+FCI
+D2h
+1Ag
+1853578
+1B1u
+1846246
+1B2u
+1844485
+1B3u
+1802190
+Be+
+2
+Linear
+FCI
+C1
+2A
+419868
+Be+
+3
+Linear
+QCI
+D2h
+2Ag
+7393226
+2B1u
+7393210
+2B2u
+7286869
+2B3u
+7286869
+B+
+2
+Linear
+FCI
+D2h
+2Ag
+6365216
+2B1u
+6365216
+2B2u
+6323328
+2B3u
+6323328
+B+
+3
+Equilateral triangle
+MRSDCI
+C1
+1A
+8422193
+11
+
+Geometry
+The simplest cluster of lithium is lithium dimer with the D∞h point group symmetry. We
+obtained the optimized bond length of Li2 cluster to be 2.70 Å, as shown in Fig. 2(a). This
+result is in excellent agreement with the bond length 2.68 Å reported by Wheeler et al.44,
+who performed the calculations at the CCSD/CCSD(T) level of theory using the Dunning
+correlation-consistent polarized core-valence triple/quadruple-zeta cc-pwCVXZ basis sets.
+Florez et al.45 performed density functional theory (DFT) calculations using the B3LYP
+and BLYP functionals, and reported the bond lengths to be 2.70 Å, and 2.71 Å, respectively,
+again in excellent agreement with our result. Furthermore, our calculated bond length is
+also in a very good agreement with the experimentally measured value 2.67 Å, reported by
+Huber46.
+As far as Li3 cluster is concerned, two isomers namely linear and isosceles triangle were
+found to be stable. The equilateral triangular structure of Li3 cluster is not stable, and
+undergoes Jahn-Teller distortion to acquire the isosceles triangular structure. The linear
+structure has the D∞h point-group symmetry, with the optimized equal bond lengths of 2.90
+Å (see Fig. 2(b)), in excellent agreement with the value 2.89 Å, reported by Jones et al.47.
+The lowest-energy geometry of the Li3 cluster is an isosceles triangle with the C2v point-
+group symmetry. The CCSD-level optimized bond lengths for this structure are found to be
+2.68 and 3.07 Å, with the bond angles 51.73◦ and 64.13◦(see Fig. 2(c)). We note that by
+performing DFT calculations, Jones et al.47 obtained the bond lengths of 2.82 and 3.37 Å,
+that are significantly different as compared to our results.
+12
+
+Figure 2: Optimized geometry of (a) Li2, (b) Li3 linear, (c) Li3 isosceles triangular, (d) Li4,
+(e) Be+
+2 , (f) Be+
+3 linear, (g) B+
+2 , and (h) B+
+3 equilateral triangular clusters considered in
+this work. The geometry optimization has been performed using the CCSD method, and
+aug-cc-pVTZ basis sets. All the listed bond lengths are in Å units.
+The lowest-energy structure for the Li4 cluster has a rhombus shape, with D2h point
+group44,47, as shown in Fig.
+2(d).
+Our optimized bond lengths of the side and minor
+diagonal of the rhombus structure are 3.02 and 2.65 Å, respectively, which are in excellent
+agreement with the values 3.04 and 2.62 Å reported by Jones et al.47.
+The optimized bond length of the Be+
+2 cluster with the D∞h point group is found to be
+2.25 Å (see Fig. 2(e)), in good agreement with the reported bond length 2.21 Å, obtained
+from DFT calculations by Srinivas et al.48. Our lowest-energy optimized structure of Be+
+3
+cluster also has a linear geometry, with two equal bond lengths 2.22 Å, as shown in Fig. 2(f).
+This value of the bond length is in very good agreement with the value 2.19 Å, computed
+by Srinivas et al.48 using DFT.
+As far as B+
+2 cluster is concerned, we computed its minimum-energy bond length to be
+2.18 Å (see Fig. 2(g)), which is 0.18 Å larger than the value 2 Å reported by Hanley et al.49.
+13
+
+(a)
+(b)
+2.70
+2.90
+Liz
+Lis chain
+(c)
+(d)
+3.07
+2.65
+2.68
+3.02
+Li3
+(f)
+Li4
+(e)
+2.25
+2.22
+Be2*
+Be3+
+(g)
+(μ)
+2.18
+1.58
+B,
+B3We attribute this difference to two factors, namely, smaller basis set (6-31G∗), coupled with a
+lower-level CI methodology used by the authors.49. Our optimized structure of B+
+3 cluster is
+an equilateral triangle of sides 1.58 Å, with the D3h point-group symmetry, as shown in Fig.
+2(h). Hanley et al.49 using a CI approach, along with the 6-31G∗ basis set, also obtained the
+optimized structure to be an equilateral triangle for the B+
+3 , but with a bond length of 1.53
+Å, which is 0.05 Å smaller than our result. We again attribute the differences to the choice
+of a smaller basis set, coupled with a lower-level correlation methodology as compared to
+the CCSD approach used by us.
+Peak locations
+Li2 dimer, with just two active electrons within the frozen-core approximation, is the small-
+est many-electron cluster considered in this work. Therefore, very high-quality correlated-
+electron calculations using large basis sets are possible for this system, not just for its ground
+states, but also for the excited states. As a result, this case can provide us deep insights into
+the influence of the choice of basis functions on the calculated excited state properties and the
+photoabsorption spectra. For the calculations, we employed the frozen-core FCI method us-
+ing six basis sets of varying sizes, namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ,
+cc-pVTZ, aug-cc-pVDZ and aug-cc-pVTZ, and the computed spectra are presented in Fig.
+3(a). All the virtual molecular orbitals generated during the RHF calculations were used in
+the CI calculations, i.e., no unoccupied orbitals were discarded. As a result, the frozen-core
+FCI results presented here are the best ones possible for the chosen basis sets.
+For the Li2 dimer, the peak locations in the computed spectra are presented in Table S1
+of the Supporting information (SI), from which it is obvious that for the first two peaks the
+excitation energies calculated using different basis sets are in very good agreement with each
+other. This is encouraging because from Fig. 3(a) it is obvious that most of the oscillator
+strength of the absorption spectrum is confined to these two peaks. However, starting from
+the third peak onward, we start seeing differences in the excitation energies predicted by
+14
+
+different basis sets. For the third peak, the predicted peak locations can be classified in two
+groups: (a) those predicted by correlation-consistent basis sets cc-pVDZ and cc-pVTZ, and
+(b) the ones predicted by 6-311G++ and augmented correlation consistent (aug-cc-) class of
+basis sets. We note that the peak locations predicted by the former class of basis functions
+have values significantly larger than those predicted by the latter class. Another noteworthy
+point is that there is very good agreement among the peak locations predicted by the second
+class of basis sets. As far as the location of the fourth peak is concerned, there is good
+agreement among the predictions by 6-311++G(3df,3pd) and aug-cc class of basis functions,
+while the remaining three basis functions predict very different values. The case of the fifth
+peak is somewhat anomalous in that the agreement among the predictions by any of the
+basis sets is not good. However, for higher peaks we note that the results from the aug-cc
+class of basis functions are in good agreement with each other, while other basis functions
+predict widely differing results. Hong et al.50 also performed first-principles calculations of
+the photoabsorption spectra of several Lin clusters employing the time-dependent density-
+functional theory (TDDFT) methodology, and for Li2 their predicted locations of the first
+two peaks are 1.92 eV and 2.53 eV50. On comparing these with our best values of 1.83 eV
+and 2.57 eV, respectively, we note: (a) our excitation energy for peak I is about 0.09 eV
+smaller than theirs, while (b) our location for peak II is about 0.04 eV larger than theirs.
+We attribute these differences to different computational methodologies adopted in the two
+sets of calculations, and it will be interesting to compare the computational results with the
+experimental ones, whenever they are available.
+The peak locations of the photoabsorption spectra of Li3 chain are presented in Table S2
+of SI, from which it is clear that the locations of the first two peaks converge completely for
+all the basis sets, similar to the case of dimer. The third peak is the most intense peak of
+the computed spectra as shown in Fig. 3(b), whose location is in good agreement for all the
+basis sets except for cc-pVDZ, which predicts higher excitation energy as compared to the
+rest. From the fourth peak onward, the peak locations can be classified in two similar group
+15
+
+as discussed previously for the case of dimer: the peak locations predicted from correlation-
+consistent basis sets cc-pVDZ and cc-pVTZ are towards the higher energy side as compared
+to all other basis sets. It can also be seen that the peak positions corresponding to the two
+classes of basis sets are in good agreement within the class.
+Next, we examine the peak locations in the photoabsorption spectra of Li3 triangular
+cluster computed using various basis sets. We note that the peak locations corresponding
+to the first five peaks are in very good agreement with each other for different basis sets
+as is obvious from Table S3 of SI. This result is very encouraging because peak IV is the
+most intense (MI) peak of the computed spectra as presented in Fig. 3(c), and it is crucial
+for a basis set to be able to accurately describe the MI peaks. The location of this peak is
+2.43 eV computed using the aug-cc-pVTZ basis set, which is in a decent agreement with the
+experimentally detected peak at 2.58 eV by Blanc et al.51. From the sixth peak onward it
+was observed that the peak locations calculated using correlation consistent basis sets (cc-
+pVDZ and cc-pVTZ) do not match with the other classes of basis sets. However, the peak
+locations computed using the 6-31G class and the aug-cc-pVTZ continue to be in very good
+agreement with each other till peak VIII, located near 3.8 eV. The locations of higher-energy
+peaks beyond peak VIII computed using these basis sets are presented in Table S4 for the
+SI.
+16
+
+Figure 3: Optical absorption spectra of (a) Li2, (b) Li3 linear, (c) Li3 triangular, and (d) Li4
+clusters computed using various basis sets and the frozen core FCI method. The uniform
+line-width 0.1 eV is used to plot the spectrum.
+The peak positions of the photoabsorption spectra of Li4 cluster computed using various
+basis sets are presented in Table S5 of SI, while the spectra are plotted in Fig. 3(d). We note
+that for this cluster, the excitation energies of the first five peaks computed using different
+basis sets are in very good agreement with each other. The first three peaks are much more
+intense as compared to the higher energy peaks, and in peak III there are slight differences
+(≈0.1 eV) in the peak locations predicted by different basis sets. The two largest basis sets
+(6-311++G(3df,3pd), and aug-cc-pVTZ) predict the location of peak III at 2.93 eV, while
+the predictions by the rest of the basis sets are in the range 3.03–3.08 eV. From peak VI
+17
+
+SO
+(a)
+60K
+6-311++G(2d,2p)
+6-311++G(3df,3pd)
+(b)
+III
+6-311++G(2d,2p)
+400
+cc-pVDZ
+500
+6-311++G(3df,3pd)
+ZLAd-0
+cc-pVDZ
+aug-cc-pVDZ
+cC-pVTZ
+awg-cc-pVTZ
+aug-cc-pVDZ
+(s))
+400
+aug-cc-pVTZ.
+300
+200
+100
+100 
+I
+VVI
+VII
+6
+Energy (eV)
+300
+400
+(c)
+6-311++G(2d,2p)
+(d)
+6-311++G(2d,2p)
+250
+6-311++G(3df,3pd)
+cc-pVDZ
+6-311++G(3df,3pd)
+ZLAd-33
+II
+cc-pVDZ
+aug-cc-pVDZ
+300
+cc-pVTZ
+200
+ZLAd-30-8ne
+aug-cc-pVDZ
+aug-cc-pVTZ
+Intensity 
+150
+Intensity
+100
+100
+50
+VIII
+X
+2
+6
+Energy (eV)
+Energy (eV)onward we begin to observe differences among the locations predicted by different basis sets,
+with a tendency towards clustering into different classes. However, the noteworthy point
+is that the intensity corresponding to these higher energy peaks is very low. As far as the
+comparison with the experiments is concerned, the first three photoabsorption peaks of the
+Li4 cluster located at 1.87 eV, 2.65 eV, and 2.93 eV for aug-cc-pVTZ basis set are in excellent
+agreement with the experimental measurements of Blanc et al.51 who detected these peaks
+at 1.83 eV, 2.65 eV, and 2.93 eV, respectively.
+For Be+
+2 cluster, we present the spectra computed by different basis sets in Fig. 4(a),
+while the corresponding peak locations are presented in Table S6 of SI. We note excellent
+convergence of the excitation energies up to the sixth peak, beyond which results obtained
+by different basis sets do not agree much with each other. We further note that Peak V
+located near 6.30 eV is the most intense peak, and, for that, the predictions of the different
+basis sets are in a fairly narrow energy range 6.30-6.37 eV.
+Figure 4: Optical absorption spectra of (a) Be+
+2 and (b) Be+
+3 clusters computed using various
+basis sets and frozen core FCI and QCI methods, respectively. The uniform line-width of
+0.1 eV is used to plot the spectrum.
+The excited-states peak locations of Be+
+3 cluster for different basis sets are presented in
+Table S7 of SI. We notice excellent agreement of the excited-states peak locations up to the
+18
+
+600
+(a)
+6-311++G(2d,2p)
+VII
+6-311++G(2d,2p)
+300
+6-311++G(3df.3pd)
+500
+6-311++G(3df.3pd)
+cc-pVDZ
+cc-pVDZ
+cc-pVTZ
+cc-pVTZ
+aug-cc-pVDZ
+aug-cc-pVDZ
+aug-cc-pVTZ
+400
+aug-cc-pVTZ
+(sirun
+200
+Intensity (arb.
+300
+Intensity
+200
+100
+100
+2
+3
+4
+5
+6
+9
+[0
+3
+5
+Energy (eV)
+Energy (eV)seventh peak which is also the most intense peak of the spectra located near 6.5 eV, as shown
+in Fig. 4(b). Although the peak location of the sixth peak computed using cc-pVDZ basis
+set is slightly towards the higher energy region as compared to all other basis sets, but the
+difference is small. Noteworthy point is that these basis sets are able to achieve convergence
+in the peak positions in Be+
+2 and Be+
+3 photoabsorption spectra up to much higher excitation
+energies, as compared to the Li clusters.
+The excited-states peak positions corresponding to the photoabsorption spectra of B+
+2
+cluster are presented in Table S8 of SI. We notice excellent agreement of the peak energies
+corresponding to first three peaks for all the basis sets. The fourth peak is the most intense
+peak of the spectra, as shown in Fig.
+5(a) for whose location excellent agreement has
+been achieved for 6-311++G (2d, 2p), cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ basis sets,
+indicating complete convergence. However, the excitation energies for peak IV computed
+using the 6-311++G (3df, 3pd) and cc-pVDZ basis sets are about 0.1 eV higher, as compared
+to other basis sets. As far as peak V is concerned, which is a very weak shoulder of peak IV,
+we again observe excellent convergence for all the basis sets, except cc-pVDZ which fails to
+predict the peak. From the sixth peak onward, as discussed previously, the predicted peak
+locations can be classified into two groups: (a) larger basis sets of 6-311++G and aug-cc-
+type, and (b) smaller basis sets cc-pVDZ and cc-pVTZ, with the peak locations predicted
+by individual classes being in very good agreement with each other.
+19
+
+Figure 5: Optical absorption spectra of (a) B+
+2 and (b) B+
+3 cluster computed using various
+basis sets and frozen core FCI and MRSDCI methods, respectively. The uniform line-width
+0.1 eV is used to plot the spectrum.
+The peak locations corresponding to the excited-states of the photoabsorption spectra
+of B+
+3 cluster are presented in Table S9 of SI, while the calculated spectra are plotted in
+Fig. 5(b) . For this cluster, we get eight well-separated peaks in the explored energy range,
+with peak VIII located near 8.9 eV being the most intense. We note that the peak energies
+corresponding to all the basis sets converge excellently up to the peak VIII, except those
+predicted by the cc-pVDZ basis set, which are consistently higher.
+We have noticed in
+Fig.4 and Fig.5, only the deep valence excitation energies are dependent on the choice of
+basis sets. This behavior can be a consequence of the frozen-core approximation, which we
+have employed in the calculations. To verify this, we have computed the optical absorption
+spectra of Li2 and Be+
+2 clusters by also including the core excitations within the large-scale
+QCI method. We found that the optical spectra of these clusters computed by including core
+excitations agrees completely with the absorption spectra computed after employing frozen-
+core approximation, as shown in Fig.S1 and Fig.S2 of the SI. Therefore, the frozen-core
+approximation does not alter the absorption spectra of small clusters.
+Based on the peak positions of the individual clusters discussed above, we observe the
+20
+
+500
+250
+(a)
+(b)
+6-311++G(2d,2p)
+6-311++G(2d,2p)
+6-311++G(3df,3pd)
+VII
+6-311++G(3df,3pd)
+400
+cc-pVDZ
+IV
+200
+cc-pVDZ
+cc-pVTZ
+VII
+cc-pVTZ
+aug-cc-pVDZ
+aug-cc-pVDZ
+aug-cc-pVTZ
+ZLAd-03-8ne
+150
+Intensity (arb.
+200
+VI
+100
+100
+50
+III
+II
+1
+2
+3
+4
+6
+8
+9
+10
+2
+3
+1
+6
+7
+8
+10
+Energy (eV)
+Energy (eV)following general trends: (a) peak locations for all the clusters used in this study are in very
+good agreement for all the basis sets up to the most intense peak of the spectra, except
+for cc-pVDZ basis set. (b) the excited-states peak locations beyond the most intense peak
+can be classified in two groups, in which the peak locations calculated using correlation-
+consistent basis sets do not match with the peak locations computed using all other basis
+sets, and (c) for the cc-pVDZ basis sets peaks are located at higher energies as compared
+to the rest of the basis functions. As the basis-set dependence of the optical properties is
+different for the density functional theory compared to the wave function-based large-scale
+configuration-interaction method, it will be interesting to explore the optical properties of
+clusters using time-dependent density functional theory (TD-DFT) and compare it with our
+results. We found that the first peak of the optical absorption spectra of B+
+3 cluster is located
+at 0.84 eV when computed using large-scale MRSDCI calculations along with a large aug-
+cc-pVTZ basis set. However, when the calculations are performed using TD-DFT method
+with B3LYP functional and the same basis set, it is obtained at 0.95 eV. The most intense
+peak of the optical absorption spectra is located at 8.85 eV using the MRSDCI approach,
+which is found to be at 9.35 eV by employing the TD-DFT method. The calulated optical
+absorption spectra and excited-states peak locations of B+
+3 cluster corresponding to TD-
+DFT calculations are provided in the Fig.S3 and Table S10 of the SI. We also report that
+the variations in the peak locations of the photoabsorption spectra of B+
+3 computed using
+various basis sets and TD-DFT method are lesser than the wave function-based CI method.
+Oscillator strength
+In addition to the excitation energy, the next important quantity determining the profile of
+the absorption spectrum is the oscillator strength (f) corresponding to various optical tran-
+sitions, connecting the ground state to the excited state in question. The oscillator strength
+calculated using Eq. 2 is determined by the excitation energy of the state involved, and the
+corresponding transition dipole moment (TDM). The TDM being a matrix element, is, in
+21
+
+turn, determined by the many-particle wave functions of the ground and the excited state
+that it connects. Another important quantity is the polarization of the photon involved in a
+given optical transition (peak), which can be measured in oriented samples. The polarization
+is a consequence of the point-group symmetry of the concerned molecule, and hence should
+be independent of the basis set employed. In this section, we discuss the convergence of the
+oscillator strengths and photon polarizations associated with various peaks of the calculated
+spectra. In Table 3, we present the oscillator strengths corresponding to the first peak, and
+the most intense peak (peak II) of the spectra of Li2 computed using different basis func-
+tions. Additionally, the table also contains the dominant configurations contributing to the
+many-particle wave functions of the excited states involved.
+Table 3: Comparison of oscillator strengths and the dominant configurations contributing to
+the many-particle wave functions for peaks I and II of Li2 cluster calculated using different
+basis sets. In the “Polarization” column, ∥ indicates photon polarization along the direction
+of the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to
+the molecular axis (transverse polarization). Note that the transversely polarized states are
+doubly degenerate, therefore, the oscillator strength corresponding to those is the sum of
+both the contributions. ’H’ and ’L’ stand for HOMO and LUMO orbitals.
+Basis Set
+Peak I
+Peak II
+Polarization
+f
+Configurations
+Polarization
+f
+Configurations
+6-311++G(2d,2p)
+∥
+0.460
+|H → L⟩
+⊥
+0.971
+|H → L + 2⟩
+|H → L + 3⟩
+|H → L + 7⟩
+6-311++G(3df,3pd)
+∥
+0.456
+|H → L⟩
+⊥
+0.966
+|H → L + 2⟩
+|H → L + 3⟩
+|H → L + 7⟩
+cc-PVDZ
+∥
+0.463
+|H → L⟩
+⊥
+0.970
+|H → L + 1⟩
+|H → L; H → L + 2⟩
+|H → L + 1; H → L + 5⟩
+cc-pVTZ
+∥
+0.455
+|H → L⟩
+⊥
+0.969
+|H → L + 1⟩
+|H → L + 4⟩
+|H → L + 6⟩
+aug-cc-pVDZ
+∥
+0.462
+|H → L⟩
+⊥
+0.972
+|H → L + 1⟩
+|H → L + 3⟩
+|H → L + 6⟩
+aug-cc-pVTZ
+∥
+0.454
+|H → L⟩
+⊥
+0.966
+|H → L + 2⟩
+|H → L + 3⟩
+|H → L + 7⟩
+From Table 3 it is obvious that the oscillator strengths computed using various basis
+functions for both the peaks are in very good agreement with each other. We also note that
+the direction of the polarization of the photons involved in a given optical transitions are of
+22
+
+the excited-states corresponding to the first and second peak of the spectra of Li2 cluster are
+parallel and perpendicular to molecular axis, respectively, irrespective of the basis set.
+The oscillator strengths corresponding to the first and most intense peaks of the spectra
+of Li3 chain and triangular clusters are presented in Table S11 and Table S12, respectively.
+We note that the oscillator strengths of the first peaks both of Li3 chain, and the triangular
+cluster, computed using the different basis sets are in excellent agreement with each other.
+The oscillator strengths corresponding to the most intense peaks of Li3, i.e. peak III of
+the chain and peak IV for the triangular cluster, calculated using various basis sets can
+be classified into two groups: (a) those calculated using correlation-consistent (cc-pVTZ
+and cc-pVDZ) class of basis sets, and (b) those computed using 6-33++G- and augmented
+correlation-consistent (aug-cc-) class of basis sets. The oscillator strength calculated by the
+first class of basis sets is comparatively higher than the second class of basis sets. But, the
+relative maximum difference between oscillator strengths of different classes is close to 6%,
+which is fairly acceptable.
+The oscillator strengths corresponding to the first and the most intense peak (peak II)
+in the photoabsorption spectra of Li4 cluster are presented in Table S13.
+We note that
+the oscillator strengths of peak I are in good agreement with each other for all the basis
+sets except for cc-pVDZ and aug-cc-pVTZ. For these basis sets the oscillator strength is
+comparatively larger.
+For peak II, we note that the difference in the oscillator strength
+computed by cc-pVDZ and 6-311++G (3df, 3pd) basis sets is about 5%, which is again
+quite small.
+We present the oscillator strengths corresponding to the first and the most intense peaks
+of the cationic beryllium clusters Be+
+2 and Be+
+3 in tables S14, and S15, respectively. We
+note very good agreement on the oscillator strengths of both the peaks of the Be+
+2 and
+Be+
+3 clusters for all the basis sets. The maximum relative disagreement we find among the
+oscillator strengths for a given peak is around 6%.
+Finally, we discuss the oscillator strengths of the first and the most intense peaks of the
+23
+
+B+
+2 and B+
+3 clusters presented in tables S16, and S17, respectively. We note that both for
+B+
+2 and B+
+3 clusters, the oscillator strengths of the first peaks are two orders of magnitude
+smaller than those of their most intense peaks, indicating that the first peaks for both the
+clusters are relatively feeble. Nevertheless, the oscillator strengths of the first peaks of the
+photoabsorption spectra of the two clusters calculated using various basis sets are in very
+good agreement with each other. As far as the most intense peaks are concerned, both for B+
+2
+and B+
+3 we see the following pattern: oscillator strengths computed using 6-311++G- and
+aug-cc-pVTZ basis sets are in very good agreement with each other, while those computed
+using other basis sets differ from them somewhat.
+Wave function analysis
+Next, we examine the dominant configurations contributing to the CI wave functions of the
+excited states contributing to various peaks. The dominant configurations corresponding
+to the excited-states CI wave functions of peak I and peak II of Li2 are presented in Table
+3. We note that for peak I, the main contribution to the corresponding excited state wave
+function is from the singly excited configuration |H → L⟩ for all the basis sets. However,
+the next important configuration to the same wave function depends on the class of basis set
+employed: (a) it is |H → L+3⟩ single excitation when calculations are performed using larger
+basis sets of the type 6-311++ and aug-cc, but (b) for smaller basis sets, this configuration is
+found to be |H → L + 4⟩ for cc-PVTZ basis, and |H → L; H → L + 2⟩ for the cc-PVDZ set.
+Peak II is due to two degenerate excited states to which the dominant contributions are from
+configurations |H → L+2⟩ and |H → L+7⟩, for the calculations performed using 6-31G++
+and aug-cc-PVTZ type basis sets. But, for the calculations performed with smaller basis
+sets, the dominant configurations is |H → L + 1⟩, while the next important configuration
+can be |H → L + 6⟩ or |H → L + 1; H → L + 5⟩, depending on the basis set. Thus, we
+can draw the following general conclusion regarding this: (a) for large basis set calculations,
+for a given peak, the configurations are in perfect agreement with each other, and (b) the
+24
+
+configurations predicted by calculations performed using smaller basis sets such as cc-PVDZ
+are found to be different as compared to those obtained in larger basis set calculations.
+The dominant configurations for the wave functions corresponding to peak I and the
+most intense peak (peak III) of Li3 chain, computed using various basis sets are presented
+in Table S11. We find that for the first peak the dominant configuration is |H − 1 → H⟩ for
+all the basis sets except for aug-cc-pVTZ. For the aug-cc-pVTZ the dominant configurations
+contributing to the excited state wave function are different compared to other basis sets,
+because of the reversal of ground and excited states. However, because the peak energies and
+oscillator strength for the state are in excellent agreement with all other basis sets implies
+that we have obtained correct quantitative description of the excited states even with this
+basis set. For peak III the main contribution to the excited state wave function is from
+|H − 1 → L + 2⟩ for the larger 6-311++ and aug-cc class of basis sets, while it is from
+|H − 1 → L + 1⟩ for the smaller cc-pVTZ and cc-pVDZ basis sets.
+The main configurations contributing to the excited states wave functions of peak I and
+the most intense peak (peak IV) of Li3 triangular cluster, computed using various basis sets
+are presented in Table S12. We note that for the first peak, the main contribution to the
+wave function is from configurations |H → L+14⟩ or |H → L+13⟩ for the larger 6-311++G
+and aug-cc class of basis sets. For the correlation-consistent basis sets (cc-pVDZ and cc-
+pVTZ) the main contribution is due to the configuration |H → L + 2⟩. For the fourth peak,
+the dominant configuration is |H − 1 → L⟩, irrespective of the type of basis set used for the
+calculation.
+The dominant configurations corresponding to the excited states wave functions of peak
+I and the most intense peak of the spectra (peak II) of the Li4 cluster are presented in Table
+S13. For the first peak, the main contribution is from the configuration |H → L + 1⟩ for all
+the basis sets, while for peak II it is |H−1 → L⟩ for all the basis sets. Thus, we have excellent
+agreement among all the basis sets when it comes to the most important configuration for
+both the peaks of the Li4 cluster.
+25
+
+The important configurations corresponding to the excited states wave function of peak
+I and the most intense peak (peak V) of the photoabsorption spectra of Be+
+2 cluster are
+presented in Table S14. The dominant configuration contributing to peak I is |H → L⟩ for
+the 6-311++G class of basis sets, and |H → L + 2⟩ for the rest. For peak V, the main
+configuration contributing to the CI wave function is |H − 1 → L + 1⟩ for 6-311++G class
+of basis sets, and |H − 1 → L⟩ for the rest of the sets.
+The configurations dominating the excited state CI wave functions of peak I and the
+most intense peak (peak VII) of Be+
+3 cluster are listed in Table S15.
+We note that the
+most important configurations contributing to peak I can be classified in two groups: (a)
+for larger 6-311++G(3df,3pd) and aug-cc class of basis sets the dominant configuration is
+|H → L + 1⟩, (b) while for smaller basis sets dominant configuration is highly basis set
+dependent.
+For peak VII, the doubly-excited configurations |H − 2 → L; H → L + 2⟩
+and |H − 1 → L; H → L + 4⟩ dominate the excited-state wave functions for the larger
+6-311++G(3df,3pd) and aug-cc class of basis sets, but vary significantly for the rest.
+Most important configurations contributing to the wave functions for peak I and the
+most intense peak (peak IV) of B+
+2 cluster are presented in Table S16. It is obvious that the
+double-excitation |H − 1 → L; H → L⟩ contributes the most to peak I for all the basis sets.
+The dominant configurations contributing to the wave functions of peak IV are |H → L+5⟩
+and |H → L + 11⟩ for all the basis sets except the cc-pVDZ/cc-pVTZ, for which instead of
+|H → L + 11⟩, the double excitation |H − 1 → L; H → L⟩ contributes.
+Finally, we present the dominant configurations in the CI wave functions corresponding
+to peak I, and the most intense peak (peak VIII), of B+
+3 cluster in Table S17. The configura-
+tion with maximum contribution to the excited state wave functions for peak I is |H → L⟩,
+irrespective of the basis set. The next dominant configuration is basis-set dependent, how-
+ever, it is a double excitation in all the cases. The dominant configuration corresponding to
+the CI wave function of peak VIII is the double excitation |H − 1 → L; H → L + 3⟩ for all
+the basis sets.
+26
+
+The detailed wave function analysis for all the peaks of the optical absorption spectra of
+clusters considered in this work using the largest aug-cc-pVTZ basis set is provided in Table
+S18-S25 of the SI.
+Conclusion
+In this work, we presented electron-correlated calculations of the optical absorption spectra
+of small neutral and ionic clusters using various basis sets. First, the stable geometries of
+various clusters were determined at the CCSD level of theory, using the aug-cc-PVTZ basis
+set. For the ground and the excited state wave functions calculations needed to compute
+the absorption spectra, we used the FCI, QCI, and MRSDCI approaches depending upon
+the size of the clusters. The CI calculations were performed using six different basis sets,
+namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ , cc-pVTZ, aug-cc-pVDZ, and
+aug-cc-pVTZ.
+We observed that the optical absorption spectra of all these clusters exhibit excellent
+convergence for all the basis sets in the lower energy range. However, usually after the first
+two peaks, the shift in peak locations for cc-pVDZ and cc-pVTZ basis set are noted in all
+likelihood because of the lack of diffuse basis functions in these sets. If we use augmented
+basis sets, the absorption spectra show good agreement with the results computed using
+other similar basis sets. Although aug-cc-pVDZ basis set has a relatively smaller number of
+basis functions as compared to aug-cc-pVTZ basis set, the agreement between the spectra
+computed using the two basis sets is very good. Because the number of two-electron integrals
+increases as N 4 where N is the number of basis functions in basis set, we can reduce the
+computational cost significantly by using aug-cc-pVDZ basis set instead of larger Pople’s
+basis sets, and aug-cc-pVTZ basis set. Thus, our general recommendation is that for optical
+absorption calculations one should use a basis set containing diffuse functions, i.e., of the
+aug-cc- type. However, whether one should use aug-cc-pVDZ, or a larger set, should be
+27
+
+decided by the available computational resources.
+We believe that the CI calculations presented in this work are quite accurate, as is obvious
+from the fact that our obtained results are in very good agreement with the experiments for
+Li3 and Li4 clusters. Therefore, it will be of interest to compare our results on other clusters
+also with the experiments, as and when they are performed.
+Associated Content
+Supporting Information
+In the supporting information file, we have provided the peak locations, oscillator strengths,
+and dominant excited state configurations corresponding to the optical absorption spectra
+of all the clusters for all the basis sets considered in this work. The SI file also contains the
+details of the many-particle wave functions of excited states contributing to the peaks in the
+optical absorption spectrum of clusters for aug-cc-pVTZ basis set.
+Author Information
+Corresponding Author
+Alok Shukla: Department of Physics, Indian Institute of Technology Bombay, Powai, Mum-
+bai 400076, India; *E-mail: [email protected]
+Acknowledgment
+This work was supported by senior research fellowship (DST-Inspire) provided by department
+of science and technology, India.
+28
+
+Authors
+Vikram Mahamiya: Department of Physics, Indian Institute of Technology Bombay, Powai,
+Mumbai 400076, India; E-mail: [email protected]
+Pritam Bhattacharyya: Institute for Theoretical Solid State Physics, Leibniz IFW Dres-
+den, Helmholtzstr. 20, 01069 Dresden, Germany; E-mail: [email protected]
+Notes
+The authors declare no competing financial interests.
+29
+
+References
+(1) Boys, S. F. Electronic Wave Functions. I. A General Method of Calculation for the
+Stationary States of Any Molecular System. Proceedings of the Royal Society of London
+Series A 1950, 200, 542–554.
+(2) McMurchie, L. E.; Davidson, E. R. One- and two-electron integrals over cartesian gaus-
+sian functions. Journal of Computational Physics 1978, 26, 218 – 231.
+(3) Davidson, E. R.; Feller, D. Basis set selection for molecular calculations. Chemical
+Reviews 1986, 86, 681–696.
+(4) Huzinaga, S. Basis sets for molecular calculations. Computer Physics Reports 1985, 2,
+281 – 339.
+(5) Huzinaga, S. Gaussian-Type Functions for Polyatomic Systems. I. The Journal of
+Chemical Physics 1965, 42, 1293–1302.
+(6) Ruedenberg, K.; Raffenetti, R. C.; Bardo, R. D. Structure and Reactivity, Proceedings
+of the 1972 Boulder Conference; Wiley: New York, 1973.
+(7) Bardo, R. D.; Ruedenberg, K. Even-tempered atomic orbitals. VI. Optimal orbital
+exponents and optimal contractions of Gaussian primitives for hydrogen, carbon, and
+oxygen in molecules. The Journal of Chemical Physics 1974, 60, 918–931.
+(8) Huzinaga, S.; Klobukowski, M.; Tatewaki, H. The well-tempered GTF basis sets and
+their applications in the SCF calculations on N2, CO, Na2, and P2. Canadian Journal
+of Chemistry 1985, 63, 1812–1828.
+(9) Tatewaki, H.; Huzinaga, S. A systematic preparation of new contracted Gaussian type
+orbital set. I. Transition metal atoms from Sc to Zn. The Journal of Chemical Physics
+1979, 71, 4339–4348.
+30
+
+(10) Tatewaki, H.; Huzinaga, S. A systematic preparation of new contracted Gaussian-type
+orbital sets. III. Second-row atoms from Li through ne. Journal of Computational Chem-
+istry 1980, 1, 205–228.
+(11) Hehre, W. J.; Stewart, R. F.; Pople, J. A. Self-Consistent Molecular-Orbital Methods. I.
+Use of Gaussian Expansions of Slater-Type Atomic Orbitals. The Journal of Chemical
+Physics 1969, 51, 2657–2664.
+(12) Ditchfield, R.; Hehre, W. J.; Pople, J. A. Self-Consistent Molecular-Orbital Meth-
+ods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic
+Molecules. The Journal of Chemical Physics 1971, 54, 724–728.
+(13) Binkley, J. S.; Pople, J. A.; Hehre, W. J. Self-consistent molecular orbital methods. 21.
+Small split-valence basis sets for first-row elements. Journal of the American Chemical
+Society 1980, 102, 939–947.
+(14) Binkley, J. S.; Pople, J. A. Self-consistent molecular orbital methods. XIX. Split-valence
+Gaussian-type basis sets for beryllium. The Journal of Chemical Physics 1977, 66, 879–
+880.
+(15) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. Self-consistent molecular or-
+bital methods. XX. A basis set for correlated wave functions. The Journal of Chemical
+Physics 1980, 72, 650–654.
+(16) Hariharan, P. C.; Pople, J. A. The influence of polarization functions on molecular
+orbital hydrogenation energies. Theoretica chimica acta 1973, 28, 213–222.
+(17) Collins, J. B.; von R. Schleyer, P.; Binkley, J. S.; Pople, J. A. Self-consistent molecular
+orbital methods. XVII. Geometries and binding energies of second-row molecules. A
+comparison of three basis sets. The Journal of Chemical Physics 1976, 64, 5142–5151.
+31
+
+(18) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; DeFrees, D. J.;
+Pople, J. A. Self-consistent molecular orbital methods. XXIII. A polarization-type basis
+set for second-row elements. The Journal of Chemical Physics 1982, 77, 3654–3665.
+(19) Pietro, W. J.; Francl, M. M.; Hehre, W. J.; DeFrees, D. J.; Pople, J. A.; Binkley, J. S.
+Self-consistent molecular orbital methods. 24. Supplemented small split-valence basis
+sets for second-row elements. Journal of the American Chemical Society 1982, 104,
+5039–5048.
+(20) Frisch, M. J.; Pople, J. A.; Binkley, J. S. Self-consistent molecular orbital methods
+25. Supplementary functions for Gaussian basis sets. The Journal of Chemical Physics
+1984, 80, 3265–3269.
+(21) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations.I The
+atoms boron through neon and hydrogen. The Journal of Chemical Physics 1989, 90,
+1007–1023.
+(22) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron affinities of the first-row atoms
+revisited.Systematic basis sets and wave functions. The Journal of Chemical Physics
+1992, 96, 6796–6806.
+(23) Woon, D. E.; Dunning, T. H. J. The Pronounced Effect of Microsolvation on Diatomic
+Alkali Halides: Ab Initio Modeling of MX(H2O)n (M = Li, Na; X=F, Cl; n = 1-3).
+Journal of the American Chemical Society 1995, 117, 1090–1097.
+(24) Balakina, M.; Nefediev, S. The choice of basis set for calculations of linear and nonlinear
+optical properties of conjugated organic molecules in gas and in dielectric medium by the
+example of p-nitroaniline. Computational Materials Science 2007, 38, 467–472, Selected
+papers from the International Conference on Computational Methods in Sciences and
+Engineering 2004.
+32
+
+(25) Parsons, T.; Balduf, T.; Cheeseman, J. R.; Caricato, M. Basis Set Dependence of
+Optical Rotation Calculations with Different Choices of Gauge. The Journal of Physical
+Chemistry A 2022, 126, 1861–1870, PMID: 35271772.
+(26) Reis, H.; Papadopoulos, M. G. Nonlinear optical properties of the rhombic B4-cluster.
+Journal of Computational Chemistry 1999, 20, 679–687.
+(27) Lauderdale, W. J.; Coolidge, M. B. Basis set effects on the nonlinear optical properties
+of selected linear diacetylenes. The Journal of Physical Chemistry 1995, 99, 9368–9373.
+(28) Jabłoński, M.; Palusiak, M. Basis Set and Method Dependence in Quantum Theory of
+Atoms in Molecules Calculations for Covalent Bonds. The Journal of Physical Chem-
+istry A 2010, 114, 12498–12505, PMID: 21049895.
+(29) Frisch, M. J. et al. Gaussian 16 Revision C.01. 2016.
+(30) McMurchie, L. E.; Elbert, S. T.; Langhoff, S. R.; Davidson, E. R. MELD package from
+Indiana University. It has been modified by us to handle bigger systems.
+(31) Shinde, R.; Shukla, A. Large-scale first principles configuration interaction calculations
+of optical absorption in aluminum clusters. Phys. Chem. Chem. Phys. 2014, 16, 20714–
+20723.
+(32) Rai, D. K.; Chakraborty, H.; Shukla, A. Tunable Optoelectronic Properties of Triply
+Bonded Carbon Molecules with Linear and Graphyne Substructures. The Journal of
+Physical Chemistry C 2018, 122, 1309–1317.
+(33) Chakraborty, H.; Shukla, A. Pariser - Parr - Pople Model Based Investigation of Ground
+and Low - Lying Excited States of Long Acenes. The Journal of Physical Chemistry A
+2013, 117, 14220–14229.
+(34) Aryanpour, K.; Shukla, A.; Mazumdar, S. Electron correlations and two-photon states
+33
+
+in polycyclic aromatic hydrocarbon molecules: A peculiar role of geometry. The Journal
+of Chemical Physics 2014, 140, 104301.
+(35) Chakraborty, H.; Shukla, A. Theory of triplet optical absorption in oligoacenes: From
+naphthalene to heptacene. The Journal of Chemical Physics 2014, 141, 164301.
+(36) SHINDE, R.; SHUKLA, A. LARGE-SCALE FIRST PRINCIPLES CONFIGURA-
+TION INTERACTION CALCULATIONS OF OPTICAL ABSORPTION IN BORON
+CLUSTERS. Nano LIFE 2012, 02, 1240004.
+(37) Shukla, A. Correlated theory of triplet photoinduced absorption in phenylene-vinylene
+chains. Phys. Rev. B 2002, 65, 125204.
+(38) Shukla, A. Theory of nonlinear optical properties of phenyl-substituted polyacetylenes.
+Phys. Rev. B 2004, 69, 165218.
+(39) Sony, P.; Shukla, A. Large-scale correlated calculations of linear optical absorption and
+low-lying excited states of polyacenes: Pariser-Parr-Pople Hamiltonian. Phys. Rev. B
+2007, 75, 155208.
+(40) Basak, T.; Chakraborty, H.; Shukla, A. Theory of linear optical absorption in diamond-
+shaped graphene quantum dots. Phys. Rev. B 2015, 92, 205404.
+(41) Priya, P. K.; Rai, D. K.; Shukla, A. Photoabsorption in sodium clusters: first principles
+configuration interaction calculations. The European Physical Journal D 2017, 71, 116.
+(42) Shinde, R.; Shukla, A. First principles electron-correlated calculations of optical ab-
+sorption in magnesium clusters. The European Physical Journal D 2017, 71, 301.
+(43) Bhattacharyya, P.; Rai, D. K.; Shukla, A. Systematic First-Principles Configuration-
+Interaction Calculations of Linear Optical Absorption Spectra in Silicon Hydrides:
+Si2H2n (n = 1-3). The Journal of Physical Chemistry A 2019, 123, 8619–8631.
+34
+
+(44) Wheeler, S. E.; Sattelmeyer, K. W.; Schleyer, P. v. R.; Schaefer, H. F. Binding energies
+of small lithium clusters (Lin) and hydrogenated lithium clusters (LinH). The Journal
+of Chemical Physics 2004, 120, 4683–4689.
+(45) Florez, E.; Fuentealba, P. A theoretical study of alkali metal atomic clusters: From Lin
+to Csn (n = 2-8). International Journal of Quantum Chemistry 2009, 109, 1080–1093.
+(46) HUBER, K. P. Molecular Structure Constants of Diatomic molecules. Molecular Spectra
+and molecular Structure Constants of Diatomic molecules 1979,
+(47) Jones, R. O.; Lichtenstein, A. I.; Hutter, J. Density functional study of structure and
+bonding in lithium clusters Lin and their oxides LinO. The Journal of Chemical Physics
+1997, 106, 4566–4574.
+(48) Srinivas, S.; Jellinek, J. Structural and electronic properties of small beryllium clusters:
+A theoretical study. The Journal of Chemical Physics 2004, 121, 7243–7252.
+(49) Hanley, L.; Whitten, J. L.; Anderson, S. L. Collision-induced dissociation and ab initio
+studies of boron cluster ions: determination of structures and stabilities. The Journal
+of Physical Chemistry 1988, 92, 5803–5812.
+(50) Hong, X.; Wang, F. TDDFT calculation for photoabsorption spectra of Lin (n=2-11,20)
+clusters. Physics Letters A 2011, 375, 1883 – 1888.
+(51) Blanc, J.; Broyer, M.; Chevaleyre, J.; Dugourd, P.; Kühling, H.; Labastie, P.; Ul-
+bricht, M.; Wolf, J. P.; Wöste, L. High resolution spectroscopy of small metal clusters.
+Zeitschrift für Physik D Atoms, Molecules and Clusters 1991, 19, 7–12.
+35
+
+For Table of Contents Only
+36
+
+400
+Optical absorption spectra of Li4
+cluster using various basis sets
+6-311++G(2d,2p)
+6-311++G(3df,3pd)
+III
+cC-pVDZ
+300
+cc-pVTZ
+aug-cc-pVDZ
+(arb. units)
+aug-cc-pVTZ
+200
+Intensity
+100
+VI
+VII
+0
+0
+2
+4
+Energy (eV)Supporting Information For: Benchmarking
+Gaussian Basis Sets in Quantum-Chemical
+Calculations of Photoabsorption Spectra of
+Light Atomic Clusters
+Vikram Mahamiya,∗,† Pritam Bhattacharyya,∗,†,‡ and Alok Shukla∗,†
+†Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076,
+India
+‡Present Address: Institute for Theoretical Solid State Physics, Leibniz IFW Dresden,
+Helmholtzstr. 20, 01069 Dresden, Germany
+E-mail: [email protected]; [email protected]; [email protected]
+Table S1: Comparison of the peak locations of the optical absorption spectra of Li2 cluster
+computed using various basis sets.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+Peak VIII
+Peak IX
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+1.82
+2.61
+3.85
+4.87
+5.88
+6.50
+7.06
+-
+-
+6-311++G(3df,3pd)
+1.83
+2.57
+3.85
+4.61
+4.84
+5.64
+5.88
+6.08
+7.00
+cc-pVDZ
+1.82
+2.65
+4.43
+5.95
+7.12
+-
+-
+-
+-
+cc-pVTZ
+1.83
+2.58
+4.26
+5.43
+5.98
+6.64
+7.03
+-
+-
+aug-cc-pVDZ
+1.82
+2.60
+3.84
+4.58
+5.06
+5.94
+6.67
+6.96
+-
+aug-cc-pVTZ
+1.83
+2.57
+3.87
+4.59
+5.43
+5.93
+6.66
+7.03
+-
+S1
+arXiv:2301.02413v1  [physics.chem-ph]  6 Jan 2023
+
+Table S2: Comparison of the peak locations of the optical absorption spectra of Li3 chain
+computed using various basis sets.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+0.72
+1.27
+2.58
+3.39
+3.91
+4.28
+6-311++G(3df,3pd)
+0.72
+1.27
+2.54
+3.38
+3.90
+4.14
+cc-pVDZ
+0.72
+1.26
+2.65
+3.47
+4.37
+4.92
+cc-pVTZ
+0.72
+1.28
+2.57
+3.45
+4.39
+4.58
+aug-cc-pVDZ
+0.72
+1.26
+2.56
+3.38
+3.88
+-
+aug-cc-pVTZ
+0.72
+1.27
+2.53
+3.37
+3.89
+-
+Table S3: The peak locations of the optical absorption spectra of Li3 isosceles triangular
+cluster computed using various basis sets are compared.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+1.09
+1.41
+2.14
+2.41
+2.66
+2.97
+3.21
+6-311++G(3df,3pd)
+1.07
+1.42
+2.12
+2.39
+2.65
+2.96
+3.19
+cc-pVDZ
+1.11
+1.40
+2.18
+2.43
+2.61
+2.81
+3.13
+cc-pVTZ
+1.08
+1.42
+2.15
+2.40
+2.70
+3.02
+4.27
+aug-cc-pVDZ
+1.09
+1.41
+2.13
+2.40
+2.65
+2.96
+3.20
+aug-cc-pVTZ
+1.07
+1.42
+2.11
+2.43
+2.65
+2.95
+3.20
+Table S4: High energy peak locations of the optical absorption spectra of Li3 Isosceles
+triangular cluster computed using various basis sets are compared.
+Li3 Cluster
+Peak VIII
+Peak IX
+Peak X
+Peak XI
+Peak XII
+Basis Set
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+3.77
+4.25
+5.29
+5.99
+-
+6-311++G(3df,3pd)
+3.75
+4.17
+4.91
+5.25
+5.79
+cc-pVDZ
+3.57
+4.24
+4.94
+5.33
+6.05
+cc-pVTZ
+4.69
+5.64
+-
+-
+-
+aug-cc-pVDZ
+4.16
+4.44
+5.35
+5.59
+5.94
+aug-cc-pVTZ
+3.77
+4.11
+5.39
+5.60
+-
+S2
+
+Table S5: The peak locations of the optical absorption spectra of Li4 rhombus cluster com-
+puted using various basis sets. The higher energy peak locations are presented in the Table
+I of the Supporting Information.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+1.84
+2.65
+3.05
+3.64
+4.24
+4.52
+5.15
+6-311++G(3df,3pd)
+1.83
+2.64
+2.93
+3.63
+4.21
+4.51
+5.11
+cc-pVDZ
+1.87
+2.67
+3.08
+3.65
+4.27
+4.86
+5.42
+cc-pVTZ
+1.84
+2.65
+3.03
+3.64
+4.22
+4.54
+5.30
+aug-cc-pVDZ
+1.85
+2.65
+3.05
+3.61
+4.23
+4.62
+-
+aug-cc-pVTZ
+1.87
+2.65
+2.93
+3.66
+4.22
+4.61
+5.30
+Table S6: The peak locations of the optical absorption spectra of Be+
+2 cluster computed
+using various basis sets are compared.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+Peak VIII
+Peak IX
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+1.75
+3.66
+4.18
+6.04
+6.32
+8.32
+8.90
+9.33
+9.65
+6-311++G(3df,3pd)
+1.74
+3.68
+4.18
+6.04
+6.30
+8.33
+8.84
+9.31
+9.63
+cc-pVDZ
+1.76
+3.69
+4.20
+6.12
+6.37
+8.28
+9.61
+-
+-
+cc-pVTZ
+1.74
+3.68
+4.19
+6.05
+6.31
+8.44
+9.50
+-
+-
+aug-cc-pVDZ
+1.77
+3.69
+4.18
+6.08
+6.36
+8.38
+9.09
+9.47
+-
+aug-cc-pVTZ
+1.74
+3.68
+4.18
+6.03
+6.30
+8.34
+8.92
+9.38
+-
+Table S7: The peak locations of the optical absorption spectra of Be+
+3 cluster computed
+using various basis sets are compared.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+1.05
+3.16
+3.66
+4.90
+5.39
+5.86
+6.53
+6-311++G(3df,3pd)
+1.02
+3.17
+3.63
+4.84
+5.40
+5.83
+6.47
+cc-pVDZ
+1.04
+3.16
+3.67
+4.91
+5.39
+5.89
+6.52
+cc-pVTZ
+1.01
+3.14
+3.60
+4.87
+5.37
+5.83
+6.50
+aug-cc-pVDZ
+1.02
+3.16
+3.66
+4.87
+5.41
+5.87
+6.51
+aug-cc-pVTZ
+1.02
+3.16
+3.66
+4.87
+5.40
+5.85
+6.52
+S3
+
+Table S8: The peak locations of the optical absorption spectra of B+
+2 cluster computed using
+various basis sets are compared.
+Basis Set
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+Peak VIII
+Peak IX
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+3.65
+4.86
+6.01
+7.00
+7.63
+9.68
+10.26
+11.42
+12.71
+6-311++G(3df,3pd)
+3.65
+4.77
+5.98
+7.14
+7.62
+9.65
+10.22
+11.40
+12.61
+cc-pVDZ
+3.64
+4.79
+6.03
+7.16
+-
+9.88
+11.29
+12.74
+cc-pVTZ
+3.66
+4.80
+5.99
+7.05
+7.63
+9.87
+11.12
+12.77
+-
+aug-cc-pVDZ
+3.63
+4.78
+5.98
+7.03
+7.60
+9.66
+10.27
+11.31
+12.52
+aug-cc-pVTZ
+3.66
+4.80
+5.99
+7.04
+7.65
+9.65
+10.21
+11.41
+12.20
+Table S9: The peak locations of the optical absorption spectra of B+
+3 cluster computed using
+various basis sets are compared.
+B+
+3 Cluster
+Peak I
+Peak II
+Peak III
+Peak IV
+Peak V
+Peak VI
+Peak VII
+Peak VIII
+Basis Set
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+0.82
+3.24
+5.00
+5.38
+6.02
+7.12
+8.09
+8.85
+6-311++G(3df,3pd)
+0.83
+3.21
+4.99
+5.44
+6.06
+7.17
+8.07
+8.87
+cc-pVDZ
+0.96
+3.28
+5.17
+5.59
+6.11
+7.39
+8.21
+8.95
+cc-pVTZ
+0.82
+3.23
+4.98
+5.35
+6.03
+7.01
+8.06
+8.79
+aug-cc-pVDZ
+0.84
+3.22
+5.00
+5.41
+6.02
+7.10
+8.03
+8.80
+aug-cc-pVTZ
+0.84
+3.22
+5.01
+5.45
+6.03
+7.16
+8.08
+8.85
+Table S10: The peak locations of the optical absorption spectra of B+
+3 cluster employing TD-
+DFT method with B3LYP functional and computed using various basis sets are compared.
+B+
+3 Cluster
+Peak I
+Peak II
+Peak III
+Peak IV
+Basis Set
+(eV)
+(eV)
+(eV)
+(eV)
+6-311++G(2d,2p)
+0.94
+2.95
+5.70
+9.35
+6-311++G(3df,3pd)
+0.95
+2.94
+5.70
+9.35
+cc-pVDZ
+0.96
+3.00
+5.73
+9.52
+cc-pVTZ
+0.96
+2.95
+5.70
+9.40
+aug-cc-pVDZ
+0.96
+2.99
+5.73
+9.40
+aug-cc-pVTZ
+0.95
+2.94
+5.70
+9.36
+S4
+
+Table S11: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak III)
+of Li3 chain calculated using different basis sets. In the “Polarization” column, ∥ indicates
+photon polarization along the direction of the molecule (longitudinal polarization), while ⊥
+indicates polarization perpendicular to the molecular axis (transverse polarization). ’H’ and
+’L’ stand for HOMO and LUMO orbitals.
+Basis Set
+Peak I
+Peak III
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+∥
+0.106
+|H − 1 → H⟩
+⊥
+0.591
+|H − 1 → L + 2⟩
+|H → L + 6⟩
+6-311++G(3df,3pd)
+∥
+0.106
+|H − 1 → H⟩
+⊥
+0.595
+|H − 1 → L + 2⟩
+|H → L + 5⟩
+|H → L + 11⟩
+cc-PVDZ
+∥
+0.097
+|H − 1 → H⟩
+⊥
+0.631
+|H − 1 → L + 1⟩
+|H → L⟩
+|H → L + 3⟩
+cc-pVTZ
+∥
+0.105
+|H − 1 → H⟩
+⊥
+0.618
+|H − 1 → L + 1⟩
+|H → L⟩
+|H → L + 3⟩
+aug-cc-pVDZ
+∥
+0.101
+|H − 1 → H⟩
+⊥
+0.603
+|H − 1 → L + 2⟩
+|H → L + 7⟩
+|H → L + 10⟩
+aug-cc-pVTZ
+∥
+0.106
+|HF⟩
+⊥
+0.594
+|H − 1 → L + 2⟩
+|H − 1 → H;
+|H − 1 → H;
+H − 1 → L + 5⟩
+H − 1 → L + 11⟩
+Table S12: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak IV)
+of Li3 triangular calculated using different basis sets. The rest of the information is same as
+in the caption of Table S11.
+Basis Set
+Peak I
+Peak IV
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+∥
+0.127
+|H → L + 14⟩
+∥
+0.465
+|H − 1 → L⟩
+|H → L + 1⟩
+|H → L + 9⟩
+6-311++G(3df,3pd)
+∥
+0.131
+|H → L + 14⟩
+∥
+0.459
+|H − 1 → L⟩
+|H → L + 1⟩
+|H → L + 5⟩
+cc-PVDZ
+∥
+0.121
+|H → L + 2⟩
+∥
+0.488
+|H − 1 → L⟩
+|H − 1 → L + 2⟩
+|H → L + 4⟩
+cc-pVTZ
+∥
+0.129
+|H → L + 2⟩
+∥
+0.478
+|H − 1 → L⟩
+|H − 1 → L + 2⟩
+|H → L + 4⟩
+aug-cc-pVDZ
+∥
+0.126
+|H → L + 14⟩
+∥
+0.469
+|H − 1 → L⟩
+|H → L + 10⟩
+|H → L + 16⟩
+aug-cc-pVTZ
+∥
+0.129
+|H → L + 13⟩
+∥
+0.462
+|H − 1 → L⟩
+|H → L + 1⟩
+|H → L + 5⟩
+S5
+
+Table S13: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak II)
+of Li4 cluster calculated using different basis sets. The rest of the information is same as in
+the caption of Table S11.
+Basis Set
+Peak I
+Peak II
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+∥
+0.628
+|H → L + 1⟩
+∥
+0.648
+|H − 1 → L⟩
+|H → L + 8⟩
+|H − 1 → L + 5⟩
+6-311++G(3df,3pd)
+∥
+0.615
+|H → L + 1⟩
+∥
+0.683
+|H − 1 → L⟩
+|H → L + 9⟩
+|H − 1 → L + 6⟩
+cc-PVDZ
+∥
+0.659
+|H → L + 1⟩
+∥
+0.649
+|H − 1 → L⟩
+|H → L + 4⟩
+|H → L + 3⟩
+cc-pVTZ
+∥
+0.624
+|H → L + 1⟩
+∥
+0.678
+|H − 1 → L⟩
+|H → L + 5⟩
+|H → L + 4⟩
+aug-cc-pVDZ
+∥
+0.634
+|H → L + 1⟩
+∥
+0.654
+|H − 1 → L⟩
+|H → L + 9⟩
+|H − 1 → L + 5⟩
+aug-cc-pVTZ
+∥
+0.656
+|H → L + 1⟩
+∥
+0.660
+|H − 1 → L⟩
+|H → L + 9⟩
+|H − 1 → L + 5⟩
+Table S14: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak V)
+of Be+
+2 cluster calculated using different basis sets. The rest of the information is same as in
+the caption of Table S11
+Basis Set
+Peak I
+Peak V
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+∥
+0.113
+|H → L⟩
+⊥
+0.745
+|H − 1 → L + 1⟩
+|H − 1 → H⟩
+|H − 1 → L; H → L + 2⟩
+6-311++G(3df,3pd)
+∥
+0.119
+|H → L⟩
+⊥
+0.749
+|H − 1 → L + 1⟩
+|H − 1 → H⟩
+|H − 1 → L; H → L + 2⟩
+cc-PVDZ
+∥
+0.114
+|H → L + 2⟩
+⊥
+0.736
+|H − 1 → L⟩
+|H → L; H − 1 → L⟩
+|H − 1 → L + 2; H → L + 3⟩
+cc-pVTZ
+∥
+0.120
+|H → L + 2⟩
+⊥
+0.749
+|H − 1 → L⟩
+|H → L; H − 1 → L⟩
+|H − 1 → L + 2; H → L + 3⟩
+aug-cc-pVDZ
+∥
+0.114
+|H → L + 2⟩
+⊥
+0.768
+|H − 1 → L⟩
+|H → L; H − 1 → L⟩
+|H − 1 → L + 2; H → L + 3⟩
+aug-cc-pVTZ
+∥
+0.120
+|H → L + 2⟩
+⊥
+0.757
+|H − 1 → L⟩
+|H → L; H − 1 → L⟩
+|H − 1 → L + 2; H → L + 3⟩
+S6
+
+Table S15: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak VII)
+of Be+
+3 cluster calculated using different basis sets. The rest of the information is same as in
+the caption of Table S11
+Basis Set
+Peak I
+Peak VII
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+∥
+0.172
+|HF⟩
+⊥
+0.655
+|H − 2 → L + 1⟩
+|H → L⟩
+|H − 1 → L + 2⟩
+6-311++G(3df,3pd)
+∥
+0.184
+|H → L + 1⟩
+⊥
+0.647
+|H − 2 → L; H → L + 2⟩
+|H − 1 → L; H → L⟩
+|H − 1 → L; H → L + 4⟩
+cc-PVDZ
+∥
+0.178
+|H → L⟩
+⊥
+0.640
+|H − 2 → L + 1⟩
+|H − 1 → H⟩
+|H − 1 → L + 2⟩
+cc-pVTZ
+∥
+0.169
+|HF⟩
+⊥
+0.635
+|H − 2 → L; H → L + 1⟩
+|H − 1 → L; H → L⟩
+|H − 1 → L; H → L + 2⟩
+aug-cc-pVDZ
+∥
+0.180
+|H → L + 1⟩
+⊥
+0.646
+|H − 2 → L; H → L + 2⟩
+|H − 1 → L; H → L⟩
+|H − 1 → L; H → L + 4⟩
+aug-cc-pVTZ
+∥
+0.179
+|H → L + 1⟩
+⊥
+0.657
+|H − 2 → L; H → L + 2⟩
+|H − 1 → L; H → L⟩
+|H − 1 → L; H → L + 4⟩
+Table S16: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak IV)
+of B+
+2 cluster calculated using different basis sets. The rest of the information is same as in
+the caption of Table S11
+Basis Set
+Peak I
+Peak IV
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+∥
+0.026
+|H − 1 → L; H → L⟩
+∥
+0.910
+|H → L + 5⟩
+|H → L + 11⟩
+|H → L + 11⟩
+6-311++G(3df,3pd)
+∥
+0.025
+|H − 1 → L; H → L⟩
+∥
+0.915
+|H → L + 5⟩
+|H → L + 11⟩
+|H → L + 11⟩
+cc-PVDZ
+∥
+0.025
+|H − 1 → L; H → L⟩
+∥
+0.965
+|H → L + 5⟩
+|H → L + 5⟩
+|H − 1 → L; H → L⟩
+cc-pVTZ
+∥
+0.024
+|H − 1 → L; H → L⟩
+∥
+0.942
+|H → L + 5⟩
+|H → L + 5⟩
+|H − 1 → L; H → L⟩
+aug-cc-pVDZ
+∥
+0.028
+|H − 1 → L; H → L⟩
+∥
+0.895
+|H → L + 11⟩
+|H → L + 11⟩
+|H → L + 5⟩
+aug-cc-pVTZ
+∥
+0.026
+|H − 1 → L; H → L⟩
+∥
+0.918
+|H → L + 11⟩
+|H → L + 11⟩
+|H → L + 5⟩
+S7
+
+Table S17: Comparison of oscillator strengths and the dominant configurations contributing
+to the many-particle wave functions for peaks I and the maximum intensity peak (peak VIII)
+of B+
+3 cluster calculated using different basis sets. The rest of the information is same as in
+the caption of Table S11
+Basis Set
+Peak I
+Peak VIII
+Polarization
+f
+Wave-function
+Polarization
+f
+Wave-function
+6-311++G(2d,2p)
+⊥
+0.005
+|H → L⟩
+∥
+0.508
+|H − 1 → L; H → L + 3⟩
+|H − 2 → L; H → L + 1⟩
+|H − 1 → L; H → L + 2⟩
+6-311++G(3df,3pd)
+⊥
+0.005
+|H → L⟩
+∥
+0.514
+|H − 1 → L; H → L + 3⟩
+|H − 1 → L; H → L + 1⟩
+|H − 1 → L; H → L + 2⟩
+cc-PVDZ
+⊥
+0.007
+|H → L⟩
+∥
+0.480
+|H − 1 → L; H → L + 3⟩
+|H − 1 → L; H → L + 1⟩
+|H − 1 → L; H → L + 3⟩
+cc-pVTZ
+⊥
+0.005
+|H → L⟩
+∥
+0.491
+|H − 1 → L; H → L + 3⟩
+|H − 1 → L; H → L + 1⟩
+|H − 1 → L; H → L + 2⟩
+aug-cc-pVDZ
+⊥
+0.005
+|H → L⟩
+∥
+0.467
+|H − 1 → L; H → L + 3⟩
+|H − 1 → L; H → L + 2⟩
+|H − 1 → L; H → L + 3⟩
+aug-cc-pVTZ
+⊥
+0.006
+|H → L⟩
+∥
+0.519
+|H − 1 → L; H → L + 3⟩
+|H − 1 → L; H → L + 2⟩
+|H − 1 → L; H → L + 3⟩
+S8
+
+Table S18: Many-particle wave functions of excited states contributing to the peaks in
+the optical absorption spectrum of Li2 cluster for aug-cc-pVTZ basis set. ’E’ corresponds
+to excitation energy (in eV) of an excited state, f denotes the oscillator strength for a
+particular electric dipole transition. In the “|TDM|” column,we present the magnitudes of the
+transition dipole moments (TDMs) to understand the extent of coupling between the relevant
+excited state and the ground state. ∥ indicates photon polarization along the direction of
+the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to the
+molecular axis (transverse polarization). ’H’ and ’L’ stand for HOMO and LUMO orbitals.
+In the “Wave function” column, each number inside the parentheses denotes the coefficient
+of the corresponding configuration in the CI wave function. GS indicates the ground states
+wave function of the cluster.
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|HF⟩(0.9520)
+|H → L + 7; H → L + 15⟩(0.0879)
+I
+1.83
+0.454
+3.184
+∥
+|H → L⟩(0.7523)
+|H → L + 3⟩(0.3959)
+II
+2.57
+0.966
+2.770
+⊥
+|H → L + 2⟩(0.7046)
+|H → L + 7⟩(0.5914)
+III
+3.87
+0.049
+0.509
+⊥
+|H → L + 7⟩(0.6291)
+|H → L + 2⟩(0.5439)
+V
+5.42
+0.024
+0.303
+⊥
+|H → L + 16⟩(0.8509)
+|H → L + 8; H → L + 16⟩(0.1702)
+VI
+5.93
+0.026
+0.421
+∥
+|H → L + 12⟩(0.2911)
+|H → L; H → L + 8⟩(0.2523)
+S9
+
+Table S19: Many-particle wave functions of excited states contributing to the peaks in the
+optical absorption spectrum of Li3 linear cluster for aug-cc-pVTZ basis set. The rest of the
+information is same as in the caption of Table S18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|H − 1 → H⟩(0.9246)
+|H − 1 → L + 13⟩(0.0919)
+I
+0.72
+0.106
+2.453
+∥
+|HF⟩(0.8814)
+|H − 1 → H; H − 1 → L + 5⟩(0.1436)
+II
+1.27
+0.503
+4.023
+∥
+|H − 1 → H; H − 1 → L⟩(0.5569)
+|H − 1 → H; H − 1 → L + 5⟩(0.5102)
+III
+2.53
+0.594
+3.096
+⊥
+|H − 1 → L + 2⟩(0.6262)
+|H − 1 → H; H − 1 → L + 11⟩(0.3628)
+IV
+3.37
+0.215
+1.141
+⊥
+|H − 1 → L + 7⟩(0.4255)
+|H − 1 → L + 14⟩(0.4237)
+V
+3.89
+0.011
+0.343
+∥
+|H − 1 → L + 8⟩(0.4214)
+|H − 1 → H; H − 1 → L + 13⟩(0.3520)
+S10
+
+Table S20: Many-particle wave functions of excited states contributing to the peaks in the
+optical absorption spectrum of Li3 isosceles triangular cluster for aug-cc-pVTZ basis set.
+The rest of the information is same as in the caption of TableS18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|HF⟩(0.9102)
+|H − 1 → H⟩(0.0933)
+I
+1.07
+0.129
+2.222
+∥
+|H → L + 13⟩(0.4241)
+|H → L + 1⟩(0.4114)
+II
+1.42
+0.020
+0.767
+∥
+|H → L + 16⟩(0.5005)
+|H − 1 → L⟩(0.4115)
+III
+2.11
+0.352
+2.611
+∥
+|H − 1 → H⟩(0.5806)
+|H − 1 → L + 15⟩(0.2483)
+IV
+2.43
+0.462
+2.885
+∥
+|H − 1 → L⟩(0.5537)
+|H → L + 5⟩(0.3869)
+V
+2.65
+0.088
+1.163
+⊥
+|H → L + 2⟩(0.5541)
+|H → L + 12⟩(0.3375)
+VI
+2.95
+0.267
+1.922
+⊥
+|H − 1 → L + 2⟩(0.4766)
+|H − 1 → L + 12⟩(0.4269)
+VII
+3.20
+0.117
+1.223
+⊥
+|H − 1 → L + 2⟩(0.3983)
+|H → L + 12⟩(0.3579)
+VIII
+3.77
+0.016
+0.417
+∥
+|H → L + 10⟩(0.3841)
+|H → L + 19⟩(0.3349)
+IX
+4.11
+0.029
+0.533
+∥
+|H − 1 → L + 14⟩(0.4375)
+|H − 1 → L⟩(0.3910)
+X
+5.39
+0.004
+0.171
+∥
+|H → L + 35⟩(0.2813)
+|H → L + 37⟩(0.2744)
+S11
+
+Table S21: Many-particle wave functions of excited states contributing to the peaks in
+the optical absorption spectrum of Li4 cluster for aug-cc-pVTZ basis set. The rest of the
+information is same as in the caption of Table S18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|HF⟩(0.8913)
+|(H − 1) → L; H → L + 18⟩(0.0791)
+I
+1.87
+0.656
+3.785
+∥
+|H → L + 1⟩(0.5922)
+|H → L + 9⟩(0.4041)
+II
+2.65
+0.660
+3.188
+∥
+|H − 1 → L⟩(0.6193)
+|H − 1 → L + 5⟩(0.2910)
+III
+2.93
+0.316
+2.098
+⊥
+|H → L + 7⟩(0.4577)
+|H → L + 20⟩(0.4174)
+IV
+3.43
+0.045
+0.736
+⊥
+|H − 1 → L + 3⟩(0.3080)
+|H → L + 7⟩(0.2299)
+V
+3.66
+0.103
+1.070
+∥
+|H → L + 6⟩(0.5653)
+|H → L + 18⟩(0.3856)
+VI
+4.22
+0.077
+0.862
+⊥
+|H − 1 → L + 3⟩(0.2380)
+|H → L + 20; H → L + 21⟩(0.2227)
+VII
+4.61
+0.063
+0.746
+⊥
+|H − 1 → L + 3⟩(0.4044)
+|H − 1 → L + 12⟩(0.4026)
+VIII
+5.30
+0.012
+0.300
+∥
+|H → L + 33⟩(0.3681)
+|H → L; H → L + 6⟩(0.2294)
+Table S22: Many-particle wave functions of excited states contributing to the peaks in the
+optical absorption spectrum of Be+
+2 cluster for aug-cc-pVTZ basis set.
+The rest of the
+information is same as in the caption of Table S18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|H → L⟩(0.9351)
+|H − 1 → L; H → L + 2⟩(0.1684)
+I
+1.74
+0.120
+1.680
+∥
+|H → L + 2⟩(0.8403)
+|H − 1 → L; H → L⟩(0.3820)
+II
+3.67
+0.121
+0.821
+⊥
+|H → L + 3⟩(0.6705)
+|H → L + 4⟩(0.6375)
+III
+4.19
+0.386
+1.940
+∥
+|H − 1 → L; H → L⟩(0.7268)
+|H → L + 2⟩(0.3897)
+IV
+6.01
+0.201
+0.827
+⊥
+|H − 1 → L⟩(0.6418)
+|H − 1 → L; H → L + 1⟩(0.6116)
+V
+6.30
+0.757
+1.566
+⊥
+|H − 1 → L⟩(0.8762)
+|H − 1 → L + 2; H → L + 3⟩(0.8573)
+S12
+
+Table S23: Many-particle wave functions of excited states contributing to the peaks in the
+optical absorption spectrum of Be+
+3 cluster for aug-cc-pVTZ basis set.
+The rest of the
+information is same as in the caption of Table S18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|H → L⟩(.8776)
+|H − 1 → L + 1; H → L⟩(.1913)
+I
+1.02
+0.179
+2.678702
+∥
+|H → L + 1⟩(0.8200)
+|H − 1 → L; H → L⟩(0.3257)
+II
+3.16
+0.433
+2.365586
+∥
+|H − 1 → L; H → L⟩(0.6152)
+|H − 1 → L + 1; H → L + 1⟩(0.4714)
+III
+3.66
+0.028
+0.563211
+⊥
+|H − 2 → L; H → L + 2⟩(0.5209)
+|H → L + 17⟩(0.4125)
+IV
+4.87
+0.020
+0.407823
+⊥
+|H − 1 → L + 1; H → L + 2⟩(0.5211)
+|H → L + 17⟩(0.3217)
+V
+5.40
+0.204
+1.242734
+∥
+|H − 2 → L; H → L + 1⟩(0.6337)
+|H − 1 → L; H → L⟩(0.3104)
+VI
+5.85
+0.054
+0.612656
+⊥
+|H − 1 → L; H → L + 4⟩(0.4315)
+|H − 1 → L; H − 1 → L + 2; H → L⟩(0.2864)
+VII
+6.52
+0.657
+2.028291
+⊥
+|H − 2 → L; H → L + 2⟩(0.4960)
+|H − 1 → L; H → L + 4⟩(0.4653)
+Table S24: Many-particle wave functions of excited states contributing to the peaks in
+the optical absorption spectrum of B+
+2 cluster for aug-cc-pVTZ basis set. The rest of the
+information is same as in the caption of Table S18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|H → L⟩(.9033)
+|H − 2 → L; H − 1 → L + 2⟩(.1271)
+I
+3.66
+0.026
+0.537
+∥
+|H − 1 → L; H → L⟩(0.7250)
+|H → L + 11⟩(0.2622)
+II
+4.80
+0.029
+0.496
+∥
+|H − 1 → L + 1; H → L + 1⟩(0.5092)
+|H − 1 → H⟩(0.4944)
+III
+5.99
+0.019
+0.364
+⊥
+|H − 1 → L; H → L + 2⟩(0.6138)
+|H − 2 → L⟩(0.4445)
+IV
+7.04
+0.918
+2.307
+∥
+|H → L + 11⟩(0.4808)
+|H → L + 5⟩(0.4713)
+V
+7.65
+0.009
+0.216
+⊥
+|H − 2 → L⟩(0.5322)
+|H − 1 → L; H → L + 2⟩(0.4844)
+S13
+
+Table S25: Many-particle wave functions of excited states contributing to the peaks in
+the optical absorption spectrum of B+
+3 cluster for aug-cc-pVTZ basis set. The rest of the
+information is same as in the caption of Table S18
+Peak
+E (eV)
+f
+|TDM|
+Polarization
+Wave function
+GS
+|HF⟩(0.8535)
+|H − 1 → L + 1⟩(0.1393)
+I
+0.84
+0.006
+0.524
+⊥
+|H → L⟩(0.8572)
+|H − 1 → L; H → L + 2⟩(0.1319)
+II
+3.22
+0.083
+0.726
+∥
+|H − 1 → L⟩(0.8246)
+|H − 1 → L; H − 1 → L + 1⟩(0.1642)
+III
+5.01
+0.053
+0.463
+∥
+|H − 1 → L; H − 1 → L⟩(0.5482)
+|H → L; H → L + 1⟩(0.3184)
+IV
+5.45
+0.013
+0.217
+∥
+|H → L; H → L + 1⟩(0.5721)
+|H → L; H → L + 2⟩(0.5675)
+V
+6.03
+0.026
+0.295
+∥
+|H → L + 3⟩(0.3937)
+|H − 1 → L + 2⟩(0.3723)
+VI
+7.16
+0.026
+0.382
+∥
+|H → L + 1; H → L + 2⟩(0.5866)
+|H → L; H → L + 1⟩(0.4308)
+VII
+8.08
+0.059
+0.380
+∥
+|H − 1 → L; H − 1 → L + 1⟩(0.4529)
+|H − 1 → L; H − 1 → L + 2⟩(0.3162)
+VIII
+8.85
+0.519
+1.091
+∥
+|H → L + 3; H − 1 → L⟩(0.3762)
+|H → L + 3; H − 1 → L⟩(0.3748)
+Figure S1: Validation of frozen core approximation for the optical absorption spectra of Li2
+cluster employing QCI method.
+S14
+
+400
+With core-electrons
+Frozen core approximated
+300
+(arb. units)
+200
+Intensity
+100
+0
+0
+2
+6
+8
+10
+Energy (eV)Figure S2: Validation of frozen core approximation for the optical absorption spectra of Be+
+2
+cluster employing QCI method.
+Figure S3: Optical absorption spectra of B+
+3 cluster computed using various basis sets em-
+ploying B3LYP functional and TD-DFT method.
+S15
+
+400
+With core-electrons
+Frozen core approximated
+300
+(arb. units)
+200
+Intensity
+100
+0
+0
+2
+4
+6
+8
+10
+Energy (eV)250
+IV
+6-311++G(2d,2p)
+6-311++G(3df,3pd)
+200
+cc-pVDZ
+cc-pVTZ
+alug-cc-pVDZ
+Intensity (arb. units)
+aug-cc-pVTZ
+150
+100
+50
+II
+III
+II
+0
+0
+1
+2
+3
+4
+5
+6
+8
+9
+10
+Energy (eV)

NdAyT4oBgHgl3EQfs_n5/content/tmp_files/2301.00589v1.pdf.txt

@@ -0,0 +1,702 @@
+Tailoring the escape rate of a Brownian particle by combining a vortex
+flow with a magnetic field
+I. Abdoli,1, a) H. Löwen,2 J.-U. Sommer,1, a) and A. Sharma1, a)
+1)Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden,
+Germany
+2)Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, 40225,
+Germany
+(*Electronic mail: [email protected].)
+The probability per unit time for a thermally activated Brownian particle to escape over a potential well is in general
+well-described by Kramers theory. Kramers showed that the escape time decreases exponentially with increasing
+barrier height. The dynamics slow down when the particle is charged and subjected to a Lorentz force due to an
+external magnetic field. This is evident via a rescaling of the diffusion coefficient entering as a prefactor in the Kramers
+escape rate without any impact on the barrier-height-dependent exponent. Here we show that the barrier height can
+be effectively changed when the charged particle is subjected to an external vortex flow. While the external vortex
+alone does not affect the mean escape time of the particle, when combined with a magnetic field it effectively pushes
+the fluctuating particle either radially outside or inside depending on its sign relative to that of the magnetic field. In
+particular, the effective potential over which the particle escapes can be changed to a flat, a stable, and an unstable
+potential by tuning the signs and magnitudes of the external vortex and the applied magnetic field. Notably, the last
+case corresponds to enhanced escape dynamics.
+I.
+INTRODUCTION
+A Brownian particle undergoes erratic motion as a result
+of its collisions with the solvent molecules. If the particle is
+being initially put at the bottom of a potential well, the ther-
+mal activation of the particle may cause an escape from the
+potential well over an energetic barrier. Using the flux-over-
+population method1, Kramers first derived the escape rate of a
+Brownian particle over an energy barrier moving in a bistable
+potential, regardless of what happens after this escape2. He
+showed that the probability per unit time for the particle to es-
+cape the potential well exponentially decays with the height
+of the energy barrier. Kramers derived limiting expressions
+for weak friction and strong damping and realized a global
+maximum at some intermediate value of the damping, which
+is known as Kramers turnover3–5. The problem has been gen-
+eralized to include memory friction6–8 and athermal fluctua-
+tions9–13 and was extended to quantum field theory14,15.
+While Kramers’ framework and its extensions have been
+thoroughly studied with the relevant deterministic potential
+force fields16–21, much less is known when the deterministic
+force is nonconservative, namely when it is not of potential
+type22–24. Recently, by taking into account a nonconservative
+force in the form of Lorentz force, we have studied the escape
+dynamics of a two-dimensional Brownian system with broken
+spatial symmetry via two noises with different strengths25. We
+have shown that while the escape process becomes anisotropic
+(i.e. particles tend to escape the potential well more along the
+axis with larger noise strength) due to two different noises,
+when subjected to an external magnetic field, the spatial sym-
+metry can be restored25. However, to our knowledge, it is
+a)Also at Technische Universität Dresden, Institut für Theoretische Physik,
+01069 Dresden, Germany
+expected that the escape process is reduced (or unaffected in
+the direction of the applied magnetic field) by external con-
+stant magnetic fields24,25 which is evident via a rescaling of
+the diffusion coefficient. It has been shown that the combined
+influence of a nonconservative force and a magnetic field may
+cause an instability in the system26. Here, taking advantage
+of such an instability, we show that the Lorentz force due to a
+constant magnetic field can result in enhanced escape dynam-
+ics.
+In this work, we study the escape dynamics of a Brown-
+ian particle from a harmonic trap which is cut-off at a certain
+distance in the presence of an external vortex and the Lorentz
+force due to an external constant magnetic field. Taking ad-
+vantage of the spatial isotropy in the system we derive an ex-
+act expression for the mean first passage time. While the ex-
+ternal vortex alone does not affect the escape dynamics, we
+observe a nontrivial result when an external magnetic field is
+present: the mean first passage time can be reduced or en-
+hanced. This is attributed to the shape change of the effec-
+tive potential well. By tuning the external magnetic field or
+alternatively the strength of the external vortex the effective
+potential can change shape to a flat, a stable, or an unstable
+potential. This means that by tuning either parameters the
+barrier energy over which the particle may escape can be ef-
+fectively altered to a smaller or larger one whose origin can
+be understood as follows: the combination of the vortex flow
+and the magnetic field effectively pushes the fluctuating parti-
+cle either radially outside or inside depending on their signs.
+In other words, the combination of the two fields, which indi-
+vidually induces no radial force, gives rise to a radial force.In
+what follows, we first introduce the model. Next we calculate
+the mean first passage time, which can be written in terms of
+an effective potential. We then study the trends of the escape
+time with respect to the magnetic field strength and the vortex
+flow and finally we discuss several experimental realizations
+of the set-up considered here.
+arXiv:2301.00589v1  [cond-mat.stat-mech]  2 Jan 2023
+
+2
+FIG. 1. A single charged particle diffusing in a two-dimensional har-
+monic potential U(x,y) = k(x2 + y2)/2, shown by concentric con-
+tours, with k being its stiffness. The particle is subjected to an ex-
+ternal magnetic field B in the −ˆz direction and a nonconservative
+force Fnc = ε(−y,x) with ε being its strength. The nonconserva-
+tive force is shown for ε > 0. The particle can escape the trap when
+reaches the boundary, truncated at r = a, shown by dashed circle,
+where r =
+�
+x2 +y2 is the distance from the origin.
+II.
+MODEL
+We consider an overdamped charged Brownian particle
+with the charge q subjected to an external magnetic field B
+in the −ˆz direction. Since the Lorentz force due to the field
+does not affect the motion of the particle in the z direction, we
+effectively reduce the system to a two-dimensional one and
+study the motion of the particle in the xy plane. The parti-
+cle is trapped in an isotropic potential U(x,y) = k(x2 +y2)/2
+and undergoes a vortex flow due to the nonconservative force
+Fnc = ε(−y,x)⊤. Here k and ε are the stiffness of the poten-
+tial and the strength of the nonconservative force, respectively.
+A schematic of the system is shown in Fig. 1. It is experi-
+mentally and theoretically known that even statically optically
+trapped Brownian particles in the overdamped limit represent
+nonequilibrium behavior characterized by Brownian vortices.
+This is due to the nonconservative forces generated by opti-
+cal scattering forces27–30. Moreover, by applying a prescribed
+external vortex flow field such as a rotating bucket to an un-
+derdamped Brownian particle one can induce similar terms to
+the nonconservative force, i.e. −εy and εx31.
+It has been shown that the overdamped dynamics of the par-
+ticle derived by simply setting the inertia term to zero can
+yield an incorrect description in the presence of a magnetic
+field32. In this case, the overdamped Langevin equation de-
+scribing dynamics of the system can be derived using the low-
+mass approach25,33,34, which can be written as
+˙x =
+1
+γ(1+κ2) [−kx−εy+kκy−εκx]+ξx(t),
+(1)
+˙y =
+1
+γ(1+κ2) [−ky+εx−kκx−εκy]+ξy(t),
+(2)
+FIG. 2. The effective potential from Eq. (4) for different values of
+the stiffness ke f f = k + εκ. By varying the parameter κ (or ε) the
+effective potential can change shape to a stable one if ke f f > 0, a flat
+one if ke f f = 0, or to an unstable one if ke f f < 0.
+where γ is the friction coefficient and κ = qB/γ is the diffu-
+sive Hall parameter quantifying the strength of the Lorentz
+force relative to the frictional force.
+We note that κ can
+be positive or negative depending on the sign of the applied
+magnetic field. Here ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite
+noise with zero mean and time correlation ⟨ξ(t)ξ⊤(t′)⟩ =
+TG−1δ+(t − t′) + T(G−1)⊤δ−(t − t′) where T is the tem-
+perature, G = γ
+� 1
+κ
+−κ 1
+�
+, and the notations δ±(s = t − t′)
+are the modified Dirac delta functions which are zero for
+s ̸= 0 while
+� ∞
+0 dsδ+(s) =
+� 0
+−∞ dsδ−(s) = 1 and
+� ∞
+0 dsδ−(s) =
+� 0
+−∞ dsδ+(s) = 0. Throughout this work we set the Boltzmann
+constant kB to unity. Length and time are measured in units of
+�
+T/k and γ/k, respectively.
+We use Itô calculus to reduce the Langevin equations in
+Eq. (1) and Eq. (2) to a one-dimensional problem for the vari-
+able r =
+�
+x2 +y2, which is given as35
+dr =
+1
+1+κ2
+�
+−k +εκ
+γ
+r + D
+r
+�
+dt +
+�
+2D
+1+κ2 η(t)dt,
+(3)
+where D = T/γ is the coefficient of a freely diffusing particle
+and η(t) is Gaussian white noise with zero mean and the Dirac
+delta time correlation ⟨η(t)η(t′)⟩ = δ(t −t′). The terms in the
+square brackets on the right hand side of Eq.(3) describe the
+force on the the particle due to an effective potential, given as
+Uef f (r) =
+kef f
+2γ(1+κ2)r2 −
+D
+1+κ2 log(r),
+(4)
+where kef f = k +εκ is the stiffness of the effective potential.
+As it is evident from the effective stiffness, in the absence of
+the vortex flow the potential simply gets rescaled by the factor
+1/(1 + κ2), as we have shown in the supplemental informa-
+tion of Ref.25. Moreover, in the absence of the magnetic field
+there is no effect of the vortex flow on the effective potential
+and Eq. (4) reduces to the well known results in Ref.35 for a
+rotationally symmetric Ornstein-Uhlenbeck process in two di-
+mensions. The second term on the right and side comes from
+
+1.0
+a.
+y 0.0
+-1.0
+-1.0
+0.0
+1.0keff= - 0.6
+keff = 0.6
+30
+ keff= 0.0
+keff= k
+keff= 2.0
+20
+10
++
+0
+-10
+0
+2
+3
+4
+5
+1
+63
+FIG. 3. The mean escape time as a function of the diffusive Hall pa-
+rameter κ from Eq. (5) and Eq. (7) for different values of the scaled
+barrier height β∆E with β = 1.0, γ = 1.0 and ε = 0.2. Obviously
+the mean escape time increases with increasing the barrier height. It
+can increase or decrease by tuning the parameter κ: the presence of
+an external vortex field can work together with the applied magnetic
+field to effectively push the fluctuating particle either radially out-
+side, if κ < −k/ε, or inside, if κ > −k/ε. The former corresponds
+to the case in which the combination helps the particle to escape.
+The point κ = 0 corresponds to unaffected escape time by the exter-
+nal vortex (i.e. ke f f = k). In the inset, we show the mean escape time
+which is scaled by the mean escape time in the absence of the exter-
+nal vortex ⟨t0⟩ where the subscript 0 indicates zero strength length
+of the vortex flow. It implies that the mean escape time can decrease
+with increasing κ as compared to the mean escape time without the
+vortex flow.
+the transformation to r and corresponds to an extremely re-
+pulsive potential at the origin due to reduced number of states
+on the circle of radius r. This term influences the motion of
+the particle only near the origin and is negligible for larger
+distances as compared to the first term.
+Figure. 2 represents the scaled effective potential from
+Eq. (4) for different values of the parameter ke f f without the
+logarithmic term. By tuning the diffusive Hall parameter or al-
+ternatively the strength of the nonconservative force, the effec-
+tive potential changes shape: the potential is stable if kef f > 0,
+flat if kef f = 0, and unstable if ke f f < 0. It becomes simple
+quadratic potential in the absence of ε or/and κ.
+III.
+MEAN ESCAPE TIME
+We consider a particle which is trapped in an isotropic po-
+tential U(x,y) which taking advantage of the spatial symmetry
+whose distance from the origin, r = |r|, can be described by
+Eq. (3). We are interested in the mean time at which the par-
+ticle reaches the boundary, truncated at r = a, as shown in
+Fig. 1. As we show in the Appendix A, the mean escape time
+can be exactly calculated from Eq. (3) which reads
+⟨t⟩ = γ(1+κ2)
+2kef f
+�
+Ei
+�
+β∆Ee f f
+�
+−log
+�
+β∆Ee f f
+�
+−γEM
+�
+, (5)
+FIG. 4.
+The mean escape time with respect to the strength of the
+conservative force ε from Eq. (5) and Eq. (7) for different values of
+the scaled barrier heights with β = 1.0 and γ = 1.0. The lines with
+circles and squares correspond to the results with κ = 2.0 and κ =
+−2.0, respectively. The mean escape time can increase or decrease
+with increasing the strength of the vortex flow, which depends on
+its sign relative to that of κ and their magnitude compared to the
+stiffness of the potential k.
+if kef f > 0 corresponding to the effective stable potential and
+⟨t⟩ = γ(1+κ2)
+2kef f
+�
+−Ei
+�
+−β|∆Eef f |
+�
++log
+�
+β|∆Eef f |
+�
++γEM
+�
+,
+(6)
+if kef f < 0 corresponding to the unstable effective potential
+where β is the inverse of the temperature, γEM is the Euler-
+Mascheroni constant, and Ei(x) is the exponential integral.
+Here ∆Eef f = ∆E + εκa2/2 is the effective barrier energy
+which is the real barrier height ∆E = ka2/2 augmented by the
+coupling between the magnitude of the applied magnetic field
+and the strength of the external vortex. Using the series ex-
+pansion of the exponential integral at kef f = 0 for Eq. (5) and
+Eq. (6), the mean escape time for the effective flat potential
+reads
+⟨t⟩ ∼ (1+κ2)
+4D
+a2,
+(7)
+which is the mean escape time for a freely diffusing parti-
+cle scaled by 1 + κ2.
+In the limit of large barrier heights
+the exponential integral in Eq. (5) can be expanded and as
+a consequence the mean escape time reduces to ⟨t⟩ ∼ γ(1 +
+κ2)exp(β∆Eef f )/(2kef f β∆Eef f ). In the absence of the exter-
+nal vortex, which corresponds to ε = 0, the result reduces to
+the Kramers result rescaled by 1 + κ2 arising from the trivial
+rescaling of the diffusion coefficient. The expression becomes
+the same as the Kramers one when the magnetic field is absent
+κ = 0. This confirms that the external vortex field alone does
+not affect the mean escape time. The intuitive reason for that
+is that, for κ = 0, the presence of a vortex field only changes
+the azimuthal motion but not the radial one which leaves the
+redial particle escape unaffected.
+Figure 3 shows the mean escape time with respect to the
+diffusive Hall parameter κ. Obviously it takes the particle
+longer time to escape over larger barrier heights as is evident
+in the figure. The magnetic field together with the vortex flow
+
+103
+102
+106
+101
+105
+100
+10-1
+104
+10-2
+(t)
+103
+-8
+9-
+-4
+-2 
+0
+2
+4
+102
+101
+β△E = 0.5
+β△E = 2.0
+100
+β△E = 4.5
+β△E = 8.0
+-8
+-6
+0
+f2
+4
+K106
+β△E = 0.5
+β△E = 2.0
+0
+K=2.0
+口
+K= - 2.0
+βE = 4.5
+105
+β△E = 8.0
+104
+0
+103
+(t)
+102
+Q
+Q
+N
+2
+101
+100
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.54
+creates additional fluctuations in radial direction which can
+be directed either outwards or inwards depending on its sign.
+The former corresponds to the case in which the combination
+of the vortex flow and the magnetic field helps the particle to
+escape. The inset shows the mean escape time scaled by the
+mean escape time in the absence of the external vortex, which
+is indicated by the subscript 0. The mean escape time can
+decrease with increasing magnetic field as compared to the
+mean escape time without the vortex flow and remains almost
+constant for small barrier height.
+In Fig.4, we show that tuning the strength of the vortex flow
+is an alternative way to vary the mean escape time which is
+evident in Eq.(5) and Eq.(6) via the production of the two pa-
+rameters, i.e. εκ. Therefore the similar trends are expected.
+The figure represents the mean escape time with respect to the
+parameter ε for a system with κ = 2.0, denoted by lines with
+circles, and a system with κ = −2.0, denoted by lines with
+squares. Our results imply that the mean escape time can be
+decreased or increased by tuning the vortex flow strength de-
+pending on its sign relative to that of the magnetic field and
+their magnitude compared to the stiffness of the potential k.
+IV.
+DISCUSSION
+In this work, we studied the effect of a vortex flow on the es-
+cape dynamics of a Brownian magneto-system made of single
+charged Brownian particle subjected to an external magnetic
+field. We expressed the potential in an effective form which
+can change shape to a stable, a flat, or an unstable potential
+depending on the stiffness of the effective potential. Taking
+advantage of the spatial isotropy in the system we obtained an
+exact expression for the mean escape time. In the absence of
+the external vortex, exerted by the nonconservative force, the
+Lorentz force due to the external magnetic field slows down
+the dynamics of the system without any qualitative change,
+which is evident via the trivial rescaling of the diffusion coef-
+ficient. We showed that while the external vortex alone does
+not affect the mean escape time, when coupled to the mag-
+netic field it can enhance or reduce the escape time: this is
+intuitive as the magnetic field together with the vortex flow
+creates additional fluctuations in radial direction which can be
+directed either outwards or inwards depending on its sign. In
+other words, the combination of the two fields, which individ-
+ually induces no radial force, gives rise to a radial force. We
+showed that the barrier over which the particle escapes can be
+effectively changed to a larger or smaller one depending on
+the relative signs of the strength of the vortex flow and the ap-
+plied magnetic field as well as their magnitude compared to
+the stiffness of the potential in which the particle is trapped.
+Moreover, the trap can be effectively switched-off by an ap-
+propriate sign and value of the magnetic field.
+A possible experimental realisation is to trap the particle us-
+ing optical tweezers either in a radio-frequency plasma sheath
+with a vertical magnetic field37,38 or in a rotating frame of
+reference. By rotating the reference frame a Coriolis force
+can be induced which acts the same as the Lorentz force due
+to an external magnetic field39–41.
+As it has been shown
+that even statically optically trapped Brownian particles un-
+dergoe a nonconservative force induced by optical scattering
+forces27–30,36, we expect that the study of the enhanced es-
+cape dynamics does not require an additional external vortex.
+Another possibility is to apply a rotating bucket to an under-
+damped Brownian particle which induces similar terms to the
+nonconservative force in the overdamped limit31.
+From a future perspective, it could be interesting to study
+the escape dynamics of an opposite charged dimer42. In the
+limit of low persistence length, an active chiral particle fol-
+lows curved trajectories, similar to the Brownian motion of a
+charged particle43,44 under a magnetic field. Therefore, an-
+other study of interest would be the escape dynamics of a
+chiral active Brownian particle in the presence of an external
+vortex. Finally, it could be interesting to study how an exter-
+nal magnetic field can affect an active turnover for an active
+particle in a bistable potential45–an optimal correlation time
+where the transition rate is maximized– and how an external
+vortex influences new turnovers observed in the presence of a
+fluctuating magnetic field22,23.
+ACKNOWLEDGMENTS
+A. Sharma and H. Löwen acknowledge the support by the
+Deutsche Forschungsgemeinschaft (DFG) within the projects
+SH 1275/3-1 (A.S.) and LO 418/25-1 (H.L.).
+DATA AVAILABILITY STATEMENT
+The data that support the findings of this study are available
+from the corresponding author upon reasonable request.
+AUTHOR DECLARATIONS
+The authors have no conflicts to disclose.
+Appendix A: Derivation of the mean escape time
+The main purpose of this section is to derive the mean
+escape time in Eq. (5) to Eq. (7).
+We start with the un-
+derdamped Langevin equation describing the dynamics of a
+charged Brownian particle with mass m and charge q sub-
+jected to a magnetic field B in the −ˆz direction. The veloc-
+ity Langevin equation for the position r = (x,y)⊤ and the
+velocity v = (vx,vy)⊤ of the particle under the effect of the
+linear nonconservative force Fnc = ε(−y,x)⊤ and the con-
+servative force Fc = −k(x,y)⊤ due to the isotropic potential
+U(x,y) = k(x2 +y2)/2, can be written as
+m ˙v = −Kr −Gv(t)+
+�
+2γTη(t),
+(A1)
+where η(t) = (ηx(t),ηy(t))⊤ is the Gaussian white noise with
+zero mean and Dirac delta correlation ⟨η(t)η⊤(t′)⟩ = δ(t −t′)
+
+5
+with γ being the friction coefficient and T the temperature.
+The matrices G and K are defined as
+G = γ
+�
+1
+κ
+−κ 1
+�
+, K =
+�
+k
+ε
+−ε k
+�
+,
+(A2)
+with κ = qB/γ being the diffusive Hall parameter which quan-
+tifies the strength of the Lorentz force relative to the frictional
+force. Using the low-mass approach, the corresponding over-
+damped Langevin equation can be written as25,33,34
+˙r = Ar +ξ(t),
+(A3)
+where A = G−1K and ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite
+noise with
+⟨ξ(t)⟩ = 0,
+(A4)
+⟨ξ(t)ξ⊤(t′)⟩ = TG−1δ+(t −t′)+T(G−1)⊤δ−(t −t′), (A5)
+where δ±(s = t − t′) are the modified Dirac delta functions
+which are zero for s ̸= 0 while
+� ∞
+0 dsδ+(s) =
+� 0
+−∞ dsδ−(s) = 1
+and
+� ∞
+0 dsδ−(s) =
+� 0
+−∞ dsδ+(s) = 0.
+Equation (A3) can be rewritten as Eq. (1) and Eq. (2) in
+the Cartesian coordinates and thereafter using Itô calculus can
+be reduced to a one-dimensional equation for the variable r,
+which is the distance from the origin and is given by Eq. (3).
+The mean time for the particle to escape the trap, truncated at
+r = a, can be obtained by the following equation
+⟨t⟩ = 1+κ2
+D
+� a
+0 y−1 exp
+�ke f f
+2γDy2
+�
+dy
+� y
+0 zexp
+�
+−kef f
+2γDz2
+�
+dz,
+(A6)
+where D = T/γ is the diffusion coefficient for a freely moving
+particle. This equation can be exactly solved: using a change
+of variables the second integral on the right hand side gives
+(γD/kef f )
+�
+1−exp
+�
+−ke f f y2/2γD
+��
+. By substitution of this
+solution into Eq. (A6), the resulting integral can be exactly
+solved which gives Eq. (5) and Eq. (6).
+REFERENCES
+1L. Farkas, “Keimbildungsgeschwindigkeit in übersättigten Dämpfen,”
+Zeitschrift für physikalische Chemie 125, 236–242 (1927).
+2H. A. Kramers, “Brownian motion in a field of force and the diffusion
+model of chemical reactions,” Physica 7, 284–304 (1940).
+3H. Grabert and S. Linkwitz, “Effect of time-delayed friction on the escape
+from a metastable well,” Physical Review A 37, 963 (1988).
+4L. I. McCann, M. Dykman, and B. Golding, “Thermally activated tran-
+sitions in a bistable three-dimensional optical trap,” Nature 402, 785–787
+(1999).
+5L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago, and L. Novotny,
+“Direct measurement of Kramers turnover with a levitated nanoparticle,”
+Nature Nanotechnology 12, 1130–1133 (2017).
+6R. F. Grote and J. T. Hynes, “The stable states picture of chemical reactions.
+ii. rate constants for condensed and gas phase reaction models,” The Journal
+of Chemical Physics 73, 2715–2732 (1980).
+7B. Carmeli and A. Nitzan, “Non-Markoffian theory of activated rate pro-
+cesses,” Physical Review Letters 49, 423 (1982).
+8R. Ianconescu and E. Pollak, “A study of Kramers’ turnover theory in the
+presence of exponential memory friction,” The Journal of Chemical Physics
+143, 104104 (2015).
+9P. Hänggi, F. Marchesoni,
+and P. Grigolini, “Bistable flow driven by
+coloured Gaussian noise: a critical study,” Zeitschrift für Physik B Con-
+densed Matter 56, 333–339 (1984).
+10P. Jung and P. Hänggi, “Bistability and colored noise in nonequilibrium
+systems: theory versus precise numerics,” Physical Review Letters 61, 11
+(1988).
+11A. Sharma, R. Wittmann, and J. M. Brader, “Escape rate of active parti-
+cles in the effective equilibrium approach,” Physical Review E 95, 012115
+(2017).
+12A. Scacchi, J. M. Brader, and A. Sharma, “Escape rate of transiently ac-
+tive Brownian particle in one dimension,” Physical Review E 100, 012601
+(2019).
+13L. Caprini, U. Marini Bettolo Marconi, A. Puglisi, and A. Vulpiani, “Ac-
+tive escape dynamics: The effect of persistence on barrier crossing,” The
+Journal of Chemical Physics 150, 024902 (2019).
+14A. Berera, J. Mabillard, B. W. Mintz, and R. O. Ramos, “Formulating the
+Kramers problem in field theory,” Physical Review D 100, 076005 (2019).
+15L. Darmé, J. Jaeckel, and M. Lewicki, “Generalized escape paths for dy-
+namical tunneling in qft,” Physical Review D 100, 096012 (2019).
+16G. R. Fleming and P. Hänggi, Activated Barrier Crossing: applications in
+physics, chemistry and biology, Vol. 4 (World Scientific, 1993).
+17P. Talkner and P. Hänggi, New trends in Kramers’ reaction rate theory,
+Vol. 11 (Springer Science & Business Media, 1995).
+18P. Hänggi and F. Mojtabai, “Thermally activated escape rate in presence of
+long-time memory,” Physical Review A 26, 1168 (1982).
+19E. Pollak, “Theory of activated rate processes:
+A new derivation of
+Kramers’ expression,” The Journal of Chemical Physics 85, 865–867
+(1986).
+20E. Pollak, H. Grabert, and P. Hänggi, “Theory of activated rate processes
+for arbitrary frequency dependent friction: Solution of the turnover prob-
+lem,” The Journal of Chemical Physics 91, 4073–4087 (1989).
+21E. Pollak and J. Ankerhold, “Improvements to Kramers turnover theory,”
+The Journal of Chemical Physics 138, 164116 (2013).
+22A. Baura, S. Ray, and B. C. Bag, “Tuning of barrier crossing time of a par-
+ticle by time dependent magnetic field,” The Journal of Chemical Physics
+138, 244110 (2013).
+23S. Mondal, A. Baura, S. Das, and B. C. Bag, “A generic signature of a fluc-
+tuating magnetic field: An additional turnover prior to the Kramers’ one,”
+Physica A: Statistical Mechanics and its Applications 502, 58–76 (2018).
+24R. Filliger and P. Reimann, “Kramers escape rate for a charged particle in a
+magnetic field,” EPL (Europhysics Letters) 77, 30008 (2007).
+25I. Abdoli, J.-U. Sommer, H. Löwen, and A. Sharma, “Escape dynamics
+in an anisotropically driven Brownian magneto-system,” EPL (Europhysics
+Letters) 139, 21003 (2022).
+26S. Lee and C. Kwon, “Nonequilibrium driven by an external torque in the
+presence of a magnetic field,” Physical Review E 99, 052142 (2019).
+27B. Sun, D. G. Grier, and A. Y. Grosberg, “Minimal model for Brownian
+vortexes,” Physical Review E 82, 021123 (2010).
+28B. Sun, J. Lin, E. Darby, A. Y. Grosberg,
+and D. G. Grier, “Brownian
+vortexes,” Physical Review E 80, 010401 (2009).
+29Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of non-
+conservative optical forces on the dynamics of optically trapped colloidal
+spheres: the fountain of probability,” Physical Review Letters 101, 128301
+(2008).
+30H. W. Moyses, R. O. Bauer, A. Y. Grosberg, and D. G. Grier, “Perturbative
+theory for Brownian vortexes,” Physical Review E 91, 062144 (2015).
+31B. Liebchen and H. Löwen, “Optimal navigation strategies for active parti-
+cles,” EPL (Europhysics Letters) 127, 34003 (2019).
+32H. D. Vuijk, J. M. Brader, and A. Sharma, “Anomalous fluxes in over-
+damped Brownian dynamics with Lorentz force,” Journal of Statistical Me-
+chanics: Theory and Experiment 2019, 063203 (2019).
+33H.-M. Chun, X. Durang, and J. D. Noh, “Emergence of nonwhite noise in
+Langevin dynamics with magnetic Lorentz force,” Physical Review E 97,
+032117 (2018).
+34I. Abdoli, R. Wittmann, J. M. Brader, J.-U. Sommer, H. Löwen,
+and
+A. Sharma, “Tunable Brownian magneto heat pump,” Scientific Reports
+12, 1–10 (2022).
+
+6
+35C. Gardiner, Stochastic methods, Vol. 4 (Springer Berlin, 2009).
+36M. Mangeat, Y. Amarouchene, Y. Louyer, T. Guérin, and D. S. Dean, “Role
+of nonconservative scattering forces and damping on Brownian particles in
+optical traps,” Physical Review E 99, 052107 (2019).
+37J. Carstensen, F. Greiner, L.-J. Hou, H. Maurer, and A. Piel, “Effect of
+neutral gas motion on the rotation of dust clusters in an axial magnetic
+field,” Physics of Plasmas 16, 013702 (2009).
+38A. Piel, Plasma physics: an introduction to laboratory, space, and fusion
+plasmas (Springer, 2017).
+39H. Kählert, J. Carstensen, M. Bonitz, H. Löwen, F. Greiner, and A. Piel,
+“Magnetizing a complex plasma without a magnetic field,” Physical Review
+Letters 109, 155003 (2012).
+40P. Hartmann, Z. Donkó, T. Ott, H. Kählert,
+and M. Bonitz, “Magneto-
+plasmons in rotating dusty plasmas,” Physical Review Letters 111, 155002
+(2013).
+41P. Hartmann, J. C. Reyes, E. G. Kostadinova, L. S. Matthews, T. W. Hyde,
+R. U. Masheyeva, K. N. Dzhumagulova, T. S. Ramazanov, T. Ott, H. Käh-
+lert, et al., “Self-diffusion in two-dimensional quasimagnetized rotating
+dusty plasmas,” Physical Review E 99, 013203 (2019).
+42R. Shinde, J. U. Sommer, H. Löwen, and A. Sharma, “Strongly enhanced
+dynamics of a charged rouse dimer by an external magnetic field,” PNAS
+Nexus 1, pgac119 (2022).
+43S. Van Teeffelen and H. Löwen, “Dynamics of a Brownian circle swimmer,”
+Physical Review E 78, 020101 (2008).
+44C. Scholz, A. Ldov, T. Pöschel, M. Engel, and H. Löwen, “Surfactants and
+rotelles in active chiral fluids,” Science Advances 7, eabf8998 (2021).
+45A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjohanns, L. Novotny,
+and C. Dellago, “Escape dynamics of active particles in multistable poten-
+tials,” Nature Communications 12, 1–6 (2021).
+

NdAyT4oBgHgl3EQfs_n5/content/tmp_files/load_file.txt

@@ -0,0 +1,465 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf,len=464
+page_content='Tailoring the escape rate of a Brownian particle by combining a vortex  flow with a magnetic field  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Abdoli,1, a) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sommer,1, a) and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma1, a)  1)Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden,  Germany  2)Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, 40225,  Germany  (*Electronic mail: abdoli@ipfdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=')  The probability per unit time for a thermally activated Brownian particle to escape over a potential well is in general  well-described by Kramers theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kramers showed that the escape time decreases exponentially with increasing  barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The dynamics slow down when the particle is charged and subjected to a Lorentz force due to an  external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' This is evident via a rescaling of the diffusion coefficient entering as a prefactor in the Kramers  escape rate without any impact on the barrier-height-dependent exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Here we show that the barrier height can  be effectively changed when the charged particle is subjected to an external vortex flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' While the external vortex  alone does not affect the mean escape time of the particle, when combined with a magnetic field it effectively pushes  the fluctuating particle either radially outside or inside depending on its sign relative to that of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In  particular, the effective potential over which the particle escapes can be changed to a flat, a stable, and an unstable  potential by tuning the signs and magnitudes of the external vortex and the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Notably, the last  case corresponds to enhanced escape dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  INTRODUCTION  A Brownian particle undergoes erratic motion as a result  of its collisions with the solvent molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' If the particle is  being initially put at the bottom of a potential well, the ther-  mal activation of the particle may cause an escape from the  potential well over an energetic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Using the flux-over-  population method1, Kramers first derived the escape rate of a  Brownian particle over an energy barrier moving in a bistable  potential, regardless of what happens after this escape2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' He  showed that the probability per unit time for the particle to es-  cape the potential well exponentially decays with the height  of the energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kramers derived limiting expressions  for weak friction and strong damping and realized a global  maximum at some intermediate value of the damping, which  is known as Kramers turnover3–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The problem has been gen-  eralized to include memory friction6–8 and athermal fluctua-  tions9–13 and was extended to quantum field theory14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  While Kramers’ framework and its extensions have been  thoroughly studied with the relevant deterministic potential  force fields16–21, much less is known when the deterministic  force is nonconservative, namely when it is not of potential  type22–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Recently, by taking into account a nonconservative  force in the form of Lorentz force, we have studied the escape  dynamics of a two-dimensional Brownian system with broken  spatial symmetry via two noises with different strengths25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' We  have shown that while the escape process becomes anisotropic  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' particles tend to escape the potential well more along the  axis with larger noise strength) due to two different noises,  when subjected to an external magnetic field, the spatial sym-  metry can be restored25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' However, to our knowledge, it is  a)Also at Technische Universität Dresden, Institut für Theoretische Physik,  01069 Dresden, Germany  expected that the escape process is reduced (or unaffected in  the direction of the applied magnetic field) by external con-  stant magnetic fields24,25 which is evident via a rescaling of  the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' It has been shown that the combined  influence of a nonconservative force and a magnetic field may  cause an instability in the system26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Here, taking advantage  of such an instability, we show that the Lorentz force due to a  constant magnetic field can result in enhanced escape dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  In this work, we study the escape dynamics of a Brown-  ian particle from a harmonic trap which is cut-off at a certain  distance in the presence of an external vortex and the Lorentz  force due to an external constant magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Taking ad-  vantage of the spatial isotropy in the system we derive an ex-  act expression for the mean first passage time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' While the ex-  ternal vortex alone does not affect the escape dynamics, we  observe a nontrivial result when an external magnetic field is  present: the mean first passage time can be reduced or en-  hanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' This is attributed to the shape change of the effec-  tive potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' By tuning the external magnetic field or  alternatively the strength of the external vortex the effective  potential can change shape to a flat, a stable, or an unstable  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' This means that by tuning either parameters the  barrier energy over which the particle may escape can be ef-  fectively altered to a smaller or larger one whose origin can  be understood as follows: the combination of the vortex flow  and the magnetic field effectively pushes the fluctuating parti-  cle either radially outside or inside depending on their signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  In other words, the combination of the two fields, which indi-  vidually induces no radial force, gives rise to a radial force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='In  what follows, we first introduce the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Next we calculate  the mean first passage time, which can be written in terms of  an effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' We then study the trends of the escape  time with respect to the magnetic field strength and the vortex  flow and finally we discuss several experimental realizations  of the set-up considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='00589v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='stat-mech]  2 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' A single charged particle diffusing in a two-dimensional har-  monic potential U(x,y) = k(x2 + y2)/2, shown by concentric con-  tours, with k being its stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The particle is subjected to an ex-  ternal magnetic field B in the −ˆz direction and a nonconservative  force Fnc = ε(−y,x) with ε being its strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The nonconserva-  tive force is shown for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The particle can escape the trap when  reaches the boundary, truncated at r = a, shown by dashed circle,  where r =  �  x2 +y2 is the distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  MODEL  We consider an overdamped charged Brownian particle  with the charge q subjected to an external magnetic field B  in the −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Since the Lorentz force due to the field  does not affect the motion of the particle in the z direction, we  effectively reduce the system to a two-dimensional one and  study the motion of the particle in the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The parti-  cle is trapped in an isotropic potential U(x,y) = k(x2 +y2)/2  and undergoes a vortex flow due to the nonconservative force  Fnc = ε(−y,x)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Here k and ε are the stiffness of the poten-  tial and the strength of the nonconservative force, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  A schematic of the system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' It is experi-  mentally and theoretically known that even statically optically  trapped Brownian particles in the overdamped limit represent  nonequilibrium behavior characterized by Brownian vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  This is due to the nonconservative forces generated by opti-  cal scattering forces27–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Moreover, by applying a prescribed  external vortex flow field such as a rotating bucket to an un-  derdamped Brownian particle one can induce similar terms to  the nonconservative force, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' −εy and εx31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  It has been shown that the overdamped dynamics of the par-  ticle derived by simply setting the inertia term to zero can  yield an incorrect description in the presence of a magnetic  field32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In this case, the overdamped Langevin equation de-  scribing dynamics of the system can be derived using the low-  mass approach25,33,34, which can be written as  ˙x =  1  γ(1+κ2) [−kx−εy+kκy−εκx]+ξx(t),  (1)  ˙y =  1  γ(1+κ2) [−ky+εx−kκx−εκy]+ξy(t),  (2)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The effective potential from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (4) for different values of  the stiffness ke f f = k + εκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' By varying the parameter κ (or ε) the  effective potential can change shape to a stable one if ke f f > 0, a flat  one if ke f f = 0, or to an unstable one if ke f f < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  where γ is the friction coefficient and κ = qB/γ is the diffu-  sive Hall parameter quantifying the strength of the Lorentz  force relative to the frictional force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  We note that κ can  be positive or negative depending on the sign of the applied  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Here ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite  noise with zero mean and time correlation ⟨ξ(t)ξ⊤(t′)⟩ =  TG−1δ+(t − t′) + T(G−1)⊤δ−(t − t′) where T is the tem-  perature, G = γ  � 1  κ  −κ 1  �  , and the notations δ±(s = t − t′)  are the modified Dirac delta functions which are zero for  s ̸= 0 while  � ∞  0 dsδ+(s) =  � 0  −∞ dsδ−(s) = 1 and  � ∞  0 dsδ−(s) =  � 0  −∞ dsδ+(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Throughout this work we set the Boltzmann  constant kB to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Length and time are measured in units of  �  T/k and γ/k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  We use Itô calculus to reduce the Langevin equations in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (2) to a one-dimensional problem for the vari-  able r =  �  x2 +y2, which is given as35  dr =  1  1+κ2  �  −k +εκ  γ  r + D  r  �  dt +  �  2D  1+κ2 η(t)dt,  (3)  where D = T/γ is the coefficient of a freely diffusing particle  and η(t) is Gaussian white noise with zero mean and the Dirac  delta time correlation ⟨η(t)η(t′)⟩ = δ(t −t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The terms in the  square brackets on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (3) describe the  force on the the particle due to an effective potential, given as  Uef f (r) =  kef f  2γ(1+κ2)r2 −  D  1+κ2 log(r),  (4)  where kef f = k +εκ is the stiffness of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  As it is evident from the effective stiffness, in the absence of  the vortex flow the potential simply gets rescaled by the factor  1/(1 + κ2), as we have shown in the supplemental informa-  tion of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Moreover, in the absence of the magnetic field  there is no effect of the vortex flow on the effective potential  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (4) reduces to the well known results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='35 for a  rotationally symmetric Ornstein-Uhlenbeck process in two di-  mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The second term on the right and side comes from  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0keff= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='6  keff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='6  30  keff= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  keff= k  keff= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  20  10  +  0  10  0  2  3  4  5  1  63  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The mean escape time as a function of the diffusive Hall pa-  rameter κ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (7) for different values of the scaled  barrier height β∆E with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Obviously  the mean escape time increases with increasing the barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' It  can increase or decrease by tuning the parameter κ: the presence of  an external vortex field can work together with the applied magnetic  field to effectively push the fluctuating particle either radially out-  side, if κ < −k/ε, or inside, if κ > −k/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The former corresponds  to the case in which the combination helps the particle to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  The point κ = 0 corresponds to unaffected escape time by the exter-  nal vortex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' ke f f = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In the inset, we show the mean escape time  which is scaled by the mean escape time in the absence of the exter-  nal vortex ⟨t0⟩ where the subscript 0 indicates zero strength length  of the vortex flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' It implies that the mean escape time can decrease  with increasing κ as compared to the mean escape time without the  vortex flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  the transformation to r and corresponds to an extremely re-  pulsive potential at the origin due to reduced number of states  on the circle of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' This term influences the motion of  the particle only near the origin and is negligible for larger  distances as compared to the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 2 represents the scaled effective potential from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (4) for different values of the parameter ke f f without the  logarithmic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' By tuning the diffusive Hall parameter or al-  ternatively the strength of the nonconservative force, the effec-  tive potential changes shape: the potential is stable if kef f > 0,  flat if kef f = 0, and unstable if ke f f < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' It becomes simple  quadratic potential in the absence of ε or/and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  MEAN ESCAPE TIME  We consider a particle which is trapped in an isotropic po-  tential U(x,y) which taking advantage of the spatial symmetry  whose distance from the origin, r = |r|, can be described by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' We are interested in the mean time at which the par-  ticle reaches the boundary, truncated at r = a, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' As we show in the Appendix A, the mean escape time  can be exactly calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (3) which reads  ⟨t⟩ = γ(1+κ2)  2kef f  �  Ei  �  β∆Ee f f  �  −log  �  β∆Ee f f  �  −γEM  �  , (5)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  The mean escape time with respect to the strength of the  conservative force ε from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (7) for different values of  the scaled barrier heights with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The lines with  circles and squares correspond to the results with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0 and κ =  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The mean escape time can increase or decrease  with increasing the strength of the vortex flow, which depends on  its sign relative to that of κ and their magnitude compared to the  stiffness of the potential k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  if kef f > 0 corresponding to the effective stable potential and  ⟨t⟩ = γ(1+κ2)  2kef f  �  −Ei  �  −β|∆Eef f |  �  +log  �  β|∆Eef f |  �  +γEM  �  ,  (6)  if kef f < 0 corresponding to the unstable effective potential  where β is the inverse of the temperature, γEM is the Euler-  Mascheroni constant, and Ei(x) is the exponential integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Here ∆Eef f = ∆E + εκa2/2 is the effective barrier energy  which is the real barrier height ∆E = ka2/2 augmented by the  coupling between the magnitude of the applied magnetic field  and the strength of the external vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Using the series ex-  pansion of the exponential integral at kef f = 0 for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (6), the mean escape time for the effective flat potential  reads  ⟨t⟩ ∼ (1+κ2)  4D  a2,  (7)  which is the mean escape time for a freely diffusing parti-  cle scaled by 1 + κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  In the limit of large barrier heights  the exponential integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) can be expanded and as  a consequence the mean escape time reduces to ⟨t⟩ ∼ γ(1 +  κ2)exp(β∆Eef f )/(2kef f β∆Eef f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In the absence of the exter-  nal vortex, which corresponds to ε = 0, the result reduces to  the Kramers result rescaled by 1 + κ2 arising from the trivial  rescaling of the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The expression becomes  the same as the Kramers one when the magnetic field is absent  κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' This confirms that the external vortex field alone does  not affect the mean escape time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The intuitive reason for that  is that, for κ = 0, the presence of a vortex field only changes  the azimuthal motion but not the radial one which leaves the  redial particle escape unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Figure 3 shows the mean escape time with respect to the  diffusive Hall parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Obviously it takes the particle  longer time to escape over larger barrier heights as is evident  in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The magnetic field together with the vortex flow  103  102  106  101  105  100  10-1  104  10-2  (t)  103  8  9-  4  2  0  2  4  102  101  β△E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  β△E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  100  β△E = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  β△E = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  8  6  0  f2  4  K106  β△E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  β△E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  0  K=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  口  K= - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  βE = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  105  β△E = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  104  0  103  (t)  102  Q  Q  N  2  101  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='54  creates additional fluctuations in radial direction which can  be directed either outwards or inwards depending on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  The former corresponds to the case in which the combination  of the vortex flow and the magnetic field helps the particle to  escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The inset shows the mean escape time scaled by the  mean escape time in the absence of the external vortex, which  is indicated by the subscript 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The mean escape time can  decrease with increasing magnetic field as compared to the  mean escape time without the vortex flow and remains almost  constant for small barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='4, we show that tuning the strength of the vortex flow  is an alternative way to vary the mean escape time which is  evident in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (6) via the production of the two pa-  rameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' εκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Therefore the similar trends are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  The figure represents the mean escape time with respect to the  parameter ε for a system with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0, denoted by lines with  circles, and a system with κ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='0, denoted by lines with  squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Our results imply that the mean escape time can be  decreased or increased by tuning the vortex flow strength de-  pending on its sign relative to that of the magnetic field and  their magnitude compared to the stiffness of the potential k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  DISCUSSION  In this work, we studied the effect of a vortex flow on the es-  cape dynamics of a Brownian magneto-system made of single  charged Brownian particle subjected to an external magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' We expressed the potential in an effective form which  can change shape to a stable, a flat, or an unstable potential  depending on the stiffness of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Taking  advantage of the spatial isotropy in the system we obtained an  exact expression for the mean escape time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In the absence of  the external vortex, exerted by the nonconservative force, the  Lorentz force due to the external magnetic field slows down  the dynamics of the system without any qualitative change,  which is evident via the trivial rescaling of the diffusion coef-  ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' We showed that while the external vortex alone does  not affect the mean escape time, when coupled to the mag-  netic field it can enhance or reduce the escape time: this is  intuitive as the magnetic field together with the vortex flow  creates additional fluctuations in radial direction which can be  directed either outwards or inwards depending on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In  other words, the combination of the two fields, which individ-  ually induces no radial force, gives rise to a radial force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' We  showed that the barrier over which the particle escapes can be  effectively changed to a larger or smaller one depending on  the relative signs of the strength of the vortex flow and the ap-  plied magnetic field as well as their magnitude compared to  the stiffness of the potential in which the particle is trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Moreover, the trap can be effectively switched-off by an ap-  propriate sign and value of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  A possible experimental realisation is to trap the particle us-  ing optical tweezers either in a radio-frequency plasma sheath  with a vertical magnetic field37,38 or in a rotating frame of  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' By rotating the reference frame a Coriolis force  can be induced which acts the same as the Lorentz force due  to an external magnetic field39–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  As it has been shown  that even statically optically trapped Brownian particles un-  dergoe a nonconservative force induced by optical scattering  forces27–30,36, we expect that the study of the enhanced es-  cape dynamics does not require an additional external vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Another possibility is to apply a rotating bucket to an under-  damped Brownian particle which induces similar terms to the  nonconservative force in the overdamped limit31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  From a future perspective, it could be interesting to study  the escape dynamics of an opposite charged dimer42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' In the  limit of low persistence length, an active chiral particle fol-  lows curved trajectories, similar to the Brownian motion of a  charged particle43,44 under a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Therefore, an-  other study of interest would be the escape dynamics of a  chiral active Brownian particle in the presence of an external  vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Finally, it could be interesting to study how an exter-  nal magnetic field can affect an active turnover for an active  particle in a bistable potential45–an optimal correlation time  where the transition rate is maximized– and how an external  vortex influences new turnovers observed in the presence of a  fluctuating magnetic field22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen acknowledge the support by the  Deutsche Forschungsgemeinschaft (DFG) within the projects  SH 1275/3-1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=') and LO 418/25-1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  DATA AVAILABILITY STATEMENT  The data that support the findings of this study are available  from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  AUTHOR DECLARATIONS  The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Appendix A: Derivation of the mean escape time  The main purpose of this section is to derive the mean  escape time in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  We start with the un-  derdamped Langevin equation describing the dynamics of a  charged Brownian particle with mass m and charge q sub-  jected to a magnetic field B in the −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' The veloc-  ity Langevin equation for the position r = (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='y)⊤ and the  velocity v = (vx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='vy)⊤ of the particle under the effect of the  linear nonconservative force Fnc = ε(−y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='x)⊤ and the con-  servative force Fc = −k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='y)⊤ due to the isotropic potential  U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='y) = k(x2 +y2)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' can be written as  m ˙v = −Kr −Gv(t)+  �  2γTη(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  (A1)  where η(t) = (ηx(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='ηy(t))⊤ is the Gaussian white noise with  zero mean and Dirac delta correlation ⟨η(t)η⊤(t′)⟩ = δ(t −t′)  5  with γ being the friction coefficient and T the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  The matrices G and K are defined as  G = γ  �  1  κ  −κ 1  �  , K =  �  k  ε  −ε k  �  ,  (A2)  with κ = qB/γ being the diffusive Hall parameter which quan-  tifies the strength of the Lorentz force relative to the frictional  force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Using the low-mass approach, the corresponding over-  damped Langevin equation can be written as25,33,34  ˙r = Ar +ξ(t),  (A3)  where A = G−1K and ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite  noise with  ⟨ξ(t)⟩ = 0,  (A4)  ⟨ξ(t)ξ⊤(t′)⟩ = TG−1δ+(t −t′)+T(G−1)⊤δ−(t −t′), (A5)  where δ±(s = t − t′) are the modified Dirac delta functions  which are zero for s ̸= 0 while  � ∞  0 dsδ+(s) =  � 0  −∞ dsδ−(s) = 1  and  � ∞  0 dsδ−(s) =  � 0  −∞ dsδ+(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  Equation (A3) can be rewritten as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (2) in  the Cartesian coordinates and thereafter using Itô calculus can  be reduced to a one-dimensional equation for the variable r,  which is the distance from the origin and is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  The mean time for the particle to escape the trap, truncated at  r = a, can be obtained by the following equation  ⟨t⟩ = 1+κ2  D  � a  0 y−1 exp  �ke f f  2γDy2  �  dy  � y  0 zexp  �  −kef f  2γDz2  �  dz,  (A6)  where D = T/γ is the diffusion coefficient for a freely moving  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' This equation can be exactly solved: using a change  of variables the second integral on the right hand side gives  (γD/kef f )  �  1−exp  �  −ke f f y2/2γD  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' By substitution of this  solution into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (A6), the resulting integral can be exactly  solved which gives Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  REFERENCES  1L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Farkas, “Keimbildungsgeschwindigkeit in übersättigten Dämpfen,”  Zeitschrift für physikalische Chemie 125, 236–242 (1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kramers, “Brownian motion in a field of force and the diffusion  model of chemical reactions,” Physica 7, 284–304 (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grabert and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Linkwitz, “Effect of time-delayed friction on the escape  from a metastable well,” Physical Review A 37, 963 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  4L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' McCann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Dykman, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Golding, “Thermally activated tran-  sitions in a bistable three-dimensional optical trap,” Nature 402, 785–787  (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  5L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Rondin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Gieseler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ricci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Quidant, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Dellago, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Novotny,  “Direct measurement of Kramers turnover with a levitated nanoparticle,”  Nature Nanotechnology 12, 1130–1133 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  6R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grote and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hynes, “The stable states picture of chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' rate constants for condensed and gas phase reaction models,” The Journal  of Chemical Physics 73, 2715–2732 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  7B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Carmeli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Nitzan, “Non-Markoffian theory of activated rate pro-  cesses,” Physical Review Letters 49, 423 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  8R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ianconescu and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Pollak, “A study of Kramers’ turnover theory in the  presence of exponential memory friction,” The Journal of Chemical Physics  143, 104104 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  9P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hänggi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Marchesoni,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grigolini, “Bistable flow driven by  coloured Gaussian noise: a critical study,” Zeitschrift für Physik B Con-  densed Matter 56, 333–339 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  10P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Jung and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hänggi, “Bistability and colored noise in nonequilibrium  systems: theory versus precise numerics,” Physical Review Letters 61, 11  (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  11A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Wittmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Brader, “Escape rate of active parti-  cles in the effective equilibrium approach,” Physical Review E 95, 012115  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  12A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Scacchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Brader, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma, “Escape rate of transiently ac-  tive Brownian particle in one dimension,” Physical Review E 100, 012601  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  13L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Caprini, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Marini Bettolo Marconi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Puglisi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Vulpiani, “Ac-  tive escape dynamics: The effect of persistence on barrier crossing,” The  Journal of Chemical Physics 150, 024902 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  14A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Berera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Mabillard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Mintz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ramos, “Formulating the  Kramers problem in field theory,” Physical Review D 100, 076005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  15L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Darmé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Jaeckel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Lewicki, “Generalized escape paths for dy-  namical tunneling in qft,” Physical Review D 100, 096012 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  16G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Fleming and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hänggi, Activated Barrier Crossing: applications in  physics, chemistry and biology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 4 (World Scientific, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  17P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Talkner and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hänggi, New trends in Kramers’ reaction rate theory,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 11 (Springer Science & Business Media, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  18P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hänggi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Mojtabai, “Thermally activated escape rate in presence of  long-time memory,” Physical Review A 26, 1168 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  19E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Pollak, “Theory of activated rate processes:  A new derivation of  Kramers’ expression,” The Journal of Chemical Physics 85, 865–867  (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  20E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Pollak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grabert, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hänggi, “Theory of activated rate processes  for arbitrary frequency dependent friction: Solution of the turnover prob-  lem,” The Journal of Chemical Physics 91, 4073–4087 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  21E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Pollak and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ankerhold, “Improvements to Kramers turnover theory,”  The Journal of Chemical Physics 138, 164116 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  22A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Baura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ray, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Bag, “Tuning of barrier crossing time of a par-  ticle by time dependent magnetic field,” The Journal of Chemical Physics  138, 244110 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  23S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Mondal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Baura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Das, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Bag, “A generic signature of a fluc-  tuating magnetic field: An additional turnover prior to the Kramers’ one,”  Physica A: Statistical Mechanics and its Applications 502, 58–76 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  24R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Filliger and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Reimann, “Kramers escape rate for a charged particle in a  magnetic field,” EPL (Europhysics Letters) 77, 30008 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  25I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Abdoli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sommer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma, “Escape dynamics  in an anisotropically driven Brownian magneto-system,” EPL (Europhysics  Letters) 139, 21003 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  26S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Lee and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kwon, “Nonequilibrium driven by an external torque in the  presence of a magnetic field,” Physical Review E 99, 052142 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  27B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grier, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grosberg, “Minimal model for Brownian  vortexes,” Physical Review E 82, 021123 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  28B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Lin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Darby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grosberg,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grier, “Brownian  vortexes,” Physical Review E 80, 010401 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  29Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Roichman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Stolarski, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grier, “Influence of non-  conservative optical forces on the dynamics of optically trapped colloidal  spheres: the fountain of probability,” Physical Review Letters 101, 128301  (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  30H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Moyses, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Bauer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grosberg, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Grier, “Perturbative  theory for Brownian vortexes,” Physical Review E 91, 062144 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  31B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Liebchen and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen, “Optimal navigation strategies for active parti-  cles,” EPL (Europhysics Letters) 127, 34003 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  32H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Vuijk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Brader, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma, “Anomalous fluxes in over-  damped Brownian dynamics with Lorentz force,” Journal of Statistical Me-  chanics: Theory and Experiment 2019, 063203 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  33H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Chun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Durang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Noh, “Emergence of nonwhite noise in  Langevin dynamics with magnetic Lorentz force,” Physical Review E 97,  032117 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  34I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Abdoli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Wittmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Brader, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sommer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen,  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma, “Tunable Brownian magneto heat pump,” Scientific Reports  12, 1–10 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  6  35C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Gardiner, Stochastic methods, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' 4 (Springer Berlin, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  36M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Mangeat, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Amarouchene, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Louyer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Guérin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Dean, “Role  of nonconservative scattering forces and damping on Brownian particles in  optical traps,” Physical Review E 99, 052107 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  37J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Carstensen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Greiner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Maurer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Piel, “Effect of  neutral gas motion on the rotation of dust clusters in an axial magnetic  field,” Physics of Plasmas 16, 013702 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  38A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Piel, Plasma physics: an introduction to laboratory, space, and fusion  plasmas (Springer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  39H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kählert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Carstensen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Bonitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Greiner, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Piel,  “Magnetizing a complex plasma without a magnetic field,” Physical Review  Letters 109, 155003 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  40P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hartmann, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Donkó, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ott, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kählert,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Bonitz, “Magneto-  plasmons in rotating dusty plasmas,” Physical Review Letters 111, 155002  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  41P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hartmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Reyes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Kostadinova, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Matthews, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Hyde,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Masheyeva, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Dzhumagulova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ramazanov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ott, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Käh-  lert, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=', “Self-diffusion in two-dimensional quasimagnetized rotating  dusty plasmas,” Physical Review E 99, 013203 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  42R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Shinde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sommer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Sharma, “Strongly enhanced  dynamics of a charged rouse dimer by an external magnetic field,” PNAS  Nexus 1, pgac119 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  43S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Van Teeffelen and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen, “Dynamics of a Brownian circle swimmer,”  Physical Review E 78, 020101 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  44C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Scholz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Ldov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Pöschel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Engel, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Löwen, “Surfactants and  rotelles in active chiral fluids,” Science Advances 7, eabf8998 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content='  45A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Militaru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Innerbichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Frimmer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Tebbenjohanns, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Novotny,  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}
+page_content=' Dellago, “Escape dynamics of active particles in multistable poten-  tials,” Nature Communications 12, 1–6 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'}

SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0d9beba5969b94a7c81b23bd93fffb62ca4ddf5b85903b87717c908605ba0c06
+size 910742

SdFLT4oBgHgl3EQfPS8r/content/tmp_files/2301.12027v1.pdf.txt

@@ -0,0 +1,1777 @@
+arXiv:2301.12027v1  [eess.SP]  27 Jan 2023
+1
+Lp Quasi-norm Minimization: Algorithm and
+Applications
+Omar M.Sleem⋆
+M.E. Ashour‡
+N.S. Aybat†§
+Constantino M. Lagoa⋆
+⋆Dep. of Electrical Engineering, Pennsylvania State University, State College, PA 16801, USA.
+‡Wireless R&D Department, Qualcomm Technologies, Inc, San Diego, CA 92121, USA.
+†Department of Industrial and Manufacturing Engineering, Pennsylvania State University, State College, PA
+16801, USA.
+Email: { [email protected], [email protected], [email protected], [email protected] }
+Abstract
+Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing.
+Computations over sparse structures are less complex compared to their dense counterparts, and their
+storage consumes less space. This paper proposes a heuristic method for retrieving sparse approximate
+solutions of optimization problems via minimizing the ℓp quasi-norm, where 0 < p < 1. An iterative
+two-block ADMM algorithm for minimizing the ℓp quasi-norm subject to convex constraints is proposed.
+For p = s/q < 1, s, q ∈ Z+, the proposed algorithm requires solving for the roots of a scalar
+degree 2q polynomial as opposed to applying a soft thresholding operator in the case of ℓ1. The
+merit of that algorithm relies on its ability to solve the ℓp quasi-norm minimization subject to any
+convex set of constraints. However, it suffers from low speed, due to a convex projection step in
+each iteration, and the lack of mathematical convergence guarantee. We then aim to vanquish these
+shortcomings by relaxing the assumption on the constraints set to be the set formed due to convex
+and differentiable, with Lipschitz continuous gradient, functions, i.e. specifically, polytope sets. Using a
+proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm speed
+while proving its convergence. We then present various applications where the proposed algorithm excels,
+namely, matrix rank minimization, sparse signal reconstruction from noisy measurements, sparse binary
+classification, and system identification. The results demonstrate the significant gains obtained by the
+proposed algorithm compared to those via ℓ1 minimization.
+Keywords— Sparsity, compressed sensing, rank minimization, ADMM, Proximal gradient method.
+
+2
+I. INTRODUCTION
+A. Motivation
+In numerical analysis and scientific computing, a sparse matrix/array is the one with many of its elements being
+zeros. The number of zeros divided by the total number of elements is called sparsity. Sparse data is often easier to
+store and process. Hence, techniques for deriving sparse solutions and exploiting them have attracted the attention
+of many researchers in various engineering fields like machine learning, signal processing, and control theory.
+The taxonomy of sparsity can be studied through the rank minimization problem (RMP). It has been lately
+considered in many engineering applications including control design and system identification. This is because the
+notions of complexity and system order can be closely related to the matrix rank. The RMP can be formulated as
+follows:
+min
+X
+Rank(X),
+s.t.
+X ∈ M,
+(1)
+where X ∈ Rm×n and M ⊂ Rm×n is a convex set. The problem (1) in its generality is NP-hard [1]. Therefore,
+polynomial time algorithms for solving large-scale problems of the form in (1) are not currently known. Hence,
+currently adopted methods for solving such problems are approximate and structured heuristics.
+A special case of RMP is the sparse vector recovery (SVR) problem involving ℓ0 pseudo-norm minimization
+given by:
+min
+x
+∥x∥0 ,
+s.t.
+x ∈ V,
+(2)
+where x ∈ Rn, V ⊂ Rn is a closed convex set and ∥·∥0 counts the number of the non-zero elements of its argument.
+From the definition of the rank being the number of non-zero singular values of a matrix, it can be easily realized
+that (1) is a generalized form of (2).
+Various works – which will be discussed in the next section in more detail – have explored efficient solution
+techniques for the problems in (1) and (2) independently using Schatten-p and ℓp quasi-norm relaxations respectively.
+However, all of these methods either assume a specific structure for the convex set M in (1) or only work for the
+special case in (2); hence, they lack generality. Using the fact that the Schatten-p quasi-norm is the ℓp quasi-norm
+of the matrix singular values, we aim to design an efficient heuristic method based on Schatten-p relaxation for
+solving both problems in a unified manner. To achieve this, we first propose an algorithm for solving ℓp quasi-norm
+relaxation of the SVR problem in (2), and next, exploiting the fact that (2) is a special case of (1), we then employ
+the derived ℓp quasi-norm minimization algorithm as a building block for the desired generalized algorithm for the
+rank minimization problem.
+B. Related work
+1) Sparse Vector Recovery: As discussed, a sparse solution is defined as the one having the minimum
+number of non-zero components while satisfying certain constraints such as a system of linear equations
+[2].
+Since many signals are either sparse or compressible, SVR problem has found applications in object recognition,
+classification and compressed sensing problems, see, e.g., [3]–[5]. In [6], the authors discussed the concept of the
+sparse representation of signals and systems, where they reviewed the theoretical and empirical results on sparse
+
+3
+optimization, and discussed sufficient conditions needed for uniqueness, stability and computational practicability.
+Different applications for the SVR problem are explored in [6] and it is argued that in certain denoising and
+compression tasks, the methods for sparse optimization provide state of the art solutions.
+The problem of constructing a sparse solution to undetermined linear systems has received great attention. In
+[7], the authors surveyed the existing algorithms for sparse approximation, namely; greedy methods [8], [9], the
+methods based on convex relaxation [4], [5], [10], [11], those involving non-convex optimization [12], [13], Bayesian
+framework [14], [15], and requiring brute force [16]. They also discussed the computational requirements of each
+algorithm and their relation to each other.
+Sparse optimization problems of the form min{f(x) + µg(x)} have been extensively studied in the literature,
+where g(x) is a sparsity inducing function, e.g., ℓ1-norm, f is a loss function on measurement errors, e.g., f(x) =
+∥Ax − b∥2, and µ > 0 is a trade-off parameter between data-fidelity and sparsity. In [2], the authors considered
+sparse recovery problem from a set of corrupted measurements. For g(·) = ∥·∥1, they established a sufficient
+condition for the exact sparse signal recovery, i.e., restricted isometry property (RIP). Motivated by the fact that
+∥x∥p
+p → ∥x∥0 as p → 0, it is natural to consider the above problem with g set to ℓp quasi-norm for p ∈ (0, 1).
+Hence, the authors in [17] presented theoretical results demonstrating the ability of the ℓp quasi-norm to recover
+sparse signals from noisy measurements. Under more relaxed RIP conditions, they showed that the ℓp quasi-norm
+provides better theoretical guarantees in terms of stability and robustness than the ℓ1 minimization. In [12], the
+problem of SVR via the ℓp quasi-norm minimization from small number of linear measurements of the target
+signal was considered. This setting is important in applications where data acquisition is difficult or expensive.
+However, the proposed approach in [12] has limited applicability due to its long reconstruction time compared
+to the ℓ1 norm. In [18], the authors exploited Fourier-based algorithms for convex optimization to solve sparse
+signals reconstruction problem via the ℓp quasi-norm minimization. They showed that their approach combines the
+construction abilities of the non-convex methods with the speed of the convex ones. In [19] the authors proposed
+an approach, for sparse reconstruction, replacing the non-convex function with a quadratic convex one. In [20],
+an alternating direction method of multipliers (ADMM) based algorithm that enforces both sparsity and group
+sparsity using non-convex regularization is presented. An iterative half thresholding algorithm for fast solution of
+the ℓ0.5 regularization is proposed in [21]. The authors proved the existence of the resolvent of gradient of ||x||0.5
+0.5
+, calculated its analytic expression, and derived a thresholding representation of solutions for ℓ0.5 regularization. In
+[22], the convergence of the iterative half thresholding algorithm is studied, where it was shown that, under certain
+conditions, the half thresholding algorithm converges, to a local minimizer of the regularized problem, with a linear
+convergence rate. Conditions for the convergence of an ADMM algorithm that solves the problem of minimizing
+the sum of a smooth function with a bounded Hessian and a non-smooth function are derived in [23]. In [24], the
+convergence of ADMM for minimizing a non-convex and possibly non-smooth objective function subject to equality
+constraints is analyzed. The developed convergence guarantee covers a variety of non-convex objectives including
+piece-wise linear functions, ℓp quasi-norm and Schatten-p quasi-norm (0 < p < 1), while allowing non-convex
+constraints as well.
+
+4
+2) Rank minimization: In [25], the authors aimed at determining the least order dynamic output feedback,
+using same formulation as in (1), which stabilizes a given linear time invariant system. They found that minimizing
+the trace instead of the rank results in a Semi-Definite Program (SDP) that can be solved efficiently. However, their
+solution was only applicable for symmetric and square matrices. In [26], a generalization of the latter approach
+was introduced which is based on replacing the rank in the objective function with the summation of the singular
+values of the matrix, i.e., the nuclear norm. They showed that this leads to the convex envelope of the non-
+convex rank objective and boils down to the original trace heuristic when the decision matrix is a symmetric
+positive semi-definite (PSD) matrix. Finally, effectiveness of the approach was shown using a frequency domain
+system identification problem. In [27], another heuristic based on the logarithm of the determinant was presented
+as a surrogate for the rank minimization over the subspace of PSD matrices, and the authors showed that this
+formulation can be solved using a sequence of trace minimization problems. The authors also extended their
+heuristic to handle matrices that are not necessarily PSD. In [28], the authors also studied the existing trace and log
+determinant heuristics for approximating (1). They discussed the applications of these heuristics for computing a
+low-rank approximation of 1) covariance matrices for a given dataset so that one can obtain a simple data model,
+easy to interpret, which is especially important in statistics and signal processing; 2) Hankel matrices arising in
+system identification of a time invariant, low-order system for given output realizations; and 3) matrices appearing
+in various other problems including H∞ and reduced-order µ-synthesis with constant scaling and problems with
+inertia constraints.
+Although the nuclear norm is the tightest convex substitute for the non-convex rank function, one of its major
+shortcomings is that it treats all the singular values equally in order to be able to preserve the convexity. Therefore,
+this restricts its performance in applications where the singular values need to be treated differently, e.g., particularly
+in image denoising. In [29], the authors proposed an iterative re-weighted nuclear norm heuristic to avoid this
+problem and analyzed its convergence. They also proposed a gradient-based algorithm and applied it to a low-order
+system identification problem. Experimental results showed that the re-weighted nuclear norm leads to a lower order
+model than the nuclear norm itself. In [30], the solution of weighted nuclear norm (WNN) problem was analyzed
+under different circumstances where the weights could be in a non-ascending, arbitrary or a non-descending order.
+The authors applied their proposed WNN algorithm to an image denoising problem by exploiting the image non-
+local self-similarity. Numerical results showed that the proposed WNN algorithm outperforms many of the state of
+the art denoising methods in terms of both quantitative measures and visual quality.
+Another method, inspired by the success of the ℓp quasi-norm (0 < p < 1) for sparse signal reconstruction,
+is to enforce low-rank structure by using the Schatten-p quasi-norm, which is defined as the ℓp quasi-norm of
+the singular values. In [31], the authors considered the matrix completion problem, which deals with constructing
+a low-rank matrix, given a subset of its entries. Instead of minimizing the nuclear norm, the authors proposed
+a Schatten-p quasi-norm formulation, for which they came up with an algorithm and studied its convergence
+properties. In each iteration, the sub-problem that needs to be solved has a closed-form solution, which makes
+it fast and suitable for large-scale problems. To improve the robustness of the solutions to matrix completion
+problem, in [32] Schatten-p quasi-norm for low-rank recovery was combined with ℓp quasi-norm (0 < p ≤ 1)
+
+5
+of the prediction errors on the observed entries. The authors proposed an algorithm based on the ADMM, which
+performed better in their numerical experiments than other completion methods like fixed-point continuation and
+accelerated proximal gradient singular-value thresholding. In [33], the authors extended the theoretical recovery
+results previously developped for the nuclear norm to Schatten-p quasi-norm using a weaker version of the RIP
+assumption; they showed that the minimum rank solution can be recovered by solving the Schatten-p quasi-norm
+minimization problem. In [34], the authors developed an iterative re-weighted least squares algorithm to solve an
+unconstrained ℓp minimization problem. The algorithm, and analysis, are extended to include the low rank recovery
+problem. Another non-convex approaches for matrix optimization problems involving sparsity are developed, by
+means of a generalized shrinkage operation, in [35]. These approaches are applied to the decomposition of video
+into low rank and sparse components, which is able to separate moving objects from the stationary background
+better than in the convex case.
+C. Contributions
+Despite the good performance of the various algorithms discussed in I-B1 and I-B2 for solving different
+relaxations of (2) and (1), they are all based on a specific structure of the considered convex constraint set;
+therefore, they are problem specific and lack generality. In this work, we present a general-purpose method based
+on projections onto the constraint set, for which we assume only closed convexity as the specific structure. [36]
+and [37] examine the characteristics of the projection on the constraint set. In the latter, the projection technique
+of every given point on ℓp balls is assumed to be known a priori, whereas in the former, the issue is solved while
+omitting an important coupling condition for the polynomial equations.
+First, based on our previous work in [38], we propose an ADMM algorithm (pQN-ADMM) to solve the ℓp
+quasi-norm relaxation of (2). At each iteration, the bottleneck operation is to compute Euclidean projections on
+to some particular convex and non-convex sets. The proposed algorithm possesses two important properties: 1)
+Its computational complexity is similar to ℓ1 minimization algorithms except for the additional effort of solving
+for the roots of a polynomial; 2) No specific structure for the convex set is required. Numerical results, using
+a SVR and binary classification examples, are presented to show the competitive performance of our algorithm
+against the ℓ1 minimization approach. We then extend the proposed algorithm to solve the relaxation of (1) based
+on the Schatten-p quasi-norm – here, we exploit the fact that minimizing the ℓp-norm of the vector of singular
+values is equivalent to minimizing the Schatten-p quasi-norm. We consider two different numerical examples: 1)
+We formulate time domain system identification problem for minimum order system detection and solve it using the
+pQN-ADMM approach and compare our recovery results against the nuclear norm minimization approach of [26];
+2) We consider a matrix completion problem, where the goal is to recover an unknown low-rank matrix based on a
+small fraction of observed entries. Numerical results in both examples show that our method is competitive against
+some of the state-of-the-art algorithms in terms of both the detected system order (for the system identification
+problem) and the rank of the matrix recovered (for the matrix completion problem).
+Finally, since the derived algorithm depends on a computationally expensive convex projection step in every
+iteration, we aim to develop a faster algorithm with a mathematical convergence guarantee. Considering only a
+
+6
+subset of problems where the constraint set is a polytope, we utilize concepts from the proximal gradient (PG)
+method to derive a fast algorithm and prove that it converges with a rate O( 1
+K ), where K is the iteration budget
+given to the algorithm.
+II. NOTATIONS AND BASIC DEFINITIONS
+Unless otherwise specified, we denote vectors with lowercase boldface letters, i.e., x, with i-th entry as xi, while
+matrices are in uppercase, i.e. X, with (i, j)-th entry as xi,j. For an integer n ∈ Z+, [n]
+∆= {1, . . ., n}. 1 represents
+a vector of all entries equal to 1, while
+1G(.) is an indicator function to the set G, i.e., it evaluates to zero if its
+argument belongs to the set G and is +∞ otherwise.
+For a vector x ∈ Rn, the general ℓp norm is defined as
+∥x∥p
+∆= (
+�
+i∈[n]
+|xi|p)
+1
+p .
+(3)
+For convenience, we let ∥x∥ be the well known euclidean norm, i.e., p = 2. When 0 < p < 1, the expression in
+(3) is termed as the quasi-norm satisfying the same axioms of the norm except the triangular inequality making it
+a non-convex function.
+Definition 1. Let X : V −→ W be a linear operator between two normed spaces equipped with ℓp norm,
+p ∈ [1, ∞). The induced p-norm is defined as,
+∥X∥p
+∆= sup
+v̸=0
+�∥Xv∥p
+∥v∥p
+�
+.
+(4)
+A special case of (4) is when p = 2, known as the spectral radius, which can be shown to be the square root of
+the maximum eigen value of XHX, where XH is the complex conjugate of the transpose of X, i.e., X⊤. In the
+rest of the analysis, we will drop the subscript 2 in the spectral norm notation and only refer to it with ∥.∥.
+For a matrix X ∈ Rm×n, the Lp,q entry-wise norm is defined as,
+∥X∥p,q
+∆= (
+�
+j∈[n]
+(
+�
+i∈[m]
+|xij|p)
+q
+p )
+1
+q .
+(5)
+A special case of (5) is when p = q = 2, known as the Frobenius norm, which were refer to by ∥.∥f.
+Definition 2. Let H1 ⊂ Rn and H2 ⊂ Rm be two separable Hilbert spaces and X ∈ Rm×n be a linear compact
+operator from H1 to H2, the Schatten-p norm of X is then defined as,
+∥X∥p
+∆= (
+�
+i∈[min{m,n}]
+σi(X)p)
+1
+p ,
+(6)
+where σi(X) is the i-th singular value of the matrix X.
+When p = 1, equation (6) yields to the nuclear norm which is the convex envelope of the rank function.
+Throughout the paper, we consider a non-convex relaxation for the rank function, specifically p = 1/2, and compare
+its performance with the nuclear norm case in the results section.
+
+7
+We define ⌈·⌉ as the ceiling operator, vec(X) ∈ Rmn as a vector formed by stacking the columns of the
+matrix X ∈ Rm×n and Hankel(.) as an operator that outputs a Hankel matrix constructed from the applied vector
+arguments.
+III.
+SPARSE VECTOR RECOVERY ALGORITHM
+A. Problem Formulation
+This section develops a method for approximating the solution of (2) using the following relaxation,
+min
+x
+∥x∥p
+p ,
+s.t.
+x ∈ V,
+(7)
+where p ∈ (0, 1] and V is a closed convex set. Problem (7) is convex for p = 1; hence, can be solved to optimality
+efficiently. However, the problem becomes non-convex when p < 1. An epigraph equivalent formulation of (7) is
+obtained by introducing the variable t = [ti]i∈[n]:
+min
+x,t
+1⊤t,
+s.t.
+ti ≥ |xi|p,
+i ∈ [n],
+x ∈ V.
+(8)
+Let X ⊂ R2 denote the epigraph of the scalar function |x|p, i.e., X = {(x, t) ∈ R2 : t ≥ |x|p}, which is a
+non-convex set for p < 1. Then, (8) can be cast as
+min
+x,t
+�
+i∈[n]
+1X (xi, ti) + 1⊤t,
+s.t.
+x ∈ V.
+(9)
+ADMM exploits the structure of the problem to split the optimization over the variables via iteratively solving fairly
+simple subproblems. In particular, we introduce auxiliary variables y = [yi]i∈[n] and z = [zi]i∈[n] and obtain an
+ADMM equivalent formulation of (9) given by:
+min
+x,t,y,z
+�
+i∈[n]
+1X (xi, ti) +
+1Y(y) + 1⊤z,
+s.t.
+x = y : λ,
+t = z : θ,
+(10)
+where Y is the 0-sublevel set of f, i.e., Y = {y ∈ Rn : y ∈ V}. The dual variables associated with the constraints
+x = y and t = z are λ and θ, respectively. Hence, the Lagrangian function corresponding to (10) augmented with
+a quadratic penalty on the violation of the equality constraints with penalty parameter ρ > 0, is given by:
+Lρ(x, t, y, z, λ, θ) =
+�
+i∈[n]
+1X (xi, ti) +
+1Y(y) + 1⊤z
++ λ⊤(x − y) + θ⊤(t − z) + ρ
+2
+�
+∥x − y∥2 + ∥t − z∥2�
+.
+(11)
+Considering the two block variables (x, t) and (y, z), ADMM [39] consists of the following iterations:
+(x, t)k+1
+=
+argmin
+x,t
+Lρ(x, t, yk, zk, λk, θk)
+(12)
+(y, z)k+1
+=
+argmin
+y,z
+Lρ(xk+1, tk+1, y, z, λk, θk)
+(13)
+λk+1
+=
+λk + ρ(xk+1 − yk+1)
+(14)
+θk+1
+=
+θk + ρ(tk+1 − zk+1).
+(15)
+
+8
+Algorithm 1 ADMM (ρ > 0)
+1: Initialize: y0, z0, λ0, θ0
+2: for k ≥ 0 do
+3:
+(xi, ti)k+1 ← ΠX
+�
+yk
+i − λk
+i
+ρ , zk
+i − θk
+i
+ρ
+�
+, ∀i ∈ [n]
+4:
+yk+1 ← ΠY
+�
+xk+1 + λk
+ρ
+�
+5:
+zk+1 ← tk+1 + θk−1
+ρ
+6:
+λk+1 ← λk + ρ(xk+1 − yk+1)
+7:
+θk+1 ← θk + ρ(tk+1 − zk+1).
+According to the expression of the augmented Lagrangian function in (11), it follows from (12) that the variables
+x and t are updated via solving the following non-convex problem
+min
+x,t
+∥x − yk + λk
+ρ ∥2 + ∥t − zk + θk
+ρ ∥2
+s.t.
+(xi, ti) ∈ X,
+i ∈ [n].
+(16)
+Exploiting the separable structure of (16), one immediately concludes that (16) can be split into n independent
+2-dimensional problems that can be solved in parallel, i.e., for each i ∈ [n],
+(xi, ti)k+1 = ΠX
+�
+yk
+i − λk
+i
+ρ , zk
+i − θk
+i
+ρ
+�
+,
+(17)
+where ΠX (.) denotes the Euclidean projection operator onto the set X. Furthermore, (11) and (13) imply that y
+and z are independently updated as follows:
+yk+1
+=
+ΠY
+�
+xk+1 + λk
+ρ
+�
+(18)
+zk+1
+=
+tk+1 + θk − 1
+ρ
+.
+(19)
+Algorithm 1 summarizes the proposed ADMM algorithm. It is clear that z, λ, and θ merit closed-form updates.
+However, updating (x, t) requires solving n non-convex problems. Our strategy for dealing with this issue is
+presented in the section that follows.
+B. Non-convex Projection
+In this part, we present the method used to tackle the non-convex projection problem required to update x and
+t.
+Among the advantages of the proposed algorithm is that it is amenable to decentralization. As it is clear from
+(17), x and t can be updated element-wise via performing a projection operation onto the non-convex set X, one
+for each i ∈ [n]. The n projection problems can be run independently in parallel. We now outline the proposed
+idea for solving one such projection, i.e., we suppress the dependence on the index of the entry of x and t. For
+(¯x, ¯t) ∈ R2, ΠX (¯x, ¯t) entails solving
+min
+x,t
+g(x, t) ≜ (t − ¯t)2 + (x − ¯x)2,
+s.t.
+t ≥ |x|p.
+(20)
+
+9
+If ¯t ≥ |¯x|p, then trivially ΠX (¯x, ¯t) = (¯x, ¯t). Thus, we focus on the case in which ¯t < |¯x|p. The following proposition
+states the necessary optimality conditions for (20).
+Proposition 1. Let ¯t < |¯x|p, and (x∗, t∗) be an optimal solution of (20). Then, the following properties are satisfied
+(a) sign(x∗) = sign(¯x),
+(b) t∗ ≥ ¯t,
+(c) |x∗|p ≥ ¯t,
+(d) t∗ = |x∗|p.
+Proof. We prove the statements by contradiction as follows:
+(a) Suppose that sign(x∗) ̸= sign(¯x), then
+|x∗ − ¯x|=|x∗ − 0|+|¯x − 0| > |¯x − 0|,
+(21)
+i.e., (x∗−¯x)2 >(0−¯x)2. Hence, g(x∗, t∗)−g(0, t∗)>0. Moreover, the feasibility of (x∗, t∗) implies that t∗ > 0.
+Thus, (0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗). This contradicts the
+optimality of (x∗, t∗).
+(b) Assume that t∗ < ¯t. Then,
+g(x∗, t∗) − g(x∗, ¯t) = (t∗ − ¯t)2 > 0.
+(22)
+Furthermore, by the feasibility of (x∗, t∗), we have |x∗|p ≤ t∗ < ¯t. Thus, (x∗, ¯t) is feasible and attains a lower
+objective value than that attained by (x∗, t∗). This contradicts the optimality of (x∗, t∗).
+(c) Suppose that |x∗|p < ��t, i.e.,
+− ¯t
+1
+p < x∗ < ¯t
+1
+p .
+(23)
+We now consider two cases, ¯x > 0 and ¯x < 0. First, let ¯x > 0. Then, we have by (a) and (23) that 0 < x∗ < ¯t
+1
+p .
+Since ¯t < |¯x|p, i.e., (¯x, ¯t) /∈ X, therefore ¯t
+1
+p < ¯x and hence, 0 < x∗ < ¯t
+1
+p < ¯x. Pick x0 > 0 such that |x0|p = ¯t,
+i.e., x0 = ¯t
+1
+p . Then clearly, x∗ < x0 < ¯x. Thus, we have
+g(x∗, t∗) − g(x0, t∗) = (x∗ − ¯x)2 − (x0 − ¯x)2 > 0,
+(24)
+where the last inequality follows the just proven identity that x∗ < x0 < ¯x. Moreover, we have |x0|p = ¯t ≤ t∗ by
+(b). Thus, (x0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗). This contradicts
+the optimality of (x∗, t∗). On the other hand, let ¯x < 0. Then, we have by (a) and (23) that −¯t
+1
+p < x∗ < 0.
+Since ¯t < |¯x|p, i.e., (¯x, ¯t) /∈ X, then ¯t
+1
+p < |¯x|, i.e., ¯x < −¯t
+1
+p . Therefore, ¯x < −¯t
+1
+p < x∗. Pick x0 < 0 such that
+|x0|p = ¯t, i.e., x0 = −¯t
+1
+p . Then, (24) also holds when ¯x < 0. Note that |x0|p = ¯t ≤ t∗ by (b). Thus, (x0, t∗)
+is feasible and attains a lower objective value than that attained by (x∗, t∗). This contradicts the optimality of
+(x∗, t∗).
+(d) The feasibility of (x∗, t∗) eliminates the possibility that t∗ < |x∗|p. Now let t∗ > |x∗|p and pick t0 = |x∗|p.
+Then, ¯t ≤ |x∗|p = t0 < t∗, where the first inequality follows from (c). Then, 0 ≤ t0 − ¯t < t∗ − ¯t. Thus, we
+have
+g(x∗, t∗) − g(x∗, t0) = (t∗ − ¯t)2 − (t0 − ¯t)2 > 0,
+(25)
+
+10
+Algorithm 2 Non-convex projection (p = s
+q < 1)
+1: R ← roots{a2q + s
+q(a2s − ¯tas) − |¯x|aq}
+2: ¯R ← R \ {complex numbers and negative reals in R}
+3: T ← {(rq, rs) : r ∈ ¯R}
+4: (ˆx, t∗) ← argmin {g(x, t) : (x, t) ∈ T }
+5: x∗ ← sign(¯x)ˆx
+Furthermore, the feasibility of (x∗, t0) follows trivially from the choice of t0. Thus, (x∗, t0) is feasible and
+attains a lower objective value than that attained by (x∗, t∗). This contradicts the optimality of (x∗, t∗).
+This concludes the proof.
+We now make use of the fact that for (20), an optimal solution (x∗, t∗) satisfies t∗ = |x∗|p and hence, (20)
+reduces to solving
+min
+x
+(|x|p − ¯t)2 + (x − ¯x)2.
+(26)
+The first order necessary optimality condition for (26) implies the following:
+p|x∗|p−1sign(x∗)(|x∗|p − ¯t) + x∗ − ¯x = 0.
+(27)
+By the symmetry of the function |x|p, without loss of generality, assume that x∗ > 0 and let 0 < p = s
+q < 1 for
+some s, q ∈ Z+. A change of variables aq = x∗ plugged in (27) shows that finding an optimal solution for (20)
+reduces to finding a root of the following scalar degree 2q polynomial:
+a2q + s
+q
+�
+a2s − ¯tas�
+− ¯xaq.
+(28)
+Thus, to find ΠX (¯x, ¯t), solve for a root a∗ of the polynomial in (28) such that (a∗q, a∗s) minimizes g(x, t). Algorithm
+2 summarizes the method we use to solve problem (20). In case ¯x = 0, we set x∗ = t∗ = 0. If the set ¯R is empty,
+we set x∗ = 0 and t∗ = (¯t)+.
+C. Convex Projection
+The convex projection for y-update in (18) can be formulated as the following convex optimization problem
+yk+1 = argmin
+y
+�����y − (xk+1 + λk
+ρ )
+�����
+2
+,
+s.t.
+y ∈ V,
+(29)
+where ∥.∥ is the euclidean norm. Convex problems can be solved by a variety of contemporary methods including
+bundle methods [40], sub-gradient projection [41], interior point methods [42], and ellipsoid methods [43]. The
+efficiency of optimization techniques rely mainly on exploiting the structure of the constraint set. As mentioned
+in I-C, to be general, we aim to solve the problem in (7) with no assumptions on the set V, other than it being
+closed and convex. That said, if possible, through exploiting the structure of V, one should be able to reduce the
+computational complexity of solving (29).
+
+11
+Remark 1. As per our knowledge, none of the existing literature considered the convergence of an ADMM algorithm
+for solving the general problem in (7). As discussed in I-B1, on one hand, the work in [23] studied the convergence
+of ADMM under mild assumptions. However, assuming V has a particular form, these assumptions hold only if the
+function f defining the the constraint set V = {x : f(x) ≤ 0} in (7) is Lipschitz differentiable. On the other hand,
+[44] studied the convergence of a non ADMM algorithm to solve (7) while assuming that the global optimal for
+each update step can be found efficiently.
+IV. RANK MINIMIZATION ALGORITHM
+We consider the same problem as in (1) and propose a method for approximating its solution efficiently. The
+Schatten-p heuristic of (1) can be written as
+min
+X
+∥X∥p
+p
+∆=
+L
+�
+i=1
+|σi(X)|p,
+s.t.
+X ∈ M,
+(30)
+where L = min(m, n) and σi(X) is the ith singular value of X. When p = 1, problem (30) is a convex one
+which is eventually the nuclear norm heuristic. We consider a non-convex case where 0 < p < 1, which has the
+corresponding epi-graph form,
+min
+X,t
+1⊤t,
+s.t.
+|σi(X)|p ≤ ti,
+i ∈ {1, . . .L},
+X ∈ M,
+(31)
+such that t = [ti]i∈[L]. Defining the epi-graph set ˚
+X for the function σ(X), where ˚
+X
+∆= {(σ(X), t)∈R2 :|σ(X)|p ≤
+t} ⊆ R2, the problem in (31) can be written as,
+min
+X,t
+1⊤t +
+1M(X) +
+L
+�
+i=1
+1 ˚
+X (σi(X), ti).
+(32)
+In order to structure the problem in a from that ADMM can exploit, we introduce the auxiliary variables
+Y ∈ Rm×n and z = [zi]i∈[L] which makes the problem in (32) be,
+min
+X,t,Y,z
+1⊤z +
+1V(Y) +
+L
+�
+i=1
+1 ˚
+X (σi(X), ti),
+s.t.
+X = Y : Λ,
+t = z : θ,
+(33)
+such that Λ, θ are the dual variables associated with X and t respectively. Similar to (11), the Lagrangian function
+associated with (33) augmented with a quadratic penalty for the equality constraint violation with a parameter
+ρ > 0, is
+Lρ(X, Y, t, z, Λ, θ)=1⊤z+
+1M(Y)+
+L
+�
+i=1
+1 ˚
+X (σi(X), ti)
++T r{Λ⊤(X−Y)}+θ⊤(t−z)+ ρ
+2(∥X−Y∥2
+f +∥t−z∥2),
+(34)
+
+12
+where T r{.} is the trace operator. Considering the 2-tuples (X, t) and (Y, z), the ADMM iterations is,
+(X, t)k+1 = argmin
+X,t
+Lρ(X, Yk, t, zk, Λk, θk),
+(35)
+Yk+1 = argmin
+Y
+Lρ(Xk+1, Y, tk+1, zk, Λk, θk),
+(36)
+zk+1 = argmin
+z
+Lρ(Xk+1, Yk+1, tk+1, z, Λk, θk),
+(37)
+Λk+1 = Λk + ρ(Xk+1 − Yk+1),
+(38)
+θk+1 = θk + ρ(tk+1 − zk+1).
+(39)
+A. (X, t) update
+By completing the square and with some simple algebra, it can be shown that the problem in (35) is equivalent
+to
+min
+X,t
+��X − ¯Xk��2
+f +
+��t − ¯tk��2 ,
+s.t.
+|σi(X)|p ≤ ti,
+i ∈ {1, . . . L},
+(40)
+where ¯Xk ∆= Yk − Λk
+ρ
+and ¯tk ∆= zk − θk
+ρ . For an ease of notations, we will drop the iteration index k. Assume
+that X = PΣQ⊤ and ¯X = U∆V⊤ is the singular value decomposition (SVD) of X and ¯X respectively. Where
+Σ, ∆ ∈ RL×L are diagonal matrices with the singular values associated X and ¯X while P, U ∈ Rm×L and
+Q, V ∈ Rn×L are the unitary matrices. By applying the same steps as in Theorem 3 of [45], we can write the first
+term of (40) after dropping k as,
+��X − ¯X
+��2
+f =
+��PΣQ⊤ − U∆V⊤��2
+f
+=
+��PΣQ⊤��2
+f +
+��U∆V⊤��2
+f − 2T {X⊤ ¯X}
+(a)
+= T r{Σ⊤Σ}+T r{∆⊤∆}−2T r{QΣ⊤P⊤U∆V⊤}
+(b)
+≥ T r{Σ⊤Σ}+T r{∆⊤∆}−2T r{Σ⊤∆}=∥Σ−∆∥2
+f ,
+(41)
+where (a) is because P⊤P = Q⊤Q = U⊤U = V⊤V = IL×L with IL×L being an identity matrix of size L,
+and exploiting the circular property of the trace while (b) holds is from the main result of [46]. In order to make
+��X − ¯Xk��2
+f achieve its derived lower bound, we set P = U and Q = V.
+Henceforth, the problem in (40) will be equivalent to,
+min
+X,t
+∥x − ¯x∥2 + ∥t − ¯t∥2 ,
+s.t.
+|xi|p ≤ ti,
+i ∈ {1, . . . L},
+(42)
+where x = [xi]i∈[L] and ¯x = [¯xi]i∈[L] are the vectors of singular values of the matrices X and ¯X respectively.
+The optimal solution X∗ for (40) can be calculated by finding the optimal x∗ of (42) and then X∗ = UΣ∗VT ,
+where Σ∗ =diag(x∗) and diag(.) is an operator that converts a vector to its corresponding diagonal matrix. Since
+the problem in (42) is separable, we drop the index i and only consider solving
+min
+x,t
+(x − ¯x)2 + (t − ¯t)2,
+s.t.
+|x|p ≤ t.
+(43)
+
+13
+It can be realized that (43) is the same as (20), hence, its optimal solution can be found by applying algorithm
+2.
+B. (Y, z) update
+After updating (X, t) while fixing Λ and θ, the problem in (36) can be written as,
+Yk+1 =argmin
+Y
+�����Y−(Xk+1+ Λk
+ρ )
+�����
+2
+f
+, s.t.
+Y ∈ M,
+(44)
+which is clearly a convex optimization problem representing the projection of the point Xk+1 + Λk
+ρ on the set M
+and can be solved by various known class of algorithms as discussed in section III-C.
+Upon updating Y, the z update in (37) is
+zk+1 = argmin
+z
+1⊤z + ρ
+2
+�����z − (tk+1 + θk
+ρ )
+�����
+2
+,
+(45)
+which has the closed-form solution z = tk+1 + θk−1
+ρ
+.
+V. PROXIMAL GRADIENT ALGORITHM
+The SVR algorithm deals with the ℓp relaxation of (2) without assuming any specific structure for V, other than
+being closed and convex. Indeed, the algorithm only requires the Euclidean projections onto V as in (29). However,
+this approach suffers from two pitfalls: 1) high computational complexity per iteration as a result of solving (29)
+in every iteration, and 2) the lack of convergence guarantees.
+In this section, we consider a sub-class of problems with a specific structure for the convex set of the form
+V = {x : f(x) ≤ 0}, where f(x) = ∥Ax − b∥2 − ǫ for some given ǫ ≥ 0, A ∈ Rm×n and b ∈ Rm. Note that f(x)
+is a convex function with Lipschitz continuous gradient. i.e., f is L-smooth: ∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥ for
+all x, y ∈ Rn and L ≜ ∥A∥2. Specifically, in order to solve
+min
+x
+∥x∥p
+p ,
+s.t.
+f(x) ≤ 0,
+(46)
+we aim to develop an efficient algorithm with some convergence guarantees for the following Lagrangian relaxation:
+min
+x
+F(x)
+∆= ∥x∥p
+p + µ
+2 f(x),
+(47)
+where µ ≥ 0 is the dual multiplier that captures the trade-off between solution sparsity and fidelity.
+A canonical problem for the regularized risk minimization has the following form:
+min
+x
+g(x) + h(x)
+(48)
+where h is an L-smooth loss function and g is a a regularizer term. When both g and h are convex, the proximal
+gradient (PG) algorithm [47] can compute a solution to (48) through iteratively taking PG steps, i.e., xk+1 =
+proxg/λ(xk − ∇h(xk)/L) where proxg/λ(.)
+∆= argminx g(x) + λ
+2 ∥x − ·∥2, for some constant λ. When g is
+convex, prox operation is well-defined; thus, the PG step can be computed.
+
+14
+Comparing both (47) and (48), the convexity assumption of g(x) in (48) is not satisfied for ∥x∥p
+p in (47). When
+the regularizer is a continuous nonconvex function, the proximal map proxg/λ may not exist, let alone it can be
+computed in closed form. On the other hand, for ∥x∥p
+p, using similar arguments for the non-convex projection step
+introduced in subsection III-B, we aim to derive an analytical solution that can be computed efficiently. Indeed,
+assuming p ∈ (0, 1) is a positive rational number, the proposed method for computing the proximal map of ∥x∥p
+p
+involves finding the roots of a polynomial of order 2q, where q ∈ Z+ such that p = s/q for some s ∈ Z+.
+Since f is L-smooth, for all x, y ∈ Rn, we have
+f(x) ≤ f(y) + ∇f(y)⊤(x − y) + L
+2 ∥x − y∥2 .
+(49)
+Given xk, replacing f(x) with the upper bound in (49) for y = xk, the prox-gradient operation naturally arises as
+follows:
+xk+1 = argmin
+X
+∥x∥p
+p + µ
+2 [f(xk) + ∇f(xk)⊤(x − xk)
++ L
+2
+��x − xk��2].
+(50)
+By completing the square, (50) yields to
+xk+1 = argmin
+X
+∥x∥p
+p + µL
+4
+����x −
+�
+xk − 1
+L∇f(xk)
+�����
+2
+.
+(51)
+Defining ¯xk ∆= xk − 1
+L∇f(xk), (51) can be rewritten as
+xk+1 = argmin
+X
+∥x∥p
+p + µL
+4
+��x − ¯xk��2
+= argmin
+X
+n
+�
+i=1
+|xi|p + µL
+4 (xi − ¯xk
+i )2,
+(52)
+which is clearly a separable structure in the entries of x. Therefore, for each i ∈ [n], we have
+xk+1
+i
+=argmin
+xi
+|xi|p+ µL
+4 (xi−¯xk
+i )2 = prox¯g/ µL
+2 (¯xk
+i ),
+(53)
+where ¯g : R → R+ such that ¯g(t) = |t|p for some positive rational p ∈ (0, 1).
+Next, we consider a generic form of (53), i.e., given some ¯t ∈ R, we would like to compute
+t∗ = argmin
+t
+{|t|p + µL
+4 (t − ¯t)2}.
+(54)
+The first-order optimality condition for (54) can be written as
+p|t∗|p−1sign(t∗) + µL
+2 (t∗ − ¯t) = 0.
+(55)
+Using similar arguments with those in section III-B for Proposition 1, we can conclude that the optimal solution t∗
+attains the property that sign(t∗) = sign(¯t). Without loss of generality, exploiting the symmetry of the function ¯g,
+we only consider the case when ¯t > 0; hence, the optimal solution t∗ is the smallest positive root of the following
+polynomial:
+p|t∗|p−1 + µL
+2 (t∗ − ¯t) = 0.
+(56)
+
+15
+Algorithm 3 Accelerated PG algorithm
+1: Initialize: µ, s = 1, q = 2, l, x0, x1, k = 1.
+2: repeat
+3:
+yk = xk + k−1
+k+2(xk − xk−1)
+4:
+∆k = maxt=max{1,k−l},...,k F(xt)
+5:
+if F(yk) ≤ ∆k then:
+6:
+vk = yk
+7:
+else:
+8:
+vk = xk
+9:
+¯xk = vk − 1
+L∇f(vk)
+10:
+for i ∈ [n] do:
+11:
+solve a2q − ¯xiaq +
+2s
+qµLas = 0
+12:
+xk+1
+i
+= a∗q
+13:
+k = k + 1
+14: until convergence
+As in (28), suppose 0 < p = s
+q < 1 for some s, q ∈ Z+. Using the change of variables a ≜ (t∗)
+1
+q , (56) reduces to
+finding the roots of a polynomial of degree 2q:
+a2q − ¯taq + 2s
+qµLas = 0.
+(57)
+To efficiently solve (46), we will use Algorithm 3, which is an implementation of nonconvex inexact accelerated
+proximal gradient (APG) descent method proposed in [48, Algorithm 2]. To summarize, [48, Algorithm 2] is
+designed to solve composite problems of the form in (48) assuming that h is L-smooth and g is proper lower-
+semicontinuous such that F ≜ h + g is bounded from below and coercive, i.e., lim∥∥→∞ F() = +∞ – note that
+there is no assumption regarding neither h nor g to be convex. The key points enhancing both practical behavior
+of and theoretical guarantees for [48, Algorithm 2] can be summarized as given below:
+• An extrapolation yk is generated as introduced in [49] for the APG algorithm (step 3).
+• Steps 4 through 9 allow non monotone update of the objective. F(yk) is checked with respect to the maximum
+of the latest l objective values. The gradient step is adjusted according to this (step 9). This permits yk to
+occasionally increase the objective and makes F(yk) be less than the maximum of the objective value of the
+latest l iterations.
+• Steps 11 and 12 are the solution of the PG step using the non-convex projection method.
+In the next part, we show that algorithm 3 converges to a critical point and it exhibits a convergence rate of O( 1
+k),
+where k is the iteration budget that is given to the algorithm.
+
+16
+Definition 3. ( [50]) The Frechet sub-differential of F at x is
+ˆ∂F(x)
+∆=
+�
+u : lim
+y̸=x lim
+y→x
+F(y) − F(x) − u⊤(y − x)
+∥y − x∥
+≥ 0
+�
+.
+(58)
+The sub-differential of F at x is
+∂F(x)
+∆= {u : ∃xk → x, F(xk) → F(x) and uk ∈ ˆ∂F(xk)
+→ u as k → ∞}.
+(59)
+Definition 4. ( [50]) x is a critical point of F if 0 ∈ ∂g(x) + ∇h(x).
+By comparing (48) and (47), it can be realized that the functions g(x) and h(x) in definition 4 are equal to
+∥x∥p
+p and µ
+2 f(x) respectively.
+Theorem 1. The sequence xk generated from algorithm 3 has at least one limit point and all the generated limit
+points are critical points of (47). Moreover, the algorithm converges with rate O( 1
+K ), where K is the iteration
+budget given to the algorithm.
+Proof. It can easily be verified that our problem in (47) satisfies all required assumptions for Algorithm 3. Indeed,
+1) The function g(x) = ∥x∥p
+p is a proper and lower semi-continuous function.
+2) The gradient of h(x) = µ
+2 f(x) is ¯L-Lipschitz smooth, i.e., ∥∇h(x) − ∇h(y)∥ ≤ ¯L ∥x − y∥ for all x, y ∈ Rn,
+with ¯L = µ
+2 L.
+3) F(x)=g(x)+h(x) is bounded from below, i.e., F(x)≥0.
+4) lim∥x∥→∞ F(x) = ∞.
+5) The introduced non-convex projection method is an exact solution for the proximal gradient step. This is
+because it is based on finding the roots of a polynomial of order 2q in equation (57).
+Therefore, the assumptions required for theorem 4.1 for critical point convergence and proposition 4.3 for the rate
+of convergence in [48] are satisfied which then completes the proof.
+Remark 2. The global convergence of several exact iterative methods that solve (48) has been explored, under the
+framework of Kurdyka–Lojasiewicz (KL) theory, in various additional literature including [50]–[54]. Other work
+(see [55] and references therein) considered the linear convergence of non-exact algorithms with relaxations on
+the assumptions of KL theory, however, it is difficult to verify that the sequence generated by algorithm 3 satisfies
+the relaxed assumptions stated in [55].
+VI. NUMERICAL RESULTS FOR SVR PROBLEM
+In this section, we present two numerical examples for the p-quasi-norm ADMM (pQN-ADMM) from algorithm 1
+and the non-convex projection from algorithm 2. For both examples, the pQN-ADMM algorithm result is compared
+with the ℓ1 objective function solution from MOSEK solver [56]. The two examples include; i) Sparse signal
+reconstruction from noisy measurements, where the pQN-ADMM algorithm is also compared with another ℓ0.5
+quasi-norm minimization based algorithm, named ℓ0.5-FL, described in [57]. ii) Binary classification using support
+vector machines (SVM).
+
+17
+10-4
+10-3
+10-2
+10-1
+100
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0.26
+Fig. 1: Effect of noise variance on the sparsity of solutions obtained by pQN-ADMM algorithm,
+ℓ0.5-FL algorithm and ℓ1 norm minimization.
+A. Sparse Signal Reconstruction
+Let n = 210 and m = n/4, randomly construct the sparse binary matrix, M ∈ Rm× n
+2 , with a few number of
+ones in each column. The number of ones in each column of M is generated independently and randomly in the
+range of integers between 10 and 20, and their locations are randomly chosen independently for each column. Let
+U = [M, −M], which is the vertical concatenation of the matrix M and its negative. Following the same setup in
+[58], the column orthogonality in U is not satisfied. Let xopt ∈ Rn be a reference signal with ∥xopt∥0 = ⌈0.2n⌉,
+where the non-zero locations are chosen uniformly at random with the values following a zero mean, unit variance
+Gaussian distribution. Let v = Uxopt+n be the allowable measurement, where n ∈ Rm is a Gaussian random vector
+with zero mean and co-variance matrix σ2Im×m, where I is the identity matrix. The sparse vector is reconstructed
+from v by solving (7) with V = {x : ∥Ux − v∥/∥v∥ − ǫ ≤ 0}. Figure 1 plots the relation between the sparsity
+level and the noise variance for ℓ1 norm minimization, ℓ0.5-FL quasi-norm and pQN-ADMM solutions. A threshold
+value of 10−6 was used where the threshold is a value below which the entry of the solution vector is considered to
+be zero. Depending on the noise variance σ2, the value of ǫ was chosen to make the problem feasible. The reported
+result is the average of 100 independent random runs. It can be realized that pQN-ADMM algorithm produces a
+sparser solution than its counter baselines for different values of σ2. On increasing σ2, the sparsity level for all
+methods decreases. This is due to the increased scarcity of information on the original signal in the realization
+vector which makes the reconstruction process less accurate.
+
+18
+B. Binary Classification
+In this part, we build an email spam classifier based on support vector machines. We use a subset of the
+training set used in the SpamAssassin Public Corpus [59]. Let {(uj, vj)}j∈[m] be the training set of feature vectors
+uj ∈ {0, 1}n with corresponding labels vj ∈ {−1, 1} identifying whether the email is spam or not. We highlight
+the effectiveness of our method in designing an email spam detector using the least number of words. Following
+[60], we maintain a dictionary of n = 1899 words. For a given email j ∈ [m], the ith entry of uj is 1 if word
+wi, i ∈ [n] of the dictionary is in email j, and is 0 otherwise. We aim to build a linear classifier with the decision
+rule ˆv = sign(u⊤x), where u is the feature vector of the email in question and x is a vector of the classifier
+coefficients with the first entry being the bias term. The main aim is to build a classifier that detects whether an
+email is a spam or not, using the least number of words from the dictionary and achieving a high training data
+accuracy. To achieve this objective, we solve (7) with M = {x : 1
+m
+�
+j∈[m]
+�
+1 − vju⊤
+j x
+�+ − ǫ ≤ 0}.
+It can be clearly realized that the training set accuracy is controlled by ǫ. Algorithm 1 was run for p = 0.5, 2000
+training emails and various values for ǫ. For each value of ǫ, the algorithm was terminated after 100 iterations and
+performance tested on 1000 emails. For comparison purpose, the problem was also solved with the ℓ1 norm convex
+relaxation under the same setup. In figure 2a, we plot the number of non-zero entries in the optimal classifier from
+both the pQN-ADMM and ℓ1 solutions vs different values of ǫ.
+We used a threshold of 10−4, where the threshold is defined as in section VI-A. It can be realized from figure 2a
+that the pQN-ADMM solution outperforms the ℓ1 in terms of the number of words used for legitimacy detection.
+When the value of ǫ increases, the number of required words decreases for both ℓ0.5 and ℓ1 problems. This outlines
+the trade-off between the sparsity level of the classifier and its accuracy, i.e., small values of ǫ enforces a low
+classification error in expense of a less sparse solution. The corresponding training and test set accuracies for the
+obtained classifiers are plotted in Fig. 2b. Both figures 2a and 2b depict the performance of the pQN-ADMM
+solution from algorithm 1 in terms of the sparsity level while maintaining nearly the same level of accuracy as the
+ℓ1 solution for both the training and test sets.
+VII. NUMERICAL RESULTS FOR RMP PROBLEM
+A. Time domain system identification
+In this part, we apply the derived pQN-ADMM approach on a time domain system identification example. In
+that example, input is applied to randomly generated systems with a known order. Using the outputs corresponding
+to these systems, the minimum rank/order system is derived and results are compared to nuclear norm heuristic in
+[26].
+We consider a discrete time stable Single Input Single Output (SISO) system with an input u ∈ RT, where T
+represents the number of input samples, i.e., input time span. We assume an impulse response of a fixed number
+of samples n. The corresponding system output is y ∈ Rm. However, we assume that only noisy realizations, ˆy, of
+the output can be considered, such that; ˆy
+∆= y + z = h ⊛ u + z , where h ∈ Rn is the system’s original impulse
+response, z ∈ Rm is a random vector with entries drawn independently from samples of a uniform distribution
+on the range [−0.25, 0.25], i.e., zi ∼ U[−0.25, 0.25], while ⊛ denotes the convolution operator. From the window
+
+19
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+50
+100
+150
+200
+250
+Number of selected words
+(a) Number of words selected for classification
+versus ǫ for pQN-ADMM and ℓ1 norm.
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+65
+70
+75
+80
+85
+90
+95
+100
+Correct percentage 
+(b) Training and test set accuracies versus ǫ
+for pQN-ADMM and ℓ1 norm.
+Fig. 2: Binary classification numerical results.
+property of the convolution, m = n + T − 1. Assume that ui, hi and yi are the ith components of the vectors u, h
+and y respectively. The three components are related to each other by convolution through yi = �∞
+j=−∞ hjui−j
+which is a linear relation. Hence, let T ∈ Rm×n be the Toeplitz matrix formed by the input u , it can be easily
+seen that h ⊛ u = hT ⊤. Assume that x ∈ Rn is an impulse response variable and let X ∈ Rn×n be a Hankel
+matrix formed by the entries of x. From [29], [61]–[63], the minimum order time domain system identification
+problem can be formulated as,
+min
+x,X
+RankX,
+(60a)
+s.t.
+X = Hankel(x),
+(60b)
+��ˆy − xT ⊤��2 ≤ ǫ,
+(60c)
+(60b) ensures that X is a Hankel matrix and (60c) holds to make the result by applying the input, u, to the optimal
+impulse response, x, fit the available noisy data, ˆy, in a non-trivial sense. Defining the convex set C
+∆={X∈Rn×n :
+��ˆy−hT ⊤��2−ǫ ≤ 0, X=Hankel(x)}, (60) can be cast as,
+min
+x,X
+Rank(X),
+s.t.
+X ∈ C,
+(61)
+which is clearly identical to the problem in (1). The problem was solved using the same pQN-ADMM approach
+discussed in section IV.
+We let T = m = 50 and n = 40. Note that m < T + n − 1, which is a reasonable assumption as in some
+practical applications, one is allowed only a specific window to realize the output. We consider the simulation for
+10 different original system orders, i.e., η = 2 : 2 : 10. An input vector, u, is generated, where the elements of u
+are independent and follow a uniform distribution on the interval [−5, 5]. For each η; 1) 50 random stable systems
+are generated using the command ’drss’ in MATLAB. 2) The generated input is applied to each system to get the
+corresponding noisy output ˆy. 3) Given the output ˆy, the problem in (60) is solved and the corresponding system’s
+
+20
+2
+4
+6
+8
+10
+Original system order
+0
+5
+10
+15
+20
+Average rank
+Threshold=10-4
+2
+4
+6
+8
+10
+Original system order
+0
+5
+10
+15
+20
+Average rank
+Threshold=10-5
+Fig. 3: Average rank vs original system order. Red and cyan colors are for the nuclear norm and
+pQN-ADMM algorithm respectively.
+η=2
+η=6
+η=10
+Nuclear norm
+2.3907
+6.6668
+7.2572
+pQN-ADMM
+0.5292
+0.9042
+1.0861
+TABLE I: Standard deviation for threshold=10−4
+rank is calculated using singular value decomposition. 4) The results are averaged out to get the corresponding
+average rank to each original η.
+Figure 3 shows the average rank for the the nuclear norm and pQN-ADMM heuristics. The results are for
+two different values of thresholds, where the threshold is defined as the value below which the singular value is
+considered to be zero. It can be realized that the introduced pQN-ADMM approach outperforms the nuclear norm
+one for both values of thresholds. Moreover, when the threshold value decreases from 10−4 to 10−5, the behavior
+of the pQN-ADMM remains the same. However, the average rank for the nuclear norm increases. This proves the
+robustness of the derived pQN-ADMM in comparison to the nuclear norm one. Tables I and II show the standard
+deviation of the algorithms. It can be seen that the standard deviation is the same for the pQN-ADMM when
+η=2
+η=6
+η=10
+Nuclear norm
+6.9877
+11.2638
+11.7854
+pQN-ADMM
+0.5325
+0.9113
+1.0861
+TABLE II: Standard deviation for threshold=10−5
+
+21
+changing the threshold, however, it increases for the nuclear norm as the threshold value decreases.
+B. Matrix Completion Example
+In this section, we apply our algorithm (pQN-ADMM) to a matrix completion example and compare the
+result to the matrix iterative re-weighted least squares (MatrixIRLS) [64], [65], truncated iterative re-weighted
+unconstrained Lq (tIRucLq) [34] and iterative re-weighted least squares (sIRLS-p & IRLS-p) [66] algorithms. The
+matrix completion problem is a special case of the low rank minimization where a linear transform takes a few
+random entries of an ambiguous matrix X ∈ Rm×n. Given only these entries, the goal is to approximate X and
+find the missing ones. The matrix completion problem with low rank recovery can be approximated by,
+min
+X
+∥X∥p
+p ,
+s.t.
+∥A(X) − b∥ ≤ ǫ,
+(62)
+where A : Rm×n → Rq is a linear map with q ≪ mn and b ∈ Rq. In order to apply the mentioned algorithms, the
+linear transform A(X) will be rewritten as Avec(X), where A ∈ Rq×mn and vec(X) ∈ Rmn is a vector formed
+by stacking the columns of the matrix X.
+A random matrix M ∈ Rm×n with rank r is created using the following method: 1) M = MLM⊤
+R, where
+ML ∈ Rm×r and MR ∈ Rn×r. 2) The entries of both ML and MR are i.i.d Gaussian random variables with zero
+mean and unit variance. Let ˆ
+M = M + Z, where Z ∈ Rm×n is a Gaussian noise with each entry being an i.i.d
+Gaussian random variable with zero mean and variance σ2. The vector b is then created by selecting random q
+elements from vec( ˆ
+M). Since b = Avec( ˆ
+M), one can easily construct the matrix A which is a sparse matrix where
+each row is composed of a value 1 at the index of the corresponding selected entry in the vector b while the rest
+are zeros. We set m = n = 100, r = 5 and p = 0.5. Let dr = r(m+n−r) denotes the dimension of the set of rank
+r matrices and define s =
+q
+mn as the sampling ratio. We assume that s = 0.195 which yields to q = 1950. It can
+be realized that dr
+q < 1. We set σ = 0.1 and let the algorithms terminate if a budget of 1000 iterations is reached.
+In order to compare the results from different algorithms, we consider the average of 50 runs for two measures: a)
+the relative Frobenius distance (RFD) to the matrix M, b) the relative error to singular (REtS) values of M.
+In figures 4a and 4b, we report the average RFD and REtS values for all the algorithms. Despite that all the
+baselines are designed to exploit the specific structure of the matrix completion problem, described in (62), while
+the proposed pQN-ADMM doesn’t, it is competitive against them all. This in turns shows the effectiveness of the
+pQN-ADMM algorithm in solving the rank minimization problems without requiring any prior information about
+the structure of the associated convex set.
+VIII. NUMERICAL RESULTS FOR THE NONCONVEX ACCELERATED PROXIMAL
+GRADIENT (APG) ALGORITHM
+In this subsection, we present numerical results for the APG method, displayed in Algorithm 3. Following the
+same procedure in [67], we first generate the target signal x∗ through
+x∗
+i =
+
+
+
+Θ(1)
+i 103Θ(2)
+i ,
+∀ i ∈ Λ,
+0,
+∀ i ∈ [n] \ Λ;
+(63)
+
+22
+0
+200
+400
+600
+800
+1000
+Number of iterations
+10-2
+10-1
+100
+IRLS-p
+sIRLS-p
+MatrixIRLS
+tIRucLq
+pQN-ADMM
+(a) RFD to M.
+1
+2
+3
+4
+5
+Index of singular value
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+IRLS-p
+sIRLS-p
+MatrixIRLS
+tIRucLq
+pQN-ADMM
+(b) REtS values of M.
+Fig. 4: The RFD and REtS average values.
+where the design parameters Λ ⊂ [n], and Θ(1)
+i , Θ(2)
+i
+for i ∈ Λ are chosen as follows:
+1) the index set Λ ⊂ [n] is constructed by selecting a subset of [n] with cardinality s uniformly at random;
+2) {Θ(1)
+i }i∈Λ are independent, identically distributed (IID) Bernoulli random variables taking values ±1 with
+equal probability;
+3) {Θ(2)
+i }i∈Λ are IID uniform [0, 1] random variables.
+The measurement matrix A ∈ Rm×n is a partial Discrete Cosine Transform (DCT) matrix with rows correspond-
+ing to m < n frequency, where these m indices are chosen uniformly at random from [n]. The noisy measurement
+vector b ∈ Rm is then set to be b = A(x∗ + ǫ1) + ǫ2, where ǫ1 ∼ N(0, σ2
+1) and ǫ2 ∼ N(0, σ2
+2) are the input and
+realization noises.
+In our experiments, n = 4096, s = ⌈0.5m⌉ and the PG algorithm memory to 5, i.e., l = 5. Following the
+medium noise setup in [68], we set σ1 = 0.005, σ2 = 0.001.
+For f(x) = ∥Ax − b∥2, we have L = 2∥A∥2. We perform our experiment for various values of m, i.e., number
+of noisy measurements, and µ, i.e., trade-off parameter, see (47). For each (m, µ) selection, in order to capture
+the inherent statistical variation of the problem, we generate 20 random instances of the triplet (x∗, A, b) and each
+random instance is solved by Algorithm 3. We reported the average performance. We terminated Algorithm 3 when
+the relative error between consecutive iterates satisfies
+��xk − xk−1�� /
+��xk−1�� ≤ 10−5 for the first time.
+In our experiments, we compared solving (47) for p = 0.5 against p = 1, i.e., against ℓ1-optimization for sparse
+recovery. On one hand, when p = 0.5, i.e., for ℓ0.5 minimization, we solve (47) using Algorithm 3, called ℓ0.5 exact,
+and using the algorithm 2 of [69], which we call ℓ0.5 approx. On the other hand, when p = 1, ℓ1-minimization
+problem is a convex one and we adopt the FISTA algorithm of [70]. The solution is denoted by ¯x while the target
+signal, from (63), by x∗. In Algorithm 3, x0 is set to a zero vector while x1 is the ℓ1 norm solution.
+Figures 5 and 6 highlight the relation between the average error and sparsity vs µ for different values of n/m. It
+can be realized that the average error (sparsity) decreases (increases) on increasing µ. For small values of µ, more
+weight is given to the loss function, which emphasizes the ℓ0 quasi-norm minimization, and hence the sparsity level,
+
+23
+Fig. 5: Average error vs µ for different values of n/m. Yellow and cyan shades are the standard
+deviations for the exact and approximate ℓ0.5 quasi-norms respectively.
+as in figure 6, is low. However, for high values of µ, more weight is assigned to the minimization of the regularization
+term, which solves ∥Ax − b∥2, and hence the error decreases, as shown in figure 3, with a corresponding increase
+in the sparsity. It can be realized that the ℓ0.5 solutions always outperforms the ℓ1 one with very slight difference
+between the exact and the approximate ones.
+Figure 7 highlights the statistics of the number of iterations used until convergence for both the ℓ0.5 exact and
+approximate algorithms. It can be realized that with a sufficient number of available realizations, n/m = 8 and
+n/m = 16, both algorithms approximately consume the same number of iterations. However, when the number
+of available realizations decreases, n/m = 32 and higher, our exact proximal solution requires significantly less
+number of iterations to converge. This conclusion, along with figures 5 and 6 findings, indicates that our algorithm
+not only finds a similar solution to the approximate method, but also converges with a fewer number of iterations.
+
+n/m=8
+n/m=16
+n/m=32
+0
+0
+0
+l1
+l1
+-1
+- l0.5 exact
+-1
+lo0.5 exact
+-1
+- l0.5 exact
+l0.5 approx
+l0.5approx
+l0.5 approx
+2
+-2
+-2
+all
+-3
+-3
+-3
+-4
+-4
+5
+-5
+-5
+-6
+-6
+-6
+-7
+-7
+-7
+0
+10
+20
+30
+0
+10
+20
+30
+0
+10
+20
+30
+u
+n/m=64
+n/m=128
+n/m=256
+0
+0
+l1
+l1
+-1
+lo.5exact
+-1
+l0.5 exact
+-1
+l0.5 exact
+l0.5approx
+0.5 approx
+0.5approx
+-2
+-2
+-2
+-3
+gl
+-3
+3
+-4
+-4
+4
+-5
+-5
+.5
+-6
+-6
+-6
+-7
+-7
+-7
+0
+10
+20
+30
+0
+10
+20
+30
+0
+10
+20
+30
+u24
+Fig. 6: Sparsity vs µ for different values of n/m. Yellow and cyan shades are the standard
+deviations for the exact and approximate ℓ0.5 quasi-norms respectively.
+IX. CONCLUSION
+In this study, we presented a non-convex ADMM algorithm (pQN-ADMM) to solve the ℓp norm minimization
+problem. The algorithm has a similar complexity to that of the ℓ1 minimization in addition to solving the roots of a
+polynomial for the non-convex projection. Our algorithm can also be considered as a general procedure for solving
+ℓp problems as no specific structure for the convex constraint set was assumed and a convex projection on that set
+was done for variables update. Applying sparse signal recover and binary classification examples, our method was
+found to outperform the ℓ1 minimization in terms of the sparsity of the generated solution. In addition, we studied
+the problem of solving a non-convex relaxation of RMPs using Schatten-p quasi-norm. This relaxation was shown
+to be the ℓp minimization of the singular values of the variable matrix and hence the primary developed algorithm
+could be used. Showing the numerical results, the pQN-ADMM was found to be less sensitive to the threshold
+decrease in time domain system identification problems. Additionally, the pQN-ADMM method was shown to be
+
+n/m=8
+n/m=16
+n/m=32
+5
+5
+5
+l1
+lo.5 exact
+l0.5exact
+4.
+l0.5 exact
+C0.5approx
+0.5 approx
+-0.5 approx
+3
+3
+3
+x*lo
+2
+区
+2
+[x*
+2
+区
+1
+0
+0
+0
+-1
+-1
+-1
+0
+10
+20
+30
+0
+10
+20
+30
+0
+10
+20
+30
+μ
+n/m=64
+n/m=128
+n/m=256
+6
+6
+10
+5
+l0.5 exact
+5
+lo.5exact
+8
+lo0.5 exact
+L0.5approx
+0.5approx
+0.5 approx
+4
+4
+6
+[x o
+3
+3
+[x*10
+重
+[x*|
+4
+2
+区
+2
+2
+0
+0
+0
+-1
+-2
+7
+0
+10
+20
+30
+0
+10
+20
+30
+0
+10
+20
+30
+u25
+Fig. 7: Iterations count vs µ for different values of n/m.
+competitive against various other baselines when solving the matrix completion problem.
+REFERENCES
+[1] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, no. 1, pp. 49–95, 1996.
+[2] E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12,
+pp. 4203–4215, 2005.
+[3] J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE
+Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210–227, 2009.
+[4] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete
+frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006.
+[5] D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006.
+[6] A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of
+signals and images,” SIAM review, vol. 51, no. 1, pp. 34–81, 2009.
+
+n/m=8
+n/m=16
+n/m=32
+6000
+6000
+6000
+l0.5 exact
+l0.5 exact
+l0.5 exact
+5000
+0.5 approx
+5000
+l0.5approx
+5000
+0.5approx
+count
+4000
+count
+4000
+count
+4000
+Iterations 
+erations
+Iterations
+3000
+3000
+3000
+2000
+2000
+2000
+1000
+1000
+1000
+0
+0
+0
+0
+10
+20
+30
+0
+10
+20
+30
+0
+10
+20
+30
+u
+n/m=64
+n/m=128
+n/m=256
+6000
+6000
+6000
+l0.5 exact
+l0.5 exact
+l0.5 exact
+5000
+l0.5approx
+5000
+l0.5 approx
+5000
+lo.5 approx
+A
+Iterations count
+4000
+4000
+count
+4000
+rations
+3000
+3000
+rations
+3000
+2000
+2000
+2000
+1000
+1000
+1000
+0
+0
+0
+0
+10
+20
+30
+0
+10
+20
+30
+0
+10
+20
+30
+u
+u26
+[7] J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proceedings of the
+IEEE, vol. 98, no. 6, pp. 948–958, 2010.
+[8] S. G. Mallat and Zhifeng Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Transactions on Signal
+Processing, vol. 41, no. 12, pp. 3397–3415, 1993.
+[9] J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Transactions on Information Theory,
+vol. 50, no. 10, pp. 2231–2242, 2004.
+[10] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM review, vol. 43, no. 1,
+pp. 129–159, 2001.
+[11] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society: Series B
+(Methodological), vol. 58, no. 1, pp. 267–288, 1996.
+[12] R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Processing Letters, vol. 14,
+no. 10, pp. 707–710, 2007.
+[13] R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in 2008 IEEE International
+Conference on Acoustics, Speech and Signal Processing, pp. 3869–3872, IEEE, 2008.
+[14] D. P. Wipf and B. D. Rao, “Sparse bayesian learning for basis selection,” IEEE Transactions on Signal Processing, vol. 52,
+no. 8, pp. 2153–2164, 2004.
+[15] P. Schniter, L. C. Potter, and J. Ziniel, “Fast bayesian matching pursuit,” in 2008 Information Theory and Applications
+Workshop, pp. 326–333, 2008.
+[16] A. Miller, Subset selection in regression. CRC Press, 2002.
+[17] R. Saab, R. Chartrand, and O. Yilmaz, “Stable sparse approximations via nonconvex optimization,” in 2008 IEEE
+International Conference on Acoustics, Speech and Signal Processing, pp. 3885–3888, 2008.
+[18] R. Chartrand, “Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data,” in 2009 IEEE
+International Symposium on Biomedical Imaging: From Nano to Macro, pp. 262–265, 2009.
+[19] N. Mourad and J. P. Reilly, “Minimizing nonconvex functions for sparse vector reconstruction,” IEEE Transactions on
+Signal Processing, vol. 58, no. 7, pp. 3485–3496, 2010.
+[20] R. Chartrand and B. Wohlberg, “A nonconvex ADMM algorithm for group sparsity with sparse groups,” in 2013 IEEE
+International Conference on Acoustics, Speech and Signal Processing, pp. 6009–6013, 2013.
+[21] Z. Xu, X. Chang, F. Xu, and H. Zhang, “l1/2 regularization: A thresholding representation theory and a fast solver,” IEEE
+Transactions on Neural Networks and Learning Systems, vol. 23, no. 7, pp. 1013–1027, 2012.
+[22] J. Zeng, S. Lin, Y. Wang, and Z. Xu, “l1/2 regularization: Convergence of iterative half thresholding algorithm,” IEEE
+Transactions on Signal Processing, vol. 62, no. 9, pp. 2317–2329, 2014.
+[23] G. Li and T. K. Pong, “Global convergence of splitting methods for nonconvex composite optimization,” SIAM Journal
+on Optimization, vol. 25, no. 4, pp. 2434–2460, 2015.
+[24] Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” Journal of Scientific
+Computing, vol. 78, no. 1, pp. 29–63, 2019.
+[25] M. Mesbahi, “On the semi-definite programming solution on the least order dynamic output feedback synthesis,” 1999.
+[26] M. Fazel, H. Hindi, and S. P. Boyd, “A rank minimization heuristic with application to minimum order system
+approximation,” in Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148), vol. 6, pp. 4734–4739
+vol.6, 2001.
+[27] M. Fazel, H. Hindi, and S. P. Boyd, “Log-det heuristic for matrix rank minimization with applications to hankel and
+euclidean distance matrices,” in Proceedings of the 2003 American Control Conference, 2003., vol. 3, pp. 2156–2162
+vol.3, 2003.
+
+27
+[28] M. Fazel, H. Hindi, and S. Boyd, “Rank minimization and applications in system theory,” in Proceedings of the 2004
+American Control Conference, vol. 4, pp. 3273–3278 vol.4, 2004.
+[29] K. Mohan and M. Fazel, “Reweighted nuclear norm minimization with application to system identification,” in Proceedings
+of the 2010 American Control Conference, pp. 2953–2959, 2010.
+[30] S. Gu, L. Zhang, W. Zuo, and X. Feng, “Weighted nuclear norm minimization with application to image denoising,” in
+2014 IEEE Conference on Computer Vision and Pattern Recognition, pp. 2862–2869, 2014.
+[31] F. Nie, H. Huang, and C. Ding, “Low-rank matrix recovery via efficient Schatten p-norm minimization,” in Twenty-sixth
+AAAI conference on artificial intelligence, 2012.
+[32] F. Nie, H. Wang, H. Huang, and C. Ding, “Joint schatten p-norm and lp norm robust matrix completion for missing value
+recovery,” Knowledge and Information Systems, vol. 42, no. 3, pp. 525–544, 2015.
+[33] L. Liu, W. Huang, and D.-R. Chen, “Exact minimum rank approximation via Schatten p-norm minimization,” Journal of
+Computational and Applied Mathematics, vol. 267, pp. 218–227, 2014.
+[34] M.-J. Lai, Y. Xu, and W. Yin, “Improved iteratively reweighted least squares for unconstrained smoothed ℓq minimization,”
+SIAM Journal on Numerical Analysis, vol. 51, no. 2, pp. 927–957, 2013.
+[35] R. Chartrand, “Nonconvex splitting for regularized low-rank + sparse decomposition,” IEEE Transactions on Signal
+Processing, vol. 60, no. 11, pp. 5810–5819, 2012.
+[36] M. D. Gupta and S. Kumar, “Non-convex p-norm projection for robust sparsity,” in 2013 IEEE International Conference
+on Computer Vision, pp. 1593–1600, 2013.
+[37] S. Bahmani and B. Raj, “A unifying analysis of projected gradient descent for ℓp-constrained least squares,” Applied and
+Computational Harmonic Analysis, vol. 34, no. 3, pp. 366–378, 2013.
+[38] M. E. Ashour, C. M. Lagoa, and N. S. Aybat, “Lp quasi-norm minimization,” in 2019 53rd Asilomar Conference on
+Signals, Systems, and Computers, pp. 726–730, 2019.
+[39] S. Boyd, N. Parikh, and E. Chu, Distributed optimization and statistical learning via the alternating direction method of
+multipliers. Now Publishers Inc, 2011.
+[40] C. Helmberg and F. Rendl, “A spectral bundle method for semidefinite programming,” SIAM Journal on Optimization,
+vol. 10, no. 3, pp. 673–696, 2000.
+[41] A. Beck and M. Teboulle, “Mirror descent and nonlinear projected subgradient methods for convex optimization,”
+Operations Research Letters, vol. 31, no. 3, pp. 167–175, 2003.
+[42] Y. Nesterov and A. Nemirovskii, Interior-point polynomial algorithms in convex programming. SIAM, 1994.
+[43] A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization: analysis, algorithms, and engineering applications.
+SIAM, 2001.
+[44] Y. Xu and W. Yin, “A globally convergent algorithm for nonconvex optimization based on block coordinate update,”
+Journal of Scientific Computing, vol. 72, no. 2, pp. 700–734, 2017.
+[45] Z. Zha, X. Zhang, Y. Wu, Q. Wang, X. Liu, L. Tang, and X. Yuan, “Non-convex weighted lp nuclear norm based ADMM
+framework for image restoration,” Neurocomputing, vol. 311, pp. 209–224, 2018.
+[46] L. Mirsky, “A trace inequality of John von Neumann,” Monatshefte f¨ur mathematik, vol. 79, no. 4, pp. 303–306, 1975.
+[47] N. Parikh and S. Boyd, “Proximal algorithms,” Foundations and Trends in optimization, vol. 1, no. 3, pp. 127–239, 2014.
+[48] Q. Yao, J. T. Kwok, F. Gao, W. Chen, and T.-Y. Liu, “Efficient inexact proximal gradient algorithm for nonconvex
+problems,” arXiv preprint arXiv:1612.09069, 2016.
+[49] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM journal
+on imaging sciences, vol. 2, no. 1, pp. 183–202, 2009.
+
+28
+[50] H. Attouch, J. Bolte, and B. F. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: proximal
+algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Mathematical Programming, vol. 137,
+no. 1, pp. 91–129, 2013.
+[51] H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, “Proximal alternating minimization and projection methods for
+nonconvex problems: An approach based on the Kurdyka-Lojasiewicz inequality,” Mathematics of operations research,
+vol. 35, no. 2, pp. 438–457, 2010.
+[52] J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth
+problems,” Mathematical Programming, vol. 146, no. 1, pp. 459–494, 2014.
+[53] M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for
+nonsmooth optimization,” SIAM Journal on Optimization, vol. 23, no. 2, pp. 1126–1153, 2013.
+[54] P. Tseng and S. Yun, “A coordinate gradient descent method for nonsmooth separable minimization,” Mathematical
+Programming, vol. 117, no. 1, pp. 387–423, 2009.
+[55] Y. Hu, C. Li, K. Meng, and X. Yang, “Linear convergence of inexact descent method and inexact proximal gradient
+algorithms for lower-order regularization problems,” Journal of Global Optimization, vol. 79, no. 4, pp. 853–883, 2021.
+[56] M. ApS, The MOSEK optimization toolbox for MATLAB manual. Version 9.0., 2019.
+[57] S. Foucart and M.-J. Lai, “Sparsest solutions of under-determined linear systems via ℓq-minimization for 0 < q ≤ 1,”
+Applied and Computational Harmonic Analysis, vol. 26, no. 3, pp. 395–407, 2009.
+[58] D. Ge, X. Jiang, and Y. Ye, “A note on the complexity of ℓp minimization,” Mathematical programming, vol. 129, no. 2,
+pp. 285–299, 2011.
+[59] SpamAssassin Public Corpus, http://spamassassin.apache.org.
+[60] Andrew Ng, Machine learning MOOC, https://www.coursera.org/learn/machine-learning.
+[61] M. Sznaier, M. Ayazoglu, and T. Inanc, “Fast structured nuclear norm minimization with applications to set membership
+systems identification,” IEEE Transactions on Automatic Control, vol. 59, no. 10, pp. 2837–2842, 2014.
+[62] Z. Liu and L. Vandenberghe, “Semidefinite programming methods for system realization and identification,” in Proceedings
+of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference,
+pp. 4676–4681, 2009.
+[63] M. Fazel, T. K. Pong, D. Sun, and P. Tseng, “Hankel matrix rank minimization with applications to system identification
+and realization,” SIAM Journal on Matrix Analysis and Applications, vol. 34, no. 3, pp. 946–977, 2013.
+[64] C. K¨ummerle and C. Mayrink Verdun, “Escaping saddle points in ill-conditioned matrix completion with a scalable
+second order method,” in Workshop on Beyond First Order Methods in ML Systems at the 37th International Conference
+on Machine Learning, 2020.
+[65] C. K¨ummerle and C. Mayrink Verdun, “A scalable second order method for ill-conditioned matrix completion from few
+samples,” in International Conference on Machine Learning (ICML), 2021.
+[66] K. Mohan and M. Fazel, “Iterative reweighted algorithms for matrix rank minimization,” Journal of Machine Learning
+Research, vol. 13, no. 110, pp. 3441–3473, 2012.
+[67] N. S. Aybat and G. Iyengar, “A first-order augmented Lagrangian method for compressed sensing,” SIAM Journal on
+Optimization, vol. 22, no. 2, pp. 429–459, 2012.
+[68] E. T. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for l1-regularized minimization with applications
+to compressed sensing,” CAAM TR07-07, Rice University, vol. 43, p. 44, 2007.
+[69] C. O’Brien and M. D. Plumbley, “Inexact proximal operators for ℓp-Quasi-norm minimization,” in 2018 IEEE International
+Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4724–4728, 2018.
+
+29
+[70] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM journal
+on imaging sciences, vol. 2, no. 1, pp. 183–202, 2009.
+

StE3T4oBgHgl3EQfzQsi/content/tmp_files/2301.04726v1.pdf.txt

@@ -0,0 +1,2041 @@
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR
+COGNATE SEQUENCES
+G.-S. CHEON1, T. FORGÁCS2, AND K. TRAN2
+Abstract. Given a Sheffer sequence of polynomials, we introduce the notion of an associated
+sequence called the cognate sequence. We study the relationship between the zeros of this pair
+of associated sequences and show that in case of an Appell sequence, as well as a more general
+family of Sheffer sequences, the zeros of the members of each sequence (for large n) are either
+real, or lie on a line Re z = c. In addition to finding the zero locus, we also find the limiting
+probability distribution function of such sequences.
+MSC: 05A15, 05A40, 30C15, 30E15
+Key words: Sheffer sequence, cognate sequence, zero locus, limiting distribution
+1. Introduction
+Sequences of polynomials play a fundamental role in several fields of mathematics, including
+enumerative combinatorics, functional analysis, applied mathematics, and differential equations.
+Polynomial sequences have been studied extensively from many different points of view [3, 8]. Some
+of the aspects recent research has focused on include their explicit formulas, generating functions,
+recurrence relations, and zero distributions.
+By a Sheffer sequence [12, 13] we shall mean a polynomial sequence indexed by the nonnegative
+integers 0, 1, 2, . . ., in which the index of each polynomial equals its degree, satisfying conditions
+related to the umbral calculus [11] in combinatorics. In this paper, given a Sheffer sequence we
+introduce the notion of its cognate sequence, and study zeros of the cognate sequence of certain
+Sheffer sequences. This direction of research is largely motivated by polynomial pairs defined using
+recurrence relations, such as the Lucas polynomial sequences [4, 7] for example.
+A Sheffer sequence {Gn(s)}∞
+n=0 is characterized by its exponential generating function
+∞
+�
+n=0
+Gn(s)zn
+n! = g(z)esf(z)
+for some (formal) power series g and f in the variable z satisfying the conditions g(0) ̸= 0, f(0) =
+0 and f ′(0) ̸= 0.
+By convention, we call {Gn(s)}∞
+n=0 the Sheffer sequence for the pair (g, f).
+In particular, the Sheffer sequence for a pair (g, az) with a constant a ̸= 0 is called an Appell
+sequence. There are a number of classical polynomial sequences that are Appell sequences, including
+the Bernoulli polynomials Bn(s) for the pair (
+z
+ez−1, z), the Euler polynomials En(s) for the pair
+G.-S. Cheon was partially supported by the National Research Foundation of Korea (NRF) grant funded by the
+Korean government (MSIP) (2016R1A5A1008055 and 2019R1A2C1007518).
+The third author thanks the organizers and participants of the workshop on Optimal Point Configurations on
+Manifolds hosted by the Erwin Schrödinger International Institute for Mathematics and Physics.
+1
+arXiv:2301.04726v1  [math.CV]  11 Jan 2023
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+2
+(
+2
+ez+1, z), and the Hermite polynomials Hn(s) for the pair (e−z2, 2z).
+Sheffer sequences form a group called the Sheffer group with the operation of umbral composition,
+defined as follows (see [6]). Suppose {Gn(s)}∞
+n=0 and {Hn(s)}∞
+n=0 are Sheffer sequences for the pairs
+(g, f) and (h, ℓ) respectively, given by
+Gn(s) =
+n
+�
+k=0
+an,ksk
+and
+Hn(s) =
+n
+�
+k=0
+bn,ksk.
+(1.1)
+Then the umbral composition of Gn(s) with Hn(s), denoted by Gn ◦ Hn(s), is the sequence of
+polynomials defined by
+Gn ◦ Hn(s) =
+n
+�
+k=0
+an,kHk(s) =
+�
+0≤ℓ≤k≤n
+an,kbk,ℓsℓ.
+It is shown in[6] that the Sheffer group is isomorphic to the Riordan group of exponential Riordan
+matrices defined in terms of exponential generating functions as follows. Let D = [di,j]i,j≥0 be an
+infinite lower triangular matrix with complex entries. If there exists a pair of exponential generating
+functions
+g =
+�
+k≥0
+gk
+zk
+k! and f =
+�
+k≥1
+fk
+zk
+k!
+with g0 ̸= 0 and f1 ̸= 0, such that the k-th column of D has exponential generating function gf k/k!
+for k = 0, 1, 2, . . ., then D is called an exponential Riordan matrix, and is denoted by [g, f]. Let R
+be the set of all exponential Riordan matrices. R is a group called the (exponential) Riordan group
+under usual matrix multiplication. In terms of generating functions the product is expressed by
+[g, f][h, ℓ] = [gh(f), ℓ(f)]
+(1.2)
+where h(f) denotes composition of power series with f(0) = 0.
+By definition, we see that the coefficient matrices [an,k] of {Gn(s)}∞
+n=0 and [bn,k] of {Hn(s)}∞
+n=0
+in (1.1) are exponential Riordan matrices [g, f] and [h, ℓ] respectively. Moreover, {Gn ◦ Hn(s)}∞
+n=0
+is the Sheffer sequence for the pair (gh(f), ℓ(f)).
+We claim that the Riordan group R is isomorphic to the group R′ of exponential Riordan matrices
+of the form [f ′/g, f]. To see this, consider the map φ : R → R′ given by φ([g, f]) = [f ′/g, f]. Then
+for any A = [g, f] and B = [h, ℓ] in R, we have
+φ(AB) = φ([gh(f), ℓ(f)]) =
+�f ′ℓ′(f)
+gh(f) , ℓ(f)
+�
+=
+�f ′
+g , f
+� �ℓ′
+h , ℓ
+�
+= φ(A)φ(B).
+Hence φ is a group homomorphism. In addition, ker(φ) = {(1, z)} and clearly, φ is onto. Thus, φ
+is a group isomorphism. We may thus associate to the Sheffer sequence {Gn(s)}∞
+n=0 for the pair
+(g, f), its cognate sequence {Gc
+n(s)}∞
+n=0 generated by the relation
+f ′(z)
+g(z) esf(z) =
+�
+n≥0
+Gc
+n(s)zn
+n! .
+For each n, we call Gc
+n(s) the cognate polynomial of Gn(s). It is natural to ask how the zeros of
+Gn(s) relate to the zeros of the cognate polynomial Gc
+n(s). After all, the map
+{Gn(s)}∞
+n=0
+Φ
+−→ {Gc
+n(s)}∞
+n=0
+is a transformation on R[x], and the properties of such transformations, as they relate to the
+preservation of zero locus, have been a central problem of study in the context of the Pólya-Schur
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+3
+program (see [1]) and beyond. In this paper we study a subset of such maps – or equivalently pairs
+of Sheffer sequences and their cognate sequence – which preserve the symmetry type of the zero
+locus of a Sheffer sequence.
+The paper is organized as follows. In Section 2 we discuss Appell sequences and their cognate
+sequences, building on the example of the Bernoulli polynomials. We also provide a characterization
+of all Appell sequneces whose zeros exhibit the same type of symmetry as those of the Bernoulli
+polynomials. In Section 3 we show that the Sheffer sequences considered in [3] along with their
+cognate sequence are generated by a pair of functions of the same general form. We show that
+any Sheffer sequence generated by functions of this kind consist of polynomials Hn whose zeros are
+either real or lie on a line of the form Re z = c for n ≫ 1. We accomplish this in two subsections:
+the first (subsection 3.1) develops the necessary asymptotic formulas for the integral representation
+of the polynomials under investigation, while the second (subsection 3.2) finds the precise location
+of the zeros of this sequence. The paper concludes with Section 4, in which we discuss the limiting
+distribution of the zeros of the family of sequences defined in Theorem 8.
+2. The zeros of Appell sequences and their cognate sequences
+We begin our investigations with the zeros of the cognate sequence of the Bernoulli polynomials.
+By definition, the cognate sequence {Bc
+n(s)}∞
+n=0 of the Bernoulli polynomials {Bn(s)}∞
+n=0 is the
+Appell sequence for the pair ( ez−1
+z
+, z). It is known that all the zeros of Bernoulli polynomials Bn(s)
+(n ≥ 1) are symmetrical with respect to the line Re s = 1
+2.
+Our first theorem (c.f. Theorem 2) demonstrates that the location of the zeros of Bc
+n(s) is closely
+related to that of the zeros of Bn(s). In the proof of this result we need to make use of the following
+lemma.
+Lemma 1. [2, 14] Let G(s) ∈ C[s] be a polynomial all of whose zeros have positive imaginary part,
+and let ¯G(s) be the polynomial whose coefficients are the complex conjugates of those of G(s). Then
+all zeros of G(s) + ¯G(s) ∈ R[s] are real.
+Theorem 2. For n ≥ 1 let Bc
+n(s) be the cognate polynomial of the Bernoulli polynomial Bn(s).
+Then all zeros of Bc
+n(s) lie on the line Re s = − 1
+2.
+Proof. Define Gn(s) = Bc
+n
+�
+− 1
+2 + is
+�
+. Then all zeros of Gn(s) are real if and only if the real part
+of every zero of Bc
+n(s) is − 1
+2, or equivalently, all zeros of Bc
+n(s) lie on the line Re s = − 1
+2. Thus it
+suffices to show that all zeros of Gn(s) are real. By the definition of Gn(s) we have
+�
+n≥0
+Gn(s)zn
+n! = ez − 1
+z
+e(− 1
+2 +is)z = e
+1
+2 z − e− 1
+2 z
+z
+eisz = 1
+z
+�
+e( 1
+2 +is)z +
+�
+−e(− 1
+2 +is)z��
+.
+Let
+e( 1
+2 +is)z =
+�
+n≥0
+fn(s)zn
+n!
+and
+− e(− 1
+2 +is)z =
+�
+n≥0
+gn(s)zn
+n! .
+Since fn(s) =
+� 1
+2 + is
+�n, all zeros of fn(s) (and also −fn(s)) have positive imaginary part, namely
+1
+2. It is easy to see that gn(s) = (−1)n+1 ¯fn(s). Hence by Lemma 1, we obtain that all zeros of
+Gn(s) = fn(s) + (−1)n+1 ¯fn(s) are real, which completes the proof.
+□
+The above connection between the zeros of Bernoulli polynomials and their cognate sequence
+does not extended to arbitrary Appell sequences and their cognate sequences. Thus, one is naturally
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+4
+led to the problem of finding and characterizing all Sheffer sequences and their cognate sequence
+whose zeros exhibit symmetries akin to that displayed by the zeros of {Bn(s)}∞
+n=0 and {Bc
+n(s)}∞
+n=0.
+Generally, it would be of interest to understand the relationship between the zeros of a Sheffer
+sequence and its cognate sequence.
+To obtain some information about the zeros of the cognate sequences of Appell sequences, we
+begin with the following lemma.
+Lemma 3. Let G(s) ∈ R[s] with degree n ≥ 1. Then all zeros of G(s) are symmetrical with respect
+to the line Re s = − m
+2 (m ∈ R) if and only if G(−s) = (−1)nG(s − m).
+Proof. Suppose that all zeros of G(s) are symmetrical with respect to the line Re s = − m
+2 for some
+m ∈ R. First let n be even. Then we may assume that n zeros of G(s) are of the form − m
+2 ± qk
+(qk ∈ C, k = 1, 2, . . . , n
+2 ) so that G(s) can be written as
+G(s) =
+n/2
+�
+k=1
+�
+s + m
+2 + qk
+� �
+s + m
+2 − qk
+�
+.
+Hence
+G(−s)
+=
+n/2
+�
+j=1
+�
+−
+�
+s − m
+2 − qk
+�� �
+−
+�
+s − m
+2 + qk
+��
+=
+(−1)n
+n/2
+�
+k=1
+�
+(s − m) + m
+2 − qk
+� �
+(s − m) + m
+2 + qk
+�
+= (−1)nG(s − m).
+Now let n be odd. Then G(s) is of the form
+G(s) =
+�
+s + m
+2
+�j (n−j)/2
+�
+k=1
+�
+s + m
+2 + qk
+� �
+s + m
+2 − qk
+�
+where j ≥ 1 and j is odd. A simple computation shows that G(−s) = (−1)nG(s − m) also holds
+for this case.
+Conversely, suppose that G(−s) = (−1)nG(s − m) holds for some m ∈ R. Let a + bi (a, b ∈ R)
+be a zero of G(s). Then a − bi = − m
+2 + ((a + m
+2 ) − bi) is also a zero of G(s). It follows from
+0 = G(a − bi) = G
+�
+−m
+2 +
+��
+a + m
+2
+�
+− bi
+��
+= (−1)nG
+�
+−m
+2 −
+��
+a + m
+2
+�
+− bi
+��
+that −(a + m) + bi = − m
+2 −
+��
+a + m
+2
+�
+− bi
+�
+is a zero of G(s), which implies that all zeros of G(s)
+are symmetrical with respect to the line Re s = − m
+2 . This completes the proof.
+□
+Theorem 4. Let {Gn(s)}n≥0 be the Appell sequence for the pair (g, az). Then the following are
+equivalent:
+(i) For n ≥ 1, the zeros of Gn(s) are symmetric with respect to the line Re s = − g′(0)
+2a
+;
+(ii) For n ≥ 1, Gn(−s) = (−1)nGn(s − g′(0)
+a ) ;
+(iii) g(z) = g(−z)eg′(0)z.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+5
+Proof. It follows from Lemma 3 that (i) and (ii) are equivalent. Moreover, (ii) holds if and only if
+g(z)e−asz
+=
+�
+n≥0
+Gn(−s)zn
+n! =
+�
+n≥0
+(−1)nGn
+�
+s − g′(0)
+a
+� zn
+n! = g(−z)e−a(s− g′(0)
+a
+)z
+=
+g(−z)eg′(0)ze−asz,
+which is equivalent to (iii).
+□
+Theorem 5. Let {Gn(s)}n≥0 be the Appell sequence for the pair (g, az). If all zeros of Gn(s) for
+n ≥ 1 are symmetrical with respect to the line Re s = − g′(0)
+2a , then all zeros of Gc
+n(s) are symmetrical
+with respect to the line Re s = g′(0)
+2a .
+Proof. It suffices to show that Gc
+n(−s) = (−1)nGc
+n(s + g′(0)
+a ) holds for all n ≥ 1. By Theorem 4 we
+have
+�
+n≥0
+Gc
+n(−s)zn
+n!
+=
+a
+g(z)e−asz =
+a
+g(−z)e(−as−g′(0))z =
+a
+g(−z)e−a(s+ g′(0)
+a
+)z
+=
+�
+n≥0
+(−1)nGc
+n
+�
+s + g′(0)
+a
+� zn
+n! ,
+which implies that for all n ≥ 1, Gc
+n(−s) = (−1)nGc
+n(s + g′(0)
+a ). The proof is complete.
+□
+The following theorem shows that there are infinitely many Appell sequences satisfying the
+assumptions of Theorem 5.
+Theorem 6. Let {Gn(s)}n≥0 be the Appell sequence for the pair (g, az). If
+g(z) = 2ρ(z)(1 + tanh(kz)),
+(2.1)
+where ρ(z) is any even function with ρ(0) = 1 and k ∈ R, then all the zeros of Gn(s) are symmetrical
+with respect to the line Re s = − k
+a for n ≥ 1. Conversely, if all the zeros of Gn(s) (n ≥ 1) are
+symmetrical with respect to a vertical line in C then there exist an even function ρ(z) with ρ(0) = 1
+and k ∈ R satisfying (2.1).
+Proof. Suppose g(z) = 2ρ(z)(1 + tanh(kz)) for some even function ρ(z) such that ρ(0) = 1 and
+k ∈ R. Then
+g(−z)e2kz
+=
+2ρ(z)(1 + tanh(−kz))e2kz = 2ρ(z)
+�
+1 + e−kz − ekz
+e−kz + ekz
+�
+e2kz
+=
+2ρ(z)
+�
+1 + ekz − e−kz
+ekz + e−kz
+�
+= 2ρ(z)(1 + tanh(kz)) = g(z),
+and g′(0) = 2k.
+Thus, Theorem 4 (iii) holds, and hence so does (i), i.e.
+the zeros of Gn(s)
+(n ≥ 1) are symmetric with respect to the line Re s = − k
+a. Conversely, suppose that all the zeros
+of Gn(s) are symmetric with respect to a vertical line in C. Let g(z) = 2
+�
+1 + �
+n≥1 gnzn�
+. Since
+G1(s) = 2(g1 +as), this vertical line is Re s = − g1
+a . By Theorem 4, g(z) satisfies g(z) = g(−z)e2g1z.
+A simple computation shows that
+g(z) = g(z) + g(−z)
+2
+(1 + tanh(g1z)).
+Setting ρ(z) = g(z)+g(−z)
+2
+yields an even function, and k = g1 ∈ R, as desired.
+□
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+6
+Remark 7. We note that if {Gn(s)}n≥0 is the Appell sequence for the pair (g(z), z) then the Appell
+sequence for the pair (g(z), az) is {Gn(as)}n≥0. Thus it suffices to explore the Appell sequence for
+the pair (g(z), z) when studying the zeros of the Appell sequence for the pair (g(z), az).
+3. The zeros of a certain family of Sheffer sequences and their cognate sequences
+We now turn our attention to the cognate sequences of Sheffer sequences previously treated in [3].
+Note – as a preview – that the symmetry of the zeros of the cognate sequence about a line remains,
+and that our main result (c.f. Theorem 8) also exploits the fact that (a suitable modification of)
+the non-exponential factor of the generating function is even.
+In order to set up the statement of the main result, suppose z2 > z1 > 0, and let Log(·) denote the
+principle logarithm. Set
+f(z) = Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z)
+g(z) = (z1 + z)(z2 + z).
+The sequence of Sheffer polynomials {Gn(s)}∞
+n=0 for the pair (g(z), f(z)) is generated by
+∞
+�
+n=0
+Gn(s)zn
+n! = g(z)esf(z) = (z1 − z)s(z2 − z)s(z1 + z)1−s(z2 + z)1−s.
+The corresponding cognate sequence {Gc
+n(s)}∞
+n=0 is the Sheffer sequence for the pair (f ′(z)/g(z), f(z)),
+generated by
+∞
+�
+n=0
+Gc
+n(s)zn
+n! = f ′(z)
+g(z) esf(z),
+where
+f ′(z)
+g(z) = −
+1
+(z1 + z)(z2 + z)
+�
+1
+z1 − z +
+1
+z2 − z +
+1
+z1 + z +
+1
+z2 + z
+�
+= −
+2(z1 + z2)
+(z1 + z)2(z2 + z)2(z1 − z)(z2 − z)(z1z2 − z2).
+The reader will note that both g and f ′
+g are of the form
+(z1 − z)p(z1 + z)p∗(z2 − z)q(z2 + z)q∗ m
+�
+i=1
+(αi − z2)pi
+for appropriate values of the constants. As Theorem 8 shows, both the Sheffer sequence for the
+pair (g, f), and its cognate sequence have zeros that are symmetric about a line Re z = k. This fact
+about the Sheffer sequence was already addressed in [3, Theorem 5]. Theorem 8 is a generalization
+of that result.
+Theorem 8. Suppose m ∈ N and p, p∗, q, q∗,αi, pi, 1 ≤ i ≤ m, are real numbers such that αi > z2
+1.
+Let
+(3.1)
+h(z) = (z1 − z)p(z1 + z)p∗(z2 − z)q(z2 + z)q∗ m
+�
+i=1
+(αi − z2)pi,
+let
+f(z) = Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z),
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+7
+and {Hn(s)}∞
+n=0 be the Sheffer sequence for the pair (h(z), f(z)). Assume that p∗−p = q∗−q := 2c.
+If c + p < 0, then all the zeros of Hn(s), n ≫ 1, lie on the line s = c + it. If c + p ≥ 0, then the
+same conclusion holds except for 2 ⌈c + p⌉ real zeros, each of which approaches c ± (c + p + 1 − k),
+0 < k ≤ ⌈c + p⌉as n → ∞.
+-4
+-3
+-2
+-1
+1
+-0.4
+-0.2
+0.2
+0.4
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-0.5
+0.5
+Figure 3.1. Zeros of Hn(s)
+Example 1. Let z1 = 1, z2 = 7. The left side graph in Figure 3.1 shows the zeros of H20(s) when
+p = 4, p∗ = 1, q = 2, q∗ = −1, p − p∗ = 3 = q − q∗, c + p = 11
+2 and
+h(z) = (1 − z)4(1 + z)(7 − z)2(7 + z)−1(2 − z2)−1.
+The right side graph shows the zeros of H20(s) when p = −4, p∗ = −1, q = −2, q∗ = 1, p − p∗ =
+−3 = q − q∗, c + p = − 11
+2 and
+h(z) = (1 − z)−4(1 + z)−1(7 − z)−2(7 + z)1(2 − z2)−1.
+The proof of Theorem 8 is presented in the next two sections, the first of which develops the
+asymptotics for an integral representation of the Hn(s)s, followed by a section on the counting of
+the zeros of these polynomials on the designated locus.
+3.1. The asymptotic formulas. In this section we find an integral representation for the Sheffer
+polynomials Hn(c + int) described in Theorem 8, and we develop of an asymptotic formula for said
+integral representation, which is uniform on the parameter range e− ln4 n/n ≪ t ≪ ln4 n/n. To
+begin, let {Hn}∞
+n=0 be the Sheffer sequence for the pair (h, f) as in the statement of Theorem 8.
+The substitution s = c + int and the Cauchy integral formula gives
+Hn (c + int) = n!
+2πi
+‰
+|z|=ϵ
+h(z)e(c+int)f(z)
+zn+1
+dz
+= n!
+2πi
+‰
+|z|=ϵ
+ψ(z)e−nφ(z,t)dz,
+where
+φ(z, t) = Log z − it (Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z)) ,
+and
+ψ(z) = h(z)
+z
+(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c .
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+8
+It follows from the definition of h(z) that, as a function of z, ψ(z)e−nφ(z,t) is analytic on the
+complement of
+{0} ∪ [z1, ∞) ∪ (−∞, −z1].
+The defintions of h(z) and f(z) also imply that on any circle arc CR with large radius R,
+lim
+R→∞
+ˆ
+CR
+h(z)e(c+int)f(z)
+zn+1
+dz = 0,
+for
+n ≫ 1.
+Thus,
+(3.2)
+Hn (c + int) = n!
+2πi
+ˆ
+Γ1∪Γ2
+ψ(z)e−nφ(z,t)dz,
+where Γ1 and Γ2 are two halves of a loop around infinity and the cuts (−∞, −z1] and [z1, ∞) with
+the counter clockwise orientation.
+The assumption p∗ − p = q∗ − q implies that
+Figure 3.2. The loop around the cuts and infinity
+h(z)(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c
+is even. Using the substitution z �→ −z we see that the part of the integral in (3.2) over Γ1 is equal
+to
+(−1)n
+ˆ
+Γ2
+ψ(z)e−nφ(z,−t)dz.
+Making the subsequent substitution z �→ z, this integral becomes the conjugate of
+(−1)n+1
+ˆ
+Γ2
+ψ(z)e−nφ(z,t)dz.
+We deduce that πHn(c + int) is imaginary part, or −i times the real part of the integral
+(3.3)
+ˆ
+Γ2
+ψ(z)e−nφ(z,t)dz,
+depending on whether n is even or odd.
+For the sake of completeness we now present the setup and definitions developed in [3] in order to
+help establish the asymptotic formula we see. To this end, let
+T1 := z2 − z1
+z1 + z2
+,
+T2 := z1 + z2
+4√z1z2
+,
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+9
+and
+T =
+�
+T1
+if z2
+1 − 6z1z2 + z2
+2 ≥ 0
+T2
+if z2
+1 − 6z1z2 + z2
+2 < 0.
+(3.4)
+For t ∈ (0, T), the function
+ζ = ζ(t) : = z1 + z2
+2
+�
+it −
+�
+T 2
+1 − t2 +
+�
+1 − 2t2 − 2it
+�
+T 2
+1 − t2
+�
+for
+0 ≤ t < T1,
+(3.5)
+ζ = ζ(t) : = z1 + z2
+2
+�
+it + i
+�
+t2 − T 2
+1 +
+�
+1 − 2t2 − 2t
+�
+t2 − T 2
+1
+�
+for
+T1 ≤ t ≤ T2,
+(3.6)
+is a solution of φz(z, t) = 0.
+Set
+g(ζ)
+:=
+2πψ2(ζ)e−2nφ(ζ,t)
+nφz2(ζ, t)
+,
+(3.7)
+p(ζ)
+:=
+ˆ
+Γ2
+ψ(z)e−nφ(z,t)dz,
+and consider the following intervals
+I1 =
+�
+t| ln4 n/n ≪ t < T − ln2 n/n2/3�
+,
+I2 =
+�
+t| ln4 n/n ≪ t < T1 − ln2 n/n2/3�
+,
+I3 = [T1 + ln2 n/n2/3, T2 − ln2 n/n2/3],
+I =
+�
+t|1/n2/3 ≪ T − t < ln2 n/n2/3�
+.
+With these definitions, the results in [3] show that as n → ∞,
+p2(ζ) ∼ g(ζ)
+uniformly on t ∈ I2 ∪ I3 if T = T2, and on t ∈ I1 if T = T1. Furthermore, if t ∈ I, then
+p(ζ) ∼
+4cψ(ζ(T))e−nφ(ζ,t)
+√
+6
+�
+C3φz3(ζ(T), T)
+αn(t)
+uniformly in t, where
+C =
+�
+�
+�
+�
+�
+z1+z2
+2
+�
+−√2T1 +
+T1
+√2T1
+√
+2T 2
+1 −1
+�
+if T = T1
+√
+32(z1z2)3/4
+√
+(z1+z2)(−z2
+1+6z1z2−z2
+2)
+if T = T2
+and Re αn(t) ≥ 0. If T = T2, then
+p(ζ) ∼
+4dψ(ζ(T1))e−nφ(ζ,t)
+√
+6
+�
+D3φz3(ζ(T1), T1)
+βn(t)
+uniformly on 1
+n ≪ |T1 − t| ≤ ln2 n/n2/3, where
+D = −(z1 + z2)√T1
+√
+2
+�
+1 +
+iT1
+�
+1 − 2T 2
+1
+�
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+10
+and Re βn(t) ≥ 0. Before we state and prove our main estimate, we note that (3.1) implies that as
+z → z1,
+h(z) = (z1 − z)p (hp + O (z1 − z))
+for some hp ∈ R. In the remainder of this section, we prove the following asymptotic equivalence.
+Proposition 9. Let hp and p(ζ) be as defined in the beginning of the section, and let Γ denote the
+Gamma function. As n → ∞ the following asymptotic formula holds uniformly on e− ln4 n/n ≪
+t ≪ ln4 n/n:
+p(ζ) ∼
+2iπhp(z2 − z1)c+int
+zn−p
+1
+2c+int(z2 + z1)c+intnc+p+1+intΓ(−c − p − int)
+.
+Proof. We rewrite p(ζ) as
+p(ζ) =
+ˆ (z−
+1 )
++∞
+ψ(z)e−nφ(z,t)dz,
+where path of integration is the Hankel contour which loops around the ray [z1, ∞). The notation
+z−
+1 means the path goes around z1 in the negative direction. We make the substitution w = z/z1
+followed by the subsitution ez = w to arrive at the expression
+p(ζ(t)) = z1
+ˆ (1−)
++∞
+ψ(wz1)e−nφ(wz1,t)dw = z1
+ˆ (0−)
++∞
+ψ(z1ez)e−nφ(z1ez,t)ezdz.
+Suppose ϵ > 0 is small such that nϵ = o(1). We break the last integral into three pieces:
+z1
+ˆ (ln5n/n)±iϵ
+(0−)
+ψ(z1ez)e−nφ(z1ez,t)ezdz
++z1
+ˆ +∞+iϵ
+(ln5 n/n)+iϵ
+ψ(z1ez)e−nφ(z1ez,t)ezdz
++z1
+ˆ (ln5 n/n)−iϵ
++∞−iϵ
+ψ(z1ez)
+2iπhp(z2 − z1)c+int
+zn−p
+1
+2c+int(z2 + z1)c+intnc+p+1+intΓ(−c − p − int)
+e−nφ(z1ez,t)ezdz.
+(3.8)
+Recall that
+(3.9)
+ψ(z)e−nφ(z,t) = h(z)
+zn+1
+(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c
+�(z1 − z)(z2 − z)
+(z1 + z)(z2 + z)
+�nit
+,
+and
+h(z) = (z1 − z)p (hp + O (z1 − z))
+for z1 − z = o(1). Thus, the first integral in (3.8) is asymptotic to
+zc+p+int
+1
+hp(z2 − z1)c+int
+zn
+1 2c+intzc+int
+1
+(z2 + z1)c+int
+ˆ (ln5 n/n)±iϵ
+(0−)
+(1 − ez)c+p+int
+enz
+dz
+∼
+zc+p+int
+1
+hp(z2 − z1)c+int
+zn
+1 2c+intzc+int
+1
+(z2 + z1)c+int
+ˆ (ln5 n/n)±iϵ
+(0−)
+(−z)c+p+inte−nzdz
+=
+zc+p+int
+1
+hp(z2 − z1)c+int
+zn
+1 2c+intzc+int
+1
+(z2 + z1)c+intnc+p+1+int
+ˆ (ln5 n)±inϵ
+(0−)
+(−z)c+p+inte−zdz.
+(3.10)
+The following auxiliary lemma helps us further refine this estimate.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+11
+Lemma 10. Suppose, as in the statement of Proposition 9, that e− ln4 n/n ≪ t ≪ ln4 n/n. If
+nϵ = o(1), then the following asymptotic equivalence holds:
+ˆ ln5 n±inϵ
+(0−)
+(−z)c+p+inte−zdz ∼ 2i sin π(c + p + 1 + int)Γ(c + p + 1 + int).
+Proof. We apply the Hankel contour representation (see for example [9]) for the Gamma function
+Γ(s) =
+1
+2i sin πs
+ˆ (0−)
++∞
+e−z(−z)s−1dz
+(s ̸= 0, −1, −2, . . .)
+to rewrite
+ˆ ln5 n±inϵ
+(0−)
+(−z)c+p+inte−zdz
+as
+(3.11)
+2i sin π(c+p+1+int)Γ(c+p+1+int)−
+ˆ +∞+inϵ
+ln5 n+inϵ
+(−z)c+p+inte−zdz +
+ˆ +∞−inϵ
+ln5 n−inϵ
+(−z)c+p+inte−zdz.
+In the case e− ln4 n ≪ nt = O(1), we have
+(3.12)
+2i sin π(c + p + 1 + int)Γ(c + p + 1 + int) ≫ e− ln4 n.
+Moreover, in this case, the second and the third terms in (3.11) are bounded by
+(3.13)
+ˆ +∞±inϵ
+ln5 n±inϵ
+|z|c+pe−nt Arg(−z)e− Re zd|z|.
+Since nϵ = o(1), we have |z| = Re z + o(1), whence by the asymptotic behavior of the upper
+incomplete Gamma function (see [5, 10]), the integral in (3.13) is O(e− ln5 n ln5(c+p) n). The result
+in this case now follows.
+If, on the other hand, nt ≫ 1, then the Stirling formula
+Γ(s) = exp
+�
+(s − 1/2) Log s − s + 1
+2 ln 2π + O(s−1)
+�
+(|s| ≫ 1)
+and the fact nt ≪ ln4 n imply that
+|Γ(c + p + 1 + int)|
+= exp
+��
+c + p + 1
+2
+�
+ln |c + p + 1 + int| − nt Arg(c + p + 1 + int) − (c + p + 1) + 1
+2 ln 2π + O
+� 1
+nt
+��
+≫ exp(−π ln4 n),
+from which we deduce
+|2i sin π(c + p + 1 + int)Γ(c + p + 1 + int)| ≫ exp
+�
+nπt − π ln4 n
+�
+.
+The claim follows from the fact that
+ˆ +∞±inϵ
+ln5 n±inϵ
+|z|c+pe−nt Arg(−z)e− Re zd|z| = O
+�
+entπ−ln5 n ln5(c+p) n
+�
+.
+The proof of Lemma 10 is complete.
+□
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+12
+We continue with the proof of Proposition 9 by finding a bound for the second integral in (3.8):
+z1
+ˆ ∞+iϵ
+ln5n/n+iϵ
+ψ(z1ez)e−nφ(z1ez,t)ezdz.
+Recall from equation (3.9) that
+ψ(z1ez)e−nφ(z1ez,t) =
+h(z1ez)
+zn+1
+1
+e(n+1)z
+(z1 − z1ez)c(z2 − z1ez)c
+(z1 + z1ez)c(z2 + z1ez)c
+�(z1 − z1ez)(z2 − z1ez)
+(z1 + z1ez)(z2 + z1ez)
+�nit
+.
+With z = u + iϵ, ln5 n/n ≤ u < ∞, we have
+|z2 − z1ez| ≥ | Im(z2 − z1ez)|
+= z1eu sin ϵ
+≫ ϵ,
+and consequently,
+(z2 − z1ez)c = O
+� 1
+ϵ|c| + eu|c|
+�
+.
+We conclude that
+h(z1ez)(z2 − z1ez)c(z1 − z1ez)c
+(z1 + z1ez)c(z2 + z1ez)c
+=
+�
+O
+�
+1
+ϵB +
+n|c+p|
+ln5(c+p) n
+�
+if z = O(1)
+O(eAu)
+if z ≫ 1
+for some constants A(depending on the degree of h) and B(depending on c and the number of poles
+of h on [z1, ∞)).
+We also note that
+�����
+�(z1 − z1ez)(z2 − z1ez)
+(z1 + z1ez)(z2 + z1ez)
+�nit����� = expnt (Arg(z1 − z1ez) + Arg(z2 − z1ez) − Arg(z1 + z1ez) − Arg(z2 + z1ez)) ,
+and that for z = u + iϵ and |z| ≪ 1,
+Arg(z1 − z1ez) + Arg(z2 − z1ez) − Arg(z1 + z1ez) − Arg(z2 + z1ez) = Arg(−z) + O(z).
+Thus, there exists a small δ (independent of n) such that if u < δ then
+�����
+�(z1 − z1ez)(z2 − z1ez)
+(z1 + z1ez)(z2 + z1ez)
+�nit����� < enπt.
+For other values of u, we note that geometrically
+|Arg(z1 − z1ez) − Arg(z1 + z1ez)|
+is the sum of two angles of the triangle with vertices 0, z1, and (z1 + z1ez)/2, which is less than π.
+The same inequality holds for
+|Arg(z2 − z1ez) − Arg(z2 + z1ez)| .
+We conclude that for u ≥ δ,
+�����
+�(z1 − z1ez)(z2 − z1ez)
+(z1 + z1ez)(z2 + z1ez)
+�nit����� < e2nπt.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+13
+In order to utilize these estimates, we break the range of integration of
+ˆ +∞+iϵ
+ln5 n/n+iϵ
+z1ψ(z1ez)e−nφ(z1ez,t)ezdz
+into three pieces: (i) ln5 n/n < Re z < δ, (ii) δ ≤ Re z and Re z = O(1) and (iii) Re z ≫ 1. The
+integral over the first range is
+enπt
+zn
+1
+O
+�� 1
+ϵB +
+n|c|
+ln5c n
+� ˆ δ
+ln5 n/n
+e−nudu
+�
+= enπt
+nzn
+1
+O
+�
+e− ln5 n
+ϵB
++
+n|c|
+ln5c ne− ln5 n
+�
+,
+the integral over the second range is
+e2πnt
+zn
+1
+O
+�� 1
+ϵB +
+n|c|
+ln5c n
+� e−nδ
+n
+�
+,
+while the integral over the third range is
+e2πnt
+zn
+1
+O
+�ˆ ∞
+C
+e−(n+1)u+Audu
+�
+= e2πnt
+zn
+1
+O
+� 1
+ne−nC
+�
+for some large constant C. We recall that nt ≪ ln4 n. If we choose ϵ so that in addition to satisfying
+the condition nϵ = o(1) we also have
+1
+ϵB = O
+�
+exp
+�ln5 n
+2
+��
+,
+then
+z1
+ˆ +∞+iϵ
+ln5 n/n+iϵ
+ψ(z1ez)e−nφ(z1ez,t)ezdz = eπnt
+zn
+1
+O
+�
+exp
+�
+−ln5 n
+2
+��
+.
+With a similar argument we obtain
+z1
+ˆ ln5 n/n−iϵ
++∞−iϵ
+ψ(z1ez)e−nφ(z1ez,t)ezdz = eπnt
+zn
+1
+O
+�
+exp
+�
+−ln5 n
+2
+��
+.
+From equations (3.8), (3.10), Lemma 10, and the fact that
+2i sin π(c + p + 1 + int)Γ(c + p + 1 + int)
+�
+≫ e− ln4 n
+if e− ln4 n ≪ nt = O(1)
+≫ exp
+�
+nπt − π ln4 n
+4
+�
+if 1 ≪ nt ≪ ln4 n
+,
+we conclude that
+ˆ (z−
+1 )
++∞
+ψ(z)e−nφ(z,t)dz ∼ 2i sin π(c + p + 1 + int)Γ(c + p + 1 + int)zc+p+int
+1
+hp(z2 − z1)c+int
+zn
+1 2c+intzc+int
+1
+(z2 + z1)c+intnc+p+1+int
+uniformly on e− ln4 n ≪ nt ≪ ln4 n. The proof of Proposition 9 is complete.
+□
+Remark 11. If c+p /∈ Z+, we do not require the condition e− ln4 n ≪ nt for the estimate in equation
+(3.12). Thus the asymptotics in Proposition 9 hold for nt ≪ ln4 n if c + p /∈ Z+.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+14
+3.2. The location of the zeros of {Hn}∞
+n=0. With the asymptotic analysis complete, in this
+section we establish that for all n ≫ 1, the zeros fo Hn lie on the line Re s = c (c.f. Theorem 8)
+except perhaps a finite number of real zeros, whose asymptotic locations we also identify. We begin
+with a technical, but crucial result.
+Lemma 12. Suppose τ1 and τ2 are constant multiples of e− ln4 n/n and ln4 n/n and α is the unique
+angle such that −π < α ≤ π and (c + p + 1/2)π = 2kπ + α for k ∈ Z. Let g and p be defined as in
+equation (3.7). Then
+(i) p(ζ(t)) ̸= 0 for τ1 ≤ t ≤ τ2, and
+(ii) ∆ argτ1≤t≤τ2 p(ζ(t)) = 1
+2 lim
+ξ→0 ∆ argξ≤t≤τ2 g(ζ(t)) + |c + p|π
+2
++ η,
+where
+(3.14)
+η =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−π/2
+if c + p < 0
+0
+if c + p ≥ 0 and α = ± π
+2
+−α
+if c + p ≥ 0 and − π/2 < α < π/2
+−α − π
+if c + p ≥ 0 and − π < α < −π/2
+−α + π
+if c + p ≥ 0 and π/2 < α ≤ π.
+.
+Proof. The fact that p(ζ(t)) ̸= 0 for τ1 ≤ t ≤ τ2 follows immediately from Proposition 10. To
+establish the claim regarding the change of arguments, we start by recalling that
+g(ζ) = 2πψ2(ζ)e−2nφ(ζ)
+nφz2(ζ, t)
+=
+2π
+nφz2(ζ, t) · h2(ζ)
+z2n+2
+(z1 − z)2c(z2 − z)2c
+(z1 + z)2c(z2 + z)2c
+�(z1 − z)(z2 − z)
+(z1 + z)(z2 + z)
+�2nit
+.
+Since z1 − z = −iz1t + O(t2), on ξ ≤ t ≤ τ2 the change in the argument of
+h2(ζ)(z1 − z)2c(z2 − z)2c
+(z1 + z)2c(z2 + z)2c = (z1 − z)2p∗H(t) ∼ (−z2
+1t2)p∗ + O(t3)
+is o(1). Thus, using the same computations in the proof of Lemma 39 in [3], we conclude that
+(3.15)
+lim
+ξ→0 ∆ argξ≤t≤τ2 g(ζ(t)) = 2nτ2 ln τ2(z2 − z1)
+2(z1 + z2) − 2nτ2 + π
+2 + o(1).
+For τ1 ≤ t ≤ τ2, the change of the arguments of the factors 2c+int, (z2 − z1)c+int, (z2 + z1)c+int,
+and e(c+p+1+int) ln n in the expression
+2iπhp(z2 − z1)c+int
+zn−p
+1
+2c+int(z2 + z1)c+intnc+p+1+intΓ(−c − p − int)
+are n(τ2 − τ1) ln 2, n(τ2 − τ1) ln(z2 − z1), n(τ2 − τ1) ln(z2 + z1), and n(τ2 − τ1) ln n respectively.
+We next compute the change in argument of the factor Γ(−c − p − int) , τ1 ≤ t ≤ τ2, which is
+given by the expression
+Im Log Γ(−c − p − int)|τ2
+τ1 ,
+where the function Log Γ(s) is defined as
+Log Γ(s) = −γs − Log s +
+∞
+�
+k=1
+� s
+k − Log(1 + s/k)
+�
+.
+Using the Stirling formula,
+Log Γ(s) ∼ (s − 1/2) Log s − s + 1/2 Log(2π) + O(1/s),
+for |s| → ∞ and | Arg s| ≤ π − δ,
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+15
+we conclude that
+Im Log Γ(−c − p − inτ2)
+= Im ((−c − p − 1/2 − inτ2) Log(−c − p − inτ2) + nτ2 + O(1/ ln4 n)
+= − nτ2 ln |c + p + inτ2| + (c + p + 1/2)π
+2 + nτ2 + O(1/ ln4 n).
+Employing the estimate
+ln |c + p + inτ2| = ln
+����inτ2
+�
+1 + c + p
+inτ2
+����� = ln(nτ2) + O
+�
+1
+n2τ 2
+2
+�
+,
+the last expression becomes
+−nτ2 ln(nτ2) + (c + p + 1/2)π
+2 + nτ2 + O
+�
+1
+ln4 n
+�
+.
+If −c − p > 0, then the fact that nτ1 ≍ e− ln4 n implies that
+Im Log Γ(−c − p − inτ1) = O
+�
+e− ln4 n�
+,
+and consequently
+∆ argτ1≤t≤τ2 Γ(−c − p − int) = −nτ2 ln(nτ2) + (c + p + 1/2)π
+2 + nτ2 + O
+�
+1
+ln4 n
+�
+.
+If, on the other hand, if c + p ≥ 0, then the identity
+Γ(−c − p − int) =
+π
+sin π(c + p + 1 + int)Γ(c + p + 1 + int)
+implies that
+∆ argτ1≤t≤τ2 Γ(−c−p−int) = −∆ argτ1≤t≤τ2 sin π(c+p+1+int)−∆ argτ1≤t≤t2 Γ(c+p+1+int).
+Using the conjugate of the gamma function, we write
+−∆ argτ1≤t≤t2 Γ(c+p+1+int) = ∆ argτ1≤t≤τ2 Γ(c+p+1−int) = −nτ2 ln(nτ2)−(c+p+1/2)π
+2 +nτ2+O
+�
+1
+ln4 n
+�
+.
+Analyzing the change in the argument of sin π(c + p + int) requires further considerations. To this
+end, recall that
+sin π(c + p + 1 + int) = e−πnt+iπ(c+p+1) − eπnt−iπ(c+p+1)
+2i
+= −eπnt−iπ(c+p+1)
+2i
+�
+e−2πnt + 1
+�
+,
+and whence
+Im sin π(c + p + 1 + int) = 1
+2
+�
+e−πnt − eπnt�
+cos π(c + p + 1).
+It is immediate that if c + p + 1/2 ∈ Z, then
+∆ argτ1≤t≤τ2 sin π(c + p + 1 + int) = 0.
+On the other hand, if c + p + 1/2 /∈ Z, then sin π(c + p + 1 + int) /∈ R, and
+∆ argτ1≤t≤τ2 sin π(c + p + 1 + int) = Arg sin π(c + p + 1 + inτ2) − Arg sin π(c + p + 1 + inτ1).
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+16
+We write
+Arg sin π(c + p + 1 + inτ2) = Arg
+�−eπnτ2−iπ(c+p+1)
+2i
+�
++ O
+�
+e−2πnτ2�
+= −α + O(e−2π ln4 n),
+where α is the unique angle such that −π < α ≤ π and (c + p + 1/2)π = 2kπ + α for k ∈ Z. Note
+that α is given explicitly by the formula
+α = ((c + p + 3/2)π
+mod 2π) − π.
+If c + p ∈ Z, then the Taylor expansion of the sine function yields
+Arg sin π(c + p + 1 + inτ1) = (−1)c+p+1 π
+2 + O
+�
+e− ln4 n�
+.
+If c + p /∈ Z, then
+Arg sin π(c + p + 1 + inτ1) =
+�
+�
+�
+�
+�
+�
+�
+O
+�
+e− ln4 n�
+if sin π(c + p + 1) > 0
+π
+if sin π(c + p + 1) < 0 and cos π(c + p + 1) > 0
+−π
+if sin π(c + p + 1) < 0 and cos π(c + p + 1) < 0.
+Combining these cases we conclude for any c, p ∈ R,
+∆ argτ1≤t≤τ2 Γ(−c − p − int) = −nτ2 ln(nτ2) + nτ2 − |c + p|π
+2
+− π
+4 − η + O
+�
+1
+ln4 n
+�
+,
+and finally,
+∆ argτ1<t≤τ2 p(ζ(t)) = nτ2 ln (z2 − z1)τ2
+2(z2 + z1) − nτ2 + |c + p|π
+2
++ π
+4 + η + O
+�
+1
+ln4 n
+�
+.
+Given equation (3.15), the result now follows.
+□
+We next identify a suitable curve on which we will compute the change of argument of g. Let γ be
+the simple closed curve with counter clockwise orientation formed by the traces of ζ(t), ζ(t), −ζ(t),
+and −ζ(t) for 0 ≤ t ≤ T and small deformations around
+(3.16)
+�
+±i√z1z2, ±ζ(T1), ±ζ(T1)
+if z2
+1 �� 6z1z2 + z2
+2 < 0
+±iζ(T)
+if z2
+1 − 6z1z2 + z2
+2 ≥ 0
+such that the region enclosed by γ contains the points defined in (3.16). We also deform γ around
+±z1 so that the cuts (−∞, −z1] and [z1, ∞) lie outside this region (see Figure 3.3). Using the
+residue theorem (for detailed computations, see [3] equation (2.86)), we find that
+1
+2πi
+ˆ
+γ
+g′(ζ)
+g(ζ) dζ =
+�
+−2n − 6
+if z2
+1 − 6z1z2 + z2
+2 < 0
+−2n − 2
+if z2
+1 − 6z1z2 + z2
+2 ≥ 0 ,
+since the values of c and p do not affect the integral.
+Let γ1 be the portion of γ in the first quadrant. Exploiting the symmetry g(ζ) = g(ζ) and
+g(−ζ) = g(ζ), we conclude that
+∆γ1 arg g(ζ) = ∆γ arg g(ζ)
+4
+=
+�
+−(n + 3)π
+if z2
+1 − 6z1z2 + z2
+2 < 0
+−(n + 1)π
+if z2
+1 − 6z1z2 + z2
+2 ≥ 0 .
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+17
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+1.5
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+1.5
+-1.0
+-0.5
+0.5
+1.0
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+1.5
+Figure 3.3. The curve γ for (z1, z2) = (1, 3) (left) and (1, 7) (right)
+Remark 13. In the case t → T and t → T1 (when T ̸= T1) , the values c and p only affect the
+change in argument of g(ζ(t)) by o(1). Thus the following results follow directly from [3].
+(1) If τ < T such that T − τ ≪ 1/n2/3, then by Lemma 40 in [3],
+lim
+ξ→0 ∆ argτ≤t≤T −ξ g(ζ(t)) ≪ 1.
+(2) If T = T2 and |T1 − τ| ≪ 1/n, then by Lemmas 41 and 42 in [3],
+lim
+ξ→0 ∆T1+ξ<t<τ arg g(ζ(t)) ≪ 1,
+and
+lim
+ξ→0 ∆τ≤t<T1−ξ arg g(ζ(t)) ≪ 1.
+(3) Lemma 37 in [3] shows that there exists some |C| < π/2 + o(1), such that
+∆ln2 n/n2/3<T −t≪1/n2/3 arg p(ζ(t) = 1
+2∆ln2 n/n2/3<T −t≪1/n2/3 arg g(ζ(t)) + C.
+(4) If T = T2 and p(ζ(T1)) ̸= 0, then p(ζ(t)) ̸= 0 on (T1 − ln2 n/n2/3, T1 + ln2 n/n2/3) and by
+Lemma 38 in [3], there exists some |C| < π + o(1) such that
+∆T1−ln2 n/n2/3<t<T1+ln2 n/n2/3 arg p(ζ(t))
+=1
+2∆ln2 n/n2/3<T1−t≪1/n arg g(ζ) + 1
+2∆1/n≪t−T1<ln2 n/n2/3 arg g(ζ) + C.
+(5) An argument completely analogous to that on p.55 in [3] shows that the change in the
+argument of g(ζ(t)) on the small arcs of γ1 around ζ(T) and ζ(T1) (when T ̸= T1) are −π/2
+and −3π/2 respectively .
+Finally, as ζ → z1,
+g(ζ) = 2πψ2(ζ)e−2nφ(ζ)
+nφz2(ζ)
+∼ c1(z1 − ζ)2c+2p+1,
+(c1 ∈ C \ {0}),
+and
+exp (2n(ζ − z1)(z1 − z2) Log(z1 − ζ)/z1) → 1.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+18
+Thus the change in argument of g(ζ) on the small arc of γ1 around z1 is −(c + p + 1/2)π + o(1).
+In contrast to the polynomials discussed in [3] – which have at most two real zeros – the family of
+polynomials we are treating in the current paper can have several real zeros, whose location changes
+with n. The next result gives a lower bound on the number of real zeros of Hn if c + p > 0, and
+also describes the asymptotic behavior of these zeros as n → ∞.
+Lemma 14. Let {Hn}∞
+n=0 be as in the statement of Theorem 8. If c + p > 0, then Hn(s) has at
+least 2 ⌈c + p⌉ many real zeros which approach c ± (c + p + 1 − k), 0 < k ≤ ⌈c + p⌉ as n → ∞.
+Proof. For x ∈ R, the Cauchy integral formula yields
+Hn(c + x) = n!
+2πi
+‰
+|z|=ϵ
+h(z)
+zn+1
+(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c
+�(z1 − z)(z2 − z)
+(z1 + z)(z2 + z)
+�x
+dz
+(3.17)
+= n!
+2πi
+ˆ
+Γ1∪Γ2
+h(z)
+zn+1
+(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c
+�(z1 − z)(z2 − z)
+(z1 + z)(z2 + z)
+�x
+dz,
+where Γ1 and Γ2 are two loops around two cuts (−∞, −z1] and [z1, ∞) oriented counter clockwise.
+Using the substitution z �→ −z and the fact that
+h(z)(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c
+is an even function, we see that the integral over Γ1 is equal to
+(−1)n
+ˆ
+Γ2
+h(z)
+zn+1
+(z1 − z)c−x(z2 − z)c−x
+(z1 + z)c−x(z2 + z)c−x dz.
+We apply Remark 11 to t = 0 and c replaced by c + x to conclude that if c + p + x /∈ Z+, then the
+integral over Γ2 is asymptotic to
+2ihp(z2 − z1)c+x sin π(c + p + x + 1)Γ(c + p + x + 1)
+zn−p
+1
+2c+x(z2 + z1)c+xnc+p+x+1
+.
+With the same application to the case when c replaced by c − x, we conclude that
+Hn(c + x) ∼ n!hp(z2 − z1)c+x sin π(c + p + x + 1)Γ(c + p + x + 1)
+πzn−p
+1
+2c+x(z2 + z1)c+xnc+p+x+1
++ (−1)n n!hp(z2 − z1)c−x sin π(c + p − x + 1)Γ(c + p − x + 1)
+πzn−p
+1
+2c−x(z2 + z1)c−xnc+p−x+1
+(3.18)
+if c + p ± x /∈ Z+ and x ̸= 0. For any small fixed δ > 0 (independent of n), we consider the intervals
+(3.19)
+Jk = [c + p + 1 − k − δ, c + p + 1 − k + δ],
+0 < k < c + p + 1 − δ.
+For each k, the values of sin(c + p − x + 1) when x is at the endpoints of Jk are (−1)k−1 sin δ and
+(−1)k sin δ. Also at these endpoints c + p ± x /∈ Z+ (for small δ), Γ(c + p − x + 1) > 0, and the
+second term of (3.18) dominates the first term when n is large. Thus, by the Intermediate Value
+Theorem, each interval Jk contain at least one zero of Hn(c + x). We deduce that Hn(c + x) has at
+least ⌈c + p⌉positive real zeros. The substitutions z by −z and x by −x in equation (3.17) yield
+Hn(c − x) = (−1)nHn(c + x).
+The result now follows from the fact that if x is a real zero of Hn(c + x), then so is −x.
+□
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+19
+We now turn our attention to the proof of Theorem 8. In addition to the number of real zeros
+of Hn(s) given in Lemma 14, we will count the number of zeros of Hn(c + nit) on t ∈ (0, T) and
+compare this number with the degree of Hn(s). We start with a lemma concerning the degree of
+Hn(s).
+Lemma 15. Let {Hn}∞
+n=0 be defined as in Theorem 8. Then for each n ≥ 0, polynomial Hn(c + x)
+has degree n and the sign of its leading coefficient is (−1)n.
+Proof. Since
+Hn(c − x) = (−1)nHn(c + x),
+it suffices to prove that Hn(c − x) has degree n, and that its leading coefficient is positive. The
+generating function for Hn(c − x) is given by is
+h(z)(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c (1 − z/z1)−x(1 − z/z2)−x(1 + z/z1)x(1 + z/z2)x.
+For each k ∈ N, the coefficient of zk in the binomial expansion of each factor (1 − z/z1)−x, (1 −
+z/z2)−x, (1 + z/z1)x, and (1 + z/z2)x is a polynomial of degree k in x with a positive leading
+coefficient. Thus, given an n ∈ N, the coefficient of zn in the product
+(1 − z/z1)−x(1 − z/z2)−x(1 + z/z1)x(1 + z/z2)x
+is of the form
+�
+i+j+k+ℓ=n
+pi(x)pj(x)pk(x)pℓ(x),
+where each factor of each summand – and hence the entire expression – has a positive leading
+coefficient, and degree equal to its index. We expand
+h(z)(z1 − z)c(z2 − z)c
+(z1 + z)c(z2 + z)c
+as a power series in z (with constant coefficients) and deduce that Hn(c − x) has degree n, and the
+sign of its leading coefficient is the same as the sign of the constant coefficient of this series which
+is h(0) > 0.
+□
+The final piece in accounting for all of the zeros of Hn(s) is provided by the fact (to be proven
+in short order) that the total number of real zeros of Hn(s) and those on c + it (except the possible
+zero at c) is at least
+(3.20)
+�
+n − 2
+if 2 | n
+n − 3
+if 2 ∤ n .
+Assuming this fact, we now provide an argument to complete the proof of Theorem 8. If n is odd,
+then we let x = 0 in Hn(c − x) = (−1)nHn(c + x) to conclude that c is a zero of Hn(s). It thus
+remains to account for the two possible missing zeros of Hn(s) regardless of the parity of n. Since
+the degree of Hn(s) is n, and the zeros of Hn(s) are symmetric about the real line and the line
+c + it, it suffices to show that the possible two remaining zeros of Hn(s) are not real. Note that
+Hn(c + x) has opposite signs at the endpoints of each Jk (as defined in (3.19)). Hence, Hn(c + x)
+must have exactly one zero on each Jk and consequently the two remaining zeros cannot lie on any
+Jk. Since on the set
+(0, c + p + δ)\
+�
+0<k<c+p+1−δ
+Jk
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+20
+the second term of expression (3.18) dominates the first, Hn(c + x) does not have zero there.
+Moreover, it follows from hp > 0 and the asymptotic expression in (3.18) that the sign Hn(c + x)
+at x = c + p + δ is (−1)n. By Lemma 15, this is the same as the sign of limx→∞ Hn(c + x), and we
+conclude that Hn(c + x) has no zero on [c + p + δ, ∞). It follows that the remaining two possible
+zeros must lie on the line Re z = c, completing the proof of Theorem 8.
+Remark 16. In the case c + p ∈ Z+, (3.18) implies that Hn(c + x) is nonzero on (0, 1 − δ) and its
+sign is (−1)n+c+p there.
+We now present the proof of the zero count of Hn(s) claimed in expression (3.20) above. Since
+a lower bound for the number of real zeros of Hn(s) is provided by Lemma 14, it remains to count
+the number of zeros of Hn(c + int) on |t| ∈ (0, T). We recall that for t ∈ (0, T), πHn(c + int) is
+the imaginary part of −i times the real part of p(ζ(t)). It therefore suffices to compute the change
+in the argument of p(ζ(t)) in order to get a lower bound on the zero count of Hn(s) on the line
+Re z = c. We proceed by case analysis, depending on whether T = T2 or T = T1 (c.f. equation
+(3.4)).
+Case T = T2. If T = T2 and p(ζ(T1)) ̸= 0, then for some |C| < 3π/2 + o(1) and c2 ∈ R+
+∆e−ln4n/n≪t<T −c2/n2/3 arg p(ζ(t)) = 1
+2∆ arge−ln4n/n≪t<T −c2/n2/3 g(ζ(t)) + |c + p|π
+2
++ η + C
+= 1
+2 (∆γ1 arg g(ζ) + (c + p + 5/2)π) + |c + p|π
+2
++ η + C
+= −nπ
+2 + c + p + |c + p|
+2
+π − π
+4 + η + C,
+(3.21)
+where η is defined as in equation (3.14) in Lemma 12. In the case c + p < 0, the equation above
+implies that the number of zeros of Hn(c + int) on (0, T) is at least
+�n
+2 + 3
+4 − C
+π
+�
+.
+It follows from |C| < 3π/2 + o(1) that Hn(s) has at least
+�
+n − 3
+if 2 ∤ n
+n − 2
+if 2 | n
+nonreal zeros on the line Re s = c + it.
+On the other hand, if c + p ≥ 0, then the number of zeros of Hn(c + int) on (0, T) is at least
+�n
+2 − (c + p) + π
+4 − η + C
+π
+�
+.
+We conclude from Lemma 18 that the total number of real zeros and those on c + it (except the
+possible zero at c) is at least
+(3.22)
+2
+�n
+2 − (c + p) + 1
+4 − η + C
+π
+�
++ 2 ⌈c + p⌉ ,
+where c + p = 2k + α/π − 1/2.
+If −π < α < −π/2, then η = −α − π. Consequently,
+−(c + p) − η + C
+π
++ 1
+4 = −2k + 7
+4 − C
+π ,
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+21
+and the expression (3.22) is at least
+�
+2
+�� n
+2
+�
+− 2k
+�
++ 2(2k − 1) = n − 2
+if 2 | n
+2
+�� n
+2
+�
+− 2k
+�
++ 2(2k − 1) = n − 3
+if 2 ∤ n .
+If π/2 < α ≤ π, then η = −α + π and
+−(c + p) − η + C
+π
++ 1
+4 = −2k − 1
+4 − C
+π ,
+from which we see that the expression in (3.22) is at least
+�
+2
+�� n
+2
+�
+− 2k − 2
+�
++ 2(2k + 1) = n − 2
+if 2 | n
+2
+�� n
+2
+�
+− 2k − 2
+�
++ 2(2k + 1) = n − 3
+if 2 ∤ n .
+If −π/2 < α < π/2, then η = −α and
+−(c + p) − η + C
+π
++ 1
+4 = −2k + 3
+4 − C
+π .
+Computing the expression in (3.22) again we find that it is at least
+�
+2
+�� n
+2
+�
+− 2k − 1
+�
++ 4k = n − 2
+if 2 | n
+2
+�� n
+2
+�
+− 2k − 1
+�
++ 4k = n − 3
+if 2 ∤ n .
+If α = π/2, then η = 0 and
+∆e−ln4n/n≪t<T −c2/n2/3 arg p(ζ(t)) = −nπ
+2 + (c + p)π − π
+4 + C,
+and the expression in (3.22) computes to be at least
+�
+2
+�� n
+2
+�
+− 2k − 2
+�
++ 4k = n − 4
+if 2 | n
+2
+�� n
+2
+�
+− 2k − 1
+�
++ 4k = n − 3
+if 2 ∤ n.
+If α = −π/2, then η = 0 and
+∆e−ln4n/n≪t<T −c2/n2/3 arg p(ζ(t)) = −nπ
+2 + (2k − 1)π − π
+4 + C.
+In this case we find that the expression in (3.22) is at least
+�
+2
+�� n
+2
+�
+− 2k − 1
+�
++ 2(2k − 1) = n − 4
+if 2 | n
+2
+�� n
+2
+�
+− 2k
+�
++ 2(2k − 1) = n − 3
+if 2 ∤ n.
+The reader will note that if α = ±π/2 and 2 | n, we need to find two more zeros in order to increase
+the the lower bound we have thus far, i.e. n − 4, to the claimed lower bound of n − 2. Suppose thus
+that 2 | n. The identity
+Hn(c − x) = (−1)nHn(c + x)
+implies that if c is a zero of Hn(z), then it is a double zero. On the other hand, if c is not a
+zero of this polynomial, then from Remark 16 we conclude that the sign of Hn(c) is (−1)c+p and
+consequently
+lim
+t→0 Arg p(ζ(t)) = (−1)c+p π
+2 .
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+22
+Moreover, Proposition 9 yields that for t ≍ e− ln4 n/n,
+Arg p(ζ(t)) = Arg(i sin(c + p + 1 + int)) + o(1) =
+�
+o(1)
+if 2 | c + p
+±π + o(1)
+if 2 ∤ c + p .
+This implies that
+(3.23)
+∆0<t≪e−ln4n/n arg p(ζ(t)) = −π
+2 or 3π
+2 .
+If the change of argument in (3.23) is 3π/2, then we have at least two zeros of Hn(c ± int) on
+the range 0 < t ≪ e−ln4n/n, since πHn(c + int) is the imaginary part of p(ζ(t)). If the change of
+argument in (3.23) is −π/2, we deduce from equation (3.21) that
+∆0<t<T −c2/n2/3 arg p(ζ(t)) = −nπ
+2 + (c + p)π − 3π
+4 + C.
+Thus, the number of real zeros of Hn(s) and those on c + it (except the possible zero at c) is at
+least
+2
+�n
+2 − 2k − 1
+�
++ 4k = n − 2
+when α = π/2, and at least
+2
+�n
+2 − 2k
+�
++ 2(2k − 1) = n − 2
+when α = −π/2. This complete the case α = ±π/2 when n is even.
+If T = T2 and p(ζ(T1)) = 0, then for some c2 ∈ R+ and small ξ > 0, the number of real zeros of
+Hn(c + int) on (0, T)\{T1} is at least
+�
+∆e−ln4n/n≪t<T1−ξ/n arg p(ζ(t))
+π
+�
++
+�∆T1+ξ/n<t<T −c2/n2/3 arg p(ζ(t))
+π
+�
+≥
+�
+∆e−ln4n/n≪t<T1−ξ/n arg p(ζ(t))
+π
++ ∆T1+ξ/n<t<T −c2/n2/3 arg p(ζ(t))
+π
+�
+− 1.
+Counting T1 as an additional zero of Hn(1/2 + int) on (0, T), we obtain the same number of zeros
+of this polynomial as in the case p(ζ(T1)) ̸= 0.
+Case T = T1. We conclude from Lemma 14 that for some |C| < π/2 + o(1) and c2 ∈ R+
+∆e−ln4n/n≪t<T −c2/n2/3 arg p(ζ(t)) = 1
+2∆ arge−ln4n/n≪t<T −c2/n2/3 g(ζ(t)) + |c + p|π
+2
++ η + C
+= 1
+2 (∆γ1 arg g(ζ) + (c + p + 1)π) + |c + p|π
+2
++ η + C
+= −nπ
+2 + c + p + |c + p|
+2
+π + η + C.
+We compare the last expression with the one in equation (3.21) and conclude this case, as well as
+the proof of Theorem 8.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+23
+4. The limiting zero distribution density function
+While we have found the zero locus of the cognate sequences under investigation, we can extract
+further information about the limiting behavior of the zeros in terms of their distribution. We do
+so by compute the limiting probability density function of the zeros of Hn(c + int) on t ∈ (0, T).
+To this end, for each x ∈ (0, T) and ϵ > 0, we let Nn,ϵ(x) denote the number of zeros of Hn(c + int)
+on the interval t ∈ (x, x + ϵ). It follows that the limiting probability density function at x ∈ (0, T)
+is given by
+lim
+ϵ→0
+1
+ϵ lim
+n→∞
+Nn,ϵ(x)
+n
+.
+We note that for any x ∈ (0, T) and x ̸= T1 (if T = T2),
+p2(ζ(t)) ∼ g(ζ(t)) = 2πψ2(ζ)e−2nφ(ζ,t)
+nφz2(ζ, t)
+uniformly on t ∈ (x, x + ϵ), and consequently
+∆ argx<t<x+ϵ p(ζ(t)) = 1
+2∆ argx<t<x+ϵ g(ζ(t))
+= −n∆ Imx<t<x+ϵ φ(ζ, t) + O(ϵ).
+It is immediate from the Taylor expansion of φ(ζ, ·) about x that
+φ(ζ, t)|t=x+ϵ
+t=x
+= dφ(ζ, t)
+dt
+����
+t=x
+ϵ + O(ϵ2),
+where
+dφ
+dt
+����
+t=x
+= ∂φ
+∂ζ
+����
+t=x
+dζ
+dt
+����
+t=x
++ ∂φ
+∂t
+����
+t=x
+= ∂φ
+∂t
+����
+t=x
+= −i (Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z)) .
+Thus, using the fact that
+πHn(c + int) = Im(−i Re(p(ζ(t)))),
+we conclude that
+lim
+ϵ→0
+1
+ϵ lim
+n→∞
+Nn,ϵ(x)
+n
+= lim
+ϵ→0
+1
+ϵ lim
+n→∞
+��∆ argx<t<x+ϵ p(ζ(t))
+��
+πn
+.
+= 1
+π ln
+����
+(z1 + ζ(x))(z2 + ζ(x))
+(z1 − ζ(x))(z2 − ζ(x))
+���� .
+(4.1)
+In the case x = T1 when T = T2, we note from the previous section that the number of zeros of
+Hn(c + int) on t ∈ (T1, T1 + ξ/n), for small ξ > 0, is O(1). Since
+∆ argT1+ξ/n<t<T1+ϵ p(ζ(t)) = 1
+2∆ argT1+ξ/n<t<T1+ϵ g(ζ(t)) + C
+for |C| < π/2 + o(1), the same argument above also shows that (4.1) holds for x = T1 as well.
+
+ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES
+24
+-0.6
+-0.4
+-0.2
+0.2
+0.4
+0.6
+0.5
+1.0
+1.5
+2.0
+-0.6
+-0.4
+-0.2
+0.2
+0.4
+0.6
+0.5
+1.0
+1.5
+Figure 4.1. Limiting probability density function for (z1, z2) = (1, 3) (left) and
+(1, 7) (right)
+References
+[1] J. Borcea and P. Brändén, Pólya-Schur master theorems for circular domains and their boundaries, Annals of
+Math., 170 (2009), 465-492.
+[2] D. Bump, Eugene K.-S. Ng, On Riemann’s zeta funcion, Math. Z. 192 (1986), 195-204.
+[3] G. Cheon, T. Forgács, H. Kim, K. Tran, On combinatorial properties and the zero distribution of certain Sheffer
+sequences, J. of Mathematical Analysis and Applications, 514 (2022), 126273.
+[4] G.-S. Cheon, H. Kim, L. W. Shapiro, A generalization of Lucas polynomial sequence, Discrete Applied Mathe-
+matics, 157 (2009), 920-927.
+[5] Dingle, R. B., Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London and New
+York, 1973.
+[6] Tian-Xiao He, Leetsch C. Hsu, Peter J.-S. Shiue, The Sheffer group and the Riordan group, Discrete Applied
+Mathematics, 155 (2007), 1895-1909.
+[7] A. F. Horadam, Extension of a Synthesis for a Class of Polynomial Sequences, Fibonacci Quart., 34(1) (1996),
+68-74.
+[8] A. Leibman, Polynomial Sequences in Groups, J. of Algebra, 201 (1998), 189-206.
+[9] G. Moretti, Functions of a Complex Variable. Englewood Cliffs, N.J.: Prentice-Hall, Inc. pp. 179-184. 1964.
+[10] Olver, F. W. J., Asymptotics and Special Functions, Academic Press, London and New York, 1974.
+[11] S. Roman, The umbral calculus, Academic Press, New York, 1984.
+[12] Gian-Carlo Rota, D. Kahaner, A. Odlyzko, On the foundations of combinatorial theory. VIII. Finite operator
+calculus, J. of Mathematical Analysis and Applications, 42 (1973), 684-760.
+[13] L. Shapiro, R. Sprugnoli, P. Barry, G.-S. Cheon, T.-X. He, D. Merlini, W. Wang, The Riordan Group and
+Applications, Springer Monographs in Mathematics, 2022.
+[14] E. C. Titchmarsh, The theory of the Riemann-zeta function, Oxford Univ. Press, New York, 1951.
+1Department of Mathematics/ Applied Algebra and Optimization Research Center, Sungkyunkwan
+University, Suwon 16419, Rep. of Korea
+Email address: [email protected]
+2Department of Mathematics, California State University, Fresno, Fresno, CA 93740-8001, USA
+Email address: [email protected]
+Email address: [email protected]
+

UdE4T4oBgHgl3EQfMAzo/content/tmp_files/2301.04944v1.pdf.txt

@@ -0,0 +1,1290 @@
+ViTs for SITS: Vision Transformers for Satellite Image Time Series
+Michail Tarasiou
+Imperial College London
+[email protected]
+Erik Chavez
+Imperial College London
+[email protected]
+Stefanos Zafeiriou
+Imperial College London
+[email protected]
+Abstract
+In this paper we introduce the Temporo-Spatial Vision
+Transformer (TSViT), a fully-attentional model for general
+Satellite Image Time Series (SITS) processing based on the
+Vision Transformer (ViT). TSViT splits a SITS record into
+non-overlapping patches in space and time which are tok-
+enized and subsequently processed by a factorized temporo-
+spatial encoder. We argue, that in contrast to natural im-
+ages, a temporal-then-spatial factorization is more intu-
+itive for SITS processing and present experimental evidence
+for this claim. Additionally, we enhance the model’s dis-
+criminative power by introducing two novel mechanisms
+for acquisition-time-specific temporal positional encodings
+and multiple learnable class tokens. The effect of all novel
+design choices is evaluated through an extensive ablation
+study. Our proposed architecture achieves state-of-the-art
+performance, surpassing previous approaches by a signifi-
+cant margin in three publicly available SITS semantic seg-
+mentation and classification datasets. All model, training
+and evaluation codes are made publicly available to facili-
+tate further research.
+1. Introduction
+The monitoring of the Earth surface man-made impacts
+or activities is essential to enable the design of effective in-
+terventions to increase welfare and resilience of societies.
+One example is the sector of agriculture in which monitor-
+ing of crop development can help design optimum strategies
+aimed at improving the welfare of farmers and resilience of
+the food production system. The second of United Nations
+Sustainable Development Goals (SDG) of Ending Hunger
+relies on increasing the crop productivity and revenues of
+farmers in poor and developing countries [35] - approxi-
+mately 2.5 billion people’s livelihoods depend mainly on
+producing crops [10]. Achieving SDG 2 goals requires to be
+Figure 1. Model and performance overview. (top) TSViT archi-
+tecture. A more detailed schematic is presented in Fig.4. (bottom)
+TSViT performance compared with previous arts (Table 2).
+able to accurately monitor yields and the evolution of culti-
+vated areas in order to measure the progress towards achiev-
+ing several goals, as well as to evaluate the effectiveness of
+different policies or interventions. In the European Union
+(EU) the Sentinel for Common Agricultural Policy program
+(Sen4CAP) [2] focuses on developing tools and analytics to
+support the verification of direct payments to farmers with
+underlying environmental conditionalities such as the adop-
+tion of environmentally-friendly [50] and crop diversifica-
+tion [51] practices based on real-time monitoring by the
+European Space Agency’s (ESA) Sentinel high-resolution
+arXiv:2301.04944v1  [cs.CV]  12 Jan 2023
+
+ClassiticationHead
+Segmentation Head
+Spatial Encoder
+Temporal Encoder
+Token embeddingGermany
+PASTIS
+T31TFM1618
+86
++7.7%
+66
++2.7%
+64
++4.3%
+segmentation
+428864
+63
+64
+(%)nojw
+62
+62
+61
+60
+60
+58
+59
+58
+72
+56
+90
++3.8%
++3.7%
+76
+88
++2.5%
+74
+classification
+mAcc(%)
+86
+72
+84
+70
+82
+68
+66
+80
+69
+64
+78
+68
+62
+sota architectures
+TSViT (ours)satellite constellation [1] to complement on site verifica-
+tion.
+Recently, the volume and diversity of space-borne
+Earth Observation (EO) data [63] and post-processing tools
+[18, 61, 70] has increased exponentially.
+This wealth of
+resources, in combination with important developments in
+machine learning for computer vision [20,28,53], provides
+an important opportunity for the development of tools for
+the automated monitoring of crop development.
+Towards more accurate automatic crop type recognition,
+we introduce TSViT, the first fully-attentional1 architecture
+for general SITS processing. An overview of the proposed
+architecture can be seen in Fig.1 (top). Our novel design
+introduces some inductive biases that make TSViT particu-
+larly suitable for the target domain:
+• Satellite imagery for monitoring land surface variabil-
+ity boast a high revisit time leading to long temporal
+sequences, for example Sentinel-2 (S2) satellites have
+an average revisit time of 5 days resulting in 60-70
+acquisitions per year. To reduce the amount of com-
+putation we factorize input dimensions into their tem-
+poral and spatial components, providing intuition (sec-
+tion 3.4) and experimental evidence (section 4.2) about
+why the order of factorization matters.
+• TSViT uses a Transformer backbone [64] following
+the recently proposed ViT framework [13]. As a result,
+every TSViT layer has a global receptive field in time
+or space, in contrast to previously proposed convolu-
+tional and recurrent architectures [14,24,40,45,49].
+• To make our approach more suitable for SITS mod-
+elling we propose a tokenization scheme for the in-
+put image timeseries and propose acquisition-time-
+specific temporal position encodings in order to extract
+date-aware features and to account for irregularities in
+SITS acquisition times (section 3.6).
+• We make modifications to the ViT framework (sec-
+tion 3.2) to enhance its capacity to gather class-specific
+evidence which we argue suits the problem at hand
+and design two custom decoder heads to accommodate
+both global and dense predictions (section 3.5).
+Our provided intuitions are tested through extensive abla-
+tion studies on design parameters presented in section 4.2.
+Overall, our architecture achieves state-of-the-art perfor-
+mance in three publicly available datasets for classification
+and semantic segmentation presented in Table 2 and Fig.1.
+2. Related work
+2.1. Crop type recognition
+Crop type recognition is a subcategory of land use recog-
+nition which involves assigning one of K crop categories
+1without any convolution operations
+(classes) at a set of desired locations on a geospatial grid.
+For successfully doing so modelling the temporal patterns
+of growth during a time period of interest has been shown
+to be critical [15, 44]. As a result, model inputs are time-
+series of T satellite images of spatial dimensions H × W
+with C channels, X ∈ RT ×H×W ×C rather than single ac-
+quisitions. There has been a significant body of work on
+crop type identification found in the remote sensing liter-
+ature [8, 9, 19, 39, 41, 55]. These works typically involve
+multiple processing steps and domain expertise to guide
+the extraction of features, e.g.
+NDVI [25], that can be
+separated into crop types by learnt classifiers.
+More re-
+cently, Deep Neural Networks (DNN) trained on raw op-
+tical data [22,26,29,46,47,62] have been shown to outper-
+form these approaches. At the object level, (SITS classi-
+fication) [24, 40, 48] use 1D data of single-pixel or parcel-
+level aggregated feature timeseries, rather than the full SITS
+record, learning a mapping f : RT ×C → RK. Among
+these works, TempCNN [40] employs a simple 1D convo-
+lutional architecture, while [48] use the Transformer archi-
+tecture [64]. DuPLo [24] consists of an ensemble of CNN
+and RNN streams in an effort to exploit the complementar-
+ity of extracted features. Finally, [16] view satellite images
+as un-ordered sets of pixels and calculate feature statistics
+at the parcel level, but, in contrast to previously mentioned
+approaches, their implementation requires knowledge of the
+object geometry. At the pixel level (SITS semantic seg-
+mentation), models learn a mapping f(X) ∈ RH×W ×K.
+For this task, [47] show that convolutional RNN variants
+(CLSTM, CGRU) [54] can automatically extract useful fea-
+tures from raw optical data, including cloudy images, that
+can be linearly separated into classes.
+The use of CNN
+architectures is explored in [45] who employ two models:
+a UNET2D feature extractor, followed by a CLSTM tem-
+poral model (UNET2D-CLSTM); and a UNET3D fully-
+convolutional model. Both are found to achieve equivalent
+performances. In a similar spirit, [7] use a FPN [31] feature
+extractor, coupled with a CLSTM temporal model (FPN-
+CLSTM). The UNET3Df architecture [60] follows from
+UNET3D but uses a different decoder head more suited to
+contrastive learning. The U-TAE architecture [14] follows
+a different approach, in that it employs the encoder part of
+a UNET2D, applied on parallel on all images, and a sub-
+sequent temporal attention mechanism which collapses the
+temporal feature dimension. These spatial-only features are
+further processed by the decoder part of a UNET2D to ob-
+tain dense predictions.
+2.2. Self-attention in vision
+Convolutional [20, 28, 57] and fully-convolutional net-
+works (FCN) [52, 53] have been the de-facto model of
+choice for vision tasks over the past decade. The convo-
+lution operation extracts translation-equivariant features via
+
+application of a small square kernel over the spatial extent
+of the learnt representation and grows the feature recep-
+tive field linearly over the depth of the network. In con-
+trast, the self-attention operation, introduced as the main
+building block of the Transformer architecture [64], uses
+self-similarity as a means for feature aggregation and can
+have a global receptive field at every layer. Following the
+adoption of Transformers as the dominant architecture in
+natural language processing tasks [6,12,64], several works
+have attempted to exploit self-attention in vision architec-
+tures. Because the time complexity of self-attention scales
+quadratically with the size of the input, its naive implemen-
+tation on image data, which typically contain more pixels
+than text segments contain words, would be prohibitive. To
+bypass this issue, early works focused on improving effi-
+ciency by injecting self-attention layers only at specific lo-
+cations within a CNN [5, 67] or by constraining their re-
+ceptive field to a local region [38,42,65], however, in prac-
+tice, these designs do not scale well with available hard-
+ware leading to slow throughput rates, large memory re-
+quirements and long training times. Following a different
+approach, the Vision Transformer (ViT) [13], presented in
+further detail in section 3.1, constitutes an effort to apply
+a pure Transformer architecture on image data, by propos-
+ing a simple, yet efficient image tokenization strategy. Sev-
+eral works have drawn inspiration from ViT to develop
+novel attention-based architectures for vision. For image
+recognition, [32, 69] re-introduce some of the inductive bi-
+ases that made CNNs successful in vision, leading to im-
+proved performances without the need for long pre-training
+schedules, [43, 59] employ Transformers for dense predic-
+tion, [11,58,71] for object detection and [3,36,68] for video
+processing. Among these works, our framework is more
+closely related to [3] who also use a spatio-temporal fac-
+torization of input dimensions, and [59] who use multiple
+learnable tokens for semantic segmentation. However, we
+deviate significantly from [3] by introducing acquisition-
+time-specific temporal encodings to accommodate an un-
+even distribution of images in time, reverse the order of fac-
+torization and are interested in both global and dense pre-
+dictions (section 3.4). Additionally, we differ from [59] in
+that we introduce the cls tokens as an input to the encoder
+in order to collapse the time dimension and obtain class-
+specific features, while they use them as class queries inputs
+to the decoder similar to their use in [11]. We also differ sig-
+nificantly from [59] in terms of the decoder design as they
+resize the output of the penultimate layer to match the in-
+put size and further process that to obtain pixel-level logits,
+while we decode each token directly into a region matching
+input patch dimensions and reassemble these into a dense
+probability map (section 3.5).
+Figure 2. Backbone architectures. (a) Transformer backbone,
+(b) ViT architecture, (c) TSViT backbone employs additional cls
+tokens (red), each responsible for predicting a single class.
+3. Method
+In this section we present the TSViT architecture in de-
+tail. First, we give a brief overview of the ViT (section 3.1)
+which provided inspiration for this work. In section 3.2 we
+present our modified TSViT backbone, followed by our to-
+kenization scheme (section 3.3), encoder (section 3.4) and
+decoder (section 3.5) modules. Finally, in section 3.6, we
+discuss several considerations behind the design of our po-
+sition encoding scheme.
+3.1. Primer on ViT
+Inspired by the success of Transformers in natural lan-
+guage processing tasks [64] the ViT [13] is an application
+of the Transformer architecture to images with the fewest
+possible modifications. Their framework involves the to-
+kenization of a 2D image X ∈ RH×W ×C to a set of
+patch tokens Z ∈ RN×d by splitting it into a sequence of
+N = ⌊ H
+h ⌋⌊ W
+w ⌋ same-size and non-overlapping patches of
+spatial extent (h × w) which are flattened into 1D tokens
+xi ∈ RhwC and linearly projected into d dimensions. Over-
+all, the process of token extraction is equivalent to the appli-
+cation of 2D convolution with kernel size (h × w) at strides
+(h, w) across respective dimensions. The extracted patches
+are used to construct model inputs as follows:
+Z0 = concat(zcls, Z + P) ∈ RN+1×d
+(1)
+A set of learned positional encoding vectors P ∈ RN×d,
+added to Z, are employed to encode the absolute posi-
+tion information of each token and break the permutation
+invariance property of the subsequent Transformer layers.
+A separate learned class (cls) token zcls ∈ Rd [12] is
+prepended to the linearly transformed and positionally aug-
+
+Layer
+Layer
+MSA
+MLP:
+Norm
+Norm
+pp
+eq.(2)
+eq.(3)
+xL
+(a) Transformer backbone
+MLP:
+=
+d-→K
+concat(zcls, Z+P)
+Transformer
+eq.(1)
+:
+(b) ViT
+MLP:
+..+
+d-→1
+concat(Zcls,Z+P)
+Transformer
+eq.(4)
+(c) TsViT backboneFigure 3. SITS Tokenization. We embed each satellite image
+independently following ViT [13]
+mented patch tokens leading to a length N + 1 sequence
+of tokens Z0 which are used as model inputs. The Trans-
+former backbone consists of L blocks of alternating lay-
+ers of Multiheaded Self-Attention (MSA) [64] and residual
+Multi-Layer Perceptron (MLP) (Fig.2(a)).
+Yl = MSA(LN(Zl)) + Zl
+(2)
+Zl+1 = MLP(LN(Yl)) + Yl
+(3)
+Prior to each layer, inputs are normalized following Lay-
+ernorm (LN) [4]. MLP blocks consist of two layers of linear
+projection followed by GELU non-linear activations [21].
+In contrast to CNN architectures, in which spatial dimen-
+sions are reduced while feature dimensions increase with
+layer depth, Transformers are isotropic in that all feature
+maps Zl ∈ R1+N×d have the same shape throughout the
+network. After processing by the final layer L, all tokens
+apart from the first one (the state of the cls token) are dis-
+carded and unormalized class probabilities are calculated by
+processing this token via a MLP. A schematic representation
+of the ViT architecture can be seen in Fig.2(b).
+3.2. Backbone architecture
+In the ViT architecture, the cls token progressively re-
+fines information gathered from all patch tokens to reach
+a final global representation used to derive class probabil-
+ities. Our TSViT backbone, shown in Fig.2(c), essentially
+follows from ViT, with few modifications in the tokeniza-
+tion and decoder layers. More specifically, we introduce K
+(equal to the number of object classes) additional learnable
+cls tokens Zcls ∈ RK×d, compared to ViT which uses a
+single token.
+Z0 = concat(Zcls, Z + P) ∈ RN+K×d
+(4)
+Without deviating from ViT, all cls and positionally aug-
+mented patch tokens are concatenated and processed by the
+L layers of a Transformer encoder. After the final layer,
+we discard all patch tokens and project each cls token into
+a scalar value. By concatenating these values we obtain a
+length K vector of unormalised class probabilities. This de-
+sign choice brings the following two benefits: 1) it increases
+the capacity of the cls token relative to the patch tokens, al-
+lowing them to store more patterns to be used by the MSA
+operation; 2) it allows for more controlled handling of the
+interactions between evidence gathered for each class. Re-
+garding the first point, introducing multiple cls tokens can
+be seen as equivalent to increasing the dimension of a sin-
+gle cls token to an integer multiple of the patch token di-
+mension dcls = kdpatch and split the cls token into k sepa-
+rate subspaces prior to the MSA operation. In this way we
+can increase the capacity of the cls tokens while avoiding
+issues such as the need for asymmetric MSA weight matri-
+ces for cls and patch tokens, which would effectively more
+than double our model’s parameter count. Furthermore, by
+choosing k = K and enforcing a bijective mapping from
+cls tokens to class predictions, the state of each cls token be-
+comes more focused to a specific class with network depth.
+In TSViT we go a step further and explicitly separate cls
+tokens by class after processing with the temporal encoder
+to allow only same-cls-token interactions in the spatial en-
+coder. In section 3.4 we argue why this is a very useful in-
+ductive bias for modelling spatial relationships in crop type
+recognition.
+3.3. Tokenization of SITS inputs
+A SITS record X ∈ RT ×H×W ×C consists of a series
+of T satellite images of spatial dimensions H × W with
+C channels. For the tokenization of our 3D inputs, we can
+extend the tokenization-as-convolution approach to 3D data
+and apply a 3D kernel with size (t×h×w) at stride (t, h, w)
+across temporal and spatial dimensions.
+In this manner
+N = ⌊ T
+t ⌋⌊ H
+h ⌋⌊ W
+w ⌋ non-overlapping tokens xi ∈ RthwC
+are extracted, and subsequently projected to d dimensions.
+Using t > 1, all extracted tokens contain spatio-temporal
+information.
+For the special case of t = 1 each token
+contains spatial-only information for each acquisition time
+and temporal information is accounted for only through the
+encoder layers. Since the computation cost of global self-
+attention layers is quadratic w.r.t. the length of the token se-
+quence O(N 2), choosing larger values for t, h, w can lead
+to significantly reduced number of FLOPS. In our experi-
+ments, however, we have found small values for t, h, w to
+work much better in practice. For all presented experiments
+we use a value of t = 1 motivated in part because this choice
+simplifies the implementation of acquisition-time-specific
+temporal position encodings, described in section 3.6. With
+regards to the spatial dimensions of extracted patches we
+
+H/h
+Conv2d(in channels=C, out channels=d
+kernel size=(h, w), stride=(h, w))
+H
+h
+-M→
+W
+t1
+tT
+timehave found small values to work best for semantic segmen-
+tation, which is reasonable given that small patches retain
+additional spatial granularity. In the end, our tokenization
+scheme is similar to ViT’s applied in parallel for each acqui-
+sition as shown in Fig.3, however, at this stage, instead of
+unrolling feature dimensions, we retain the spatial structure
+of the original input as reshape operations will be handled
+by the TSViT encoder submodules.
+3.4. Encoder architecture
+In the previous section we presented a motivation for us-
+ing small values t, h, w for the extracted patches. Unless
+other measures are taken to reduce the model’s computa-
+tional cost this choice would be prohibitive for process-
+ing SITS with multiple acquisition times. To avoid such
+problems, we choose to factorize our inputs across their
+temporal and spatial dimensions, a practice commonly em-
+ployed for video processing [17,27,37,56,66,72]. We note
+that all these works use a spatial-temporal factorization or-
+der, which is reasonable when dealing with natural images,
+given that it allows the extraction of higher level, semanti-
+cally aware spatial features, whose relationship in time is
+useful for scene understanding. However, we argue that in
+the context of SITS, reversing the order of factorization is
+a meaningful design choice for the following reasons: 1) in
+contrast to natural images in which context can be useful for
+recognising an object, in crop type recognition context can
+provide little information, or can be misleading. This arises
+from the fact that the shape of agricultural parcels, does not
+need to follow its intended use, i.e. most crops can gener-
+ally be cultivated independent of a field’s size or shape. Of
+course there exist variations in the shapes and sizes of agri-
+cultural fields [34], but these depend mostly on local agri-
+cultural practices and are not expected to generalize over
+unseen regions. Furthermore, agricultural parcels do not in-
+herently contain sub-components or structure. Thus, know-
+ing what is cultivated in a piece of land is not expected to
+provide information about what grows nearby. This is in
+contrast to other objects which clearly contain structure, e.g.
+in human face parsing there are clear expectations about the
+relative positions of various face parts. To test this hypoth-
+esis we enumerate over all agricultural parcels belonging to
+the most popular crop types in the T31TFM S2 tile in France
+for year 2018 and take crop-type-conditional pixel counts
+over a 1km square region from their centers. Then, we cal-
+culate the cosine similarity of these values with uncondi-
+tional pixel counts over the extent of the T31TFM tile and
+find a high degree of similarity, suggesting that there are no
+significant variations between these distributions; 2) a small
+region in SITS is far more informative than its equivalent in
+natural images, as it contains more channels than regular
+RGB images (S2 imagery contains 13 bands in total) whose
+intensities are averaged over a relatively large area (high-
+est resolution of S2 images is 10 × 10 m2); 3) SITS for
+land cover recognition do not typically contain moving ob-
+jects. As a result, a timeseries of single pixel values can be
+used for extracting features that are informative of a spe-
+cific object part found at that particular location. Therefore,
+several objects can be recognised using only information
+found in a single location; plants, for example, can be recog-
+nised by variations of their spectral signatures during their
+growth cycle. Many works performing crop classification
+do so using only temporal information in the form of time-
+series of small patches [47], pixel statistics over the extent
+of parcels [46] or even values from single pixels [40, 48].
+On the other hand, the spatial patterns in a single image
+are uninformative of the crop type, as evidenced by the low
+performance of systems relying on single images [14]. Our
+encoder architecture can be seen in Fig.4(a,b). We now de-
+scribe the temporal and spatial encoder submodules.
+Temporal encoder Thus, we tokenize a SITS record
+X ∈ RT ×H×W ×C into a set of tokens of size (NT × NH ×
+NW × d), as described in section 3.3 and subsequently re-
+shape to ZT ∈ RNHNW ×NT ×d, to get a list of token time-
+series for all patch locations. The input to the temporal en-
+coder is:
+Z0
+T = concat(ZTcls, ZT + PT[t, :]) ∈ RNHNW ×K+NT ×d
+(5)
+where PT[t, :] ∈ RNT ×d and ZTcls ∈ RK×d are re-
+spectively added and prepended to all NHNW timeseries
+and t ∈ RT is a vector containing all T acquisition times.
+All samples are then processed in parallel by a Transformer
+module. Consequently, the final feature map of the tempo-
+ral encoder becomes ZL
+T ∈ RNHNW ×K+NT ×d in which the
+first K tokens in the temporal dimension correspond to the
+prepended cls tokens. We only keep these tokens, discard-
+ing the remaining NT vectors.
+Spatial encoder We now transpose the first and second
+dimensions in the temporal encoder output, to obtain a list
+of patch features ZS ∈ RK×NHNW ×d for all output classes.
+In a similar spirit, the input to the spatial encoder becomes:
+Z0
+S = concat(ZScls, ZS + PS) ∈ RK×1+NHNW ×d
+(6)
+Where PS ∈ RNHNW ×d are respectively added to all
+K spatial representations and each element of ZScls ∈
+RK×1×d is prepended to each class-specific feature map.
+We note, that while in the temporal encoder cls tokens were
+prepended to all patch locations, now there is a single cls
+token per spatial feature map such that ZScls are used to
+gather global SITS-level information. Processing with the
+spatial encoder leads to a similar size output feature map
+ZL
+S ∈ RK×1+NHNW ×d.
+3.5. Decoder architecture
+The TSViT encoder architecture described in the pre-
+vious section is designed as a general backbone for SITS
+
+Figure 4. TSViT submodules. (a) Temporal encoder. We reshape tokenized inputs, retaining the spatio-temporal structure of SITS, into
+a set of timeseries for each spatial location, add temporal position encodings PT[t, :] for acquisition times t, concatenate local cls tokens
+ZTcls (eq.5) and process in parallel with a Transformer. Only the first K output tokens are retained. (b) Spatial encoder. We reshape
+the outputs of the temporal encoder into a set of spatial feature maps for each cls token, add spatial position encodings PS, concatenate
+global cls tokens ZScls (eq.6) and process in parallel with a Transformer. (c) Segmentation head. Each local cls token is projected into hw
+values denoting class-specific evidence for every pixel in a patch. All patches are then reassembled into the original image dimensions. (d)
+Classification head. Global cls tokens are projected into scalar values, each denoting evidence for the presence of a specific class.
+processing. To accommodate both global and dense pre-
+diction tasks we design two decoder heads which feed on
+different components of the encoder output. We view the
+output of the encoder as ZL
+S = [ZL
+Sglobal|ZL
+Slocal] respec-
+tively corresponding to the states of the global and local
+cls tokens. For image classification, we only make use of
+ZL
+Sglobal ∈ RK×d. We proceed, as described in sec.3.2, by
+projecting each feature into a scalar value and concatenate
+these values to obtain global unormalised class probabilities
+as shown in Fig.4(d). Complementarily, for semantic seg-
+mentation we only use ZL
+Slocal ∈ RK×NHNW ×d. These
+features encode information for the presence of each class
+over the spatial extent of each image patch. By project-
+ing each feature into hw dimensions and further reshaping
+the feature dimension to (h × w) we obtain a set of class-
+specific probabilities for each pixel in a patch. It is pos-
+sible now to merge these patches together into an output
+map (H × W × K) which represents class probabilities for
+each pixel in the original image. This process is presented
+schematically in Fig.4(c).
+3.6. Position encodings
+As described in section 3.4, positional encodings are
+injected in two different locations in our proposed net-
+work. First, temporal position encodings are added to all
+patch tokens before processing by the temporal encoder
+as shown in eq.(5). This operation aims at breaking the
+permutation invariance property of MSA by introducing
+time-specific position biases to all extracted patch tokens.
+For crop recognition encoding the absolute temporal po-
+sition of features is important as it helps identifying a
+plant’s growth stage within the crop cycle. Furthermore,
+the time interval between successive images in SITS varies
+depending on acquisition times and other factors, such as
+the degree of cloudiness or corrupted data. To introduce
+acquisition-time-specific biases into the model, our tempo-
+ral position encodings PT[t, :] depend directly on acquisi-
+tion times t. More specifically, we make note of all the
+dates t′ = [t1, t2, ..., tT ′] corresponding to the acquisition
+times found in the training data and construct a lookup ta-
+ble PT ∈ RT ′×d containing all learnt temporal position
+encodings indexed by date. Finding the date-specific en-
+codings that need to be added to patch tokens (eq.5) re-
+duces to looking up appropriate indices from PT. In this
+way temporal position encodings introduce a dynamic prior
+of where to look at in the models’ global temporal receptive
+field, rather than simply encoding the order of SITS acquisi-
+tions which would discard valuable information. Following
+token processing by the temporal encoder, spatial position
+embeddings PS are added to the extracted cls tokens. These
+are not dynamic in nature and are similar to the position en-
+codings used in the original ViT architecture, with the dif-
+ference that these biases are now added to K feature maps
+instead of a single one.
+
+00·0000...
+00·0
+location in time
+Transformer
+Transformer
+location in space
+patch (local) cls tokens
+0-000-00
+000-0
+000-[
+0.000
+global cls tokens
+Reassemble
+concat(Zcls, Z + PT(t)
+concat(Zcls, Z + Ps)
+Zrels 
+Zscls 
+Z + PT(t)
+Z+ Ps
+eq.(5)
+Pr[t,: 
+.00
+00:0
+Ps
+Reshape: (K x NHNw x hw)
+→(NHNw x h x w x K)
+concat(*)
+Reshape: (T × NH × Nw × d) -→ (NHNw × T × d)
+Reshape: (NHNw × K × d) → (K × NHNw × d)
+MLP: d → hw
+MLP: d→1
+...
+0:00
+0·0
+00-0
+(a) Temporal Encoder
+(b) Spatial Encoder
+(c) Segmentation Head
+(d) Classification Head4. Experiments
+We apply TSViT to two tasks using SITS records X ∈
+RT ×H×W ×C as inputs: classification and semantic seg-
+mentation. At the object level, classification models learn
+a mapping f(X) ∈ RK for the object occupying the center
+of the H × W region. Semantic segmentation models learn
+a mapping f(X) ∈ RH×W ×K, predicting class probabili-
+ties for each pixel over the spatial extent of the SITS record.
+We use an ablation study on semantic segmentation to guide
+model design and hyperparameter tuning and proceed with
+presenting our main results on three publicly available SITS
+semantic segmentation and classification datasets.
+4.1. Training and evaluation
+Datasets To evaluate the performance of our proposed
+semantic segmentation model we are using three publicly
+available S2 land cover recognition datasets. The dataset
+presented in [47] covers a densely cultivated area of interest
+of 102×42 km2 north of Munich, Germany and contains 17
+distinct classes. Individual image samples cover a 240×240
+m2 area (24×24 pixels) and contain 13 bands. The PASTIS
+dataset [14] contains images from four different regions in
+France with diverse climate and crop distributions, span-
+ning over 4000 km2 and including 18 crop types. In total,
+it includes 2.4k SITS samples of size 128 × 128, each con-
+taining 33-61 acquisitions and 10 image bands. Because
+the PASTIS sample size is too large for efficiently train-
+ing TSViT with available hardware, we split each sample
+into 24 × 24 patches and retain all acquisition times for a
+total of 57k samples. To accommodate a large set of ex-
+periments we only use fold 1 among the five folds provided
+in PASTIS. Finally, we use the T31TFM-1618 dataset [60]
+which covers a densely cultivated S2 tile in France for years
+2016-18 and includes 20 distinct classes. In total, it includes
+140k samples of size 48 × 48, each containing 14-33 ac-
+quisitions and 13 image bands. For the SITS classification
+experiments, we construct the datasets from the respective
+segmentation datasets. More specifically, for PASTIS we
+use the provided object instance ids to extract 24 × 24 pixel
+regions whose center pixel falls inside each object and use
+the class of this object as the sample class. The remain-
+ing two datasets contain samples of smaller spatial extent,
+making the above strategy not feasible in practice. Here, we
+choose to retain the samples as they are and assign the class
+of the center pixel as the global class. We note that this
+strategy forces us to discard samples in which the center
+pixels belongs to the background class. Additional details
+are provided in the supplementary material.
+Implementation details For all experiments presented
+we train for the same number of epochs using the provided
+data splits from the respective publications for a fair com-
+parison. More specifically, we train on all datasets using
+the provided training sets and report results on the valida-
+Ablation
+Settings
+mIoU
+Factorization order
+Spatial & Temporal
+48.8
+Temporal & Spatial
+78.5
+#cls tokens
+1
+78.5
+K
+83.6
+Position encodings
+Static
+80.8
+Date lookup
+83.6
+Interactions between
+cls tokens
+Temporal
+Spatial
+✓
+✓
+✓
+XXX
+81.5
+✓
+✓
+83.6
+Patch size
+2 × 2
+84.8
+3 × 3
+83.6
+4 × 4
+81.5
+6 × 6
+79.6
+Table 1. Ablation on design choices for TSViT. All proposed
+design choices are found to have a positive effect on performance.
+tion sets for Germany and T31TFM-1618, and on the test
+set for PASTIS. For training TSViT we use the AdamW op-
+timizer [23] with a learning rate schedule which includes a
+warmup period starting from zero to a maximum value 10−3
+at epoch 10, followed by cosine learning rate decay [33]
+down to 5 ∗ 10−6 at the end of training. For Germany and
+T31TFM-1618 we train with the above settings and report
+the best performances between what we achieve and the
+original studies. Since we split PASTIS, we are training
+with both settings and report the best results. Overall, we
+find that our settings improve model performance. We train
+with a batch size of 16 or 32 and no regularization on ×2
+Nvidia Titan Xp gpus in a data parallel fashion. All mod-
+els are trained with a Masked Cross-Entropy loss, masking
+the effect of the background class in both training loss and
+evaluation metrics. We report overall accuracy (OA), aver-
+aged over pixels, and mean intersection over union (mIoU)
+averaged over classes. For SITS classification, in addition
+to the 1D models presented in section 2 we modify the best
+performing semantic segmentation models by aggregating
+extracted features across space prior to the application of a
+classifier, thus, outputing a single prediction. Classification
+models are trained with Focal loss [30]. We report OA and
+mean accuracy (mAcc) averaged over classes.
+4.2. Ablation studies
+We perform an ablation study on design parameters of
+our framework using the Germany dataset [47]. Starting
+with a baseline TSViT with L = 4 for both encoder net-
+works, a single cls token, h = w = 3, t = 1, d =
+128 we successively update our design after each ablation.
+Here, we present the effect of the most important design
+choices; additional ablations are presented in the supple-
+mentary material. Overall, we find that the order of factor-
+
+Dataset
+Germany [47]
+PASTIS [14]
+T31TFM-1618 [60]
+Model
+Semantic segmentation (OA / mIoU)
+BiCGRU [47]
+91.3 / 72.3
+80.5 / 56.2
+88.6 / 57.7
+FPN-CLSTM [7]
+91.8 / 73.7
+81.9 / 59.5
+88.4 / 57.8
+UNET3D [45]
+92.4 / 75.2
+82.3 / 60.4
+88.4 / 57.6
+UNET3Df [60]
+92.4 / 75.4
+82.1 / 60.2
+88.6 / 57.7
+UNET2D-CLSTM [45]
+92.9/ 76.2
+82.7 / 60.7
+89.0 / 58.8
+U-TAE [14]
+93.1 / 77.1
+82.9 / 62.4
+88.9 / 58.5
+TSViT (ours)
+95.0 / 84.8
+83.4 / 65.1 (83.4/ 65.4)
+90.3 / 63.1
+Model
+Object classification (OA / mAcc)
+TempCNN∗ [40]
+89.8 / 78.4
+84.8 / 69.1
+84.7 / 62.6
+DuPLo∗ [24]
+93.1 / 82.2
+84.8 / 69.4
+83.9 / 69.5
+Transformer∗ [48]
+92.4/ 84.3
+84.4 / 68.1
+84.3 / 71.4
+UNET3D [45]
+92.7 / 83.9
+84.8 / 70.2
+84.8 / 71.4
+UNET2D-CLSTM [45]
+93.0 / 84.0
+84.7 / 70.3
+84.7 / 71.6
+U-TAE [14]
+92.6 / 83.7
+84.9 / 71.8
+84.8 / 71.7
+TSViT (ours)
+94.7 / 88.1
+87.1 / 75.5
+87.8 / 74.2
+Table 2. Comparison with state-of-the-art models from literature. (top) Semantic segmentation. (bottom) Object classification. ∗1D
+temporal only models. We report overall accuracy (OA), mean intersection over union (mIoU) and mean accuracy (mAcc). For PASTIS
+we report results for fold-1 only; average test set performance across all five folds is shown in parenthesis for direct comparison with [14].
+ization is the most important design choice in our proposed
+framework.
+Using a spatio-temporal factorization from
+the video recognition literature performs poorly at 48.8%
+mIoU. Changing the factorization order to temporo-spatial
+raises performance by an absolute +29.7% to 78.5% mIoU.
+Including additional cls tokens increases performance to
+83.6%mIoU (+5.1%), so we proceed with using K cls to-
+kens in our design. We test the effect of our date-specific
+position encodings compared to a fixed set of values and
+find a significant −2.8% performance drop from using fixed
+size PT compared to our proposed lookup encodings. As
+discussed in section 3.4 our spatial encoder blocks cross cls-
+token interactions. Allowing interactions among all tokens
+comes at a significant increase in compute cost, O(K2) to
+O(K), and is found to decrease performance by −2.1%
+mIoU. Finally, we find that smaller patch sizes generally
+work better, which is reasonable given that tokens retain a
+higher degree of spatial granularity and are used to predict
+smaller regions. Using 2 × 2 patches raises performance by
++1.2%mIoU to 84.8% compared to 3 × 3 patches. Our fi-
+nal design which is used in the main experiments presented
+in Table 2 employs a temporo-spatial design with K cls to-
+kens, acquisition-time-specific position encodings, 2×2 in-
+put patches and four layers for both encoders.
+4.3. Comparison with SOTA
+In Table 2 and Fig.1, we present performance results of
+our final TSViT design compared to state-of-the-art mod-
+els presented in section 2. For semantic segmentation, we
+find that all models from literature perform similarly, with
+the BiCGRU being overall the worst performer, match-
+ing CNN-based architectures only in T31TFM-1618. For
+Figure 5. Visualization of predictions in Germany. The back-
+ground class is shown in white, ”x” indicates a false prediction.
+all datasets, TSViT outperforms previously suggested ap-
+proaches by a very large margin. A visualization of predic-
+tions in Germany for the top-3 performers is shown in Fig.5.
+In object classification, we observe that 1D temporal mod-
+els are generally outperformed by spatio-temporal models,
+with the exception of the Transformer [48]. All 1D models
+perform poorly in PASTIS. Again, TSViT trained for clas-
+sification consistently outperforms all other approaches by
+a large margin across all datasets. In both tasks, we find
+smaller improvements for the pixel-averaged compared to
+class-averaged metrics, which is reasonable given the large
+class imbalance that characterizes the datasets.
+
+Ground truth
+UNET3Df
+UNET2D-CLSTM
+TSViT
+×x
+×x
+X
+XXXXXXXX
+XXX
+X×>
+XXXXX
+××××
+××5. Conclusion
+In this paper we proposed TSViT, the first fully-
+attentional architecture for general SITS processing.
+By
+taking advantage of the Transformer’s global receptive field,
+capacity to learn a rich feature space and by incorporating
+inductive biases that suit SITS data, we surpass the state-of-
+the-art performance by a large margin in object classifica-
+tion and semantic segmentation using three publicly avail-
+able land cover recognition datasets.
+References
+[1] European Space Agency. The sentinel missions. https://
+www.esa.int/Applications/Observing_the_
+Earth/Copernicus/The_Sentinel_missions.
+Accessed: 2022-11-11. 2
+[2] European Space Agency. Sentinels for common agriculture
+policy. http://esa-sen4cap.org/. Accessed: 2022-
+11-11. 1
+[3] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen
+Sun, Mario Luˇci´c, and Cordelia Schmid. Vivit: A video vi-
+sion transformer. In 2021 IEEE/CVF International Confer-
+ence on Computer Vision (ICCV), pages 6816–6826, 2021.
+3
+[4] Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer
+normalization. ArXiv, abs/1607.06450, 2016. 4
+[5] I. Bello, B. Zoph, Q. Le, A. Vaswani, and J. Shlens. Atten-
+tion augmented convolutional networks. In 2019 IEEE/CVF
+International Conference on Computer Vision (ICCV), pages
+3285–3294, 2019. 3
+[6] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Sub-
+biah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakan-
+tan, Pranav Shyam, Girish Sastry, Amanda Askell, Sand-
+hini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom
+Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler,
+Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric
+Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack
+Clark, Christopher Berner, Sam McCandlish, Alec Radford,
+Ilya Sutskever, and Dario Amodei.
+Language models are
+few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell,
+M.F. Balcan, and H. Lin, editors, Advances in Neural Infor-
+mation Processing Systems, volume 33, pages 1877–1901.
+Curran Associates, Inc., 2020. 3
+[7] Jorge Andres Chamorro Martinez, Laura Elena Cu´e La
+Rosa, Raul Queiroz Feitosa, Ieda Del’Arco Sanches, and
+Patrick Nigri Happ. Fully convolutional recurrent networks
+for multidate crop recognition from multitemporal image se-
+quences.
+ISPRS Journal of Photogrammetry and Remote
+Sensing, 171:188–201, 2021. 2, 8
+[8] Christopher Conrad, Stefan Dech, Olena Dubovyk, Sebas-
+tian Fritsch, Doris Klein, Fabian L¨ow, Gunther Schorcht, and
+Julian Zeidler. Derivation of temporal windows for accurate
+crop discrimination in heterogeneous croplands of Uzbek-
+istan using multitemporal rapideye images. Computers and
+Electronics in Agriculture, 103:63–74, 2014. 2
+[9] Christopher Conrad, Sebastian Fritsch, Julian Zeidler, Gerd
+R¨ucker, and Stefan Dech. Per-field irrigated crop classifica-
+tion in arid central asia using spot and aster data. Remote
+Sensing, 2(4):1035–1056, 2010. 2
+[10] Gordon Conway.
+One Billion Hungry: Can we Feed the
+World? Cornell University Press, 2012. 1
+[11] Xiyang Dai, Yinpeng Chen, Jianwei Yang, Pengchuan
+Zhang, Lu Yuan, and Lei Zhang. Dynamic detr: End-to-end
+object detection with dynamic attention. In 2021 IEEE/CVF
+International Conference on Computer Vision (ICCV), pages
+2968–2977, 2021. 3
+[12] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina
+Toutanova. BERT: Pre-training of deep bidirectional trans-
+formers for language understanding. In Proceedings of the
+2019 Conference of the North American Chapter of the As-
+sociation for Computational Linguistics: Human Language
+Technologies, Volume 1 (Long and Short Papers), pages
+4171–4186, Minneapolis, Minnesota, June 2019. Associa-
+tion for Computational Linguistics. 3
+[13] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov,
+Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner,
+Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl-
+vain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is
+worth 16x16 words: Transformers for image recognition at
+scale. In International Conference on Learning Representa-
+tions, 2021. 2, 3, 4
+[14] Vivien Sainte Fare Garnot and Loic Landrieu.
+Panoptic
+segmentation of satellite image time series with convolu-
+tional temporal attention networks. In Proceedings of the
+IEEE/CVF International Conference on Computer Vision
+(ICCV), pages 4872–4881, October 2021. 2, 5, 7, 8
+[15] V. S. F. Garnot, L. Landrieu, S. Giordano, and N. Chehata.
+Time-space tradeoff in deep learning models for crop clas-
+sification on satellite multi-spectral image time series.
+In
+IGARSS 2019 - 2019 IEEE International Geoscience and Re-
+mote Sensing Symposium, pages 6247–6250, 2019. 2
+[16] Vivien Sainte Fare Garnot, Loic Landrieu, Sebastien Gior-
+dano, and Nesrine Chehata. Satellite image time series clas-
+sification with pixel-set encoders and temporal self-attention.
+In Proceedings of the IEEE/CVF Conference on Computer
+Vision and Pattern Recognition (CVPR), June 2020. 2
+[17] Rohit Girdhar and Deva Ramanan. Attentional pooling for
+action recognition. In I. Guyon, U. Von Luxburg, S. Bengio,
+H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, ed-
+itors, Advances in Neural Information Processing Systems,
+volume 30. Curran Associates, Inc., 2017. 5
+[18] Noel
+Gorelick,
+Matt
+Hancher,
+Mike
+Dixon,
+Simon
+Ilyushchenko, David Thau, and Rebecca Moore.
+Google
+earth engine: Planetary-scale geospatial analysis for every-
+one. Remote Sensing of Environment, 202:18–27, 2017. Big
+Remotely Sensed Data: tools, applications and experiences.
+2
+[19] Pengyu Hao, Yulin Zhan, Li Wang, Zheng Niu, and Muham-
+mad Shakir. Feature selection of time series modis data for
+early crop classification using random forest: A case study
+in Kansas, USA. Remote Sensing, 7(5):5347–5369, 2015. 2
+[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image recognition. In 2016 IEEE
+Conference on Computer Vision and Pattern Recognition
+(CVPR), pages 770–778, 2016. 2
+
+[21] Dan Hendrycks and Kevin Gimpel.
+Gaussian error linear
+units (gelus). arXiv: Learning, 2016. 4
+[22] Dino Ienco, Raffaele Gaetano, Claire Dupaquier, and Pierre
+Maurel. Land cover classification via multitemporal spatial
+data by deep recurrent neural networks. IEEE Geoscience
+and Remote Sensing Letters, 14(10):1685–1689, 2017. 2
+[23] Loshchilov Ilya, Hutter Frank, et al. Decoupled weight decay
+regularization. Proceedings of ICLR, 2019. 7
+[24] Roberto Interdonato, Dino Ienco, Raffaele Gaetano, and
+Kenji Ose.
+DuPLO: A dual view point deep learning ar-
+chitecture for time series classification. ISPRS Journal of
+Photogrammetry and Remote Sensing, 149:91 – 104, 2019.
+2, 8
+[25] Xue Jinru and Baofeng Su. Significant remote sensing veg-
+etation indices: A review of developments and applications.
+Journal of Sensors, 2017:1–17, 01 2017. 2
+[26] Andreas Kamilaris and Francesc X. Prenafeta-Bold´u. Deep
+learning in agriculture: A survey. Computers and Electronics
+in Agriculture, 147:70–90, 2018. 2
+[27] Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas
+Leung, Rahul Sukthankar, and Li Fei-Fei. Large-scale video
+classification with convolutional neural networks. In 2014
+IEEE Conference on Computer Vision and Pattern Recogni-
+tion, pages 1725–1732, 2014. 5
+[28] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.
+Imagenet classification with deep convolutional neural net-
+works. In F. Pereira, C.J. Burges, L. Bottou, and K.Q. Wein-
+berger, editors, Advances in Neural Information Processing
+Systems, volume 25. Curran Associates, Inc., 2012. 2
+[29] Nataliia Kussul, Mykola Lavreniuk, Sergii Skakun, and An-
+drii Shelestov. Deep learning classification of land cover and
+crop types using remote sensing data. IEEE Geoscience and
+Remote Sensing Letters, 14(5):778–782, 2017. 2
+[30] Tsung-Yi Lin, Priya Goyal, Ross B. Girshick, Kaiming He,
+and Piotr Doll´ar.
+Focal loss for dense object detection.
+CoRR, abs/1708.02002, 2017. 7
+[31] Tsung-Yi Lin, Piotr Dollar, Ross Girshick, Kaiming He,
+Bharath Hariharan, and Serge Belongie.
+Feature pyramid
+networks for object detection. In Proceedings of the IEEE
+Conference on Computer Vision and Pattern Recognition
+(CVPR), July 2017. 2
+[32] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei,
+Zheng Zhang, Stephen Lin, and Baining Guo. Swin trans-
+former: Hierarchical vision transformer using shifted win-
+dows. In Proceedings of the IEEE/CVF International Con-
+ference on Computer Vision (ICCV), pages 10012–10022,
+October 2021. 3
+[33] Ilya Loshchilov and Frank Hutter. SGDR: Stochastic gradi-
+ent descent with warm restarts. In International Conference
+on Learning Representations, 2017. 7
+[34] NASA.
+Agricultural
+patterns.
+https : / /
+earthobservatory.nasa.gov/images/6605/
+agricultural-patterns. last accessed on 2022-9-22.
+5
+[35] United Nations. Goal 2: Zero hunger. https://www.un.
+org/sustainabledevelopment/hunger/.
+Ac-
+cessed: 2022-11-11. 1
+[36] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan Assel-
+mann. Video transformer network. In 2021 IEEE/CVF In-
+ternational Conference on Computer Vision Workshops (IC-
+CVW), pages 3156–3165, 2021. 3
+[37] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan As-
+selmann.
+Video transformer network.
+In Proceedings of
+the IEEE/CVF International Conference on Computer Vision
+(ICCV) Workshops, pages 3163–3172, October 2021. 5
+[38] Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz
+Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Im-
+age transformer. In Jennifer Dy and Andreas Krause, ed-
+itors, Proceedings of the 35th International Conference on
+Machine Learning, volume 80 of Proceedings of Machine
+Learning Research, pages 4055–4064. PMLR, 10–15 Jul
+2018. 3
+[39] Charlotte Pelletier, Silvia Valero, Jordi Inglada, Nicolas
+Champion, and G´erard Dedieu. Assessing the robustness of
+random forests to map land cover with high resolution satel-
+lite image time series over large areas. Remote Sensing of
+Environment, 187:156–168, 2016. 2
+[40] Charlotte Pelletier, Geoffrey I. Webb, and Franc¸ois Petitjean.
+Temporal convolutional neural network for the classification
+of satellite image time series. Remote Sensing, 11(5), 2019.
+2, 5, 8
+[41] Jos´e M. Pe˜na-Barrag´an, Moffatt K. Ngugi, Richard E. Plant,
+and Johan Six. Object-based crop identification using multi-
+ple vegetation indices, textural features and crop phenology.
+Remote Sensing of Environment, 115(6):1301–1316, 2011. 2
+[42] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan
+Bello, Anselm Levskaya, and Jon Shlens. Stand-alone self-
+attention in vision models. In H. Wallach, H. Larochelle,
+A. Beygelzimer, F. d'Alch´e-Buc, E. Fox, and R. Garnett, ed-
+itors, Advances in Neural Information Processing Systems,
+volume 32, pages 68–80. Curran Associates, Inc., 2019. 3
+[43] Ren´e Ranftl, Alexey Bochkovskiy, and Vladlen Koltun. Vi-
+sion transformers for dense prediction. In 2021 IEEE/CVF
+International Conference on Computer Vision (ICCV), pages
+12159–12168, 2021. 3
+[44] Bradley C. Reed, Jesslyn F. Brown, Darrel VanderZee,
+Thomas R. Loveland, James W. Merchant, and Donald O.
+Ohlen. Measuring phenological variability from satellite im-
+agery. Journal of Vegetation Science, 5(5):703–714, 1994.
+2
+[45] Rose Rustowicz, Robin Cheong, Lijing Wang, Stefano Er-
+mon, Marshall Burke, and David B. Lobell. Semantic seg-
+mentation of crop type in Africa: A novel dataset and analy-
+sis of deep learning methods. In CVPR Workshops, 2019. 2,
+8
+[46] M. Rußwurm and M. K¨orner.
+Temporal vegetation mod-
+elling using long short-term memory networks for crop iden-
+tification from medium-resolution multi-spectral satellite im-
+ages. In 2017 IEEE Conference on Computer Vision and Pat-
+tern Recognition Workshops (CVPRW), pages 1496–1504,
+2017. 2, 5
+[47] Marc Rußwurm and Marco K¨orner.
+Multi-temporal land
+cover classification with sequential recurrent encoders. IS-
+PRS International Journal of Geo-Information, 7(4):129,
+Mar 2018. 2, 5, 7, 8
+
+[48] Marc Rußwurm and Marco K¨orner. Self-attention for raw
+optical satellite time series classification. ISPRS Journal of
+Photogrammetry and Remote Sensing, 169:421 – 435, 2020.
+2, 5, 8
+[49] Marc Rußwurm, Charlotte Pelletier, M. Zollner, S´ebastien
+Lef`evre, and Marco K¨orner.
+Breizhcrops:
+A time se-
+ries dataset for crop type mapping. ISPRS - International
+Archives of the Photogrammetry, Remote Sensing and Spa-
+tial Information Sciences, XLIII-B2-2020:1545–1551, 08
+2020. 2
+[50] Sen4CAP.
+Agricultural practices.
+http : / / esa -
+sen4cap . org / content / agricultural -
+practices. Accessed: 2022-11-11. 1
+[51] Sen4CAP. Crop diversification. http://esa-sen4cap.
+org/content/crop-diversification. Accessed:
+2022-11-11. 1
+[52] Pierre Sermanet, David Eigen, Xiang Zhang, Michael Math-
+ieu, Rob Fergus, and Yann LeCun.
+Overfeat: Integrated
+recognition, localization and detection using convolutional
+networks.
+2014.
+Publisher Copyright: © 2014 Interna-
+tional Conference on Learning Representations, ICLR. All
+rights reserved.; 2nd International Conference on Learning
+Representations, ICLR 2014 ; Conference date: 14-04-2014
+Through 16-04-2014. 2
+[53] Evan Shelhamer, Jonathan Long, and Trevor Darrell. Fully
+convolutional networks for semantic segmentation.
+IEEE
+Transactions on Pattern Analysis and Machine Intelligence,
+39(4):640–651, 2017. 2
+[54] Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Ye-
+ung, Wai-kin Wong, and Wang-chun WOO. Convolutional
+LSTM network: A machine learning approach for precipi-
+tation nowcasting. In C. Cortes, N. Lawrence, D. Lee, M.
+Sugiyama, and R. Garnett, editors, Advances in Neural In-
+formation Processing Systems, volume 28, pages 802–810.
+Curran Associates, Inc., 2015. 2
+[55] Sofia Siachalou, Giorgos Mallinis, and Maria Tsakiri-Strati.
+A hidden Markov models approach for crop classification:
+Linking crop phenology to time series of multi-sensor re-
+mote sensing data. Remote Sensing, 7(4):3633–3650, 2015.
+2
+[56] Karen Simonyan and Andrew Zisserman. Two-stream con-
+volutional networks for action recognition in videos.
+In
+NIPS, 2014. 5
+[57] Karen Simonyan and Andrew Zisserman. Very deep convo-
+lutional networks for large-scale image recognition. In In-
+ternational Conference on Learning Representations, 2015.
+2
+[58] Hwanjun Song, Deqing Sun, Sanghyuk Chun, Varun Jam-
+pani, Dongyoon Han, Byeongho Heo, Wonjae Kim, and
+Ming-Hsuan Yang. ViDT: An efficient and effective fully
+transformer-based object detector. In International Confer-
+ence on Learning Representations, 2022. 3
+[59] Robin Strudel, Ricardo Garcia, Ivan Laptev, and Cordelia
+Schmid. Segmenter: Transformer for semantic segmenta-
+tion. In 2021 IEEE/CVF International Conference on Com-
+puter Vision (ICCV), pages 7242–7252, 2021. 3
+[60] Michail Tarasiou, Riza Alp G¨uler, and Stefanos Zafeiriou.
+Context-self contrastive pre-training for crop type semantic
+segmentation.
+IEEE Transactions on Geoscience and Re-
+mote Sensing, pages 1–1, 2022. 2, 7, 8
+[61] Michail Tarasiou and Stefanos Zafeiriou.
+Deepsatdata:
+Building large scale datasets of satellite images for training
+machine learning models.
+In IGARSS 2022 - 2022 IEEE
+International Geoscience and Remote Sensing Symposium,
+pages 4070–4073, 2022. 2
+[62] Michail Tarasiou and Stefanos Zafeiriou. Embedding earth:
+Self-supervised contrastive pre-training for dense land cover
+classification. 2022. 2
+[63] Andrew Tatem, Scott Goetz, and Simon Hay. Fifty years of
+earth observation satellites. American Scientist, 96:390–398,
+09 2008. 2
+[64] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko-
+reit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Il-
+lia Polosukhin.
+Attention is all you need.
+In I. Guyon,
+U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vish-
+wanathan, and R. Garnett, editors, Advances in Neural In-
+formation Processing Systems 30, pages 5998–6008. Curran
+Associates, Inc., 2017. 2, 3, 4
+[65] Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam,
+Alan Yuille, and Liang-Chieh Chen. Axial-DeepLab: Stand-
+alone axial-attention for panoptic segmentation. In ECCV,
+2020. 3
+[66] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua
+Lin, Xiaoou Tang, and Luc Van Gool. Temporal segment net-
+works: Towards good practices for deep action recognition.
+In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling,
+editors, Computer Vision – ECCV 2016, pages 20–36, Cham,
+2016. Springer International Publishing. 5
+[67] X. Wang, R. Girshick, A. Gupta, and K. He. Non-local neu-
+ral networks. In 2018 IEEE/CVF Conference on Computer
+Vision and Pattern Recognition, pages 7794–7803, 2018. 3
+[68] Yuqing Wang, Zhaoliang Xu, Xinlong Wang, Chunhua Shen,
+Baoshan Cheng, Hao Shen, and Huaxia Xia.
+End-to-end
+video instance segmentation with transformers.
+In 2021
+IEEE/CVF Conference on Computer Vision and Pattern
+Recognition (CVPR), pages 8737–8746, 2021. 3
+[69] Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi,
+Zi-Hang Jiang, Francis E.H. Tay, Jiashi Feng, and Shuicheng
+Yan. Tokens-to-token vit: Training vision transformers from
+scratch on imagenet. In Proceedings of the IEEE/CVF In-
+ternational Conference on Computer Vision (ICCV), pages
+558–567, October 2021. 3
+[70] Peng Yue, Boyi Shangguan, Lei Hu, Liangcun Jiang, Chenx-
+iao Zhang, Zhipeng Cao, and Yinyin Pan. Towards a train-
+ing data model for artificial intelligence in earth observation.
+International Journal of Geographical Information Science,
+36(11):2113–2137, 2022. 2
+[71] Zixiao Zhang, Xiaoqiang Lu, Guojin Cao, Yuting Yang,
+Licheng Jiao, and Fang Liu.
+Vit-yolo:transformer-based
+yolo for object detection. In 2021 IEEE/CVF International
+Conference on Computer Vision Workshops (ICCVW), pages
+2799–2808, 2021. 3
+[72] Bolei Zhou, Alex Andonian, Aude Oliva, and Antonio Tor-
+ralba.
+Temporal relational reasoning in videos.
+In Pro-
+ceedings of the European Conference on Computer Vision
+(ECCV), September 2018. 5
+

UdFKT4oBgHgl3EQfli6N/content/tmp_files/2301.11854v1.pdf.txt

@@ -0,0 +1,1840 @@
+MNRAS 000, 1–14 (2022)
+Preprint 30 January 2023
+Compiled using MNRAS LATEX style file v3.0
+GrGadget: an N-body TreePM relativistic code for cosmological
+simulations
+Eduardo Quintana-Miranda,1,2,3★ Pierluigi Monaco1,2,3,4 and Luca Tornatore1,2,3
+1 Dipartimento di Fisica, Sezione di Astronomia, via G.B. Tiepolo 11, I-34143 Trieste, Italy
+2 INAF – Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, I-34143 Trieste, Italy
+3 IFPU – Institute for the Fundamental Physics of the Universe, via Beirut 2, I-34100 Trieste, Italy
+4 INFN – Istituto Nazionale di Fisica Nucleare, Via Valerio 2, I-34127 Trieste, Italy
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present the merging of the Particle-Mesh (PM) relativistic Gevolution code with the TreePM Gadget-4 code, with the
+aim of studying general relativity effects in cosmology. Our code, called GrGadget, is able to track the evolution of metric
+perturbations in the weak field limit by using Gevolution’s implementation of a relativistic PM in the Poisson gauge. To
+achieve this, starting from Gevolution we have written a C++ library called Libgevolution, that allows a code to access and
+use the same abstractions and resources that Gevolution uses for its PM-only N-body simulations. The code works under the
+assumption that particle interactions at short distances can be approximated as Newtonian, so that we can combine the forces
+computed with a Newtonian Tree with those computed with a relativistic PM. The result is a TreePM simulation code that
+represents metric perturbations at the scales where they are relevant, while resolving non-linear structures. We validate our code
+by closely matching Gadget-4 forces, computed with the Tree switched off, with those computed with Libgevolution in the
+Newtonian limit. With GrGadget we obtain a matter power spectrum that is compatible with Newtonian Gadget-4 at small
+scales and contains GR features at large scales that are consistent with results obtained with Gevolution. We demonstrate that,
+due to the better resolution of the highly non-linear regime, the representation of the relativistic fields sampled on the mesh
+improves with respect to the PM-only simulations.
+Key words: cosmology: theory – large-scale structure of the Universe
+1 INTRODUCTION
+The state of the art of precision cosmology provides a standard cos-
+mological model, ΛCDM, that is consistent with most observational
+evidence on large scales, but relies on the existence of a dark sector
+populated by Dark Matter (DM) and Dark Energy (DE). The first
+is responsible for the formation of cosmological structures such as
+galaxies and their large-scale density field, while the second causes
+the observed accelerated expansion of the universe in the present
+epoch. Their physical nature is an open problem, since the only ev-
+idence of their existence comes from their gravitational interaction
+with visible matter. A possible explanation is that the dark sector is
+due to a misrepresentation of gravity, that on large scales does not
+follow Einstein’s General Relativity (GR), at the basis of the ΛCDM
+model.
+This fact has triggered a wave of interest in modifications of GR,
+that can lead to extra terms that explain dark energy or dark matter
+(see, e.g., Silvestri & Trodden 2009; Capozziello & De Laurentis
+2012, and references therein). Such modifications must be significant
+only on large scales or low density, because GR is very accurate in
+predicting planetary orbits, light deflection and Doppler effects in
+solar system tests and has more recently been successfully tested
+★ E-mail: [email protected]
+with the detection of gravitational waves (Abbott et al. 2016) and the
+direct imaging of black hole event horizons (Event Horizon Telescope
+Collaboration et al. 2019).
+In order to characterize dark energy in the age of its dominance,
+many projects have been planned to survey large parts of the sky and
+probe the large-scale distribution of matter using galaxy clustering
+and galaxy lensing, both from the ground (DES1, Krause et al. 2017;
+DESI2, DESI Collaboration et al. 2016; Rubin’s LSST3, Ivezić et al.
+2019; SKAO4 surveys) and from space (Euclid5, Laureijs et al. 2011;
+Roman6, Spergel et al. 2015; SphereX7, Doré et al. 2014). Some of
+these surveys have already started to produce a flood of data that will
+soon lead to a precise characterization of the galaxy and matter den-
+sity fields. A comparison of these observations to model predictions,
+either using summary statistics or field-level inference, will lead to
+unprecedented tests not only of the cosmological model but also
+of the gravity theory behind it. With precision being guaranteed by
+1 www.darkenergysurvey.org
+2 www.desi.lbl.gov
+3 www.lsst.org
+4 www.skao.int
+5 sci.esa.int/web/euclid
+6 roman.gsfc.nasa.gov
+7 spherex.caltech.edu
+© 2022 The Authors
+arXiv:2301.11854v1  [astro-ph.CO]  27 Jan 2023
+
+2
+E. Quintana-Miranda et al.
+the amount of available high-quality data, accuracy will be achieved
+only by rigorous control of systematics, both in the data and in theory
+predictions.
+The highly non-linear nature of the observed density field and the
+non-locality of gravity make cosmological simulations necessary to
+compare the predictions of current theories with the observations
+at an increasing level of accuracy. Yet, most of the widely adopted
+simulation codes, like e.g. Gadget-4 (Springel et al. 2021), use New-
+tonian dynamics for the evolution of matter perturbations. This is not
+the ideal configuration to pass from the unobservable distribution
+of matter in a periodic comoving box to the observable distribution
+of light in the past light cone. Relativistic corrections can be added
+a posteriori by post-processing Newtonian simulation outputs; one
+specific example of this approach is the modeling of lensing due
+to the distortion of null geodesics (Bartelmann & Schneider 2001),
+while a more comprehensive approach to adding relativistic effects is
+presented by Borzyszkowski et al. (2017). However, even though the
+biases introduced by this approach are expected to be small, a fully
+self-consistent approach is necessary to convincingly demonstrate
+our ability of controlling theory systematics. For instance, galaxy
+clustering is affected by magnification bias due to lensing, and ne-
+glecting this effect induces a non-negligible bias in parameter esti-
+mation (Lepori et al. 2020; Alam et al. 2021). This is even more true
+when modified gravity theories are used: extensions of gravity are
+typically derived in a full relativistic context, and while they influence
+the Newtonian limit of gravity, the small but measurable relativistic
+effects may provide smoking-gun signals of a specific class of gravity
+theories. In this sense, restricting to the treatment of the Newtonian
+limit of modified gravity theories (as, e.g., in Puchwein et al. 2013)
+may leave out crucial observable signatures.
+Two examples of fully relativistic N-body codes for the evolution of
+cosmic perturbations, that integrate Einstein’s equations to follow the
+motion of massive particles along their geodesics, are the Adaptive
+Mesh Refinement (AMR) code Gramses (Barrera-Hinojosa & Li
+2020) and the Particle-Mesh (PM) code Gevolution (Adamek et al.
+2016). These have proven to be precious tools to produce accurate
+cosmological predictions, like a self-consistent treatment of massive
+neutrinos (Adamek et al. 2022), and to explore phenomena that were
+previously overlooked, like the strength of the frame dragging field
+acting on dark matter haloes (Barrera-Hinojosa et al. 2020). These
+codes sample the fields in a mesh that fills the simulated volume,
+but while Gramses uses an AMR scheme to increase resolution only
+where it is needed, PM schemes working on a single non-adaptive
+mesh are well known to be limited by memory, so they are unable to
+achieve the large dynamic range required, e.g., to resolve DM halos in
+large cosmological volumes. The integration of Newtonian particle
+trajectories has historically been addressed with the introduction of an
+oct-tree data structure (Barnes & Hut 1986), that provides a 𝑁 log 𝑁
+scaling for the computation of gravity without compromising its
+accuracy. Because the integration of large-scale perturbations is very
+slow in this scheme, such an oct-tree is used to compute short-range
+forces, and is complemented by a Particle-Mesh (PM) code on large
+scales. The resulting algorithm is commonly called TreePM, and it
+is the standard gravity solver for Gadget-4.
+As we will show in next Section, deviations from a pure Newtonian
+approach become significant on scales that are comparable with the
+Hubble horizon, so a Newtonian treatment of small-scale clustering,
+performed by the Tree algorithm, would introduce a negligible error
+if large scales are treated by a fully relativistic gravity solver. This
+can be achieved, in a TreePM scheme, by using a relativistic PM code
+for large-scale gravity, where relativistic potentials are sampled on a
+small enough mesh so as to be effectively Newtonian on the scales
+where the Tree code gets in.
+In this paper we present an implementation of Gadget-4 that
+uses a PM library, based on Gevolution relativistic code, as the
+PM part of the TreePM solver. This is a step toward the construc-
+tion of an ecosystem of codes and post-processing tools to perform
+end-to-end simulations of future surveys, with the aim of achieving
+optimal control of all systematics, including theoretical ones. The
+paper is organized as follows: Section 2 gives an overview of the the-
+ory of relativistic perturbations, with a focus on the approach used
+in Gevolution. Section 3 gives a description of the Gadget-4 and
+Gevolution codes, and describes the implementation of Libgevo-
+lution and GrGadget. Section 4 presents the tests performed to
+validate GrGadget, while Section 5 gives our conclusions.
+2 THEORY OF RELATIVISTIC PERTURBATIONS
+The success of Newtonian simulations in describing the large-scale
+structure of the universe follows from the fact that, for an observer at
+rest with respect to the CMB, the metric of spacetime is very close to
+Friedmann-Lemaitre-Robertson-Walker’s (FLRW). Deviations from
+the Newtonian approach are expected to be significant, albeit small,
+on scales near the Hubble horizon, or when the energy-momentum
+tensor has relativistic components like radiation or fast massive neu-
+trinos. Deviations from FLRW metric are expected to be strong in
+the proximity of compact objects, but this happens on scales that are
+far smaller than the resolution that can be afforded in simulations of
+large comoving volumes. It is thus fair to assume that the perturba-
+tions to the metric are small and can be described in a weak-field
+regime. This does not imply that deviations of the components of
+the energy-momentum tensor from homogeneity are assumed to be
+small, density perturbations can be highly non-linear: what we re-
+quire is that the size of self-gravitating objects is much larger than
+their gravitational radius.
+The Gevolution code (see Adamek et al. 2016) models the space-
+time metric with a perturbed FLRW metric in the weak field regime.
+In the Poisson gauge the metric can be written as:
+𝑑𝑠2 =𝑎2�
+− 𝑐2 𝑑𝜏2(1 + 2Ψ) − 2𝑐 𝑑𝜏𝑑𝑥𝑖𝐵𝑖+
++ 𝑑𝑥𝑖𝑑𝑥 𝑗 �𝛾𝑖 𝑗 (1 − 2Φ) + ℎ𝑖 𝑗
+��
+,
+(1)
+where 𝑎(𝜏) is the scale factor of the FLRW background, 𝜏 is the
+conformal time and 𝑥𝑖 are the space coordinates. It is possible to
+exploit the residual degrees of freedom of the metric to impose
+the conditions 𝐵𝑖|𝑖 = 0, ℎ𝑖𝑖 = 0 and ℎ𝑖 𝑗 | 𝑗 = 0. In our notation,
+repeated latin indexes denote Einstein’s summation over the spatial
+coordinates 1, 2, 3 and the vertical bar subscript, e.g. 𝐵𝑖| 𝑗, denotes a
+covariant derivative with respect to the affine connection that emerges
+from the background spatial metric 𝛾𝑖 𝑗.
+The choice of the Poisson gauge is convenient because the two
+potentials Ψ and Φ are the gauge-invariant Bardeen potentials, and
+in the Newtonian limit the the field Ψ can be interpreted as the
+gravitational potential. In other words, this is the gauge in which the
+standard N-body solver is integrating the right equations of motion
+in the Newtonian limit (Chisari & Zaldarriaga 2011).
+2.1 Field equations
+The background, characterized by 𝑎(𝜏), is by construction a solution
+of the Einstein’s equations in the presence of a homogeneous and
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+3
+isotropic energy-momentum tensor ¯𝑇 𝜇𝜈:
+¯𝐺𝜇𝜈 = −8𝜋𝐺
+𝑐4
+¯𝑇 𝜇𝜈 ,
+(2)
+where ¯𝐺𝜇𝜈 is Einstein’s tensor constructed from the metric (1) with
+the perturbations Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 set to zero. Applying equation (2) to
+the FLRW metric one obtains Friedmann’s equations.
+To solve for the perturbations of the metric, the usual procedure
+consist in subtracting (2) from the full Einstein’s equations:
+𝐺𝜇𝜈 − ¯𝐺𝜇𝜈 = −8𝜋𝐺
+𝑐4 (𝑇 𝜇𝜈 − ¯𝑇 𝜇𝜈) .
+(3)
+The right hand side now contains the perturbation of the energy-
+momentum tensor due to inhomogeneities in mass and energy dis-
+tributions, while the left hand side is a very complicated non-linear
+expression containing the potentials Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 and their space-
+time derivatives up to second order.
+To reach a tractable set of equations that we can interpret and solve
+numerically, we apply the weak field assumption. The perturbations
+Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 are assumed to be of order 𝜖 ≪ 1. Spatial derivatives
+are known to increase their amplitude by a factor of 𝜖−1/2, accounting
+for the presence of shortwave fluctuations induced by the non-linear
+structure in the energy-momentum tensor, while time derivatives are
+assumed to preserve the perturbation order. Then one can expand
+𝐺𝜇𝜈 − ¯𝐺𝜇𝜈 in terms of the metric perturbations, neglecting contri-
+butions with order higher than 𝜖. For example: Φ is a term of order
+O(𝜖), Φ,𝑖 has order O(𝜖1/2), Φ|𝑛𝑛 is a leading term (order 1, because
+of the second derivative), quadratic terms like Φ,𝑛Φ,𝑛 are O(𝜖), and
+a term like Φ,00 is considered as O(𝜖). This type of expansion is
+known as the shortwave correction (Adamek et al. 2014).
+Furthermore, experience has shown that the scalar perturbations Φ
+and Ψ are generally larger than the vector and tensor perturbations 𝐵𝑖
+and ℎ𝑖 𝑗. Indeed, the scalar potentials, that are sourced by the density
+perturbation Δ𝑇00, become the Newtonian potential in the Newtonian
+limit, while the vector perturbation 𝐵𝑖 is sourced by Δ𝑇0𝑖, that is
+small by a factor of 𝑣/𝑐 for non-relativistic matter perturbations,
+and ℎ𝑖 𝑗 by Δ𝑇𝑖 𝑗, that is suppressed by a (𝑣/𝑐)2 factor. Hence, it is
+fair to drop quadratic terms of 𝐵𝑖 and ℎ𝑖 𝑗 in this weak field limit
+approximation.
+In this approximation, from Eq. (3) it descends that its time-time
+component yields a Poisson-like equation for the scalar Φ:
+Φ|𝑛𝑛(1 + 4Φ) − 3H
+𝑐2 Φ,0 + 3H2
+𝑐2 (𝜒 − Φ) + 3
+2Φ|𝑛Φ|𝑛
+= 4𝜋𝐺𝑎2
+𝑐4
+Δ𝑇00 ,
+(4)
+where H = 𝑎−1 𝑑𝑎
+𝑑𝜏 and 𝜒 = Φ − Ψ. From the time-space section of
+eq. (3) we obtain:
+−
+𝐵𝑖|𝑛𝑛
+4𝑐
+− Φ,𝑖0
+𝑐2
+− H
+𝑐2 (Φ,𝑖 − 𝜒,𝑖) = −4𝜋𝐺𝑎2
+𝑐4
+Δ𝑇0𝑖 ,
+(5)
+that, taking advantage of the condition 𝐵𝑛|𝑛 = 0, can be reduced to:
+−
+𝐵𝑖|𝑛𝑛
+4𝑐
+= −4𝜋𝐺𝑎2
+𝑐4
+𝑃⊥Δ𝑇0𝑖 ,
+(6)
+where 𝑃⊥ is a linear operator that selects from a vector field its
+divergenceless component.
+The traceless part of the spatial section of eq. 3 leads to:
+�
+𝛿 𝑗 𝑏𝛿𝑎𝑖 − 1
+3𝛿𝑎𝑏𝛿 𝑗𝑖
+� �
+𝜒| 𝑗𝑖 − 2Φ| 𝑗𝑖 𝜒 + 4ΦΦ| 𝑗𝑖 + 2Φ| 𝑗Φ|𝑖
++
+1
+2𝑐2 ℎ𝑖 𝑗,00 + H
+𝑐2 ℎ𝑖 𝑗,0 − 1
+2 ℎ𝑖 𝑗 |𝑛𝑛
++ 1
+2𝑐
+� 𝜕
+𝜕𝜏 + 2H
+� �
+𝐵𝑖 | 𝑗 + 𝐵 𝑗 |𝑖� �
+=
+�
+𝛿 𝑗 𝑏𝛿𝑎𝑖 − 1
+3𝛿𝑎𝑏𝛿 𝑗𝑖
+� �
+−8𝜋𝐺
+𝑐4 Δ𝑇𝑖 𝑗
+�
+,
+(7)
+from which we can determine the rest of the metric degrees of free-
+dom 𝜒 and ℎ𝑖 𝑗. Since the source of 𝜒 and ℎ𝑖 𝑗 are the perturbation
+of the of the energy-momentum tensor Δ𝑇𝑖 𝑗, their amplitude in a
+matter dominated universe is suppressed by a factor (𝑣/𝑐)2. That
+is equivalent to say: since dark matter is non-relativistic, 𝜒 and ℎ𝑖 𝑗
+must be very small with respect to Φ or even 𝐵𝑖.
+As a matter of fact, V1.2 of Gevolution implements an improved
+expansion of the metric perturbations, that has been presented in
+Adamek et al. (2017). For our tests we used the implementation
+of the original expansion, the one presented above. However, the
+improved expansion has been ported to Libgevolution and will be
+used when analysing result on the past light cone. We do not expect
+the results presented in this paper to depend on the specific expansion
+used.
+2.2 Motion of particles along geodesics
+Massive particles move along geodesics, whose equation can be
+expressed as:
+𝑑𝑥𝑖
+𝑑𝜏 =
+𝑐𝑝𝑖
+√︁
+(𝑚𝑐𝑎)2 + 𝑝2 + 𝑐𝐵𝑖
++
+𝑐𝑝𝑖
+√︁
+(𝑚𝑐𝑎)2 + 𝑝2
+�
+Ψ + Φ2(𝑚𝑎𝑐)2 + 𝑝2
+(𝑚𝑎𝑐)2 + 𝑝2
+�
+,
+(8)
+𝑑𝑝𝑖
+𝑑𝜏 = − 𝑐
+�
+𝑝𝑛𝐵𝑛|𝑖 + Ψ,𝑖
+√︃
+(𝑚𝑐𝑎)2 + 𝑝2 +
+𝑝2Φ,𝑖
+√︁
+(𝑚𝑐𝑎)2 + 𝑝2
+�
+,
+(9)
+where 𝑝𝑖 is the space part of the particle momentum and 𝑝 its
+norm. The right hand side in the last equation is the generalized
+force acting on the particles. The term proportional to Ψ,𝑖 becomes
+the Newtonian force in the limit of small velocities, while 𝑝𝑛𝐵𝑛|𝑖
+represent the corrections due to frame dragging and the third term in
+parenthesis is a further relativistic correction.
+The energy-momentum tensor is constructed from the knowledge
+of particle positions and momenta, but its computation depends on the
+perturbed metric. This means that, in Eqs. (4), (6), and (7), the source
+terms on the right hand sides depend on the potentials themselves.
+These implicit equations may be solved starting from the potentials
+at the previous time step and solving the equations iteratively until
+convergence. The integration scheme that Gevolution implements
+is simpler: at each time step the energy momentum tensor is computed
+using the potentials from the previous step, then the Poisson equations
+are solved to find the updated potentials, that will be used in the next
+time step to compute the energy-momentum tensor.
+The Newtonian limit is recovered when we consider Fourier modes
+larger than H/𝑐 and we further neglect 𝐵𝑖 and consider Φ ≪ 1; then
+equation (4) becomes:
+Φ|𝑛𝑛 = 4𝜋𝐺𝑎2
+𝑐4
+Δ𝑇00
+(10)
+MNRAS 000, 1–14 (2022)
+
+4
+E. Quintana-Miranda et al.
+while (8) and (9) become:
+𝑑𝑥𝑖
+𝑑𝜏 = 𝑝𝑖
+𝑚𝑎 ,
+(11)
+𝑑𝑝𝑖
+𝑑𝜏 = −Φ,𝑖𝑚𝑐2𝑎 .
+(12)
+3 ALGORITHMS AND CODE INFRASTRUCTURE
+3.1 Gevolution
+Gevolution8 (Adamek et al. 2016) is an N-body relativistic cosmo-
+logical code, written in C++ and parallelized with the MPI paradigm.
+The physical theory behind this code has been described at length in
+Section 2. Numerically, this code implements a PM scheme to follow
+the evolution of energy-momentum tensor perturbations. As in PM
+codes, the advantage of working with a single grid and using Fast
+Fourier Transforms (FFTs) to solve the Poisson-like equations for the
+fields is paid with a high cost in memory, of O(𝑁3) where 𝑁 is the
+number of grid points per dimension.
+Gevolution, can run in either Newton or General Relativity
+modes. The Newtonian gravity solver inverts the Laplace operator
+in the Poisson equation for the Newtonian potential, Eq. 10. When
+running the General Relativity mode, the code solves Eqs. 4, 6 and
+7, that require the computation of the perturbed energy-momentum
+tensor. This is performed using a Cloud-In-Cell (CIC) scheme both
+for the density and for particle velocities; details are given in the
+presentation paper. Then the Hamiltonian forces to which particles
+are subjected are computed from Eqs. 8 and 9.
+Gevolution solves the field equations in Fourier space, using a
+C++ library called LATfield2 to operate FFTs on classical fields in
+massively parallel applications with distributed memory. LATfield2
+provides a programming interface to perform operations on the fields,
+either in their real or Fourier space representations. This library im-
+plements FFTs of 3-dimensional fields whose memory is distributed
+among parallel processes following a 2-dimensional uniform decom-
+position of space, in which each process owns in memory a portion
+of the grid with a rod shape (Daverio et al. 2015). In this way LAT-
+field2 overcomes the scaling limitations of a simpler 1-dimensional
+domain (slab) decomposition provided by the mainstream FFTW3
+library9. FFTW3 is used, however, to compute 1D FFTs.
+3.2 Gadget-4
+Gadget-410 is a state-of-the-art TreePM N-body hydrodynamical
+cosmological code written in C++ (see Springel et al. 2021); it is
+massively parallelized in a distributed-memory paradigm using MPI.
+As in most N-body codes, gravity in Gadget-4 is represented
+in the Newtonian limit, but the equations of motion are modified to
+take into account the Universe expansion, obtained by integrating the
+Friedmann equations separately. As mentioned above, this approach
+is consistent with General Relativity in the Poisson gauge, and gives
+the leading-order term of weak field expansion. This amounts to
+neglecting the metric degrees of freedom 𝐵𝑖, 𝜒 and ℎ𝑖 𝑗, and is
+valid on scales much smaller than the Hubble horizon. In a typical
+configuration that is convenient for large cosmological volumes, the
+8 https://github.com/gevolution-code
+9 http://fftw.org/
+10 https://wwwmpa.mpa-garching.mpg.de/gadget4
+code solves for the forces acting on each particle, representing them
+as the sum of two contributions, one due to the interactions with
+nearby particles, computed with a Tree algorithm, and one due to
+long-range interactions, computed with a PM algorithm.11
+The Tree algorithm works by partitioning the space into cubic
+cells, called nodes; in turn, each node is recursively partitioned into
+8 children nodes down to a pre-determined maximum refinement
+level. A tree structure tracks the list of particles that are located
+within each node. This structure is used to speed up the computation
+of gravitational force on a particle: in a particle-particle integration
+scheme, this force is computed by adding up a series of �𝑟 𝑚/𝑟3 terms,
+one for each particle pair, but we know that the accuracy of force
+evaluation does not depend strongly on the small-scale distribution
+of distant particles, so in the Tree scheme the evaluation of gravity
+is performed by grouping particles that belong to the same node,
+under the condition that the node subtends a given aperture angle
+𝜃. Particle-particle computation is then used only for the nearest
+neighbours. This is equivalent to considering the leading order in a
+multipole expansion of the gravity force from particles belonging to a
+distant cell. While the construction of the Tree is expensive in terms
+of computing time, it allows to achieve O(𝑁𝑝 log 𝑁𝑝) scaling for
+the force computation, where 𝑁𝑝 is the total number of particles in
+the simulation. Thus the Tree is able to compute with high accuracy
+the short wavelength modes of the gravitational interaction, while
+keeping the computational time low for large simulations. However,
+the Tree code is slow in integrating particle motions near the initial
+conditions, when the departures from homogeneity are small. This
+is why it is often coupled with a PM code to speed up the first time
+steps of a cosmological box.
+The PM algorithm represents gravity through the gravitational po-
+tential field Φ, evaluated on a Cartesian cubic mesh of fixed size. The
+potential is found from the density field by solving the Poisson equa-
+tion in Fourier space, while the force is computed from the gradient
+of the potential, obtained with a finite differences scheme. Accord-
+ing to the Nyquist-Shannon theory, this implies that the information
+handled by the PM is limited to the long modes, up to the Nyquist
+frequency.
+To combine the forces provided by the PM and Tree codes, the
+gravitational potential is split into the sum of two fields:
+Φ = Φ(𝐿) + Φ(𝑆) ,
+(13)
+where Φ(𝐿) represents long-range modes from the PM, and Φ(𝑆)
+represents short-range modes from the Tree. Written in Fourier space
+(tilde on top of symbols denotes a Fourier transform), the Poisson
+equation reads:
+˜Φ𝑘 = −4𝜋
+𝑘2 ˜𝜌𝑘 ,
+(14)
+where �� denotes the mass density. We can split the density as a sum
+of short-range and long-range terms, using Gaussian filters:
+˜Φ𝑘 = −4𝜋
+𝑘2 ˜𝜌𝑘
+�
+1 − exp(−𝑘2𝑟2
+𝑎)
+�
+− 4𝜋
+𝑘2 ˜𝜌𝑘 exp(−𝑘2𝑟2
+𝑎) .
+(15)
+The scale 𝑟𝑎 is the one at which we split long- and short-range modes.
+We can obtain Φ(𝑆) by solving the modified Poisson equation for
+short modes:
+˜Φ(𝑆)
+𝑘
+= −4𝜋
+𝑘2 ˜𝜌𝑘
+�
+1 − exp(−𝑘2𝑟2
+𝑎)
+�
+,
+(16)
+11 The code can work in other configurations (a non-cosmological volume,
+switching off the PM, enhancing the Tree part using multipole expansion)
+that are however not relevant for this paper.
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+5
+and Φ(𝐿) by solving the modified Poisson equation for long modes
+˜Φ(𝐿)
+𝑘
+= −4𝜋
+𝑘2 ˜𝜌𝑘 exp(−𝑘2𝑟2
+𝑎) .
+(17)
+The long-mode Poisson equation (17) is solved by the PM in
+Fourier space, so the convolution with the kernel is a simple mul-
+tiplication. The Tree on the other hand works in real space, hence
+equation (16) has to be transformed; this can be done analytically,
+yielding:
+Φ(𝑆) (�𝑥) = −𝐺
+∑︁
+𝑖
+𝑚𝑖
+|�𝑥 − �𝑟𝑖| erfc
+� |�𝑥 − �𝑟𝑖|
+2𝑟2𝑎
+�
+.
+(18)
+3.3 GrGadget
+3.3.1 Libgevolution library
+In order to have a relativistic PM code working in Gadget-4, we
+developed a library that implements both the Newtonian and the rel-
+ativistic PM algorithms of the monolithic Gevolution code. This
+was done by forking the Gevolution github repository into Libgevo-
+lution, a library that is publicly available on github12 under MIT
+license.
+The rationale behind the development of Libgevolution is to
+encapsulate Gevolution’s resources and methods into abstract ob-
+jects. This yields several benefits. Firstly, Gevolution maintenance
+is eased by the logical modularization of the code, i.e. instead of a
+monolitic code with a unique workflow we can divide Gevolution
+into components (C++ classes and/or namespaces) with well defined
+purposes. Secondly, we are allowed to re-use Gevolution compo-
+nents within other applications, such as we do within Gadget-4 in
+the present paper.
+We give here an overview of the library; the precise signature of
+all the defined functions, methods and data structures is described
+in the technical documentation of the code. Libgevolution is based
+on three cornerstones: (i) a particle container implemented through
+the class Particles_gevolution; (ii) a PM data structure named
+particle_mesh, templated on the particle container type, that can
+be used either as a relativistic_pm or a newtonian_pm; (iii) an
+executable application that uses the previous components to produce
+N-body simulations as the original code does. particle_mesh has
+to be understood as a container that is aware of the parallelization of
+the tasks and distribution of memory; it holds the gravitational fields
+and it allows the user to compute the forces acting on the simulation
+particles. The user interface declared in particle_mesh consists of
+the following functions:
+• sample(...), that builds the sources (density field or energy-
+momentum tensor) by sampling particle properties in the mesh;
+• compute_potential(...), that solves Poisson equations to
+compute the potential fields;
+• compute_forces(...), that computes the forces acting on
+particles.
+particle_mesh is specialized to solve the Newtonian problem
+or the General Relativistic problem using class inheritance; Figure 1
+illustrates the class hierarchy of Libgevolution’s particle_mesh.
+The expert user will be able to specialize particle_mesh to his/her
+own needs, for example by deriving a PM that solves a modified
+gravity problem.
+newtonian_pm is the specialization of particle_mesh that
+12 https://github.com/GrGadget/gevolution-1.2
+particle_mesh
+relativistic_pm
+newtonian_pm
+Figure 1. PM class hierarchy in Libgevolution.
+contains a real LATfield2::Field scalar field ΦNewton and its
+complex LATField2::Field Fourier transform
+˜ΦNewton, plus
+a LATField2::PlanFFT that connects ΦNewton with
+˜ΦNewton
+through discrete Fourier transform. relativistic_pm is the spe-
+cialization of particle_mesh that contains the above quoted de-
+grees of freedom of the perturbed FLRW metric, Φ, 𝐵𝑖 and 𝜒.
+These are represented as real LATfield2::Field, with complex
+LATField2::Field counterparts to represent their Fourier trans-
+forms and a LATField2::PlanFFT for each field.
+As a first testing phase, we run Libgevolution, called with a sim-
+ple wrapper, and the native Gevolution code, applying them to the
+same set of initial conditions, checking that the results were identical
+both in the Newtonian and relativistic cases. Then we stripped down
+Gadget-4 by switching off the Tree code, and compared its results
+to the Newtonian results of Libgevolution. It is necessary that this
+comparison gives nearly identical results if we want Libgevolution
+to substitute the native PM code of Gadget-4 without loss of accu-
+racy. To achieve a satisfactory match of the two PM codes we had to
+change the Gevolution scheme in a few points.
+We started from V1.2 of Gevolution, that implemented a first-
+order version of finite differences instead of the fourth-order scheme
+of Gadget-4. This resulted in a difference with Gadget-4 run on
+the same initial conditions, and in a percent-level offset of the matter
+power spectrum on large scales at low redshift. We upgraded the
+computation of spatial derivatives to fourth order, in parallel with
+the Gevolution developers that had noticed the same problem; our
+implementation is equivalent the most recent issue of Gevolution
+(used, e.g., in Adamek et al. 2022). The upgrade is the following: let’s
+consider the gravitational potential along one direction of the mesh,
+and let’s call its values Φ𝑖, where the index𝑖 denotes its position along
+that direction. Its first derivative is computed with finite differences
+at the first order as:
+𝜕Φ𝑖
+𝜕𝑥 = Φ𝑖+1 − Φ𝑖
+ℎ
++ O(ℎ),
+(19)
+where ℎ is the size of the mesh cell. Fourth-order Taylor expansion
+gives:
+𝜕Φ𝑖
+𝜕𝑥 = 8Φ𝑖+1 − Φ𝑖−1
+12ℎ
+− Φ𝑖+2 − Φ𝑖−2
+12ℎ
++ O(ℎ4) .
+(20)
+This has a smaller error of order O(ℎ4), so it achieves higher pre-
+cision than (19) with the little cost of knowing the potential value
+at the second-nearest cell, that implies a negligible communication
+overhead.
+Another improvement with respect to V1.2 of Gevolution, that
+follows an implementation of Gadget-4, was the application of cor-
+recting filters to the density in Fourier space to compensate for cloud-
+in-cell (CIC) interpolation. Indeed, as discussed e.g. in Springel
+(2005) or Sefusatti et al. (2016), CIC interpolation at some finite or-
+der leads to some loss of power that can be compensated for in Fourier
+space using suitable kernels. This was applied both to the compu-
+tation of the density and to the computation of energy-momentum
+tensor components in the relativistic case.
+Lastly, to make the Newtonian PM scheme equivalent to that of
+MNRAS 000, 1–14 (2022)
+
+6
+E. Quintana-Miranda et al.
+Gadget-4 we changed the form of the discrete Laplacian operator in
+the Poisson equation solver from its original form
+∇2 → −4𝑁2
+𝐿2
+�
+sin2 𝜋𝑘𝑥
+𝑁
++ sin2 𝜋𝑘𝑦
+𝑁
++ sin2 𝜋𝑘𝑧
+𝑁
+�
+,
+(21)
+described in Adamek et al. (2016), equation (C.5), to the form used
+in Gadget-4:
+∇2 → −4𝜋2
+𝐿2
+�
+𝑘2
+𝑥 + 𝑘2
+𝑦 + 𝑘2
+𝑧
+�
+.
+(22)
+3.3.2 Calling Libgevolution from Gadget-4
+The implementation of Libgevolution in Gadget-4 was performed
+as follows. We created a new PM class with a similar interface as
+the original one in Gadget-4, so that it is initialized and executed
+with the same functions as Gadget-4, i.e. init_periodic() and
+pmforce_periodic(). A new class relativistic_pm was imple-
+mented within an gadget::gevolution_api namespace, avoiding
+to use the wider gadget namespace to make a clear distinction of
+purpose between the original Gadget-4 code and our additional fea-
+tures. This relativistic_pm class acts much like a mediator taking
+information in and out of gadget simulation particles, processing
+the correct units conversion and calling the methods on gevolution
+namespace. Figure 2 shows a diagram that summarizes the contents
+of this PM class, its relation with Gadget-4’s resources and the entry
+points for gevolution’s api.
+relativistic_pm consists of:
+• A variable of type simparticle_handler that acts as
+a wrapper for providing particle information from Gadget-4’s
+simparticles global variable and writing back the data produced
+by gevolution’s PM.
+• A variable of type latfield_handler that takes care of cor-
+rectly initializing LATfield global state. Indeed, while Gadget-4 can
+run with any number of MPI processes, LATfield has limitations that
+depend on the number of grid points in the PM. latfield_handler
+also takes care of creating a sub-communicator from Gadget-4’s MPI
+global communicator that satisfies the constraints set by LATfield.
+• A variable of type gevolution::cosmology that contains the
+parameters for the background evolution.
+• A container of type gevolution::Particles_gevolution
+that holds particle information, stored according to their location on
+the PM grid.
+• Variables of type gevolution::relativistic_pm and
+gevolution::newtonian_pm that perform the actual PM compu-
+tations, i.e. construct the sources, either density or the components
+of the energy-momentum tensor, compute the gravitational potential
+or the metric perturbation fields and the forces that act upon the
+particles.
+• The methods pm_init_periodic and pmforce_periodic,
+for initialization and execution of the PM, respectively.
+3.3.3 Kick and drift operators
+In order to keep the Hamiltonian character of the equations of motion
+in Gadget-4, we have to describe the state of each particle through its
+position and momentum, not velocity. Following a leap-frog scheme,
+the momentum should be updated with a kick operation using the full
+relativistic Eqs. (8) and (9). However, velocities in Gadget-4 are to be
+interpreted as momenta (per unit mass) of non-relativistic particles in
+the Newtonian limit. Then we redefine the Gadget-4 kick and drift
+operators assuming non-relativistic matter, 𝑝 ≪ 𝑚𝑐𝑎, and further
+neglecting the very small contribution coming from 𝜒:
+𝑑𝑥𝑖
+𝑑𝜏 = 𝑝𝑖
+𝑚𝑎 (1 + 3Φ) + 𝑐𝐵𝑖 ,
+(23)
+𝑑𝑝𝑖
+𝑑𝜏 = − 𝑐𝑝𝑛𝐵𝑛|𝑖 − Φ,𝑖𝑚𝑐2𝑎 .
+(24)
+The right hand side of (24) is what we call force.
+3.3.4 Adding long-range and short-range forces
+To combine the forces computed with the relativistic PM and
+Gadget-4’s Newtonian Tree we have extended the idea of the TreePM
+coupling. From equation (13) one obtains that the force acting on a
+particle in a TreePM scheme consists of two terms:
+�𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree
+Newton] + 𝐿𝑟𝑎 [ �𝐹PM
+Newton].
+(25)
+The first term is the force computed using the Tree on which an
+exponential high-pass filter 𝑆𝑟𝑎 is applied, leaving short-wavelength
+modes. The second term corresponds to the PM force on which
+the complementary low-pass filter 𝐿𝑟𝑎 is applied to leave long-
+wavelength modes. The symbols 𝑆𝑎 and 𝐿𝑎 formally denote these
+linear operators:
+𝑆𝑟𝑎 [ 𝑓 ](�𝑟) = 1
+𝑁
+∑︁
+�𝑘
+˜𝑓�𝑘 (1 − exp(−𝑘2𝑟𝑎2)) exp(−𝑖�𝑘 · �𝑟) ,
+(26)
+and
+𝐿𝑟𝑎 [ 𝑓 ](�𝑟) = 1
+𝑁
+∑︁
+�𝑘
+˜𝑓�𝑘 exp(−𝑘2𝑟𝑎2) exp(−𝑖�𝑘 · �𝑟) .
+(27)
+The grid smoothing scale 𝑟𝑎 scales with the PM mesh size, and
+its value is optimized in Gadget-4, in a way that will be tested
+below, to minimize the impact of the two different treatments of the
+gravitational force.
+In order to account for the relativistic dynamics while preserving
+the match between tree and PM contributions that is valid in the New-
+tonian case, we choose the following strategy: Gadget-4 calls both
+newtonian_pm and relativistic_pm, the Newtonian value of the
+force is added to the Tree force as in a standard Newtonian simula-
+tion, while the difference between the Newtonian and the relativistic
+forces is added on top as a correction, but filtered on a different scale
+𝑟𝑏, that we call gr-smoothing scale. Eq. (25) then becomes:
+�𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree
+Newton] + 𝐿𝑟𝑎 [ �𝐹PM
+Newton] + 𝐿𝑟𝑏 [ �𝐹PM
+GR − �𝐹PM
+Newton] .
+(28)
+The case 𝑟𝑎 = 𝑟𝑏 would correspond to simply adding the relativistic
+force to the Tree:
+�𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree
+Newton] + 𝐿𝑟𝑏 [ �𝐹PM
+GR ] .
+(29)
+However, while the size of 𝑟𝑎, that regulates the match between
+Newtonian Tree and PM forces, is very well tested within Gadget-4,
+the optimal value of 𝑟𝑏 is to be found; we will show in the next
+Section that using 𝑟𝑏 larger than 𝑟𝑎 allows us to achieve percent
+accuracy at small scales.
+4 VALIDATION
+The GrGadget code has been validated by running it on a few real-
+izations of initial conditions, listed in table 1. These were generated
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+7
+gadget::
+simparticles
+LATfield2::
+parallel
+particle_handler
+read/write
+latfield_handler
+read/write
+sim::begrun1()
+sim::gravity_long_range_force()
+init_periodic()
+pmforce_periodic()
+execute
+execute
+gevolution::cosmology
+gevolution::Particles_gevolution
+gevolution::relativistic_pm
+gevolution::newtonian_pm
+gadget::gevolution_api::relativistic_pm::
+Figure 2. Diagram of resource ownership and relations for Libgevolution integrated into Gadget-4’s workflow. Each solid box represent a memory resource
+(an instantiation of a variable type) while the dashed boxes indicate ownership. The newly developed code, represented in the right part of the diagram denoted
+with the namespace gadget::gevolution_api, consists in a class named relativistic_pm that owns a particle_handler object that reads and writes
+directly into gadget::simparticles, a latfield_handler that takes care of setting up and inspect the state of LATfield2::parallel, and some types
+defined in Libgevolution, that are defined in gevolution namespace, like cosmology, Particles_gevolution and relativistic_pm. The methods
+sim::begrun1() and sim::gravity_long_range_force() in gadget:: interact with the relativistic_pm through their interface init_periodic()
+and pmforce_periodic().
+name
+𝑁𝑝 (particles)
+𝑁 (PM grid points)
+𝐿 (box size)
+N64
+643
+64
+1 Gpc/ℎ
+N256
+2563
+256
+1 Gpc/ℎ
+high_res
+5123
+512
+500 Mpc/ℎ
+Table 1. Cosmological simulation configurations used to validate GrGadget.
+with Gadget-4’s ngenic code at 𝑧 = 19, starting from a linear power
+spectrum generated with CAMB13 and with cosmological parame-
+ters consistent with Planck 2018 result (Planck Collaboration et al.
+2020): Ω𝑏ℎ2 = 0.0223, Ω𝑐ℎ2 = 0.120, 𝐻0 = 67.3 km s−1 Mpc−1,
+𝐴𝑠 = 2.097 × 10−9 and 𝑛𝑠 = 0.965.
+4.1 Gevolution and Gadget-4 original codes
+As already discussed in Section 3.3.1, the newtonian_pm imple-
+mentation in V1.2 of Gevolution computes the Newtonian forces
+differently from those obtained with Gadget-4’s PM. Before imple-
+menting Libgevolution as the PM engine of Gadget-4, we need to
+make the two algorithms work in the same way.
+To this aim, we have run a set of simulations with the configuration
+N64 (described in table 1) with a small number of particles 𝑁𝑝 = 643
+to be able to compute forces using a straightforward particle-particle
+(PP) scheme, that can be taken as the true force that we are trying to
+approximate. The same initial conditions at 𝑧 = 19 have been fed to
+both Gadget-4 (with Tree either on or switched off to have a pure PM
+run) and Gevolution (in Newtonian mode) codes. At later times,
+𝑧 = 8 and 𝑧 = 0, we have written snapshots of the forces that the
+simulation particles experience, separating the PM and the TreePM
+components; we have then compared those to the true Newtonian
+force computed with the PP scheme. The data we have obtained
+are summarized in the plots shown in figure 3. We have binned
+particles according to the value of the true force, then for each bin
+we have computed the mean (colored lines) and standard deviation
+13 https://camb.info/
+(shaded regions) of the difference between the force computed with
+approximate methods (PM or TreePM) and the true value. Forces
+are given in Gadget-4’s default units, which is actually acceleration,
+measured in units of 10𝐻0 km/s = ℎ km2 s−2 kpc−1. The green
+line shows the PM result using the original Gevolution code (the
+true force is anyway computed with Gadget-4 and matched particle
+by particle) while the red line is obtained from a pure PM using
+Gadget-4’s original code. The black line gives the TreePM method
+precision, obtained using Gadget-4.
+Looking at the red and green lines (and their shaded areas) we find
+two known results. Firstly, the TreePM method produces far less bias
+and dispersion when estimating forces; for instance, in the left panel
+of Fig. 3 the error is of the order14 of 0.1 ℎ km2 s−2 kpc−1, while in
+the right panel it is larger but barely visible when compared with the
+other curves. Secondly, while the PM force has low bias but a much
+larger variance than the TreePM one at high redshift, at low redshift,
+i.e. at higher level of non-linearity, it underestimates the value of the
+Newtonian force as its magnitude increases. This underestimation is
+due to the failure of PM in resolving interaction at scales smaller
+than the grid resolution.
+When comparing Gevolution PM and true forces, we notice an
+S-shaped feature in the plot, much more visible at high redshift.
+As anticipated in Section 3.3.1, this is mostly due to the first-order
+interpolation used to find the gradient of the potential in the code
+version that we tested.
+In Fig. 4 we show the matter power spectra15 obtained at 𝑧 = 0
+from a set of larger simulations with the configuration N256 (see
+table 1). The red solid line shows the result obtained with the orig-
+inal Gadget-4 code with its TreePM method, while the red dotted
+line shows the results obtained by switching off the Tree so that the
+14 This quantification is in code units, we can take this value as a reference
+for a high accuracy gravity solver.
+15 In this paper all particle power spectra were computed using PowerI4
+code presented in Sefusatti et al. (2016). Unless otherwise stated, all power
+spectra are computed up to the the Nyquist frequency of the PM mesh.
+MNRAS 000, 1–14 (2022)
+
+8
+E. Quintana-Miranda et al.
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Force (direct summation)
+2
+1
+0
+1
+2
+Force (estimated) - Force (direct summation)
+Force Test (z=8)
+TreePM (Gadget)
+Gevolution PM
+Gadget PM
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Force (direct summation)
+100
+50
+0
+50
+100
+Force (estimated) - Force (direct summation)
+Force Test (z=0)
+TreePM (Gadget)
+Gevolution PM
+Gadget PM
+Figure 3. Difference of gravity force with respect to the true PP value, binned according to the true force, for N64 initial conditions, at 𝑧 = 8 (left panel) and
+𝑧 = 0 (right panel). Lines represent the mean value of force difference in the bins, with colours explained in the legend; the shaded regions give the standard
+deviation of the corresponding force difference.
+forces are computed using the PM alone. The green lines show re-
+sults obtained with the latest develop version of Gevolution that
+implements higher order schemes for finite differences; the dotted
+line gives results obtained with GRADIENT_ORDER=1 and is identi-
+cal to the result obtained with V1.2 of Gevolution, the green solid
+line uses GRADIENT_ORDER=2, that corresponds to a second-order
+scheme. These power spectra show that the matter distribution in
+Gevolution using first-order gradients loses power in what seems
+to be a uniform trend for large-scale modes. This is a behaviour
+which is not inherent to the PM nature of the code, since that type of
+numerical approximation should predict very well the linear evolu-
+tion at large scales; indeed, the higher-order scheme recovers power
+on large scales to sub-percent accuracy. Conversely, Gadget-4’s PM
+and TreePM agree very well at wavenumbers below 𝑘 ∼ 0.1ℎ/Mpc
+scale,
+The higher-order differentiation worsens the loss of power of
+Gevolution for high values of 𝑘, that is not present in Gadget-4.
+This can be explained as a consequence of the particle-to-mesh sam-
+pling and mesh-to-particle interpolation described in section 3.3.1.
+As discussed there, Gadget-4’s PM corrects for these effects, result-
+ing in a power spectrum that degrades only at very high values of 𝑘 as
+we approach the Nyquist frequency, while producing a ∼ 2 percent
+overcorrection at 𝑘 ∼ 0.4 ℎ/Mpc.
+After implementing the higher-order differentiation scheme, the
+correction for the loss of power discussed above and the change in
+the discrete Laplacian operator (Section 3.3.1), the results of native
+Gadget-4 and Libgevolution PMs become indistinguishable.
+4.2 Newtonian forces
+We have tested our implementation of the GrGadget code by run-
+ning a standard test in Gadget-4: we create an N-body configuration
+in which there is a single massive particle in the entire simulation
+box, while other massless test particles are placed at different dis-
+tances from the first. In this setting the exact value of the force on
+each particle is known, hence one can compare the numerical results
+coming from the TreePM algorithm to the analytical solution.
+The results are shown in figure 5, where each dot represents a
+test particle. The x-axis gives the distance to the massive particle
+that sources the gravitational field, in units of the PM resolution
+(𝐿/𝑁), while the y-axis gives the corresponding absolute value of
+the relative difference of the true and estimated forces acting on the
+test particle. The red and blue lines correspond to the mean value of
+force residuals, for particles binned into distance bins; the red line
+denotes the statistics obtained from a simulation using Gadget-4’s
+original TreePM implementation and the blue line was produced
+using GrGadget, in this case with the Newtonian gravity engine.
+This figure shows that the accuracy with which the TreePM code
+reproduces the gravitational force is at worst at percent level on scales
+of a few mesh cells, corresponding to the scale where the PM and Tree
+contributions are matched, and gets very accurate in the limits where
+either the Tree (small scales) or the PM (large scales) dominates.
+Gadget-4’s and GrGadget’s Newtonian PMs show basically the
+same accuracy, even though their PM implementations are very dif-
+ferent.
+In Fig. 6 we show the matter power spectra of a set of N256
+simulations (see table 1). In this case we are comparing the matter
+clustering of GrGadget, in blue (with Newtonian forces for testing
+purposes), against Gadget-4, in red. In agreement with the previous
+test of force differences, we find that both codes produce the same
+matter power spectrum up to floating point errors. This is verified
+both in the case of simulations computing forces using a pure PM
+and in the case of TreePM.
+4.3 Relativistic simulations with GrGadget.
+We present here results obtained by running GrGadget with
+relativistic_pm, comparing them with the corresponding rela-
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+9
+102
+103
+104
+P(k)/(Mpc3h−3)
+camb
+Gevolution Newton PM
+Gevolution Newton PM (2nd order)
+Gadget4 PM
+Gadget4 TreePM
+10−2
+10−1
+100
+k/(hMpc−1)
+−0.10
+−0.05
+0.00
+0.05
+0.10
+(P(k) − P(k)Gadget4)/P(k)Gadget4
+Matter power spectrum z = 0.00
+Figure 4. Matter power spectrum of N256 cosmological simulations. The
+lower panel shows residuals with respect to Gadget-4’s original code (in red),
+used as baseline. The black line shows the linear power spectrum obtained with
+CAMB. Red lines show results obtained with Gadget-4, with the Tree part
+on (solid line) or switched off (dotted line). Green lines show results obtained
+with Gevolution in Newtonian configuration, with finite differences at first
+order (dotted line) or second order (solid line).
+Figure 5. Forces due to a point source: the points are test particles located at
+different distances (in units of the mesh resolution 𝐿/𝑁 ) from the source and
+the lines represent the RMS of the difference between real and TreePM forces
+in different distance bins. The red line corresponds to Gadget-4 original
+TreePM while the blue line was obtained with GrGadget in Newtonian
+mode. As for the the grid smoothing scale, the default value was used: 𝑟𝑎 =
+1.25𝐿/𝑁 . For this test we have used 𝑁 = 256 and 𝐿 = 1 Gpc/ℎ.
+102
+103
+104
+P(k)/(Mpc3h−3)
+camb
+GrGadget (Newton) PM
+GrGadget (Newton) TreePM
+Gadget4 PM
+Gadget4 TreePM
+10−2
+10−1
+100
+k/(hMpc−1)
+−0.10
+−0.05
+0.00
+0.05
+0.10
+(P(k) − P(k)Gadget4)/P(k)Gadget4
+Matter power spectrum z = 0.00
+Figure 6. Matter power spectrum of four simulations starting from the same
+initial conditions high_res: blue lines give results for Gadget-4 original
+code, red lines give results for GrGadget. In both cases dotted lines refer to
+runs with PM-only, solid lines refer to runs with full TreePM.
+tivistic version of Gevolution. We expect that the power spectrum
+of the matter density displays some relativistic features at large scales
+due to terms preceded by H in the field equation (4), while at small
+scales results should be compatible with Gadget-4’s Newtonian sim-
+ulations. However, the matter power spectrum shown here is not an
+observable quantity, so this comparison is just meant to give a first
+validation of the results. A more thorough comparison of observ-
+ables reconstructed on the past light cone will be presented in a
+future paper.
+Figure 7 shows the matter power spectra for a series of N256 sim-
+ulations (see table 1). In this case Gevolution and GrGadget are
+run in GR mode. The parameter that regulates the scale of the rela-
+tivistic correction (Eqs. 28 and 29) is set to 𝑟𝑏 = 6 𝐿/𝑁 ≈ 23Mpc/ℎ,
+i.e. the relativistic corrections of the PM method are smoothed at a
+distances below 6 grid cells. The plot shows that relativistic PM-only
+simulations, GrGadget (blue dotted line) and Gevolution (green
+lines) are compatible on large scales (𝑘 < 0.03ℎ/Mpc) up to a small
+percent-level difference that it is likely caused by the use of differ-
+ent orders for finite difference gradient; indeed, going from first- to
+second-order differences (from dotted to solid green line) the power
+spectrum gets nearer to GrGadget’s fourth-order one. The plot also
+confirms that our combination of Tree and PM forces in the relativis-
+tic weak field limit with GrGadget (blue solid line) reproduces the
+Newtonian non-linear features to sub-percent level at small scales,
+that is for 𝑘 > 0.1ℎ/Mpc; here Gadget-4 (red solid line) is again our
+reference for the non-linear clustering.
+Being designed for the use of Fourier methods from the beginning,
+Libgevolution offers an interface for the computation of the power
+spectrum of the fields defined through the library’s interface. Thus
+we can also extract and analyse the power spectra of the individual
+components of the metric perturbations from the relativistic sim-
+ulations. Figures 8 and 9 show the power spectra of the relativistic
+potentials, Φ, 𝐵𝑖 and 𝜒, for a high resolution configuration high_res
+(see table 1). These plots show a comparison of PM (blue lines) and
+TreePM (red lines) simulations. The power spectrum of the gravita-
+MNRAS 000, 1–14 (2022)
+
+Force Test
+0.0200
+Gadget TreePM
+GrGadget (Newton) TreePM
+0.0175
+0.0150
+0.0125
+0.0100
+0.0075
+0.0050
+0.0025
+0.0000
+100
+101
+distance*N/L10
+E. Quintana-Miranda et al.
+102
+103
+104
+P(k)/(Mpc3h−3)
+camb
+GrGadget TreePM
+GrGadget PM
+Gevolution Gr PM
+Gevolution Gr PM (2nd order)
+Gadget4 PM
+Gadget4 TreePM
+10−2
+10−1
+100
+k/(hMpc−1)
+−0.10
+−0.05
+0.00
+0.05
+0.10
+(P(k) − P(k)Gadget4)/P(k)Gadget4
+Matter power spectrum z = 0.00
+Figure 7. Matter power spectrum of Gadget-4, Gevolution and GrGadget
+runs, the last code being run in relativistic mode. The upper panel shows
+the absolute value and the lower panel the relative difference with respect to
+Gadget-4’s TreePM. The black line gives the linear matter power spectrum;
+red and blue lines give Gadget-4 and GrGadget results, with full TreePM
+forces (solid lines) or with the Tree switched off (dotted lines). Green lines
+give Gevolution results, dotted line referring to first-order finite differences
+(GRADIENT_ORDER=1) and solid line referring to second-order calculation
+(GRADIENT_ORDER=2).
+tional potentials converge for both methods on large scales. However,
+below 1 Mpc/ℎ the PM-only simulation loses power with respect to
+the TreePM one; the differences can reach up to 40% as we approach
+the Nyquist frequency. This pattern is equally found for the scalar
+fields Φ and 𝜒, as well as for the individual components of 𝐵𝑖.
+The right plot in Fig. 8 helps to understand the reason behind
+this result. Generally speaking, energy density, momentum density
+and their respective density current (the components of the Energy-
+Momentum tensor) are sources of the metric perturbations. Even
+though those quantities, as fields, are found at discrete positions of
+space defined by the mesh, their values are computed by sampling
+the energy and momentum carried by the particle distribution, which
+contain information on the clustering due to the short range inter-
+actions (through the Tree) that goes well below the mesh resolution
+𝐿/𝑁. Therefore, TreePM simulations, having power on scales well
+smaller than the PM mesh, give a better representation of the source
+of metric perturbation, and thus allow to recover power at frequency
+modes right below Nyquist. Fig. 8 highlights the particular case of
+𝑇00 (the matter density) as a source for Φ; by comparing 𝑇0
+0 with
+𝑘2Φ, we are verifying the Poisson equation 𝑘2 ˜Φ ≈ ˜𝑇00 that is valid
+for wavelengths below the Hubble horizon. This confirms that the
+presence of small-scale clustering in the particle distribution prop-
+agates to the gravitational fields up to the maximum resolution that
+the PM allows. The same thing is visible in the vector modes 𝐵𝑖 and
+in 𝜒 (Figure 9), where we also notice a small, few-percent mismatch
+on large scales. These fields are known to give sub-percent effects on
+observables, so this difference, that is likely due to some degree of
+numerical mode coupling, is non considered as a problem.
+In figure 10 we show how the matter power spectrum obtained
+using GrGadget is affected by the choice of the gr-smoothing scale
+parameter 𝑟𝑏. We have used an N256 box configuration to perform
+this test, and tested values of 𝑟𝑏 = 1.5, 3, 6 in units of 𝐿/𝑁 ≈
+4 Mpc/ℎ. We find that large-scales power is independent of the value
+of 𝑟𝑏 parameter; structures one scales below the PM resolution are
+resolved by the Tree algorithm, hence for 𝑘 > 𝑘Nyquist there is
+a convergence of all simulations to a common non-linear power
+spectrum tail. It is in the medium to small scales 𝑘Nyquist > 𝑘 >
+0.2 Mpc−1ℎ that we notice differences in the power spectrum above
+the ∼ 1% (dashed grey line). For small values of 𝑟𝑏 (∼ 1.5 𝐿/𝑁), we
+obtain discrepancies in the power spectrum at 𝑘 ∼ 0.5 Mpc−1ℎ that
+can be as large as 5 percent and indicate the limitations of our force
+summation scheme, Eq. (28). A value of 𝑟𝑏 = 3 𝐿/𝑁 or possibly
+higher is needed to obtain a good compatibility of GrGadget and
+Gadget-4 for all modes greater than 0.1 Mpc−1ℎ, where relativistic
+features in the matter clustering is negligible.
+The last test we present here regards the convergence of the nu-
+merical results for increasing resolution. Figure 11 shows the matter
+power spectrum obtained from running Gadget-4’s TreePM (red
+lines), GrGadget with PM-only (blue dotted lines) and GrGadget
+with TreePM (blue continuous line). These various code configu-
+rations were run with different combinations of the number of grid
+points per dimension 𝑁 = 256, 𝑁 = 512 and box length 𝐿 = 250, 500,
+1000, 2000 Mpc/ℎ; the number of particles was fixed as 𝑁𝑝 = 𝑁3. In
+all cases we have set the PM smoothing scale to 𝑟𝑎 = 1.5 𝐿/𝑁 and the
+gr-smoothing scale to 𝑟𝑏 = 3 𝐿/𝑁. It can be observed with the finest
+resolution, in the top plots, that there is a matching between General
+Relativity and Newtonian dynamics in the small scales. Then as the
+mesh size becomes coarser, in the middle plots, some discrepancies
+in the power spectrum start to appear which become more evident
+for even coarser meshes, in the bottom plots. This mismatch may
+be caused by 𝑟𝑏 = 3 𝐿/𝑁 moving towards larger scales, so that the
+assumption that PM forces are Newtonian on the small scales breaks.
+Indeed, while with 𝐿/𝑁 = 1 ℎ−1 Mpc (𝑟𝑏 = 3 ℎ−1 Mpc) the scales
+where relativistic effects become evident in the matter power spec-
+trum and the scales where the pure PM prediction starts to deviate
+from TreePM are well separated, for larger 𝐿/𝑁 values the two scales
+get nearer, indicating that the assumption of pure Newtonian forces
+on the mesh scale may not be very good. This conclusion is appar-
+ently at variance with the discussion of Figure 10, where a larger
+value of 𝑟𝑏 was preferred; however, that figure refers to 𝐿/𝑁 = 1
+and is shown at 𝑧 = 0.5, where clustering is a bit weaker. swe thus
+recommend to work with mesh sizes of 𝐿/𝑁 ∼ 1 Mpc/ℎ.
+5 CONCLUSIONS
+We have constructed a relativistic TreePM code, that we call Gr-
+Gadget, where the large-scale contribution to the gravitational force
+is computed using the relativistic C++ PM library Libgevolution,
+based on Gevolution code, while gravity coming from small scales
+is computed by the Tree code of Gadget-4. The code works under
+the assumption that, in the context of cosmological simulations, dark
+matter can be treated non-relativistically and then the equations of
+motion of tracer particles tend to the Newtonian limit at scales well
+below the Hubble horizon. Following the Gevolution approach, we
+use a weak field approximation of GR, where the perturbations of the
+space-time metric with respect to FLRW background are encoded as
+fields and simulated by the PM. Comparing the matter power spec-
+trum from GrGadget simulations with that of original Gadget-4
+and Gevolution codes, we conclude that the code produces consis-
+tent results as long as the PM cell size 𝐿/𝑁 is smaller than 2 Mpc/ℎ
+and the gr-smoothing parameter is 𝑟𝑏 ≈ 3 𝐿/𝑁.
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+11
+10
+30
+10
+28
+10
+26
+10
+24
+10
+22
+10
+20
+10
+18
+Pk
+ TreePM
+ PM
+10
+2
+10
+1
+100
+k/(h Mpc
+1)
+0.4
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+(Pk
+Pref)/Pref
+101
+102
+103
+104
+Pk
+k2
+ TreePM
+k2
+ PM
+T00 TreePM
+10
+2
+10
+1
+100
+k/(h Mpc
+1)
+0.4
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+(Pk
+Pref)/Pref
+Figure 8. In the left plot: power spectrum of the metric perturbation Φ in a high_res simulation obtained with GrGadget. In the right plot: power spectrum
+of 𝑘2Φ and 𝑇 00. For modes well below the Hubble horizon and small perturbations it should be verified that 𝑘2 ˜Φ ≈ ˜𝑇 00.
+10
+2
+10
+1
+100
+10
+29
+10
+27
+10
+25
+10
+23
+10
+21
+10
+19
+Pk
+B0 TreePM
+B0 PM
+10
+2
+10
+1
+100
+k/(h Mpc
+1)
+0.4
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+(Pk
+Pref)/Pref
+10
+39
+10
+37
+10
+35
+10
+33
+10
+31
+10
+29
+10
+27
+Pk
+ TreePM
+ PM
+10
+2
+10
+1
+100
+k/(h Mpc
+1)
+0.4
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+(Pk
+Pref)/Pref
+Figure 9. In the left plot: power spectrum of the metric perturbation 𝐵𝑖 (the 𝑥 component) in a high_res simulation obtained with GrGadget. In the right
+plot: power spectrum of 𝜒.
+With respect to the pure PM implementation of Gevolution,
+the predictive power of GrGadget gives an improvement even on
+the scales sampled by the mesh. This is due to the fact that the
+energy-momentum tensor, that sources the equations of the fields
+that represent the perturbations of the metric, is computed from a
+fully non-linear distribution of particles, with gravity being resolved
+down to a much smaller softening length and not down to the mesh
+size. This may be very useful, e.g., when assessing the possibility of
+detecting the frame-dragging effect of a rotating dark-matter halo, if
+not of a spiral galaxy (Bruni et al. 2014). Furthermore, this code is a
+development of the widely used Gadget-4 code, and because the PM
+sector of the code is called only by the computation of the gravity
+force, our code can be easily extended to simulations of galaxies or
+galaxy clusters by switching on the hydrodynamics, star formation
+and feedback sectors. All the physics described by these sectors can
+safely be treated in the Newtownian limit; one should in principle
+MNRAS 000, 1–14 (2022)
+
+12
+E. Quintana-Miranda et al.
+102
+103
+104
+P(k)/(Mpc3h−3)
+camb
+GrGadget TreePM (rb=6.0)
+GrGadget TreePM (rb=3.0)
+GrGadget TreePM (rb=1.5)
+Gadget4 TreePM
+10−2
+10−1
+100
+k/(hMpc−1)
+−0.10
+−0.05
+0.00
+0.05
+0.10
+(P(k) − P(k)Gadget4)/P(k)Gadget4
+Nyquist freq.
+Matter power spectrum z = 0.50
+Figure 10. Power spectrum of matter density for Gadget-4 and GrGadget,
+on a N256 simulation configuration. The upper panel shows the absolute value
+and the lower panel the relative difference with respect to Gadget-4’s TreePM.
+Different shades of blue indicate different values of the gr-smoothing scale
+parameter 𝑟𝑏 = 1.5, 3, 6 in units of 𝐿/𝑁 . The PM smoothing scale is 𝑟𝑎 =
+1.5 𝐿/𝑁 . The power spectra in this plot are computed beyond the Nyquist
+frequency to show the convergence of the matter distribution correlations for
+distances below the grid resolution, the Tree regime.
+add thermal energy of gas particles to the energy-momentum tensor,
+but while this extension is straightforward, it is likely to provide a
+negligible contribution.
+This is, for our group, a further step in the construction of an
+ecosystem of simulation codes and post-processing tools for model-
+ing the evolution of structure in the Universe, with the aim of making
+predictions for precision cosmology. Sub-percent accuracy in cos-
+mological predictions, that matches the smallness of the statistical
+error that will be obtained with forthcoming galaxy surveys men-
+tioned in the Introduction, can only be obtained taking into account
+relativistic effect (e.g. Lepori et al. 2020), and we can foresee that
+a self-consistent treatment of these effects (to within the required
+accuracy) will soon become the standard in cosmological simula-
+tions. These effects can also be added by post-processing Newtonian
+simulations, but a validation of these procedures requires validation
+against a more self-consistent approach. Conversely, a large com-
+munity is developing Gevolution in the direction of adding modi-
+fications of gravity, whose formulation is typically worked out in a
+general relativistic context. This line of development, coupled with
+a Newtonian treatment of modified gravity in the Tree code, would
+be precious in the formulation of tests of gravity, because relativistic
+effects may hide smoking-gun features of specific classes of modified
+gravity theories.
+APPENDIX A: CODE SCALING
+The code we presented in this work is the merging of two codes
+whose behaviour in terms of run-time scaling is well-known and
+characterized; since we did not modify the underlying algorithms, it
+is expected that the run-time scaling of our code follows that of the
+parent codes.
+However, the Libgevolution’s PM is obviously different from
+Gadget-4’s, and we added the translation of particles data from the
+host code to the target relativistic PM. Both this facts require that
+we establish the overall scaling of GrGadget in its fully-relativistic
+configuration and the overhead associated to both the relativistic PM
+and the interface between the two codes.
+In figure A1 we show the fraction of time spent in the PM in
+both the original and relativistic configurations as a function of the
+grid cell size (see the caption for details). The relativistic PM is an
+order of magnitude more expensive than the original Gadget-4’s
+Newtonian PM, although in absolute sense it is still either negligible
+or secondary in the simulation sets that have been tested (it reaches a
+maximum value of 16% at highest resolution, i.e. in the 𝑁 = 512,𝐿 =
+250 Mpc/ℎ). However, it scales with both the resolution and the grid
+number as the original Newtonian PM does.
+Figures A2 and A3 report the scaling of run time in strong and
+weak scaling tests respectively for the total run time, the tree time
+and the PM time (left. middle and right panels in both figures; see the
+captions for details). As inferred from A1, the run-time and hence its
+scaling, are dominated by the Gadget-4’s Tree section.
+ACKNOWLEDGEMENTS
+We thank Julian Adamek for many fruitful discussions on gevolu-
+tion, Volker Springel for his comments on an early draft, Francesca
+Lepori, Marco Bruni, Marco Baldi and Emilio Bellini for discus-
+sions. Simulations were performed with the HOTCAT system of
+INAF (Taffoni et al. 2020; Bertocco et al. 2020). PM acknowledges
+partial support by a Fondo di Ricerca di Ateneo grant of University
+of Trieste.
+DATA AVAILABILITY
+The simulation codes presented in this paper are publicly available
+on github in the following path: https://github.com/GrGadget.
+REFERENCES
+Abbott B. P., et al., 2016, Phys. Rev. Lett., 116, 061102
+Adamek J., Durrer R., Kunz M., 2014, Classical and Quantum Gravity, 31,
+234006
+Adamek J., Daverio D., Durrer R., Kunz M., 2016, Journal of Cosmology
+and Astroparticle Physics, 2016, 053
+Adamek J., Durrer R., Kunz M., 2017, J. Cosmology Astropart. Phys., 2017,
+004
+Adamek J., et al., 2022, arXiv e-prints, p. arXiv:2211.12457
+Alam S., et al., 2021, J. Cosmology Astropart. Phys., 2021, 050
+Barnes J., Hut P., 1986, Nature, 324, 446
+Barrera-Hinojosa C., Li B., 2020, Journal of Cosmology and Astroparticle
+Physics, 2020, 007
+Barrera-Hinojosa C., Li B., Bruni M., hua He J., 2020, Vector modes in
+ΛCDM: the gravitomagnetic potential in dark matter haloes from rela-
+tivistic 𝑁 -body simulations (arXiv:2010.08257)
+Bartelmann M., Schneider P., 2001, Phys. Rep., 340, 291
+Bertocco S., et al., 2020, in Pizzo R., Deul E. R., Mol J. D., de Plaa J.,
+Verkouter H., eds, Astronomical Society of the Pacific Conference Series
+Vol. 527, Astronomical Data Analysis Software and Systems XXIX.
+p. 303 (arXiv:1912.05340)
+Borzyszkowski M., Bertacca D., Porciani C., 2017, MNRAS, 471, 3899
+Bruni M., Thomas D. B., Wands D., 2014, Phys. Rev. D, 89, 044010
+Capozziello S., De Laurentis M., 2012, Annalen der Physik, 524, 545
+Chisari N. E., Zaldarriaga M., 2011, Physical Review D, 83
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+13
+0.100
+0.075
+0.050
+0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+(P(k)
+P(k)ref)/P(k)ref
+N=256
+L/N = 1 Mpc/h
+L/N = 1 Mpc/h
+L/N = 1 Mpc/h
+L/N = 1 Mpc/h
+L/N = 1 Mpc/h
+L/N = 1 Mpc/h
+N=512
+0.100
+0.075
+0.050
+0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+(P(k)
+P(k)ref)/P(k)ref
+L/N = 2 Mpc/h
+L/N = 2 Mpc/h
+L/N = 2 Mpc/h
+L/N = 2 Mpc/h
+L/N = 2 Mpc/h
+L/N = 2 Mpc/h
+Gadget4 TreePM
+GrGadget PM
+GrGadget TreePM
+10
+2
+10
+1
+100
+k/(Mpc
+1h)
+0.100
+0.075
+0.050
+0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+(P(k)
+P(k)ref)/P(k)ref
+10
+2
+10
+1
+100
+k/(Mpc
+1h)
+L/N = 4 Mpc/h
+L/N = 4 Mpc/h
+L/N = 4 Mpc/h
+L/N = 4 Mpc/h
+L/N = 4 Mpc/h
+L/N = 4 Mpc/h
+Figure 11. Matter power spectrum from cosmological simulations at 𝑧 = 0 using GrGadget (the blue lines) and compared to Gadget-4 (the red line) at 𝑧 = 0.
+The dotted line is obtained with a simulation in which only the PM is used to compute forces. The plots show the relative difference with respect to the power
+spectrum obtained with Gadget-4. The left column corresponds to simulations with 𝑁 = 256 grid points per dimension while for the right column 𝑁 = 512.
+The boxsize changes along the ranks so that for the top plots the resolution is the highest 𝐿/𝑁 ≈ 1 Mpc/ℎ, in the middle 𝐿/𝑁 ≈ 2 Mpc/ℎ and the bottom plots
+correspond to 𝐿/𝑁 ≈ 4 Mpc/ℎ. In all cases 𝑟𝑎 = 1.5 𝐿/𝑁 and 𝑟𝑏 = 3 𝐿/𝑁 . The grey dashed line indicate a 1% error.
+DESI Collaboration et al., 2016, arXiv e-prints, p. arXiv:1611.00036
+Daverio D., Hindmarsh M., Bevis N., 2015, Latfield2: A c++ library for
+classical lattice field theory (arXiv:1508.05610)
+Doré O., et al., 2014, arXiv e-prints, p. arXiv:1412.4872
+Event Horizon Telescope Collaboration et al., 2019, ApJ, 875, L1
+Ivezić Ž., et al., 2019, ApJ, 873, 111
+Krause E., et al., 2017, arXiv e-prints, p. arXiv:1706.09359
+Laureijs R., et al., 2011, arXiv e-prints, p. arXiv:1110.3193
+Lepori F., Adamek J., Durrer R., Clarkson C., Coates L., 2020, Monthly
+Notices of the Royal Astronomical Society, 497, 2078–2095
+MNRAS 000, 1–14 (2022)
+
+14
+E. Quintana-Miranda et al.
+1
+2
+3
+4
+5
+6
+7
+8
+L/N (Mpc/h)
+10
+2
+10
+1
+time_pm / time_total
+GR (N=256)
+Newton (N=256)
+GR (N=512)
+Newton (N=512)
+Figure A1. The fraction of PM time to the total running time. Relativistic runs
+are shown in blue while Newtonian runs are shown in red, whereas symbols
+distinguish the value of grid points per dimension 𝑁 (squares and circles for
+𝑁 = 256 and 512 respectively). We plot the time fraction on the 𝑦–axis (log
+scale) vs the mesh resolution 𝐿/𝑁 on the 𝑥–axis.
+Planck Collaboration et al., 2020, A&A, 641, A6
+Puchwein E., Baldi M., Springel V., 2013, MNRAS, 436, 348
+Sefusatti E., Crocce M., Scoccimarro R., Couchman H. M. P., 2016, mnras,
+460, 3624
+Silvestri A., Trodden M., 2009, Reports on Progress in Physics, 72, 096901
+Spergel D., et al., 2015, arXiv e-prints, p. arXiv:1503.03757
+Springel V., 2005, Monthly Notices of the Royal Astronomical Society, 364,
+1105
+Springel V., Pakmor R., Zier O., Reinecke M., 2021, MNRAS, 506, 2871
+Taffoni G., Becciani U., Garilli B., Maggio G., Pasian F., Umana G., Smareglia
+R., Vitello F., 2020, in Pizzo R., Deul E. R., Mol J. D., de Plaa J.,
+Verkouter H., eds, Astronomical Society of the Pacific Conference Series
+Vol. 527, Astronomical Data Analysis Software and Systems XXIX.
+p. 307 (arXiv:2002.01283)
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–14 (2022)
+
+GrGadget
+15
+25
+50
+75
+100
+125
+150
+175
+200
+# process
+25
+50
+75
+100
+125
+150
+175
+200
+speedup x processes_pivot
+Strong Scalability (Total time)
+N = 128, L=250
+N = 128, L=500
+N = 128, L=1000
+N = 128, L=2000
+N = 256, L=250
+N = 256, L=500
+N = 256, L=1000
+N = 256, L=2000
+N = 512, L=250
+N = 512, L=500
+N = 512, L=1000
+N = 512, L=2000
+ideal
+25
+50
+75
+100
+125
+150
+175
+200
+# process
+25
+50
+75
+100
+125
+150
+175
+200
+speedup x processes_pivot
+Strong Scalability (PM time)
+N = 128, L=250
+N = 128, L=500
+N = 128, L=1000
+N = 128, L=2000
+N = 256, L=250
+N = 256, L=500
+N = 256, L=1000
+N = 256, L=2000
+N = 512, L=250
+N = 512, L=500
+N = 512, L=1000
+N = 512, L=2000
+ideal
+25
+50
+75
+100
+125
+150
+175
+200
+# process
+25
+50
+75
+100
+125
+150
+175
+200
+speedup x processes_pivot
+Strong Scalability (Tree time)
+N = 128, L=250
+N = 128, L=500
+N = 128, L=1000
+N = 128, L=2000
+N = 256, L=250
+N = 256, L=500
+N = 256, L=1000
+N = 256, L=2000
+N = 512, L=250
+N = 512, L=500
+N = 512, L=1000
+N = 512, L=2000
+ideal
+Figure A2. Strong-scaling test. We present the code scaling as the number 𝑃 of MPI tasks is increased while running the same simulation set-up. All the results
+refer to GrGadget, i.e. to the configuration with fully-relativistic PM. On the 𝑥–axis 𝑃 increases from 24 to 192, by ×2 steps. On the 𝑦–axis we report the
+speed-up (normalized so that the ideal speed-up for 𝑃 = 1 is 1) for the total running time, the time spent in the PM and the time spent in the Tree on the Left,
+Middle and Right panels respectively. Note that the ideal behaviour (black dotted line) would result in a linear speed-up. The PM data includes the translation of
+particles data from Gadget-4 to Libgevolution. We show the results for 𝑁 = 128, 256 and 512 (solid, dashed and dot–dashed lines respectively) for 4 different
+box sizes (i.e. mass resolutions), 𝐿 = 250, 500, 1000 and 2000 Mpc/ℎ (circles, squares and stars respectively). See the discussion in Appendix A for details.
+25
+50
+75
+100
+125
+150
+175
+200
+# process
+1.0
+1.1
+1.2
+1.3
+1.4
+time / time_pivot
+Weak scalability (Total time)
+N = 128->256, L = 250->500
+N = 128->256, L = 500->1000
+N = 128->256, L = 1000->2000
+N = 256->512, L = 250->500
+N = 256->512, L = 500->1000
+N = 256->512, L = 1000->2000
+ideal
+25
+50
+75
+100
+125
+150
+175
+200
+# process
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+time / time_pivot
+Weak scalability (PM time)
+N = 128->256, L = 250->500
+N = 128->256, L = 500->1000
+N = 128->256, L = 1000->2000
+N = 256->512, L = 250->500
+N = 256->512, L = 500->1000
+N = 256->512, L = 1000->2000
+ideal
+25
+50
+75
+100
+125
+150
+175
+200
+# process
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+1.30
+1.35
+time / time_pivot
+Weak scalability (Tree time)
+N = 128->256, L = 250->500
+N = 128->256, L = 500->1000
+N = 128->256, L = 1000->2000
+N = 256->512, L = 250->500
+N = 256->512, L = 500->1000
+N = 256->512, L = 1000->2000
+ideal
+Figure A3. Weak-scaling test. We present the code scaling as the number 𝑃 of MPI tasks is increased for a proportionally increasing problem, then keeping
+constant the particles–per–task occupancy. All the results refer to GrGadget, i.e. to the configuration with fully-relativistic PM. On the 𝑥–axis 𝑃 increases from
+24 to 192 with only 2 test cases. On the 𝑦–axis we report the speed-up for the total running time, the time spent in the PM and the time spent in the Tree on the
+Left, Middle and Right panels respectively. Note that the ideal behaviour would result in a constant running time (horizontal dotted black line). The PM data
+includes the translation of particles data from Gadget-4 to Libgevolution. We show the results for two cases: from 𝑁 = 128, to 𝑁 = 256 (solid lines with
+circles), and from 𝑁 = 256, to 𝑁 = 512 (dashed lines with squares). Each of the two cases has been run for three different box sizes (i.e. mass resolutions):
+𝐿 = 250 → 𝐿 = 500 Mpc/ℎ, 𝐿 = 500 → 𝐿 = 1000 Mpc/ℎ and 𝐿 = 1000 → 𝐿 = 2000 Mpc/ℎ (red, green and blue colors respectively). See the discussion in
+Appendix A for details.
+MNRAS 000, 1–14 (2022)
+

V9E2T4oBgHgl3EQfYAcW/content/tmp_files/2301.03849v1.pdf.txt

@@ -0,0 +1,3000 @@
+A UNIVERSAL FRAMEWORK FOR ENTANGLEMENT
+DETECTION UNDER GROUP SYMMETRY
+SANG-JUN PARK, YEONG-GWANG JUNG, JEONGEUN PARK,
+AND SANG-GYUN YOUN
+ABSTRACT. One of the most fundamental questions in quantum infor-
+mation theory is PPT-entanglement of quantum states, which is an NP-
+hard problem in general. In this paper, however, we prove that all PPT
+(πA ⊗ πB)-invariant quantum states are separable if and only if all ex-
+tremal unital positive (πA, πB)-covariant maps are decomposable where
+πA, πB are unitary representations of a compact group and πA is irre-
+ducible. Moreover, an extremal unital positive (πB, πA)-covariant map
+L is decomposable if and only if L is completely positive or completely
+copositive. We apply the results to prove that all PPT quantum channels
+of the form
+Φ(ρ) = aρ + bρT + cTr(ρ)
+d
+Idd + (1 − a − b − c)diag(ρ)
+are entanglement-breaking, and that all A-BC PPT (U⊗U⊗U)-invariant
+tripartite quantum states are A-BC separable. The former resolves some
+open questions raised in [DFV08, KMS20], and the latter is a strong
+contrast to the fact that there exist PPT-entangled (U ⊗U ⊗U)-invariant
+tripartite Werner states [EW01].
+1. INTRODUCTION
+Quantum entanglement is one of the most non-classical manifestations of
+quantum formalism and is considered a key resource for quantum communi-
+cation. Indeed, quantum entanglement plays crucial roles in the existence of
+Bell correlations [Bel64, Wer89], quantum cryptography [Eke91, JSW+00,
+TBZG00, NPW+00], superdense coding [BW92, MWKZ96], quantum tele-
+portation [BBC+93, BPM+97], entanglement-assisted classical communi-
+cation [BSST99], and computational supremacy for communication com-
+plexity problems [Bra03, BvDHT99, C¯D13].
+The question of whether a given quantum state is entangled or separa-
+ble is of fundamental importance in quantum information theory(QIT). It
+turned out that this question is NP-hard in general [Gur03, Gha10], so it is
+unnatural to expect an efficient general scheme to characterize quantum en-
+tanglement. Nevertheless, there have been numerous efforts to characterize
+separability in some subclasses of quantum states. For example, a quantum
+1
+arXiv:2301.03849v1  [math-ph]  10 Jan 2023
+
+2
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+state ρ ∈ D(HA ⊗ HB) of positive partial transpose (PPT) is separable if
+dim(HA) × dim(HB) ≤ 6 [HHH96, Wor76b] and, moreover, PPT implies
+separability in some bipartite or multipartite systems of low-rank [KCKL00,
+HLVC00, EW01, C¯D13]. Classification of entanglement of GHZ states has
+also been studied in various contexts [Kay11, G¨11, HK16a, HK16b]. Note
+that any PPT entangled state is bound entangled, which is applicable to
+perform nonclassical tasks [HHH99, VW02, Mas06] and to produce secure
+cryptographic key [HHHO05, HHHO09, HPHH08].
+In this paper, we restrict our interests to the so-called invariant quantum
+states in a general context of compact group symmetries. More precisely,
+for unitary representations πA : G → B(HA) and πB : G → B(HB) of
+a compact group G, a bipartite quantum state ρ ∈ D(HA ⊗ HB) is called
+(πA ⊗ πB)-invariant if
+(πA(x) ⊗ πB(x))ρ = ρ(πA(x) ⊗ πB(x))
+(1.1)
+for all x ∈ G. Werner states and isotropic states are standard examples of in-
+variant quantum states for fundamental unitary group symmetries, and their
+separability was characterized in [Wer89] and [HH99], respectively. Sep-
+arability of invariant quantum states has been studied extensively for vari-
+ous group symmetries [EW01, UDUPR07, DPR07, KCL05, TG09, AN14,
+SN21, CKK+21].
+The dual objects of invariant quantum states are the so-called (πA, πB)-
+covariant quantum channels, which are completely positive trace-preserving
+(CPTP) maps L : B(HA) → B(HB) satisfying
+L(πA(x)XπA(x)T) = πB(x)L(X)πB(x)∗
+(1.2)
+for all X ∈ B(HB) and x ∈ G. Indeed, for a linear map L : B(HA) →
+B(HB) and its normalized Choi matrix CL =
+1
+dA
+�
+i,j=1 eij ⊗ L(eij), the
+given map L is (πA, πB)-covariant if and only if CL is (πA ⊗ πB)-invariant
+(Corollary 2.4).
+One of the key observations of this paper is that all PPT (πA ⊗ πB)-
+invariant quantum states are separable if and only if all (πB, πA)-covariant
+positive maps are decomposable. If πA is irreducible, then the following
+three statements are equivalent (Corollary 3.8):
+• All PPT (πA⊗πB)-invariant quantum states are separable (PPT=SEP).
+• All positive (πB, πA)-covariant maps are decomposable (POS=DEC).
+• All PPT (πA, πB)-covariant quantum channels are entanglement-
+breaking (PPT=EB).
+Moreover, it is enough to consider only extremal elements (Theorem 3.9)
+and, in particular, an extremal positive unital (πB, πA)-covariant linear map
+L is decomposable if and only if L is completely positive (CP) or com-
+pletely copositive (CCP) (Theorem 3.11).
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+3
+Our framework focusing on decomposability of extremal positive unital
+(πB, πA)-covariant maps is applicable to study PPT entanglement for con-
+crete examples. In Section 4, we prove that EB property coincides with PPT
+property, i.e. PPT=EB holds for any quantum channels of the form
+Φ(ρ) = aρ + bρT + cTr(ρ)
+d
+Idd + (1 − a − b − c)diag(ρ).
+(1.3)
+where diag(X) =
+d
+�
+i=1
+Xiieii for X = (Xij)1≤i,j≤d. An important observa-
+tion is that the quantum channels of the form (1.3) are irreducibly covariant
+channels with respect to the standard representation of the signed symmetric
+group (or the hyperoctahedral group) Hd (Lemma 4.3). Furthermore, we
+characterize all positive unital covariant maps of the form (1.3) (Theorem
+4.5) and our main theorem (Theorem 3.9) allows us to focus only on eight
+extremal positive unital covariant maps to detect EB property for quantum
+channels of the form (1.3). Our results strengthen Section 5 and Section 6
+of [KMS20] to a larger class, and resolve some of open questions posed in
+[KMS20] and [DFV08].
+In Section 5, we focus on the question of whether all PPT quantum states
+are separable, i.e. PPT=SEP problem for some tripartite quantum states
+with unitary group symmetries. In Section 5.1, we present explicit positive
+non-decomposable covariant linear maps L : Md(C) → Md2(C) satisfying
+L(UXU T) = (U ⊗ U)L(X)(U ⊗ U)∗
+(1.4)
+for all d×d unitary matrices U ∈ Ud and X ∈ Md(C). This result gives an-
+other explanation of the fact PPT̸=SEP for tripartite Werner states [EW01],
+which implies the existence of PPT entangled quantum states ρ ∈ Md3(C)
+satisfying
+(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U)
+(1.5)
+for all U ∈ U(d). On the other hand, in Section 5.2, we show that a
+strong contrast PPT=SEP holds for quantum orthogonally invariant quan-
+tum states. More generally, we prove that any PPT tripartite quantum state
+ρ ∈ Md3(C) satisfying
+(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U)
+(1.6)
+for all unitary matrices U ∈ U(d) is separable (Theorem 5.6).
+2. PRELIMINARIES
+2.1. Separability and PPT property. In this paper, we focus only on finite-
+dimensional complex Hilbert spaces H = Cd, HA = CdA, HB = CdB, and
+their direct sums and tensor products. Recall that a quantum state ρ ∈ B(H)
+is a positive matrix with Tr(ρ) = 1 and the set of all quantum states in B(H)
+
+4
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+is denoted by D(H). A bipartite positive operator X ∈ B(HA⊗HB) is said
+to be of positive partial transpose (PPT) if
+(idA ⊗ TB)(X) ≥ 0
+(2.1)
+where TB is the transpose map on B(HB), and X is called separable if
+there exist families of positive operators (XA
+i )n
+i=1 and (XB
+i )n
+i=1 such that
+X =
+n
+�
+i=1
+XA
+i ⊗ XB
+i .
+(2.2)
+In particular, if ρ ∈ D(HA ⊗ HB) is a separable quantum state, then there
+exists a probability distribution (pi)n
+i=1 and a family of product quantum
+states (ρA
+i ⊗ ρB
+i )n
+i=1 such that
+ρ =
+n
+�
+i=1
+piρA
+i ⊗ ρB
+i .
+(2.3)
+It is clear that separability implies PPT property, but the converse is not
+true in general. More precisely, all PPT quantum states in B(HA ⊗ HB)
+are separable if and only if dA · dB ≤ 6 [Per96, HHH96, Wor76a, Cho82].
+Moreover, it is known that the separability question is NP-hard [Gur03,
+Gha10].
+For v ∈ H, we define linear maps |v⟩ : C → H given by λ �→ λv and
+⟨v| : H → C given by w �→ ⟨v|w⟩ where ⟨v|w⟩ is the inner product of
+v, w ∈ H whose first variable is the anti-linear part. In particular, |Ω⟩ =
+�d
+i=1
+1
+√
+d|i⟩⊗|i⟩ ∈ H⊗H is called the maximally entangled Bell state where
+{|1⟩, |2⟩, · · · , |d⟩} is the standard orthonormal basis of H. The matrix unit
+|i⟩⟨j| and the product vector |i1⟩ ⊗ |i2⟩ ⊗ · · · ⊗ |ik⟩ are also denoted by eij
+and |i1i2 · · · ik⟩ respectively.
+The (normalized) Choi matrix of a linear map L : B(HA) → B(HB) is
+defined by
+CL = (idA ⊗ L)(|ΩA⟩⟨ΩA|) = (idA ⊗ L)
+�
+1
+dA
+dA
+�
+i,j=1
+eij ⊗ eij
+�
+(2.4)
+= 1
+dA
+dA
+�
+i,j=1
+eij ⊗ L(eij) ∈ B(HA ⊗ HB).
+(2.5)
+Recall that L is completely positive (CP) if and only if the Choi matrix CL
+is positive, and L is trace-preserving (TP) if and only if (idA ⊗TrB)(CL) =
+1
+dA
+IdA. In particular, if Φ : B(HA) → B(HB) is a CPTP linear map, i.e. a
+quantum channel in the Schr¨odinger’s picture, then the Choi matrix CΦ is a
+quantum state in D(HA ⊗ HB). We call this channel-state duality.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+5
+Let L : B(HA) → B(HB) be a linear map. Then L is called completely
+copositive (CCP) if TB ◦L is completely positive, L is called decomposable
+if there exist a CP map L1 and a CCP map L2 such that L = L1 + L2, and
+L is called PPT if L is both CP and CCP. Thus, L is PPT if and only if CL
+is PPT.
+Another important property of quantum channels is the entanglement-
+breaking (EB) property. A quantum channel Φ : B(HA) → B(HB) is
+called EB if the Choi matrix CΦ is a separable quantum state. Note that any
+EB quantum channel is PPT, but the converse is not true in general.
+2.2. Invariance and covariance. In this section, we introduce two impor-
+tant objects to discuss conservation of symmetry, namely invariant opera-
+tors and covariant linear maps. Let us suppose that G is a compact group
+throughout this paper. A continuous function π : G → U(Hπ) is called a
+(finite-dimensional) unitary representation of G if it is a group homomor-
+phism, i.e.,
+π(xy) = π(x)π(y)
+(2.6)
+for all x, y ∈ G. In this case, an operator X ∈ B(Hπ) is called π-invariant
+if
+π(x)Xπ(x)∗ = X
+(2.7)
+for all x ∈ G. The set of all π-invariant operators, the set of all π-invariant
+quantum states, and the set of π-invariant PPT quantum states in B(Hπ) are
+denoted by Inv(π), InvQS(π), and InvPPTQS(π), respectively. A unitary
+representation π : G → B(Hπ) is called irreducible if Inv(π) = C · IdHπ.
+If π is irreducible, so is the contragredient representation π : G → U(Hπ)
+of π which is defined by π(x) = π(x) for all x ∈ G.
+For unitary representations πA : G → U(HA) and πB : G → U(HB), the
+tensor representation πA ⊗ πB : G → U(HA ⊗ HB) is given by
+(πA ⊗ πB)(x) = πA(x) ⊗ πB(x)
+(2.8)
+for all x ∈ G. The tensor representation πA ⊗ πB is not irreducible in
+general, but admits the so-called irreducible decomposition, i.e. irreducible
+representations σ1, σ2, · · · , σk of G such that
+πA ⊗ πB ∼= σ1 ⊕ σ2 ⊕ · · · ⊕ σk.
+(2.9)
+Here, (σ1 ⊕ σ2 ⊕ · · · ⊕ σk)(x) is the block diagonal matrix of σ1(x), σ2(x),
+· · · , σk(x) for all x ∈ G, and π ∼= π′ means that there exists a unitary V
+such that π(x) = V π′(x)V ∗ for all x ∈ G. We say that the irreducible
+decomposition (2.9) is multiplicity-free if σi ≇ σj for all i ̸= j. This prop-
+erty was highlighted in [GBW21] in view of programmability of covariant
+
+6
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+quantum channels. More generally, we may rewrite (2.9) as
+πA ⊗ πB ∼=
+l
+�
+i=1
+σi ⊗ Idmi
+(2.10)
+meaning that σi ̸∼= σj for all i ̸= j and each irreducible representation σi
+appears mi times in the irreducible decomposition (2.9). In this case, we
+have the following identification of Inv(πA ⊗ πB) as ∗-algebras [GBW21,
+Lemma 6]:
+Inv(πA ⊗ πB) ∼=
+l
+�
+i=1
+Idni ⊗ Mmi(C),
+(2.11)
+where ni = dim Hσi.
+For unitary representations πA : G → B(HA) and πB : G → B(HB), a
+linear map L : B(HA) → B(HB) is called (πA, πB)-covariant if
+L(πA(x)Y πA(x)∗) = πB(x)L(Y )πB(x)∗
+(2.12)
+for all x ∈ G and Y ∈ B(HA), and let us denote by Cov(πA, πB) the space
+of all (πA, πB)-covariant linear maps. Some subclasses of Cov(πA, πB) in
+our interest are as follows:
+• CovPos(πA, πB) is the set of all (πA, πB)-covariant positive maps,
+• CovPos1(πA, πB) is the set of all (πA, πB)-covariant positive unital
+maps,
+• CovPosTP(πA, πB) is the set of all (πA, πB)-covariant positive TP
+maps,
+• CovQC(πA, πB) is the set of all (πA, πB)-covariant CPTP maps,
+• CovPPTQC(πA, πB) is the set of all (πA, πB)-covariant PPT quan-
+tum channels.
+2.3. Twirling operation. An averaging technique called the twirling op-
+eration is a standard method to analyze invariant operators and covariant
+linear maps. For a unitary representation π : G → U(H), we define a
+twirling map Tπ : B(H) → Inv(π) by
+Tπ(X) =
+�
+G
+π(x)Xπ(x)∗dx
+(2.13)
+for all X ∈ B(H), where dx denotes the normalized Haar measure on G.
+Then Tπ is unital CPTP, and its well-definedness, i.e. Tπ(X) ∈ Inv(π),
+is thanks to the translation-invariance property of the Haar measure. Fur-
+thermore, we have X ∈ Inv(π) if and only if Tπ(X) = X, which means
+that Tπ is a projection (more precisely, a conditional expectation) onto the
+∗-subalgebra Inv(π) of B(H). Note that for any finite dimensional von
+Neumann algebra M ⊂ Md(C), there is a unique TP conditional expec-
+tation of Md(C) onto M [BO08, Lemma 1.5.11]. For example, the map
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+7
+X ∈ Mn ⊗ Mn �→
+1
+nIdn ⊗ (Trn ⊗ idm)(X) is the unique TP conditional
+expectation onto M = Idn ⊗ Mm. This observation allow us to get the
+following explicit formula of the twirling map Tπ for the case M = Inv(π).
+Proposition 2.1. Suppose that a unitary representation π : G → U(H)
+has an irreducible decomposition of the form (2.10) with the identification
+Inv(π) ∼= �l
+i=1 Idni ⊗ Mmi(C) ⊆ B
+��l
+i=1 Hi
+�
+. Let Πi be the orthogonal
+projection from H onto Hi = Cni ⊗ Cmi. Then the twirling Tπ(X) of
+X ∈ B(H) is given by
+Tπ(X) =
+l
+�
+i=1
+1
+ni
+Idni ⊗
+�
+(Trni ⊗ idmi)(ΠiXΠi)
+�
+.
+(2.14)
+In particular, if the irreducible decomposition of π is multiplicity-free, i.e.,
+if mi ≡ 1 for all i = 1, 2, · · · , l, then
+Tπ(X) =
+l
+�
+i=1
+Tr(ΠiX)
+ni
+Πi.
+(2.15)
+For unitary representations πA : G → U(HA) and πB : G → U(HB), the
+twirling TπA,πBL of L : B(HA) → B(HB) is defined by
+(TπA,πBL)(X) =
+�
+G
+πB(x)∗L(πA(x)XπA(x)∗)πB(x) dx
+(2.16)
+for all X ∈ B(HA). Then similarly, the twirling operation TπA,πB is a well-
+defined projection from B(B(HA), B(HB)) onto Cov(πA, πB).
+Let us collect some useful properties of the twirling operations for the
+next section.
+Proposition 2.2. For any unitary representations πA and πB of G, the
+twirling map TπA⊗πB preserves separability and PPT property of bipartite
+operators. Furthermore, the twirling operation TπA,πB preserves positivity,
+CP, TP, CCP, PPT, decomposability, and EB property of linear maps.
+Proof. It is straightforward from the definitions and closedness of the spaces
+associated with each of the properties mentioned above. For example, the
+set of all decomposable linear maps L : B(HA) → B(HB) is closed in
+B(B(HA), B(HB)) with respect to the natural (Euclidean) topology.
+□
+For a linear map L : B(HA) → B(HB), the adjoint map L∗ : B(HB) →
+B(HA) of L is a linear map satisfying
+Tr(L(X) Y ) = Tr(X L∗(Y ))
+(2.17)
+for all X ∈ B(HA) and Y ∈ B(HB). Recall that the adjoint operation
+L �→ L∗ preserves positivity, CP, CCP, PPT, and decomposability.
+
+8
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+Proposition 2.3. Let π : G → U(H), πA : G → U(HA) and πB : G →
+U(HB) be unitary representations of G. Then we have the following.
+(1) Tr((TπX) Y ) = Tr(X(TπY )) for any X, Y ∈ B(H).
+(2) TπA⊗πB◦(TA⊗idB) = (TA⊗idB)◦TπA⊗πB where TA is the transpose
+on B(HA).
+(3) (TπA,πBL)∗ = TπB,πA(L∗) for any linear map L : B(HA) → B(HB).
+(4) The Choi matrix of TπA,πBL is given by TπA⊗πB (CL) for any linear
+map L : B(HA) → B(HB).
+Proof. (1) Since the Haar measure on the compact group G is invariant
+under the inverse x �→ x−1, we have
+Tr((TπX) Y ) =
+�
+G
+Tr(π(x)Xπ(x−1)Y )dx
+(2.18)
+= Tr
+�
+X
+�
+G
+π(x−1)Y π(x)dx
+�
+(2.19)
+= Tr
+�
+X
+�
+G
+π(x)Y π(x−1)dx
+�
+= Tr(X(TπY ))
+(2.20)
+for any X, Y ∈ B(H).
+(2) It suffices to show the equality for product operators X = P ⊗Q, and
+the conclusion follows immediately from the observation
+πA(x)P TπA(x)∗ =
+�
+πA(x)PπA(x)T�T
+.
+(2.21)
+(3) For any X ∈ B(HA) and Y ∈ B(HB), we have
+Tr(X (TπB,πAL∗) (Y ))
+(2.22)
+=
+�
+G
+Tr(XπA(x)∗L∗(πB(x)Y πB(x)∗)πA(x))dx
+(2.23)
+=
+�
+G
+Tr(πB(x)∗L(πA(x)XπA(x)∗)πB(x) Y )dx
+(2.24)
+= Tr((TπA,πBL) (X) Y ),
+(2.25)
+which gives us the desired conclusion.
+(4) First of all, note that
+dA
+�
+i,j=1
+(πA(x)eijπA(x)t) ⊗ (πB(x)L(eij)πB(x)∗)
+(2.26)
+=
+dA
+�
+i,j=1
+eij ⊗ (πB(x)L(πA(x)∗eijπA(x))πB(x)∗).
+(2.27)
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+9
+for each x ∈ G. Indeed, the LHS (2.26) can be understood as
+dA(idA ⊗ (AdπB(x) ◦ L))
+�
+(πA(x) ⊗ IdA)|ΩA⟩⟨ΩA|(πA(x)t ⊗ IdA)
+�
+,
+(2.28)
+and the RHS (2.27) can be understood as
+dA(idA ⊗ (AdπB(x) ◦ L)) ((IdA ⊗ πA(x)∗)|ΩA⟩⟨ΩA|(IdA ⊗ πA(x)))
+(2.29)
+where AdV (Y ) = V Y V ∗. Moreover, the so-called ricochet property
+(X ⊗ IdA)|ΩA⟩ = (IdA ⊗ Xt)|ΩA⟩, X ∈ B(HA),
+(2.30)
+implies (2.28) = (2.29). Finally, taking the Haar integral on both sides com-
+pletes the proof.
+□
+Combining Proposition 2.3 (2), (3), and (4) with the fact that both Inv(πA⊗
+πB) and Cov(πA, πB) are the images of the twirling projections, we obtain
+the following useful properties.
+Corollary 2.4. Let X ∈ B(HA ⊗ HB) be a bipartite operator and L :
+B(HA) → B(HB) be a linear map. Then
+(1) X ∈ Inv(πA ⊗ πB) if and only if (TA ⊗ id)(X) ∈ Inv(πA ⊗ πB).
+(2) L ∈ Cov(πA, πB) if and only if L∗ ∈ Cov(πB, πA).
+(3) L ∈ Cov(πA, πB) if and only if CL ∈ Inv(πA ⊗ πB).
+Remark 2.5. The results in Corollary 2.4 have been noted in various con-
+texts, [EW01, Lemma 6], [GBW21, Lemma 11], and [LY22, Proposition
+5.1, Theorem 3.5] for examples. Moreover, extendibility to more general
+contexts of compact quantum group symmetry was proved in [LY22].
+3. A FRAMEWORK TO CHARACTERIZE ENTANGLEMENT UNDER GROUP
+SYMMETRY
+Let us recall a result of Horodecki on the characterization of entangle-
+ment [HHH96]: a bipartite quantum state ρ ∈ D(HA ⊗ HB) is separable if
+and only if (idA ⊗ L)(ρ) ≥ 0 for all positive linear maps L : B(HB) →
+B(HA). Indeed, by duality arguments, the authors showed that they are
+also equivalent to seemingly a weaker condition ‘⟨ρ, L⟩ ≥ 0’. Here, the
+dual pairing ⟨·, ·⟩ is defined by
+⟨X, N⟩ = Tr((idA ⊗ N)(X) |ΩA⟩⟨ΩA|) = Tr(XCN ∗)
+(3.1)
+for any operator X ∈ B(HA ⊗HB) and linear map N : B(HB) → B(HA).
+In other words, positive linear maps play a crucial role as detectors for the
+bipartite entanglement, in the sense that there should exist a positive linear
+map L such that (id ⊗ L)(ρ) is non-positive whenever ρ is entangled.
+
+10
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+One technical issue in this characterization is that verifying whether a
+linear map is positive or not is computationally intractable [Las10, NZ16].
+However, one of the main purposes of this paper is to develop a universal
+framework to characterize separability of invariant quantum states. The
+key ideas is that, for ρ ∈ InvQS(πA ⊗ πB), it is enough to consider only
+L ∈ CovPos(πB, πA) to investigate separability of ρ. Let us begin with a
+simple and useful lemma.
+Lemma 3.1. For any bipartite operator X ∈ B(HA ⊗ HB) and linear map
+L : B(HB) → B(HA), we have
+⟨TπA⊗πBX, L⟩ = ⟨X, TπB,πAL⟩.
+(3.2)
+Proof. Thanks to Proposition 2.3, we have
+⟨TπA⊗πBX, L⟩ = Tr((TπA⊗πBX)CL∗)
+(3.3)
+= Tr(X(TπA⊗πBCL∗)) = Tr(XCL∗) = ⟨X, L⟩,
+where L = (TπA,πBL∗)∗ = TπB,πAL.
+□
+Then Lemma 3.1 allows us to conclude that covariant positive linear
+maps are enough to characterize separability of bipartite invariant quantum
+states. We remark that the ideas of the following proof appeared already for
+some specified symmetries [Kay11, G¨11, SN21].
+Theorem 3.2. Let πA : G → B(HA) and πB : G → B(HB) be unitary
+representations, and let ρ ∈ InvQS(πA⊗πB). The following are equivalent.
+(1) ρ is a separable quantum state.
+(2) (idA ⊗ L)(ρ) ≥ 0 in B(HA ⊗ HA) for any (πB, πA)-covariant pos-
+itive linear map L : B(HB) → B(HA).
+(3) ⟨ρ, L⟩ ≥ 0 for any (πB, πA)-covariant positive linear map L :
+B(HB) → B(HA).
+Proof. Two directions (1) ⇒ (2) and (2) ⇒ (3) are clear. For the last
+direction (3) ⇒ (1), let us show that ⟨ρ, L⟩ ≥ 0 for all positive linear
+maps L : B(HB) → B(HA). Indeed, since ρ is πA ⊗ πB-invariant and
+TπB,πAL ∈ CovPos(πB, πA), we can apply Lemma 3.1 to obtain
+⟨ρ, L⟩ = ⟨TπA⊗πBρ, L⟩ = ⟨ρ, TπB,πAL⟩ ≥ 0.
+(3.4)
+□
+From now on, let us focus on the question of whether PPT property coin-
+cides with separability, i.e. PPT=SEP problem for invariant quantum states.
+Recall that the dual notions of PPT property and separability correspond
+to decomposability and positivity respectively. Indeed, many duality argu-
+ments [Kye23] carry over into our framework, and the PPT=SEP problem
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+11
+in InvQS(πA ⊗ πB) is equivalent to the question whether all positive maps
+are decomposable, i.e. POS=DEC problem in Cov(πB, πA).
+Proposition 3.3. Let L : B(HB) → B(HA) be (πB, πA)-covariant. Then
+(1) L is positive if and only if (idA ⊗ L)(ρ) ≥ 0 for any separable
+ρ ∈ InvQS(πA ⊗ πB).
+(2) L is decomposable if and only if (idA ⊗ L)(ρ) ≥ 0 for any PPT
+ρ ∈ InvQS(πA ⊗ πB).
+Proof. (1) Suppose (idA⊗L)(ρ) ≥ 0 for any separable ρ ∈ InvQS(πA⊗πB).
+Then for every separable state ρ ∈ D(HA ⊗ HB), we have
+⟨ρ, L⟩ = ⟨ρ, TπB,πAL⟩ = ⟨TπA⊗πBρ, L⟩ ≥ 0
+(3.5)
+by Lemma 3.1 and by the separability of TπA⊗πBρ. Now positivity of L
+follows from [EK00, Theorem 3.1]. The converse direction is clear.
+(2) It is enough to repeat the arguments from (1) based on the following
+duality result [Sr82]: L is decomposable if and only if ⟨ρ, L⟩ ≥ 0 for every
+PPT state ρ.
+□
+Corollary 3.4. The following are equivalent:
+(1) PPT=SEP in InvQS(πA ⊗ πB).
+(2) POS=DEC in Cov(πB, πA).
+Proof. ((1) ⇒ (2)) If L ∈ CovPos(πB, πA), then (idA ⊗ L)(ρ) ≥ 0 for
+every separable (hence every PPT) state ρ ∈ InvQS(πA ⊗ πB). Thus, L is
+decomposable by Proposition 3.3.
+((2) ⇒ (1)) If ρ ∈ InvQS(πA ⊗πB) is a PPT state, then (idA ⊗L)(ρ) ≥ 0
+for every decomposable (hence every positive) linear map L ∈ Cov(πB, πA).
+Thus, ρ is separable by Theorem 3.2.
+□
+Note that PPT=SEP in InvQS(πA ⊗ πB), or equivalently POS=DEC in
+Cov(πB, πA), implies that PPT property coincides with the entanglement-
+breaking property, i.e. PPT=EB in CovQC(πA, πB). Moreover, we have
+the following characterization of EB property for Φ ∈ CovQC(πA, πB) by
+Corollary 2.4 (3).
+Corollary 3.5. Let Φ ∈ CovQC(πA, πB). Then the following are equiva-
+lent.
+(1) Φ is entanglement-breaking.
+(2) CL◦Φ = (id ⊗ L)(CΦ) ≥ 0 for any L ∈ CovPos(πB, πA).
+(3) L ◦ Φ is completely positive for any L ∈ CovPos(πB, πA).
+To summarize, we have
+PPT=SEP in InvQS(πA ⊗ πB) ⇔ DEC=POS in Cov(πB, πA),
+
+12
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+and these conditions imply PPT=EB in CovQC(πA, πB). One might ask
+whether all these three problems are equivalent, but one technical issue is
+that CovQC(πA, πB) is not identified with InvQS(πA ⊗πB) in general. This
+leads us to question whether the (reduced) channel-state duality
+�C : CovQC(πA, πB) → InvQS(πA ⊗ πB)
+(3.6)
+is bijective. The channel-state duality �C is not surjective in general, but it
+is known to be the case if πA is irreducible, as already noted in [GBW21,
+Lemma 15]. Moreover, we prove that the converse is also true, i.e. the
+channel-state duality �C is bijective if and only if πA is irreducible. Let us
+start with the following lemma.
+Lemma 3.6. Let πA : G → U(HA) and πB : G → U(HB) be unitary
+representations of G and let L ∈ Cov(πA, πB).
+(1) If πB is irreducible, then L(IdA) = c IdB for some constant c.
+(2) If πA is irreducible, then there is a constant c such that Tr(L(X)) =
+c Tr(X) for X ∈ B(HA).
+Proof.
+(1) From the irreducibility of πB and the identity
+πB(x)L(IdA)πB(x)∗ = L(πA(x)πA(x)∗) = L(IdA),
+(3.7)
+we have L(IdA) ∈ Inv(πB) = C · IdB.
+(2) The adjoint map L∗ is (πB, πA)-covariant by Corollary 2.4 (2), so
+L∗(IdB) = c IdA for some c by (1). In this case, we have
+Tr(L(X)) = Tr(L(X) IdB) = Tr(X L∗(IdB)) = c Tr(X)
+(3.8)
+for any X ∈ B(HA).
+□
+Now, let us apply Lemma 3.6 (2) to prove that the channel-state duality
+�C : CovQC(πA, πB) → InvQS(πA ⊗ πB) should be bijective if and only if
+πA is irreducible.
+Proposition 3.7. Let πA : G → U(HA) and πB : G → U(HB) be unitary
+representations of G and let L ∈ Cov(πB, πA). Then the channel-state
+duality
+�C : CovQC(πA, πB) → InvQS(πA ⊗ πB)
+(3.9)
+is bijective if and only if πA is irreducible.
+Proof. Let us prove the if part first. For any ρ ∈ InvQS(πA ⊗ πB) there
+exists completely positive L ∈ Cov(πA, πB) such that CL = ρ by Corollary
+2.4 (3). Moreover, L should be trace-preserving. Indeed, irreducibility of
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+13
+πA implies that there exists a constant c such that Tr(L(X)) = cTr(X) for
+all X ∈ B(HA) by Lemma 3.6 (2), and we have
+c = c
+dA
+dA
+�
+i=1
+Tr(eii) = 1
+dA
+dA
+�
+i=1
+Tr(eii ⊗ L(eii)) = Tr(CL) = 1.
+(3.10)
+Conversely, if we assume that πA = π(1)
+A ⊕ π(2)
+A with HA = H(1)
+A ⊕ H(2)
+A
+and if Π1 is the orthogonal projection from HA onto H(1)
+A , then we can take
+a CP non-TP map L : B(HA) → B(HB) given by
+L(X) =
+dA
+dB · dim H(1)
+A
+Tr(Π1X)IdB
+(3.11)
+whose Choi matrix is
+CL =
+�
+1
+dim H(1)
+A
+Π1
+�
+⊗
+� 1
+dB
+IdB
+�
+∈ InvQS(πA ⊗ πB).
+(3.12)
+□
+Corollary 3.8. Let πA : G → U(HA) and πB : G → U(HB) be unitary
+representations of G and suppose that πA is irreducible. Then the following
+are equivalent.
+(1) PPT=SEP in InvQS(πA ⊗ πB).
+(2) PPT=EB in CovQC(πA, πB).
+(3) POS=DEC in CovPos1(πB, πA).
+Proof. It is enough to note that Lemma 3.6 allows us to focus on a smaller
+convex set CovQC(πA, πB) and CovPos1(πB, πA) rather than Cov(πA, πB)
+and CovPos(πB, πA), respectively, in Corollary 3.4.
+□
+Moreover, we prove that the extreme points of CovPos1(πB, πA) are enough
+for the entanglement detection, which we propose as a universal machinery
+to characterize entangled invariant quantum states with general compact
+group symmetries. Let us denote by Ext(S) the set of all extreme points of
+a convex set S.
+Theorem 3.9. Let πA : G → B(HA) and πB : G → B(HB) be unitary
+representations, and suppose that πA is irreducible. Let ρ ∈ InvQS(πA ⊗
+πB) and Φ ∈ CovQC(πA, πB) such that CΦ = ρ from Proposition 3.7. The
+following are equivalent.
+(1) ρ is separable.
+(2) (id ⊗ L)(ρ) ≥ 0 for any L ∈ CovPos1(πB, πA).
+(3) (id ⊗ L)(ρ) ≥ 0 for any L ∈ Ext (CovPos1(πB, πA)).
+(4) Φ is entanglement-breaking.
+(5) L ◦ Φ is completely positive for any L ∈ CovPos1(πB, πA).
+
+14
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+(6) L ◦ Φ is completely positive for any L ∈ Ext (CovPos1(πB, πA)).
+Proof. The equivalences (1) ⇔ (4), (2) ⇔ (5), (3) ⇔ (6) and one-side impli-
+cations (1) ⇒ (2) ⇒ (3) are clear. Moreover, the direction (2) ⇒ (1) follows
+from Lemma 3.6 (1). For the proof of (3) ⇒ (2), note that CovPos1(πB, πA)
+is a compact subset of
+{L ∈ B(B(HB), B(HA)) : ∥L∥op ≤ 1} ,
+(3.13)
+where ∥ · ∥op denotes the operator norm with respect to Hilbert-Schmidt
+norms on B(HB) and B(HA), since the positivity of L implies ∥L∥op =
+∥L(IdB)∥ = 1 [Pau02, Corollary 2.9]. Therefore, CosPos1(πB, πA) can be
+written as a convex hull of its extreme points, which completes the proof.
+□
+Remark 3.10. If πB is irreducible instead of irreducibility of πA, then Φ
+is chosen to be a unital CP map up to constant, and CovPosTP(πB, πA)
+replaces the role of CovPos1(πB, πA) in (2), (3), (5), (6). Note that compact-
+ness of CovPosTP(πB, πA) comes from the identification with CovPos1(πA, πB)
+(up to constant) via taking the adjoint operation.
+Finally, we claim that the decomposability of the extremal elements in
+CovPos1(πB, πA) is much easier to check thanks to the following theorem.
+Theorem 3.11. Suppose that πA (resp. πB) is irreducible, and let L ∈
+Ext(CovPos1(πB, πA)) (resp. L ∈ Ext(CovPosTP(πB, πA)). Then L is de-
+composable if and only if L is CP or CCP.
+Proof. Let us focus only on the case where πA is irreducible since the other
+case is analogous. If L is decomposable, then there exist a CP map L1 and
+a CCP map L2 such that L = L1 + L2. By taking the twirling operation
+TπB,πA, we have L = L′
+1 + L′
+2 where L′
+i = TπB,πA(Li) ∈ Cov(πB, πA).
+Note that L′
+1 is CP, L′
+2 is CCP, and we can write L′
+i = λiL′′
+i for some
+λi ≥ 0, λ1 + λ2 = 1, and L′′
+i ∈ CovPos1(πB, πA) by Lemma 3.6 (1). Then
+extremality of L allows us to conclude that L = L′′
+1 or L = L′′
+2, which
+proves the assertion. The other direction is immediate.
+□
+To summarize, our strategy to study PPT=SEP and PPT=EB problems
+consists of the following three independent steps, assuming πA is irreducible.
+[Step 1] The first step is to characterize all elements in CovPos1(πB, πA) for
+given specific unitary representations πA and πB.
+[Step 2] The next step is to solve POS=DEC problem in CovPos1(πB, πA).
+In particular, for extremal elements L ∈ Ext(CovPos1(πB, πA)), the
+given L is decomposable if and only if L is CP or CCP. If POS=DEC
+holds, then PPT=SEP problem in InvQS(πA ⊗ πB) and PPT=EB
+problem in CovQC(πA, πB) has the affirmative answer.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+15
+[Step 3] If there exists a non-decomposable element in CovPos1(πB, πA),
+then the last step is to find the following objects:
+– Φ ∈ CovPPTQC(πA, πB) for which L ◦ Φ is non-CP,
+– ρ ∈ InvPPTQS(πA ⊗ πB) for which (id ⊗ L)(ρ) ≱ 0.
+4. PPT=EB HOLDS FOR (H, H)-COVARIANT QUANTUM CHANNELS
+One of the main applications of the results in Section 3 is a complete char-
+acterization of EB property for quantum channels Φ : Md(C) → Md(C) of
+the form
+Φ(X) = aTr(X)
+d
+Idd + bX + cXT + (1 − a − b − c) diag(X).
+(4.1)
+The main result of this section is as follows.
+Theorem 4.1. Let Φ be a quantum channel of the form (4.1). Then Φ is
+entanglement-breaking if and only if Φ is PPT.
+Remark 4.2.
+(1) The quantum channels of the form (4.1) include two
+important one-parameter families of quantum channels, namely de-
+polarizing channels
+∆b(X) = (1 − b)Tr(X)
+d
+Idd + bX
+(4.2)
+with −
+1
+d2−1 ≤ b ≤ 1, and transpose depolarizing channels
+Λc(X) = (1 − c)Tr(X)
+d
+Idd + cXT
+(4.3)
+with −
+1
+d−1 ≤ c ≤
+1
+d+1. It was already known that PPT=EB for these
+channels from various perspectives [Wer89, HH99, Wat18, SN21],
+and our Theorem 4.1 covers these classes.
+(2) Moreover, Theorem 4.1 gives an affirmative answer to PPT=EB
+problem for the so-called generalized Werner-Holevo quantum chan-
+nels, which was once conjectured to be false [DFV08]. See Appen-
+dix A for more details on the generalized Werner-Holevo channels.
+A starting point for a proof of Theorem 4.1 is to observe that any quantum
+channel of the form (4.1) is covariant with respect to the signed symmetric
+group Hd. One of the equivalent ways to realize the signed symmetric group
+is to define Hd as a subgroup of the orthogonal group Od generated by
+permutation matrices and diagonal orthogonal matrices. In other words,
+every element in Hd is written as an orthogonal matrix
+d
+�
+i=1
+si|σ(i)⟩⟨i| for
+s1, s2, . . . , sn ∈ {±1} and σ ∈ Sd. We define Inv(H ⊗ H) and Cov(H, H)
+
+16
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+with respect to the fundamental representation H ∈ Hd �→ H ∈ Od, which
+is irreducible as proved below.
+Lemma 4.3. The fundamental representation H ∈ Hd �→ H ∈ Od is
+irreducible.
+Proof. The identity
+HXHT =
+d
+�
+i,j=1
+sisjXij|σ(i)⟩⟨σ(j)| =
+d
+�
+i,j=1
+sσ−1(i)sσ−1(j)Xσ−1(i)σ−1(j)|i⟩⟨j|
+(4.4)
+and the invariance property HXHT = X for all H ∈ Hd tell us that
+sσ(i)sσ(j)Xσ(i)σ(j) = Xij
+(4.5)
+for all s1, . . . , sd ∈ {±1} and σ ∈ Sd. This implies that Xii ≡ X11 for all
+1 ≤ i ≤ d and Xij = 0 for all i ̸= j, i.e., X = X11 Idd ∈ C · Idd.
+□
+Let us denote by W the space of linear maps spanned by the following
+four unital TP maps ψ0, ψ1, ψ2, ψ3 : Md(C) → Md(C), where
+�
+�
+�
+�
+�
+�
+�
+ψ0(X) = Tr(X)
+d
+Idd,
+ψ1(X) = X,
+ψ2(X) = XT,
+ψ3(X) = diag(X) = �d
+i=1 Xii|i⟩⟨i|.
+(4.6)
+It is straightforward to check ψi ∈ Cov(H, H) for i = 0, . . . , 3, so we have
+W ⊆ Cov(H, H). To prove Cov(H, H) = W, let us note the fact that
+any L ∈ Cov(H, H) satisfies the so-called diagonal orthogonal covariance
+(DOC) property, i.e.
+L(ZXZT) = ZL(X)ZT
+(4.7)
+for all X ∈ Md(C) and diagonal orthogonal matrices Z. This class of
+channels has been analyzed recently in [SN21, SN22, SDN22]. In partic-
+ular, it is shown that any DOC map L can be parameterized by a triple
+(A, B, C) ∈ Md(C)3 satisfying diag(A) = diag(B) = diag(C) such that
+L(X) = diag(A|diag X⟩) + �B ⊙ X + �C ⊙ XT,
+(4.8)
+where |diag Y ⟩ = �d
+i=1 Yii|i⟩, �Y = Y − diag(Y ), and ⊙ denotes the Schur
+product (or Hadamard product) between matrices. In this case, let us denote
+by L = LA,B,C.
+Proposition 4.4. The space Cov(H, H) is spanned by the four unital TP
+positive maps ψ0, ψ1, ψ2, and ψ3 from (4.6).
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+17
+Proof. We already know W ⊆ Cov(H, H), and let us pick an arbitrary
+L ∈ Cov(H, H). Since L is DOC, there exists (A, B, C) ∈ Md(C)3 such
+that L = LA,B,C of the form (4.8). Note that L further satisfies
+L(PσXP T
+σ ) = PσL(X)P T
+σ
+(4.9)
+for all X ∈ Md(C) and σ ∈ Sd. Here, Pσ = �d
+i=1 |σ(i)⟩⟨i| is the permuta-
+tion matrix associated with σ.
+Let us take X = eij. If i = j, then (4.9) implies
+d
+�
+k=1
+Akσ(i)|k⟩⟨k| =
+d
+�
+k=1
+Aki|σ(k)⟩⟨σ(k)|,
+(4.10)
+which means that Aik = Aσ(i)σ(k) for all 1 ≤ i, k ≤ d and σ ∈ Sd. There-
+fore, Aii ≡ A11 for all i and Aik ≡ A12 for all i ̸= k. On the other hand, if
+i ̸= j, then (4.9) becomes
+Bσ(i)σ(j)|σ(i)⟩⟨σ(j)| + Cσ(j)σ(i)|σ(j)⟩⟨σ(i)|
+(4.11)
+= Bij|σ(i)⟩⟨σ(j)| + Cji|σ(j)⟩⟨σ(i)|,
+(4.12)
+which gives Bij ≡ B12 and Cij ≡ C12 for all i ̸= j. Consequently, the
+formula (4.8) now gives
+L = dA12ψ0 +B12ψ1 +C12ψ2 +(A11 −A12 −B12 −C12)ψ3 ∈ W, (4.13)
+which in turn shows Cov(H, H) ⊆ W.
+□
+From now, let us denote (H, H)-covariant unital (and TP) maps by
+ψa,b,c = aψ0 + bψ1 + cψ2 + (1 − a − b − c)ψ3
+(4.14)
+for simplicity, where ψ0, . . . , ψ3 are from (4.6). Note that ψa,b,c can be
+understood as a DOC map LA,B,C under the correspondence
+�
+�
+�
+�
+�
+A = a
+dJd + (1 − a)Idd,
+B = b(Jd − Idd) + ( a
+d + (1 − a))Idd,
+C = c(Jd − Idd) + ( a
+d + (1 − a))Idd,
+(4.15)
+where Jd = �d
+i,j=1 eij. According to [SN21, Section 6], LA,B,C is CPTP if
+and only if
+�
+�
+�
+�
+�
+Aij ≥ 0, �d
+k=1 Akj = 1 for all i, j
+B ≥ 0,
+Cij = Cji, |Cij|2 ≤ AijAji for all i, j.
+(4.16)
+
+18
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+In terms of the parameters a, b, c, the map ψa,b,c is CPTP if and only if
+�
+�
+�
+0 ≤ a ≤
+d
+d−1,
+a
+d −
+1
+d−1 ≤ b ≤ 1 − d−1
+d a,
+− a
+d ≤ c ≤ a
+d.
+(4.17)
+Note that the set of (a, b, c) ∈ R3 satisfying (4.17) is a tetrahedral depicted
+in Figure 1.
+FIGURE 1. The region of CovQC(H, H)
+In particular, there are exactly four extremal (H, H)-covariant quantum
+channels corresponding to the four vertices given by
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ψ1 = ψ1,
+Ψ2 =
+d
+d−1ψ0 +
+1
+d−1ψ2 −
+2
+d−1ψ3,
+Ψ3 = −
+1
+d−1ψ1 +
+d
+d−1ψ3,
+Ψ4 =
+d
+d−1ψ0 −
+1
+d−1ψ1.
+(4.18)
+whose Choi matrices are (up to normalization) four mutually orthogonal
+projections. On the other hand, it is easy to see that
+Td ◦ ψa,b,c = ψa,b,c ◦ Td = ψa,c,b, a, b, c ∈ C.
+(4.19)
+Therefore, ψa,b,c is a PPT quantum channel if and only if both ψa,b,c and
+ψa,c,b are CPTP. Let us denote by CovPPTQC(H, H) the set of all PPT
+quantum channels ψa,b,c. Then CovPPTQC(H, H) can be realized as a con-
+vex set in R3 as in the following Figure 2.
+If d ≥ 3, the eight vertices of the polytope CovPPTQC(H, H) are explic-
+itly given by ψvi (i = 1, . . . , 8), where
+• v0 = (0, 0, 0),
+
+亚4
+亚2
+1
+1
+-
+-
+-
+1
+-
+-
+-
+-
+1
+1
+亚ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+19
+FIGURE 2. The region of CovPPTQC(H, H)
+• v1 =
+�
+d
+2(d−1),
+1
+2(d−1), −
+1
+2(d−1)
+�
+, v2 =
+�
+d
+2(d−1), −
+1
+2(d−1),
+1
+2(d−1)
+�
+,
+v3 =
+�
+d
+2(d−1), −
+1
+2(d−1), −
+1
+2(d−1)
+�
+,
+• v4 =
+�
+1, 1
+d, 1
+d
+�
+, v5 =
+�
+1, 1
+d, −
+1
+d(d−1)
+�
+, v6 =
+�
+1, −
+1
+d(d−1), 1
+d
+�
+,
+• v7 =
+�
+d
+d−1, 0, 0
+�
+.
+[Step 1+Step 2] One of the main steps in this section is to characterize
+all elements in CovPos1(H, H). Indeed, CovPos1(H, H) is given as follows
+with eight extreme points (see Figure 3).
+FIGURE 3. The region of CovPos1(H, H)
+
+-
+1
+1
+V5
+V6
+.
+1
+-
+-
+1
+V3
+1
+V2
+1
+-
+1
+-
+1
+4
+1
+1
+1
+-
+-
+1
+-
+o
+W亚
+亚2
+亚3
+I3 0 Td
+ioTd
+2
+亚20
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+Theorem 4.5. Let d ≥ 3. Then the convex set CovPos1(H, H) has exactly
+8 extreme points
+Ψ1, Ψ2, Ψ3, Ψ4,
+Ψ1 ◦ Td, Ψ2 ◦ Td, Ψ3 ◦ Td, Ψ4 ◦ Td,
+(4.20)
+where Ψ1, . . . , Ψ4 are given by (4.18). In particular, all positive (H, H)-
+covariant maps are decomposable.
+Proof. Since Ψ1, . . . , Ψ4 are CP and Ψ1◦Td, . . . , Ψ4◦Td are CCP, the convex
+hull Vd of these 8 maps is obviously contained in CovPos1(H, H). To show
+the reverse inclusion CovPos1(H, H) ⊆ Vd, we observe that the set
+Vd :=
+�
+(a, b, c) ∈ R3 : ψa,b,c ∈ Vd
+�
+⊂ R3
+(4.21)
+is the convex hull of 8 points
+� (0, 1, 0) ,
+�
+d
+d−1, 0,
+1
+d−1
+�
+,
+�
+0, −
+1
+d−1, 0
+�
+,
+�
+d
+d−1, 0, −
+1
+d−1
+�
+,
+(0, 0, 1) ,
+�
+d
+d−1,
+1
+d−1, 0
+�
+,
+�
+0, 0, −
+1
+d−1
+�
+,
+�
+d
+d−1, −
+1
+d−1, 0
+�
+,
+(4.22)
+which are got from (4.18) and (4.19). Therefore, Vd can be understood as
+the region of (a, b, c) ∈ R3 satisfying the following inequalities:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(1)
+0 ≤ a ≤
+d
+d−1,
+(2)
+d−2
+d a + b + c ≤ 1,
+(3)
+d−2
+d a + |b − c| ≤ 1,
+(4)
+b + c ≥ −
+1
+d−1,
+(5)
+b − (d − 1) c ≤ 1,
+(6)
+c − (d − 1) b ≤ 1.
+(4.23)
+Now if ψa,b,c /∈ Vd (which is equivalent to (a, b, c) /∈ Vd, and hence violates
+at least one of the inequalities (1) - (6) in (4.23)), we can choose a unit
+vector ξ ∈ Cd such that ψa,b,c(|ξ⟩⟨ξ|) is not positive semidefinite as in Table
+1. This shows CovPos1(H, H) ⊆ Vd.
+TABLE 1. Non-positivity outside Vd
+(a, b, c) violates (1)
+|ξ⟩ = |1⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0
+(a, b, c) violates (2)
+|ξ⟩ =
+1
+√
+2(|1⟩ + |2⟩) =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0
+(a, b, c) violates (3)
+|ξ⟩ =
+1
+√
+2(|1⟩ + i|2⟩) =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0
+(a, b, c) violates (4)
+|ξ⟩ =
+1
+√
+d
+d
+�
+k=1
+|k⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0
+(a, b, c) violates (5) or (6)
+|ξ⟩ =
+1
+√
+d
+d
+�
+k=1
+e
+2πik
+d |k⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0
+□
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+21
+Proof of Theorem 4.1. Now, the conclusion is straightforward from Corol-
+lary 3.8 and Theorem 4.5.
+□
+Remark 4.6.
+(1) Note that Theorem 4.5 gives a complete characteri-
+zation of all positive linear maps ψ spanned by ψ0, ψ1, ψ2, ψ3. This
+strengthens the results in Section 5 of [KMS20] focusing on positive
+linear maps spanned only by ψ0, ψ1, ψ3 without ψ2.
+(2) Theorem 4.5 tells us not only POS=DEC, but also explicit decom-
+positions of our positive covariant maps into sums of CP and CCP
+maps. Note that this was one of the open questions raised in Section
+6.c of [KMS20]. We refer to Appendix B for more details.
+5. PPT=SEP PROBLEMS IN TRIPARTITE SYSTEMS WITH UNITARY
+GROUP SYMMETRIES
+Recall that a tripartite quantum state ρ ∈ D(HA ⊗ HB ⊗ HC) is called
+A-BC separable (resp. A-BC PPT) if ρ is separable (resp. PPT) in the
+situation where B(HA ⊗ HB ⊗ HC) is understood as the bipartite system
+B(HA) ⊗ B(HB ⊗ HC). Furthermore, C-AB or B-AC separability (resp.
+PPT) is defined similarly. We will focus on the situation where HA = HB =
+HC = Cd, and let us denote by
+�
+�
+�
+XTA = (Td ⊗ idd2)(X),
+XTB = (idd ⊗ Td ⊗ idd)(X),
+XTC = (idd2 ⊗ Td)(X),
+(5.1)
+the three partial transposes of X ∈ B(HA ⊗ HB ⊗ HC) = Md3(C).
+The main purpose of this section is to apply our results in Section 3 as
+new sources to study PPT=SEP problems, equivalently POS=DEC prob-
+lems for some tripartite invariant quantum states. In Section 5.1, we exhibit
+positive non-decomposable covariant maps L : Md(C) → Md2(C) satisfy-
+ing
+L(UXU T) = (U ⊗ U)L(X)(U ⊗ U)∗
+(5.2)
+for all unitary matrices U ∈ Ud and X ∈ Md(C). This result is parallel
+to the fact PPT̸=SEP for tripartite Werner states [EW01], i.e. tripartite
+quantum states ρ ∈ Md3(C) satisfying
+(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U)
+(5.3)
+for all unitary matrices U ∈ U(d).
+On the other hand, in Section 5.2, we show that a strong contrast PPT=SEP
+holds for quantum orthogonally invariant quantum states. More generally,
+we prove that PPT=SEP holds for any tripartite quantum states ρ ∈ Md3(C)
+satisfying
+(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U)
+(5.4)
+
+22
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+for all unitary matrices U ∈ U(d).
+5.1. Tripartite Werner states. Let πA, πBC be unitary representations of
+the unitary group Ud given by πA(U) = U and πBC(U) = U ⊗ U. Then
+the elements in InvQS(πA ⊗ πBC) are called tripartite Werner states. Let
+us write Inv(U ⊗3) = Inv(πA ⊗ πBC) and Cov(U, UU) = Cov(πA, πBC)
+for simplicity. The application of Schur-Weyl duality [EW01] or von Neu-
+mann’s bicommutant theorem [Wat18, Theorem 7.15] implies that the space
+Inv(U ⊗3) is spanned by six unitary operators {Vσ : σ ∈ S3}. Here, Vσ :
+(Cd)⊗3 → (Cd)⊗3 is determined by Vσ(ξ1 ⊗ ξ2 ⊗ ξ3) = ξσ−1(1) ⊗ ξσ−1(2) ⊗
+ξσ−1(3) for any ξ1, ξ2, ξ3 ∈ Cd and σ ∈ S3, or equivalently,
+Vσ =
+d
+�
+j1,j2,j3=1
+|j1j2j3⟩⟨jσ(1)jσ(2)jσ(3)|.
+(5.5)
+Recall that A-BC PPT property and separability of ρ ∈ InvQS(U ⊗3)
+were already characterized in [EW01], and it was shown that PPT=SEP if
+and only if d = 2. Therefore, a direct application of Corollary 3.4 gives us
+the following result.
+Theorem 5.1. All positive (UU, U)-covariant maps are decomposable if
+and only if d = 2. By taking the adjoint operation L �→ L∗, the same
+conclusion holds for positive (U, UU)-covariant maps.
+In the remaining of this section, we will assume d ≥ 3 and exhibit posi-
+tive non-decomposable (U, UU)-covariant maps.
+[Step 1] First of all, let us characterize all elements in CovPos(U, UU).
+Note that Corollary 2.4 (3) implies that the space Cov(U, UU) is spanned
+by the following six linear maps Lσ whose unnormalized Choi matrices are
+the operators Vσ ∈ Inv(U ⊗3) in (5.5):
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Le(X) = (Tr X) · Idd ⊗ Idd,
+L(12)(X) = XT ⊗ Idd,
+L(13)(X) = Idd ⊗ XT,
+L(23)(X) = (Tr X) · �d
+j2,j3=1 |j3j2⟩⟨j2j3|,
+L(123)(X) = �d
+j1,j2,j3=1 Xj1j2|j2j3⟩⟨j3j1|,
+L(132)(X) = �d
+j1,j2,j3=1 Xj1j3|j2j3⟩⟨j1j2|.
+(5.6)
+Lemma 5.2. Let L = �
+σ∈S3 aσLσ ∈ Cov(U, UU). Then L is positive if
+and only if
+�
+�
+�
+�
+�
+�
+�
+(1)
+ae, a(12), a(13), a(23) ∈ R and a(132) = a(123),
+(2)
+ae ≥ max
+�
+−a(12), −a(13), |a(23)|
+���
+,
+(3)
+ae + a(12) + a(13) + a(23) + a(123) + a(132) ≥ 0,
+(4)
+�
+ae + a(12)
+� �
+ae + a(13)
+�
+≥
+��a(23) + a(123)
+��2 .
+(5.7)
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+23
+Proof. Since every unit vector ξ ∈ Cd can be written as |ξ⟩ = U|1⟩ for
+some U ∈ Ud, the (U, UU)-covariance property implies that L is positive if
+and only if L(e11) ≥ 0. Moreover, L(e11) has a matrix decomposition
+L(e11) ∼= (ae + a(12) + a(13) + a(23) + a(123) + a(132))1 ⊕ (ae + a(23))Idd−1
+⊕
+�
+�
+d
+�
+j=2
+� ae + a(12)
+a(23) + a(123)
+a(23) + a(132)
+ae + a(13)
+��
+� ⊕
+�
+�
+�
+2≤i<j≤d
+� ae
+a(23)
+a(23)
+ae
+��
+�
+(5.8)
+with respect to the bases {|11⟩}, {|22⟩, |33⟩, . . . , |dd⟩}, {|1j⟩, |j1⟩} for j =
+2, . . . , d, and {|ij⟩, |ji⟩} for 2 ≤ i < j ≤ d, respectively. Therefore,
+L(e11) ≥ 0 if only if (5.7) holds.
+□
+The next step is to classify CP and CCP conditions in Cov(U, UU) to
+find all PPT elements in InvQS(U ⊗3).
+Lemma 5.3. Let L = �
+σ aσLσ and let X = �
+σ aσVσ. Then
+(1) L is CP if and only if X ≥ 0 if and only if
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ae, a(12), a(13), a(23) ∈ R and a(123) = a(132),
+ae + a(123) + a(132) ≥ |a(12) + a(13) + a(23)|,
+2ae − a(123) − a(132) ≥ 0,
+(ae + ωa(123) + ωa(132))(ae + ωa(123) + ωa(132))
+≥
+��ωa(12) + ωa(13) + a(23)
+��2 .
+(5.9)
+(2) L is CCP if and only if XTA ≥ 0 if and only if
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ae, a(12), a(13), a(23) ∈ R and a(123) = a(132),
+ae ≥ |a(23)|,
+2ae + a(123) + a(132) + d(a(12) + a(13)) ≥ 0,
+�
+ae + a(23) + d+1
+2 (a(12) + a(13) + a(123) + a(132))
+�
+×
+�
+ae − a(23) + d−1
+2 (a(12) + a(13) − a(123) − a(132))
+�
+≥ d2−1
+4 (|a(12) − a(13)|2 + |a(123) − a(132)|2).
+(5.10)
+These characterizations were already known from [EW01, Lemma 2 and
+Lemma 8], but with a different parametrization. An elaboration on Lemma
+5.3 is attached in Appendix C using the following identifications
+span {Vσ : σ ∈ S3} ∼= C ⊕ C ⊕ M2(C),
+(5.11)
+span
+�
+V TA
+σ
+: σ ∈ S3
+� ∼= C ⊕ C ⊕ M2(C)
+(5.12)
+as ∗-algebras.
+[Step 2] All extremal elements in CovPosTP(U, UU) are completely
+characterized in the following lemma. Our proof is straightforward but
+rather cumbersome, so we attach the proof in Appendix D.
+
+24
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+Lemma 5.4. Let L = �
+σ aσLσ ∈ Cov(U, UU). Then the following are
+equivalent.
+(1) L ∈ Ext(CovPosTP(U, UU))
+(2) ae, a(12), a(13), a(23) ∈ R, a(123) = a(132), and the associated 6-tuple
+(ae, a(12), a(13), a(23), Re(a(123)), Im(a(123))) ∈ R6
+(5.13)
+is one of the following three types:
+Type I
+c1(1, −1, −1, −1, 1, 0)
+Type II
+c2(0, A, B, 0, C, ±
+√
+AB − C2)
+Type III
+c3( A+B+2C
+2
+, A−B−2C
+2
+, −A+B−2C
+2
+, A+B+2C
+2
+, − A+B
+2 , ±
+√
+AB − C2)
+where A, B ≥ 0, C ∈ R, AB ≥ C2, and the normalizing constants
+c1, c2, and c3 are chosen to satisfy the TP condition
+d2ae + d(a(12) + a(13) + a(23)) + (a(123) + a(132)) = 1.
+(5.14)
+Then, combining Lemma 5.3 and Lemma 5.4, we can check that
+• Every L ∈ Ext(CovPosTP(U, UU)) of Type I is CP,
+• Every L ∈ Ext(CovPosTP(U, UU)) of Type II is CCP,
+• Let L ∈ Ext(CovPosTP(U, UU)) of Type III. Then
+– L is CP if and only if A = B = C,
+– L is CCP if and only if A = B = −C.
+Thus, Type III (with neither A = B = C nor A = B = −C) provides
+explicit positive non-decomposable maps in CovPos(U, UU) by Theorem
+3.11. For example, we can choose A = 1, B = 0, and C = 0 to obtain a
+specific extremal element
+L0 = Le+L(12)−L(13)+L(23)−L(123)−L(132) ∈ Ext(CovPosTP(U, UU))
+(5.15)
+up to a normalizing constant.
+[Step 3] On the dual side, the chosen positive non-decomposable map
+L∗
+0 ∈ CovPos(UU, U) should play a role as an entanglement detector. In-
+deed, if we take
+ρt =
+1
+d3 + (t + 1)d2 + 2t
+�d + t
+d
+Ve + V(13) + t
+dV(123) + t
+dV(132)
+�
+(5.16)
+with 0 < t ≤ 3.89 and d ≥ 3, then ρt ∈ InvQS(U ⊗3) is A-BC PPT by
+Lemma 5.3. Moreover, it is straightforward to see that
+(d3 + (t + 1)d2 + 2t) · (id ⊗ L∗
+0)(ρt)
+=
+�
+d2 + (t + 2)d + 3t − 2t
+d
+�
+Idd ⊗ Idd −
+�
+d2 + (2t + 2)d
+�
+|Ωd⟩⟨Ωd|
+(5.17)
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+25
+has a negative eigenvalue −t
+�
+d + 2
+d − 3
+�
+< 0. Consequently, the quan-
+tum state ρt is A-BC PPT entangled by Theorem 3.9 or by Horodecki’s
+criterion.
+Remark 5.5. Note that ρt is also C-AB PPT entangled since V(13)ρtV(13) =
+ρt. On the other hand, ρt is not B-AC PPT (and hence entangled). Indeed,
+we can observe that
+ρTB
+t
+= V(12)(V(12)ρtV(12))TAV(12),
+(5.18)
+but V(12)ρtV(12) is not A-BC PPT since
+V(12)ρtV(12) =
+1
+d3 + (t + 1)d2 + 2t
+�d + t
+d
+Ve + V(23) + t
+dV(123) + t
+dV(132)
+�
+(5.19)
+does not satisfy the CCP condition (5.10).
+It might be interesting if we can find a tripartite PPT-entangled Werner
+state with respect to all the three partitions A-BC, B-AC, and C-AB. How-
+ever, Lemma 7 of [EW01] implies that there is no such an example ρ =
+�
+σ aσVσ if one of the following conditions is satisfied:
+• a(12) = a(13) and a(123) = a(132),
+• a(13) = a(23) and a(123) = a(132),
+• a(23) = a(12) and a(123) = a(132),
+We leave the general situation as an open question.
+5.2. Tripartite quantum orthogonally invariant quantum states. Within
+the framework of compact quantum groups, it is well-known that the or-
+thogonal group Od allows a universal object, namely the free orthogonal
+quantum group O+
+d [Wan95, Tim08]. In other words, the invariance prop-
+erty with respect to O+
+d is a stronger notion than the (classical) orthogonal
+group invariance. See [LY22] for a general discussion on invariant quantum
+states and covariant quantum channels with quantum group symmetries.
+In this section, we focus on the space Inv(O⊗3
++ ) of the tripartite quantum
+orthogonally invariant operators spanned by five tripartite operators
+Tσ = V TB
+σ
+=
+d
+�
+j1,j2,j3=1
+|j1jσ(2)j3⟩⟨jσ(1)j2jσ(3)|
+(5.20)
+for σ ∈ S3 \ {(13)} = {e, (12), (23), (123), (132)}.
+See Appendix F
+for more discussions on (5.20) and O+
+d . Although Theorem 3.9 does not
+cover quantum group symmetries, any X ∈ Inv(O⊗3
++ ) satisfies the follow-
+ing group invariance property
+(U ⊗ U ⊗ U)X(U ⊗ U ⊗ U)∗ = X
+(5.21)
+
+26
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+for all U ∈ Ud thanks to Corollary 2.4 (1). This transfers our problem to the
+realm of classical group symmetries. More precisely, we have
+Inv(O⊗3
++ ) ⊆ Inv(U ⊗ U ⊗ U) = Inv(πA ⊗ πBC)
+(5.22)
+where πA(U) = U and πBC(U) = U ⊗U. The main theorem of this section
+is the following.
+Theorem 5.6. Let ρ ∈ InvQS(U ⊗ U ⊗ U). Then ρ is A-BC separable if
+and only if ρ is A-BC PPT. In particular, A-BC PPT= A-BC SEP holds in
+InvQS(O⊗3
++ ).
+Remark 5.7. Note that, for any unitary representation π of a compact
+group, the following three problems
+• A-BC PPT = A-BC SEP in Inv(π ⊗ π ⊗ π)
+• B-AC PPT = B-AC SEP in Inv(π ⊗ π ⊗ π)
+• C-AB PPT = C-AB SEP in Inv(π ⊗ π ⊗ π)
+are equivalent. However, Theorem 5.6 implies that this equivalence is no
+longer true when π is replaced by a unitary representation of a compact
+quantum group. Indeed, a B-AC PPT entangled state V(12)ρtV(12) ∈ InvQS(U ⊗3)
+from (5.19) is transferred to the following B-AC PPT entangled state in
+InvQS(O⊗3
++ ):
+(V(12)ρtV(12))TB
+=
+1
+d3 + (t + 1)d2 + 2t
+�d + t
+d
+Te + T(23) + t
+dT(123) + t
+dT(132)
+�
+. (5.23)
+In other words, A-BC PPT=SEP problem is not equivalent to B-AC PPT=SEP
+problem in InvQS(O⊗3
++ ). A reason for this genuine quantum phenomenon is
+that the associated C∗-algebra of O+
+d is non-commutative.
+[Step 1] Let us apply Corollary 3.8 to prove that POS=DEC holds in
+CovPos1(UU, U) = CovPos1(πBC, πA), or equivalently, POS=DEC holds
+in CovPosTP(U, UU). Recall that the space Cov(U, UU) is spanned by six
+linear maps
+Mσ = (Td ⊗ idd) ◦ Lσ
+(5.24)
+for σ ∈ S3, where Lσ is given by (5.6). Then the unnormalized Choi matrix
+of Mσ is given by Tσ.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+27
+Lemma 5.8. Let d ≥ 3 and M = �
+σ aσMσ ∈ Cov(U, UU). Then M is
+positive if and only if
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(1)
+ae, a(12), a(13), a(23) ∈ R and a(132) = a(123),
+(2)
+ae ≥ max
+�
+0, −a(12), −a(13)
+�
+,
+(3)
+ae + a(12) + a(13) + a(23) + a(123) + a(132) ≥ 0,
+(4)
+ae + (d − 1)a(23) ≥ 0,
+(5)
+(ae + a(12) + a(13) + a(23) + a(123) + a(132))(ae + (d − 1)a(23))
+≥ (d − 1)|a(23) + a(123)|2.
+(5.25)
+Proof. As in the proof of Lemma 5.2, the positivity of M is equivalent to
+M(e11) ≥ 0. Moreover, M(e11) ∈ Md(C) ⊗ Md(C) has a matrix decom-
+position
+M ⊕ (ae + a(12)) Idd−1 ⊕ (ae + a(13)) Idd−1 ⊕ ae Id(d−1)(d−2),
+(5.26)
+where
+M =
+�
+c
+(a(23) + a(132))⟨v|
+(a(23) + a(123))|v⟩
+a(23)|v⟩⟨v|
+�
++ ae Idd
+(5.27)
+with c = a(12) +a(13) +a(23) +a(123) +a(132) and v = (1, 1, . . . , 1)t ∈ Cd−1.
+The four matrices in the matrix decomposition (5.26) are with respect to the
+bases {|11⟩, |22⟩, . . . , |dd⟩}, {|12⟩, |13⟩, . . . , |1d⟩}, {|21⟩, |31⟩, . . . , |d1⟩},
+and {|ij⟩ : i, j ̸= 1 and i ̸= j}, respectively. Thus, M(e11) ≥ 0 if and only
+if the conditions (1) and (2) in (5.25) hold and M ≥ 0. Moreover, we can
+rewrite (5.27) as
+M − ae Idd = V
+�
+c
+√
+d − 1α
+√
+d − 1α
+(d − 1)a(23)
+�
+V ∗,
+(5.28)
+where α = a(23) + a(123) and V =
+�1
+0
+0
+1
+√
+d−1|v⟩
+�
+∈ Md,2(C) is an isome-
+try. Thus, the nonzero eigenvalues of M − aeIdd are the same with those of
+�
+c
+√
+d − 1α
+√
+d − 1α
+(d − 1)a(23)
+�
+. Consequently, the condition M ≥ 0 is equiva-
+lent to the conditions (3), (4), and (5) of (5.25).
+□
+Thanks to Lemma 5.3, it is easy to derive CP and CCP conditions in
+Cov(U, UU) or A-BC PPT condition in InvQS(U ⊗ U ⊗ U).
+Lemma 5.9. Let d ≥ 3 and M = �
+σ∈S3 aσMσ ∈ CovPos(U, UU). Then
+(1) M is CP if and only if the operator
+V(12)
+��
+σ∈S3
+aσVσ
+�
+V(12) =
+�
+σ∈S3
+a(12)σ(12)Vσ
+(5.29)
+
+28
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+satisfies the condition (5.10).
+(2) M is CCP if and only if the operator
+V(13)
+��
+σ∈S3
+aσVσ
+�
+V(13) =
+�
+σ∈S3
+a(13)σ(13)Vσ
+(5.30)
+satisfies the condition (5.10).
+Proof. Let X = �
+σ∈S3 aσVσ ∈ Inv(U ⊗3) and X′ = V(12)XV(12). Then M
+is CP if and only if
+�
+σ∈S3
+aσTσ = XTB = V(12)(X′)TAV(12) ≥ 0,
+(5.31)
+which is equivalent to (X′)TA ≥ 0. On the other hand, let X′′ = V(13)XV(13).
+Then M is CCP if and only if
+��
+σ∈S3
+aσTσ
+�TA
+=
+�
+XTC�T =
+�
+V(13)(X′′)TAV(13)
+�T ≥ 0,
+(5.32)
+and this is equivalent to (X′′)TA ≥ 0.
+□
+[Step 2] We refer the reader to Appendix D for a proof of the following
+Lemma 5.10 classifying all extremal elements in CovPosTP(U, UU).
+Lemma 5.10. Let d ≥ 3 and M = �
+σ∈S3 aσMσ ∈ CovPos(U, UU). Then
+the following are equivalent.
+(1) M ∈ Ext(CovPosTP(U, UU)),
+(2) ae, a(12), a(13), a(23) ∈ R, a(123) = a(132), and the associated 6-tuple
+(ae, a(12), a(13), a(23), Re(a(123)), Im(a(123))) ∈ R6
+(5.33)
+is one of the following four types:
+Type I
+c1(d − 1, −1, 1 − d, −1, 1, 0),
+Type II
+c2(d − 1, 1 − d, −1, −1, 1, 0),
+Type III
+c3(0, A + B − 2C, 0, B, C − B, ±
+√
+AB − C2),
+Type IV
+c4(0, 0, A + B − 2C, B, C − B, ±
+√
+AB − C2),
+where A, B ≥ 0, C ∈ R, AB ≥ C2, and ci (i = 1, 2, 3, 4) are
+normalizing constants chosen to satisfy the TP condition
+d2ae + d(a(12) + a(13) + a(23)) + (a(123) + a(132)) = 1.
+(5.34)
+Proof of Theorem 5.6. Let us assume d ≥ 3. According to Theorem 3.11
+and Lemma 5.10, it suffices to show that M = �
+σ∈S3 aσMσ is either CP
+or CCP whenever (aσ)σ∈S3 is one of the four Types I - IV. Now, by applying
+Lemma 5.9, we can check that
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+29
+• Every L ∈ Ext(CovPosTP(U, UU)) of Type I or Type III is CP,
+• Every L ∈ Ext(CovPosTP(U, UU)) of Type II or Type IV CCP,
+thanks to the conditions A, B ≥ 0 and AB ≥ C2. When d = 2, we refer to
+Appendix E for the complete proof.
+□
+Acknowledgements: The authors thank Professor Hun Hee Lee for the
+helpful discussions and comments. S-J.Park, Y-G.Jung and S-G.Youn were
+supported by the National Research Foundation of Korea (NRF) grant funded
+by the Ministry of Science and ICT (MSIT) (No.2021K1A3A1A21039365).
+Y-G.Jung, J.Park and S-G.Youn were also supported by Samsung Science
+and Technology Foundation under Project Number SSTF-BA2002-01. S-
+J.Park and S-G.Youn were supported by the National Research Foundation
+of Korea (NRF) grant funded by the Ministry of Science and ICT (MSIT)
+(No. 2020R1C1C1A01009681).
+APPENDIX A. GENERALIZED WERNER-HOLEVO CHANNELS
+Note that quantum channels of the form
+Φb,c(X) = (1 − b − c)Tr(X)
+d
+Idd + bX + cXT
+(A.1)
+with proper choices of b and c form a subclass of CovQC(H, H). These
+channels are called generalized Werner-Holevo channels, and they include
+all mixtures of the depolarizing and transpose depolarizing channels. It is
+known in [Has18] that the generalized Werner-Holevo channels are charac-
+terized by the canonical orthogonal group covariance property:
+Φb,c(OXOT) = OΦb,c(X)OT,
+(A.2)
+for all X ∈ Md(C) and O ∈ Od. These channels were studied in connec-
+tion with additivity problems [Mic07, DFV08]. In particular, it was conjec-
+tured PPT̸=EB for generalized Werner-Holevo channels in [DFV08], but an
+immediate Corollary of Theorem 4.1 is that a generalized Werner-Holevo
+channel Φb,c = ψ1−b−c,b,c is PPT if and only if it is EB.
+Furthermore, there is another approach to explain this fact. Note that, by
+Theorem 4.5, a generalized Werner-Holevo channel Φb,c is a PPT quantum
+channel if and only if Φb,c is a convex combination of the following four
+extremal PPT channels Φw1, · · · , Φw4, where
+�
+w1 =
+�
+1
+d+2,
+1
+d+2
+�
+,
+w2 =
+�
+−
+2
+d2+d−2,
+d
+d2+d−2
+�
+,
+w3 =
+�
+−
+1
+d2+d−2, −
+1
+d2+d−2
+�
+,
+w4 =
+�
+d
+d2+d−2, −
+2
+d2+d−2
+�
+.
+(A.3)
+As DOC maps, w1 corresponds to a triple (A1, B1, C1) where A1 = B1 =
+C1 =
+1
+d+2(Jd + 2Idd), and w3 corresponds to a triple (A3, B3, C3) where
+
+30
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+A3 =
+1
+d2+d−2((d + 1)Jd − 2Idd) and B3 = C3 =
+1
+d2+d−2(−Jd + dIdd), by
+the correspondence (4.15). Since
+A1 =
+1
+d + 2|v⟩⟨v| +
+2
+d + 2
+d
+�
+i=1
+|i⟩⟨i|,
+(A.4)
+where |v⟩ = �d
+i=1 |i⟩ ∈ Rd
++, A1 is a completely positive (CP) matrix. Thus,
+(A1, A1, A1) is triplewise completely positive (TCP), so Φw1 = LA1,A1,A1
+is EB (see [NS21] for more detail). Moreover, (A3, B3, C3) is also TCP by
+[NS21, Corollary B.10]. Since Φw2 = Td ◦ Φw4 from (4.19), the only thing
+to do is to prove that
+CΦw2 =
+2
+d2 + d − 2
+�Idd + F
+2
+− |Ωd⟩⟨Ωd|
+�
+,
+(A.5)
+is separable, where F = �d
+i,j=1 eij ⊗ eji is the flip matrix. We remark that
+[DFV08] suggested Φw2 as a possible example of a PPT non-EB map.
+Let us denote by InvQS(O ⊗ O) the set of all O ⊗ O-invariant states.
+Then InvQS(O ⊗ O) has three extreme points, namely the scalar multiples
+of mutually orthogonal projections
+Π1 = |Ωd⟩⟨Ωd|, Π2 = Idd + F
+2
+− |Ωd⟩⟨Ωd|, Π3 = Idd − F
+2
+.
+(A.6)
+Then Proposition 2.3 gives us a formula of the O ⊗ O-twirling map
+TO⊗O(X) =
+�
+Od
+(O ⊗O)X(O ⊗O)T dO =
+3
+�
+i=1
+Tr(ΠiX)
+Πi
+rank(Πi). (A.7)
+Lastly, let us take a product unit vector |ψ⟩ = |ξ⟩ ⊗ |ξ⟩ with |ξ⟩ =
+1
+√
+2(|1⟩ + i|2⟩).
+Then |ψ⟩ ∈ ran(Π2), which implies TO⊗O(|ψ⟩⟨ψ|) =
+Π2
+rank(Π2).
+Thus, CΦw2 = TO⊗O(|ψ⟩⟨ψ|) is a separable quantum state by
+Proposition 2.2.
+APPENDIX B. COVARIANT POSITIVE MAPS WITH RESPECT TO
+MONOMIAL UNITARY GROUPS
+In Section 5 and Section 6 of [KMS20], the authors analyzed the general
+structure of irreducibly covariant linear maps under some natural symme-
+tries of the symmetric group S4 and the monomial unitary group MU(d, n),
+and presented new examples of positive irreducibly covariant maps. In this
+section, we elaborate on how our Theorem 4.5 strengthens their results and
+resolves several open questions raised in [KMS20].
+On one side, they considered irreducibly (τ3, τ3)-covariant maps, where
+τ3 : S4 → U3 is a 3-dimensional irreducible component of the canonical
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+31
+representation σ ∈ S4 �→ σ ∈ U(4), where σ is identified with the per-
+mutation matrix �4
+k=1 |σ(k)⟩⟨k|. More precisely, the canonical representa-
+tion is not irreducible, and it allows one invariant 1-dimensional subspace
+C·|v⟩ with |v⟩ = �d
+k=1 |k⟩, and the other 3-dimensional invariant subspace
+V = (C · |v⟩)⊥. Then the fundamental representation τ3 : S4 → U(3) is de-
+fined by τ3(σ) = ΠV σ
+��
+V ∈ U(3) for all x ∈ S4, where ΠV is the orthogonal
+projection from C4 onto V .
+The authors characterized all (τ3, τ3)-covariant maps and suggested a suf-
+ficient condition for positivity using the so-called inverse reduction map cri-
+terion [MRS15]. On the other hand, it was shown in [LY22, Sectio 6.1.1]
+that, up to a change of basis, the (τ3, τ3)-covariant maps are precisely the
+linear combinations of Ψ1, Ψ2, Ψ3, Ψ4 : M3(C) → M3(C) from (4.18) with
+d = 3. In other words, we have Cov(τ3, τ3) = Cov(H, H) up to a uni-
+tary equivalence. Thus, Theorem 4.5 gives the solution to the open ques-
+tion of the characterization of all positive (τ3, τ3)-covariant maps raised in
+[KMS20].
+On the other side, recall that the monomial unitary group MU(d) is
+a subgroup of Ud generated by all permutation matrices and all diagonal
+unitary matrices. Moreover, the subgroup MU(d, n) of MU(d) is gen-
+erated by all permutation matrices and all diagonal matrices of the form
+�d
+i=1 ωi|i⟩⟨i| where ωi ∈
+�
+1, e2πi/n, . . . , e2(n−1)πi/n�
+. For a closed sub-
+group G of Ud, we denote by πG : x ∈ G �→ x ∈ Ud the fundamental
+representation of G, temporarily in this section. It was shown in [KMS20]
+that, if n ≥ 3, then
+Cov(πMU(d,n), πMU(d,n)) = span {ψ0, ψ1, ψ3}
+(B.1)
+where ψ0, ψ1, ψ3 are from (4.6). Moreover, the authors characterized all
+positive maps in this class, and proved that all (πMU(d,n), πMU(d,n))-covariant
+positive maps are decomposable for n ≥ 3. However, explicit decomposi-
+tions were left as an open question, and the authors conjectured that a non-
+decomposable positive map may arise under (πMU(d), πMU(d))-covariance.
+Our results in this paper resolve their open questions in the sense that
+(πMU(d), πMU(d))-covariance does not make a difference, but a weaker con-
+dition (πMU(d,2), πMU(d,2))-covariance does. Furthermore, their POS=DEC
+result in Cov(πMU(d,n), πMU(d,n)) with n ≥ 3 (Section 6.c of [KMS20]) ex-
+tends to a more general result POS=DEC in Cov(πMU(d,2), πMU(d,2)) with
+explicit decompositions into the sum of CP and CCP maps.
+(1) More precisely, it is clear that
+Cov(πMU(d), πMU(d)) ⊆ Cov(πMU(d,n), πMU(d,n)) = span {ψ0, ψ1, ψ3} , (B.2)
+
+32
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+and all the three maps ψ0, ψ1, and ψ3 are covariant with respect to
+general diagonal unitary matrices. Therefore, for n ≥ 3 we have
+Cov(πMU(d), πMU(d)) = Cov(πMU(d,n), πMU(d,n)) = span {ψ0, ψ1, ψ3} . (B.3)
+Therefore, there is no positive non-decomposable element inside
+Cov(πMU(d), πMU(d)).
+(2) On the other hand, since MU(d, 2) = Hd, we have
+Cov(πMU(d,2), πMU(d,2)) = Cov(H, H) = span {ψ0, ψ1, ψ2, ψ3} ,
+(B.4)
+by Proposition 4.4. Moreover, POS=DEC in Cov(πMU(d), πMU(d))
+from Section 6.c of [KMS20] is strengthened to POS=DEC in the
+larger space Cov(πMU(d,2), πMU(d,2)) with explicit decompositions
+by Theorem 4.5.
+APPENDIX C. CHARACTERIZATION OF INVPPTQS(U ⊗3)
+Recall that if d ≥ 3, there exist ∗-algebra isomorphisms
+F : Inv(U ⊗ U ⊗ U) → C ⊕ C ⊕ M2(C),
+(C.1)
+G : Inv(U ⊗ U ⊗ U) → C ⊕ C ⊕ M2(C)
+(C.2)
+by the representation theory of the unitary group Ud and (2.11). Moreover,
+the authors in [EW01] proposed specific choice of ∗-isomorphisms F and
+G that can be used to characterize the PPT condition of X = �
+σ∈S3 aσVσ ∈
+Inv(U ⊗3). For the convenience of the reader, we again present explicit
+maps F and G in terms of the bases {Vσ : σ ∈ S3} and
+�
+V TA
+σ
+: σ ∈ S3
+�
+of Inv(U ⊗3) and Inv(U ⊗ U ⊗ U), respectively, in Table 2.
+Proof of Lemma 5.3. Let X = �
+σ∈S3 aσVσ. Then X∗ = X is equivalent
+to ae, a(12), a(13), a(23) ∈ R and a(123) = a(132) since {Vσ}σ∈S3 is linearly
+independent. Now X ≥ 0, or equivalently, F(X) ≥ 0 holds if and only if
+(ae + a(123) + a(132)) ± (a(12) + a(13) + a(23)) ≥ 0 and
+(C.3)
+�
+ae + ωa(123) + a(123)ω
+ωa(12) + ωa(13) + a(23)
+ωa(12) + ωa(13) + a(23)
+ae + ωa(123) + ωa(132)
+�
+≥ 0,
+(C.4)
+which is equivalent to the condition (5.9). Similarly, we get the equivalence
+between the condition XTA ≥ 0 and (5.10).
+□
+APPENDIX D. EXTREMAL POSITIVE MAPS IN COVPOSTP(U, UU) AND
+COVPOSTP(U, UU)
+This section is to give detailed proofs of Lemma 5.4 and Lemma 5.10.
+For convenience, we assume ae, a(12), a(13), a(23) ∈ R, a(123) = a(132), and
+write r = Re(a(123)) and s = Im(a(123)) throughout this section.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+33
+TABLE 2. The isomorphisms F and G (ω = e2πi/3)
+X
+F(X)
+XTA
+G(XTA)
+Ve
+(1, 1,
+�
+1
+0
+0
+1
+�
+)
+V TA
+e
+(1, 1,
+�
+1
+0
+0
+1
+�
+)
+V(12)
+(1, −1,
+�
+0
+ω
+ω
+0
+�
+)
+V TA
+(12)
+(0, 0,
+�
+d+1
+2
+√
+d2−1
+2
+√
+d2−1
+2
+d−1
+2
+�
+)
+V(13)
+(1, −1,
+�
+0
+ω
+ω
+0
+�
+)
+V TA
+(13)
+(0, 0,
+�
+d+1
+2
+−
+√
+d2−1
+2
+−
+√
+d2−1
+2
+d−1
+2
+�
+)
+V(23)
+(1, −1,
+�
+0
+1
+1
+0
+�
+)
+V TA
+(23)
+(1, −1,
+�
+1
+0
+0
+−1
+�
+)
+V(123)
+(1, 1,
+�
+ω
+0
+0
+ω
+�
+)
+V TA
+(123)
+(0, 0,
+�
+d+1
+2
+−
+√
+d2−1
+2
+√
+d2−1
+2
+− d−1
+2
+�
+)
+V(132)
+(1, 1,
+�
+ω
+0
+0
+ω
+�
+)
+V TA
+(132)
+(0, 0,
+�
+d+1
+2
+√
+d2−1
+2
+−
+√
+d2−1
+2
+− d−1
+2
+�
+)
+Let P0 be the set of all tuples (ae, a(12), a(13), a(23), r, s) satisfying (5.7)
+and the TP condition
+d2ae + d(a(12) + a(13) + a(23)) + (a(123) + a(132)) = 1.
+(D.1)
+Then P0 describes the condition �
+σ aσLσ ∈ CovPosTP(U, UU) exactly,
+so P0 must be a convex and compact subset of R6. For simplification of the
+condition (5.7), let us consider a linear isomorphism
+α : (ae, a(12), a(13), a(23), r, s) �→ (ae, A, B, C, r, s)
+(D.2)
+of R6, where A = ae + a(12), B = ae + a(13), and C = a(23) + r. Then
+P = α(P0) is the set of all tuples (ae, A, B, C, r, s) ∈ R6 satisfying
+�
+�
+�
+�
+�
+�
+�
+(1)
+A, B ≥ 0,
+(2)
+AB ≥ C2 + s2,
+(3)
+A + B + C + r ≥ ae ≥ |C − r|,
+(4)
+d(d − 2)ae + d(A + B + C) − (d − 2)r = 1.
+(D.3)
+Note that we have Ext(P0) = α−1(Ext(P)). That is, it suffices to find the
+extreme points of P and restore the coefficients (aσ)σ∈S3 to get the corre-
+sponding extremal positive (U, UU)-covariant maps.
+Lemma D.1. Let S be the set of tuples (A, B, C, s) satisfying (1) and (2) of
+(D.3). Then S is a convex cone in R4. Moreover, if x0 = x1 + x2 in S with
+xi = (Ai, Bi, Ci, si) and if A0B0 = C2
+0 + s2
+0, then x1 = λ1x0, x2 = λ2x0
+for some λ1, λ2 ≥ 0. That is, the half-line R+ · x0 is an extremal ray of S.
+
+34
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+Proof. It is straightforward that x ∈ S implies λx ∈ S for all λ ≥ 0. Let
+us write xi = (Ai, Bi, Ci, si) ∈ S for i = 1, 2. Then A1 + A2 ≥ 0 and
+B1 + B2 ≥ 0, so the last thing to check is
+(A1 + A2) · (B1 + B2) ≥ (C1 + C2)2 + (s1 + s2)2.
+(D.4)
+Let us choose C′
+i ≥ |Ci|, s′
+i ≥ |si| such that AiBi = (C′
+i)2 + (s′
+i)2. Then,
+indeed, we have
+(A1 + A2)(B1 + B2) ≥ A1B1 + 2
+�
+A1B1A2B2 + A2B2
+(D.5)
+= (C′
+1)2 + (s′
+1)2 + 2
+�
+((C′
+1)2 + (s′
+1)2)((C′
+2)2 + (s′
+2)2) + (C′
+2)2 + (s′
+2)2
+(D.6)
+≥ (C′
+1)2 + (s′
+1)2 + 2(C′
+1C′
+2 + s′
+1s′
+2) + (C′
+2)2 + (s′
+2)2
+(D.7)
+= (C′
+1 + C′
+2)2 + (s′
+1 + s′
+2)2 ≥ (C1 + C2)2 + (s1 + s2)2.
+(D.8)
+by applying the AM-GM inequality and the Cauchy-Schwarz inequality.
+Therefore, x1 + x2 ∈ S, which proves that S is a convex cone. The latter
+statement follows by investigating the equality condition carefully in the
+above inequality, which is left to the reader.
+□
+Proof of Lemma 5.4. It is sufficient to show that all the extreme points x =
+(ae, A, B, C, r, s) of P are classified into the following three types up to
+normalizing constants: for A, B ≥ 0 and AB ≥ C2,
+Type I′
+(1, 0, 0, 0, 1, 0),
+Type II′
+(0, A, B, C, C, ±
+√
+AB − C2),
+Type III′
+( A+B+2C
+2
+, A, B, C, − A+B
+2 , ±
+√
+AB − C2).
+If AB > C2+s2, then we can choose s′ > |s| such that AB = C2+(s′)2.
+In this case, x(0)
+± = (ae, A, B, C, r, ±s′) ∈ P, and x is a (nontrivial) convex
+combination of x(0)
++ and x(0)
+− . Thus, x is not extremal in P.
+From now on, we assume AB = C2 + s2 (i.e., s = ±
+√
+AB − C2) and
+divide the condition (3) of (D.3) into the following cases.
+[Case 1] A + B + C + r ≥ ae > |C − r|. Then for sufficiently small
+δ > 0,
+x(1)
+± =
+�
+ae ∓ 2(A + B + C)
+d − 2
+δ, A ± Aδ, B ± Bδ, C ± Cδ, r ∓ d(A + B + C)
+d − 2
+δ, s ± sδ
+�
+(D.9)
+are elements of P, and x = (x(1)
++ + x(1)
+− )/2. Therefore, x /∈ Ext(P).
+[Case 2] A + B + C + r > ae = |C − r| > 0. Here we consider only the
+case C > r since the other case C < r can be argued similarly. Then for
+sufficiently small δ > 0,
+x(2)
+± = (ae ± (C − k)δ, A ± Aδ, B ± Bδ, C ± Cδ, r ± kδ, s ± sδ)
+(D.10)
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+35
+are elements of P, where k ∈ R satisfies
+d(d − 2)(C − k) + d(A + B + C) − (d − 2)k = 0
+(D.11)
+so that the condition (4) of (D.3) holds for x(2)
+± . Since x = (x(2)
++ + x(2)
+− )/2,
+it is not extremal.
+[Case 3] A + B + C + r ≥ ae = |C − r| = 0, so C = r. We claim that
+x = (0, A, B, C, C, s) ∈ Ext(P) corresponding to Type II′. Suppose that x
+is a convex combination of x(3)
+± = (a±, A±, B±, C±, r±, s±) ∈ P. Then the
+condition ae = 0 and a± ≥ 0 imply a± = 0, which again forces |C±−r±| =
+0. Therefore, Lemma D.1 implies that x(3)
+± = (0, A±, B±, C±, C±, s±) =
+λ±x for some λ± ≥ 0. Now the TP condition (D.3) (4) implies λ± = 1, so
+x = x(3)
+± .
+[Case 4] A + B + C + r = ae = C − r ≥ 0. Then r = − A+B
+2
+and
+x = ( A+B+2C
+2
+, A, B, C, − A+B
+2 , s). Here we claim that x ∈ Ext(P) which
+corresponds to Type III′ (note that A + B ≥ 2
+√
+AB = 2
+√
+C2 + s2 ≥ 2|C|,
+so r = − A+B
+2
+conversely implies C ≥ r). If x is a convex combination
+of x(4)
+±
+= (a±, A±, B±, C±, r±, s±) ∈ P, then the condition (D.3) (3) for
+x(4)
+± implies A± + B± + C± + r± = a± = |C± − r±|. We may assume
+A+ ≥ A ≥ A− without loss of generality, so Lemma D.1 implies
+x(4)
++ =
+�A + B + 2C
+2
++ δ′, A + Aδ, B + Bδ, C + Cδ, −A + B
+2
++ δ′′, s + sδ
+�
+(D.12)
+for some δ ≥ 0, δ′, δ′′ ∈ R, and δ′ = (A + B + C)δ + δ′′ from A+ + B+ +
+C+ + r+ = a+. Now for the case a+ = r+ − C+ ≥ 0, we have
+0 ≤ A + B + 2C = −(A + B + 2C)δ ≤ 0.
+(D.13)
+However, this says A+B = −2C and x = (0, A, B, C, C, s), which can be
+absorbed into Case 3. For the case a+ = C+ − r+, we have δ′′ = − A+B
+2 δ
+and δ′ = A+B+2C
+2
+δ. However, then the TP condition (D.3) (4) implies
+�
+d(d − 2)A + B + 2C
+2
++ d(A + B + C) + (d − 2)A + B
+2
+�
+δ = 0,
+(D.14)
+which is possible only if δ = 0. Therefore, x = x(4)
++ = x(4)
+− .
+[Case 5] A+B +C +r = ae = r−C ≥ 0. Then A+B = −2C, and the
+previous inequality A + B ≥ 2
+√
+AB ≥ 2|C| implies A = B = −C ≥ 0
+and s = 0. Thus, x = (r − C, −C, −C, C, r, 0) with C ≤ 0 and r ≥ C.
+Then our problem is divided into the following three subcases.
+
+36
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+• If C < 0 and r > C, then x /∈ Ext(P) since x = (x(5)
++ + x(5)
+− )/2,
+where
+x(5)
+± =
+�
+r − C ∓
+2
+d − 2δ, −C ± δ, −C ± δ, C ∓ δ, r ∓
+d
+d − 2δ, 0
+�
+∈ P
+(D.15)
+for sufficiently small δ > 0.
+• If r = C, then x = (0, −C, −C, C, C, 0) is extremal since it can be
+absorbed into Case 3.
+• If C = 0, then x = r(1, 0, 0, 0, 1, 0) is indeed extremal (corre-
+sponding to Type I′) since the point (A, B, C, s) = (0, 0, 0, 0) is an
+extreme point of S in Lemma D.1 and since r is uniquely deter-
+mined by the TP condition (D.3) (4).
+□
+Now we shall prove Lemma 5.10 using similar arguments. Let Q0 be the
+set of all tuples (ae, a(12), a(13), a(23), r, s) satisfying (5.25) and (D.1), and
+then consider a linear isomorphism
+β : (ae, a(12), a(13), a(23), r, s) �→ (A, B, C, p, q, s)
+(D.16)
+of R6, where
+�
+A = �
+σ∈S3 aσ, B =
+ae
+d−1 + a(23), C = a(23) + r,
+p = ae + a(12), q = ae + a(13).
+Then
+Q = β(Q0) becomes the set of all tuples (A, B, C, p, q, s) ∈ R6 satisfying
+�
+�
+�
+�
+�
+�
+�
+(1)
+A, B, p, q ≥ 0,
+(2)
+AB ≥ C2 + s2,
+(3)
+A + B − 2C ≤ p + q,
+(4)
+(−d2 + d + 1)A − (d − 1)2B + 2d(d − 1)C + (d2 − 1)(p + q) = 1.
+(D.17)
+Proof of Lemma 5.10. It is sufficient to show that the extreme points y =
+(A, B, C, p, q, s) of Q are classified into the following four types up to nor-
+malizing constants: for A, B ≥ 0 and AB ≥ C2,
+Type I′
+(0, 0, 0, 1, 0, 0),
+Type II′
+(0, 0, 0, 0, 1, 0),
+Type III′
+(A, B, C, A + B − 2C, 0, ±
+√
+AB − C2),
+Type IV′
+(A, B, C, 0, A + B − 2C, ±
+√
+AB − C2).
+As in the proof of Lemma 5.4, we may assume AB = C2+s2. Furthermore,
+we may assume p = 0 or q = 0 since y is a convex combination of y(0)
+± ∈ Q,
+where y(0)
++ = (A, B, C, p + q, 0, s) and y(0)
+− = (A, B, C, 0, p + q, s). Let us
+first assume q = 0, and divide the condition (3) of (D.17) into the following
+three cases.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+37
+[Case 1] (A, B) ̸= (0, 0) and A + B − 2C < p. Then for sufficiently
+small δ > 0,
+y(1)
+± = (A ± Aδ, B ± Bδ, C ± Cδ, p ± δ′, 0, s ± sδ) ∈ Q
+(D.18)
+where δ′ ∈ R satisfies
+�
+(−d2 + d + 1)A − (d − 1)2B + 2d(d − 1)C
+�
+δ+(d2−1)δ′ = 0, (D.19)
+so that the condition (4) of (D.17) holds for y(1)
+± . Since y = (y(1)
++ + y(1)
+− )/2
+and y(1)
++ ̸= y(1)
+− , we have y /∈ Ext(Q).
+[Case 2] A = B = 0 (hence C = s = 0). Then y = p(0, 0, 0, 1, 0, 0) is
+extremal in Q (corresponding to Type I′) since (A, B, C, s) = (0, 0, 0, 0) is
+an extreme point of S in Lemma D.1 and since p is uniquely determined by
+(D.17) (4).
+[Case 3] A+B −2C = p. In this case, we claim that y = (A, B, C, A+
+B − 2C, 0, s) ∈ Ext(Q), which corresponds to Type III′. Indeed, if y is
+a convex combination of y(2)
+±
+= (A±, B±, C±, p±, q±, s±) ∈ Q, then the
+conditions q = 0 and q± ≥ 0 imply q± = 0. Moreover, the conditions
+A + B − 2C = p and A± + B± − 2C± ≤ p± imply A± + B± − 2C± = p±.
+Now applying Lemma D.1, we can write
+y(2)
++ = (A(1 + δ), B(1 + δ), C(1 + δ), (A + B − 2C)(1 + δ), 0, s(1 + δ))
+(D.20)
+for some δ ∈ R. On the other hand, the TP condition (D.17) (4) for y(2)
++
+gives
+(dA + 2(d − 1)B − 2(d − 1)C) δ = 0.
+(D.21)
+However,
+dA + 2(d − 1)B = A + (d − 1)B + (d − 1)(A + B) ≥ 2(d − 1)C (D.22)
+since A + B ≥ 2C, and the equality above holds only if A = B = C =
+p = s = 0 which is impossible. Therefore, (D.21) holds only if δ = 0, and
+hence we have y = y(2)
++ = y(2)
+− .
+Finally, we can proceed analogously when p = 0 and get the tuples of
+Type II′ and Type IV′ as extreme points of Q.
+□
+APPENDIX E. PROOF OF THEOREM 5.6 WHEN d = 2
+When d = 2, we have an additional relation
+Ve − V(12) − V(13) − V(23) + V(123) + V(132) = 0.
+(E.1)
+Therefore, {Vσ}σ∈S3 is no longer linearly independent, and both the spaces
+Inv(U ⊗3) = span {Vσ : σ ∈ S3} and Inv(U ⊗U ⊗U) = span {Tσ : σ ∈ S3}
+are 5-dimensional. In particular, we have Inv(O⊗3
++ ) = Inv(U ⊗ U ⊗ U) in
+this case.
+
+38
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+We can write a general element in Cov(U, UU) as M = �
+σ∈S3\{e} aσMσ.
+Then M is positive if and only if
+�
+�
+�
+a(12), a(13), a(23) ≥ 0 and a(132) = a(123),
+a(12) + a(13) + a(23) + a(123) + a(132) ≥ 0,
+(a(12) + a(13) + a(23) + a(123) + a(132))a(23) ≥ |a(23) + a(123)|2,
+(E.2)
+by following the same proof in Lemma 5.8. Now let us write (r, s) =
+(Re(a(123)), Im(a(123))) for convenience and consider a linear isomorphism
+˜β : (a(12), a(13), a(23), r, s) �→ (a(12), A, B, C, s),
+(E.3)
+of R5, where A = a(12) + a(13) + a(23) + 2r, B = a(23), and C = a(23) + r.
+Then the set �Q =
+�
+˜β(a(12), a(13), a(23), r, s) : M ∈ CovPosTP(U, UU)
+�
+is
+equal to the set of tuples (a(12), A, B, C, s) ∈ R5 satisfying
+�
+�
+�
+�
+�
+�
+�
+(1)
+A, B ≥ 0,
+(2)
+AB ≥ C2 + s2,
+(3)
+0 ≤ a(12) ≤ A + B − 2C,
+(4)
+A + B − C = 1
+2.
+(E.4)
+In order to find the extreme points y = (a(12), A, B, C, s) of �Q, note that
+we still have AB = C2 + s2 as in the proof of Lemma 5.10. Moreover,
+we have a(12) = 0 or A + B − 2C since y is a convex combination of
+y+ = (A + B − 2C, A, B, C, s) and y− = (0, A, B, C, s). Therefore, we
+can list all possible extreme points of �Q in the following two types:
+Type I′
+(A + B − 2C, A, B, C, ±
+√
+AB − C2),
+Type II′
+(0, A, B, C, ±
+√
+AB − C2),
+for A, B ≥ 0, AB ≥ C2, and A + B − C = 1
+2. Moreover, any extreme
+point of Type I′ corresponds to a tuple
+(a(12), a(13), a(23), r, s) = (A+B −2C, 0, B, C −B, ±
+√
+AB − C2), (E.5)
+so the associated linear map M = �
+σ∈S3\{e} aσMσ is CP by Lemma 5.9
+(note that Lemma 5.9 (1) still gives a sufficient condition for M to be CP
+when d = 2). Similarly, any extreme point of Type II′ corresponds a CCP
+map. In other words, every element in Ext(CovPosTP(U, UU)) is CP or
+CCP, thus POS=DEC holds in CovPos1(UU, U). This completes the proof
+of Theorem 5.6 by Theorem 3.9.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+39
+APPENDIX F. QUANTUM ORTHOGONAL SYMMETRY
+Within the framework of compact quantum groups, the orthogonal group
+Od is understood as the space C(Od) of continuous functions on Od en-
+dowed with the co-multiplication ∆ : C(Od) → C(Od × Od) given by
+(∆f)(x, y) = f(xy)
+(F.1)
+for all x, y ∈ Od and f ∈ C(Od). Moreover, there exists a family of
+continuous functions (πij)1≤i,j≤d generating C(Od) and
+∆(πij) =
+d
+�
+k=1
+πik ⊗ πkj ∈ C(Od) ⊗min C(Od) ∼= C(Od × Od)
+(F.2)
+for all 1 ≤ i, j ≤ d, where ⊗min means the minimal tensor product of
+C∗-algebras.
+The free orthogonal quantum group O+
+d is a liberation of Od in the sense
+that the space C(O+
+d ) of ‘non-commutative’ continuous functions on O+
+d
+is the universal unital C∗-algebra generated by d2 self-adjoint operators uij
+satisfying that u =
+d
+�
+i,j=1
+eij ⊗ uij is a unitary, i.e. u∗u = uu∗ = Idd ⊗ 1
+in Md(C) ⊗ C(O+
+d ). The quantum group structure is encoded in the unital
+∗-homomorphism ∆ : C(O+
+d ) → C(O+
+d ) ⊗min C(O+
+d ) determined by
+∆(uij) =
+d
+�
+k=1
+uik ⊗ ukj.
+Then u =
+d
+�
+i,j=1
+eij ⊗ uij is the standard unitary representation of O+
+d satis-
+fying uc =
+d
+�
+i,j=1
+eij ⊗ u∗
+ij = u in the sense of [Wor87, Ban96]. The 3-fold
+tensor representation of u is defined by
+u ⊤ u ⊤ u =
+d
+�
+i1,j1,i2,j2,i3,j3=1
+ei1j1 ⊗ ei2j2 ⊗ ei3j3 ⊗ ui1j1ui2j2ui3j3.
+(F.3)
+Then the space Inv(O⊗3
++ ) in Section 5.2 is understood as the space Inv(u ⊤ u ⊤ u)
+of operators X ∈ Md(C)⊗3 satisfying
+(u ⊤ u ⊤ u) · (X ⊗ 1) = (X ⊗ 1) · (u ⊤ u ⊤ u)
+(F.4)
+in view of [LY22]. To sketch a proof of this fact, we can observe that the
+5 operators Tσ (σ ∈ S3\ {(13)}) in (5.20) are linearly independent, and the
+
+40
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+operators Tσ satisfy (F.4) using the identity
+(u ⊤ u)(|Ωd⟩ ⊗ 1) = |Ωd⟩ ⊗ 1.
+(F.5)
+Thus, Inv(O⊗3
++ ) ⊆ Inv(u ⊤ u ⊤ u). Moreover, the space Inv(u ⊤ u ⊤ u)
+should be of dimension five thanks to the representation theory of O+
+d (see
+Corollary 6.4.12 and Corollary 5.3.5 of [Tim08]). Hence, we have Inv(O⊗3
++ ) =
+Inv(u ⊤ u ⊤ u).
+REFERENCES
+[AN14]
+Muneerah Al Nuwairan. The extreme points of SU(2)-irreducibly covariant
+channels. Internat. J. Math., 25(6):1450048, 30, 2014.
+[Ban96]
+Teodor Banica. Th´eorie des repr´esentations du groupe quantique compact
+libre O(n). C. R. Acad. Sci. Paris S´er. I Math., 322(3):241–244, 1996.
+[BBC+93]
+Charles H. Bennett, Gilles Brassard, Claude Cr´epeau, Richard Jozsa, Asher
+Peres, and William K. Wootters. Teleporting an unknown quantum state
+via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett.,
+70(13):1895–1899, 1993.
+[Bel64]
+J. S. Bell. On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz.,
+1(3):195–200, 1964.
+[BO08]
+Nathanial P. Brown and Narutaka Ozawa. C∗-algebras and finite-
+dimensional approximations, volume 88 of Graduate Studies in Mathemat-
+ics. American Mathematical Society, Providence, RI, 2008.
+[BPM+97]
+Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald We-
+infurter, and Anton Zeilinger. Experimental quantum teleportation. Nature,
+390(6660):575–579, 1997.
+[Bra03]
+Gilles Brassard. Quantum communication complexity. volume 33, pages
+1593–1616. 2003. Special issue dedicated to David Mermin, Part II.
+[BSST99]
+Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal.
+Entanglement-assisted classical capacity of noisy quantum channels. Phys.
+Rev. Lett., 83:3081–3084, Oct 1999.
+[BvDHT99] Harry Buhrman, Wim van Dam, Peter Høyer, and Alain Tapp. Multiparty
+quantum communication complexity. Phys. Rev. A, 60:2737–2741, Oct 1999.
+[BW92]
+Charles H. Bennett and Stephen J. Wiesner. Communication via one- and
+two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett.,
+69(20):2881–2884, 1992.
+[C¯D13]
+Lin Chen and Dragomir ˇZ ¯Dokovi´c. Separability problem for multipartite
+states of rank at most 4. J. Phys. A, 46(27):275304, 24, 2013.
+[Cho82]
+M.D. Choi. Positive linear-maps. In in Operator Algebras and Applica-
+tions (Kingston, 1980), Proc.Sympos.Pure.Math., volume 38, pages 583–
+590. Amer.Math.Soc., 1982.
+[CKK+21]
+Euijung Chang, Jaeyoung Kim, Hyesun Kwak, Hun Hee Lee, and Sang-
+Gyun Youn. Irreducibly su(2)-covariant quantum channels of low rank. to
+appear in Rev. Math. Phys., arXiv preprint arXiv:2105.00709, 2021.
+[DFV08]
+B. Dierckx, M. Fannes, and C. Vandenplas. Additivity of the Renyi entropy
+of order 2 for positive-partial-transpose-inducing channels. Phys. Rev. A (3),
+77(6):060302, 4, 2008.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+41
+[DPR07]
+A. R. Usha Devi, R. Prabhu, and A. K. Rajagopal. Characterizing multipar-
+ticle entanglement in symmetric n-qubit states via negativity of covariance
+matrices. Phys. Rev. Lett., 98:060501, Feb 2007.
+[EK00]
+Myoung-Hoe Eom and Seung-Hyeok Kye. Duality for positive linear maps
+in matrix algebras. Math. Scand., 86(1):130–142, 2000.
+[Eke91]
+Artur K. Ekert. Quantum cryptography based on Bell’s theorem. Phys. Rev.
+Lett., 67(6):661–663, 1991.
+[EW01]
+T. Eggeling and R. F. Werner. Separability properties of tripartite states with
+U � U � U symmetry. Phys. Rev. A, 63:042111, Mar 2001.
+[G¨11]
+Otfried G¨uhne. Entanglement criteria and full separability of multi-qubit
+quantum states. Phys. Lett. A, 375(3):406–410, 2011.
+[GBW21]
+Martina Gschwendtner, Andreas Bluhm, and Andreas Winter. Programma-
+bility of covariant quantum channels. Quant., 5(6):488, 2021.
+[Gha10]
+Sevag Gharibian. Strong NP-hardness of the quantum separability problem.
+Quantum Inf. Comput., 10(3-4):343–360, 2010.
+[Gur03]
+Leonid Gurvits. Classical deterministic complexity of Edmond’s problem
+and quantum entanglement. In Proceedings of the Thirty-Fifth Annual ACM
+Symposium on Theory of Computing, pages 10–19. ACM, New York, 2003.
+[Has18]
+Anna-Lena K. Hashagen. Symmetry Methods in Quantum Information The-
+ory. 2018. Thesis (Ph.D.)–Technische Universit¨at M¨unchen Lehrstuhl f¨ur
+mathematische Physik.
+[HH99]
+Michał Horodecki and Paweł Horodecki. Reduction criterion of separability
+and limits for a class of distillation protocols. Phys. Rev. A, 59:4206–4216,
+Jun 1999.
+[HHH96]
+MichałHorodecki, PawełHorodecki, and Ryszard Horodecki. Separability of
+mixed states: necessary and sufficient conditions. Phys. Lett. A, 223(1-2):1–
+8, 1996.
+[HHH99]
+Paweł Horodecki, Michał Horodecki, and Ryszard Horodecki. Bound entan-
+glement can be activated. Phys. Rev. Lett., 82:1056–1059, Feb 1999.
+[HHHO05]
+Karol Horodecki, Michał Horodecki, Paweł Horodecki, and Jonathan Op-
+penheim. Secure key from bound entanglement. Phys. Rev. Lett., 94:160502,
+Apr 2005.
+[HHHO09]
+Karol Horodecki, MichałHorodecki, PawełHorodecki, and Jonathan Oppen-
+heim. General paradigm for distilling classical key from quantum states.
+IEEE Trans. Inform. Theory, 55(4):1898–1929, 2009.
+[HK16a]
+Kil-Chan Ha and Seung-Hyeok Kye. Construction of three-qubit gen-
+uine entanglement with bipartite positive partial transposes. Phys. Rev. A,
+93:032315, Mar 2016.
+[HK16b]
+Kyung Hoon Han and Seung-Hyeok Kye. Construction of multi-qubit opti-
+mal genuine entanglement witnesses. J. Phys. A, 49(17):17503, 16, 2016.
+[HLVC00]
+Paweł Horodecki, Maciej Lewenstein, Guifr´e Vidal, and Ignacio Cirac. Op-
+erational criterion and constructive checks for the separability of low-rank
+density matrices. Phys. Rev. A, 62:032310, Aug 2000.
+[HPHH08]
+Karol Horodecki, Łukasz Pankowski, Michał Horodecki, and Paweł
+Horodecki. Low-dimensional bound entanglement with one-way distillable
+cryptographic key. IEEE Trans. Inform. Theory, 54(6):2621–2625, 2008.
+
+42
+S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN
+[JSW+00]
+Thomas Jennewein, Christoph Simon, Gregor Weihs, Harald Weinfurter, and
+Anton Zeilinger. Quantum cryptography with entangled photons. Phys. Rev.
+Lett., 84:4729–4732, May 2000.
+[Kay11]
+Alastair Kay. Optimal detection of entanglement in greenberger-horne-
+zeilinger states. Phys. Rev. A, 83:020303, Feb 2011.
+[KCKL00]
+B. Kraus, J. I. Cirac, S. Karnas, and M. Lewenstein. Separability in 2 × N
+composite quantum systems. Phys. Rev. A (3), 61(6):062302, 10, 2000.
+[KCL05]
+J. K. Korbicz, J. I. Cirac, and M. Lewenstein. Spin squeezing inequalities
+and entanglement of n qubit states. Phys. Rev. Lett., 95:120502, Sep 2005.
+[KMS20]
+Piotr Kopszak, Marek Mozrzymas, and MichałStudzi´nski. Positive maps
+from irreducibly covariant operators. J. Phys. A, 53(39):395306, 33, 2020.
+[Kye23]
+Seung-Hyeok Kye. Compositions and tensor products of linear maps be-
+tween matrix algebras. Linear Algebra Appl., 658:283–309, 2023.
+[Las10]
+Jean Bernard Lasserre. Moments, positive polynomials and their applica-
+tions, volume 1 of Imperial College Press Optimization Series. Imperial Col-
+lege Press, London, 2010.
+[LY22]
+Hun Hee Lee and Sang-Gyun Youn. Quantum channels with quantum group
+symmetry. Comm. Math. Phys., 389(3):1303–1329, 2022.
+[Mas06]
+Llu´ıs Masanes. All bipartite entangled states are useful for information pro-
+cessing. Phys. Rev. Lett., 96:150501, Apr 2006.
+[Mic07]
+S. Michalakis. Multiplicativity of the maximal output 2-norm for depolarized
+Werner-Holevo channels. J. Math. Phys., 48(12):122102, 5, 2007.
+[MRS15]
+Marek Mozrzymas,
+Adam Rutkowski,
+and MichałStudzi´nski. Using
+non-positive maps to characterize entanglement witnesses. J. Phys. A,
+48(39):395302, 11, 2015.
+[MWKZ96] Klaus Mattle, Harald Weinfurter, Paul G. Kwiat, and Anton Zeilinger. Dense
+coding in experimental quantum communication. Phys. Rev. Lett., 76:4656–
+4659, Jun 1996.
+[NPW+00]
+D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat.
+Entangled state quantum cryptography: Eavesdropping on the ekert protocol.
+Phys. Rev. Lett., 84:4733–4736, May 2000.
+[NS21]
+Ion Nechita and Satvik Singh. A graphical calculus for integration over ran-
+dom diagonal unitary matrices. Linear Algebra and its Applications, 613:46–
+86, 2021.
+[NZ16]
+Jiawang Nie and Xinzhen Zhang. Positive maps and separable matrices.
+SIAM J. Optim., 26(2):1236–1256, 2016.
+[Pau02]
+Vern Paulsen. Completely bounded maps and operator algebras, volume 78
+of Cambridge Studies in Advanced Mathematics. Cambridge University
+Press, Cambridge, 2002.
+[Per96]
+Asher Peres. Separability criterion for density matrices. Phys. Rev. Lett.,
+77(8):1413–1415, 1996.
+[SDN22]
+Satvik Singh, Nilanjana Datta, and Ion Nechita. Ergodic theory of diagonal
+orthogonal covariant quantum channels, 2022.
+[SN21]
+Satvik Singh and Ion Nechita. Diagonal unitary and orthogonal symmetries
+in quantum theory. Quantum, 5:519, August 2021.
+[SN22]
+Satvik Singh and Ion Nechita. The PPT2 Conjecture Holds for All Choi-Type
+Maps. Annales Henri Poincare, 23(9):3311–3329, 2022.
+
+ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES
+43
+[Sr82]
+Erling Stø rmer. Decomposable positive maps on C∗-algebras. Proc. Amer.
+Math. Soc., 86(3):402–404, 1982.
+[TBZG00]
+W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. Quantum cryptography using
+entangled photons in energy-time bell states. Phys. Rev. Lett., 84:4737–4740,
+May 2000.
+[TG09]
+G´eza T´oth and Otfried G¨uhne. Entanglement and permutational symmetry.
+Phys. Rev. Lett., 102:170503, May 2009.
+[Tim08]
+Thomas Timmermann. An Invitation to Quantum Groups and Duality. Euro-
+pean Mathematical Society, 2008.
+[UDUPR07] A. R. Usha Devi, M. S. Uma, R. Prabhu, and A. K. Rajagopal. Constraints on
+the uncertainties of entangled symmetric qubits. Phys. Lett. A, 364(3-4):203–
+207, 2007.
+[VW02]
+Karl Gerd H. Vollbrecht and Michael M. Wolf. Activating distillation with
+an infinitesimal amount of bound entanglement. Phys. Rev. Lett., 88:247901,
+May 2002.
+[Wan95]
+Shuzhou Wang. Free products of compact quantum groups. Comm. Math.
+Phys., 167(3):671–692, 1995.
+[Wat18]
+John Watrous. The Theory of Quantum Information. Cambridge University
+Press, 2018.
+[Wer89]
+Reinhard F. Werner. Quantum states with einstein-podolsky-rosen correla-
+tions admitting a hidden-variable model. Phys. Rev. A, 40:4277–4281, Oct
+1989.
+[Wor76a]
+S. L. Woronowicz. Nonextendible positive maps. Comm. Math. Phys.,
+51(3):243–282, 1976.
+[Wor76b]
+S. L. Woronowicz. Positive maps of low dimensional matrix algebras. Rep.
+Math. Phys., 10(2):165–183, 1976.
+[Wor87]
+S. L. Woronowicz. Compact matrix pseudogroups. Comm. Math. Phys.,
+111(4):613–665, 1987.
+SANG-JUN PARK, DEPARTMENT OF MATHEMATICAL SCIENCES, SEOUL NATIONAL
+UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF KOREA
+Email address: [email protected]
+YEONG-GWANG JUNG, DEPARTMENT OF MATHEMATICS EDUCATION, SEOUL NA-
+TIONAL UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF
+KOREA
+Email address: [email protected]
+JEONGEUN PARK, DEPARTMENT OF MATHEMATICS EDUCATION, SEOUL NATIONAL
+UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF KOREA
+Email address: [email protected]
+SANG-GYUN YOUN, DEPARTMENT OF MATHEMATICS EDUCATION, SEOUL NA-
+TIONAL UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF
+KOREA
+Email address: [email protected]
+

V9E5T4oBgHgl3EQfBw5N/content/tmp_files/2301.05389v1.pdf.txt

@@ -0,0 +1,1057 @@
+Thermal-dephasing-tolerant generation of Schr¨odinger cat states with Rydberg
+dressed blockade
+Ri-Hua Zheng,1 S.-L. Su,2, ∗ Jie Song,3 Weibin Li,4, † and Yan Xia1, ‡
+1Fujian Key Laboratory of Quantum Information and Quantum Optics,
+College of Physics and Information Engineering,
+Fuzhou University, Fuzhou, Fujian 350108, China
+2School of Physics, Zhengzhou University, Zhengzhou 450001, China
+3Department of Physics, Harbin Institute of Technology, Harbin 150001, China
+4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
+Multipartite entangled states involving non-locality are one of the most fascinating characteristics
+of quantum mechanics. In this work, we propose a generation of Schr¨odinger cat states in Rydberg
+atom arrays against the thermal dephasing.
+Unlike the previous work for producing large-scale
+entanglement [A. Omran et al, Science 365, 570 (2019)], we encode logical 1 on dressed states
+rather than Rydberg states. Such treatment can increase the lifetime of multipartite entanglement
+coherence to around 3 times compared to the previous work [A. Omran et al, Science 365, 570
+(2019)], at the same system size, and therefore induce solid fidelities of Schr¨odinger cat states
+generation.
+The current work theoretically verifies the advantages of Rydberg dressed state in
+many-body quantum entanglement, which is meaningful for large-scale quantum computation and
+many-body Rydberg quantum simulation.
+Introduction.—Multipartite entanglement lies at the
+heart of quantum information processing.
+Among di-
+verse types of entangled states, the Greenberger-Horne-
+Zeilinger (GHZ) states [1] (or called two-component
+atomic Schr¨odinger cat states [2, 3]), have attracted a
+lot of research interest because of their characteristics
+of the maximum entanglement. Recently, two outstand-
+ing experimental works [4, 5] simultaneously report the
+generation of 20-qubit GHZ states with similar fidelity
+F ∼ 0.55 on the Rydberg atoms platform and the super-
+conducting qubits platform, respectively.
+One of the major obstacles to achieving higher fidelity
+in the work of Ref. [4], as well as other extensive quantum
+simulation and quantum computation based on Rydberg
+atoms [6–14], is thermal dephasing.
+Due to the finite
+temperature of neutral atoms (∼10 µK), the Doppler
+shift (∼ 2π × 43 kHz) occurs on each site of the Ry-
+dberg atom arrays [4, 15].
+Additionally, the thermal
+noise causes fluctuations of atomic distance (for exam-
+ple, δR/R ∼ 6% [6]), i.e., disturbing the desired elec-
+tric dipole-dipole interaction (EDDI) strength.
+Above
+two disturbances together constitute the thermal dephas-
+ing mechanisms, which have hindered the high-precision
+completion of many quantum works [4, 6, 16] in Rydberg
+atoms. Especially when multiple Rydberg atoms are ex-
+cited to Rydberg states, the thermal dephasing, quan-
+tified by the lifetime of coherence, T2, becomes shorter
+with the increase in the number of atoms N (specifically,
+T2 ∼0.6 µs for N = 20 [4]).
+The root cause of the thermal dephasing error is Ryd-
+berg state excitation. Thus, an intuitive idea is to dilute
+the percentage of Rydberg states Pr to reduce the ther-
+mal dephasing error. Rydberg dressed states are efficient
+to realize this goal while still retains some of the Ryd-
+berg interaction strength that enough to realize the de-
+sired quantum tasks [17–19]. Particularly, the Rydberg
+dressed state method has been utilized for the produc-
+tion of exotic quantum phases [18, 20, 21], spin squeez-
+ing [17, 22], quantum computation [23, 24], and entan-
+glement generation [25–27]. For Rydberg dressed states,
+the percentage of Rydberg states Pr can be appropriately
+adjusted by modulating the Rabi frequency and detuning
+of the dressed laser on each single atom. The lower value
+of Pr will bring less affection from the thermal dephas-
+ing, but weaker EDDI strength, which is an interesting
+trade-off.
+And one can flexibly adjust the strength of
+the Rydberg interaction and the percentage of Rydberg
+dressed states.
+In this work, we propose to generate Schr¨odinger cat
+states through Rydberg dressing induced blockade. We
+dress the Rydberg atoms probability Pr = 25%, leading
+to coherence lifetimes T2 ={11.0, 8.2, 7.2, 6.2, 5.7, 5.1}
+µs for {4, 6, 8, 10, 12, 14}-atom GHZ states, around
+3 times of T2 in the work [4], at the same system size.
+For larger scale systems, the dressed lifetime T2 decreases
+with the system size N as 1/
+√
+N, which is the same as
+the scaling rule in the work without Rydberg dressing [4].
+The fidelities are {97.2%, 95.1%, 91.1%, 84.6%} for {4,
+6, 8, 10}-atom GHZ states when considering the thermal
+dephasing. Based on the above fidelities data, we give
+a prediction fidelity of 80% for the 20-atom GHZ state.
+The balance between dressing low-percentage Rydberg
+states and shortening the evolution time in large-scale
+neutral atoms systems in the present work may provide
+a reference for quantum computation and quantum sim-
+ulation based on multiple dressed atoms. The Rydberg-
+dressing method for many-body entanglement generation
+is scalable and dephasing-tolerant, which will contribute
+to realizing near-term quantum simulation and quan-
+tum computation with Rydberg atom arrays, such as
+arXiv:2301.05389v1  [quant-ph]  13 Jan 2023
+
+2
+|������������⟩
+|1⟩
+|0⟩
+Δ
+Ω
+Ωmw
+������������mw
+������������
+(b)
+(a)
+(c)
+FIG. 1. (a) Envisioned experimental setup. All atoms fixed
+on the one-dimensional array are driven by Rydberg dressed
+lasers (780 and 480 nms) and Raman (microwave) drivings
+with the same detunings and Rabi frequencies. Two address-
+ing lasers (not shown) are added on the two edge atoms to
+differentiate microwave detunings from other atomic ones. (b)
+Energy levels and couplings of each atom. (c) Performance
+of the entanglement generation versus dressed percentage Pr.
+This performance index combines the impact of the lifetime,
+dressed interaction, and dressed interaction bandwidth (See
+the text for details).
+entanglement-enhanced sensing [28], quantum metrology
+[29], and quantum error correction [30] based on GHZ
+states.
+Model.—Consider a one-dimensional array with an
+even number N of neutral 87Rb atoms, trapped in the op-
+tical tweezers, as shown in Fig. 1. For all the atoms, we
+encode two ground states |0⟩ = |5S1/2, F = 1, mF = 1⟩
+and |1⟩ = |5S1/2, F = 2, mF = 2⟩ and one excited Ry-
+dberg state |r⟩ = |70S, J = 1/2, mJ = −1/2⟩ [4, 31].
+The coupling between |1⟩ and |r⟩ is driven by a two-
+photon transition with effective coupling strength Ω and
+detuning ∆. Additionally, the microwave-frequency fields
+with strength Ωmw(t) and detuning δn
+mw(t) are added
+for driving the rotations between the ground states |0⟩
+and |1⟩. We choose the nearest-neighbor EDDI energy,
+V/(2π) = 21 MHz, i.e., equal adjacent atomic intervals
+of 5.87 µm and C6 = 858 GHz µm6 · h. The Hamiltonian
+here is given by (ℏ = 1 and n, i, j ≤ N)
+H = HRDB + Hmw,
+(1a)
+HRDB =
+N
+�
+n=1
+Ω
+2 σn
+x,r1 + ∆σn
+rr +
+�
+i<j
+V
+|i − j|6 σi
+rrσj
+rr, (1b)
+Hmw =
+N
+�
+n=1
+[Ωmw(t)σn
+x,0d + [U1 − δn
+mw(t)]σn
+00,
+(1c)
+with σn
+αα
+=
+|α⟩n⟨α|,
+σn
+x,αβ
+=
+|α⟩n⟨β| + |β⟩n⟨α|
+(α = r, 0; β = 1, d), where |d⟩ is the dressed state,
+given by |d⟩ = −sign(∆) sin(θ/2)|1⟩ + cos(θ/2)|r⟩ with
+sin θ = Ω/
+√
+Ω2 + ∆2 and cos θ = −sign(∆)∆/
+√
+Ω2 + ∆2.
+Therefore, the percentage Pr of Rydberg states in the
+dressed states is cos2(θ/2).
+Note the driving between
+|0⟩ and |r⟩ is also a two-photon transition.
+For each
+pair of adjacent atoms, after abandoning decoupled anti-
+symmetric state (|1r⟩ − |r1⟩)/
+√
+2, the Hamiltonian can
+be reduced to
+H(2)
+RDB =
+�
+�
+�
+0
+Ω
+√
+2
+0
+Ω
+√
+2
+∆
+Ω
+√
+2
+0
+Ω
+√
+2 2∆ + V
+�
+�
+� ,
+(2)
+in basis {|11⟩, (|1r⟩ + |r1⟩)/
+√
+2, |rr⟩}.
+Equation 2 is
+insightful in that it allows us to understand the energy
+scales directly.
+The performance of the entanglement generation is
+quantified as per. = J · T2 · bwJ, shown in Fig.
+1(c)
+with free units, where J and bwJ are the dressed en-
+ergy and dressed energy bandwidth, respectively.
+The
+dressed energy, arising from the Stark shifts caused
+by the reciprocity among the EDDI and dressed lasers
+drivings, is given by J
+= |2U1 − U2|, with U1
+=
+(∆ − sign(∆)
+√
+Ω2 + ∆2)/2
+and
+U2
+=
+−sign(∆) ·
+min |eig(H(2)
+RDB)| representing single- and double-atom
+Stark shifts on the dressed states, respectively. When the
+Rydberg blockade effect is strong enough (V ≫ {Ω, ∆}),
+one can calculate that U2 = (∆−sign(∆)
+√
+2Ω2 + ∆2)/2,
+which is coincide with the results in [18, 19, 25]. For illu-
+mination, we plot the dressed energy J versus detuning
+∆ and EDDI strength V in Fig. 2.
+As exhibited in Fig. 2, the stick-shaped area around
+the green dashed line processes higher dressed energy
+J. This area demonstrates an interesting physical phe-
+nomenon, called Rydberg anti-blockade [32–37]. In prin-
+ciple, in all the regions of Fig. 2, the dressed energy J
+on the two-atom dressed states |dd⟩i(i+1) blockade the
+associated transitions from the ground state to |dd⟩i(i+1)
+(adjusting J ≫ Ωmw, dropping (t) hereafter). That is an-
+other phenomenon with rich physical connotations, called
+Rydberg dressed blockade (RDB) [17–19, 25–27], which
+acts like the Rydberg blockade, however with a differ-
+ence that the transitions to adjacent atoms in the dressed
+state |dd⟩i(i+1) rather than the two-atom Rydberg states
+|rr⟩i(i+1) are forbidden.
+To shorten the evolution time of the dynamics induced
+by Hmw, we prefer a larger dressed energy J for RDB.
+On the other hand, to reduce the thermal dephasing of
+atoms, the composition of Rydberg states Pr should be
+small (resisting atomic Doppler shifts). Additionally, the
+dressed energy J needs to be stable within a certain
+fluctuation range of V (resisting atomic position float-
+ing).
+Ergo, we choose (V, ∆)/(2π) = (21, −4.5) MHz
+with Pr = 25%.
+Up to now, the light shift (caused by the dressed
+
+Raman
+driving3
+-2
+-3
+4
+5
+ 
+－
+－
+ 
+(Z
+H艺
+飞N)
+<1 －6
+-7
+-85 
+J (21r MHz) 
+Chosen point 
+一一 飞一 一一一一一一一 女一一
+Percentage of 
+Rydberg<25% 
+5
+4
+3
+2
+1
+ 
+10 
+15 
+20 
+V (21r MHz) 
+25 
+FIG. 2. (Color online) Dressed energy J versus detuning ∆
+and EDDI strength V . The effective coupling strength Ω is
+fixed at 2π × 8 MHz. The area where the percentage of Ry-
+dberg states Pr is less than 25%, is marked below the red
+dotted line.
+The stick-shaped area represents the Rydberg
+anti-blockade with weak regimes (not satisfying ∆ ≫ Ω). The
+selected point (V, ∆)/(2π) = (21, −4.5) MHz is marked by a
+cross.
+laser) for each atom can be canceled by the U1|0⟩n⟨0|
+term of the Hamiltonian of microwave-frequency fields,
+Hmw.
+Then we use Hmw to flexibly manipulate the
+dynamics of multiple atoms in subspace S1 ={|0⟩, |d⟩}
+(equal to an N-qubit quantum system). Moreover, due
+to the RDB effect, the subspace S2 = {|dd⟩i(i+1)} are
+forbidden. The calculation space is further locked into
+S3 = ∁S1S2. Therefore we successfully reduce the cal-
+culation dimension of the N-qubit quantum system from
+2N to �N/2
+m=0 Cm
+N+1−m (see Supplemental Material [38] for
+details), analogous to the multiple-qubit work by Omran
+et al. [4]. Remarkably, such a reduction of the dimen-
+sion of the system subspace induces a non-local effect to
+generate entanglement [4, 25].
+Further, by appropriately adjusting the addressing
+drivings applied on edge atoms, we can make their mi-
+crowave detunings (δe
+mw) different from other atomic
+ones (δne
+mw), i.e., δ1
+mw = δN
+mw = δe
+mw ̸= δp
+mw = δne
+mw
+(p = 2, 3, ..., N − 1).
+When δe
+mw is closed to δne
+mw but
+differs certain values, the initial state |0000 · · · ⟩ will be
+driven to the GHZ state |GHZN⟩ = (|d0d0 · · · d0⟩ +
+|0d0d · · · 0d⟩)/
+√
+2 by adiabatically increasing detuning
+(−δn
+mw) from negative large values to positive large values
+(see Supplemental Material [38] for deviations). Such an
+adiabatic method has been proved in the work of Ref. [4],
+and they optimize their pulses through a remote dressed
+chopped-random basis algorithm [39, 40]. Here we utilize
+another effective optimal control method, GRAPE [41]
+(recently proved to be a precise algorithm in the experi-
+ment [42–45]), to shorten the evolution time of the above
+adiabatic process for the multipartite GHZ states gener-
+Number of atoms
+Time (µs)
+Ideal fidelity
+4
+2.1
+0.9956
+6
+4.2
+0.9956
+8
+4.8
+0.9873
+10
+12.5
+0.9799
+TABLE I. Ideal fidelities and preparation times optimized by
+the GRAPE. The ideal fidelity is calculated in the effective
+subspace S3 (no {|dd⟩i(i+1)} subspace).
+ation. The optimization results are exhibited in Table I.
+Note that the maximum absolute value of Ωmw does not
+exceed 2π ×0.14 MHz to ensure the stability of the RDB
+dynamics (J ≫ Ωmw). Detailed driving pulses for {4,
+6, 8, 10}-atom systems can be seen in the Supplemental
+material [38].
+Thermal dephasing.—We next consider a very critical
+experimental imperfection, the thermal dephasing noise,
+in the numerical simulation. The finite temperature of
+atoms induces their nonzero spread velocity and position
+uncertainty. Such that the thermal dephasing mechanism
+is determined by these two factors.
+(i) The position uncertainty can be modeled by the
+Gaussian distribution with the probability density func-
+tion being N(xn, σ2) = e−(x−xn)2/(2σ2)/(
+√
+2πσ), where
+xn the ideal position of the nth atom and σ2 the variance.
+The adjacent distance of the atoms is R = N(xi, σ2) −
+N(xi+1, σ2) = N(R0, 2σ2) with R0 = xi − xi+1 =
+5.87 µm. Specifically, the standard deviation σ of Gaus-
+sian distribution for an atom in a finite temperature ∼10
+µK is around 0.1 µm [4, 46]. As V = C6/R6, a spread
+in interaction strength can be calculated by the posi-
+tion distribution and further simulated with the original
+Hamiltonian in Eq. (1).
+(ii) The spread velocity causes fluctuating Doppler
+shifts.
+According to concrete reports [15, 47, 48], the
+Doppler shift leads to a random detuning δD on each
+atom-array site, viz., δD ∈ N(0, σ2
+D), simulated as an
+error term HD = δD|r⟩⟨r| added to the original Hamil-
+tonian in Eq. (1). The value of σD is given by σD =
+keff∆v with keff = |k|1⟩↔|e⟩ + k|e⟩↔|r⟩| the effective
+wave vector of two-photon transition |1⟩ ↔ |r⟩ and
+∆v =
+�
+kBT/m the one-dimensional root-mean-square
+(rms) velocity spread of atoms. Bringing in the values of
+Boltzmann constant kB, atomic temperature T = 10 µK
+[4], and atomic mass m = 87 × 1.66 × 10−27 kg, we can
+give the rms velocity spread ∆v =
+�
+kBT/m = 0.031
+m/s.
+On the other hand, one can drive |1⟩ ↔ |e���
+(|e⟩ ↔ |r⟩) by a 780 nm (480 nm) laser with |k|1⟩↔|e⟩| =
+2π/780 nm−1 (|k|e⟩↔|r⟩| = 2π/480 nm−1). These two
+lasers can be focused on the atom array by opposite
+directions [15] to minimize the Doppler shifts, result-
+ing in keff = 2π/480 nm−1 − 2π/780 nm−1 and further
+σD/(2π) = 24.77 kHz. Since the energy of ground states
+|0⟩ and |1⟩ is closed, the Doppler shifts are almost the
+
+Rydberg
+anti-blockade4
+4
+6
+8
+10
+12
+14
+16
+18
+20
+0
+0.2
+0.4
+0.6
+0.8
+1
+Simulated 
+data & Error
+fit line
+4
+6
+8
+10
+12
+FIG. 3. Fidelities of GHZ states in the systems at different
+scales. The colors of dots scale the corresponding preparation
+time for GHZ states. The green dashed line exhibits the fitting
+results [based on log(F − 1/2) ∝ −
+√
+N] induced from the
+fidelities for {4, 6, 8, 10}-atom systems.
+This line gives a
+prediction of the fidelities for larger N ≥ 12 and also more
+difficult to compute systems, marked by a gray area.
+same for |1⟩ ↔ |r⟩ and |0⟩ ↔ |r⟩. Therefore the error
+term HD = δD|r⟩⟨r| can well describe the Doppler shifts
+effect in the present atomic systems.
+Scaling of fidelities.—Considering the above two nega-
+tive factors, the simulation results for GHZ states prepa-
+ration in {4, 6, 8, 10}-atom systems are shown in Fig. 3.
+The full Hamiltonian in Eq. (1) is utilized to calculate
+the fidelities for {4, 6, 8, 10}-atom systems. The ther-
+mal dephasing (including the position uncertainty and
+the Doppler shifts) is considered during the numerical
+simulation by repeatedly calculating fidelities based on
+different values of R and δD (obeying the normal dis-
+tribution), which induces the error bars (standard devi-
+ation) in Fig. 3. For each simulated final density ma-
+trix ρf, we deduce fidelity F = ⟨GHZN|ρf|GHZN⟩ =
+1
+2(pAN + p ¯
+AN + cN + c∗
+N) [4], where pAN (p ¯
+AN ) is pop-
+ulation on |AN⟩ = |d0d0 · · · d0⟩ (| ¯AN⟩ = |0d0d · · · 0d⟩)
+and cN = ⟨ ¯AN|ρf|AN⟩ describes the coincidence of off-
+diagonal terms. The fidelity of the 4-atom GHZ state is
+higher than 97%. Even in the 10-atom system, the fi-
+delity is still close to 85%. Additionally, according to the
+prediction line in Fig. 3, the fidelity of 20-atom will be
+around 80%.
+Scaling of T2.—To further benchmark the dephasing
+effect of present dressed-atom systems, we assume the
+initial state is |GHZN⟩ and turn off all the driving lases
+for a specific time τ and calculate the corresponding den-
+sity matrix ρ′(τ). By counting 2|⟨ ¯AN|ρ′(τ)|AN⟩| as the
+coherence and defining the coherence lifetime T2 satisfy-
+ing 2|⟨ ¯AN|ρ′(T2)|AN⟩| = e−1, we show T2 versus system
+size N in Fig. 4. These T2 data are around 3 times of the
+observed data in the previous work [4] without dressing
+atoms, at the same system size.
+The above results come from the current cooling lim-
+4
+6
+8
+10
+12
+14
+16
+18
+20
+2
+4
+6
+8
+10
+12
+0
+5
+10
+0
+1
+Simulated 
+data & Error
+fit line
+FIG. 4. Coherence lifetime (T2) versus system size N. The
+circles show the T2 data inferred by Gaussian fittings of the co-
+herence damping for systems with different sizes and the blue
+dashed line gives a predicted trend (T2 ∝ 1/
+√
+N) for larger
+systems (N ≥ 16). As an example, the inset exhibits the co-
+herence damping of an 8-atom system with a green dashed
+line obeying Gaussian fitting. Similar to Fig. 3, the dots and
+error bars are mean values and stand deviations of coherence.
+ited temperature T = 10 µK of cold atoms [4, 49]. In the
+future, with the development of experimental techniques,
+the atoms may be cooled to lower temperatures. We here
+give an estimate of fidelity F versus atomic temperature
+T to show the prospect of our scheme, with the simula-
+tion results of {4, 6, 8, 10}-atom systems shown in Fig.
+5. The relationship between fidelity F and decoherence
+time T2 is log(F − 1/2) ∝ −1/T2. The coherence T2 re-
+sults from to thermal motion described by the Boltzmann
+distribution and therefore satisfies T2 ∝ 1/
+√
+T. The final
+relationship between fidelity F and atomic temperature
+T is log(F − 1/2) ∝ −
+√
+T, which coincides well with the
+simulation results in Fig.
+5, and also gives prediction
+that the fidelities of {4, 6, 8, 10}-atom GHZ states can
+reach around 95% when the atomic temperature is lower
+than 10 µK.
+It is worth noting that the Rydberg state percentage
+Pr of the dressed state is 25%. According to the lifetime
+of 70S Rydberg state of 146 µs [15, 50], the lifetime of
+the dressed state could extend to 573 µs, long enough
+to ignore the depopulation effect factor as the prepara-
+tion times of GHZ states around 10 µs. Additionally, we
+only consider the near-neighbor and next-neighbor inter-
+actions.
+Conclusion.—We
+have
+explored
+the
+thermal-
+dephasing-tolerant
+generation
+of
+GHZ
+states
+in
+dressed-atom
+systems.
+The
+thermal
+dephasing
+is
+suppressed here to 1/3 of the original performance
+in multi-body entangled systems [4], leading to solid
+fidelities of multi-body GHZ states.
+As the dephasing
+factor becomes an important obstacle to large-scale
+
+5
+0
+2
+4
+6
+8
+0.7
+0.8
+0.9
+1
+Fidelity
+4 atoms
+6 atoms
+8 atoms
+10 atoms
+FIG. 5. Fidelities of GHZ states versus atomic temperature.
+The circles, diamonds, triangles, and squares represent the
+fidelity of {4, 6, 8, 10}-atom GHZ states, respectively. The
+dash lines correspondingly show the fitting results based on
+log(F − 1/2) ∝ −
+√
+T for different scales systems.
+quantum computation and quantum simulation based
+on neutral atoms, the present work may be referable to
+building dephasing-tolerant quantum tasks in large-scale
+systems. Combining the Rydberg dressed state method
+with GPAPE optimal control techniques also provides
+an enlightening new perspective for robust handling of
+many-body Rydberg quantum simulations and quantum
+computation.
+This work was supported by the National Natural Sci-
+ence Foundation of China under Grants No. 11575045,
+No. 11874114, No. 11674060, No. 11805036, and the
+Natural Science Funds for Distinguished Young Scholar
+of Fujian Province under Grant 2020J06011, Project from
+Fuzhou University under Grant JG202001-2. S. L. S. ac-
+knowledges support from National Natural Science Foun-
+dation of China under Grant No. 12274376 and Major
+science and technology project of Henan Province under
+Grant No. 221100210400. W. L. acknowledges support
+from the EPSRC through Grant No. EP/W015641/1 and
+the British Council through an Industry Academia Col-
+laborative Grant (No. IND/CONT/G/22-23/26).
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+[1] D. M. Greenberger, M. A. Horne, A. Shimony, and
+A. Zeilinger, Bell’s theorem without inequalities, Am. J
+Phys. 58, 1131 (1990).
+[2] G. S. Agarwal, R. R. Puri, and R. P. Singh, Atomic
+Schr¨odinger cat states, Phys. Rev. A 56, 2249 (1997).
+[3] C. C. Gerry and R. Grobe, Generation and properties
+of collective atomic Schr¨odinger-cat states, Phys. Rev. A
+56, 2390 (1997).
+[4] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T.
+Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler,
+S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Mon-
+tangero, T. Calarco, M. Endres, M. Greiner, V. Vuleti´c,
+and M. D. Lukin, Generation and manipulation of
+Schr¨odinger cat states in Rydberg atom arrays, Science
+365, 570 (2019).
+[5] C. Song, K. Xu, H. Li, Y.-R. Zhang, X. Zhang, W. Liu,
+Q. Guo, Z. Wang, W. Ren, J. Hao, H. Feng, H. Fan,
+D. Zheng, D.-W. Wang, H. Wang, and S.-Y. Zhu, Gen-
+eration of multicomponent atomic Schr¨odinger cat states
+of up to 20 qubits, Science 365, 574 (2019).
+[6] S. Ravets, H. Labuhn, D. Barredo, L. B´eguin, T. Lahaye,
+and A. Browaeys, Coherent dipole–dipole coupling be-
+tween two single Rydberg atoms at an electrically-tuned
+F¨orster resonance, Nat. Phys. 10, 914 (2014).
+[7] I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale,
+A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and
+M. Endres, High-fidelity entanglement and detection of
+alkaline-earth Rydberg atoms, Nat. Phys. 16, 857 (2020).
+[8] Y.-H. Kang, Y.-H. Chen, Z.-C. Shi, B.-H. Huang, J. Song,
+and Y. Xia, Nonadiabatic holonomic quantum computa-
+tion using Rydberg blockade, Phys. Rev. A 97, 042336
+(2018).
+[9] Y. Wang, C.-S. Hu, Z.-C. Shi, B.-H. Huang, J. Song,
+and Y. Xia, Accelerated and noise-resistant protocol of
+dissipation-based Knill–Laflamme–Milburn state gener-
+ation with Lyapunov control, Ann. Phys. (Berlin) 531,
+1900006 (2019).
+[10] S.-L. Su, F.-Q. Guo, L. Tian, X.-Y. Zhu, L.-L. Yan, E.-
+J. Liang, and M. Feng, Nondestructive Rydberg parity
+meter and its applications, Phys. Rev. A 101, 012347
+(2020).
+[11] R.-H. Zheng, Y.-H. Kang, D. Ran, Z.-C. Shi, and Y. Xia,
+Deterministic interconversions between the Greenberger-
+Horne-Zeilinger states and the W states by invariant-
+based pulse design, Phys. Rev. A 101, 012345 (2020).
+[12] Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, Her-
+alded atomic nonadiabatic holonomic quantum compu-
+tation with Rydberg blockade, Phys. Rev. A 102, 022617
+(2020).
+[13] S.-L. Su and W. Li, Dipole-dipole-interaction–driven an-
+tiblockade of two Rydberg atoms, Phys. Rev. A 104,
+033716 (2021).
+[14] S. Liu, J.-H. Shen, R.-H. Zheng, Y.-H. Kang, Z.-C. Shi,
+J. Song, and Y. Xia, Optimized nonadiabatic holonomic
+quantum computation based on F¨orster resonance in Ry-
+dberg atoms, Frontiers of Physics 17, 21502 (2022).
+[15] H.
+Levine,
+A.
+Keesling,
+A.
+Omran,
+H.
+Bernien,
+S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner,
+V. Vuleti´c, and M. D. Lukin, High-fidelity control and
+entanglement of Rydberg-atom qubits, Phys. Rev. Lett.
+121, 123603 (2018).
+[16] I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale,
+A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and
+M. Endres, High-fidelity entanglement and detection of
+alkaline-earth Rydberg atoms, Nat. Phys. 16, 857 (2020).
+[17] I. Bouchoule and K. Mølmer, Spin squeezing of atoms by
+the dipole interaction in virtually excited Rydberg states,
+Phys. Rev. A 65, 041803(R) (2002).
+[18] J. E. Johnson and S. L. Rolston, Interactions between
+Rydberg-dressed atoms, Phys. Rev. A 82, 033412 (2010).
+[19] J. B. Balewski, A. T. Krupp, A. Gaj, S. Hofferberth,
+R. L¨ow, and T. Pfau, Rydberg dressing: understanding
+of collective many-body effects and implications for ex-
+periments, New J. Phys. 16, 063012 (2014).
+[20] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and
+
+6
+P. Zoller, Strongly correlated gases of Rydberg-dressed
+atoms:
+Quantum and classical dynamics, Phys. Rev.
+Lett. 104, 223002 (2010).
+[21] S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground
+state of the S = 1/2 kagome Heisenberg antiferromagnet,
+Science 332, 1173 (2011).
+[22] L. I. R. Gil, R. Mukherjee, E. M. Bridge, M. P. A. Jones,
+and T. Pohl, Spin Squeezing in a Rydberg Lattice Clock,
+Phys. Rev. Lett. 112, 103601 (2014).
+[23] T. Keating, K. Goyal, Y.-Y. Jau, G. W. Biedermann,
+A. J. Landahl, and I. H. Deutsch, Adiabatic quantum
+computation with Rydberg-dressed atoms, Phys. Rev. A
+87, 052314 (2013).
+[24] T. Keating, R. L. Cook, A. M. Hankin, Y.-Y. Jau, G. W.
+Biedermann, and I. H. Deutsch, Robust quantum logic
+in neutral atoms via adiabatic Rydberg dressing, Phys.
+Rev. A 91, 012337 (2015).
+[25] Y.-Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch,
+and G. W. Biedermann, Entangling atomic spins with
+a Rydberg-dressed spin-flip blockade, Nat. Phys. 12, 71
+(2015).
+[26] X. Q. Shao, D. X. Li, Y. Q. Ji, J. H. Wu, and X. X. Yi,
+Ground-state blockade of Rydberg atoms and applica-
+tion in entanglement generation, Phys. Rev. A 96, 012328
+(2017).
+[27] D.-X. Li, T.-Y. Zheng, and X.-Q. Shao, Adiabatic prepa-
+ration of multipartite GHZ states via Rydberg ground-
+state blockade, Opt. Express 27, 20874 (2019).
+[28] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum
+sensing, Rev. Mod. Phys. 89, 035002 (2017).
+[29] W. D¨ur, M. Skotiniotis, F. Fr¨owis, and B. Kraus, Im-
+proved quantum metrology using quantum error correc-
+tion, Phys. Rev. Lett. 112, 080801 (2014).
+[30] L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu,
+D. Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina,
+and C. Monroe, Fault-tolerant operation of a quantum
+error-correction code, arXiv:2009.11482v2 (2020).
+[31] T. Wilk, A. Ga¨etan, C. Evellin, J. Wolters, Y. Mirosh-
+nychenko, P. Grangier, and A. Browaeys, Entanglement
+of two individual neutral atoms using Rydberg blockade,
+Phys. Rev. Lett. 104, 010502 (2010).
+[32] C. Ates, T. Pohl, T. Pattard, and J. M. Rost, Antiblock-
+ade in Rydberg excitation of an ultracold lattice gas,
+Phys. Rev. Lett. 98, 023002 (2007).
+[33] T. Amthor, C. Giese, C. S. Hofmann, and M. Wei-
+dem¨uller, Evidence of antiblockade in an ultracold Ry-
+dberg gas, Phys. Rev. Lett. 104, 013001 (2010).
+[34] Z. Zuo and K. Nakagawa, Multiparticle entanglement in
+a one-dimensional optical lattice using Rydberg-atom in-
+teractions, Phys. Rev. A 82, 062328 (2010).
+[35] T. E. Lee, H. H¨affner, and M. C. Cross, Collective quan-
+tum jumps of Rydberg atoms, Phys. Rev. Lett. 108,
+023602 (2012).
+[36] W. Li, C. Ates, and I. Lesanovsky, Nonadiabatic motional
+effects and dissipative blockade for Rydberg atoms ex-
+cited from optical lattices or microtraps, Phys. Rev. Lett.
+110, 213005 (2013).
+[37] S.-L. Su, E. Liang, S. Zhang, J.-J. Wen, L.-L. Sun,
+Z. Jin, and A.-D. Zhu, One-step implementation of the
+Rydberg-Rydberg-interaction gate, Phys. Rev. A 93,
+012306 (2016).
+[38] See Supplemental Material for additional details of
+derivation and calculation, which includes Refs. [4, 28].
+[39] N. Rach, M. M. M¨uller, T. Calarco, and S. Mon-
+tangero, Dressing the chopped-random-basis optimiza-
+tion: A bandwidth-limited access to the trap-free land-
+scape, Phys. Rev. A 92, 062343 (2015).
+[40] R. Heck, O. Vuculescu, J. J. Sørensen, J. Zoller, M. G.
+Andreasen, M. G. Bason, P. Ejlertsen, O. El´ıasson,
+P. Haikka, J. S. Laustsen, L. L. Nielsen, A. Mao,
+R. M¨uller, M. Napolitano, M. K. Pedersen, A. R.
+Thorsen, C. Bergenholtz, T. Calarco, S. Montangero, and
+J. F. Sherson, Remote optimization of an ultracold atoms
+experiment by experts and citizen scientists, Proc. Natl.
+Acad. Sci. U.S.A. 115, E11231 (2018).
+[41] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨uggen,
+and S. J. Glaser, Optimal control of coupled spin dynam-
+ics: design of NMR pulse sequences by gradient ascent
+algorithms, J. Magn. Reson. 172, 296 (2005).
+[42] L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu,
+H. Wang, Y. P. Song, C.-L. Zou, S. M. Girvin, L.-M.
+Duan, and L. Sun, Quantum error correction and univer-
+sal gate set operation on a binomial bosonic logical qubit,
+Nat. Phys. 15, 503 (2019).
+[43] Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan,
+H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, Error-
+transparent operations on a logical qubit protected by
+quantum error correction, Nat. Phys. 16, 827 (2020).
+[44] Y. Ma, X. Pan, W. Cai, X. Mu, Y. Xu, L. Hu, W. Wang,
+H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and
+L. Sun, Manipulating complex hybrid entanglement and
+testing multipartite Bell inequalities in a superconduct-
+ing circuit, Phys. Rev. Lett. 125, 180503 (2020).
+[45] Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu,
+X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B.
+Zheng, and L. Sun, Demonstration of controlled-phase
+gates between two error-correctable photonic qubits,
+Phys. Rev. Lett. 124, 120501 (2020).
+[46] A. Omran, Personal communication, (2022).
+[47] S. de L´es´eleuc, D. Barredo, V. Lienhard, A. Browaeys,
+and T. Lahaye, Analysis of imperfections in the coher-
+ent optical excitation of single atoms to Rydberg states,
+Phys. Rev. A 97, 053803 (2018).
+[48] J.-L. Wu, Y. Wang, J.-X. Han, S.-L. Su, Y. Xia, Y. Jiang,
+and J. Song, Resilient quantum gates on periodically
+driven Rydberg atoms, Phys. Rev. A 103, 012601 (2021).
+[49] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Se-
+meghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pich-
+ler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner,
+V. Vuleti´c, and M. D. Lukin, Quantum phases of matter
+on a 256-atom programmable quantum simulator, Nature
+595, 227 (2021).
+[50] I. I. Beterov, I. I. Ryabtsev, D. B. Tretyakov, and
+V. M. Entin, Quasiclassical calculations of blackbody-
+radiation-induced depopulation rates and effective life-
+times of Rydberg nS, nP, and nD alkali-metal atoms
+with n ≤ 80, Phys. Rev. A 79, 052504 (2009).
+
+Supplementary Material for “Thermal-dephasing-tolerant generation of Schr¨odinger
+cat states with Rydberg dressed blockade”
+Ri-Hua Zheng,1 S.-L. Su,2, ∗ Jie Song,3 and Yan Xia1, †
+1Fujian Key Laboratory of Quantum Information and Quantum Optics,
+College of Physics and Information Engineering,
+Fuzhou University, Fuzhou, Fujian 350108, China
+2School of Physics, Zhengzhou University, Zhengzhou 450001, China
+3Department of Physics, Harbin Institute of Technology, Harbin 150001, China
+CONTENTS
+S1. Calculation dimension reduction through the Rydberg dressed blockade
+1
+S2. Adiabatic passage for the GHZ states generation
+2
+S3. GRAPE-optimized pulses for the GHZ states generation
+2
+S4. Parameter choice of the simulation for Fig. 5 in the main text
+3
+References
+3
+S1.
+CALCULATION DIMENSION REDUCTION THROUGH THE RYDBERG DRESSED BLOCKADE
+In the main text, we use Hmw to manipulate the dynamics of multiple atoms in subspace S1 ={|0⟩, |d⟩}. Further,
+the subspace S2 = {|dd⟩i(i+1)} are forbidden because of the Rydberg dressed blockade (RDB) effect. Therefore, the
+calculation space is locked to S3 = ∁S1S2, with the calculation dimension reduced from 2N to �N/2
+m=0 Cm
+N+1−m (see
+Fig. S1).
+4
+6
+8
+10
+12
+14
+16
+0
+2
+4
+6
+8
+Calculation dimension
+104
+with the RDB
+without the RDB
+14
+22
+30
+105
+1010
+FIG. S1. Calculation dimension versus the number of atoms N (a) with RDB and (b) without RDB effects. The inset shows
+the case of logarithmic coordinates when N ∈ [14, 30].
+∗ [email protected]
+† [email protected]
+arXiv:2301.05389v1  [quant-ph]  13 Jan 2023
+
+2
+|
+⟩
+|
+⟩
+|
+⟩
+quench
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+quench
+quench
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+|
+⟩
+quench
+Detuning −𝛿mw
+𝑛
+(2𝜋 MHz)
+Detuning −𝛿mw
+𝑛
+(2𝜋 MHz)
+𝑁 = 4
+𝑁 = 6
+𝑁 = 8
+𝑁 = 10
+𝛿𝑑
+𝛿𝑑
+𝛿𝑑
+𝛿𝑑
+FIG. S2. (Color online) Energy gaps versus the microwave detuning −δn
+mw for the system size (a) N = 4, (b) N = 6, (c)
+N = 8, and (d) N = 10. We choose Ωmw/(2π) = 0.2 MHz and (δe
+mw − δne
+mw)/(2π) = δd/(2π) = 0.5 MHz. Formats |◦⟩ and |•⟩
+represent |0⟩ and |d⟩ for visual intuition, respectively. Some unimportant eigenstates are plotted as gray. This graph refers to
+the presentation form of Fig. 1(b) in Ref. [S1], with data differences.
+S2.
+ADIABATIC PASSAGE FOR THE GHZ STATES GENERATION
+In the subspace S2 restricted by the RDB, there are at most N/2 atoms in dressed states, when the detuning
+−δn
+mw be large negative values (compare to Ωmw), the system ground state is clearly |0000 · · · ⟩. While −δn
+mw be large
+positive values, the system ground state becomes degenerate with N/2 + 1 states with N/2 dressed-state excitation,
+for examples, ground states {|d00d⟩, |0d0d⟩, |d0d0⟩} when N = 4 and {|d0d00d⟩, |d00d0d⟩, |0d0d0d⟩, |d0d0d0⟩} when
+N = 6. Further, by differing the detuning −δn
+mw between the edge atoms and other atoms, one can separate such
+degenerate ground states with energy δd/(2π) = (δe
+mw − δne
+mw)/(2π) = 0.5 MHz. Subsequently, the ground state when
+−δn
+mw be large positive values becomes only the GHZ state |GHZN⟩. That is why we can adiabatically change the
+detuning −δn
+mw from large negative values to large positive values to prepare the GHZ states from the initial state
+|0000 · · · ⟩. For visualization, we have drawn the diagram of adiabatic passages for systems with different sizes in Fig.
+S2. Such an adiabatic method has been proved in the work of Ref. [S1].
+S3.
+GRAPE-OPTIMIZED PULSES FOR THE GHZ STATES GENERATION
+The optimized method we used is call gradient ascent pulse engineering (GRAPE) [S2]. We preset the control
+Hamiltonians as Hh (h = 1, 2, 3) with H1 = �N
+n=1 |0⟩n⟨d| + H.c., H2 = �N−1
+p=2 |0⟩p⟨0|, and H3 = �
+l=1,N |0⟩l⟨0|. The
+corresponding control pulses are set as fh (clearly, Ωmw = f1, δne
+mw = −f2, and δe
+mw = −f3). The detailed derivation
+can be found in Ref. [S2], and here we only give the specific optimization steps:
+(1) Guess initial control pulses f1/(2π) = 0.2 MHz, f2/(2π) = (−2 + 4t/tf) MHz, and f3 = f2 + 2π × 0.5 MHz,
+where t is the evolution time and tf the final evolution time depended on the system size N.
+
+3
+0
+0.5
+1
+1.5
+-2
+0
+2
+0
+2
+-2
+-1
+0
+1
+0
+2
+-4
+-2
+0
+2
+0
+5
+-2
+-1
+0
+1
+FIG. S3. Optimized microwave pulses by the GRAPE for systems with different sizes N = {4, 6, 8, 10}. The dashed, solid, and
+dash-dotted lines in each subplot represent Ωmw, δne
+mw, and δe
+mw, respectively.
+(2) Calculate the evolution operator U(t).
+(3) Change new control pulses as f ′
+h = fh − iϵh · ∆t · Tr
+�
+[Hk, U(t)ρ0U(t)†] · U(t)U(tf)†ρfU(tf)U(t)†�
+, with ρf
+the desire state, i.e., the GHZ state. Here ϵh and ∆t = tf/200 are the correction strength and the time derivative,
+respectively. Specifically, {ϵ1, ϵ2, ϵ3} · ∆t/(2π) = {0.02, 0.2, 0.2} MHz.
+(4) Go to step (2) until the fidelity Tr[U(t)ρ0U(t)†ρf] reaches a certain requirement.
+The Optimization results are shown in Table I in the main text. Note that the maximum absolute value of Ωmw
+does not exceed 2π × 0.2 MHz to ensure the stability of the RDB dynamics (J ≫ Ωmw). Detailed driving pulses for
+systems at different scales can be seen in Fig. S3.
+S4.
+PARAMETER CHOICE OF THE SIMULATION FOR FIG. 5 IN THE MAIN TEXT
+We have shown Fig. 5 in the main text to demonstrate the relationship between fidelities of GHZ states F and
+atomic temperature T.
+The selection of parameters for each temperature is different to ensure high fidelity F.
+Specifically, we choose {V, ∆, |Ωmw|max}/(2π) = {22, 7, 0.1}, {20, 6, 0.1}, {20, 6, 0.1}, {20, 6, 0.1}, and {22, 6, 0.1} MHz
+for T = 0, 2, 4, 6, and 8 µK, respectively.
+[S1] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi,
+J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuleti´c, and M. D. Lukin,
+Generation and manipulation of Schr¨odinger cat states in Rydberg atom arrays, Science 365, 570 (2019).
+[S2] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨uggen, and S. J. Glaser, Optimal control of coupled spin dynamics:
+design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005).
+

WdE4T4oBgHgl3EQfNAwx/content/tmp_files/2301.04952v1.pdf.txt

@@ -0,0 +1,1428 @@
+Sensing Diamagnetic Electrolytes with Spin
+Defects in Diamond
+Fabian A. Freire-Moschovitis,† Roberto Rizzato,† Anton Pershin,‡ Moritz R.
+Schepp,† Robin D. Allert,† Lina M. Todenhagen,¶ Martin S. Brandt,¶ ´Ad´am
+Gali,‡,§ and Dominik B. Bucher∗,†
+†TUM School of Natural Sciences, Department of Chemistry, Technical University of
+Munich, Lichtenbergstraße 4, 85748 Garching bei M¨unchen, Germany
+‡Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, PO.
+Box 49, Budapest H-1525, Hungary
+¶Walter Schottky Institut and Physik-Department, Technical University of Munich, Am
+Coulombwall 4, 85748 Garching bei M¨unchen, Germany
+§Department of Atomic Physics, Institute of Physics, Budapest University of Technology
+and Economics, M˝uegyetem rakpart 3, Budapest H-1111, Hungary
+E-mail: [email protected]
+Abstract
+Quantum sensing with spin defects in diamond, such as the nitrogen vacancy (NV)
+center, enables the detection of various chemical species on the nanoscale. Molecules or
+ions with unpaired electronic spins are typically probed by their influence on the NV-
+center’s spin relaxation. Whereas it is well-known that paramagnetic ions reduce the
+NV-center’s relaxation time (T1), here we report on the opposite effect for diamagnetic
+ions.
+We demonstrate that millimolar concentrations of aqueous diamagnetic elec-
+trolyte solutions increase the T1 time of near-surface NV-center ensembles compared
+1
+arXiv:2301.04952v1  [physics.app-ph]  12 Jan 2023
+
+to pure water. To elucidate the underlying mechanism of this surprising effect, single
+and double quantum NV experiments are performed, which indicate a reduction of
+magnetic and electric noise in the presence of diamagnetic electrolytes. In combination
+with ab initio simulations, we propose that a change in the interfacial band bending
+due to the formation of an electric double layer leads to a stabilization of fluctuating
+charges at the interface of an oxygen-terminated diamond. This work not only helps
+to understand noise sources in quantum systems but also broadens the application
+space of quantum sensors towards electrolyte sensing in cell biology, neuroscience and
+electrochemistry.
+Introduction
+Nitrogen vacancy (NV) centers in diamond offer a broad platform for quantum sensing ap-
+plications ranging from the measurement of basic physical properties, such as temperature,1
+pressure,2 strain,3 electric4 and magnetic fields5,6 down to single cells,7 single molecules,8
+or even single nuclear spins.9,10 This unprecedented sensitivity is achieved due to the atomic
+size of the qubit enabling its location only a few nanometer away from the diamond interface
+(see Figure 1a).11 Near surface NV-centers (≤ 20 nm below the surface) can be used to sense
+nuclear magnetic resonance (NMR)8,12–15 and electron spin resonance (ESR) signals16 from
+nanoscale chemical and biological samples. The NV-center translates these magnetic field
+fluctuations directly to an optical signal, detected by a change in the fluorescence intensity.
+Together with optical spin state initialization with green laser light and coherent spin state
+manipulation with microwave pulses, these key features make the NV-center a unique tool
+for (bio)chemical analysis.11,15,17,18
+NV-centers are also perceptive to electric fields.4,19 Even a single elementary charge
+located ∼ 150 nm away from the quantum sensor produces a strong enough static electric field
+to affect an NV-center’s ODMR (optically detected magnetic resonance) transitions.4 This
+high sensitivity of NV-centers to magnetic or electric fields marks a narrow ridge: On the
+2
+
+one hand, it allows for single nuclear spin or elementary charge detection, on the other hand
+it makes the qubit prone to magnetic or electronic spin noise in its close environment. This
+is particularly relevant for NV-centers close to the surface, where interfacial processes and
+defects cause additional noise and reduce the performance of the NV-quantum sensors.20–23
+Due to the NV-center’s susceptibility to a broad range of frequencies (from DC to GHz),24
+noise can be measured by applying different sensing protocols.18 High frequency noise (∼
+GHz) can be probed by the longitudinal spin-lattice relaxation time (T1) with a protocol
+that is usually referred to as NV-relaxometry.17,18,24–26 The T1 time defines the time constant
+of the spin-state-dependent fluorescence decay from the magnetic sublevel ms = 0 (bright
+state) to thermal equilibrium (mixed state) and is on the order of a few milliseconds for near-
+surface NV-ensembles in bulk diamonds,11 or on the order of a few hundred microseconds for
+nanodiamonds (depending on their size).17 Generally, any magnetic noise overlapping with
+the NV-center’s Larmor frequency (on the order of the zero-field splitting D = 2.87 GHz) will
+decrease the T1 relaxation time (see Figure 1b).18,25,27,28 Relaxometry has been successfully
+applied to map high frequency magnetic noise originating from inside the diamond (e.g.
+paramagnetic impurities29), at the diamond interface (e.g.
+dangling bonds20,30) or from
+samples on top of the diamond (e.g.
+organic radicals27,31 or paramagnetic ions, such as
+Mn2+,32 Fe3+,33 or Gd3+ 25,28). Furthermore, nanoscale NV-relaxometry has also been used
+to determine the pH value34 or to monitor chemical reactions in situ.35
+While the increase of the relaxation rate due to paramagnetic species has been studied
+extensively, herein we detect the opposite effect for diamagnetic ions. When near-surface
+NV-centers in oxygen-terminated diamonds are exposed to aqueous diamagnetic electrolytes,
+we observe a systematic extension of the T1 relaxation time compared to pure (deionized)
+water. We show that this unexpected effect is proportional to the electrolyte concentration,
+reversible and dependent on the NV-center’s implantation depth. In order to shed light
+on the underlying sensing mechanism, we perform single and double quantum relaxometry
+experiments which indicate a reduction of electric as well as magnetic noise. Furthermore,
+3
+
+double electron electron resonance (DEER) spectroscopy experiments show a similar effect
+on surface dark spins, which possibly act as surface reporter spins. In combination with
+theoretical methods including ab initio simulations of the diamond/electrolyte interface, we
+propose that diamagnetic ions alter the interfacial band bending. This leads to a stabilization
+of fluctuating charges at the interface and to the increase of the T1 relaxation time.
+Results
+T1 Relaxometry on Electrolytes with Near-Surface NV-Center En-
+sembles
+In this study we use near-surface high dense NV-center ensembles (implanted with 15N
+at an energy of 2.5 keV and a fluence of 2 × 1012 cm−2), distributed ∼ 5 nm underneath the
+diamond surface,13,36,37 to investigate the effect of aqueous electrolyte solutions on the spin-
+lattice relaxation time T1 probed by NV-relaxometry. Before experiments are conducted,
+we prepare the diamond surface with a tri-acid clean procedure according to Brown et al.38
+This procedure not only ensures to remove non-diamond carbon material from the interface
+but also creates an oxygen-terminated surface comprised of mixed carbon oxide species
+including hydroxyl groups, ethers, ketones, aldehydes and carboxylic acids.39 We position
+the diamond in a microfluidic device that guarantees controllable in- and output of the
+applied liquids, prevents sample evaporation and provides a constant and defined volume
+for following measurements (see Figure 1a).15 Importantly, the microfluidic device avoids a
+direct contact between the liquid and the microwave delivery.
+The NV-relaxometry protocol is depicted in Figure 1b and essentially consists of two
+532 nm laser pulses of 5 µs duration for optical spin state initialization and readout separated
+by a sweep time t. A subsequent measurement with a π0,-1-pulse (where the subscripts 0 and
+-1 indicate transitions between ms = 0 ↔ ms = −1) at the start is used for normalization and
+noise cancellation purposes.18 The T1 time can be extracted from the (bi)exponential fit of
+4
+
+Figure 1: Scheme of NV-relaxometry experiments with aqueous electrolyte so-
+lutions. a) Top (left): The NV-center in the diamond crystal lattice. A nitrogen atom
+(blue) replaces a carbon atom (black) in the crystal. Together with an adjacent vacancy
+an NV-center is formed. The orbitals (petrol) indicate four possible NV-center orienta-
+tions within the diamond lattice. Top (right): Scheme of an oxygen-terminated diamond
+surface with an ensemble of near-surface NV-centers (dNV ∼ 5 nm). The oxygen termi-
+nation consists of hydroxyl groups, ethers, ketones, aldehydes and carboxylic acids. We
+probe pure water, paramagnetic (purple) and diamagnetic (yellow) ions. Bottom: A mi-
+crofluidic device placed on top of the diamond. In- and outlet allow for adding and remov-
+ing aqueous electrolyte solutions or pure water from the diamond surface. b) Top (left):
+NV-relaxometry pulse sequence. Two laser pulses for spin state initialization and read-
+out are separated by a sweep time (t). A consecutive measurement with a π-pulse at the
+start is used for noise cancellation.18 Top (right): Energy levels of the NV-center’s elec-
+tronic ground state (S = 1). The zero-field splitting (D = 2.87 GHz) separates the
+ms = 0 (bright) and ms = ±1 states (dark). A bias magnetic field B0 splits the degen-
+erate ms = ±1 states according to the Zeeman effect. Bottom: T1 relaxation curves of pure
+water and solutions of NaCl (500 mM) and MnCl2 (1 µM). While paramagnetic MnCl2 re-
+duces the T1 relaxation time with respect to pure water, diamagnetic NaCl extends the
+relaxation time. Experiments are performed at fNV = 1.88 GHz.
+5
+
+a)
+b)
+Init. Read
+ms = +1
+Laser
+t
+ms = -1
+GHz
+元0,1
+MW
+ms = 0
+元0,1
+OH
+Bo = 0
+Bo > 0
+~ 5 nm
+0.25
+Inlet
+[Norm.]
+Diamagnetic
+Microfluidic
+0.5
+Device
+Contrast [
+Paramagnetic
+Near-Surface
+0.5
+1.5
+2.5
+NV-Ensemble
+3.5
+Outlet
+Pure Water
+NaCI (aq.)
+MnCl2 (ag.)
+Laser (532 nm)
+Diamond
+0
+1
+2
+4
+7
+8
+9
+Fluorescence (635 to 800 nm)
+Sweep Time t [ms]the relaxation curves depicting the contrast as a function of sweep time t (see Supplementary
+Note 1 for fitting details).
+We perform the measurements by filling the microfluidic channel and covering the di-
+amond surface either with pure water or aqueous electrolyte solutions (see Methods for
+detail). Figure 1b depicts the T1 relaxation curves when water and solutions of diamagnetic
+NaCl or paramagnetic MnCl2 cover the diamond surface. Paramagnetic MnCl2 (1 µM) on
+the diamond leads to a T1 time reduction of 0.47 ± 0.19 with respect to water, which is in
+accordance with other studies and can be ascribed to the strong dipole-dipole interaction of
+the NV-center with the paramagnetic species.25,32 In contrast, when we repeat the same ex-
+periment with diamagnetic NaCl (500 mM) solution, we observe an extension of the T1 time
+by a factor of 2.04 ± 0.45 compared to water. Experiments supporting this observation are
+also conducted with other diamagnetic salt solutions (mono-, di- and trivalent) and reveal
+similar results (see Supplementary Note 2). Therefore, we choose NaCl as a representative
+of a standard diamagnetic electrolyte for the following measurements in our work and expect
+comparable results for other diamagnetic salt solutions.
+Moreover, by tuning the magnetic field B0 and thereby the NV-center’s Larmor fre-
+quency NV0,-1 (i.e., the ms = 0 → ms = −1 transition frequency) we are able to map the
+spectral noise density. We probe water/NaCl (500 mM) solution with NV0,-1 frequencies from
+131 MHz to 2.87 GHz and observe a similar effect over the entire frequency range (see Supple-
+mentary Note 2). Consequently, the extension of the T1 time of near-surface NV-ensembles
+with exposure to diamagnetic electrolyte solutions is an effect that covers a broad range of
+(high) frequencies (i.e., from ∼ hundreds of MHz to GHz).
+Additionally, in order to exclude an impact of the solvent’s physical properties (i.e.,
+polarity) on our experiments,19 we choose typical organic solvents whose dielectric constants
+(κ) and chemical structure differ significantly from water (κ = 8040) and probe them with
+relaxometry (see Supplementary Note 3). Since the T1 time remains unaffected, we conclude
+that the herein described effect is not induced by the physical properties of water, but by
+6
+
+the diamagnetic electrolyte.
+Sensitivity of T1 Relaxometry on Electrolytes
+In order to obtain information about the sensitivity of the NV-relaxometry protocol
+to para- and diamagnetic electrolyte solutions, we perform additional measurement series
+where the electrolyte concentration is increased stepwise by one order of magnitude (from
+10−5 to 10−2 mM in the case of paramagnetic MnCl2 and from 10−4 to 103 mM in the case
+of diamagnetic NaCl).
+Paramagnetic MnCl2 shows a stepwise T1 decrease in micromolar concentrations reaching
+a decline of up to 86 ± 10% for a 10 µM solution with respect to water covering the diamond
+(see Figure 2a and Supplementary Note 4).
+Note that a further concentration increase
+(> 10 µM) is not measurable, as it leads to a collapse of the T1 time. In contrast, diamagnetic
+NaCl shows a slight T1 increase compared to water, which then fluctuates moderately from
+micromolar to lower millimolar concentrations. Importantly, a significant and gradual T1
+increase is measurable from 10 mM to 500 mM NaCl solution, where the effect saturates at
+81 ± 11% with respect to water (see Figure 2b and Supplementary Note 4).
+The decrease of the T1 time with paramagnetic species (e.g., MnCl2) is expected and well
+studied.17,25,26,30 Here, high frequency (∼ GHz) noise originates from magnetic dipole-dipole
+interactions of the NV-center’s electronic spin and the sample’s electronic spin (“spin-flips”),
+resulting in a decline of the T1 time if unpaired electrons are near the sensor. On the other
+hand, for diamagnetic ions (e.g., NaCl) these interactions are absent as only paired electrons
+without a (net) magnetic moment are present.
+Surprisingly, here we observe a gradual
+extension of the T1 time with increasing millimolar concentrations of diamagnetic NaCl
+solution.
+Importantly, both sensitivity regimes (∼ nano- to micromolar for paramagnetic and ∼
+millimolar for diamagnetic species) match the typical physiological41,43 or (for diamagnetic
+electrolytes) electrochemical concentrations,44 opening up sensing applications in cell biology
+7
+
+Figure 2: NV-relaxometry with increasing concentrations of para- and diamag-
+netic electrolyte solutions. a) Paramagnetic MnCl2 shows a stepwise T1 time decrease
+for concentrations in the micromolar regime until the effect reaches a maximum measur-
+able decline of 86 ± 10% for 10 µM solutions with respect to water covering the diamond.
+b) In contrast to that, diamagnetic NaCl (right) shows a slight increase of the T1 time
+compared to water, which then fluctuates moderately from the micromolar to the lower
+millimolar regime. For concentrations ≥ 10 mM the T1 time increases gradually along with
+the NaCl concentration until the effect saturates to 81 ± 11% for NaCl (500 mM) solutions.
+Shaded areas indicate typical physiological concentration regimes for para- and diamag-
+netic ions.41–43 Experiments are performed at fNV = 1.88 GHz.
+or electrochemistry.
+8
+
+InCl2 (aq.)
+Conc.
+Flips
+Conc.Reversibility, Passivation and NV-Center Depth
+Because of the surprising observation, that the T1 time increases with diamagnetic elec-
+trolyte solutions compared to water covering the diamond, the next experiments concentrate
+on the mechanism behind this effect. Therefore, we probe the reversibility and passivation
+of the effect along with the sensor’s response in dependence of its implantation depth. First,
+we evaluate if the extension of the T1 relaxation time is a reversible process by exposing an
+oxygen-terminated diamond alternatingly to water and NaCl (500 mM) solution. Thereby,
+we show that the T1 relaxation time is altered from “short” in case of water exposure to
+“long” when NaCl solution covers the surface (see Figure 3a in green). Alternating between
+water and electrolyte solution demonstrates a 1.83±0.35 fold increase of the T1 time with elec-
+trolyte exposure on the oxygen-terminated diamond. In a next step, we examine if the effect
+is specific to the diamond surface termination. Therefore, the formerly oxygen-terminated
+diamond is coated with an aluminium oxide (Al2O3) thin film (thickness ∼ 1 nm)13 prepared
+by Atomic Layer Deposition (ALD). The aluminium oxide thin film ensures a controllable
+and uniform surface termination with hydroxyl groups. We repeat the previous experiment,
+but this time the T1 relaxation time remains unaffected by the NaCl solution (see Figure 3a
+in black).
+Additionally, we investigate if the extent of the electrolyte’s effect is dependent on the
+depth of the embedded NV-center ensemble. Therefore, we prepare two diamonds with 15N
+implantation energies of 2.5 and 4 keV with the tri-acid clean procedure described before
+and probe them with NV-relaxometry. Near-surface NV-centers implanted with an energy
+of 2.5 keV are mainly distributed within a depth of ∼ 5 nm below the surface, while ensembles
+created with 4 keV 15N are located about ∼ 12 nm beneath the surface.37 Figure 3b shows a
+significantly larger effect of the electrolyte on the relaxation time of the shallow implanted
+NV-diamond with respect to the deeper one, although a T1 time extension is still detectable
+in the latter case (see also Supplementary Note 5). Importantly, while the effect with the
+∼ 1 nm thick aluminium oxide layer is completely passivated, a T1 time increase can still be
+9
+
+Figure 3: NV-relaxometry experiments with different surface terminations and
+NV-center ensemble implantation depths. a) An oxygen-terminated diamond sur-
+face is alternatingly exposed to water and NaCl (500 mM) solution. The T1 time increases
+with electrolyte exposure by a factor of 1.83 ± 0.35 with respect to water. Importantly,
+this behavior can be altered by either the presence or absence of water or NaCl solution.
+After coating the same diamond with an aluminium oxide (Al2O3) thin film (thickness
+∼ 1 nm) the T1 time is unaffected by the NaCl. b) T1 relaxation curves of water and NaCl
+(500 mM) solution covering the diamond surface. Diamonds were implanted with 15N at
+an energy of 2.5 keV (top) and 4 keV (bottom) resulting in different NV-center ensemble
+depths (dNV). While on the shallower implanted diamond the T1 time increases by a factor
+of around two with exposure to NaCl solution, on the deeper implanted diamond the effect
+is strongly reduced. Experiments are performed at fNV = 1.88 GHz.
+10
+
+2.2
+1.8
+1.4
+1.0
+dv ~ 5 nm
+Oxygen-Terminated Diamond
+OHO OH @ OH e
+~ 1 nm Al203
+dnv ~ 5 nm
+dnv ~ 5 nm
+dnv ~ 10-15 nmobserved with the oxygen-termination when the sensor is ∼ 5 to 10 nm further away from
+the electrolyte. Note that the same measurements conducted with LiCl (500 mM) solution
+lead to similar results (see Supplementary Note 5).
+From these experiments we conclude that the extension of the T1 relaxation time is a
+reversible and interfacial process which is dependent on the distance of the sensor to the
+sample.
+Additionally, NV-charge state alterations or changes in NV-dephasing and NV-coherence
+are not observed in our experiments (see Supplementary Note 6).
+Probing Magnetic and Electric Noise Contributions
+In the next set of experiments we investigate if the T1 relaxation time increase originates
+from a reduction of electric and/or magnetic field noise. Since we are dealing with electrolytes
+dissolved in water, it is particularly interesting to explore the influence that charged ions
+and their randomly fluctuating electric fields might have on the T1 relaxation time of the
+near-surface NV-ensembles. Whereas the typically used relaxometry experiments use single
+quantum (SQ) transitions to probe magnetic field noise (as in the experiments from the
+previous sections), double quantum (DQ) transitions are influenced by electric field noise.45
+For these transitions, the full NV-center ground state (S = 1) is considered, where an
+additional relaxation pathway between ms = −1 ↔ ms = +1 with ∆ms = 2 (see Figure 4a)
+becomes accessible, the DQ transition. Regarding the NV-center as a “qutrit” rather than a
+qubit allows to probe the effect of the diamagnetic electrolyte solution on both electric and
+magnetic field noise at the same time.
+SQ and DQ relaxometry measurements on the system water/NaCl (500 mM) solution
+reveal that the diamagnetic electrolyte has an effect on both relaxation channels, i.e., it
+reduces magnetic as well as electric field fluctuations (see Figure 4b). Here, two different T1
+times can be defined: T1,SQ for the relaxation time in the SQ and T1,DQ in the DQ channel.
+An extension of T1,SQ by a factor of 1.52 ± 0.16 and a 2.84 ± 0.31 increase of T1,DQ can
+11
+
+Figure 4: Single quantum (SQ) and double quantum (DQ) relaxation experi-
+ments. a) Energy level scheme of the NV-center ground state transitions. SQ transitions
+(∆ms = ±1) with relaxation rates Ω are susceptible to magnetic noise. The DQ relax-
+ation (∆ms = ±2) with relaxation rate γ is magnetically forbidden but susceptible to
+electric noise.45 b) Top: DQ pulse sequence. Bottom: SQ and DQ relaxation curves of
+water and NaCl (500 mM) solution covering the diamond. Experiments are performed at
+B0 = 15 G, where the NV0,-1 transition is at fNV = 2.83 GHz (corresponding to a DQ tran-
+sition frequency of 80 MHz). T1,SQ increases by a factor of 1.52±0.16 and T1,DQ by a factor
+of 2.84 ± 0.31
+when the diamond is exposed to NaCl solution.
+12
+
+SQ Relaxatior
+ating
+Spin
+bles
+~MHz
+Q ~GHzbe measured in presence of NaCl solution with respect to water (see also Supplementary
+Note 7). Thus, diamagnetic electrolytes reduce both – electric and magnetic – noise at the
+diamond surface. Importantly, when we repeat the same experiments with paramagnetic
+MnCl2, only T1,SQ reduces by 80%, whereas the DQ transition remains unaffected compared
+to water (see Supplementary Note 7). This indicates an exclusive impact of the paramagnetic
+electrolyte on magnetic field noise and could provide a possible pathway to distinguish para-
+from diamagnetic ions in solution.
+Additionally, we can exclude an influence of diamagnetic NaCl solution on the static
+electric field environment by performing zero field ESR Measurements (see Supplementary
+Note 7).
+Influence of Electrolytes on Surface Dark Spins: DEER Experi-
+ments
+So far, we have focused on the direct influence of electrolytes on the NV-centers. How-
+ever, the diamond as the NV-center’s host material provides various surface dark spins, e.g.
+dangling bonds, whose response to the electrolytes is probed in the next set of experiments.
+Intrinsic T1 times of these surface dark spins are often long (∼ a few microseconds)46 which
+allows us to probe them with NV-based double electron electron resonance (DEER) spec-
+troscopy.16,47 Figure 5a shows the pulse sequence of a typical DEER experiment: A spin-echo
+is performed on the NV-center’s electronic spin (MWNV spin-echo), at the same time, in the
+second free precession time of the echo, an additional microwave pulse (MWDEER) is ap-
+plied to drive the target electronic spins. Sweeping the MW frequency (f DEER) flips the
+surface dark spins when their Larmor frequency is matched and causes a dip in the DEER
+signal (see Figure 5b). As shown in Figure 5b, a clear dip in the DEER spectrum appears
+when the diamond interface is covered with NaCl (500 mM) solution.
+The resonance at
+f DEER = 0.887 GHz corresponds to ge ∼ 2 spins and is typically assigned to dangling bonds
+at the diamond surface.50 Interestingly, the dip is drastically reduced when the experiment
+13
+
+Figure 5: NV-DEER experiments probing the response of surface dark spins
+to electrolyte exposure. a) Pulse sequence of the double electron electron resonance
+(DEER) experiment. Sweeping the microwave frequency of the MWDEER pulse (f DEER)
+while applying a spin-echo experiment on the NV-center (MWNV spin-echo) allows for the
+detection of electronic (surface dark) spins coupled to the NV-centers.48 b) DEER ex-
+periment with water and NaCl (500 mM) solution covering the diamond surface. A pro-
+nounced dip in the DEER spectrum appears at around 0.887 GHz (where ge ∼ 2) when the
+diamond is exposed to NaCl solution. The dip gets drastically reduced when water cov-
+ers the diamond surface. c) Top: Pulse sequence of the DEER-Rabi experiment. Bottom:
+When the pulse duration of the microwave drive (MWDEER) at the surface dark spins’
+resonance frequency is swept during the spin-echo, DEER-Rabi oscillations can be ob-
+served.49 While the πds-pulse lengths remain equal when water or NaCl (500 mM) solution
+cover the diamond surface (πds ∼ 24 ns), the latter causes a by a factor of around three
+more pronounced Rabi amplitude. d) Top: Pulse sequence of the DEER-T1 experiment.
+Bottom: Varying the correlation time (tcorr) between two subsequent DEER segments,46
+shows a surface dark spin relaxation with T1,ds = 7.20±1.10 µs in case of NaCl exposure. In
+contrast, no significant relaxation decay can be observed when water covers the diamond
+surface. DEER experiments are performed at fNV = 1.98 GHz.
+14
+
+MWDEER
+Surface
+Dark Spins
++
+dNvis repeated with water.
+Once the resonance condition for the ge ∼ 2 spins is found, coherent control of the
+surface dark spin state can be demonstrated. Figure 5c shows a DEER-Rabi experiment
+on the surface dark spins. Sweeping the microwave pulse duration (MWDEER) during the
+spin-echo causes oscillations of the defect’s spin state.49 While we determine equal πds pulse
+lengths (πds ∼ 24 ns) for water and the electrolyte, the former leads to an about three
+times increased DEER-Rabi amplitude with respect to the latter.
+In the next step, we
+probe the surface dark spin relaxation time (T1,ds). Figure 5d depicts a pulse sequence from
+Sushkov et al.,46 where the T1 time of the surface dark spins can be measured by correlating
+two subsequent DEER segments and varying the correlation time tcorr. Interestingly, we
+measure a relaxation time T1,ds of 7.20 ± 1.10 µs for the surface dark spins exposed to the
+NaCl solution. In contrast, a clear relaxation decay cannot be observed in case of pure water
+covering the diamond. Although we cannot exclude that the surface dark spins vanish, the
+more likely case is that the dark spins act as reporter spins46 experiencing the same effect of
+the electrolyte solution as the NV-center: “fast” relaxation in the case of water and “slow”
+relaxation when diamagnetic electrolyte solutions cover the diamond, which is also expressed
+in the increased DEER signals (see Figure 5b-d). Accordingly, we enhance the sensitivity of
+our sensor by the proximity of the reporter spins to the electrolyte solutions.46,51
+Theoretical Modeling
+Our experimental results show an influence of diamagnetic electrolyte solutions on near-
+surface spin defects in diamond where electric as well as magnetic noise is suppressed resulting
+in an increase of the T1 relaxation time. To further study this surprising effect computational
+modeling is used. Here, as a working assumption, we focus on charge fluctuations within the
+diamond lattice. We note, that further processes such as proton hopping at the interface or
+water and ion dynamics within the electric double layer can also play a role but have not
+15
+
+been treated herein.
+To this end, we model an interface between a slab of diamond and a thin layer of water
+subsequently enriched by Na+ and Cl- ions (see Methods for detail). Then, we probe the
+interfacial structure and vacuum level shifts (VLS) based on the configurations obtained
+from the ab initio molecular dynamics (MD)(see Figure 6a). The calculated alignment of
+the electronic levels of water and the model diamond surface is shown in Figure S10. Here,
+a mismatch of the chemical potentials (defined as a center of the band gap) promotes an
+electron leakage from the diamond surface towards the water. The resulting redistribution
+of charges leads to the development of an electric field, that further rearranges the charged
+solvated Na+ and Cl- ions. The large positive VLS of 1.1 eV (see Figure S10a) causes the
+ions to rearrange with the direction of the field, facilitating the effect of band bending. By
+adding a carboxyl group, we observe a stabilization of the downwards band bending relative
+to the case of the model diamond surface in water. However, in both cases, we obtain a
+broad distribution of surface dipoles, owing to the complexity of ion dynamics within the
+Stern layer. By contrast, a sharp distribution of the dipole moments is observed between a
+dissociated carboxyl group and a solvated Na+ ion nearby. This stable configuration gives
+rise to a large VLS of ∼ -1.9 eV. This value is further used to trace the evolution of the
+electrostatic potential at the microscopic level. More specifically, we set it as a boundary
+condition for solving the Poisson equation to access the modifications of the potential inside
+a semi-infinite diamond slab. As shown in Figure 6b, we observe that the interfacial region
+of ∼ 40 nm is affected by the respective readjustments of the charges, resulting in a rapid
+decay of the potential near the interface and a slow saturation towards the bulk.
+To establish a relation between the band bending and the noise reduction, we consider the
+electric and magnetic fluctuations caused by a pair of active defects around the NV-centers.
+The charge transfer process leads to a continuous change in the charge/spin state of the
+nearby defects, which can affect the relaxation and coherence time of the NV-center when
+the rate approaches the timescale of the quantum sensing experiment. In the Marcus theory,
+16
+
+Figure 6: Ab initio MD simulations of the diamond/electrolyte interface. a)
+Representative snapshot of the diamond/electrolyte interface from the ab initio MD simu-
+lations. Color code: C (grey), O (red), H (white), Na+ (yellow), Cl- (green). b) Variations
+of the electrostatic potential in a semi-infinite diamond due the arrangement of NaCl at
+the interface. Inset: Electrostatic potential (∆Vmax) for a pair of defects in 3 nm distance
+to each other surrounding a ∼ 5 nm deep NV-center. c) Schematic representation of a con-
+tinuous charge hopping between two defects around the NV-center. HAB is the transfer
+integral, ∆G0 is calculated as ∆Vmax/2 from b). d) Calculated rate constants for pairs of
+substitutional nitrogen and vacancy defects as a function of distance between the defects.
+e) Rate constants for the forward and backward electron transfer between a pair of va-
+cancy defects as a function of bias due to the interfacial band bending. Vertical lines refer
+to an estimated difference in the effects by replacing the electrolyte with pure water, con-
+sidering a pair of defects and a NV-center in a configuration from the inset in b).
+17
+
+a)
+b)
+c)
+forward
+Electrostatic potential [eV]
+0
+A
+B
+-0.5
+-1.25
+V-
+NV
+↑ HAB
+-1
+vo
+AV
+-1.5
+-1.5
+-1.75
+△G0
+3
+4
+5
+6
+7
+-2
+back
+0
+10
+20 30
+40
+50 60
+Distance [nm]
+d)
+10 12
+10 10
+108
+_s] -
+108
+INON+
+H20
+NaCI
+107
+106
+forward
+104
+back
+10 6
+0
+5  10 15 20 25 30
+0
+0.05
+0.1
+0.15
+△G° [eV]
+Distance [A]such fluctuations are described as a sequence of thermally activated hopping events, whilst
+the rate constants are determined from the distance-dependent coupling parameters and
+the required structural reorganizations (see Figure 6c). The dipoles at the diamond/solvent
+interface affect this equilibrium by altering the onsite Gibbs energy term with a contribution
+from the electrostatic potential (∆V ). As shown in Figures 6c and 6e, the band bending
+accelerates a forward charge transfer process (until reaching the Marcus inverted region), but
+at the same time, the magnitude of the back charge transfer (BCT) rate drops exponentially.
+Hence, regardless of the defect type, large interfacial band bending can lead to a dynamical
+trapping of the charges around a site with the lower Gibbs energy. For a numerical validation,
+we focus on the electron fluctuations between a pair of carbon vacancies as well as on the hole
+fluctuations between two substitutional nitrogen defects. We note that carbon vacancies are
+often generated by the implantation and irradiation techniques used to create the NV-centers,
+and substitutional N defects are present around NV-centers to stabilize the negative charge
+state of the NV-center. After determining the relevant parameters in the periodic supercells
+(see Methods for detail), we first analyze the possible contribution of each charge transfer
+reaction to the electrical noise. To this end, we compare the fluctuation rates, calculated for
+both defect pairs in a bulk-like environment (∆V = 0). As shown in Figure 6d, the charge
+fluctuation rates for a pair of nitrogen defects is remarkably smaller than for a vacancy pair.
+This difference is attributed to a formation of an energetically unfavorable N0 configuration,
+that hinders the charge fluctuations by large structural reorganizations (λreorg = 1.89 eV).
+Hence, substitutional N0 pairs are unlikely to be the origin of electric and magnetic noise,
+affecting the T1 time due to charge fluctuation. By contrast, owing to a rather modest λreorg
+of 0.28 eV, a vacancy pair gives rise to noise in a broad frequency range, where the respective
+rate constants are controlled by the separation between the active sites.
+The relevant distance between a pair of vacancies is readily obtained from an exponential
+fit of the rate constants in Figure 6d. More specifically, we find the charge fluctuation rates,
+which would be relevant for an influence on the T1 time (∼ 0.1 GHz) at a separation of ∼
+18
+
+3 nm. The molecular dynamics simulations performed by F´avaro de Oliveira et al. show
+that such a high local density of vacancies around the region of a NV-center is achieved
+during the nitrogen implantation due to the cascade process of the “kick-out” mechanism.52
+Using the variations of electrostatic potential from Figure 6b for the implantation depths of
+5 and 12 nm, we calculate a contribution to the onsite Gibbs energies by 0.125 and 0.075 eV,
+respectively (see Methods for details).
+As shown in Figure 6e, the BCT rate at 12 nm
+reduces by a factor of ∼ 7 relative to the case of ∆V = 0. Moreover, in agreement with
+our experimental results, the faster changes in the potential at 5 nm enhances the reduction
+factor to ∼ 19 for the shallower NV-centers. Relative to pure water, the BCT rates decrease
+by factors of ∼ 10 and 3.5 for the depths of 5 nm and 12 nm, respectively. Furthermore, the
+proposed mechanism is consistent with the experimental results on dark spins (see Figure 5).
+Interestingly, our calculations point to an even larger decrease of the BCT rate at smaller
+distances from the interface which should translate to an even larger sensitivity.
+Given
+the favorable downwards band bending, these results call for a further optimization of the
+implantation parameters as well as the surface structure to fully exploit the extension of the
+T1 time by diamagnetic electrolyte solutions.
+Conclusion and Outlook
+We report on the effect of diamagnetic electrolyte solutions on highly dense near-surface
+spin defects in oxygen-terminated diamonds. Surprisingly, we observe that diamagnetic ions
+increase the T1 relaxation time of NV-centers. We demonstrate that this effect is reversible,
+surface sensitive and responsive to millimolar concentrations. We find that also interfacial
+spin defects are sensitive to diamagnetic species, anticipating their possible use as reporter
+spins for future optimization. Furthermore, we investigate the underlying mechanism by
+single and double quantum NV-relaxometry experiments in combination with ab initio sim-
+ulations. We propose that ions at the interface stabilize charge fluctuations between pairs of
+19
+
+carbon vacancies and alike deep defects, surrounding the NV-centers. This reduces magnetic
+as well as electric noise at the diamond interface by a dynamical trapping of mobile electrons
+to a site with lower Gibbs energy. These findings encourage further simulations and experi-
+ments (e.g, on other NV-diamond systems, such as nanodiamonds or single NV-centers) to
+elaborate on a comprehensive understanding of the complex processes at the solid/liquid
+interface.
+We would like to emphasize that the sensitivities of relaxometry to para- and diamagnetic
+electrolyte solutions both represent scientifically relevant concentration regimes. Paramag-
+netic species in the physiological environment, e.g., reactive oxygen species (ROS) or trace
+metals, e.g., manganese can typically be found in ∼ nanomolar to micromolar concentra-
+tions fitting the highly sensitive feedback of NV-relaxometry (see Figure 2a).41,53 However,
+diamagnetic ion concentrations are typically orders of magnitude higher in the cytoplasm
+(∼ millimolar)42,43 or in electrochemistry (∼ 0.1 to 1 molar)44 which fit very well the NV-
+center’s response reported in our work (see Figure 2b). Importantly, these two effects may
+counteract if both species are present. A possible pathway to differentiate between these
+two could be recording single and double quantum experiments, which are only affected by
+diamagnetic species in the latter case (see Figure 4 and Supplementary Note 7). Therefore,
+we propose to probe both relaxation times in future relaxometry studies. We envision ap-
+plications ranging from probing electrochemical interfaces54 to nanoscale ion sensing in cells
+or neuroscience, where changes in the membrane potential occur as a result of concentration
+gradients of diamagnetic ions.55–57
+Methods
+Sample Preparation
+Two 2 × 2 × 0.5 mm electronic grade diamond samples (natural 13C abundance, Element
+Six) were implanted with 15N at an energy of 2.5 keV or 4 keV, an off-axis tilt of 7° and a
+20
+
+fluence of 2×1012 cm−2 by Innovion and annealed according to Bucher et al.18 Before exper-
+iments are conducted, the diamonds are cleaned with a tri-acid cleaning protocol according
+to Brown et al.:38 Samples are boiled in equal parts of sulfuric, nitric and perchloric acid at
+a temperature of 280 °C for two hours. This cleaning procedure is also applied before the
+deposition of aluminium oxide (Al2O3) on the diamond.
+Preparation of Electrolyte Solutions
+For the measurements where pure water is used, deionized water with a resistivity of
+18.2 MΩ·cm at 25 °C (Merck Millipore) is utilized.
+Sodium chloride (NaCl, Merck 106404) is prepared in a 1 M stock solution, where NaCl is
+dissolved in deionized water. Before the experiments, NaCl is diluted from the stock solution
+to obtain 500, 250, 100, 50, 10 and 1 mM concentrated solutions. The other salt solutions
+used within this work are prepared in the same manner.
+Atomic Layer Deposition (ALD)
+The 2.5 keV 15N implanted diamond is coated with an aluminium oxide (Al2O3) thin
+film by ALD according to Liu et al.13 The deposition includes 10 cycles of alternated sample
+exposure to trimethyl aluminium (TMA) and H2O. This procedure results in a film thickness
+of ∼ 1 nm and ensures surface termination with hydroxyl groups by exposing the diamond
+to a remote oxygen plasma within the ALD system.13,58 The Al2O3 layer can be removed
+from the diamond surface by soaking the sample overnight in 5% NaOH solution.
+Experimental Setup
+The quantum sensing setup is based on a modified version of the experiment described
+in Bucher et al.18 Before experiments are performed, the diamond is glued to a thin glass
+cover slide (48393026, VWR) together with a microfluidic device that encloses the diamond
+21
+
+edges and covers its surface, such that a volume of ∼ 0.60 µL of the sample liquid can be
+applied in a controllable way. On the other side of the cover slide a 6 mm diameter glass
+hemisphere (TECHSPEC® N-BK7 Half-Ball Lenses, Edmund Optics) is glued, in order to
+improve the fluorescence light collection efficiency. The glass cover slide is then fixed on a
+30 mm cage plate (CP4S, Thorlabs). This whole assembly is then positioned between two
+permanent magnets, that are rotated and tilted in order to align the B0 field with one of
+the four possible NV-center orientations. The distance between the two magnets can be
+adjusted in order to correspond to the working magnetic field strengths B0 (in this work:
+15, 316, 352 and 978 G). Initialization of the NV-ensemble is realized with a 532 nm laser
+(Verdi G5, Coherent) with a power of ∼ 250 mW (CW) after the AOM. The laser light is
+focused on the diamond by a Plano-Convex Lens (LA 1986-A-M, Thorlabs) in a total internal
+reflection geometry. Laser pulses are regulated by an acousto-optic modulator (Gooch and
+Housego, model 3260-220) with pulse durations of 5 µs. Photoluminescence (PL) is collected
+and focused on a large area photodiode (OE-300-SI-10, Femto Messtechnik GmbH, Berlin,
+Germany) by two condenser lenses (ACL25416U-B, Thorlabs). The excitation light is filtered
+by a long-pass optical filter (Edge Basic 647 Long Wave Pass, Semrock) placed between
+the bottom condenser lens and the photodiode. The output voltage of the photodiode is
+digitized with a data acquisition unit (USB-6229 DAQ, National Instruments). A 500 MHz
+PulseBlaster card (ESR-Pro-II, Spincore) is utilized to trigger and to time the microwave
+and light pulses used for quantum control of the NV-centers. The microwave frequencies
+are produced by a signal source (SynthHD, Windfreak Technologies, LLC.).
+Microwave
+phase control is obtained by a combination of a phase-shifting splitter (ZX10Q-2-27-S+,
+Mini-Circuits), two switches (ZASWA-2-50dRA+, Mini-Circuits) and a combiner (ZX10-2-
+42-S+, Mini-Circuits). The amplified microwave pulses (ZHL-16W-43-S+, Mini-Circuits)
+are delivered by a homebuilt microwave loop on top of the microfluidic chip. The electron
+spin resonance (ESR) frequency is used to determine the magnetic field strength B0 as well
+as the NV0,-1 resonance frequency f NV.
+22
+
+T1 Relaxometry Experiments (Single and Double Quantum)
+Single quantum (SQ) relaxometry experiments: To obtain a signal-to-noise ratio (SNR)
+as shown in Figure 1b the sequence is repeated 5,000 times for every data point.
+Each
+experiment consists of 31 data points measured in a logarithmic increasing sweep time t to
+guarantee more sampling points at short times t. Note that these parameters are also used
+for double quantum (DQ) relaxometry. For normalization and noise cancellation, the second
+half of the sequence contains a MW π0,-1-pulse, where the subscripts 0 and -1 indicate the
+initialization of the spin state from ms = 0 to ms = −1.18 The spectra are then plotted
+as the measurement result of the first half divided by the result of the second half of the
+sequence.
+Double quantum (DQ) relaxometry experiments: For a detailed discussion of the DQ
+relaxation and pulse sequence, the reader is referred to Myers et al.45 In short, the DQ pulse
+sequence (see inset of Figure 4b) consists of two consecutive measurements where MW π-
+pulses are used to control spin state initialization and readout. In both halves of the sequence
+the NV-center is initialized in ms = −1. After a sweep time t the spin state population of
+either ms = −1 (in the first part) or ms = +1 (in the second part) is read out. Dividing the
+second by the first part yields a population ratio of the two states.
+Sensitivity of T1 Relaxometry on Electrolytes
+Experiments to determine the sensitivity of T1 relaxometry measurements on para- and
+diamagnetic electrolytes are conducted for MnCl2 and NaCl solutions using the SQ relax-
+ometry pulse sequence. Probing each concentration results in a relaxation curve of which
+the T1 time is determined. The T1 time is then normalized to the one of water covering
+the diamond. Before probing any electrolyte concentration, we wash the microfluidic device
+with water to ensure equal starting conditions, i.e. a constant T1 time for water covering
+the diamond. We perform each series three times resulting in a mean T1 value for each
+concentration (see also Supplementary Note 4). Figure 2 in the main text shows the mean
+23
+
+(normalized) T1 time along with the standard deviation.
+DEER Measurements
+DEER spectra (see Figure 5b) are recorded by performing a spin-echo sequence on the
+NV-center spins with a free evolution time of 1 µs. The duration of the MW-pulse (MWDEER)
+applied to the surface dark spins is set to 200 ns and the driving frequency (f DEER) is swept
+over 90 MHz (from fDEER = 0.84 to 0.93 GHz). To obtain a SNR as shown in Figure 5b
+the sequence is repeated 10,000 times for every data point. Each experiment consists of
+67 data points in equally separated time steps and this whole experiment is repeated four
+times. Referencing for noise cancellation is achieved by alternating the last MW-pulse of the
+spin-echo sequence from π/2 to 3/2π.
+Once the resonance condition for ge = 2 is found, DEER-Rabi experiments on the surface
+dark spins are performed by sweeping the MW-pulse duration (MWDEER) during the NV
+spin-echo (see Figure 5c) as described above. The sequence is repeated 10,000 times for
+every data point.
+Each experiment consists of 101 equally spaced data points and this
+whole experiment is repeated ten times. To account for MW (MWDEER) noise, the same
+procedure is repeated 20 MHz off the resonance condition. The outcome of both on- and
+off-resonant measurements are subtracted resulting in the spectra shown in Figure 5c. After
+that, measurements of the surface dark spin population relaxation are carried out according
+to Sushkov et al. with a πds-pulse length of 24 ns.46 The sequence shown in Figure 5d is
+repeated 10,000 times for every data point. Each experiment consists of 21 data points in
+equally separated time steps. This whole experiment is then repeated 50 times. Background
+subtraction is achieved by performing the experiment in the same procedure without the
+additional MW drive (MWDEER). Subtracting the outcome of both MW-on and MW-off
+measurements then yields the spectra shown in Figure 5d.
+24
+
+Simulation of the Diamond/Water Interface
+In our simulations, we use a slab of a model diamond surface with hydrogen, hydroxyl,
+and ether surface terminations (see Figure S10d). It is a symmetric (100) surface of ∼ 1.4 nm
+with a 2 × 1 surface reconstruction pattern, exhibiting a positive electron affinity and no
+surface states inside the band gap.59 The water layer on top of the diamond (thickness
+∼ 2 nm) was constructed as follows. First, we equilibrate 74 water molecules with the clas-
+sical molecular dynamics (MD) for 5 ns in a simulation box of commensurate lateral size
+with the diamond slab. These calculations are done with the GROMACS software in the
+canonical NVT ensemble,60 using the GROMOS 54A7 force field.61 After that, we superim-
+pose the water box and the diamond surface and allow for an additional equilibration step
+of 10 ps with the ab initio MD, as implemented in the VASP package.62 We also incorporate
+∼ 1.9 nm of vacuum together with a dipole correction scheme to eliminate the interaction
+with the periodic images. This yields the simulation supercell of 1.0097 × 1.0097 × 5.3 nm3,
+which is further used in the ab initio MD calculations. Ab initio calculations are performed
+using the PBE functional63 in conjunction with the D2 dispersion correction, using a pro-
+jector augmented wave method with the kinetic energy cutoff of 370 eV. We note that the
+PBE functional provides semi-quantitative results for the electronic structure but is able to
+accurately yield the trends in the change of the electronic structure upon different surface
+terminations and environments of diamond. Further, we note that we focus on the difference
+in the electrostatic environment due to the interaction of the electrolyte with the surface
+groups, assuming no change in the microstructure of the carbon layer.
+Charge Transfer Rates
+We calculate the charge transfer rate with an expression from the Marcus theory,64 given
+as:
+kCT = 2π
+ℏ |HAB|2
+1
+√4πλkBT exp
+�
+−(λ + ∆G)2
+4πλkBT
+�
+25
+
+where HAB is the transfer integral, λ the reorganization energy, ∆G the Gibbs energy dif-
+ference due to an external field, kB the Boltzmann constant, ℏ the reduced Planck constant,
+and T the temperature. The reorganization energy is determined for a single defect (either
+a carbon vacancy or a substitutional nitrogen) in a 1000-carbon supercell. For computing
+the transfer integrals as a function of distance, we use diamond supercells of different sizes,
+varying between 64 and 1000-carbon atoms. The reorganization energies are calculated by
+the four-point scheme, while the transfer integrals are estimated at a high symmetry con-
+figuration as 1/4 of the bandwidth along the Γ-X direction. The contribution to the Gibbs
+energy is computed by solving the one dimensional Poisson equation given the experimental
+depth of the NV center.65 Noteworthy, the effect from the band bending is governed by the
+orientation of defect pairs relative to the direction of the electric field. At a reference depth
+of the NV-center, the maximum strength, corresponding to a change in the electrostatic
+potential (∆Vmax), is reached in a parallel configuration, whilst the effect is quenched to-
+wards the orthogonal arrangement. Considering a uniform distribution of the defects in our
+samples, we compute an expectation value of ∆V as ∆Vmax/2.
+Acknowledgement
+This study was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
+Foundation) - 412351169 within the Emmy Noether program. R.R. acknowledges support
+from the DFG Walter Benjamin Programme (Project RI 3319/1-1). D.B.B. acknowledges
+support from the DFG under Germany’s Excellence Strategy—EXC 2089/1—390776260 and
+the EXC-2111 390814868. M.S.B. acknowledges support from the DFG through the Munich
+Center of Quantum Science and Technology (MCQST, EXC-2111) and by BMBF (epiNV,
+13N15702). A.G. acknowledges the Hungarian NKFIH grant No. KKP129866 of the National
+Excellence Program of Quantum-coherent materials project, the support for the Quantum
+Information National Laboratory from the Ministry of Culture and Innovation of Hungary
+26
+
+(NKFIH grant No. 2022-2.1.1-NL-2022-00004), the EU EIC Pathfinder project ”QuMicro”
+(grant No. 101046911) and the EU QuantERA for the project MAESTRO. We acknowledge
+KIF¨U for awarding us access to computational resources based in Hungary.
+Author Contributions
+D.B.B., R.R. and F.A.F.-M. discovered the effect of diamagnetic electrolytes on the
+relaxation of NV-centers. D.B.B., R.R. and F.A.F.-M. designed the experiments. D.B.B.
+supervised the study. F.A.F.-M. performed the experiments and was supported by M.R.S. for
+NV-relaxometry. F.A.F.-M., R.R. and D.B.B. analyzed the data. R.D.A. built the quantum
+sensing setup and designed the microfluidic device. A.G. and A.P. incorporated theoretical
+modeling and simulations. M.S.B. and L.M.T. helped with the charge state experiments.
+All authors discussed the results and contributed to the writing of the manuscript.
+Additional Information
+Competing interests: The authors declare no competing interests.
+Data availability
+The data supporting our findings are available within the paper and the Supplementary
+Information. Additional relevant data are available from the corresponding author upon
+reasonable request.
+Code availability
+The codes used for data acquisition and processing are available from the corresponding
+author upon reasonable request.
+27
+
+Supplementary Information
+Supplementary Note 1: Fitting of T1 Single Quantum (SQ), Double
+Quantum (DQ) and Surface Dark Spin Relaxation Curves
+Recorded single quantum (SQ) and double quantum (DQ) relaxation curves are fitted
+with a biexponential function as the T1 decay exhibited two components according to prior
+work:25,27,28,32,66
+C(t) = A · exp(− 1
+T1a
+· t) + (1 − A) · exp(− 1
+T1b
+· t)
+where C is the contrast, A is the amplitude and T1a >> T1b. For completeness, relaxation
+times in the tables are given by both time constants. In agreement with prior work,25,27,32
+values of T1 in the main text are only considering the longer component T1a. However, both
+time constants are longer in all cases where diamagnetic electrolytes are measured with NV-
+relaxometry and compared to water (see Table S1). Errors and errorbars from SQ and DQ
+relaxation curves shown in tables or figures are standard deviations from the biexponential fit
+function or in case of the sensitivity experiments (see Figure 2) the standard deviation from
+three consecutive measurements. T1 time constants in the tables are given to three significant
+digits. In case of the T1,ds relaxation measurements (see Figure 5d), the relaxation curve is
+fitted to a single exponential decay: C(t) = A · exp(−
+1
+T1,ds · t).46
+Supplementary Note 2: T1 Time Constants of Measured Electrolytes
+and T1 Time Magnetic Field Dependence for Pure Water/NaCl
+(500 mM)
+Table S1 and Figure S1 show the T1 time constants (T1a and T1b) and T1 relaxation
+curves of the measured electrolyte solutions in this work. Experiments are conducted with
+the relaxometry pulse sequence according to the main text.
+28
+
+Figure S1: T1 relaxation curves of water and a-f) diamagnetic electrolyte
+(500 mM) solutions as well as g) and h) paramagnetic electrolyte (1 µM) so-
+lutions covering the diamond surface. Experiments are performed at f NV =
+1.88 GHz.
+29
+
+Table S1: T1 time constants (T1a and T1b) of water and measured diamagnetic
+electrolyte solutions (500 mM) as well as paramagnetic electrolyte solutions
+(1 µM) covering the diamond surface. Experiments are performed at fNV = 1.88 GHz
+.
+Electrolyte [c = 500 mM]
+T1a [µs]
+T1b [µs]
+Water
+920 ± 170
+140 ± 60.0
+CsF
+1510 ± 250
+200 ± 60.0
+KCl
+1720 ± 300
+270 ± 90.0
+KNO3
+1360 ± 220
+240 ± 50.0
+LiCl
+2940 ± 720
+930 ± 40.0
+NaCl
+1920 ± 200
+170 ± 20.0
+CaCl2
+2920 ± 260
+460 ± 190
+MgSO4
+3600 ± 160
+310 ± 100
+AlCl3
+2070 ± 370
+360 ± 190
+Electrolyte [c = 1 µM]
+T1a [µs]
+T1b [µs]
+MnCl2
+430 ± 160
+140 ± 50.0
+Gd(NO3)3
+250 ± 15.0
+21 ± 6.00
+Further measurements of water/NaCl (500 mM) solution are performed in different mag-
+netic fields B0 (978, 352, 15 and 0 G), i.e., different resonance frequencies of the NV-center’s
+ms = 0 → ms = −1 transition (f NV = 0.131, 1.88, 2.83 and 2.87 GHz). Figure S2 shows the
+T1a time constants depending on f NV (see Supplementary Note 1 for details). In Table S2
+both time constants (T1a and T1b) are listed.
+Figure S2: T1a time constants for water and NaCl (500 mM) solution and their
+dependence on the NV0,-1 resonance frequency f NV.
+30
+
+Table S2: T1 time constants (T1a and T1b) for water and NaCl (500 mM) solution
+covering the diamond surface depending on the NV0,-1 resonance frequency
+f NV.
+T1a [µs]
+T1b [µs]
+f NV = 0.131 GHz
+Water
+1810±460
+400±60.0
+NaCl 500 mM
+2300±340
+670±130
+f NV = 1.88 GHz
+Water
+940±180
+130±20.0
+NaCl 500 mM
+1990±200
+170±20.0
+f NV = 2.83 GHz
+Water
+2660±110
+510±80.0
+NaCl 500 mM
+3970±400
+1210±410
+f NV = 2.87 GHz
+Water
+3460±630
+930±190
+NaCl 500 mM
+4830±740
+1830±510
+Supplementary Note 3: NV-Relaxometry Experiments with Differ-
+ent Organic Solvents
+Following measurements are performed in order to investigate the impact of the solvent’s
+physical properties on NV-relaxometry experiments. Therefore, we choose organic solvents
+with dielectric constants (κ) which differ significantly from the properties of water.40 The
+diamond is covered three times alternatingly with water and the organic solvent. T1 times
+of the solvents are then normalized to the T1 time of water. Figure S3 shows that the T1
+time remains unaffected by the solvent.
+31
+
+Figure S3: NV-relaxometry experiments showing the impact of the solvent’s di-
+electric constant (κ). Experiments are performed at f NV = 1.88 GHz.
+Supplementary Note 4: NV-Relaxometry Measurement Series of
+Para- and Diamagnetic Electrolyte Solutions in Increasing Con-
+centrations
+Figure S4: NV-relaxometry measurement series with increasing concentrations
+of a) MnCl2 and b) NaCl solutions. Data points are T1 times normalized to the
+T1 time of water for each series. Solid lines connect the mean values of three
+consecutive performed series. Experiments are performed at f NV = 1.88 GHz.
+The NV-relaxometry measurement series with paramagnetic MnCl2 and diamagnetic
+NaCl solutions are performed in order to determine the sensitivity of the protocol to increas-
+32
+
+ing electrolyte solutions in each case. Experiments are conducted using the SQ relaxometry
+pulse sequence (see Methods for detail). We perform each series three times resulting in a
+mean value for each concentration (see color codes in Figure S4). Figure 2 in the main text
+shows the mean (normalized) T1 time along with the standard deviation.
+Paramagnetic MnCl2 solutions decrease the T1 time in ∼ nano- to micromolar concen-
+trations with respect to water. In contrast to that, diamagnetic NaCl solutions increase the
+T1 time in ∼ millimolar concentrations.
+Supplementary Note 5: NV-Depth Dependence Measurements with
+Water/LiCl (500 mM)
+Figure S5: T1 relaxation curves of water and LiCl 500 mM solution covering
+the diamond surface. Diamonds were implanted with 15N at an energy of a)
+2.5 keV and b) 4 keV. Experiments are performed at f NV = 1.88 GHz.
+NV-relaxometry with water/LiCl (500 mM) covering the diamond is performed in order
+to investigate the impact of another diamagnetic electrolyte and to support the experiments
+with NaCl (500 mM) solution using differently deep NV-center ensembles (implanted with
+2.5 keV and 4 keV, see Figure 3b). Figure S5 and Table S3 show similar results for both
+NaCl and LiCl (500 mM) solution.
+33
+
+Table S3: T1 time constants (T1a and T1b) of water, NaCl (500 mM) and LiCl
+(500 mM) solution on the diamond surface depending on the nitrogen implan-
+tation energy. Experiments are performed at f NV = 1.88 GHz.
+Implantation energy [keV]
+T1a [µs]
+T1b [µs]
+2.5
+Water
+940 ± 180
+130 ± 20.0
+NaCl 500 mM
+1920 ± 200
+170 ± 20.0
+Water
+660 ± 180
+200 ± 50.0
+LiCl 500 mM
+2940 ± 720
+930 ± 40.0
+4
+Water
+1090 ± 190
+190 ± 30.0
+NaCl 500 mM
+1270 ± 180
+220 ± 40.0
+Water
+2750 ± 340
+380 ± 80.0
+LiCl 500 mM
+2880 ± 270
+440 ± 90.0
+Supplementary Note 6: NV-Charge State, Coherence and Dephas-
+ing Measurements
+We observe an increase of T1 by diamagnetic electrolyte solutions. However, it is known
+that NV-charge state alteration (i.e., NV0 ↔ NV–) can influence the outcome of NV-
+relaxometry measurements.67,68 For that reason, we perform NV-Rabi experiments with
+water/NaCl (500 mM) solution (see Figure S6a). Any change in the NV-Rabi contrast indi-
+cates an alteration of the NV-center’s charge state. For instance, an ionization of NV– would
+increase the proportion of NV0, thereby raising the background fluorescence and lowering
+the contrast. The NV-Rabi experiments show no difference in the outcome between water
+and the electrolyte implying a constant charge state distribution during the measurement.
+Secondly, to supplement the NV-Rabi experiments, we conduct NV-relaxometry with dis-
+tinct optical readout of the NV0 and NV– charge states and with three different laser powers
+(see Figure S6b and Figure S6c). Possible ionization of NV– in the dark or recombination
+processes would be visible as an alteration in the readout signal of the NV0 charge state
+(see Figure S6b).67,68 These measurements are carried out using the first half of the relax-
+ometry pulse sequence (i.e., without a π-pulse) and with two different optical filters. The
+647 nm long pass filter predominantly reads out the fluorescence from the NV– state and the
+600 ± 40 band pass filter mostly reads out the fluorescence from the NV0 state.69 While a T1
+34
+
+Figure S6: Pulse sequences and spectra of NV-charge state measurements. a)
+NV-Rabi experiments, b) NV-charge state measurements with selective read-
+out of the NV0 or the NV– state and c) T1 relaxation curves using three differ-
+ent laser powers.
+35
+
+fluorescence decay curve can be extracted from the measurements with the long pass filter,
+no decisive change in the NV0 state is visible using the band pass filter. Probable impact of
+the laser power on the NV–/NV0 ratio and a subsequent change in the T1 relaxation curves
+is probed with relaxometry experiments using laser powers of 25, 50 and 100 µW µm−2 (see
+Figure S6c). Both NV-Rabi and NV-charge state experiments do not show an impact on
+NV-charge state alteration on the relevant timescales of the relaxometry measurements we
+conduct herein.
+Figure S7: Pulse sequences and spectra of Ramsey and T2 Hahn-echo mea-
+surements. a) Ramsey oscillations performed at a 4 MHz detuned NV0,-1 reso-
+nance frequency f NV and b) T2 Hahn-echo experiments with water and NaCl
+(500 mM) solution covering the diamond surface at f NV = 1.88 GHz.
+Additionally, we perform Ramsey (NV-dephasing) and T2 (NV-coherence) Hahn-echo
+experiments, whose outcome is typically affected by changes in the low frequency components
+of the noise (see Figure S7a and S7b).30 Both experiments show no difference in the outcome
+for water or NaCl (500 mM) solution. However, we note that probable changes in this noise
+frequency regime might not be observable with the high-dense NV-center ensemble we use in
+this work, since the surrounding spin-bath (e.g. P1-centers or other paramagnetic impurities)
+is limiting the NV-dephasing and NV-coherence in this case.37,50
+36
+
+Supplementary Note 7: T1 Time Constants for Single Quantum and
+Double Quantum Experiments at B0 = 15 G and Zero Field ESR
+Measurements
+Figure S8: a) SQ and b) DQ relaxation curves of water and MnCl2 (100 µM) so-
+lution covering the diamond. Experiments are performed at B0 = 15 G, where
+the NV0,-1 transition is at fNV = 2.83 GHz (corresponding to a DQ transi-
+tion frequency of 80 MHz). T1,SQ decreases by 80%, whereas T1,DQ remains un-
+changed compared to water when MnCl2 solution covers the diamond.
+Single and double quantum T1 experiments of water/NaCl (500 mM) and water/MnCl2
+(100 µM) solution covering the diamond surface are performed in order to elucidate the effect
+of the electrolyte on magnetic and electric field noise. In the case of the NaCl solution, both
+T1 time constants (T 1a,SQ and T 1a,DQ) increase compared to water, indicating a reduction of
+both magnetic and electric field noise (see also Table S4). Importantly, MnCl2 only reduces
+the T1 time for the SQ relaxation, whereas the DQ transition remains unaffected compared to
+37
+
+Table S4: T1 time constants (T 1a,SQ and T 1a,DQ) for water/NaCl (500 mM) and
+water/MnCl2 (100 µM) solution covering the diamond surface. Experiments
+are performed at B0 = 15 G
+.
+T 1a,SQ [µs]
+T 1a,DQ [µs]
+Water
+2600 ± 280
+440 ± 24.0
+NaCl 500 mM
+3970 ± 400
+1250 ± 120
+Water
+2000 ± 340
+410 ± 71
+MnCl2 100 µM
+390 ± 71.0
+390 ± 78.0
+water (see also Table S4). This indicates an exclusive impact of the paramagnetic electrolyte
+on magnetic field noise. However, we note that probing MnCl2 in higher (> 100 µM) concen-
+trations would lead to a collapse of the NV-center’s T1 time (see also Figure 2). Therefore, a
+final statement on the impact of higher concentrated paramagnetic electrolyte solutions on
+the DQ (as well as the SQ) relaxation cannot be made.
+Additionally, we investigate the static electric field environment of the NV-center, i.e.,
+charges within the diamond and adjacent to the NV-center (e.g., N+ and NV–).70 Therefore,
+we measure ESR at zero magnetic field (here the earth’s magnetic field ∼ 0.5 G), because
+any difference in the static electric field in the proximity of the NV-center with respect to
+water or the electrolyte solution covering the surface would induce a shifting and/or splitting
+of the ms = ±1 states apparent in the ESR spectra.70 Figure S9 shows no significant change
+of the ESR resonance lines for the exposure of water or electrolyte solution, indicating that
+static electric fields do not contribute.
+38
+
+Figure S9: ESR experiments at zero magnetic field with water and NaCl
+(500 mM) solution covering the diamond surface.
+39
+
+Supplementary Note 8: Results of DFT-PBE Ab Initio Molecular
+Dynamics Simulations
+Figure S10: a) Band alignment of the water layer and the model diamond sur-
+face. b) Distribution of interfacial dipoles, sampled from the MD trajectories
+for three different compositions of the interface and solvent. c) Average elec-
+trostatic potentials and vacuum level shifts (VLS) computed for the configu-
+rations corresponding to the middle of the distributions in b). Vertical lines
+show the parts of the simulation box, spanned by diamond (C), water or aque-
+ous NaCl solution and vacuum. d) Structures of the model diamond surface
+before and after adding a COOH group.
+40
+
+a)
+Band alignment
+b)
+600
+model
+-0.86 eV
+Intensity [arb.units]
+model+COOH
+CBM
+500
+model+CoOo'Na
+LUMO
+-1.94 eV
+400
+300
+200
+VBM
+-5.20 eV
+100
+HOMO
+-6.32 eV
+Diamond
+0
+-2
+-1
+0
+c)
+Water
+1
+2
+Dipole moment [at.units]
+10
+Electrostatic potential [eV]
+model+ Na*cl
+model +COOH
+model+Coo'Nat
+5
+士
+VLS=1.1
+VLS=-0.6
+VLS=-1.9
+0
+-5
+-10
+-15
+c
+H20
+-20
+NaCI
+-25
+WW
+-10
+0
+10 20 30
+40
+50
+-10
+10 20 30
+4050
+10
+0
+10
+20 30 40
+50
+Z-coordinate [A]
+Z-coordinate [A]
+Z-coordinate [A]
+d)
+model
+model+COOHReferences
+(1) Acosta, V. M.; Bauch, E.; Ledbetter, M. P.; Waxman, A.; Bouchard, L.-S.; Budker, D.
+Temperature Dependence of the Nitrogen-Vacancy Magnetic Resonance in Diamond.
+Physical Review Letters 2010, 104, 070801.
+(2) Iv´ady, V.; Simon, T.; Maze, J. R.; Abrikosov, I. A.; Gali, A. Pressure and temperature
+dependence of the zero-field splitting in the ground state of NV centers in diamond: A
+first-principles study. Physical Review B 2014, 90, 235205.
+(3) Teissier, J.; Barfuss, A.; Appel, P.; Neu, E.; Maletinsky, P. Strain Coupling of a
+Nitrogen-Vacancy Center Spin to a Diamond Mechanical Oscillator. Physical Review
+Letters 2014, 113, 020503.
+(4) Dolde, F.; Fedder, H.; Doherty, M. W.; N¨obauer, T.; Rempp, F.; Balasubramanian, G.;
+Wolf, T.; Reinhard, F.; Hollenberg, L. C. L.; Jelezko, F.; Wrachtrup, J. Electric-field
+sensing using single diamond spins. Nature Physics 2011, 7, 459–463.
+(5) Balasubramanian, G.; Chan, I. Y.; Kolesov, R.; Al-Hmoud, M.; Tisler, J.; Shin, C.;
+Kim, C.; Wojcik, A.; Hemmer, P. R.; Krueger, A.; Hanke, T.; Leitenstorfer, A.; Brats-
+chitsch, R.; Jelezko, F.; Wrachtrup, J. Nanoscale imaging magnetometry with diamond
+spins under ambient conditions. Nature 2008, 455, 648–651.
+(6) Maze, J. R.; Stanwix, P. L.; Hodges, J. S.; Hong, S.; Taylor, J. M.; Cappellaro, P.;
+Jiang, L.; Dutt, M. V. G.; Togan, E.; Zibrov, A. S.; Yacoby, A.; Walsworth, R. L.;
+Lukin, M. D. Nanoscale magnetic sensing with an individual electronic spin in diamond.
+Nature 2008, 455, 644–647.
+(7) Glenn, D. R.; Lee, K.; Park, H.; Weissleder, R.; Yacoby, A.; Lukin, M. D.; Lee, H.;
+Walsworth, R. L.; Connolly, C. B. Single-cell magnetic imaging using a quantum dia-
+mond microscope. Nature Methods 2015, 12, 736–738.
+41
+
+(8) Lovchinsky, I.; Sushkov, A. O.; Urbach, E.; de Leon, N. P.; Choi, S.; De Greve, K.;
+Evans, R.;
+Gertner, R.;
+Bersin, E.;
+Muller, C.;
+McGuinness, L.;
+Jelezko, F.;
+Walsworth, R. L.; Park, H.; Lukin, M. D. Nuclear magnetic resonance detection and
+spectroscopy of single proteins using quantum logic. Science 2016, 351, 836–841.
+(9) M¨uller, C.; Kong, X.; Cai, J.-M.; Melentijevi´c, K.; Stacey, A.; Markham, M.;
+Twitchen, D.; Isoya, J.; Pezzagna, S.; Meijer, J.; Du, J. F.; Plenio, M. B.; Nayde-
+nov, B.; McGuinness, L. P.; Jelezko, F. Nuclear magnetic resonance spectroscopy with
+single spin sensitivity. Nature Communications 2014, 5, 4703.
+(10) Sushkov, A. O.; Lovchinsky, I.; Chisholm, N.; Walsworth, R. L.; Park, H.; Lukin, M. D.
+Magnetic Resonance Detection of Individual Proton Spins Using Quantum Reporters.
+Physical Review Letters 2014, 113, 197601.
+(11) Bucher, D. B. Principles of nano-and microscale NMR-spectroscopy with NV-diamond
+sensors. eMagRes 2019, 8, 363–370.
+(12) Lovchinsky, I.; Sanchez-Yamagishi, J. D.; Urbach, E. K.; Choi, S.; Fang, S.; Ander-
+sen, T. I.; Watanabe, K.; Taniguchi, T.; Bylinskii, A.; Kaxiras, E.; Kim, P.; Park, H.;
+Lukin, M. D. Magnetic resonance spectroscopy of an atomically thin material using a
+single-spin qubit. Science 2017, 355, 503–507.
+(13) Liu, K. S.; Henning, A.; Heindl, M. W.; Allert, R. D.; Bartl, J. D.; Sharp, I. D.;
+Rizzato, R.; Bucher, D. B. Surface NMR using quantum sensors in diamond. Proceedings
+of the National Academy of Sciences 2022, 119, e2111607119.
+(14) Allert, R. D.; Briegel, K. D.; Bucher, D. B. Advances in nano- and microscale NMR
+spectroscopy using diamond quantum sensors. Chemical Communications 2022, 58,
+8165–8181.
+(15) Allert, R. D.; Bruckmaier, F.; Neuling, N. R.; Freire-Moschovitis, F. A.; Liu, K. S.;
+42
+
+Schrepel, C.; Sch¨atzle, P.; Knittel, P.; Hermans, M.; Bucher, D. B. Microfluidic quantum
+sensing platform for lab-on-a-chip applications. Lab on a Chip 2022, 22, 4831–4840.
+(16) Shi, F.; Zhang, Q.; Wang, P.; Sun, H.; Wang, J.; Rong, X.; Chen, M.; Ju, C.; Rein-
+hard, F.; Chen, H.; Wrachtrup, J.; Wang, J.; Du, J. Single-protein spin resonance
+spectroscopy under ambient conditions. Science 2015, 347, 1135–1138.
+(17) Schirhagl, R.; Chang, K.; Loretz, M.; Degen, C. L. Nitrogen-Vacancy Centers in Dia-
+mond: Nanoscale Sensors for Physics and Biology. Annual Review of Physical Chemistry
+2014, 65, 83–105.
+(18) Bucher, D. B.; Aude Craik, D. P.; Backlund, M. P.; Turner, M. J.; Ben Dor, O.;
+Glenn, D. R.; Walsworth, R. L. Quantum diamond spectrometer for nanoscale NMR
+and ESR spectroscopy. Nature Protocols 2019, 14, 2707–2747.
+(19) Kim, M.; Mamin, H. J.; Sherwood, M. H.; Ohno, K.; Awschalom, D. D.; Rugar, D.
+Decoherence of Near-Surface Nitrogen-Vacancy Centers Due to Electric Field Noise.
+Physical Review Letters 2015, 115, 087602.
+(20) Stacey, A.; Dontschuk, N.; Chou, J.; Broadway, D. A.; Schenk, A. K.; Sear, M. J.;
+Tetienne, J.; Hoffman, A.; Prawer, S.; Pakes, C. I.; Tadich, A.; de Leon, N. P.; Gali, A.;
+Hollenberg, L. C. L. Evidence for Primal sp 2 Defects at the Diamond Surface: Can-
+didates for Electron Trapping and Noise Sources. Advanced Materials Interfaces 2019,
+6, 1801449.
+(21) Sangtawesin, S. et al. Origins of Diamond Surface Noise Probed by Correlating Single-
+Spin Measurements with Surface Spectroscopy. Physical Review X 2019, 9, 031052.
+(22) Romach, Y.; M¨uller, C.; Unden, T.; Rogers, L. J.; Isoda, T.; Itoh, K. M.; Markham, M.;
+Stacey, A.; Meijer, J.; Pezzagna, S.; Naydenov, B.; McGuinness, L. P.; Bar-Gill, N.;
+Jelezko, F. Spectroscopy of Surface-Induced Noise Using Shallow Spins in Diamond.
+Physical Review Letters 2015, 114, 017601.
+43
+
+(23) Chrostoski, P.; Sadeghpour, H. R.; Santamore, D. H. Electric Noise Spectra of a Near-
+Surface Nitrogen-Vacancy Center in Diamond with a Protective Layer. Physical Review
+Applied 2018, 10, 064056.
+(24) Degen, C. L.; Reinhard, F.; Cappellaro, P. Quantum sensing. Reviews of Modern
+Physics 2017, 89, 035002.
+(25) Steinert, S.; Ziem, F.; Hall, L. T.; Zappe, A.; Schweikert, M.; G¨otz, N.; Aird, A.; Bala-
+subramanian, G.; Hollenberg, L.; Wrachtrup, J. Magnetic spin imaging under ambient
+conditions with sub-cellular resolution. Nature Communications 2013, 4, 1607.
+(26) Mzyk, A.; Sigaeva, A.; Schirhagl, R. Relaxometry with Nitrogen Vacancy (NV) Centers
+in Diamond. Accounts of Chemical Research 2022, 55, 3572–3580.
+(27) Perona Mart´ınez, F.; Nusantara, A. C.; Chipaux, M.; Padamati, S. K.; Schirhagl, R.
+Nanodiamond Relaxometry-Based Detection of Free-Radical Species When Produced in
+Chemical Reactions in Biologically Relevant Conditions. ACS Sensors 2020, 5, 3862–
+3869.
+(28) Li, R.; Vedelaar, T.; Mzyk, A.; Morita, A.; Padamati, S. K.; Schirhagl, R. Following
+Polymer Degradation with Nanodiamond Magnetometry. ACS Sensors 2022, 7, 123–
+130.
+(29) Jarmola, A.; Acosta, V. M.; Jensen, K.; Chemerisov, S.; Budker, D. Temperature- and
+Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensem-
+bles in Diamond. Physical Review Letters 2012, 108, 197601.
+(30) Rosskopf, T.; Dussaux, A.; Ohashi, K.; Loretz, M.; Schirhagl, R.; Watanabe, H.;
+Shikata, S.; Itoh, K. M.; Degen, C. L. Investigation of Surface Magnetic Noise by
+Shallow Spins in Diamond. Physical Review Letters 2014, 112, 147602.
+44
+
+(31) Nie, L.; Nusantara, A. C.; Damle, V. G.; Baranov, M. V.; Chipaux, M.; Reyes-San-
+Martin, C.; Hamoh, T.; Epperla, C. P.; Guricova, M.; Cigler, P.; van den Bogaart, G.;
+Schirhagl, R. Quantum Sensing of Free Radicals in Primary Human Dendritic Cells.
+Nano Letters 2022, 22, 1818–1825.
+(32) Ziem, F. C.; G¨otz, N. S.; Zappe, A.; Steinert, S.; Wrachtrup, J. Highly Sensitive Detec-
+tion of Physiological Spins in a Microfluidic Device. Nano Letters 2013, 13, 4093–4098.
+(33) Ermakova, A.; Pramanik, G.; Cai, J.-M.; Algara-Siller, G.; Kaiser, U.; Weil, T.;
+Tzeng, Y.-K.; Chang, H. C.; McGuinness, L. P.; Plenio, M. B.; Naydenov, B.; Jelezko, F.
+Detection of a Few Metallo-Protein Molecules Using Color Centers in Nanodiamonds.
+Nano Letters 2013, 13, 3305–3309.
+(34) Fujisaku, T.; Tanabe, R.; Onoda, S.; Kubota, R.; Segawa, T. F.; So, F. T.-K.;
+Ohshima, T.; Hamachi, I.; Shirakawa, M.; Igarashi, R. pH Nanosensor Using Electronic
+Spins in Diamond. ACS Nano 2019, 13, 11726–11732.
+(35) Simpson, D. A.; Ryan, R. G.; Hall, L. T.; Panchenko, E.; Drew, S. C.; Petrou, S.;
+Donnelly, P. S.; Mulvaney, P.; Hollenberg, L. C. L. Electron paramagnetic resonance
+microscopy using spins in diamond under ambient conditions. Nature Communications
+2017, 8, 458.
+(36) Pham, L. M.; DeVience, S. J.; Casola, F.; Lovchinsky, I.; Sushkov, A. O.; Bersin, E.;
+Lee, J.; Urbach, E.; Cappellaro, P.; Park, H.; Yacoby, A.; Lukin, M.; Walsworth, R. L.
+NMR technique for determining the depth of shallow nitrogen-vacancy centers in dia-
+mond. Physical Review B 2016, 93, 045425.
+(37) Henshaw, J.; Kehayias, P.; Saleh Ziabari, M.; Titze, M.; Morissette, E.; Watanabe, K.;
+Taniguchi, T.; Li, J. I. A.; Acosta, V. M.; Bielejec, E. S.; Lilly, M. P.; Mounce, A. M.
+Nanoscale solid-state nuclear quadrupole resonance spectroscopy using depth-optimized
+nitrogen-vacancy ensembles in diamond. Applied Physics Letters 2022, 120, 174002.
+45
+
+(38) Brown, K. J.; Chartier, E.; Sweet, E. M.; Hopper, D. A.; Bassett, L. C. Cleaning
+diamond surfaces using boiling acid treatment in a standard laboratory chemical hood.
+Journal of Chemical Health & Safety 2019, 26, 40–44.
+(39) Li, C.; Zhang, X.; Oliveira, E. F.; Puthirath, A. B.; Neupane, M. R.; Weil, J. D.;
+Birdwell, A. G.; Ivanov, T. G.; Kong, S.; Gray, T.; Kannan, H.; Biswas, A.; Vajtai, R.;
+Galvao, D. S.; Ajayan, P. M. Systematic comparison of various oxidation treatments
+on diamond surface. Carbon 2021, 182, 725–734.
+(40) Seyferth, D. Organic Solvents. Physical Properties and Methods of Purification. Fourth
+Edition. Volume 11 of Weissberger’s ”Techniques of Chemistry”. Organometallics 1987,
+6, 1375–1376.
+(41) Flora, S. J. Biomarkers in Toxicology; Elsevier, 2014; pp 485–519.
+(42) Flowers, T. J.; Munns, R.; Colmer, T. D. Sodium chloride toxicity and the cellular
+basis of salt tolerance in halophytes. Annals of Botany 2015, 115, 419–431.
+(43) Ladenson, J. H.; Apple, F. S.; Aguanno, J. J.; Koch, D. D. Sodium measurements in
+multiple myeloma: two techniques compared. Clinical Chemistry 1982, 28, 2383–2386.
+(44) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications,
+2nd ed.; Wiley, 2001.
+(45) Myers, B. A.; Ariyaratne, A.; Jayich, A. C. B. Double-Quantum Spin-Relaxation Limits
+to Coherence of Near-Surface Nitrogen-Vacancy Centers. Physical Review Letters 2017,
+118, 197201.
+(46) Sushkov, A. O.; Lovchinsky, I.; Chisholm, N.; Walsworth, R. L.; Park, H.; Lukin, M. D.
+Magnetic Resonance Detection of Individual Proton Spins Using Quantum Reporters.
+Physical Review Letters 2014, 113, 197601.
+46
+
+(47) Schlipf, L.; Oeckinghaus, T.; Xu, K.; Dasari, D. B. R.; Zappe, A.; de Oliveira, F. F.;
+Kern, B.; Azarkh, M.; Drescher, M.; Ternes, M.; Kern, K.; Wrachtrup, J.; Finkler, A. A
+molecular quantum spin network controlled by a single qubit. Science Advances 2017,
+3.
+(48) Mamin, H. J.; Sherwood, M. H.; Rugar, D. Detecting external electron spins using
+nitrogen-vacancy centers. Physical Review B 2012, 86, 195422.
+(49) Bluvstein, D.; Zhang, Z.; McLellan, C. A.; Williams, N. R.; Jayich, A. C. B. Extending
+the Quantum Coherence of a Near-Surface Qubit by Coherently Driving the Paramag-
+netic Surface Environment. Physical Review Letters 2019, 123, 146804.
+(50) Barry, J. F.; Schloss, J. M.; Bauch, E.; Turner, M. J.; Hart, C. A.; Pham, L. M.;
+Walsworth, R. L. Sensitivity optimization for NV-diamond magnetometry. Reviews of
+Modern Physics 2020, 92, 015004.
+(51) Zhang, Z.; Joos, M.; Bluvstein, D.; Lyu, Y.; Jayich, A. C. B. Reporter-spin-assisted T1
+relaxometry. 2022, arXiv:2208.11470 [quant–ph].
+(52) F´avaro de Oliveira, F.; Antonov, D.; Wang, Y.; Neumann, P.; Momenzadeh, S. A.;
+H¨außermann, T.; Pasquarelli, A.; Denisenko, A.; Wrachtrup, J. Tailoring spin defects
+in diamond by lattice charging. Nature Communications 2017, 8, 15409.
+(53) Sies, H.; Jones, D. P. Reactive oxygen species (ROS) as pleiotropic physiological sig-
+nalling agents. Nature Reviews Molecular Cell Biology 2020, 21, 363–383.
+(54) Favaro, M.; Jeong, B.; Ross, P. N.; Yano, J.; Hussain, Z.; Liu, Z.; Crumlin, E. J. Unrav-
+elling the electrochemical double layer by direct probing of the solid/liquid interface.
+Nature Communications 2016, 7, 12695.
+(55) Kaufmann, S.; Simpson, D. A.; Hall, L. T.; Perunicic, V.; Senn, P.; Steinert, S.;
+McGuinness, L. P.;
+Johnson, B. C.;
+Ohshima, T.;
+Caruso, F.;
+Wrachtrup, J.;
+47
+
+Scholten, R. E.; Mulvaney, P.; Hollenberg, L. Detection of atomic spin labels in a lipid
+bilayer using a single-spin nanodiamond probe. Proceedings of the National Academy
+of Sciences 2013, 110, 10894–10898.
+(56) Hall, L. T.; Hill, C. D.; Cole, J. H.; St¨adler, B.; Caruso, F.; Mulvaney, P.; Wrachtrup, J.;
+Hollenberg, L. C. L. Monitoring ion-channel function in real time through quantum
+decoherence. Proceedings of the National Academy of Sciences 2010, 107, 18777–18782.
+(57) Szatm´ari, D.; S´ark´any, P.; Kocsis, B.; Nagy, T.; Miseta, A.; Bark´o, S.; Longauer, B.;
+Robinson, R. C.; Nyitrai, M. Intracellular ion concentrations and cation-dependent
+remodelling of bacterial MreB assemblies. Scientific Reports 2020, 10, 12002.
+(58) Henning, A.; Bartl, J. D.; Zeidler, A.; Qian, S.; Bienek, O.; Jiang, C.; Paulus, C.;
+Rieger, B.; Stutzmann, M.; Sharp, I. D. Aluminum Oxide at the Monolayer Limit via
+Oxidant-Free Plasma-Assisted Atomic Layer Deposition on GaN. Advanced Functional
+Materials 2021, 31, 2101441.
+(59) Kaviani, M.; De´ak, P.; Aradi, B.; Frauenheim, T.; Chou, J.-P.; Gali, A. Proper Sur-
+face Termination for Luminescent Near-Surface NV Centers in Diamond. Nano Letters
+2014, 14, 4772–4777.
+(60) Berendsen, H.; van der Spoel, D.; van Drunen, R. GROMACS: A message-passing par-
+allel molecular dynamics implementation. Computer Physics Communications 1995,
+91, 43–56.
+(61) Schmid, N.; Eichenberger, A. P.; Choutko, A.; Riniker, S.; Winger, M.; Mark, A. E.;
+van Gunsteren, W. F. Definition and testing of the GROMOS force-field versions 54A7
+and 54B7. European Biophysics Journal 2011, 40, 843–856.
+(62) Kresse, G.; Furthm¨uller, J. Efficient iterative schemes for ab initio total-energy calcu-
+lations using a plane-wave basis set. Physical Review B 1996, 54, 11169–11186.
+48
+
+(63) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made
+Simple. Physical Review Letters 1996, 77, 3865–3868.
+(64) Marcus, R. A. On the Theory of Oxidation-Reduction Reactions Involving Electron
+Transfer. I. The Journal of Chemical Physics 1956, 24, 966–978.
+(65) Broadway, D. A.; Dontschuk, N.; Tsai, A.; Lillie, S. E.; Lew, C. T.-K.; McCallum, J. C.;
+Johnson, B. C.; Doherty, M. W.; Stacey, A.; Hollenberg, L. C. L.; Tetienne, J.-P. Spatial
+mapping of band bending in semiconductor devices using in situ quantum sensors.
+Nature Electronics 2018, 1, 502–507.
+(66) Rioux, J. A.; Levesque, I. R.; Rutt, B. K. Biexponential longitudinal relaxation in white
+matter: Characterization and impact on T1 mapping with IR-FSE and MP2RAGE.
+Magnetic Resonance in Medicine 2016, 75, 2265–2277.
+(67) Bluvstein, D.; Zhang, Z.; Jayich, A. C. B. Identifying and Mitigating Charge Insta-
+bilities in Shallow Diamond Nitrogen-Vacancy Centers. Physical Review Letters 2019,
+122, 076101.
+(68) Dhomkar, S.; Jayakumar, H.; Zangara, P. R.; Meriles, C. A. Charge Dynamics in near-
+Surface, Variable-Density Ensembles of Nitrogen-Vacancy Centers in Diamond. Nano
+Letters 2018, 18, 4046–4052.
+(69) Doherty, M. W.; Manson, N. B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.; Hollen-
+berg, L. C. The nitrogen-vacancy colour centre in diamond. Physics Reports 2013,
+528, 1–45.
+(70) Mittiga, T.; Hsieh, S.; Zu, C.; Kobrin, B.; Machado, F.; Bhattacharyya, P.; Rui, N. Z.;
+Jarmola, A.; Choi, S.; Budker, D.; Yao, N. Y. Imaging the Local Charge Environment
+of Nitrogen-Vacancy Centers in Diamond. Physical Review Letters 2018, 121, 246402.
+49
+

YNFQT4oBgHgl3EQfdzaO/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f6b216a73a369bdf832b9621c567a83e4a6115a2568d6a8d1d8245870406ab55
+size 2949165

ZNAyT4oBgHgl3EQfWveE/content/tmp_files/2301.00169v1.pdf.txt

@@ -0,0 +1,2171 @@
+Generative Graph Neural Networks for Link Prediction
+Xingping Xiana, Tao Wua,∗, Xiaoke Mab, Shaojie Qiaoc, Yabin Shaod, Chao Wange, Lin Yuana, Yu Wua
+aSchool of Cybersecurity and Information Law, Chongqing University of Posts and Telecommunications, Chongqing, China.
+bSchool of Computer Science and Technology, XiDian University, XiAn, China.
+cSchool of Software Engineering, Chengdu University of Information Technology, Chengdu, China.
+dSchool of Science, Chongqing University of Posts and Telecommunications, Chongqing, China.
+eSchool of Computer and Information Science, Chongqing Normal University, Chongqing, China.
+Abstract
+Inferring missing links or detecting spurious ones based on observed graphs, known as link prediction, is a long-standing challenge
+in graph data analysis. With the recent advances in deep learning, graph neural networks have been used for link prediction and
+have achieved state-of-the-art performance. Nevertheless, existing methods developed for this purpose are typically discriminative,
+computing features of local subgraphs around two neighboring nodes and predicting potential links between them from the perspec-
+tive of subgraph classification. In this formalism, the selection of enclosing subgraphs and heuristic structural features for subgraph
+classification significantly affects the performance of the methods. To overcome this limitation, this paper proposes a novel and rad-
+ically different link prediction algorithm based on the network reconstruction theory, called GraphLP. Instead of sampling positive
+and negative links and heuristically computing the features of their enclosing subgraphs, GraphLP utilizes the feature learning abil-
+ity of deep-learning models to automatically extract the structural patterns of graphs for link prediction under the assumption that
+real-world graphs are not locally isolated. Moreover, GraphLP explores high-order connectivity patterns to utilize the hierarchical
+organizational structures of graphs for link prediction. Our experimental results on all common benchmark datasets from different
+applications demonstrate that the proposed method consistently outperforms other state-of-the-art methods. Unlike the discriminative
+neural network models used for link prediction, GraphLP is generative, which provides a new paradigm for neural-network-based link
+prediction. The code is available at https://github.com/star4455/GraphLP.
+Keywords: Graph Machine Learning, Graph Neural Networks, Link Prediction, Structural Patterns, Network Reconstruction.
+1. Introduction
+Graphs provide an elegant representation for characterizing
+entities and their interrelations in complex systems. Given that
+real-world graphs can usually only be partially observed and
+are often noisy, link prediction aimed at inferring missing and
+spurious links based on observed graphs is a paradigmatic and
+fundamental problem across many scientific domains, including
+knowledge graph completion [52], experimental design in bio-
+logical networks [6], fake account detection in online social net-
+works [24], and product recommendation on e-commerce web-
+sites [31].
+To address the link prediction problem, numerous heuristic
+methods have been proposed, including local indices such as
+Common Neighbors (CN) [2], and Resource Allocation (RA)
+[61], global indices such as Katz [25], and SimRank [23], and
+quasi-local indices such as the Local Path Index (LP) [61]. How-
+ever, heuristic methods have a strong assumption on when two
+nodes are likely to be linked in real-world graphs and lack uni-
+versal applicability to diverse areas [8]. Subsequently, statistical
+learning-based algorithms have been proposed to obtain ground-
+breaking results, such as maximum likelihood-based hierarchi-
+cal structure model [13], stochastic block model [21], matrix
+factorization-based link prediction method [37], Linear Opti-
+mization (LO) link prediction method [38], and Low Frobenius
+norm-based Link Prediction (LFLP) method [53]. With the pro-
+posal of network representation learning, various network em-
+∗Corresponding author.
+Email addresses: [email protected] (Xingping Xian),
+[email protected] (Tao Wu)
+bedding algorithms have been put forth so that the likelihood of
+a non-observed links can be estimated based on the proximity
+of nodes in low-dimensional vector space, including LINE [44],
+Node2Vec [20], and DNGR [10].
+Recently, driven by the dramatic advances in deep learning
+techniques, neural networks have gradually been used to solve
+the link prediction problem. [56] trained a fully-connected neu-
+ral network on the enclosing subgraphs of target links for link
+prediction, wherein a Weisfeiler-Lehman (WL) algorithm-based
+graph labeling mechanism was proposed to encode subgraphs.
+Based on the enclosing subgraphs extracted around links, [57]
+trained a Graph Neural Network (GNN) for link prediction to
+achieve a performance comparable to that of heuristic meth-
+ods. Along this line of research, [36] encoded subgraphs into
+random-walk transition probabilities and then computed fea-
+tures using these probabilities to classify positive and negative
+links.
+Although these subgraph classification-based methods
+have achieved state-of-the-art link prediction performance, the
+prediction results are found to be considerably affected by the
+extraction process of the k-hop enclosing subgraphs and the
+graph structure features for them. For example, in representa-
+tion learning on graphs [55], the range of enclosing subgraphs
+strongly depends on the graph structure, and the effective range
+should be different for subgraphs with varying properties.
+Typically, from the perspective of subgraph classification, link
+prediction methods treat subgraphs in real-world graphs inde-
+pendently and equivalently. That is, the global structural in-
+formation of real-world graphs is totally neglected during this
+process.
+However, extensive empirical analyses indicate that
+real-world graphs are not locally isolated but globally relevant
+Preprint submitted to Journal of LATEX Templates
+January 3, 2023
+arXiv:2301.00169v1  [cs.SI]  31 Dec 2022
+
+Figure 1: An illustrative example depicting the global and high-order organizations of real-world graphs. (a) Gene network for C.
+elegans [12]. (b) Representative hierarchical star-like structure [45]. (c) Representative hierarchical modular organization [39]. (b) and
+(c) depict the representative structural patterns of real-world graphs such as (a).
+[35, 51]; here, nodes and edges naturally portray different struc-
+tural roles and contribute differently to the global organization of
+real-world graphs [53, 54]. Moreover, subgraph classification-
+based link prediction methods assume that real-world graphs ex-
+hibit low-order connectivity patterns and can be captured at the
+level of individual nodes and edges. However, empirical studies
+have discovered that real-world graphs exhibit high-order orga-
+nizations at the level of small subgraphs, which are recursively
+grouped into a hierarchical structure [7, 3]. An illustrative ex-
+ample of the global and high-order organization in real-world
+graphs is depicted in Figure 1. Hence, two challenges need to be
+addressed for link prediction: (i) how to learn good representa-
+tion preserving both local and global graph structural features?
+and (ii) how to characterize and utilize hierarchical structure pat-
+terns?
+To address these challenges, instead of predicting poten-
+tial links through subgraph classification, this study designs a
+novel generative and multi-order GNN for link prediction, called
+GraphLP. Evidently, real-world graphs share some global prop-
+erties, such as low-rank and sparsity, that can be used to pro-
+vide guidance for graph learning.
+Hence, motivated by the
+network reconstruction theory [21], GraphLP defines a self-
+representation model-based collaborative inference operation to
+refine the observed graphs globally, which assumes that the orig-
+inal graph can be reconstructed utilizing the correlation between
+subgraph patterns.
+Assuming that the paths between a pair
+of nodes provide evidence for the existence of potential links,
+GraphLP extracts the local structural information via a high-
+order connectivity operation on the observed graphs. Thus, ev-
+ery neural network layer obtains the connectivity of node pairs
+within two-hop neighborhood, and a neural network with multi-
+ple connectivity layers captures the degree of connectivity be-
+tween node pairs with various path lengths.
+Meanwhile, the
+weighted adjacency matrices generated by the connectivity op-
+eration in every neural network layer reflect the multi-order con-
+nectivity pattern in the graphs. Further, the hierarchical organi-
+zational structure of real-world graphs is explored by applying
+a collaborative inference operation. The contributions of this
+study can be summarized as follows:
+• Generative framework.
+Rather than subgraph classi-
+fication based discriminative schemes, a novel network
+reconstruction-based generative GNN is proposed for link
+prediction, which provides a new paradigm for the applica-
+tion of neural networks in link prediction problem.
+• End-to-end learning. Instead of designing heuristic graph
+structural features for subgraph representation, local and
+global structural patterns are extracted and fused in an end-
+to-end fashion for link prediction.
+• Algorithm. A novel collaborative inference operation and
+high-order connectivity computation mechanism are devel-
+oped to characterize the structural patterns in real-world
+graphs at different scales.
+• Experiment. Extensive experiments on real-world datasets
+from different areas reveal that the proposed method,
+GraphLP, achieves promising performance and consistently
+outperforms other state-of-the-art methods.
+Paper Organization. The rest of this work is organized as
+follows. Section 2 discusses related studies. Section 3 presents
+the problem definitions and describes the preliminaries. Section
+4 describes the proposed method. Section 5 presents the experi-
+mental results, and finally, Section 6 presents the conclusion and
+discussion.
+2. Related Work
+GNNs and link prediction task have been extensively investi-
+gated in recent years. A brief review of related studies is pro-
+vided in this section.
+2.1. Graph Neural Networks
+Owing to their potential in modeling the complex structures
+of non-Euclidean graphs, GNNs have achieved state-of-the-art
+performance on almost all graph-based tasks, such as node clas-
+sification, graph classification, link prediction. Based on differ-
+ent theories and perspectives, a plethora of different GNNs have
+2
+
+(a)
+(b)
+(c)
+8.80been proposed over the years. Generally, GNNs can be divided
+into two categories: spectral-based and spatial-based methods.
+Of these, spectral-based GNNs are types of GNNs that design
+graph convolution operators in the spectral domain using Fourier
+transform. The involved convolution operation is defined as fol-
+lows:
+f1 ∗ f2 = U[(UT f1) ⊙ (UT f2)],
+(1)
+where ⊙ denotes an element-wise product. The spectral filter is
+defined as g = UT f1, and the node signal X can be processed as
+follows:
+Z = U[g(Λ) ⊙ (UTX)] = Ug(Λ)UTX.
+(2)
+where U denotes a matrix of eigenvectors of the normalized
+Laplacian graph L = I − D− 1
+2 AD− 1
+2 = UΛUT [11]. Assum-
+ing that feature representation of node should be affected only
+by its k-hop neighborhood, [16] proposed a Chebyshev poly-
+nomial based k-localized convolution and developed a convolu-
+tional neural network, ChebNet, which eliminated the need to
+compute the eigenvectors of the Laplacian. Subsequently, [50]
+simplified the Chebshev polynomial filter using its first-order
+approximation and proposed the popular spectral-based method
+called Graph Convolutional Networks (GCNs). Notably, spatial-
+based GNNs define graph convolution operator based on graph
+topology wherein the feature vectors of node’s neighbors are ag-
+gregated via a permutation-invariant function. Specifically, [22]
+proposed a GraphSAGE approach that sampled fixed size neigh-
+borhood nodes and used max pooling, mean pooling, and LSTM
+pooling scheme to aggregate neighbor information. Considering
+the different weights of node’s neighbors, [46] proposed a Graph
+Attention Network (GAT) algorithm to calculate attention coeffi-
+cient and then aggregated the neighborhood information. Other
+related models include PATCHY-SAN [34], DCNN [4], and fur-
+ther details on GNNs can be found in the review [60].
+2.2. Neural Networks based Link Prediction
+Following heuristic methods, matrix completion-based meth-
+ods and network embedding-based methods, neural networks
+have been gradually applied to link prediction problem and
+have achieved state-of-the-art results.
+Specifically, [56] pro-
+posed a link prediction method called Weisfeiler-Lehman Neural
+Machine (WLNM), which labeled nodes using the Weisfeiler-
+Lehman algorithm and encoded subgraphs to construct a feed-
+forward neural network-based classification model. Next, from
+the perspective of subgraph classification, [57] proposed a novel
+GNN-based link prediction framework, SEAL, to learn subgraph
+structures and node features from local enclosing subgraphs.
+Along this line, to directly leverage the topology features of lo-
+cal subgraphs, [36] proposed a new random-walk-based pool-
+ing scheme, WalkPool, and built features for subgraph classifi-
+cation. Moreover, [18] proposed a neural network-based link
+prediction method with only one-hop neighborhood informa-
+tion, which demonstrated almost equivalent performance to the
+WLNM and SEAL. Instead of subgraph classification, [8] con-
+verted the original graph into a corresponding line graph and
+solved the node classification problem for link prediction. To
+perform link prediction for general directed or undirected com-
+plex networks, [48] represented the adjacency matrices of net-
+works as binary images and developed a generative adversarial
+networks (GANs)-based method. In addition, because existing
+GNN-based methods do not scale appropriately to large graphs,
+[30] extracted sparse enclosing subgraphs based on multiple ran-
+dom walks and presented a scalable link prediction solution,
+called ScaLed. To reduce the time required to determine the
+distances between two nodes, [27] defined an anchor-based dis-
+tance and proposed a new distance-enhanced GNN method for
+link prediction.
+Among all existing methods for link prediction, the work clos-
+est to the one condidered in this study is the GANs-based method
+[48]. However, this method predicts potential links via image
+processing within the GANs framework, whereas the proposed
+method conducts link prediction via GNNs-based network re-
+construction.
+2.3. Network Structure Analysis
+Real-world graphs, also known as complex networks, are ab-
+stract representation of complex systems and have been exten-
+sively studied in the field of network science. Consequently,
+numerous studies have revealed that complex networks exhibit
+rich and diverse connectivity patterns. [32] augmented that the
+organization of real networks usually embodies both regularities
+and irregularities, where the former can be modeled and decides
+the extent to which the formation of a network can be explained.
+Notably, link predictability reflects the structural regularities in
+real-world networks and denotes the inherent difficulty of link
+prediction. [53] proposed a self-representation network model-
+based method, called NetSRE, for measuring and regulating link
+predictability of networks. [54] proposed a deep linear coding-
+based link prediction adversarial attack method by disturbing
+the underlying structural pattern of networks, which proved that
+links play global structural roles in network organization. More-
+over, [7] suggested that high-order connectivity patterns are es-
+sential for understanding the fundamental structures of networks
+and developed a framework that identified clusters of network
+motifs. [41] claimed that hierarchical structure plays an impor-
+tant role in complex systems. To prove the existence of hierar-
+chical organization, an unsupervised method for extracting the
+hierarchical organization of complex networks was introduced
+and validated.
+Although real-world graphs exhibit various structural pat-
+terns, most existing neural networks-based link prediction meth-
+ods simply assume that they are flattened and locally isolated,
+and these methods judge the existence of links explicitly based
+only on local enclosing subgraphs. With the exception of local
+structural features, this study focuses on integrating global and
+hierarchical structural patterns into neural networks for link pre-
+diction.
+3. Problem Definition and Preliminaries
+3.1. Problem Definition
+Notations.
+Let G = (V, E) denote an undirected and un-
+weighted graph, where V = {v1, · · · , vN} denotes the set of nodes
+and E = {e1, · · · , eM} denotes the set of edges. The adjacency
+matrix of graph G is denoted as A ∈ {0, 1}N×N, where Ai j = 1 if
+nodes i and j are connected and Ai j = 0 otherwise. Each edge
+e can be represented as a node pair (u, v, ), where u, v ∈ V. Let
+N (u) denote the neighbors of node u, N (u) = {v|(u, v) ∈ E}.
+Link Prediction. Given an observed graph Go = (V, Eo) that
+corresponds to the original graph G = (V, E), link prediction
+aims to infer the presence or absence of an edge between a
+pair of target nodes based on Go, thereby generating a recovered
+graph G∗ to approximate the original graph G. In particular, the
+prediction problem involves identifying a function that generates
+3
+
+a likelihood score for a pair of nodes (u, v) � E to infer the miss-
+ing link (u, v), or to produce a likelihood score for an existing
+edge (u, v) ∈ E to identify spurious links. Thus, the link predic-
+tion problem can be formulated as suv = f(u, v, A|θ), where θ
+denotes the parameter of link prediction model. In this work, Em
+and Es denote the identified missing and spurious links, respec-
+tively.
+Note that data augmentation is a set of techniques that in-
+creases the amount and diversity of data by creating reasonable
+virtual data points from existing data, such that better machine
+learning models can be constructed based on them. According
+to [59], this study considers graph data augmentation and adopts
+a random mapping mechanism to produce augmented graph set
+D based on the observed graph Go = (V, Eo). Specifically, the
+set of all possible edges in the graph Go is denoted as Ω, the ex-
+isting edge set is denoted as Eo, and the non-existing edge set is
+denoted as Enon = Ω − Eo. Thus, the candidate sets for random
+mapping are defined as follows:
+Ec
+del = Eo,
+Ec
+add = Enon.
+(3)
+Thereafter, samples are randomly produced from the candidate
+sets to obtain the edge sets Edel and Eadd. Finally, a new aug-
+mented graph is generated by modifying the graph Go based on
+Edel and Eadd:
+G′ = (V, (E ∪ Eadd)\Edel).
+(4)
+Each input graph can be viewed as an instance for link pre-
+diction, owing to the generative learning scheme of the models
+considered in this work. Thus, the dataset containing a series of
+augmented graphs can be denoted as D = {Gi|i = 1, ..., t} and
+split to yield disjoint training and validation sets. These can be
+denoted as Dtrain and Dval respectively, wherein the missing and
+spurious links of the validation set are guaranteed not to appear
+in the training set. The observed graph Go used to generate the
+augmented graphs is defined as test set Dtest.
+3.2. Graph Convolutional Networks
+GCNs are a class of neural networks designed to general-
+ize traditional convolution operator for non-euclidean graph-
+structured data. In essence, GCNs aim to learn new feature rep-
+resentations of nodes in graphs by exploiting their structural in-
+formation. Let adjacency matrix A ∈ {0, 1}N×N denote the struc-
+tural information of the graph G, and X ∈ RN×F denote the fea-
+ture matrix of all graph nodes. Mathematically, using the output
+of the l-th layer as the input for the next layer, each neural net-
+work layer can be formulated as a nonlinear function:
+H(l+1) = f(H(l), A)
+(5)
+where H(l) corresponds to the feature matrix of the l-th layer, and
+H(0) = X is the input feature matrix of the first layer. Specific
+GCNs models differ only in the manner in which the nonlinear
+function f(·) is instantiated. A simple example of f(·) is as fol-
+lows:
+f(H(l), A) = σ(AH(l)W(l))
+(6)
+where σ(·) denotes a nonlinear activation function, such as
+a Rectified Linear Unit (ReLU), and W(l) denotes a trainable
+weight matrix for the l-th layer. With this propagation rule, the
+neighbour’s features are aggregated to represent each node at
+every layer, and the features become increasingly abstract by
+stacking layers on top of each other. However, there exist two
+limitations: the propagation rule simply aggregates the features
+Table 1: Notations and meanings.
+Notations
+Descriptions
+G
+Original graph
+Go
+Observed graph
+A
+Adjacency matrix of graph
+Em
+Missing links
+Es
+Spurious links
+D
+Dataset that contains augmented graphs
+H(l)
+Feature matrix of l-th neural network layer
+W(l)
+Trainable weight matrix for the l-th layer
+|| · ||0
+ℓ0−norm
+|| · ||2,1
+ℓ2,1−norm
+of neighboring nodes but not the node itself, and the multiplica-
+tion with A expected to change the scale of the feature vectors.
+That is, the nodes with a high degree will have a larger value,
+and the nodes with a low degree may have smaller values. To
+address the problems, a new propagation function, f(·), is pre-
+sented as follows:
+f(H(l), A) = σ( ˆD− 1
+2 ˆA ˆD− 1
+2 H(l)W(l))
+(7)
+where ˆA is obtained by adding an identity matrix I to the adja-
+cency matrix ˆA = A + I, ˆD denotes the diagonal node degree
+matrix of ˆA, and ˆD− 1
+2 ˆA ˆD− 1
+2 denotes symmetric normalization.
+3.3. Low-rank and Sparse Modeling
+Traditionally, Principal component analysis (PCA) was pro-
+posed to determine a low-dimensional representation of data
+while retaining as much information as possible. However, the
+PCA is particularly effective when dealing with Gaussian noise,
+which is independent and identically distributed with respect to
+the original data. Hence, the Robust Principal Component Anal-
+ysis (RPCA) [9] has been proposed to eliminate the effect of
+erratic noise (outliers). PCA and RPCA methods implicitly as-
+sume that the underlying data structure is a single low-rank sub-
+space; however, real-world data may be drawn from a union of
+multiple subspaces, and therefore, modeling may be inaccurate.
+To this end, Low-Rank Representation (LRR) [28] has been pro-
+posed.
+Considering the correlation between the connectivity patterns
+of nodes in real-world graphs, the adjacency matrix of the graphs
+should be low-rank. In other words, the rows or columns of the
+adjacency matrix must not be linearly independent. Thus, as-
+suming that hidden non-zero entries representing missing links
+can be recovered according to the adjacency matrix, [37] pro-
+posed an RPCA-based link prediction method, which is formu-
+lated as the following optimization problem:
+min
+X∗,E rank(X∗) + γ||E||0 s.t., A = X∗ + E
+(8)
+where rank(X∗) denotes the rank of matrix X∗, || · ||0 is the
+ℓ0−norm, and γ denots the balancing parameter. The method
+searches for X∗ with a low rank as low as possible and E as
+sparse as possible from A. Moreover, by representing a net-
+work structure with as few representative subgraphs as possible,
+[53] proposed an LRR-based link prediction method, wherein
+networks could be modeled via a low-rank and sparse represen-
+4
+
+Figure 2: Demonstration of our link prediction method, GraphLP. (a) Link prediction method, GraphLP. The original graph is per-
+turbed using a random mapping mechanism to obtain the observed graph; after this, the observed graph is further perturbed to generate
+augmented graphs. These augmented graphs are fed into GraphLP to learn the model using the observed graph as the label. Subse-
+quently, the learned model is used to infer the original graph based on the observed graph. (b) Self-representation-based collaborative
+inference. Based on the structural regularity of graphs, the original graph can be reconstructed by utilizing the correlation between
+subgraph patterns. (c) Example of high-order connectivity. In addition to the 1-hop neighborhood, multi-hop connectivity influences
+the existence of links. The right graph represents the two-hop connectivity of the graph on the left, and the red dotted lines in the left
+graph provide an example of the two-hop connectivity path of node 2.
+tation, as follows:
+min
+Z,E rank(Z) + α||Z||0 + β||E||2,1 s.t., A = AZ + E
+(9)
+where Z denotes the representation matrix reflecting the organi-
+zation principle of the network, and || · ||2,1 is the ℓ2,1−norm.
+The notations used in this study are listed in Table 1.
+4. The Proposed Method
+This section presents the proposed link prediction method,
+GraphLP. As depicted in Figure 2, the framework of GraphLP
+consists of three main components:
+• Collaborative inference operation. There exist certain sim-
+ilarities between the connection patterns of individuals in
+a complex system such that the perturbed structure of real-
+world graphs can be recovered globally based on the corre-
+lation between subgraph patterns (Section 4.1).
+• High-order connectivity computation. The existence of a
+link between any two target nodes is intended to be primar-
+ily determined by the connectivity degree between nodes,
+i.e., the number of paths and their length. Thus, the like-
+lihood of a link can be estimated locally by computing the
+connectivity (Section 4.2).
+• Pattern fusion operation. In addition to the first-order adja-
+cency matrix, the connection patterns of nodes in the high-
+order adjacency matrix are also considered to be correlated,
+and the high-order connectivity can be reconstructed based
+on the collaborative inference. Thus, the graph topology
+can be estimated by fusing the k-order (k ≥ 1) adjacency
+matrix (Section 4.3).
+4.1. Collaborative Inference Operation
+[32] suggested that link formation in real-world graphs is usu-
+ally driven by both regular and irregular factors, and the for-
+mer can be explained based on the mixture of multiple mech-
+anisms, such as homophily, triadic closure, preferential attach-
+ment. Meanwhile, assuming that high-dimensional data are a
+mixture of simple data and are drawn from a union of multiple
+low-dimensional linear subspaces, the LRR has been proposed
+to represent the data A = [a1, a2, ..., aN] as a linear combination
+of the basis in a ”dictionary” D = [d1, d2, ..., dM]:
+min
+Z rank(Z) s.t., A = DZ,
+(10)
+5
+
+(a) Link Prediction Method GraphLP
+Target Graph for
+A* = AZ*
+(D 2AD 2H()W()
+Original Graph Observed Graph
+Model Testing
+Collaborative
+Collaborative
+Connectivity
+Connectivity
+Connectivity
+High-order
+Inference
+Inference
+Inference
+Concat
+Augmented Graph
+Target Graph for
+ Process of Model Training
+ Flatten of Adjacency Matrix
+Model Training
+-→ Process of Link Prediction
+(c) High-order Connectivity
+(b) Self-representation based Collaborative Inference
+2-order Connectivity Path
+A
+E
+A
+*
+Z
++Thus, the optimal representation matrix Z∗ uncovers the under-
+lying subspaces in the data. By using each subspace to model
+a homogeneous subset of the data, multiple subspaces in LRR
+can capture heterogeneous structures within the data. There-
+fore, considering the above ideas, the regular structure of real-
+world graphs can be described appropriately by the LRR model,
+wherein the generation mechanisms of graph organization essen-
+tially corresponds to subspaces and the low rankness constraint
+captures the global correlation in graphs.
+Meanwhile, based
+on the generation mechanisms of graph organization, individual
+nodes may have similar connection patterns, and substructures
+that follow the same generation mechanism can be represented
+by each other, as depicted in Figure 2(b). Therefore, by using
+the adjacency matrix A as the dictionary, the real-world graph
+can be represented by itself, as follows:
+min
+Z rank(Z) s.t., A = AZ.
+(11)
+In addition to their regular structure, real-world graphs also
+contain irregular components. Thus, we let matrix E denote such
+irregular connections; then, the proposed self-representation
+model can be modified as A = AZ + E. According to the LRR,
+data are considered to be ”sample specific”, and the ℓ21−norm
+is adopted to constrain the matrix E, i.e., ||E||2,1. However, al-
+though the proposed method can be used to model real-world
+graphs, the low-rank model and ℓ21−norm constraints are usu-
+ally solved using Alternating direction method (ADM), which
+requires a large number of iterations and has high complexity.
+Therefore, a reasonable strategy is to relax the constraints with
+Frobenius norm:
+min
+Z ||Z||2
+F + λ||A − AZ||2
+F s.t. , A = AZ + E
+(12)
+Let L = ||Z||2
+F + λ||A − AZ||2
+F denote the partial derivative of L
+with respect to Z is ∂L/∂Z = 2Z + λ(−2ATA + 2ATAZ). By
+setting ∂L/∂Z = 0, the optimal representation Z∗ can be obtained
+as follows:
+Z∗ = λ(λATA+I)−1ATA.
+(13)
+where I denotes the identity matrix. Thus, in the case that the
+clean data is sufficient enough to represent the graph’s struc-
+tural patterns and the irregular connections are properly char-
+acterized, the structure perturbations can be inferred using AZ∗.
+Hence, the collaborative inference operation is defined as fol-
+lows:
+CI(A) = λA(λATA+I)−1ATA
+(14)
+4.2. High-order Connectivity Computation
+According to local similarity indices for link prediction, the
+more the number of paths two nodes possess, the greater the
+similarity between them. Specifically, two nodes with a high
+mutual connectivity are more likely to generate a link between
+them. Thus, n-hop-based (n ≥ 2) paths must be explored to char-
+acterize the local structural features for link prediction. Using a
+deep learning framework, the n-hop computation can be decom-
+posed into two-hop operations on each neural layer. Hence, a
+high-order connectivity computation calculates the two hop con-
+nectivity of graph nodes in each layer, and the mutual connec-
+tivity of two nodes can be estimated by stacking the high-order
+connectivity computation mechanism. Assuming that the integer
+powers of the adjacency matrix characterizes the mutual connec-
+tivity of graph nodes, that is, [An]ij denotes the number of paths
+Figure 3: Illustration of high-order connectivity computation.
+with length n connecting nodes i and j, the high-order connectiv-
+ity computation in each neural layer can be defined based on the
+idea of the second power of adjacency matrix A. From the per-
+spective of graph convolution networks, high-order connectivity
+computation can be defined as
+HCCA(A) = ˆD− 1
+2 ˆA ˆD− 1
+2 CI(A),
+(15)
+where the weighted adjacency matrix generated by the proposed
+collaborative inference operation is viewed as the features of
+graph nodes. Figure 3 illustrates a high-order connectivity com-
+putation. As presented in Equation (15), the global and local
+structural features can be captured for link prediction at the level
+of individual nodes and edges. Thus, the nonlinear propagation
+function can be defined as follows:
+H(l+1) = ˆD− 1
+2 ˆA ˆD− 1
+2 CI(H(l))W(l).
+(16)
+Thus, the hierarchical structure of real-world graphs can be char-
+acterized by executing the nonlinear propagation function itera-
+tively, in which HCCA(H(l)) = ˆD− 1
+2 ˆA ˆD− 1
+2 CI(H(l)) represents
+the high-order connectivity of graph nodes, as depicted in Figure
+2(c), and CI(H(l+1)) denotes the collaborative inference.
+4.3. Pattern Fusion Operation
+To estimate the likelihood of potential links, the output of the
+(l − 1)-th layer, i.e., H(l), is fed as the input of the l-th layer.
+Based on CI(H(l)) and HCCA(H(l)), the shallow layers extract
+the low-order global and local structure features, while the deep
+layers extract the high-order global and local structure features.
+Meanwhile, the effective range that the local structure features
+drawn from increases as the model depth increases. Therefore,
+the structure features in different range at various order, i.e.,
+HCCA(H(l)) and CI(H(l)), 0 ≤ l ≤ L, all contribute to the
+inference of potential links, although the exact extent of their
+contribution depends on the graph data.
+To overcome the issues mentioned above, in addition to being
+used as the inputs of the next layer, the outputs of neural net-
+work layers are mapped to skip a block of several layers based
+on residual connections, as illustrated in Figure 2(a). Next, all
+outputs are concatenated and used as the input of a two-layer
+Multi-layer Perceptron (MLP), which is defined as:
+O= MLP(concat(CI(H(l)), HCCA(H(l)))),0 ≤ l ≤ L.
+(17)
+where O is a vector containing the probabilities of links between
+all possible node pairs, and missing and spurious links can be
+inferred based on it.
+6
+
+Central node
+1-hop neighbor nodes
+2-hop neighbor nodes
+Node features representing link
+weights to 2-hop neighbors
+Computing connectivity of
+central node to 2-hop neighbors
+Adjacency relations of central node
+Weighted links produced by C(A)4.4. Model Training
+To train the proposed model, augmented graphs generated
+based on the observed graph are used as training data, and the
+adjacency matrix of the observed graph is flatten as its labels Y,
+where Yi∗N+j denotes the existence of the link between nodes i
+and j. Correspondingly, O represents the prediction results ob-
+tained by the proposed model for all possible links. Here, the
+Binary Cross-Entropy (BCE) is used as the loss function:
+L = − 1
+N2
+N2
+�
+i=1
+Yi log(Oi) + (1 − Yi) log(1 − Oi).
+(18)
+The learned model is then deployed on the observed graph to
+reconstruct the original graph. The training process of GraphLP
+is outlined in Algorithm 1.
+Algorithm 1 Training Process of GraphLP
+Input: Training set Dtrain, validation set Dval, and test set Dtest,
+number of neural network layers L.
+Output: The well-trained model GraphLP.
+1: while not convergence do
+2:
+for 0 ≤ l ≤ L do
+3:
+Conduct collaborative inference operation using (14);
+4:
+Compute high-order connectivity using (16);
+5:
+end for
+6:
+Fuse the outputs based on MLP using (17);
+7:
+Update the model by minimizing the loss function (18);
+8: end while
+4.5. Model Analysis
+(1) Generalized local similarity indices. The high-order con-
+nectivity computation HCCA(H(l)) in every neural network
+layer is essentially the second power of the adjacency matrix,
+and it obtains the connectivity of node pairs within two-hop
+neighborhood. As the model depth increases, the connectivity
+of node pairs in a wider range is considered. Thus, GraphLP can
+degenerate to S i j = A2 + αA3 + βA4 + · · · when collaborative
+inference and deep learning mechanism are abolished.
+(2) Connection to WalkPool. WalkPool [36] first generates
+node representations based on GNN and encodes them into edge
+weights of the extracted enclosing subgraphs; following this, it
+uses the edge weights to compute the transition probabilities of
+random walk.
+Next, the method calculates a list of features
+based on the transition probabilities to classify the subgraphs.
+However, for an enclosing subgraph G = (V, E), its variants
+G+ = (V, E ∪ {i, j}) and G− = (V, E\{i, j}) are used as positive
+and negative samples, respectively. In essence, this method dis-
+criminates only those subgraphs that differ by a single edge and
+is not suitable for practical link prediction scenarios. In contrast,
+GraphLP can predict any potential links based on graph structure
+features.
+(3) Connection to LFLP. The LFLP [53] constructs an ad-
+jacency matrix based on a self-representation model and then
+combines it with the observed network to identify missing and
+spurious links. The collaborative inference operation CI(H(l))
+of our work is similar to that in the LFLP with respect to model-
+ing the global structure of graphs; however, the difference is that
+only low-order global structural features are considered in LFLP,
+whereas multi-order global and local structural features are char-
+acterized based on the deep-learning framework in GraphLP.
+5. Experiments
+Further, extensive experiments are conducted on real-world
+graphs to evaluate the performance of the proposed method
+GraphLP: (1) Compare GraphLP with state-of-the-art meth-
+ods; (2) Compare GraphLP with traditional baseline methods;
+(3) Model architecture analysis; (4) Model sensitivity analysis.
+Here, Area Under Curve (AUC) and Average Precision (AP) are
+adopted to evaluate the performance of the methods. Further-
+more, Precision is used to verify the superiority of GraphLP over
+traditional link prediction methods. Based on the link prediction
+results O, the scores are sorted in descending and ascending or-
+ders, and following this, their top-L links are taken as the pre-
+dicted missing and spurious links. Note that Precision is defined
+by calculating the ratio of accurately discovered links to the total
+number of links in the probe set:
+Precision = T/R
+(19)
+where T is the number of accurately identified links, and R is
+the total number of links in the probe set.
+5.1. Experimental Settings
+5.1.1. Experimental Datasets
+Herein, seven widely used graph datasets are used for link
+prediction. (1) USAir [40]. This is the transportation network
+of the United States, including 332 airports as nodes and 2,126
+airlines as edges, connecting the United States worldwide. The
+average node degree is 12.81. (2) C.ele [49]. This is a neu-
+ral network of C. elegans, with 297 neurons representing nodes
+and 2,148 synaptic connections representing edges. The average
+node degree is 14.46. (3) PB [1]. This dataset is a network of
+hyperlinks between weblogs on US political blogs, with 1,222
+blogs on US politics as nodes and 16,714 hyperlinks between
+blogs as edges. The average node degree is 27.36. (4) NS [33].
+This is an undirected co-authorship network with 1,589 nodes
+and 2,742 edges, where the nodes denote the scientists engaged
+in network science research, and the edges denote two scientists
+have co-authored a publication. The average node degree is 3.45.
+(5) Yeast [47]. This represents a protein-protein interaction net-
+work formed in yeast with 2,375 proteins as nodes and 11,693
+protein-protein interactions as edges. The average node degree
+is 9.85. (6) E.coli [58]. This is a pairwise reaction network of
+metabolites with 1,805 nodes and 14,660 edges. The average
+node degree is 12.55. (7) Router [43]. It is a snapshot of the In-
+ternet structure at the level of autonomous systems, with 5,022
+nodes and 6,258 edges, in which the nodes represent routers and
+the edges represent the data transmission between routers. The
+average node degree is 2.49. The properties of the datasets are
+listed in Table 2.
+To extensively validate the performance of the proposed
+method, 90% and 50% of the links of the original graph are se-
+lected randomly to first construct the observed graphs. There-
+after, based on the observed graph Go, 10% nonexisting links
+are add randomly as spurious links, and 10% existing links are
+removed randomly as missing links, denoted as Edel and Eadd
+respectively, to generate the augmented graph set D = {Gi|i =
+1, ..., t}. Following this, 90% and 10% graphs are randomly se-
+lect from D as the training and validation set, respectively, and
+the observed graph Go is used as the test set.
+7
+
+Table 2: Summary of the datasets. ACC is the average clustering
+coefficient, and AD is the average node degree.
+Dataset USAir
+NS
+PB
+Yeast
+C.ele
+Router
+E.coli
+Node
+332
+1589
+1222
+2375
+297
+5022
+1805
+Edges
+2126
+2742
+16714 11693
+2148
+6258
+14660
+ACC
+0.625
+0.638
+0.320
+0.306
+0.292
+0.012
+0.516
+AD
+12.81
+3.45
+27.36
+9.85
+14.46
+2.49
+12.55
+5.1.2. Comparison Methods
+The proposed method was compared with six state-of-the-art
+deep learning-based link prediction methods, including:
+(1) Weisfeiler-Lehman graph kernel (WLK) [42] is a fast fea-
+ture extraction scheme based on the WL test for graph isomor-
+phism, which maps the original graph to a graph sequence and
+adds the pair-wise similarities between the graphs.
+(2) Weifeiler-Lehmam Neural Machine (WLNM) [56] is a
+subgraph classification-based link prediction method that lever-
+age deep learning to automatically learn topological features
+from enclosing subgraphs.
+(3) Node2Vec [20] is a network embedding method that en-
+codes proximity information into low-dimensional vectors. The
+node features and low-dimensional vectors are then fed into the
+MLP for link prediction.
+(4) LINE [44] learns network embeddings that preserve the
+first-order and second-order proximity, and the resulting low-
+dimensional vectors are used for link prediction.
+(5) SEAL [57] extracts the enclosing subgraphs of positive
+and negative links and marks different roles of their nodes. The
+method then trains a GNN based on the node information matrix
+to classify subgraphs for link prediction.
+(6) WalkPool (WP) [36] is a subgraph classification-based
+link prediction method. It encodes node feature and graph topol-
+ogy into the transition probabilities of random walk, and follow-
+ing this, a list of features is computed to classify subgraphs.
+5.1.3. Parameter Settings
+GraphLP is implemented on a Pytorch platform with a
+NVIDIA GeForce RTX GPU and optimized using Adam op-
+timizer. All models are implemented using Python 3.6. The
+learning rate is set to 0.0012 for the NS dataset and 0.0005 for
+the other graphs. For all the datasets, the weight decay is set to
+0.0. The number of epochs on the E.coli and Yeast dataset is
+300, whereas it was 200 on the other datasets. Dropout is ap-
+plied to the MLP, and the dropout rate is set to 0.5 on Router
+and 0.2 on the others. The trade-off parameter λ is set to 0.13,
+and the number of neural network layers in the GraphLP is set
+to three. The detailed hyperparameter settings for the model are
+listed in Table 3.
+Table 3: Hyperparameter setting for the proposed method.
+Name
+Value
+optimizer
+Adam
+loss function
+binary cross entropy
+learning rate
+NS=0.0012, others=0.0005
+weight decay
+0.0
+epochs
+Ecoli, Yeast=300, others=200
+dropout
+Router=0.5, others=0.2
+λ
+0.13
+number of network layers
+3
+5.2. Experimental Result
+For 90% of the observed links, the results about the AUC
+and AP with standard deviations are presented in Table 4 and
+5, which indicate that GraphLP significantly outperforms other
+state-of-the-art algorithms in terms of both AUC and AP, with
+exception of the NS and Router datasets. The results demon-
+strate that the learning of local and global graph structure en-
+tirely characterizes the underlying structural patterns; thus, the
+missing links and spurious links can be better identified. Ta-
+ble 4 indicates that GraphLP significantly improves the AUC
+on the PB, C.ele, and Router datasets, with approximately 4%,
+7%, and 3% performance improvement, respectively, compared
+to the WP algorithm. In addition, the proposed method still per-
+forms better than other state-of-the-art methods on the USAir,
+Yeast and NS datasets. Moreover, the results for the AP pre-
+sented in Table 5 also indicate that GraphLP outperforms state-
+of-the-art methods on most of datasets, and GraphLP achieves a
+maximum performance enhancement of approximately 9% com-
+pared to the best performing graph neural network method WP.
+For 50% of the observed links, the results also demonstrate
+that the proposed model achieves remarkable performance com-
+pared to the methods, as described in Table 6 and 7. The results
+illustrate that, as the amount of structure perturbation increases,
+GraphLP can still appropriately learn the real graph structure,
+thus recovering the original graph effectively. Therefore, the
+values of the AUC and AP decreased to a lower extent. Fur-
+thermore, by comparing Table 4 with Table 6 and Table 5 with
+Table 7, we can infer that the AUC and AP values drop faster
+for the other state-of-the-art methods than those for GraphLP,
+which demonstrates that GraphLP can better capture the under-
+lying structural patterns to demonstrate better performance.
+5.3. Compared with Traditional Link Prediction Methods
+To further verify the proposed method, the precision of
+GraphLP and traditional link prediction methods are calculated
+based on the following datasets: (1) Macaque [15], cortical net-
+works of the macaque monkey; (2) Mangwet [5], the food web
+in Mangrove Estuary during the wet season; (3) Jazz [19], a
+collaboration network of jazz musicians; (4) Metabolic [17], a
+metabolic network of C.elegans; (5) USAir, (6) C.ele, (7) E.coli
+and (8) Yeast. Here, six representative traditional link prediction
+methods are selected for comparison.
+(1) The CN [29] metric is among the most widely used meth-
+ods for link prediction problem. It assumes that two nodes will
+be more easily connected if they share more common neighbors;
+(2) The RA [61] metric is inspired by the physical processes
+involved in resource allocation, which suppresses the contribu-
+tion of high-degree common neighbors;
+(3) The LP [61] index measures the structural similarity of
+node pairs within three-hops;
+(4) The Non-negative Matrix Factorization (NMF) [26] model
+is used for structure prediction by learning the latent features of
+real-world graphs;
+(5) The Robust Principal Component Analysis (RPCA) [37]
+represents a real-world graph based on the sparsity and low rank
+property of its adjacency matrix and infers potential links based
+on matrix completion.
+(6) The LFLP [53] uses a self-representation model to re-
+construct the original graph based on a few representative sub-
+graphs.
+The results of missing link prediction with respect to Preci-
+sion are shown in Table 8. For each network, the bold number
+8
+
+Table 4: Prediction measured by AUC (90% observed links). Bold numbers are the best results of all methods.
+Data
+USAir
+NS
+PB
+Yeast
+C.ele
+Router
+E.coli
+WLK
+96.63 ± 0.73
+98.57 ± 0.51
+93.83 ± 0.59
+95.86 ± 0.54
+89.72 ± 1.67
+87.42 ± 2.08
+96.94 ± 0.29
+WLNM
+95.95 ± 1.10
+98.61 ± 0.49
+93.49 ± 0.47
+95.62 ± 0.52
+86.18 ± 1.72
+94.41 ± 0.88
+97.21 ± 0.27
+Node2Vec
+91.44 ± 1.78
+91.52 ± 1.28
+85.79 ± 0.78
+93.67 ± 0.46
+84.11 ± 1.27
+65.46 ± 0.86
+90.82 ± 1.49
+LINE
+81.47 ± 10.71
+80.63 ± 1.90
+76.95 ± 2.76
+87.45 ± 3.33
+69.21 ± 3.14
+67.15 ± 2.10
+82.38 ± 2.19
+SEAL
+97.09 ± 0.7
+98.85 ± 0.41
+95.01 ± 0.34
+97.91 ± 0.52
+90.30 ± 1.35
+96.38 ± 1.45
+97.64 ± 0.22
+WP
+98.68 ± 0.48
+98.95 ± 0.41
+95.60 ± 0.37
+98.37 ± 0.25
+95.79 ± 1.09
+97.27 ± 0.28
+98.58 ± 0.19
+GraphLP
+99.26 ± 1.01
+99.64 ± 0.98
+99.73 ± 0.25
+99.41 ± 0.15
+99.90 ± 0.14
+99.02 ± 0.19
+99.23 ± 0.23
+Table 5: Prediction measured by AP (90% observed links). Bold numbers are the best results of all methods.
+Data
+USAir
+NS
+PB
+Yeast
+C.ele
+Router
+E.coli
+WLK
+96.82 ± 0.84
+98.79 ± 0.40
+93.34 ± 0.89
+96.82 ± 0.35
+88.96 ± 2.06
+86.59 ± 2.23
+97.25 ± 0.42
+WLNM
+95.95 ± 1.13
+98.81 ± 0.49
+92.69 ± 0.64
+96.40 ± 0.38
+85.08 ± 2.05
+93.53 ± 1.09
+97.50 ± 0.23
+Node2Vec
+89.71 ± 2.97
+94.28 ± 0.91
+84.79 ± 1.03
+94.90 ± 0.38
+83.12 ± 1.90
+68.66 ± 1.49
+90.87 ± 1.48
+LINE
+97.70 ± 11.76
+85.17 ± 1.65
+78.82 ± 2.71
+90.55 ± 2.39
+67.51 ± 2.72
+71.92 ± 1.53
+86.45 ± 1.82
+SEAL
+97.13 ± 0.80
+99.06 ± 0.37
+94.55 ± 0.43
+98.33 ± 0.37
+89.48 ± 1.85
+96.23 ± 1.71
+98.03 ± 0.20
+WP
+98.66 ± 0.55
+99.09 ± 0.29
+95.28 ± 0.41
+98.64 ± 0.28
+91.53 ± 1.33
+97.20 ± 0.38
+98.79 ± 0.21
+GraphLP
+99.91 ± 1.03
+98.94 ± 0.96
+98.32 ± 1.43
+98.74 ± 0.16
+99.41 ± 0.42
+79.30 ± 0.19
+98.96 ± 0.19
+Table 6: Prediction measured by AUC ( 50% observed links). Bold numbers are the best results of all methods.
+Data
+USAir
+NS
+PB
+Yeast
+C.ele
+Router
+E.coli
+WLK
+91.93 ± 0.71
+87.27 ± 1.71
+92.54 ± 0.33
+91.15 ± 0.35
+83.29 ± 0.89
+71.25 ± 4.37
+92.38 ± 0.46
+WLNM
+91.42 ± 0.95
+87.61 ± 1.63
+90.93 ± 0.23
+92.22 ± 0.32
+75.72 ± 1.33
+86.10 ± 0.52
+92.81 ± 0.30
+Node2Vec
+84.63 ± 1.58
+80.29 ± 1.20
+79.29 ± 0.67
+90.18 ± 0.17
+75.53 ± 1.23
+62.45 ± 0.81
+84.73 ± 0.81
+LINE
+72.51 ± 12.19
+65.96 ± 1.60
+75.53 ± 1.78
+79.44 ± 7.90
+59.46 ± 7.08
+62.43 ± 3.10
+74.50 ± 11.10
+SEAL
+93.36 ± 0.67
+90.88 ± 1.18
+93.79 ± 0.25
+93.90 ± 0.54
+82.33 ± 2.31
+86.64 ± 1.58
+94.18 ± 0.41
+WP
+95.50 ± 0.74
+90.97 ± 0.96
+94.57 ± 0.16
+95.00 ± 0.21
+87.62 ± 1.39
+88.13 ± 0.61
+95.33 ± 0.30
+GraphLP
+98.97 ± 0.15
+97.08 ± 0.14
+98.19 ± 0.10
+98.74 ± 0.25
+97.96 ± 0.14
+98.10 ± 0.15
+98.05 ± 0.14
+Table 7: Prediction measured by AP ( 50% observed links). Bold numbers are the best results of all methods.
+Data
+USAir
+NS
+PB
+Yeast
+C.ele
+Router
+E.coli
+WLK
+93.34 ± 0.51
+89.97 ± 1.02
+92.34 ± 0.34
+93.55 ± 0.46
+83.20 ± 0.90
+75.49 ± 3.43
+94.51 ± 0.32
+WLNM
+92.54 ± 0.81
+90.10 ± 1.11
+91.01 ± 0.20
+93.93 ± 0.20
+76.12 ± 1.08
+86.12 ± 0.68
+94.47 ± 0.21
+Node2Vec
+82.51 ± 2.08
+86.01 ± 0.87
+77.21 ± 0.97
+92.45 ± 0.23
+72.91 ± 1.74
+66.77 ± 0.57
+85.41 ± 0.94
+LINE
+71.75 ± 11.85
+71.53 ± 0.97
+78.72 ± 1.24
+83.06 ± 9.70
+60.71 ± 6.26
+64.87 ± 6.76
+75.98 ± 14.45
+SEAL
+94.15 ± 0.54
+92.21 ± 0.97
+93.42 ± 0.19
+95.32 ± 0.38
+81.99 ± 2.18
+87.79 ± 1.71
+95.67 ± 0.24
+WP
+95.87 ± 0.74
+92.33 ± 0.76
+94.22 ± 0.27
+96.15 ± 0.13
+86.25 ± 1.42
+89.17 ± 0.55
+96.36 ± 0.34
+GraphLP
+97.96 ± 0.09
+93.08 ± 0.08
+96.27 ± 0.10
+97.27 ± 0.09
+95.89 ± 0.11
+79.23 ± 0.14
+96.48 ± 0.13
+Table 8: The precision (90% observed links) of missing links prediction. Bold numbers are the best results of all methods.
+Data
+Macaque
+Mangwet
+Jazz
+Metabolic
+USAir
+C.ele
+E.coli
+Yeast
+RA
+0.5099
+0.1292
+0.5547
+0.2451
+0.4443
+0.093
+0.4857
+0.2609
+CN
+0.5695
+0.125
+0.5292
+0.1127
+0.3764
+0.091
+0.4399
+0.1334
+LP
+0.5483
+0.1319
+0.5109
+0.1275
+0.3821
+0.089
+0.4837
+0.1454
+NMF
+0.7316
+0.4398
+0.5309
+0.2315
+0.3981
+0.1270
+0.5013
+0.3812
+RPCA
+0.7421
+0.5421
+0.6138
+0.1842
+0.3596
+0.098
+0.3418
+0.5359
+LFLP
+0.7605
+0.5572
+0.5956
+0.3241
+0.4545
+0.2010
+0.5007
+0.60
+GraphLP
+0.7881
+0.7986
+0.8212
+0.7030
+0.8208
+0.8224
+0.7169
+0.7374
+9
+
+Table 9: The precision (90% observed links) of spurious links prediction. Bold numbers are the best results of all methods.
+Data
+Macaque
+Mangwet
+Jazz
+Metabolic
+USAir
+C.ele
+E.coli
+Yeast
+RA
+0.5490
+0.1380
+0.5410
+0.140
+0.2650
+0.2790
+0.4171
+0.1110
+CN
+0.5710
+0.2880
+0.5690
+0.1670
+0.2480
+0.2458
+0.3222
+0.073
+LP
+0.5939
+0.3280
+0.7016
+0.6911
+0.6271
+0.4780
+0.6089
+0.4585
+NMF
+0.8090
+0.5660
+0.6510
+0.2430
+0.4820
+0.4333
+0.4662
+0.2330
+RPCA
+0.810
+0.5180
+0.5920
+0.074
+0.443
+0.2609
+0.3795
+0.4250
+LFLP
+0.818
+0.583
+0.663
+0.2210
+0.5970
+0.4390
+0.4246
+0.5680
+GraphLP
+0.9073
+0.8750
+0.9197
+0.8812
+0.9057
+0.9533
+0.8452
+0.6210
+Figure 4: The topology visualization of Club dataset. The experiment performs 10% link perturbation, i.e. 10% spurious links are
+added and 10% missing links are deleted.
+Figure 5: The topology visualization of Club dataset. The experiment performs 20% link perturbation, i.e. 20% spurious links are
+added and 20% missing links are deleted.
+in the corresponding column indicates the highest accuracy. The
+results presented in Table 8 demonstrate that the proposed model
+GraphLP model performs the best among the methods. Further-
+more, the link prediction accuracy of the proposed model is far
+higher than that of the other methods, which can be at least three
+times higher than that of the best-performing method. For spuri-
+ous links prediction, the results measured by Precision are listed
+in Table 9. For all networks, GraphLP performs the best among
+the methods and is remarkably better than the second best algo-
+rithm. The results presented in Table 8 and 9 demonstrate that
+GraphLP has stronger ability to learn structural features, and can
+recover the structure of the original network more accurately.
+Based on Table 2, it can be observed that the Precision of our
+proposed model performs best, despite the large differences be-
+tween the ACC and AD across all the datasets; thus indicates
+that the proposed model performs well for heterogeneous graph
+structures.
+5.4. Recovered Graph Visualization
+To verify the effectiveness of the proposed model of miss-
+ing and spurious links inference, the topology of the recovered
+graphs in the model training process on Club dataset is visually
+compared, as depicted in Figure 4 and 5. The top half depicts the
+topology of the graphs, wherein the red links denote the missing
+10
+
+missing links
+spurious links
+missing links
+spurious links
+ spurious links
+missing links
+spurious links
+missing links
+spurious links
+missing links
+19): 0.4048
+(6 , 18): 0.4008
+(1 .
+,19): 0.2513
+(6 , 18): 0.5258
+(1
+(1 , 19): 0.2069
+(6 , 18): 0.6525
+19): 0.2090
+(6 , 18): 0.6648
+19): 0.2329
+(6 , 18): 0.6827
+(14, 22): 0.4129
+(14, 22): 0.3667
+(19, 25): 0.4614
+(14, 22): 0.2958
+(14, 22): 0.2439
+(14, 22): 0.1988
+(19, 25): 0.4028
+(19, 25): 0.6381
+(19, 25): 0.7258
+(19, 25): 0.7441
+(6 , 26): 0.4099
+(6 , 17): 0.4039
+(6 , 26): 0.2985
+(6 , 17): 0.4416
+(6 , 26): 0.1294
+(6 , 17): 0.6539
+(6 , 26): 0.1033
+(6 , 17): 0.7358
+(6,
+, 26): 0.0972
+(6 , 17): 0.7634
+(25, 27): 0.1967
+8): 0.4286
+(25, 27): 0.0915
+(25, 27): 0.4147
+(6 ，8): 0.4008
+(25, 27): 0.1181
+8): 0.6909
+(25, 27): 0.1132
+(6 ，8): 0.7282
+8): 0.7768
+(6
+(6
+(6
+，9): 0.2884
+(18, 32): 0.3396
+9): 0.1618
+(3
+9): 0.4123
+(18, 32): 0.4053
+(3 
+(3
+9): 0.1895
+(18, 32): 0.6068
+(3,
+9): 0.1828
+(18, 32): 0.7402
+(3
+(18, 32): 0.7749
+,19): 0.2591
+,19): 0.4038
+, 14): 0.4049
+14): 0.4613
+(6 , 19): 0.1851
+(2 , 14): 0.6174
+(6 , 19): 0.1719
+,14): 0.7085
+19): 0.1827
+(6
+(2.
+(2 , 14): 0.7264
+(2 
+7): 0.4149
+, 32): 0.4041
+, 17): 0.2157
+32): 0.3625
+(6
+17): 0.1525
+(6 , 32): 0.6537
+17): 0.1588
+(6
+32): 0.7956
+(3
+17): 0.1412
+(6 , 32): 0.8063
+(3
+(6 
+3 
+(3 ,
+(3
+(11, 18): 0.4092
+(1, 17): 0.4071
+(11, 18): 0.2697
+17): 0.3585
+(11, 18): 0.2444
+17): 0.4941
+(11, 18): 0.2197
+17): 0.6085
+(11, 18): 0.2203
+(1,
+17): 0.5978
+25): 0.4102
+(13, 19): 0.3980
+, 25): 0.3579
+(13, 19): 0.2835
+(2 , 25): 0.2842
+3, 19): 0.3608
+(2
+(2
+, 25): 0.2526
+(13, 19): 0.4578
+(2
+25): 0.2606
+(13, 19): 0.5659
+, 32): 0.4138
+(18, 21): 0.4040
+(8 , 32): 0.1396
+(18, 21): 0.4487
+(8
+(8 , 32): 0.0680
+, 32): 0.0656
+(18, 21): 0.7626
+(8
+,32): 0.0697
+(18, 21): 0.7722
+(18, 21): 0.6938
+(8
+(11, 14): 0.4051
+(11, 14): 0.5059
+(9 , 11): 0.4131
+(9
+, 11): 0.3098
+(9 , 11): 0.2488
+(11, 14): 0.7683
+(9 , 11): 0.2433
+(11, 14): 0.8456
+,11): 0.2229
+(9
+(11, 14): 0.8543
+, 10): 0.1805
+8): 0.3698
+10): 0.4088
+8): 0.4026
+(2,
+(2 , 10): 0.0934
+8): 0.5869
+10): 0.0770
+8): 0.6296
+(2
+10): 0.0706
+8): 0.6542
+(1 
+(1
+, 14): 0.4129
+(5 , 14): 0.3129
+(10, 17): 0.3848
+(10, 17): 0.4076
+(5 , 14): 0.2576
+(10, 17): 0.5349
+(5
+(5
+,14): 0.2454
+(10, 17): 0.5966
+(5
+14): 0.2427
+(10, 17): 0.5926
+(4 , 25): 0.2169
+,25): 0.4127
+, 25): 0.2767
+7): 0.3751
+,25): 0.2205
+(4 
+7): 0.4047
+, 25): 0.2369
+7): 0.4549
+(1，
+7): 0.5862
+(1 ,
+7): 0.6325
+(4 
+(4
+33): 0.4091
+7): 0.4049
+(9 , 33): 0.3014
+7): 0.5319
+(9 , 33): 0.2692
+7): 0.5389
+(9
+(9
+(3
+,33): 0.1888
+7): 0.6441
+(9
+, 33): 0.1894
+7): 0.6740
+(3，
+(a)The similarity of spurious and
+(b)The similarity of spurious and
+(b)The similarity of spurious and
+(b)The similarity of spurious and
+(b)The similarity of spurious and
+missing links performed 40 epochs
+missing links performed 120 epochs
+missing links performed 1 epochs
+missing links performed 80 epochs
+missing links performed 140 epochsnegative links
+ positive links
+ negative links
+positive links
+ negative links
+positive links
+negative links
+positive links
+negative links
+positive links
+28): 0.5039
+(10, 25): 0.2858
+(5,
+28): 0.5615
+(10, 25): 0.3655
+(10, 25): 0.1989
+(5.
+28): 0.8291
+(10, 25): 0.1192
+(5.
+28): 0.9795
+(10, 25): 0.0702
+(5.
+28): 0.9960
+14): 0.5108
+32): 0.2786
+(0,
+14): 0.6029
+32): 0.1386
+14): 0.7367
+32): 0.0366
+(0,
+32): 0.4057
+(0,
+(0,
+(0,
+(0,
+(0,
+(0,
+14): 0.9393
+(0,
+32): 0.0814
+(0,
+14): 0.9787
+32): 0.3874
+32): 0.5822
+(5,32): 0.2812
+32): 0.6264
+(5, 32): 0.1954
+32): 0.7374
+32): 0.1713
+32): 0.8605
+32): 0.1426
+32): 0.9424
+(5.
+(1,
+(1,
+(5,
+(1,
+(5,
+(1,
+19): 0.2930
+(18, 24): 0.4254
+19): 0.2342
+(18, 24): 0.4081
+19): 0.1049
+(18, 24): 0.5512
+19): 0.0263
+(18, 24): 0.8087
+19): 0.0058
+(18, 24): 0.8801
+(0.
+(0,
+(0,
+(0,
+(18, 33): 0.1842
+17): 0.6631
+(18, 33): 0.0816
+17): 0.8405
+(18, 33): 0.0169
+17): 0.9706
+(18, 33): 0.0029
+17): 0.9902
+(18, 33): 0.3128
+(2,
+17): 0.5981
+(2.
+(2,
+(2,
+(2,
+21): 0.2887
+23): 0.5964
+21): 0.2166
+(0,
+(0,
+21): 0.0993
+(0,
+23): 0.8907
+23): 0.9841
+(0, 21): 0.0085
+(0,
+(0,
+23): 0.6629
+(0, 21): 0.0200
+(0,
+(0,
+23): 0.9933
+(12, 18): 0.3138
+16): 0.4231
+(12, 18): 0.2342
+16): 0.4246
+(12, 18): 0.1588
+16): 0.4794
+(12, 18): 0.0733
+16): 0.6510
+(9,
+(9.
+(9,
+(9,
+(12, 18): 0.0378
+(9,
+16): 0.8059
+(a)The similarity of negative and
+(b)The similarity of negative and
+(c)The similarity of negative and
+(d) The similarity of negative and
+(c)The similarity of negative and
+positive links performed 20 epochs
+ positive links performed 40 epochs
+positive links performed 60 epochs
+ positive links performed 100 epochs
+ positive links performed 80 epochsYeast
+USAir
+C.ele
+E.coli
+PB
+NS
+Router
+Training networks
+0.4
+0.6
+0.8
+1.0
+AUC
+Layer=1
+Layer=2
+Layer=3
+Layer=4
+(a)
+Yeast
+USAir
+C.ele
+E.coli
+PB
+NS
+Router
+Training networks
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+AP
+Layer=1
+Layer=2
+Layer=3
+Layer=4
+(b)
+Figure 6: The performance of link prediction under various model depth. (a) AUC of link prediction method GraphLP with different
+layer number. (b) AP of link prediction method GraphLP with different layer number.
+Yeast
+USAir
+C.ele
+E.coli
+PB
+NS
+Router
+Training networks
+0.980
+0.985
+0.990
+0.995
+1.000
+AUC
+=0.08
+=0.10
+=0.13
+=0.15
+=0.20
+(a)
+Yeast
+USAir
+C.ele
+E.coli
+PB
+NS
+Router
+Training networks
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+AP
+=0.08
+=0.10
+=0.13
+=0.15
+=0.20
+(b)
+Figure 7: The performance of link prediction under different values of λ. (a) AUC of link prediction method GraphLP under different
+values of λ. (b) AP of link prediction method GraphLP under different values of λ.
+0
+50
+100
+150
+epoch
+0.40
+0.45
+0.50
+0.55
+train_loss
+0
+50
+100
+150
+epoch
+0.25
+0.50
+0.75
+1.00
+val_auc
+0
+50
+100
+150
+epoch
+0.0
+0.5
+1.0
+val_ap
+0
+50
+100
+150
+epoch
+0.2
+0.4
+val_loss
+(a)
+0
+50
+100
+150
+epoch
+0.4
+0.5
+0.6
+train_loss
+0
+50
+100
+150
+epoch
+0.96
+0.98
+1.00
+val_auc
+0
+50
+100
+150
+epoch
+0.85
+0.90
+0.95
+val_ap
+0
+50
+100
+150
+epoch
+0.2
+0.4
+0.6
+val_loss
+(b)
+Figure 8: Visualization of model convergence. (a) Convergence of USAir dataset. (b) Convergence of C.ele dataset.
+links, the blue links denote the spurious links, and the gray links
+denote the original links. The bottom half depicts the likelihood
+scores of missing and spurious links. Based on the results, it
+can be concluded that with an increase in epoch times, the like-
+lihood scores of missing links increases gradually, and the like-
+lihood scores of spurious links decline gradually. The widths
+of the lines indicate the following process: when the number of
+epochs reaches 100, the likelihood scores of missing links ap-
+proach 1.0, and the likelihood scores of spurious links approach
+0.0. This proves that the proposed GraphLP model can distin-
+guish between missing links and spurious links and infer them
+effectively. Moreover, to further prove the effectiveness of the
+proposed model, the topology of the recovered graphs in the
+model training process is visualized when 20% of the links of
+Club are perturbed, as depicted in Figure 5. Compared to Fig-
+ure 4, we determined that the likelihood of missing and spuri-
+ous links is weakened with an increase in the structure pertur-
+bation ratio. However, with an increase in the training epochs,
+the model is still able to distinguish and infer the missing and
+spurious links according to the structural patterns. For instance,
+when the training epoch reaches 140, the likelihood scores of
+missing links are greater than 0.5, and those of spurious links
+11
+
+are less than 0.3, indicating that the proposed model can still
+predict missing and spurious links with high accuracy.
+5.5. Impact of Model Depth
+Next, the performance of GraphLP is explored at various
+model depths. As depicted in Figure 6, the performance of the
+model in terms of the AUC and AP trained with one-layer neu-
+ral network is poor; however, its performance improves signif-
+icantly with an increase in the number of layers. In particular,
+when the model depth is equal to two, a significant performance
+improvement is noted. The primary reason for this is that the
+model with two-layers multi-order global and local structural
+features is integrated adaptively based on the MLP component,
+which considerably improves the performance of the model.
+Subsequently, as the layer number increases, a slight improve-
+ment in model performance is still noted. When the number of
+layers is four, the accuracy of GraphLP declines significantly on
+NS and fluctuates on other datasets. A possible reason for this is
+that the model with four layers becomes more complex, thereby
+requiring more training iterations or an appropriate learning rate
+[14]. In general, the performance of the proposed model is opti-
+mal when the depth is three, and a deep architecture is necessary.
+5.6. Impact of Trade-off Parameter
+To examine the sensitivity of the proposed model to the trade-
+off parameter, the AUC and AP values of link prediction meth-
+ods with different λ are presented in Figure 7. Based on the re-
+sults, it can be concluded that the performance of the proposed
+model is not sensitive to λ for most datasets. In Figure 7(a), for
+the USAir and NS datsets, the AUC value varies significantly
+under different λ, but the performance is still better than other
+of the other algorithms. In Figure 7(b), the AP value remains
+stable for different λ values, indicating that the proposed model
+is insensitive to different λ. Overall, our proposed algorithm ex-
+hibited satisfactory performance on most datasets with various
+λ.
+5.7. The Convergence Analysis
+Generally, GraphLP converges to optimal values after approx-
+imitely 200 epochs on most datasets. In particular, Figure 8 plots
+the learning curves of GraphLP on the USAir and C.ele datasets,
+including the training loss, validation AUC, validation AP, and
+validation loss. The results indicate that the AUC and AP values
+increase rapidly with the decrease in training loss and validation
+loss, and these values converge to the optimal value when the
+validation loss approaches a minimum value. Additionally, we
+discover that validation loss is lower than training loss, and the
+difference between them remains relatively stable. A possible
+reason for this is that the dropout manipulation is only applied
+to the training process.
+6. Conclusion
+This paper aims to reconstruct graph structure to improve the
+performance of link prediction. In particular, unlike existing
+subgraph-classification-based discriminative methods, this work
+achieves the aforementioned objective by developing a genera-
+tive GNN, namely GraphLP, which considered both global and
+local structure features and hierarchical structural patterns. Con-
+currently, a novel collaborative inference operation and high-
+order connectivity computation mechanism are developed. We
+also present an analysis about the relation between GraphLP and
+other classical link prediction methods. Extensive experimental
+results demonstrate the superiority of the proposed method over
+other state-of-the-art models and traditional baseline methods.
+This could be a fruitful avenue for future research aimed at ad-
+dressing graph learning tasks.
+Acknowledgment
+This work was partially supported by the National Natu-
+ral Science Foundation of China under Grant Nos. 62106030,
+61802039, 62272066; Chongqing Municipal Postdoctoral Sci-
+ence Foundation under Grant No.
+cstc2021jcyj-bsh0176;
+Chongqing Municipal Natural Science Foundation under Grant
+No. cstc2020jcyj-msxmX0804; the Chongqing Research Pro-
+gram of Basic Research and Frontier Technology under Grant
+No. cstc2021jcyj-msxmX0530.
+References
+[1] Robert Ackland et al. Mapping the us political blogosphere: Are conserva-
+tive bloggers more prominent? In BlogTalk Downunder 2005 Conference,
+Sydney. BlogTalk Downunder 2005 Conference, Sydney, 2005.
+[2] Lada A Adamic and Eytan Adar. Friends and neighbors on the web. Social
+networks, 25(3):211–230, 2003.
+[3] Yong-Yeol Ahn, James P Bagrow, and Sune Lehmann. Link communi-
+ties reveal multiscale complexity in networks. nature, 466(7307):761–764,
+2010.
+[4] James Atwood and Don Towsley.
+Diffusion-convolutional neural net-
+works. In Advances in neural information processing systems, pages 1993–
+2001, 2016.
+[5] Daniel Baird, J Luczkovich, and Robert R Christian. Assessment of spa-
+tial and temporal variability in ecosystem attributes of the st marks national
+wildlife refuge, apalachee bay, florida. Estuarine, Coastal and Shelf Sci-
+ence, 47(3):329–349, 1998.
+[6] Baruch Barzel and Albert-L´aszl´o Barab��asi. Network link prediction by
+global silencing of indirect correlations. Nature biotechnology, 31(8):720–
+725, 2013.
+[7] Austin R Benson, David F Gleich, and Jure Leskovec. Higher-order orga-
+nization of complex networks. Science, 353(6295):163–166, 2016.
+[8] Lei Cai, Jundong Li, Jie Wang, and Shuiwang Ji. Line graph neural net-
+works for link prediction. IEEE Transactions on Pattern Analysis and Ma-
+chine Intelligence, 2021.
+[9] Emmanuel J Cand`es, Xiaodong Li, Yi Ma, and John Wright. Robust prin-
+cipal component analysis? Journal of the ACM (JACM), 58(3):1–37, 2011.
+[10] Shaosheng Cao, Wei Lu, and Qiongkai Xu.
+Deep neural networks for
+learning graph representations. In Proceedings of the AAAI Conference on
+Artificial Intelligence, volume 30, 2016.
+[11] Zhiqian Chen, Fanglan Chen, Lei Zhang, Taoran Ji, Kaiqun Fu, Liang
+Zhao, Feng Chen, Lingfei Wu, Charu Aggarwal, and Chang-Tien Lu.
+Bridging the gap between spatial and spectral domains: A survey on graph
+neural networks. arXiv preprint arXiv:2002.11867, 2020.
+[12] Ara Cho, Junha Shin, Sohyun Hwang, Chanyoung Kim, Hongseok Shim,
+Hyojin Kim, Hanhae Kim, and Insuk Lee. Wormnet v3: a network-assisted
+hypothesis-generating server for caenorhabditis elegans. Nucleic acids re-
+search, 42(W1):76–82, 2014.
+[13] Aaron Clauset, Cristopher Moore, and Mark EJ Newman. Hierarchical
+structure and the prediction of missing links in networks.
+Nature, 453
+(7191):98–101, 2008.
+[14] Weilin Cong, Morteza Ramezani, and Mehrdad Mahdavi. On provable
+benefits of depth in training graph convolutional networks. Advances in
+Neural Information Processing Systems, 34:9936–9949, 2021.
+[15] Lucianoda F Costa, Marcus Kaiser, and Claus C Hilgetag. Predicting the
+connectivity of primate cortical networks from topological and spatial node
+properties. BMC systems biology, 1(1):1–17, 2007.
+[16] Micha¨el Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolu-
+tional neural networks on graphs with fast localized spectral filtering. In
+Proceedings of the 30th International Conference on Neural Information
+Processing Systems, pages 3844–3852, 2016.
+[17] Jordi Duch and Alex Arenas. Community detection in complex networks
+using extremal optimization. Physical review E, 72(2):027104, 2005.
+[18] Zhihong Fang, Shaolin Tan, Yaonan Wang, and Jinhu Lu. Elementary sub-
+graph features for link prediction with neural networks. IEEE Transactions
+on Knowledge and Data Engineering, 2021.
+12
+
+[19] Pablo M Gleiser and Leon Danon. Community structure in jazz. Advances
+in complex systems, 6(04):565–573, 2003.
+[20] Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for
+networks. In Proceedings of the 22nd ACM SIGKDD international confer-
+ence on Knowledge discovery and data mining, pages 855–864, 2016.
+[21] Roger Guimer`a and Marta Sales-Pardo. Missing and spurious interactions
+and the reconstruction of complex networks. Proceedings of the National
+Academy of Sciences, 106(52):22073–22078, 2009.
+[22] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation
+learning on large graphs. Advances in neural information processing sys-
+tems, 30, 2017.
+[23] Glen Jeh and Jennifer Widom. Simrank: a measure of structural-context
+similarity. In Proceedings of the eighth ACM SIGKDD international con-
+ference on Knowledge discovery and data mining, pages 538–543, 2002.
+[24] Meng Jiang, Peng Cui, Alex Beutel, Christos Faloutsos, and Shiqiang
+Yang. Detecting suspicious following behavior in multimillion-node so-
+cial networks. In Proceedings of the 23rd International Conference on
+World Wide Web, pages 305–306, 2014.
+[25] Leo Katz. A new status index derived from sociometric analysis. Psy-
+chometrika, 18(1):39–43, 1953.
+[26] Daniel Lee and H Sebastian Seung. Algorithms for non-negative matrix
+factorization.
+Advances in neural information processing systems, 13,
+2000.
+[27] Boning Li, Yingce Xia, Shufang Xie, Lijun Wu, and Tao Qin. Distance-
+enhanced graph neural network for link prediction. In ICML 2021 Work-
+shop on Computational Biology, 2021.
+[28] Guangcan Liu, Zhouchen Lin, Shuicheng Yan, Ju Sun, Yong Yu, and
+Yi Ma. Robust recovery of subspace structures by low-rank representa-
+tion. IEEE transactions on pattern analysis and machine intelligence, 35
+(1):171–184, 2012.
+[29] Francois Lorrain and Harrison C White. Structural equivalence of indi-
+viduals in social networks. The Journal of mathematical sociology, 1(1):
+49–80, 1971.
+[30] Paul Louis, Shweta Ann Jacob, and Amirali Salehi-Abari. Sampling en-
+closing subgraphs for link prediction. arXiv preprint arXiv:2206.12004,
+2022.
+[31] Linyuan L¨u, Mat´uˇs Medo, Chi Ho Yeung, Yi-Cheng Zhang, Zi-Ke Zhang,
+and Tao Zhou.
+Recommender systems.
+Physics reports, 519(1):1–49,
+2012.
+[32] Linyuan L¨u, Liming Pan, Tao Zhou, Yi-Cheng Zhang, and H Eugene Stan-
+ley. Toward link predictability of complex networks. Proceedings of the
+National Academy of Sciences, 112(8):2325–2330, 2015.
+[33] Mark EJ Newman. Finding community structure in networks using the
+eigenvectors of matrices. Physical review E, 74(3):036104, 2006.
+[34] Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning
+convolutional neural networks for graphs. In International conference on
+machine learning, pages 2014–2023. PMLR, 2016.
+[35] Gergely Palla, Imre Der´enyi, Ill´es Farkas, and Tam´as Vicsek. Uncovering
+the overlapping community structure of complex networks in nature and
+society. nature, 435(7043):814–818, 2005.
+[36] Liming Pan, Cheng Shi, and Ivan Dokmani´c. Neural link prediction with
+walk pooling. In International Conference on Learning Representations,
+pages 5171–5181, 2021.
+[37] Ratha Pech, Dong Hao, Liming Pan, Hong Cheng, and Tao Zhou. Link pre-
+diction via matrix completion. EPL (Europhysics Letters), 117(3):38002,
+2017.
+[38] Ratha Pech, Dong Hao, Yan-Li Lee, Ye Yuan, and Tao Zhou. Link pre-
+diction via linear optimization. Physica A: Statistical Mechanics and its
+Applications, 528:121319, 2019.
+[39] Erzs´ebet Ravasz and Albert-L´aszl´o Barab´asi. Hierarchical organization in
+complex networks. Physical review E, 67(2):026112, 2003.
+[40] Ryan Rossi and Nesreen Ahmed. The network data repository with inter-
+active graph analytics and visualization. In Twenty-ninth AAAI conference
+on artificial intelligence, 2015.
+[41] Marta Sales-Pardo, Roger Guimera, Andr´e A Moreira, and Lu´ıs A Nunes
+Amaral.
+Extracting the hierarchical organization of complex systems.
+Proceedings of the National Academy of Sciences, 104(39):15224–15229,
+2007.
+[42] Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt
+Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels.
+Journal of Machine Learning Research, 12(9), 2011.
+[43] Neil Spring, Ratul Mahajan, and David Wetherall. Measuring isp topolo-
+gies with rocketfuel. ACM SIGCOMM Computer Communication Review,
+32(4):133–145, 2002.
+[44] Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu
+Mei. Line: Large-scale information network embedding. In Proceedings
+of the 24th international conference on world wide web, pages 1067–1077,
+2015.
+[45] Ala Trusina, Sergei Maslov, Petter Minnhagen, and Kim Sneppen. Hi-
+erarchy measures in complex networks. Physical review letters, 92(17):
+178702, 2004.
+[46] Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero,
+Pietro Lio, and Yoshua Bengio. Graph attention networks. arXiv preprint
+arXiv:1710.10903, 2017.
+[47] Christian Von Mering, Roland Krause, Berend Snel, Michael Cornell,
+Stephen G Oliver, Stanley Fields, and Peer Bork. Comparative assess-
+ment of large-scale data sets of protein–protein interactions. Nature, 417
+(6887):399–403, 2002.
+[48] Xu-Wen Wang, Yize Chen, and Yang-Yu Liu. Link prediction through
+deep generative model. iScience, 23(10):101626, 2020.
+[49] Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-
+world’networks. nature, 393(6684):440–442, 1998.
+[50] Max Welling and Thomas N Kipf.
+Semi-supervised classification with
+graph convolutional networks. In J. International Conference on Learning
+Representations (ICLR 2017), 2016.
+[51] Tao Wu, Yuxiao Guo, Leiting Chen, and Yanbing Liu. Integrated struc-
+ture investigation in complex networks by label propagation. Physica A:
+Statistical Mechanics and its Applications, 448:68–80, 2016.
+[52] Tao Wu, Hongyu Ma, Chao Wang, Shaojie Qiao, Liang Zhang, and Shui
+Yu. Heterogeneous representation learning and matching for few-shot re-
+lation prediction. Pattern Recognition, page 108830, 2022.
+[53] Xingping Xian, Tao Wu, Shaojie Qiao, Xi-Zhao Wang, Wei Wang, and
+Yanbing Liu.
+Netsre:
+Link predictability measuring and regulating.
+Knowledge-Based Systems, 196:105800, 2020.
+[54] Xingping Xian, Tao Wu, Shaojie Qiao, Wei Wang, Chao Wang, Yanbing
+Liu, and Guangxia Xu. Deepec: Adversarial attacks against graph structure
+prediction models. Neurocomputing, 437:168–185, 2021.
+[55] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi
+Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs
+with jumping knowledge networks. In International conference on ma-
+chine learning, pages 5453–5462. PMLR, 2018.
+[56] Muhan Zhang and Yixin Chen. Weisfeiler-lehman neural machine for link
+prediction. In Proceedings of the 23rd ACM SIGKDD international con-
+ference on knowledge discovery and data mining, pages 575–583, 2017.
+[57] Muhan Zhang and Yixin Chen. Link prediction based on graph neural
+networks. In Proceedings of the 32nd International Conference on Neural
+Information Processing Systems, pages 5171–5181, 2018.
+[58] Muhan Zhang, Zhicheng Cui, Shali Jiang, and Yixin Chen. Beyond link
+prediction: Predicting hyperlinks in adjacency space. In Proceedings of
+the AAAI Conference on Artificial Intelligence, volume 32, 2018.
+[59] Jiajun Zhou, Jie Shen, and Qi Xuan. Data augmentation for graph clas-
+sification. In Proceedings of the 29th ACM International Conference on
+Information & Knowledge Management, pages 2341–2344, 2020.
+[60] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang,
+Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph
+neural networks: A review of methods and applications. AI Open, 1:57–
+81, 2020.
+[61] Tao Zhou, Linyuan L¨u, and Yi-Cheng Zhang. Predicting missing links
+via local information. The European Physical Journal B, 71(4):623–630,
+2009.
+13
+

_NFKT4oBgHgl3EQfUi2J/content/tmp_files/2301.11784v1.pdf.txt

@@ -0,0 +1,1418 @@
+Description of the cattle and small ruminants 
+trade network in Senegal and implication for 
+the surveillance of animal diseases 
+ 
+ 
+Mamadou Ciss1§, Alessandra Giacomini2,3,4§, Mame Nahé Diouf1, Alexis Delabouglise2,3, Asma 
+Mesdour2,3, Katherin Garcia Garcia2,3, Facundo Munoz2,3, Eric Cardinale2,3, Mbargou Lo5, Adji 
+Marème Gaye5, Mathioro Fall5, Khady Ndiaye5, Assane Guèye Fall1, Catherine Cetre-Sossah2,3, 
+Andrea Apolloni2,3 
+ 
+1 Institut Sénégalais de Recherches Agricoles/Laboratoire National de l’Elevage et de Recherches 
+Vétérinaires BP 2057 Dakar-Hann, Sénégal 
+2 CIRAD, UMR ASTRE, Montpellier, France 
+3 CIRAD, UMR ASTRE, Univ Montpellier, INRAE, Montpellier, France 
+4 Department of Biosciences, Swansea University, Swansea, SA2 8PP, UK 
+5 Direction des Services Vétérinaires, Dakar, Sénégal 
+§ M. C. and A. G. contributed equally 
+ 
+Corresponding author: Alessandra Giacomini; [email protected] 
+ 
+ 
+Abstract 
+Livestock mobility, particularly that of small and large ruminants, is one of the main pillars of 
+production and trade in West Africa: livestock is moved around in search of better grazing or sold in 
+markets for domestic consumption and for festival-related activities. These movements cover several 
+thousand kilometers and have the capability of connecting the whole West African region thus 
+facilitating the diffusion of many animal and zoonotic diseases. Several factors shape mobility 
+patterns even in normal years and surveillance systems need to account for such changes. In this 
+paper, we present a procedure based on temporal network theory to identify possible sentinel locations 
+using two indicators: vulnerability (i.e. the probability of being reached by the disease) and time of 
+infection (i.e. the time of first arrival of the disease). Using these indicators in our structural analysis 
+
+of the changing network enabled us to identify a set of nodes that could be used in an early warning 
+system. 
+As a case study we simulated the introduction of F.A.S.T. (Foot and Mouth Similar Transboundary) 
+diseases in Senegal and used data taken from 2020 Sanitary certificates (LPS – laissez-passer 
+sanitaire) issued by the Senegalese Veterinary Services to reconstruct the national mobility network. 
+Our analysis showed that a static approach can significantly overestimate the speed and the extent of 
+disease propagation, whereas temporal analysis revealed that the reachability and vulnerability of the 
+different administrative departments (used as nodes of the mobility network) change over the course 
+of the year. For this reason, several sets of sentinel nodes were identified in different periods of the 
+year, underlining the role of temporality in shaping patterns of disease diffusion.  
+ 
+Keywords: network analysis, livestock mobility, epidemiology, livestock production 
+ 
+1. Introduction 
+ 
+The West African region includes the southern part of the bulge in the African continent and is crossed 
+by the Sahel, a transitional strip between the Sahara Desert in the north and the Sudanic zone in the 
+south (Bossard, 2009). The region is composed of 18 countries and is bounded in the north by 
+Mauritania, Mali and Niger, in the east by Chad and Cameroon, in the south and west by the Atlantic 
+Ocean. The region is characterized by different climates, and hence, different agro-ecological zones 
+and different livestock farming  systems (Missohou et al., 2016). Livestock farming (particularly 
+cattle and small ruminants) is one of the most important economic activities in this area. 
+ 
+In West Africa, livestock mobility is an intrinsic component of livestock production and trade. The 
+harsh environmental conditions, as well as the absence of the facilities required to slaughter animals 
+and store meat, means livestock has to be mobile. To optimize the use of natural resources such as 
+pasture and surface water, whose availability varies throughout the year, livestock farmers are forced 
+to move their herds around: these movements occur all the year round (nomadism) or in specific 
+periods (transhumance). Because of the lack of storage facilities and infrastructure, the majority of 
+animals are sold alive at markets all year round. Most animals are concentrated in the northern part 
+of West Africa, notably in Mali, Chad, Niger, and Mauritania, where the vast uninhabited areas are 
+unsuitable for cropping but allow extensive livestock raising, and the animals are moved towards the 
+greener southern   coastal areas. These movements are seasonal, and depend both on the availability 
+of resources and on other socio-cultural factors, and the mobility patterns and the distribution of the 
+
+volume of animals involved change over the course of the year (Apolloni et al., 2019; Bouslikhane, 
+2015). These movements integrate the region, connect contrasted agro-ecological areas, and, in 
+addition, generate income for many supply chain actors, including producers, traders, transporters, 
+and vendors, and contribute to the food and nutrition security of the region (Valerio, 2020). 
+As it is, mobility in West Africa is a complex phenomenon involving different temporal scales (from 
+a few days to several months) and spatial scales (from a few kilometers to reach local markets to 
+international transhumance and/or international trade), and whose determinants range from 
+environmental factors, e.g. the availability of natural resources, commercial factors, e.g. market 
+demand and prices, to social factors, such as religious festivals (Apolloni et al., 2019).  
+ 
+In Senegal, livestock production is one of the main economic activities, it involves 28% of the 
+population (ANSD, 2013) and provides almost 4% (ANSD, 2020) of gross national domestic product. 
+Due to the different agro-ecological zones, several production systems co-exist. Senegal is located on 
+the Atlantic Coast axes of the transhumance routes (from Mauritania and Mali to Guinea and Guinea-
+Bissau) and is involved in international trade movements. Within Senegal, trade follows a strict 
+market hierarchy: from village markets to consumption markets in coastal areas. National 
+transhumance involves movements from the central and southern predominantly agricultural area 
+towards the area of Ferlo in the north. 
+Like in other West African countries, movements within and towards Senegal vary over the course 
+of the year. This is particularly true of the Tabaski religious festival, an important Muslim festival 
+characterized by the sacrifice of rams, and the Grand Magal of Touba, during which the consumption 
+of beef increases significantly. The two festivals mean imports of livestock increase enormously in a 
+short period of time (Apolloni et al., 2019; Cesaro et al., 2010). 
+Animal movements also mean pathogens can be introduced and spread at national and international 
+scales. Such pathogens spread very rapidly across national borders and have serious socio-economic 
+and public health consequences. Some of these, such as Contagious Bovine Pleuropneumonia 
+(CBPP), Foot-and-Mouth Disease (FMD), Peste des Petits Ruminants (PPR), and Rift Valley Fever 
+(RVF) are currently a major problem in West Africa (Apolloni et al., 2019; Bouslikhane, 2015; 
+Chaters et al., 2019; Di Nardo et al., 2011). 
+The porosity of the border, the absence of an animal identification system, together with the lack of 
+coordinated control and surveillance systems hinders the development of a regional surveillance 
+system and increases the risk of epidemics (Apolloni et al., 2019). Understanding mobility patterns, 
+as well as their variations, is of the uttermost importance to optimize surveillance and control systems. 
+Senegal is one of the few countries in West Africa already equipped with a system for mapping and 
+
+controlling animal movements within its borders. Movements are regulated through the use of 
+sanitary certificates (LPS – laissez-passer sanitaire), issued by the veterinary services to livestock 
+transporters every time they move animals. The certificates are also routinely collected and 
+centralized by the veterinary services. The information that can be retrieved from these data provides 
+a snapshot of the livestock mobility network at each period of the year and could be used to develop 
+tools to improve the surveillance system, adapted to the period concerned. 
+Network-based approaches are widely used in veterinary epidemiology to study the role of animal 
+mobility in the spread of diseases, with the aim of developing effective strategies for disease 
+surveillance and control (Dubé et al., 2009; Motta et al., 2017). Network-based approaches make it 
+possible to depict livestock movements as a spatial network in which the nodes represent villages, 
+administrative units, markets or herds, and a link is created each time at least one animal is moved 
+from one node to another. However, while network methods have been extensively applied to 
+engineer surveillance system in  European countries thanks to the existence of vast live animal 
+movement traceability datasets (i.e. Lentz et al. (Lentz et al., 2016) and Schirdewahn et al. 
+(Schirdewahn et al., 2021)), little has been done in West Africa due to the scarcity of such information 
+(Muwonge et al., 2021). Only a few articles that report network analysis in West Africa have been 
+published recently, including Apolloni et al. (Apolloni et al., 2018), and Nicolas et al. (Nicolas et al., 
+2018) for Mauritania, Jahel et al. (Jahel et al., 2020) for Senegal and Mauritania, and Valerio et al. 
+(Valerio et al., 2020) for the whole West Africa region. 
+Static network approaches may not be the best way to create effective surveillance and control tools 
+against the spread of infectious diseases,  as a static approach can overestimate or underestimate the 
+rate and extent of outbreaks (Masuda & Holme, 2013). The influence of temporality on the structure 
+of the network can significantly affect the spread of a disease, which consequently can only be 
+accurately predicted if the chronology of links is accurately represented (Masuda & Holme, 2013; 
+Williams & Musolesi, 2016). 
+ 
+In this work, we used a temporal network approach to assess the influence of change on the diffusion 
+of animal diseases over time. We used data collected in 2020 by the Senegalese Veterinary Services 
+to build a representation of the network, and adapted tools from complex networks to assess the risk 
+of being infected and the role of different Senegalese areas in spreading infections over the course of 
+the year. To this end, we relied on measures of the “vulnerability” and “reachability” of nodes. 
+Vulnerability gives an indication of the likelihood a node will be infected, while reachability gives 
+an indication of the time to infection. This approach takes changes in the network over time into 
+account as well as the network structure, and differs markedly from the static, individual-centric 
+
+approaches used in previous risk assessments. The objective of the present work is to provide a 
+theoretical basis for improving the Senegalese surveillance system by identifying different potential 
+geographical spots that contribute to the spread of pathogens at different times of the year and that 
+could be used as sentinel nodes. 
+  
+2. Materials and methods 
+ 
+ 
+2.1 Study area 
+Bordering Mauritania to the north, Mali to the east, Guinea and Guinea-Bissau to the south, and the 
+Gambia and the Atlantic Ocean to the west, Senegal occupies an area of 196,722 km2 and in 2020, 
+had an estimated population of more than 16.7 million (World Bank, 2022). 
+ 
+The administration of the Senegalese territory is organized in 14 regions, 45 departments, and 123 
+arrondissements (Ministère de l’Intérieur du Sénégal, 2017). There is a clear contrast between the 
+empty area in the east (hosting around 10% of the human population of Senegal) and the populated 
+and urbanized central and areas in the west, where 90% of human population is concentrated, of 
+which 25% in the Dakar area (ANSD, 2020; World Bank, 2022). 
+ 
+Senegal’s climate is very varied and distinct climatic zones are characterized by different levels of 
+rainfall and different types of vegetation. This diverse climate strongly influences the livestock 
+farming sector, whose different farming systems depending on agro-climatic gradients, among other 
+factors (Cesaro et al., 2010). 
+ 
+As mentioned above, the livestock trade is organized in a strict hierarchical system starting at village 
+weekly markets (Lumo), the animals are collected by traders to be sold at collection markets before 
+being sent on to consumer markets, where they are sold to be slaughtered. Because there are 
+practically no meat storage facilities, most trade involves live animals. Livestock trade routes 
+converge on the Dakar region, the main consumer market, with stops in smaller markets such as Saint-
+Louis, Touba, Thiès, and Kaolack. Before reaching the urban markets, the vast majority of the animals 
+originating from northern Senegal, Mauritania, and Mali, are grouped in Dahra, called the “livestock 
+capital” of Senegal. Another collection market in the southeastern part of the country also plays a 
+major role in the livestock trade: Tambacounda, the point of convergence for animals from eastern 
+
+and southern Senegal, as well as from southern Mauritania and Mali. In addition to the movement of 
+animals for sale, transhumance is widespread in Senegal, both at national scale from the central area 
+to the north (in particular the Ferlo region), and, due to its location, international, from Mali and 
+Mauritania to the Senegalese coast (Apolloni et al., 2019; Cesaro et al., 2010) (SI Figure 1). 
+ 
+2.2 Data 
+In Senegal, a certification system based on a sanitary pass named “Laissez-Passer Sanitaire” (LPS) 
+is used to track animal mobility and to map the most important axes of movement in the region. 
+Veterinary posts belonging to the Senegalese Ministry of Livestock and Animal Production provide 
+an LPS each time a herd is moved, the document states the origin of the movement (village, 
+department, region, country), the destination (village, department, region, country), the date, the 
+species and number of animals involved, and the means of transport. Copies of the LPS are centralized 
+and stored in electronic form. 
+ 
+We focused our analyses on movements of cattle and small ruminants (goats and sheep), either 
+separately or together. For analytical purposes, the two were aggregated on the spatial scale of an 
+administrative department (all 45 Senegalese departments are involved in this trade) and on a time 
+scale of one month or one week, depending on the type of analysis: we chose a month as the temporal 
+unit for the general description of the data and for cluster analysis, and a week to simulate the disease 
+spread, as a week is a more realistic unit to study disease propagation. 
+ 
+3. Methods 
+3.1 Descriptive and network analyses 
+We analyzed mobility data using a complex network approach. LPS data were used to build three 
+oriented and weighted networks, one for each species plus one species-independent network: the 
+nodes corresponded to the departments of origin and destination; a direct link existed between two 
+nodes if at least one animal was moved from the department of origin to the destination department; 
+the link was weighted according to the number of animals moved along it. 
+ 
+A cluster analysis was performed to explore the behavior of the different nodes over the study period. 
+The nodes were ranked based on their activity, defined as the effective number of animals traded each 
+month; the number being positive if the inflow of animals was greater than the outflow (importing 
+behavior), otherwise negative (exporting behavior). 
+
+ 
+Clustering was performed using HCPC (Hierarchical Clustering on Principal Components), which 
+successively applies three standard methods used in multivariate analyses: (i) Principal Component 
+Analysis (PCA), which identifies the principal components, (ii) hierarchical clustering, which defines 
+the optimal number of clusters of nodes according to their score on the principal components, and 
+(iii) non-hierarchical clustering (in particular the k-means algorithm), which associates a cluster with 
+each node (Celebi, 2015). 
+To study the structure of the livestock network, we conducted a spatio-temporal analysis of link’s 
+frequency, defined as the number of months in the year in which movements occur on the link. We 
+considered a link to be active when at least one trade movement was recorded in a given month. We 
+then categorized the links according to the number of months in which they were active. In particular, 
+we identified four frequency categories, which were, starting from the least frequent: occasional 
+(activity only occurred in one month per year), intermediate (activity occurred in two or three months 
+per year), frequent (activity in four to nine months per year), and backbone (activity in 10 to 12 
+months). 
+ 
+To compare the risk of diffusion over the course of the year, we used the epidemic threshold q 
+(Volkova et al., 2010). This measure provides information on the minimum probability for a virus to 
+spread throughout the network: the lower the value of the epidemic threshold, the higher the risk of 
+propagation. 
+For a weighted network, this parameter can be estimated as follows: 
+𝑞𝑤 = 
+〈𝑤𝑜𝑢𝑡〉
+〈𝑤𝑖𝑛  × 𝑤𝑜𝑢𝑡〉 
+where 〈 〉 indicates the average value, win and wout indicate the nodes’ in-weight and out-weight, 
+respectively (Nicolas et al., 2018). 
+ 
+Following the procedures of Lancelot et al. (Lancelot et al., 2017) and Nicolas et al. (Nicolas et al. 
+2018), for each of the three mobility networks considered (All species, Cattle, Small ruminants 
+separately), the epidemic threshold was estimated for each monthly snapshot of the network, to assess 
+the risk of an epidemic occurring over the course of the year. 
+ 
+3.2 Simulation of disease spread 
+Temporality, i.e. the variation in time of the mobility network, affects disease spread. Figure 1 – A 
+shows an example of a temporal network and its static counterpart. The network is composed of seven 
+
+nodes and eight possible links, whose direction is indicated by the arrows. In this case, the temporal 
+network is characterized by three temporal snapshots that contain the same nodes but different links. 
+A link that is present and active in a snapshot is not necessarily the same in the previous or the 
+following snapshots. If we disregard the information on timing, we obtain an aggregated/static 
+network composed of the same nodes and links as the temporal network, all present and active at the 
+same time. If we simulate an outbreak in the two networks (temporal and static) (Figure 1 – B), we 
+can see that the potential diffusion of the pathogen differs in the two situations. In this case, there is 
+significantly more propagation in the static network than in the temporal one. This happens because, 
+in the temporal network, the disease can only propagate through temporal paths. In other words, if a 
+link connecting an infected node to a susceptible one is active in a specific temporal snapshot, the 
+disease can spread to the latter; conversely, if the link is not active in the temporal window concerned, 
+disease propagation stops. 
+ 
+To study the influence of temporality on disease propagation, we simulated the spread of an animal 
+disease transmitted by direct contact through the livestock mobility networks. We used a SI 
+(Susceptible-Infected) model: the disease was transmitted from an infected node to a susceptible one 
+with a probability of 1, and the infected nodes remained infected for the entire period of analysis, and 
+were consequently able to continue to spread the disease even weeks after being infected. The aim of 
+this procedure was to estimate the number of potentially infected nodes when the underlying structure 
+varied. Because of our focus on the control of transboundary animal diseases, the departments of Mali 
+and Mauritania, which export livestock directly to Senegal, were chosen as sources of the disease, as 
+the majority of Senegalese imports of small ruminants and cattle are from these two countries 
+(Apolloni et al., 2019; Cesaro et al., 2010).   
+To explore the effect of temporality on the structure of the network, and hence on diffusion of the 
+disease, we compared results obtained with a static representation (in which the structure of the 
+network remains unchanged throughout the year) with results obtained with a temporal 
+representation. In the first case, all the links recorded in the dataset were present at the same time, the 
+time of activation was not taken into account, while in the second case, we included changes in the 
+structure in every week of the study period. To this end, we used temporal path formalism, according 
+to which a temporal path is a sequence of links connecting two nodes with each link in the path 
+coming temporally after the one before it (Masuda & Holme, 2013). This approach enabled us to 
+estimate the infection time: that is the minimum number of timesteps (i.e. weeks) needed to create a 
+temporal path between an infected node and the node under observation. 
+ 
+
+ 
+ 
+Figure 1: (A) An example of a directed temporal network and its static counterpart. The dark links are those in the temporal snapshot, 
+while the pale links are those that are possible but are not present in the temporal snapshot. (B) Simple simulation of disease spread 
+in the temporal network (on the left) and the static network (on the right). 
+Among all the possible temporal paths between the sources and the other nodes, we decided to 
+consider the “earliest arriving” paths, which represent the first time a node is infected by the disease 
+(Bender-deMoll et al., 2021; Berlingerio et al., 2013). The speed/rate at which a node became infected 
+was estimated by the infection time, i.e. the number of weeks that elapsed between the onset of the 
+disease and the time at which the department concerned was reached for the first time. For static 
+networks, the speed/rate of infection was estimated from the length of the shortest paths, converting 
+the links into temporal units, specifically, weeks. If a node was reached by more than one source, the 
+shortest infection time (for temporal networks) or the shortest path (for static networks) was chosen. 
+ 
+All descriptive analyses and static/temporal network analyses were carried out using  R software with 
+the following packages: ggplot2 for graphs (Wickham, 2016), ggplot2 and tmap for maps (Tennekes, 
+2018); FactoMineR (Lê et al., 2008) and factoextra (Kassambara & Mundt, 2020) for cluster analysis, 
+sna (Butts, 2020), and tsna (Bender-deMoll et al., 2021) for static and temporal network analysis, 
+respectively. 
+ 
+B
+A
+
+Temporal network
+Static network
+t2
+t1,3
+ti
+t2
+t3
+t1,3
+Susceptiblenode
+Infectednode
+Sourceofthe disease4. Results 
+4.1 Summary statistics 
+The database contained information on 8,861 livestock trade movements from January to December 
+2020. The network is composed of a total of 88 nodes, corresponding to an Administrative Unit of 
+level 2, of which 45 are Senegalese (Departments), and 590 unique links, i.e. origin-destination 
+combinations. The movements concerned Senegal as the origin and/or destination of 87% of the 
+movements, and eight other countries: Mali (9%), Gambia (2%), Mauritania (1%), Guinea, Guinea-
+Bissau, Burkina Faso, Niger and Nigeria (<1% each) as either the origin or as the destination of 
+movements. Focusing on Senegal, a total of 6,511 national movements and 2,350 international 
+movements, over respectively 458 and 132 unique links, involving 87,017 cattle and 553,718 small 
+ruminants were recorded in the dataset. Despite the large number of national trades, the majority of 
+animals were moved for the purpose of international trade. More than 95% of these movements were 
+in trucks, which is the most widely used means of transport for animals in the region concerned. More 
+than 600,000 animals were transported by truck, the remainder mainly on foot (Table 1). 
+ 
+The livestock network was analyzed as static but also took temporality into account, which influences 
+the presence/absence of links. 
+Table 1: summary of the characteristics of the data analyzed in the study. The number of movements, the number of animals and the 
+number of unique links are given for each species, type of trade, and means of transport. 
+ 
+ 
+Trade movements Headcount 
+Number of 
+unique links 
+Species 
+ 
+ 
+ 
+ 
+ 
+Cattle 
+3,186 
+87,017 
+328 
+ 
+Small Ruminants 5,675 
+553,718 
+502 
+Type 
+ 
+ 
+ 
+ 
+ 
+International 
+2,350 
+365,903 
+132 
+ 
+National 
+6,511 
+274,832 
+458 
+Means of 
+transport 
+ 
+ 
+ 
+ 
+ 
+Train 
+4 
+170 
+2 
+ 
+Truck 
+8,239 
+608,816 
+552 
+ 
+On foot 
+587 
+30,354 
+85 
+ 
+Boat 
+31 
+1,395 
+9 
+
+As shown in Figure 2, all Senegalese administrative departments are involved in animal trade either 
+as the origin, the destination, or both. Movements are both national and international, and, while 
+Senegal is the final destination of almost all the trade, many animals are moved not only from other 
+Senegalese departments, but also from Mali and Mauritania, the main exporters, with some 
+departments, particularly in Mali, exporting a significant number of animals (Figure 2– A). 
+ 
+The departments in north-eastern Senegal (Podor, Matam, Kanel, and Ranérou Ferlo), are notable for 
+their high level of animal “exports”. Other Senegalese departments (Tambacounda in the south, 
+Koungheul and Gossas in the center, Louga and Kébémer in the north) also export considerable 
+numbers of animals.  
+ 
+Concerning imports, the departments that import the most animals are located in the Dakar region, in 
+particular Pikine, Rufisque, Thiès, and M’bour, where the majority of consumer markets are located. 
+Other Senegalese departments that import large numbers of animals are Saint-Louis in the north, 
+Mbacké and Guinguinéo in the center, Ziguinchor in the south-west, Tambacounda and Sayara in the 
+south-east, on the border with Mali (Figure 2– B). 
+ 
+Figure 2: Distribution of the volume of animals in the administrative departments of Senegal, according to whether the department is 
+the origin (A) or the destination (B) of livestock movement. The miniature pie charts show the percentage of cattle (yellow) and small 
+ruminants (green) in the total number of animals. Quartiles were chosen for the colors representing the volume of animals traded. 
+Only countries that account for at least 1% of exports (A) or imports (B) are shown. 
+Figure 3 shows the number of movements and the volume of animals traded in each species (cattle or 
+small ruminants) per month. Overall, movements of animals for the purpose of trade were less 
+frequent in the first six months of the year, but increased in July, particularly trade in small ruminants. 
+Similarly, July was the month with the most trade in small ruminants in the study period, involving 
+N
+0
+50
+100
+150
+200 km
+Volume of animals
+0 to 9
+10 to 109
+110 to 724
+725 to 6337
+6338 to 209485
+Species
+Cattle
+Small Ruminants
+0
+50 100 150 200km
+A
+N
+0
+50
+100
+150
+200 km
+Volume of animals
+0 to 20
+21 to 560
+561 to 2438
+2439 to 8464
+8465 to 199744
+Species
+Cattle
+Small Ruminants
+B
+
+more than 300,000 animals. In August and September, the volume of small ruminants decreased, 
+while both the movement and volume of cattle traded increased, overtaking those of small ruminants. 
+In October, November and December, the number of cattle trades decreased, but remained higher 
+than in the rest of the year, while the number and volume of trade in small ruminants increased, 
+although less sharply. In 2020, two important Muslin festivals took place at the end of July (Tabaski) 
+and at the beginning of October (Grand Magal of Touba) and are represented on the chart by a dashed 
+line and a dotted line, respectively (Figure 3). 
+ 
+ 
+Figure 3: Number of trade movements (line plot) and volume of livestock traded (bar plot) recorded in 2020, per species and per 
+month. Data concerning cattle are in yellow, data concerning small ruminants are in green. The dashed line represents the Tabaski 
+festival (July 31), the dotted line represents the Grand Magal of Touba festival (October 6). 
+ 
+In the whole year, trades of small ruminants were concentrated on 503 links and trades in cattle on 
+329 links, including 242 links trades of both species. Like for small ruminants, the highest number of 
+unique trade links occurred in July, followed by, in decreasing order, December, November and 
+0
+500
+1,000
+1,500
+0
+100,000
+200,000
+300,000
+Jan.
+Feb.
+March
+April
+May
+June
+July
+Aug.
+Sept.
+Oct.
+Nov.
+Dec.
+Month
+Number of trade movements
+Volume of animals
+Number of trades
+Cattle
+Small ruminants
+Species
+Cattle
+Small ruminants
+
+October, also the months with the highest number of trade links for cattle. The links used by the two 
+species also increased in the last three months of the year (SI Table 1). 
+Concerning the means of transport, trucks were used for almost all movements of animals for sale 
+throughout the year. The number of movements peaked in July, and, then after a significant drop, 
+started to increase again in September (SI Table 2). 
+ 
+4.2 Cluster analysis 
+Figure 5 shows the nodes of the livestock network in three (3) clusters: 
+• Cluster 1, composed of 7 nodes and characterized by a “weak”1 exporting behavior; 
+• Cluster 2, composed of 61 nodes and characterized by a “strong”1 exporting behavior; 
+• Cluster 3, composed by 20 nodes and characterized by a “strong”1 importing behavior. 
+Cluster 1 (in red) aggregates seven nodes, of which four are located on the north-eastern border of 
+Senegal (Matam, Podor, Kanel, and Ranérou Ferlo) with high volumes of animals traded, while the 
+other three (Foundiougne, Kaffrine, Gossas) are located on the southern border of Dakar region 
+(Figure 4 – A). However, in September, Cluster 1 imports are “weak”, with a slightly less than 3,000 
+animals imported (Figure 4 – B). 
+ 
+Cluster 2 (in green) aggregates all the foreign nodes, except Banjul (Gambia), and several nodes 
+across Senegal, accounting for a total of 61 nodes out of 88. Cluster 2 is “strong” in terms of volume 
+of animals exported over the year, despite the fact some nodes import more than export. Some nodes 
+that export large numbers of livestock include Bamako (Mali, 208,462) and Nouakchott (Mauritania, 
+43,472), while Tambacounda (Senegal, 70,845) is a good example of an importing node (Figure 4 – 
+A). The highest number of exports by this cluster occurred in July, when the number of animals 
+exported was slightly under 200,000. (Figure 4 – B). 
+Cluster 3 (in blue) aggregates 20 nodes of which the majority is concentrated in the Dakar region but 
+includes some nodes in southern Senegal and one foreign node, Banjul (Gambia). Of the nodes 
+located in southern Senegal, two are on the border with Mali (Saraya and Kédougou), while the other 
+four are located farther west. All the nodes in this cluster were characterized by strong import trade, 
+with most imported animals via Pikine (Senegal, 199,703), but also via Thiès (Senegal, 51,939) and 
+Kaolack (Senegal, 46,432) (Figure 4 – A). Reflecting the movements of livestock for export, this 
+cluster shows a peak of imported animals in July, with a volume of around 250,000 animals, and 
+ 
+1  We introduce the terms importing and exporting behaviour to indicate those nodes whose net flow of animals (difference 
+between inflow and outflow) through them is positive and negative respectively. Weak and strong refer to magnitude of 
+the net flow (small or large). 
+
+another, 
+less 
+significant 
+increase 
+from 
+September 
+to 
+October 
+(Figure 
+4 
+– 
+B).
+ 
+Figure 4: Clustering of livestock network. (A) Spatial representation of nodes colored according to the cluster to which they belong. 
+The size of each dot indicates the volume of animals traded over the course of the year and the division is made in quantiles. (B) 
+Temporal representation of trade by the three clusters over the course of the year, in terms of the volume of animals traded. Imported 
+animals are represented as positive numbers, exported animals as negative numbers. The dashed line represents the Tabaski festival 
+on July 31, the dotted line represents the Grand Magal of Touba festival on October 6. 
+4.3 Frequency of links 
+Figure 5 shows the trade links divided by the frequency of their activity over the course of the year. 
+In general, far more links were characterized by low and very low activity than by very high activity. 
+ 
+The backbone links are three national, short-range connections between north-eastern and north-
+western nodes: Kanel – Pikine, Ranérou Ferlo – Linguère, Ranérou Ferlo – Mbacké. The majority of 
+frequent links is concentrated in the north of Senegal, where several connections link eastern and 
+western nodes, but some connections link northern and southern nodes. Moreover, some international 
+links are frequent, in particular four originating from Mali and one from Guinea-Bissau. The number 
+of intermediate links is significantly larger than that of the two previous categories, with several 
+connections between Senegal and Mali, and between Senegal and Mauritania. Occasional links are 
+extremely numerous and dense, with several connections in Senegal but also links to all its 
+neighboring countries (Figure 5). The majority of intermediate and occasional links are active in July 
+and October, due to the Tabaski and the Grand Magal of Touba religious festivals. However, 
+considering the whole study period, frequent links are the most common (SI Figure 4). 
+10°N
+12°N
+14°N
+16°N
+18°N
+20°N
+15°W
+10°W
+ 5°W
+ 0°
+ 5°E
+Longitude
+Latitude
+Cluster
+1
+2
+3
+Volume
+202
+1892
+8912
+208462
+A
+−200,000
+−100,000
+0
+100,000
+200,000
+Jan.
+Feb. March April May June July
+Aug. Sept. Oct. Nov. Dec.
+Month
+Volume of animals
+Cluster
+1
+2
+3
+B
+
+ 
+Figure 5: Geographical representation of the livestock network links, divided by the frequency of their activity over the year. Backbone 
+links were active in more than nine months, frequent links were active between four and nine months, intermediate links were active 
+for two or three months, and occasional links were active for only one month of the study period. We decided to only show the 
+administrative departments of Senegal on these maps. Therefore, concerning national trade, the origin and the destination are both 
+Senegalese departments, while for international trades, they may be a Senegalese department or a central point in a foreign country. 
+
+Backbone
+18°N
+Mauritania
+17N
+16°N
+15°N
+Mall
+No
+13°N
+12°N
+11°N
+Guinea
+Frequent
+18°N
+Mauritania
+17°N
+16°N
+15"N
+Mali
+14°N
+Gambia
+13°N
+12°N
+Guinea
+Frequency category
+Backbone
+Intermediate
+Frequent
+18°N
+Mauritania
+intermediate
+Occasional
+17°N
+16°N
+15°N
+14°N
+Gambia
+13-N
+12°N
+11°N
+Guinea
+Occasional
+18°N
+Mauritania
+17°N
+16°N
+Niger
+15°N
+Mali
+14°N
+Gambia
+13°N
+Burkina Faso
+12°N
+Nigeria
+11°N
+Guinea
+18W
+12°W
+10W
+M.8
+Mot
+Longitude4.4 Epidemic threshold 
+Overall, the values of the epidemic threshold of all three networks are extremely low, particularly for 
+the combined livestock and the small ruminants network, whose results are almost identical. April is 
+the only month with a significantly higher value than in the rest of the year. On the other hand, the 
+epidemic threshold values of the cattle network (yellow curve in Figure 6), are higher overall, with a 
+value of 1 in April. The lowest values of this network were measured in January, June, and from 
+October until the end of the year (Figure 6). 
+ 
+ 
+Figure 6: Logarithmic representation of changes in the epidemic threshold over the course of the year in the three livestock mobility 
+networks. The zero on the left extremity of the horizontal axis identifies the value calculated for the whole year. The small ruminants 
+network is in yellow, the cattle network in green, and the combined livestock network in blue. 
+ 
+4.5 Simulation of disease spread 
+ 
+Maps focused on Senegalese departments were drawn to compare the results of the simulations run 
+on the three networks in an efficient and easily understandable way (Figure 7). To assess the role of 
+changes in the structure of the networks over time, and hence changes in disease spread, we compared 
+the results of a static representation (in the column on the left) with those of a temporal representation 
+1e−04
+1e−02
+1e+00
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Epidemic Threshold
+Network
+Cattle
+Cattle and small ruminants
+Small ruminants
+
+(in the other seven columns). For the static representation, considering the time the outbreak began is 
+meaningless, whereas for the temporal one, it is important, as the network structure can change over 
+time. Therefore, each element in the seven columns representing the temporal networks corresponds 
+to the results of an outbreak that began in a specific week of the year (the number of the week is given 
+in the header of each map).  
+ 
+The departments are colored according to their infection time, i.e. the length of the period before they 
+were reached by the virus. For each scenario, the infection time was estimated as the time (number of 
+weeks) elapsed since the outbreak of the epidemic. We created four categories of infection time, each 
+category is shown in a different color: red for departments infected less than one month from the 
+beginning of the disease spread (less than 5 weeks), orange for departments infected after 1-2 months 
+(between 5 and 9 weeks), yellow for departments infected after more than two months (more than 9 
+weeks), and green for those never reached by the disease. If a node was reached by infections from 
+several sources, the shortest infection time was chosen. 
+ 
+For the static networks, we considered the number of links in the shortest path between the source 
+and the node as weeks: red for the shortest paths with less than 5 links, orange for paths with between 
+5 and 9 links, yellow for paths with more than 9 links, and green for nodes that were never reached 
+by the disease. 
+ 
+The results presented are those of the simulation of a disease spreading from Mali, the origin of most 
+animals imported into Senegal in 2020. For the temporal networks, we present only a few weeks 
+characterized by activity, in order to be able to simultaneously show changes over time and 
+differences between the three networks. The complete results of the spread of a disease from Mali 
+plus for a disease spreading from Mauritania can be found in Supplementary Information (SI Figure 
+5 – 10). 
+ 
+In general, in all three networks, maps representing aggregate networks strongly overestimated both 
+the quantity of potentially infected nodes and the earliness of infection, compared to those of temporal 
+networks. In addition, there is a difference in the potential sanitary risk between the cattle temporal 
+network and small ruminants temporal network, the latter showing on average wider and potentially 
+greater disease propagation. However, the combined network of cattle and small ruminants (the 
+livestock network) is under the greatest sanitary risk.  
+ 
+
+Figure 7 also shows that, particularly for the livestock network and the small ruminants network, the 
+periods around religious festivals (weeks 30 and 31 for the Tabaski, and weeks 40 and 41 for the 
+Grand Magal of Touba) are characterized by a large number of infected departments, some of which 
+are infected early.  
+ 
+ 
+Figure 7: Geographical representation of infection time in the case of disease propagation from Mali. For each mobility network, the 
+first column on the left represents the spread in the static network, the other seven columns represent the seven worst scenarios of 
+transmission if that specific week represents the beginning of the disease spread. The colors indicate the infection time of the disease: 
+red for less than one month, orange for less than two months, yellow for more than two months, green for nodes that have never been 
+touched in one year time. For the static networks in the first column on the left, the colors are based on the number of links in the path: 
+up to 5 in red, green for nodes that have never been touched in one year time. The squares outlined in blue identify the week of the 
+Tabaski festival (week 31) and the week of the Grand Magal of Touba festival (week 41). 
+5. Discussion and conclusions 
+In 2020, during the COVID-19 pandemic, several restrictive measures were introduced in Senegal 
+that affected both human and livestock mobility. Borders were closed for both humans and animals 
+in March 2020 and, at the same time, movements between regions were regulated. To supply markets 
+and families in preparation for the Tabaski festival on July 31, borders were reopened 45 days before 
+Tabaski and measures were eased for national and international movement (lettre circulaire n° 01806 
+PR/MESG/CT-PSS du 17 juin 2020). Similar decisions were taken on the occasion of the Grand 
+Magal of Touba, a religious pilgrimage during which a large number of cattle in particular are sold 
+and consumed. The application of these restrictions had a huge effect on the structure of the networks 
+and on the risk of introduction and diffusion of pathogens, as did re-opening the border. To assess the 
+impact of these measures on the spread of livestock diseases, and for possible future use, we used 
+tools from temporal network theory to identify the area with the highest risk of introduction. It is 
+
+Static
+All
+merimportant to note that normally, there are other religious festivals, like Gamou of Tivaouane, in 
+addition to the Grand Magal of Touba, but these were cancelled due to COVID-19 pandemic.  
+ 
+In our study, we considered the diffusion of a generic direct animal disease transmission and 
+estimated the vulnerability and the reachability of nodes when the underlined network changes over 
+time. In this way, we were able to identify Senegalese departments that could be infected at the earliest 
+stage of an epidemic. With a few modifications, our approach could be extended to include vector-
+borne diseases.  
+The structure of the Senegalese livestock network varies widely over the course of the year due to the 
+seasonality of transhumance and the effect of religious festivals (Apolloni et al., 2019; Jahel et al., 
+2020) but, in 2020, these effects were exacerbated by the restrictive measures introduced as a result 
+of Covid-19. In fact, around June and July, we noted a pickup in the movement and exchange of 
+animals (mainly small ruminants) mainly due to the easing of the restrictive measures in preparation 
+for the Tabaski festival and (mainly cattle) for the Grand Magal of Touba festival. We also noted that 
+the dynamics of the small ruminants trade strongly drive the dynamics of the network as a whole. 
+ 
+Dakar is the main consumption area of Senegal because almost a quarter of the population of the 
+country live in the city. Consequently, the main markets of Dakar and Pikine (at the entrance of 
+Dakar) are the main destination of livestock movements. In particular, regular movements occur 
+between the areas of Kanel, Ranerou Ferlo, Dahra and the Senegalese capital. In these areas, there is 
+a high concentration of collection markets (local name luma), where traders frequently buy animals 
+to be sold directly to Dakar, or to the other collection markets in Dahra or Thiès before being sent on 
+to the capital city. Overall, the majority of northern links end in the Dakar region, or in some smaller 
+but nevertheless important markets such as Saint-Louis, Thiès, Mbour, but also Ziguinchor in the 
+south. Some links in the southeast start from Tambacounda, an important point of convergence for 
+animals from eastern Senegal, as well as from Mauritania and Mali. Our analysis revealed that the 
+role played by the different departments changes over the course of the year. Locations that are idle 
+for a large part of the year become active during the Tabaski period and continue to be active until 
+the end of the year, in particular, departments that produce small ruminants. Occasional and 
+intermediate links that are active a few times a year, are usually located near festival centers, to 
+support the increased supply of livestock, thereby increasing the sanitary risk.  
+ 
+Analysis of the threshold parameters showed that the network is prone to disease spread, but that the 
+risk fluctuates over the course of the year. The risk increases significantly on the occasion of festivals 
+
+due to the introduction of large numbers of animals and the creation of new commercial routes, and 
+diseases can then spread easily across the network. However, the potential infected areas, and the 
+reachable time does not remain stable over the course of the year and this information cannot be 
+captured using a static representation of the network. In fact, a static representation of the mobility 
+patterns may largely overestimate the speed and the extent of disease diffusion: when the simple static 
+approach is used, diseases appear to spread throughout the country in less than a month, whereas 
+temporal analysis shows that reachability and vulnerability of departments varies over the course of 
+the year. In most cases, and depending on the species involved, few departments are reached in a 
+month, although during the months around Tabaski and Grand Magal, the number of departments that 
+can be reached increases drastically, and for some departments (like Dakar, Thiès, Tambacounda and 
+Dahra) where the main markets are located, and at the border, this risk is even higher. These results 
+could be of great interest not only for risk-based surveillance but also for optimizing the distribution 
+of resources and personnel needed for control at specific times of the year by focusing on the areas 
+that are most likely to be reached. The fact that departments located at the border are most prone to 
+early infection, means that sanitary control at the border should be strengthened and surveillance and 
+control measures should be harmonized at regional level. 
+ 
+Previous works already underlined the importance of mobility and of data collection as a tool to 
+improve surveillance and control in Africa (Chaters et al., 2019; Motta et al., 2017; Nicolas et al., 
+2018). Our work fits into this strand, emphasizing the importance of collecting data on animal 
+mobility on a regular basis in order to retrieve information on structural changes. The objective of the 
+present study is to provide theoretical tools to assess the importance of network dynamics when 
+planning control and surveillance policies. A more detailed analysis focused on specific diseases and 
+that accounts for volume distribution may reduce the list of departments to monitor. To this end, 
+further simulations are needed and their results will depend to a large extent on the characteristics of 
+the disease concerned, e.g. it transmissibility and incubation period, that could shape the spatio-
+temporal pattern of the epidemics and hence the involvement of the different departments. Future 
+works should thus also consider stochastic models like Kim et al. (2018) (Kim et al., 2018) for specific 
+diseases. In the model we used here, we aggregated data at the spatial scale of an administrative 
+department, based on the assumption that the diffusion within a department is homogeneous. In 
+practice, the presence of markets or transhumance corridors could attract movements in specific parts 
+of the department, thereby increasing the risk in certain locations over the risk in other parts of the 
+same department. Data on movements within departments were rare in our dataset (because of the 
+way data were collected) and further field studies are recommended to collect data at a finer scale. 
+
+ 
+ 
+6. Acknowledgments 
+The authors are grateful to Dr Mbargou Lô, head of the veterinary Services Directorate, and Dr 
+Mathioro Fall, head of Animal Health Protection Division. The authors acknowledge the support of 
+Yves Amevoin, Alioune Ka, Fallou Niakh, and Khady Ndiaye, and all those who assisted in the LPS 
+collection. We thank all the donors who supported the CGIAR Livestock research program through 
+their contributions to the CGIAR trust fund. 
+7. Funding 
+This study was partially funded by the Project Eco-PPR (European Commission through the 
+International Fund for Agricultural Development (grant number 2000002577) and the CGIAR 
+Livestock research program, RVF OIE twinning program (CIRAD-ISRA) granted through the EBO-
+SURSY project (European Union FOOD/2016/379-660). The funders had no role in study design, 
+data collection and analysis, decision to publish, or preparation of the manuscript. 
+ 
+8. Conflict of interest statement 
+All authors declare that they have no conflicts of interest. 
+ 
+9. References 
+ANSD. (2013). Recensement Général de la Population et de l’Habitat, de l’Agriculture et de l’Elevage 
+(RGPHAE). 
+ANSD. (2020). Situation économique et sociale du Sénégal Ed. 2017/2018 (p. 413). 
+Apolloni, A., Corniaux, C., Coste, C., Lancelot, R., & Toure, I. (2019). Livestock Mobility in West 
+Africa and Sahel and Transboundary Animal Diseases. In Transboundary Animal Diseases in 
+Sahelian Africa and Connected Regions (p. 31:52). Springer International Publishing. 
+https://doi.org/10.1007/978-3-030-25385-1 
+Apolloni, A., Nicolas, G., Coste, C., EL Mamy, A. B., Yahya, B., EL Arbi, A. S., Gueya, M. B., 
+Baba, D., Gilbert, M., & Lancelot, R. (2018). Towards the description of livestock mobility 
+
+in Sahelian Africa: Some results from a survey in Mauritania. PLoS ONE, 13(1). 
+https://doi.org/10.1371/journal.pone.0191565 
+Bender-deMoll, S., Morris, M., & Moody, J. (2021, aprile 23). Tools for Temporal Social Network 
+Analysis [R package tsna version 0.3.3]. Comprehensive R Archive Network (CRAN). 
+https://CRAN.R-project.org/package=tsna 
+Berlingerio, M., Coscia, M., Giannotti, F., Monreale, A., & Pedreschi, D. (2013). Evolving networks: 
+Eras and turning points. Intell. Data Anal. https://doi.org/10.3233/IDA-120566 
+Bossard, 
+L. 
+(2009). 
+Regional 
+Atlas 
+on 
+West 
+Africa. 
+Éditions 
+OCDE. 
+https://doi.org/10.1787/9789264056763-en 
+Bouslikhane, M. (2015). CROSS BORDER MOVEMENTS OF ANIMALS AND ANIMAL 
+PRODUCTS AND THEIR RELEVANCE TO THE EPIDEMIOLOGY OF ANIMAL 
+DISEASES IN AFRICA. OIE Regional Commission. 
+Butts, C. T. (2020, ottobre 6). Tools for Social Network Analysis [R package sna version 2.6]. 
+Comprehensive R Archive Network (CRAN). https://CRAN.R-project.org/package=sna 
+Celebi, M. E. (A c. Di). (2015). Partitional Clustering Algorithms. Springer International Publishing. 
+https://doi.org/10.1007/978-3-319-09259-1 
+Cesaro, J. D., Magrin, G., & Ninot, O. (2010). Atlas de l’elevage au Senegal: Commerces et 
+territoires. PRODIG. http://publications.cirad.fr/une_notice.php?dk=558823 
+Chaters, G., Johnson, P., Cleaveland, S., Crispell, J., De Glanville, W., Doherty, T., Matthews, L., 
+Mohr, S., Nyasebwa, O., Rossi, G., Salvador, L., Swai, E., & Kao, R. (2019). Analysing 
+livestock network data for infectious disease control: An argument for routine data collection 
+in emerging economies. Philosophical Transactions of the Royal Society B: Biological 
+Sciences, 374, 20180264. https://doi.org/10.1098/rstb.2018.0264 
+Di Nardo, A., Knowles, N. j, & Paton, D. j. (2011). Combining livestock trade patterns with 
+phylogenetics to help understand the spread of foot and mouth disease in sub-Saharan Africa, 
+the Middle East and Southeast Asia. 30(1), 63. https://doi.org/10.20506/rst.30.1.2022 
+
+Dubé, C., Ribble, C., Kelton, D., & McNab, B. (2009). A Review of Network Analysis Terminology 
+and its Application to Foot-and-Mouth Disease Modelling and Policy Development. 
+Transboundary and Emerging Diseases, 56(3), 73–85. https://doi.org/10.1111/j.1865-
+1682.2008.01064.x 
+Jahel, C., Lenormand, M., Seck, I., Apolloni, A., Toure, I., Faye, C., Sall, B., Lo, M., Diaw, C. S., 
+Lancelot, R., & Coste, C. (2020). Mapping livestock movements in Sahelian Africa. Scientific 
+Reports, 10. https://doi.org/10.1038/s41598-020-65132-8 
+Kassambara, A., & Mundt, F. (2020, aprile 1). Extract and Visualize the Results of Multivariate Data 
+Analyses [R package factoextra version 1.0.7]. Comprehensive R Archive Network (CRAN). 
+https://CRAN.R-project.org/package=factoextra 
+Kim, Y., Dommergues, L., M’sa, A. B., Mérot, P., Cardinale, E., Edmunds, J., Pfeiffer, D., Fournié, 
+G., & Métras, R. (2018). Livestock trade network: Potential for disease transmission and 
+implications for risk-based surveillance on the island of Mayotte. Scientific Reports, 8(1), 
+11550. https://doi.org/10.1038/s41598-018-29999-y 
+Lancelot, R., Béral, M., Rakotoharinome, V. M., Andriamandimby, S.-F., Héraud, J.-M., Coste, C., 
+Apolloni, A., Squarzoni-Diaw, C., de La Rocque, S., Formenty, P. B. H., Bouyer, J., Wint, G. 
+R. W., & Cardinale, E. (2017). Drivers of Rift Valley fever epidemics in Madagascar. 
+Proceedings 
+of 
+the 
+National 
+Academy 
+of 
+Sciences, 
+114(5), 
+938–943. 
+https://doi.org/10.1073/pnas.1607948114 
+Lê, S., Josse, J., & Husson, F. (2008). FactoMineR: A Package for Multivariate Analysis. Journal of 
+Statistical Software, 25(1), 1–18. https://doi.org/10.18637/jss.v025.i01 
+Lentz, H. H. K., Koher, A., Hövel, P., Gethmann, J., Sauter-Louis, C., Selhorst, T., & Conraths, F. J. 
+(2016). Disease Spread through Animal Movements: A Static and Temporal Network 
+Analysis 
+of 
+Pig 
+Trade 
+in 
+Germany. 
+PloS 
+One, 
+11(5), 
+e0155196. 
+https://doi.org/10.1371/journal.pone.0155196 
+
+Masuda, N., & Holme, P. (2013). Predicting and controlling infectious disease epidemics using 
+temporal networks. F1000Prime Reports, 5, 6. https://doi.org/10.12703/P5-6 
+Ministère de l’Intérieur du Sénégal. (2017). Politique de Gouvernance intérieure | Ministère de 
+l’Intérieur. https://interieur.sec.gouv.sn/administration-territoriale/politique-de-gouvernance-
+interieure 
+Missohou, A., Nahimana, G., Ayssiwede, S. B., & Sembene, M. (2016). Elevage caprin en Afrique 
+de l’Ouest: Une synthèse. Revue d’élevage et de médecine vétérinaire des pays tropicaux, 
+69(1), 3. https://doi.org/10.19182/remvt.31167 
+Motta, P., Porphyre, T., Handel, I., Hamman, S. M., Ngu Ngwa, V., Tanya, V., Morgan, K., Christley, 
+R., & Bronsvoort, B. M. deC. (2017). Implications of the cattle trade network in Cameroon 
+for 
+regional 
+disease 
+prevention 
+and 
+control. 
+Scientific 
+Reports, 
+7. 
+https://doi.org/10.1038/srep43932 
+Muwonge, A., Bessell, P. R., Porphyre, T., Motta, P., Rydevik, G., Devailly, G., Egbe, N. F., Kelly, 
+R. F., Handel, I. G., Mazeri, S., & Bronsvoort, B. M. deC. (2021). Inferring livestock 
+movement networks from archived data to support infectious disease control in developing 
+countries. https://doi.org/10.1101/2021.03.18.435930 
+Nicolas, G., Apolloni, A., Coste, C., Wint, G. R. W., Lancelot, R., & Gilbert, M. (2018). Predictive 
+gravity models of livestock mobility in Mauritania: The effects of supply, demand and cultural 
+factors. PLoS ONE, 13(7). https://doi.org/10.1371/journal.pone.0199547 
+Schirdewahn, F., Lentz, H. H. K., Colizza, V., Koher, A., Hövel, P., & Vidondo, B. (2021). Early 
+warning of infectious disease outbreaks on cattle-transport networks. PLOS ONE, 16(1), 
+e0244999. https://doi.org/10.1371/journal.pone.0244999 
+Tennekes, M. (2018). tmap: Thematic Maps in R. Journal of Statistical Software, 84(6). 
+https://doi.org/10.18637/jss.v084.i06 
+Valerio, V. C. (2020). The structure of livestock trade in West Africa (West African Papers Fasc. 29; 
+West African Papers, Vol. 29). https://doi.org/10.1787/f8c71341-en 
+
+Valerio, V. C., Walther, O. J., Eilittä, M., Cissé, B., Muneepeerakul, R., & Kiker, G. A. (2020). 
+Network analysis of regional livestock trade in West Africa. PLoS ONE, 15(5). 
+https://doi.org/10.1371/journal.pone.0232681 
+Volkova, V. V., Howey, R., Savill, N. J., & Woolhouse, M. E. J. (2010). Sheep Movement Networks 
+and 
+the 
+Transmission 
+of 
+Infectious 
+Diseases. 
+PLoS 
+ONE, 
+5(6), 
+e11185. 
+https://doi.org/10.1371/journal.pone.0011185 
+Wickham, H. (2016). ggplot2: Elegant Graphics for Data Analysis (2nd ed. 2016). Springer 
+International Publishing : Imprint: Springer. https://doi.org/10.1007/978-3-319-24277-4 
+Williams, M. J., & Musolesi, M. (2016). Spatio-temporal networks: Reachability, centrality and 
+robustness. Royal Society Open Science, 3(6), 160196. https://doi.org/10.1098/rsos.160196 
+World 
+Bank. 
+(2022). 
+Population 
+Senegal 
+[Text/HTML]. 
+World 
+Bank. 
+https://www.worldbank.org/en/country/senegal/overview 
+ 
+ 
+ 
+
+Supplementary information 
+ 
+ 
+ 
+SI Figure 1: Map of Senegal colored based on the aridity index. The inset map shows all the countries involved in livestock mobility 
+network. 
+ 
+ 
+
+V
+Mauritania
+Mali
+Niger
+SENEGAL
+MAURITANIA
+Gambia
+Burkina Faso
+Podor
+suin
+Nigeria
+Senegal
+Saint-Louis
+RiverValley
+Matam
+Louga
+Sylvopastoral
+Dahra
+zone
+Kanel
+Pikine
+Tivaouane
+Touba
+Thies
+Mbake
+DAKAR
+Rur
+oue
+Diourbel
+M'bour
+Groundnut
+-Kaolack
+basin
+Tambacounda
+Eastern
+MALI
+Senegal
+GAMBIA
+casamdne
+OKolda
+Ziguinchor
+GUINEA-BISSAU
+GUINEA
+0
+100
+200km
+Aridity Index
+Maincities(population）
+Agro-ecologicalzones
+0.03 -0.2
+Morethan500,000
+Borders
+0.2-0.35
+100,000-500,000
+0.35-0.5
+Main rivers
+50000-100,000
+0.5-0.65
+o
+Lessthan50,000
+0.65-0.8 
+ 
+Small Ruminants 
+Cattle 
+Links in common 
+Whole year 
+ 
+503 
+329 
+242 
+Months 
+ 
+ 
+ 
+ 
+ 
+January 
+42 
+27 
+13 
+ 
+February 
+52 
+27 
+22 
+ 
+March 
+59 
+30 
+19 
+ 
+April 
+26 
+18 
+12 
+ 
+May 
+27 
+22 
+11 
+ 
+June 
+71 
+28 
+20 
+ 
+July 
+202 
+47 
+37 
+ 
+August 
+29 
+21 
+6 
+ 
+September 
+24 
+41 
+14 
+ 
+October 
+131 
+123 
+66 
+ 
+November 
+163 
+113 
+69 
+ 
+December 
+184 
+128 
+82 
+SI Table 1: Number of unique trade links in the small ruminant network and in the cattle network. The values are represented divided 
+by month, while the “whole year” line corresponds to the number of unique links considering the network as static. The last column 
+shows the number of links that are used by the two species. 
+ 
+ 
+
+ 
+ 
+Truck 
+Others 
+Links in common 
+ 
+ 
+Water 
+Walking 
+Train 
+Whole year 
+ 
+552 
+9 
+85 
+2 
+55 
+Months 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+January 
+49 
+0 
+8 
+0 
+1 
+ 
+February 
+56 
+0 
+3 
+0 
+2 
+ 
+March 
+67 
+0 
+3 
+0 
+0 
+ 
+April 
+31 
+0 
+1 
+0 
+0 
+ 
+May 
+38 
+0 
+0 
+0 
+0 
+ 
+June 
+73 
+0 
+9 
+0 
+3 
+ 
+July 
+207 
+0 
+11 
+0 
+6 
+ 
+August 
+43 
+0 
+3 
+0 
+2 
+ 
+September 
+42 
+3 
+11 
+0 
+5 
+ 
+October 
+178 
+1 
+17 
+1 
+9 
+ 
+November 
+178 
+3 
+33 
+0 
+7 
+ 
+December 
+219 
+3 
+20 
+1 
+11 
+SI Table 2: Number of unique trade links according to the means of transport. The values are given per month, while the “whole year” 
+line corresponds to the number of unique links considering the network as static. The last column shows the number of links that are 
+shared by the movements made by truck and those made by all other types of transport, including on foot. 
+ 
+ 
+ 
+
+ 
+ 
+SI Figure 2: volume of livestock traded in 2020 divided by week. Cattle are shown in yellow and small ruminants are in green. The 
+black dashed line represents the day of the Tabaski festival (July 31), the black dotted line represents the day of the Grand Magal of 
+Touba (October 6). The months are indicated by the gray dashed lines. 
+ 
+ 
+ 
+SI Figure 3: Graphical visualization of methods chosen to assess the number of clusters: (A) Elbow method, (B) cluster dendrogram, 
+and (C) cluster division with the HCPC (Hierarchical Clustering on Principal Components) function of the FactoMineR package. 
+
+120.000
+B0T000
+folume of animal
+WCK
+Spedes
+Cefle
+Smel rumnantOptimalnumberofclusters
+Factormap
+5e+05
+ofSqu
+4e+05
+2e+05
+2
+3
+4
+5
+6
+10
+cluster
+Numberofclustersk
+ClusterDendrogram
+4-
+Height
+-2.
+Dim1 (50.9%) 
+ 
+SI Figure 4: Activity of the links over the course of the year. For each month, the quantity of active links is shown, colored according 
+to their frequency. Backbone links are those active more than nine months a year, frequent links are those active from four to nine 
+months, intermediate links are those active two or three months, and occasional links are only active one month. The orange line 
+represents the Tabaski festival (July 31), the violet line represents the Grand Magal of Touba festival (October 6). 
+
+200
+150
+100
+50
+Jan.
+Fob.
+April
+May
+Juy
+Aug.
+Sept.
+Ort.
+Nov.
+Dec.
+Month
+Berckbone
+I feguon
+I reguency category
+Intermeclafe
+Occasional 
+ 
+SI Figure 5: Geographical representation of infection time, in the case of a disease propagated from Mali through the livestock 
+network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the 
+path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been reached are in green. 
+
+Infectiontime
+Withinonemonth
+Withintwomonths
+Aftertwomonths
+Never
+42 
+ 
+SI Figure 6: Geographical representation of infection time, in the case of a disease propagated from Mali through the small ruminant 
+network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the path: 
+up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. 
+
+Infectiontime
+Withinonemonth
+Withintwomonths
+Aftertwomonths
+Never 
+ 
+SI Figure 7: Geographical representation of infection time in the case of a disease propagated from Mali through the cattle network. 
+Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the path: up to 5 
+in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. 
+
+Infectiontime
+Withinonemonth
+Withintwomonths
+Aftertwomonths
+Never 
+ 
+SI Figure 8: Geographical representation of infection time in the case of a disease propagated from Mauritania through the livestock 
+network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the 
+path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. 
+
+Infectiontime
+Withinonemonth
+Withintwomonths
+Aftertwomonths
+Never 
+ 
+ 
+SI Figure 9: Geographical representation of infection time in the case of a disease propagated from Mauritania through the small 
+ruminant network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links 
+in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. 
+ 
+ 
+
+Infectiontime
+Withinonemonth
+Withintwomonths
+Aftertwomonths
+Never
+45 
+ 
+SI Figure 10: Geographical representation of infection time in the case of a disease propagated from Mauritania through the cattle 
+network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the path: 
+up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. 
+
+37
+38
+39
+Infectiontime
+42
+Withinonemonth
+Withintwomonths
+Aftertwomonths
+Never
+Gt

btFAT4oBgHgl3EQf5R5J/content/tmp_files/2301.08732v1.pdf.txt

@@ -0,0 +1,2691 @@
+Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime
+Rick Bebon and Aljaˇz Godec∗
+Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 G¨ottingen, Germany
+We derive general bounds on the probability that the empirical first-passage time τ n ≡ �n
+i=1 τi/n
+of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates
+from the true mean first-passage time by more than any given amount in either direction. We
+construct non-asymptotic confidence intervals that hold in the elusive small-sample regime and
+thus fill the gap between asymptotic methods and the Bayesian approach that is known to be
+sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. Our
+concentration-of-measure-based results allow for model-free error control and reliable error estimation
+in kinetic inference, and are thus important for the analysis of experimental and simulation data in
+the presence of limited sampling.
+The first-passage time τ denotes the time a random pro-
+cess reaches a threshold a, typically referred to as the “tar-
+get”, for the first time. First-passage times [1–4] quantify
+the kinetics of chemical reactions [5–10], cell signaling and
+gene regulation in the low-copy [11–20] and “fastest en-
+counter” limits [21–29], intracellular transport [30], RNA
+biosynthesis [31], protein accumulation [32, 33] and DNA-
+binding [34], emergence of drug resistance [35], virus up-
+take [36], spreading of diseases [37, 38], and the foraging
+behavior of bacteria and animals [39]. First-passage the-
+ory was further applied to nanocluster formation [40], cell
+adhesion [41–43], gating of ion channels [44], and diffusion
+through interfaces [45] and across phase boundaries [46].
+In more abstract settings, first-passage times charac-
+terize barrier-crossing in energy landscapes [6, 23, 47–
+54], persistence properties [55–61], and the statistics of
+stochastic currents [62, 63], thermodynamic entropy pro-
+duction [64–67], and dynamical activity [68, 69] in non-
+equilibrium systems. First-passage ideas are intimately
+tied to the statistics of extremes [70–73], and were ex-
+tended to quantum systems [74, 75], additive functionals
+of stochastic paths [76–81], intermittent targets [82–85],
+active particles [86, 87], non-Markovian dynamics [88–91],
+and processes under resetting [92–101].
+Whereas theoretical studies focus on predicting first-
+passage statistics, practical applications typically aim
+at inferring kinetic rates—inverse mean first-passage
+times—from experimental [52, 102–106] or simulation
+data [51, 107–113]. The inference of empirical first-passage
+times τ n ≡ �n
+i=1 τi/n from data is, however, challenging
+because usually only a small number of realizations n
+(typically 1-10 [113–118], sometimes up to 100 [119]) is
+available, which gives rise to large uncertainties and non-
+Gaussian errors. Insufficient sampling is especially detri-
+mental in the case of broadly distributed [51, 120, 121]
+and high-dimensional data [106]. Moreover, first-passage
+times are generically not exponentially distributed [8, 9,
+17, 19, 23, 24, 122–127], which further complicates quan-
+tification of uncertainty. A systematic understanding of
+statistical deviations of the empirical from the true mean
+first-passage time (see Fig. 1a), especially in the small-
+sample n ≲ 100 regime, remains elusive.
+Computer simulations in particular often suffer from
+FIG. 1. Deviations of empirical first-passage times from the
+true mean and model systems. (a) Schematic probability den-
+sity of empirical first-passage time τ n inferred from a sample
+of n realizations of an ergodic reversible Markov process. The
+tail probability that the estimate τ n deviates from the true
+mean ⟨τ⟩ by more or equal than t upwards P(τ n ≥ ⟨τ⟩ + t)
+or downwards P(τ n ≤ ⟨τ⟩ − t) is shown in green and blue,
+respectively. (b) Brownian molecular search process in a d-
+dimensional domain (here d = 2) with outer radius R and
+target radius a. Discrete-state Markov jump models of protein
+folding for (c) a toy protein and (d) experimentally inferred
+model of calmodulin [124]. Transitions between states are in-
+dicated by arrows and obey detailed balance. For all systems
+considered the absorbing target is colored red.
+insufficient sampling, which leads to substantial errors in
+inferred rates [128–131] and, in the worst case, erroneous
+conclusions (see discussion in [113, 132]). Even extensive
+computing resources may result in only a few indepen-
+dent estimates spread over many orders of magnitude,
+rendering uncertainty quantification challenging and not
+amenable to standard error analysis [116].
+Constructing reliable confidence intervals is a fundamen-
+tal challenge in statistical inference, and many prevalent
+methods rely on asymptotic arguments that hold when
+the number of realizations tends to infinity. However, the
+applicability of asymptotic results in a finite-sample set-
+ting is, by definition, problematic. In particular, Central-
+Limit- and bootstrapping-based methods [133] may easily
+arXiv:2301.08732v1  [cond-mat.stat-mech]  20 Jan 2023
+
+2
+underestimate the uncertainty for small n and fail to guar-
+antee coverage of the confidence level [116, 134–139].
+Conversely, Bayesian methods (see e.g. [140]) do not
+rely on asymptotic arguments and are therefore often (in
+general erroneously [141, 142]) believed to readily alle-
+viate the small-sample problem. Bayesian estimates are
+sensitive to, dependent on, and potentially biased by, the
+specification of the prior distribution, especially in the
+small-sample setting [140, 143–145]. Due to the prior
+dependence of estimates and their uncertainties, Bayesian
+methods must be treated with care when applied to small
+samples [146, 147] (see [123, 129, 148–150] specifically for
+kinetic inference) and can perform worse than asymptotic
+frequentist methods [146].
+Moreover, so-called “credible intervals”—the Bayesian
+analogue to confidence intervals—have a nominally differ-
+ent meaning, as they treat the estimated parameter as a
+random variable. Bayesian posterior intervals are similarly
+affected by limited sampling [116], i.e. the constructed
+uncertainty estimates and their quality are sensitive to
+the choice of prior probability [141, 142] and may likely
+underestimate the true uncertainty and thus fail to pro-
+vide trustworthy confidence intervals [129, 151].
+On a more subtle level, the classical Bernstein-von-
+Mises theorem establishes a rigorous (frequentist) justi-
+fication of posterior-based Bayesian credible intervals as
+asymptotically correct, prior independent confidence inter-
+vals for (finite dimensional) parametric models in the large-
+sample limit [152–154]. Analogous statements for semi-
+parametric and (infinite dimensional) non-parametric
+models are more delicate [155–158] and, despite having
+received signifficant attention [159–170] (see also [171]
+for misspecified and high dimensional [172] parametric
+models), seem to remain—even in the asymptotic, large-
+sample regime—an elusive problem.
+There is thus a pressing need for understanding fluctu-
+ations of inferred empirical first-passage times, a rigorous
+error control, and reliable non-asymptotic error estima-
+tion in the small-sample regime. These are fundamental
+problems of statistical kinetics and are essential for the
+analysis of experimental and simulation data.
+Here, we present general bounds on fluctuations of
+empirical first-passage times that allow a rigorous uncer-
+tainty quantification (e.g. using confidence intervals with
+guaranteed coverage probabilities for all sample sizes)
+under minimal assumptions. We prove non-asymptotic
+lower (L) and upper (U) bounds on the deviation proba-
+bility P(τ n ≥ ⟨τ⟩ + t) and P(τ n ≤ ⟨τ⟩ − t) (see Fig. 1a),
+i.e., the probability that the empirical first-passage time
+inferred from a sample of n ≥ 1 realizations of an ergodic
+reversible Markov process, τ n, deviates from the true
+mean ⟨τ⟩ by more than t in either direction,
+L±
+n (t) ≤ P(±[τ n − ⟨τ⟩] ≥ t) ≤ U±
+n (t)
+∀t ≥ 0,
+(1)
+the upper bounds U±
+n (t) corresponding to so-called concen-
+tration inequalities [173]. The most conservative version
+of the derived upper bounds is independent of any details
+about the underlying dynamics. The validity and sharpness
+of the bounds are demonstrated by means of spatially con-
+fined Brownian molecular search processes in dimensions
+1 and 3 (Fig. 1b), and discrete-state Markov jump models
+of protein folding for a toy protein [24, 129, 174, 175]
+(Fig. 1c) and the experimentally inferred model of calmod-
+ulin [124] (Fig. 1d). We use the bounds U±
+n (t) to quantify
+the uncertainty of the inferred sample mean τ n in a gen-
+eral setting and under minimal assumptions, for all n ≥ 1.
+We conclude with a discussion of the practical implications
+of the results and further research directions.
+Setup.—We consider time-homogeneous Markov pro-
+cesses xt on a continuous or discrete state-space Ω with
+(forward) generator ˆL corresponding to a Markov rate-
+matrix or an effectively one-dimensional Fokker-Planck
+operator. Let the transition probability density to find
+xt at x at time t given that it evolved from x0 be
+pt(x|x0) ≡ eˆLtδx0(x) where δx0(x) denotes the Dirac
+or Kronecker delta for continuous and discrete state-
+spaces, respectively. We assume the process to be ergodic
+limt→∞ pt(x|x0) = peq(x), where peq(x) ≡ e−ϕ(x) denotes
+the equilibrium probability density and ϕ(x) the general-
+ized potential in units of thermal energy kBT [176]. We
+assume that ˆL obeys detailed balance [177] and is either
+(i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω,
+or (iii) Ω is infinite but ϕ(x) sufficiently confining (see
+[178]). Each of the conditions (i)-(iii) ensures that the
+spectrum of ˆL is discrete [179].
+We are interested in the first-passage time to a target
+a when xt=0 is drawn from a density p0(x)
+τ = inf
+t [ t |xt = a, p0(x0)],
+(2)
+and focus on p0(x) = ˜peq(x) where the tilde denotes that
+the absorbing state is excluded [180]. For completeness we
+also provide in [181] results for general initial conditions
+p0(x) that require more precise conditions on ϕ(x) [182].
+The probability density of τ for such processes has the
+generic form [23, 24]
+℘a(t|x0) =
+�
+k>0
+µkwx0
+k e−µkt,
+(3)
+where µk > 0 denote first-passage rates and wx0
+k
+the
+(not necessarily positive) spectral “weights” normalized
+according to �
+k>0 wx0
+k
+= 1 and wx0
+1
+> 0. The m-
+th moment of τ is given by ⟨τ m⟩ = m! �
+k>0 wx0
+k /µm
+k
+and the survival probability
+reads P(τ
+>
+t)
+≡
+Sa(t|x0)
+=
+�
+k>0 wx0
+k e−µkt.
+If x0 is drawn from
+the equilibrium density, ˜peq(x), we have ℘a(t|˜peq) ≡
+�
+Ω\a ℘a(t|x0)˜peq(x0)dx0 [183] which renders all weights
+non-negative, wk ≡
+�
+Ω\a wx0
+k ˜peqdx0 ≥ 0 (see proof in
+[181]). We henceforth abbreviate Sa(t|˜peq) ≡ Sa(t).
+To examplify the need for uncertainty bounds in Eq. (1)
+we show in Fig. 2a-d that the probability that τ n − ⟨τ⟩
+lies within a desired range of say ± 10% of the longest
+first-passage time scale µ−1
+1 , P(µ1[τ n − ⟨τ⟩] ∈ [−0.1, 0.1])
+is low even for n ≈ 50 for all models in Fig. 1b-d.
+Lower bounds on deviation probability.—There exists
+a “noise floor” for τ n for any n. Since µk ≤ µk+1 and
+
+3
+FIG. 2.
+Deviation probabilities and corresponding bounds for a spatially confined Brownian search process in (a,e) d = 1 and
+(b,f) d = 3 dimensions, and Markov-jump models of protein folding for (c,g) the experimentally inferred model of calmodulin
+and (d,h) the toy protein. (a-d) Probability that δτ n = τ n − ⟨τ⟩ lies within a range of ±10% of the longest time-scale 1/µ1,
+P(µ1δτ n ∈ [−0.1, 0.1]), as a function of n determined from the statistics of τ n for different fixed n for all model systems. (e-h)
+Scaled probabilities P1/n(sgn(t)δτ n ≥ |t|) that the sample mean τ n inferred from n realizations deviates from ⟨τ⟩ by more
+than t in either direction. Right tail areas are shown for t > 0 and left for t < 0, respectively. Lower L±
+n (t) and upper U±
+n (t; C)
+bounds are depicted as red and black lines, respectively, and the model-free upper bound U±
+n (t; 2) as the dashed yellow line.
+Symbols denote corresponding scaled empirical deviation probabilities as a function of t and are sampled for different n.
+wk are non-negative [184] and normalized [23, 24], the
+equilibrium survival probability obeys w1e−µ1t ≤ Sa(t) ≤
+e−µ1t, which directly leads to lower bounds L±
+n (t) in
+Eq. (1). Namely, τ n ≥ mini∈[1,n] τi ≡ τ min
+n
+and τ n ≤
+maxi∈[1,n] τi ≡ τ max
+n
+. Therefore, P(τ min
+n
+≥ t) ≤ P(τ n ≥
+t) ≤ P(τ max
+n
+≥ t) and we have P(τ min
+n
+≥ t) = S(t)n and
+P(τ max
+n
+≤ t) = (1 − S(t))n, leading to lower bounds
+P (τ n − ⟨τ⟩ ≥ t) ≥
+�
+w1e−µ1(⟨τ⟩+t)�n
+≡ L+
+n (t)
+P (τ n − ⟨τ⟩ ≤ −t) ≥
+�
+1 − e−µ1(⟨τ⟩−t)�n
+≡ L−
+n (t),
+(4)
+where equality is reached for n = 1 and w1 → 1. Anal-
+ogous results are obtained for upper bounds (see [181])
+which, however, are much weaker than those derived be-
+low with the Cram´er-Chernoff approach and concurrently
+require even more information about the dynamics.
+We remark that bounds on the survival probability
+consequently also bound the probability density ℘(n)
+a (t)
+of the fastest first-passage time of n independent par-
+ticles [23, 25, 26, 185] according to nw1e−µ1(n−1)t ≤
+℘(n)
+a (t)/℘a(t) ≤ ne−(n−1)µ1t. We now turn to the more
+challenging upper bounds.
+Cram´er-Chernoff bounds.—Let δτ n ≡ |τ n−⟨τ⟩| and λ ∈
+R+. We start with the obvious inequality eλt1δτ n≥t ≤
+eλτ n, where 1b is the indicator function of the set b. Tak-
+ing the expectation yields P(δτ n ≥ t) ≤ e−λt⟨eλδτ n⟩ ≡
+e−λt+ψδτn(λ), where we defined the cumulant generating
+function of δτ n, ψδτ n(λ) ≡ ln⟨eλδτ n⟩. Note that τi are sta-
+tistically independent. The bound can be optimized [186]
+to find Chernoff’s inequality, P(δτ n ≥ t) ≤ e−nψ†δτ(t),
+where ψ∗
+δτ(t) is the Cram´er transform of ψδτ(λ) [173], i.e.
+ψ∗
+δτ(t) ≡ sup
+λ
+(λt − ψδτ(λ)),
+(5)
+where δτ ≡ δτ 1. On the interval λ ∈ [0, µ1) we have the
+following bounds on ψδτ(λ) (see proof in [181])
+ψδτ(λ) ≤ φδτ(λ; C) ≡
+�
+�
+�
+�
+�
+�
+�
+λ2
+2µ2
+1
+C
+1 − λ/µ1
+τ ≥ ⟨τ⟩
+λ2
+2µ2
+1
+C
+1 − (λ/µ1)2
+τ < ⟨τ⟩,
+(6)
+which are non-negative, convex, and increasing on λ ∈
+[0, µ1), and we introduced C ≡ µ2
+1⟨τ 2⟩ [187]. The bound (6)
+further implies ψ∗
+δτ(t) ≥ φ∗
+δτ(t; C) ∀t ≥ 0, and may thus
+be optimized according to [186] to obtain the inequalities
+announced in Eq. (1) via Chernoff’s inequality:
+U+
+n (t; C) = exp (−nCh+ (µ1t/C))
+0 ≤ t ≤ ∞
+U−
+n (t; C) = exp (−nCh− (µ1t/C))
+0 ≤ t ≤ ⟨τ⟩
+(7)
+where we defined the functions
+h+(u) ≡ 1 + u −
+√
+1 + 2u
+(8)
+h−(u) ≡ Λ(u)u − 1
+2
+Λ(u)2
+1 − Λ(u)2
+(9)
+with Λ(u) ≡ 1
+2
+�
+g(u) −
+�
+4 + 2/g(u)u − g(u)2
+�
+and
+g(u) ≡
+2
+√
+3
+�
+1 + 2 cosh
+�1
+3arcosh
+�
+1 +
+33
+27u2
+���1/2
+. (10)
+
+4
+The tail behavior of δτ in Eq. (7) provides quantitative
+insight into fluctuations of τ even when ⟨τ⟩ is unknown
+or is an insufficient or non-representative observable [188–
+190].
+Deviations are readily expressed relative to the
+longest natural time scale 1/µ1 that does not need to be
+known. That is, deviations are naturally parameterized
+by the dimensionless variable ˜t = µ1t. Asymptotically
+as n → ∞, U±
+n is substantial only for ˜t/C ≪ 1 and the
+tails become symmetric and sub-Gaussian [173], h+(u) =
+u2/2 − O(u3) and h−(u) = u2/2 − O(u4) (see [181]).
+Notably, details about the underlying dynamics only
+enter the tail bounds (7) via the system-dependent con-
+stant C that, however, can be bounded. In particular, for
+equilibrium initial conditions we have 0 ≤ 2w1 ≤ C ≤ 2
+(see [181]). Since φδτ(λ; C) is monotonically increasing
+with C ∈ (0, 2], we have φδτ(λ; C) ≤ φδτ(λ; 2) which im-
+plies φ∗
+δτ(t; C) ≥ φ∗
+δτ(t; 2). Thus, we find the model-free
+bounds
+U±
+n (t; C) ≤ U±
+n (t; 2) ≡ U±
+n (t)
+(11)
+requiring no information about the system. The non-
+asymptotic bounds on deviation probabilities of τ n in
+Eqs. (7) and (11) are our first main result.
+Notably, analogous concentration inequalities were pre-
+viously derived for time-averages of Markov processes
+[191–193] (see also [194]), and were recently applied to
+bound time-averaged measurement outcomes in quantum
+Markov processes [195] and to derive inverse thermody-
+namic uncertainty relations [196].
+Illustration of bounds.—The lower L±
+n (t) and upper
+U±
+n (t) bounds on P(±[τ n − ⟨τ⟩] ≥ t) in Eqs. (4) and (7),
+respectively, are examplified in Fig. 2e-h (see red and
+black lines) for the model systems shown in Fig. 1b-d.
+Note that to illustrate all bounds, for convenience in a
+single panel, we formally let t → −t for the left tails
+L−
+n (t) and U−
+n (t), such that t (as shown) has support on
+[−⟨τ⟩, ∞). Deviation probabilities are in turn expressed
+as P(sgn(t)δτ n ≥ |t|) where sgn(x) denotes the signum
+function and δτ n = τ n − ⟨τ⟩.
+To assess the quality of our bounds for several n we
+further scale probabilities P1/n such that L±
+n (t) and U±
+n (t)
+collapse onto a master curve for all n (see also inset in
+Fig. 2f). Symbols denote empirical deviation probabili-
+ties obtained by sampling τ n for different n (see [181] for
+details), which approach the upper bound as n increases.
+For n = 1 empirical right-tail deviations are close to L+
+1 (t)
+even for w1 ≤ 1 [197]. As expected the model-free upper
+bound U±
+n (t; 2) (yellow) holds universally but is generally
+more conservative, however, it is remarkably good for
+C ≳ 1.3 (see e.g. Fig. 2e-g) but becomes weaker as C
+approaches 0 (see e.g. Fig. 2h).
+Uncertainty quantification.—The bounds (7) provide
+the elusive systematic framework to rigorously quantify
+the uncertainty of the estimate τ n for any, and especially
+for small, sample sizes. In particular, they allow us to
+construct “with high probability” guarantees such as con-
+fidence intervals, which—unlike traditional confidence
+intervals in statistics—are not only asymptotically correct
+but hold for any n. Furthermore, these concentration-
+based guarantees do not require specifying a prior belief
+as in the Bayesian context. Setting U±
+n (t±
+α±; C) = α± for
+chosen acceptable left- and right-tail error probabilities
+α± (with α+ + α− < 1) we get an implicit definition of
+the confidence interval [−t−
+α−, t+
+α+] at confidence level (or
+“coverage probability”) 1 − (α+ + α−) in the form
+P(−t−
+α− ≤ δτ n ≤ t+
+α+) ≥ 1 − α− − α+ ≡ 1 − α,
+(12)
+stating that with probability of at least 1 − α the sample
+mean τ n lies within [⟨τ⟩ − t−
+α−, ⟨τ⟩ + t+
+α+]. Confidence
+intervals are closely related to, and can be used for, statis-
+tical significance tests [198, 199]. However, they provide
+more insight; instead of mere rejection/acceptance they
+provide quantitative bounds on statistical uncertainty.
+Two-sided intervals are not uniquely determined by
+specifying a confidence level. It is customary to choose
+equal tail probabilities α+ = α− = α/2 yielding so-called
+central confidence intervals for which t±
+α± are generally
+not equidistant. Two-sided central confidence intervals
+for δτ n as a function of n for a confidence level of α = 0.1
+and models systems in Fig. 1b-d are shown (rescaled to a
+master scaling) in Fig. 3a. One may also choose symmetric
+intervals which in turn do not necessarily imply equal tail
+probabilities (i.e. α+ ̸= α−). In some situations only one-
+sided confidence intervals are required P(±δτ n ≤ t±
+α±) ≥
+1 − α± (for a discussion see [181]).
+In particular, we may now also answer the practical
+question: How many realizations are required to achieve
+a desired accuracy with a specified probability? To ensure
+with probability of at least 1 − α that δτ n∗ ∈ [−t−
+α−, t+
+α+]
+one needs n∗ realizations defined via
+U+
+n∗(t+
+α+; C) + U−
+n∗(t−
+α−; C) = α.
+(13)
+The number of samples n∗ required to guarantee that τ n∗
+falls within a symmetric interval of length ∆t = 0.2/µ1,
+(i.e. τ n∗ ∈ [⟨τ⟩ − 0.1/µ1, ⟨τ⟩ + 0.1/µ1]) with probability
+of at least 1 − α is shown in Fig. 3b for several values of C
+(intersections with the dashed line yield n∗ guaranteeing
+a coverage of at least 90%). Fig. 3c depicts the comple-
+mentary symmetric interval ∆t covering the range of δτ n
+for a given n with probability of at least 90%. Note that
+hundreds to thousands of samples may be required to
+ensure an accuracy of ±0.1/µ1 with a 90% confidence,
+which is seemingly not met in experiments [113–119].
+Eqs. (12) and (13) constitute our second main result as
+they provide rigorous error estimates in the small-sample
+regime that allow for systematic error control in kinetic
+inference and can be solved for t±
+α± and n∗, respectively,
+using standard root-finding methods (see [181]).
+Using Eq. (11) we can construct system-independent
+but more conservative universal confidence intervals (see
+yellow line in Fig. 3b,c). Interestingly, even when C ≈ 1
+the universal bound remains reasonably tight, only for
+C ≪ 1 differences become substantial.
+Conclusion.—Leveraging spectral analysis and the
+framework of concentration inequalities we derived gen-
+eral upper and lower bounds on the probability that the
+
+5
+FIG. 3. Non-asymptotic uncertainty quantification of the sam-
+ple mean τ n. (a) Relative error µ1δτ n = µ1(τ n−⟨τ⟩) (symbols)
+obtained from sampling of τ n for different model systems and
+as a function n (re-scaled to a master scaling). The correspond-
+ing two-sided central confidence interval [−µ1t−
+α/2, µ1t+
+α/2] with
+α = 0.1 is shown as black lines. (b) Required number of sam-
+ples n∗ to ensure that the relative error δτ n∗ falls within the
+symmetric interval [−0.1, 0.1] of length ∆t = 0.2/µ1 with
+probability of at least 1 − α for several values of C. (c) Cor-
+responding symmetric confidence interval [−µ1∆t/2, µ1∆t/2]
+(only the upper limit is shown) at confidence level α = 0.1 as
+a function of n for different C.
+empirical first-passage time τ n inferred from n indepen-
+dent realizations deviates from the true mean ⟨τ⟩ by any
+given amount. We used these bounds to construct non-
+asymptotic confidence intervals that hold in the elusive
+small-sample regime and thus go beyond Central-Limit-
+and bootstrapping-based methods, which are known to
+fail for small n. The results require minimal input and
+in particular do not require any prior belief as in the
+Bayesian approach that is known to be problematic and
+likely underestimates the uncertainty in the small-sample
+setting. Our concentration-based results allow for rigor-
+ous, model-free error control and reliable error estimation,
+which is essential for the analysis of experimental and
+simulation data. They may further be applied to popu-
+lation dynamics and epidemiology, e.g. in the inference
+of extinction or incubation times of diseases [200–204],
+and may be extended to the concentration around the
+typical instead of mean first-passage times [205] as well
+as non-ergodic and irreversible dynamics.
+Acknowledgments.—Financial support from Studiens-
+tiftung des Deutschen Volkes (to R. B.) and the German
+Research Foundation (DFG) through the Emmy Noether
+Program GO 2762/1-2 (to A. G.) is gratefully acknowl-
+edged.
+∗ [email protected]
+[1] S. Redner, A Guide to First-Passage Processes (Cam-
+bridge University Press, 2001).
+[2] R. Metzler, S. Redner, and G. Oshanin, First-Passage
+Phenomena and their Applications, Vol. 35 (World Scien-
+tific, 2014).
+[3] Y. Zhang and O. K. Dudko, Annu. Rev. Biophys. 45, 117
+(2016).
+[4] S. Iyer-Biswas and A. Zilman, Adv. Chem. Phys. 160,
+261 (2016).
+[5] A. Szabo, K. Schulten, and Z. Schulten, J. Chem. Phys.
+72, 4350–4357 (1980).
+[6] P. H¨anggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys.
+62, 251 (1990).
+[7] E. Ben-Naim, S. Redner, and F. Leyvraz, Phys. Rev. Lett.
+70, 1890 (1993).
+[8] D. S. Grebenkov, R. Metzler, and G. Oshanin, Phys.
+Chem. Chem. Phys. 20, 16393–16401 (2018).
+[9] D. S. Grebenkov, R. Metzler, and G. Oshanin, Commun.
+Chem. 1, 1 (2018).
+[10] D. S. Grebenkov, Phys. Rev. Lett. 117, 260201 (2016).
+[11] O. G. Berg, R. B. Winter, and P. H. Von Hippel, Bio-
+chemistry 20, 6929–6948 (1981).
+[12] E. Koslover, M. D´ıaz de la Rosa, and A. Spakowitz, Bio-
+phys. J. 101, 856 (2011).
+[13] D. Holcman and Z. Schuss, J. Phys. A: Math. Theor. 47,
+173001 (2014).
+[14] O. B´enichou, C. Chevalier, B. Meyer, and R. Voituriez,
+Phys. Rev. Lett. 106, 038102 (2011).
+[15] E. G. Marklund, A. Mahmutovic, O. G. Berg, P. Hammar,
+D. van der Spoel, D. Fange, and J. Elf, Proc. Natl. Acad.
+Sci. 110, 19796–19801 (2013).
+[16] M. Bauer and R. Metzler, PLoS ONE 8, e53956 (2013).
+[17] O. B´enichou, C. Chevalier, J. Klafter, B. Meyer, and
+R. Voituriez, Nat. Chem. 2, 472–477 (2010).
+[18] O. B´enichou and R. Voituriez, Phys. Rep. 539, 225 (2014).
+[19] A. Godec and R. Metzler, Phys. Rev. X 6, 041037 (2016).
+[20] J. Newby and J. Allard, Phys. Rev. Lett. 116, 128101
+(2016).
+[21] S. Redner and B. Meerson, J. Stat. Mech. 2014, P06019
+(2014).
+[22] B. Meerson and S. Redner, Phys. Rev. Lett. 114, 198101
+(2015).
+[23] D. Hartich and A. Godec, New J. Phys. 20, 112002 (2018).
+[24] D. Hartich and A. Godec, J. Stat. Mech. 2019, 024002
+(2019).
+[25] D. Hartich and A. Godec, Reaction kinetics in the few-
+encounter limit, in Chemical Kinetics (World Scientific,
+2019) Chap. 11, pp. 265–283.
+[26] Z. Schuss, K. Basnayake, and D. Holcman, Phys. Life Rev.
+28, 52–79 (2019).
+[27] S. D. Lawley and J. B. Madrid, J. Chem. Phys. 150,
+214113 (2019).
+[28] S. D. Lawley and J. B. Madrid, J. Nonlinear Sci. 30,
+1207–1227 (2020).
+[29] S. D. Lawley, Phys. Rev. E 102, 062118 (2020).
+[30] P. C. Bressloff and J. M. Newby, Rev. Mod. Phys. 85,
+135 (2013).
+[31] ´E. Rold´an, A. Lisica, D. S´anchez-Taltavull, and S. W.
+Grill, Phys. Rev. E 93, 062411 (2016).
+[32] K. R. Ghusinga, J. J. Dennehy, and A. Singh, Proc. Natl.
+Acad. Sci. 114, 693 (2017).
+[33] K. Rijal, A. Prasad, A. Singh, and D. Das, Phys. Rev.
+Lett. 128, 048101 (2022).
+
+6
+[34] J. J. Parmar, D. Das, and R. Padinhateeri, Nucleic Acids
+Res. 44, 1630 (2015).
+[35] D. A. Charlebois, N. Abdennur, and M. Kaern, Phys. Rev.
+Lett. 107, 218101 (2011).
+[36] F. Frey, F. Ziebert, and U. S. Schwarz, Phys. Rev. Lett.
+122, 088102 (2019).
+[37] A. L. Lloyd and R. M. May, Science 292, 1316 (2001).
+[38] L. Hufnagel, D. Brockmann, and T. Geisel, Proc. Natl.
+Acad. Sci. 101, 15124 (2004).
+[39] O. B´enichou, C. Loverdo, M. Moreau, and R. Voituriez,
+Rev. Mod. Phys. 83, 81 (2011).
+[40] F. Boccardo and O. Pierre-Louis, Phys. Rev. Lett. 128,
+256102 (2022).
+[41] T. Erdmann and U. S. Schwarz, Phys. Rev. Lett. 92,
+108102 (2004).
+[42] S. Chakrabarti, M. Hinczewski, and D. Thirumalai, Proc.
+Natl. Acad. Sci. 111, 9048–9053 (2014).
+[43] K. Blom and A. Godec, Phys. Rev. X 11, 031067 (2021).
+[44] I. Goychuk and P. H¨anggi, Proc. Natl. Acad. Sci. 99, 3552
+(2002).
+[45] T. Kay and L. Giuggioli, Phys. Rev. Res. 4, 032039 (2022).
+[46] S. Bo, L. Hubatsch, J. Bauermann, C. A. Weber, and
+F. J¨ulicher, Phys. Rev. Res. 3, 043150 (2021).
+[47] H. Kramers, Physica 7, 284 (1940).
+[48] S. Sabhapandit and S. N. Majumdar, Phys. Rev. Lett.
+125, 200601 (2020).
+[49] B. Trendelkamp-Schroer and F. No´e, Phys. Rev. X 6,
+011009 (2016).
+[50] M. Chupeau, J. Gladrow, A. Chepelianskii, U. F. Keyser,
+and E. Trizac, Proc. Natl. Acad. Sci. 117, 1383 (2019).
+[51] T. D. Swinburne, D. Kannan, D. J. Sharpe, and D. J.
+Wales, J. Chem. Phys. 153, 134115 (2020).
+[52] A. L. Thorneywork, J. Gladrow, Y. Qing, M. Rico-Pasto,
+F. Ritort, H. Bayley, A. B. Kolomeisky, and U. F. Keyser,
+Sci. Adv. 6, 1 (2020).
+[53] A. Goychuk and E. Frey, Phys. Rev. Lett. 123, 178101
+(2019).
+[54] R. Bebon and U. S. Schwarz, New J. Phys. 24, 063034
+(2022).
+[55] D. B. Dougherty, I. Lyubinetsky, E. D. Williams, M. Con-
+stantin, C. Dasgupta, and S. Sarma, Phys. Rev. Lett. 89,
+136102 (2002).
+[56] M. Constantin, S. D. Sarma, C. Dasgupta, O. Bondarchuk,
+D. B. Dougherty, and E. D. Williams, Phys. Rev. Lett.
+91, 086103 (2003).
+[57] D. B. Dougherty, C. Tao, O. Bondarchuk, W. G. Cullen,
+E. D. Williams, M. Constantin, C. Dasgupta, and S. D.
+Sarma, Phys. Rev. E 71, 021602 (2005).
+[58] J. Merikoski, J. Maunuksela, M. Myllys, J. Timonen, and
+M. J. Alava, Phys. Rev. Lett. 90, 024501 (2003).
+[59] M. Constantin, C. Dasgupta, P. P. Chatraphorn, S. N.
+Majumdar, and S. D. Sarma, Phys. Rev. E 69, 061608
+(2004).
+[60] C. Godr`eche, S. N. Majumdar, and G. Schehr, Phys. Rev.
+Lett. 102, 240602 (2009).
+[61] A. J. Bray, S. N. Majumdar, and G. Schehr, Adv. Phys.
+62, 225–361 (2013).
+[62] T. R. Gingrich and J. M. Horowitz, Phys. Rev. Lett. 119,
+170601 (2017).
+[63] S. Singh, P. Menczel, D. S. Golubev, I. M. Khaymovich,
+J. T. Peltonen, C. Flindt, K. Saito, ´E. Rold´an, and J. P.
+Pekola, Phys. Rev. Lett. 122, 230602 (2019).
+[64] E. Rold´an, I. Neri, M. D¨orpinghaus, H. Meyr, and
+F. J¨ulicher, Phys. Rev. Lett. 115, 250602 (2015).
+[65] I. Neri, E. Rold´an, and F. J¨ulicher, Phys. Rev. X 7, 011019
+(2017).
+[66] G. Falasco and M. Esposito, Phys. Rev. Lett. 125, 120604
+(2020).
+[67] I. Neri, J. Phys. A: Math. Theor. 55, 304005 (2022).
+[68] J. P. Garrahan, Phys. Rev. E 95, 032134 (2017).
+[69] K. Hiura and S. ichi Sasa, Phys. Rev. E 103, 050103
+(2021).
+[70] M. Kac, in Proceedings of the Second Berkeley Symposium
+on Mathematical Statistics and Probability (University of
+California Press, Berkeley, Calif., 1951) pp. 189–215.
+[71] G. Schehr and S. N. Majumdar, First-Passage Phenomena
+and Their Applications , 226–251 (2014).
+[72] S. N. Majumdar, G. Schehr, and G. Wergen, J. Phys. A:
+Math. Theor. 45, 355002 (2012).
+[73] D. Hartich and A. Godec, J. Phys. A: Math. Theor. 52,
+244001 (2019).
+[74] H. Friedman, D. A. Kessler, and E. Barkai, Phys. Rev. E
+95, 032141 (2017).
+[75] F. Thiel, E. Barkai, and D. A. Kessler, Phys. Rev. Lett.
+120, 040502 (2018).
+[76] M. J. Kearney and S. N. Majumdar, J. Phys. A: Math.
+Gen. 38, 4097 (2005).
+[77] M. J. Kearney, S. N. Majumdar, and R. J. Martin, J.
+Phys. A: Math. Theor. 40, F863 (2007).
+[78] M. J. Kearney and S. N. Majumdar, J. Phys. A: Math.
+Theor. 47, 465001 (2014).
+[79] M. J. Kearney and R. J. Martin, J. Phys. A: Math. Theor.
+54, 055002 (2021).
+[80] S. N. Majumdar and B. Meerson, J. Stat. Mech.: Theory
+Exp. 2021 (3), 039801.
+[81] P. Singh and A. Pal, J. Phys. A: Math. Theor. 55, 234001
+(2022).
+[82] G. Mercado-V´asquez and D. Boyer, Phys. Rev. Lett. 123,
+250603 (2019).
+[83] A. Kumar, A. Zodage, and M. S. Santhanam, Phys. Rev.
+E 104, 052103 (2021).
+[84] J. L. Spouge, A. Szabo, and G. H. Weiss, Phys. Rev. E
+54, 2248 (1996).
+[85] Y. Scher and S. Reuveni, Phys. Rev. Lett. 127, 018301
+(2021).
+[86] E. Woillez, Y. Zhao, Y. Kafri, V. Lecomte, and J. Tailleur,
+Phys. Rev. Lett. 122, 258001 (2019).
+[87] F. Mori, P. L. Doussal, S. N. Majumdar, and G. Schehr,
+Phys. Rev. Lett. 124, 090603 (2020).
+[88] P. H¨anggi and P. Talkner, Phys. Rev. Lett. 51, 2242
+(1983).
+[89] P. Hanggi and P. Talkner, Phys. Rev. A 32, 1934 (1985).
+[90] T. Gu´erin, N. Levernier, O. B´enichou, and R. Voituriez,
+Nature 534, 356 (2016).
+[91] H. Meyer and H. Rieger, Phys. Rev. Lett. 127, 070601
+(2021).
+[92] M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. 106,
+160601 (2011).
+[93] L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and
+G. Schehr, Phys. Rev. Lett. 113, 220602 (2014).
+[94] S. Reuveni, Phys. Rev. Lett. 116, 170601 (2016).
+[95] A. Pal and S. Reuveni, Phys. Rev. Lett. 118, 030603
+(2017).
+[96] A. Pal, I. Eliazar, and S. Reuveni, Phys. Rev. Lett. 122,
+020602 (2019).
+[97] M. R. Evans, S. N. Majumdar, and G. Schehr, J. Phys.
+A: Math. Theor. 53, 193001 (2020).
+
+7
+[98] B. Besga, A. Bovon, A. Petrosyan, S. N. Majumdar, and
+S. Ciliberto, Phys. Rev. Res. 2, 032029 (2020).
+[99] O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and
+Y. Roichman, J. Phys. Chem. Lett. 11, 7350 (2020).
+[100] B. D. Bruyne, J. Randon-Furling, and S. Redner, Phys.
+Rev. Lett. 125, 050602 (2020).
+[101] B. D. Bruyne, S. N. Majumdar, and G. Schehr, Phys.
+Rev. Lett. 128, 200603 (2022).
+[102] D. L. Ensign and V. S. Pande, J. Phys. Chem. B 113,
+12410–12423 (2009).
+[103] R. Satija, A. Das, S. M¨uhle, J. Enderlein, and D. E.
+Makarov, J. Phys. Chem. B 124, 3482–3493 (2020).
+[104] S. Zolaktaf, F. Dannenberg, X. Rudelis, A. Condon,
+J. M. Schaeffer, M. Schmidt, C. Thachuk, and E. Winfree,
+Inferring Parameters for an Elementary Step Model of
+DNA Structure Kinetics with Locally Context-Dependent
+Arrhenius Rates (Springer International Publishing, 2017)
+p. 172–187.
+[105] C. Weinreb, S. Wolock, B. K. Tusi, M. Socolovsky, and
+A. M. Klein, Proc. Natl. Acad. Sci. 115, E2467 (2018).
+[106] P. Pearce, F. G. Woodhouse, A. Forrow, A. Kelly,
+H. Kusumaatmaja, and J. Dunkel, Nat. Commun. 10,
+1 (2019).
+[107] Y. Zhou, C. Zhang, G. Stell, and J. Wang, J. Am. Chem.
+Soc. 125, 6300–6305 (2003).
+[108] J. O. Daldrop, J. Kappler, F. N. Br¨unig, and R. R. Netz,
+Proc. Natl. Acad. Sci. 115, 5169–5174 (2018).
+[109] D. A. Nicholson and G. C. Rutledge, J. Chem. Phys.
+144, 134105 (2016).
+[110] V. J. van Hijkoop, A. J. Dammers, K. Malek, and M.-O.
+Coppens, J. Chem. Phys. 127, 085101 (2007).
+[111] R. Belousov,
+M. N. Qaisrani,
+A. Hassanali, and
+E. Roldan, Soft Matter 16, 9202–9216 (2020).
+[112] S. Ditlevsen and O. Ditlevsen, Probabilistic Eng. Mech
+23, 170–179 (2008).
+[113] V. Gapsys and B. L. de Groot, eLife 9, e57589 (2020).
+[114] K. Lindorff-Larsen, S. Piana, R. O. Dror, and D. E.
+Shaw, Science 334, 517 (2011).
+[115] J. L. Adelman and M. Grabe, J. Chem. Phys. 138,
+044105 (2013).
+[116] B. Mostofian and D. M. Zuckerman, J. Chem. Theory
+Comput. 15, 3499 (2019).
+[117] R. Mehra and K. P. Kepp, J. Chem. Phys. 151, 085101
+(2019).
+[118] A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjo-
+hanns, L. Novotny, and C. Dellago, Nat. Commun. 12, 1
+(2021).
+[119] L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago,
+and L. Novotny, Nat. Nanotechnol. 12, 1130 (2017).
+[120] D. J. Sharpe and D. J. Wales, J. Chem. Phys. 153,
+024121 (2020).
+[121] R. M. Donovan, A. J. Sedgewick, J. R. Faeder, and D. M.
+Zuckerman, J. Chem. Phys. 139, 115105 (2013).
+[122] J. Sabelko, J. Ervin, and M. Gruebele, Proc. Nat. Acad.
+Sci. 96, 6031 (1999).
+[123] D. L. Ensign and V. S. Pande, J. Phys. Chem. B 113,
+12410 (2009).
+[124] J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt,
+and M. Rief, Science 334, 512 (2011).
+[125] A. M. Berezhkovskii and A. Szabo, J. Chem. Phys. 150,
+054106 (2019).
+[126] I. Nayak, D. Das, and A. Nandi, Phys. Rev. Res. 2,
+013114 (2020).
+[127] D. J. Wales, J. Phys. Chem. Lett. 13, 6349 (2022).
+[128] N. Singhal and V. S. Pande, J. Chem. Phys. 123, 204909
+(2005).
+[129] G. R. Bowman, V. S. Pande, and F. No´e, An Intro-
+duction to Markov State Models and their Application to
+Long Timescale Molecular Simulation, Vol. 797 (Springer
+Science & Business Media, 2013).
+[130] A. Grossfield and D. M. Zuckerman, Annu. Rep. Comput.
+Chem. 5, 23 (2009).
+[131] A. Grossfield, P. N. Patrone, D. R. Roe, A. J. Schultz,
+D. W. Siderius, and D. M. Zuckerman, Living J. Comp.
+Mol. Sci. 1 (2018).
+[132] B. Knapp, L. Ospina, and C. M. Deane, J. Chem. Theory
+Comput. 14, 6127 (2018).
+[133] Resampling methods like bootstrapping assume the data
+to be representative of the inferred statistic, which is not
+necessarily the case for small n, possibly even when n is
+large but finite for broad distributions.
+[134] A. C. Davison and D. V. Hinkley, Bootstrap Methods and
+their Application (Cambridge University Press, 1997).
+[135] J. Shao, Mathematical Statistics (Springer New York,
+2003).
+[136] A. Abadie and G. W. Imbens, Econometrica 76, 1537
+(2008).
+[137] H. Putter and W. R. van Zwet, in Selected Works of
+Willem van Zwet (Springer New York, 2011) pp. 245–266.
+[138] R. V. Hogg, J. W. McKean, and A. T. Craig, Introduction
+to Mathematical Statistics (Pearson, 2018).
+[139] N. Schenker, J. Am. Stat. Assoc. 80, 360 (1985).
+[140] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin,
+Bayesian Data Analysis (Chapman and Hall/CRC, 1995).
+[141] A. R. Brazzale, A. C. Davison, N. Reid, et al., Applied
+Asymptotics: Case Studies in Small-Sample Statistics,
+Vol. 23 (Cambridge University Press, 2007).
+[142] L. Lista, Statistical Methods for Data Analysis in Particle
+Physics (Springer International Publishing, 2017).
+[143] D. Kaplan, Bayesian Statistics for the Social Sciences
+(Guilford Publications, 2014).
+[144] R. McElreath, Statistical Rethinking: A Bayesian Course
+with Examples in R and Stan (Chapman and Hall/CRC,
+2020).
+[145] M. Tavakoli, J. N. Taylor, C.-B. Li, T. Komatsuzaki,
+and S. Press´e, in Advances in Chemical Physics (John
+Wiley & Sons, Inc., 2017) pp. 205–305.
+[146] D. McNeish, Struct. Equ. Modeling 23, 750 (2016).
+[147] S. C. Smid, D. McNeish, M. Mioˇcevi´c, and R. van de
+Schoot, Struct. Equ. Modeling 27, 131 (2019).
+[148] S. Bacallado, J. D. Chodera, and V. Pande, J. Chem.
+Phys. 131, 045106 (2009).
+[149] J.-H. Prinz, H. Wu, M. Sarich, B. Keller, M. Senne,
+M. Held, J. D. Chodera, C. Sch¨utte, and F. No´e, J. Chem.
+Phys. 134, 174105 (2011).
+[150] B. Trendelkamp-Schroer, H. Wu, F. Paul, and F. No´e, J.
+Chem. Phys. 143, 174101 (2015).
+[151] J. D. Chodera and F. No´e, J. Chem. Phys. 133, 105102
+(2010).
+[152] L. Le Cam, L. M. LeCam, and G. L. Yang, Asymptotics
+in Statistics: Some Basic Concepts (Springer Science &
+Business Media, 2000).
+[153] A. W. Van der Vaart, Asymptotic Statistics, Vol. 3 (Cam-
+bridge university press, 2000).
+[154] L. Le Cam, Asymptotic Methods in Statistical Decision
+Theory (Springer Science & Business Media, 2012).
+[155] P. Diaconis and D. Freedman, Ann. Stat. 14, 1 (1986).
+[156] D. D. Cox, Ann. Stat. 21, 1 (1993).
+
+8
+[157] P. W. Diaconis and D. Freedman, Bernoulli , 411 (1998).
+[158] D. Freedman, Ann. Stat. 27, 1119 (1999).
+[159] A. Barron, M. J. Schervish, and L. Wasserman, Ann.
+Stat. 27, 536 (1999).
+[160] S. Ghosal, J. K. Ghosh, and R. Ramamoorthi, Ann. Stat.
+27, 143 (1999).
+[161] J. K. Ghosh and R. V. Ramamoorthi, Bayesian Non-
+parametrics (Springer-Verlag, 2003).
+[162] Y. Kim and J. Lee, Ann. Stat. 32, 1 (2004).
+[163] S. Boucheron and E. Gassiat, Electron. J. Stat. 3, 114
+(2009).
+[164] P. J. Bickel and B. J. Kleijn, Ann. Stat. 40, 206 (2012).
+[165] V. Rivoirard and J. Rousseau, Ann. Stat. 40, 1489
+(2012).
+[166] I. Castillo and R. Nickl, Ann. Stat. 42, 1941 (2014).
+[167] J. Rousseau, Annu. Rev. Stat. Appl. 3, 211 (2016).
+[168] V. Rockov´a, in International Conference on Machine
+Learning (PMLR, 2020) pp. 8137–8146.
+[169] K. Ray and A. van der Vaart, Elec. J. Stat. 15, 1 (2021).
+[170] S. Ghosal and A. Van der Vaart, Fundamentals of Non-
+parametric Bayesian Inference, Vol. 44 (Cambridge Uni-
+versity Press, 2017).
+[171] B. Kleijn and A. van der Vaart, Electron. J. Stat. 6,
+none (2012).
+[172] I. M. Johnstone, in Institute of Mathematical Statistics
+Collections (Institute of Mathematical Statistics, 2010)
+pp. 87–98.
+[173] S. Boucheron, G. Lugosi, and P. Massart, Concentration
+Inequalities: A Nonasymptotic Theory of Independence
+(Oxford University Press, 2013).
+[174] J.-H. Prinz, B. Keller, and F. No´e, Phys. Chem. Chem.
+Phys. 13, 16912 (2011).
+[175] S. Olsson, H. Wu, F. Paul, C. Clementi, and F. No´e,
+Proc. Nat. Acad. Sci. 114, 8265 (2017).
+[176] G. A. Pavliotis, Stochastic Processes and Applications
+(Springer New York, 2014).
+[177] ˆL is self-adjoint in the left eigenspace with respect to
+a scalar product weighted by e−ϕ(x) and the operator
+eϕ(x)/2 ˆLe−ϕ(x)/2 is self-adjoint with respect to a flat mea-
+sure.
+[178] Precisely, we require that ϕ(x) satisfies the Poincar´e
+inequality, i.e. lim|x|→∞(|∇ϕ(x)|2/2 − ∇2ϕ(x)) = ∞.
+[179] The relaxation eigenvalue problem reads −ˆLΨk(x) =
+νkΨk(x) with ν0 = 0 and νk≥1 > 0 [176].
+[180] In a continuous state-space the absorbing state a has
+zero measure and ˜peq(x) = peq(x); In the discrete case
+˜peq(xk̸=a) ≡ peq(xk)/ �
+k̸=a peq(x).
+[181] See Supplemental Material at [...] for further details,
+mathematical proofs, and generalizations to arbitrary
+initial conditions p0(x), as well as Refs [3, 8–10, 12, 14].
+[182] When the initial condition is not sampled from ˜peq(x)
+we assume that ϕ(x) is sufficiently confining to assure a
+“nice” asymptotic growth of eigenvalues, limk→∞ νk = bkβ
+with β > 1/2 and 0 < b < ∞. The latter condition
+is automatically satisfied when Ω is finite, since regu-
+lar Sturm-Liouville problems display Weyl asymptotics
+with β = 2 [212]. The condition is in fact satisfied by
+most physically relevant processes with discrete spectra,
+incl. the Ornstein-Uhlenbeck or Rayleigh process [213]
+with β = 1.
+[183] When Ω is discrete the integral is replaced by a sum
+over states excluding the target.
+[184] wk ≥ 0 is a necessary condition for the validity of the
+lower bounds. Thus, in contrast to our Cram´er-Chernoff
+bounds U±
+n (t) that generalize to arbitrary initial condi-
+tions, L±
+n (t) hold only for p0(x0) = ˜peq(x0).
+[185] D. S. Grebenkov, R. Metzler, and G. Oshanin, New J.
+Phys. 22, 103004 (2020).
+[186] ψδτn(λ) is differentiable, convex, non-negative, and non-
+decreasing and thus ψ∗
+δτ(t) = ψδτn(λ†), where λ† solves
+ψ′
+δτ(λ†) = t.
+[187] In case of arbitrary initial conditions ⟨τ 2⟩ becomes re-
+placed by �
+i wi1wi>0 < ∞ while the rest remains un-
+changed.
+[188] C. Mej´ıa-Monasterio, G. Oshanin, and G. Schehr, J. Stat.
+Mech.: Theory Exp. 2011 (06), P06022.
+[189] G. Oshanin, Y. Holovatch, and G. Schehr, Physica A
+390, 4340 (2011).
+[190] T. G. Mattos, C. Mej´ıa-Monasterio, R. Metzler, and
+G. Oshanin, Phys. Rev. E 86, 031143 (2012).
+[191] P. Lezaud, Ann. Appl. Probab. 8, 849 (1998).
+[192] P. Lezaud, ESAIM Probab. Stat. 5, 183–201 (2001).
+[193] F. Gao, A. Guillin, and L. Wu, Theory Probab. its Appl.
+58, 358 (2014).
+[194] Ref. [192] contains an error; the Proof of Lemma 2.3 is
+only valid in the regime r < λ1/3||f||∞, but the Lemma
+may be shown to hold in the claimed regime [214].
+[195] F. Girotti, J. P. Garrahan, and M. Gut¸˘a, Concentra-
+tion inequalities for output statistics of quantum Markov
+processes (2022).
+[196] G. Bakewell-Smith, F. Girotti, M. Gut¸˘a, and J. P. Garra-
+han, Inverse thermodynamic uncertainty relations: Gen-
+eral upper bounds on the fluctuations of trajectory ob-
+servables (2022).
+[197] However, w1 can get arbitrary close to 0 in principle,
+rendering the lower bound trivial.
+[198] D. Wackerly, W. Mendenhall, and R. L. Scheaffer, Math-
+ematical Statistics with Applications (Cengage Learning,
+2014).
+[199] W. Q. Meeker, G. J. Hahn, and L. A. Escobar, Statistical
+Intervals: A Guide for Practitioners and Researchers, Vol.
+541 (John Wiley & Sons, 2017).
+[200] M. Dykman, I. Schwartz, and A. Landsman, Phys. Rev.
+Lett. 101, 078101 (2008).
+[201] J. A. Gilbert, L. A. Meyers, A. P. Galvani, and J. P.
+Townsend, Epidemics 6, 37 (2014).
+[202] B. Ottino-Loffler, J. G. Scott, and S. H. Strogatz, eLife
+6, e30212 (2017).
+[203] M. Aliee, K. S. Rock, and M. J. Keeling, J. R. Soc.
+Interface 17, 20200540 (2020).
+[204] D. Hathcock and S. H. Strogatz, Phys. Rev. Lett. 128,
+218301 (2022).
+[205] S. Belan, Phys. Rev. Res. 2, 013243 (2020).
+[14] R. L. Burden, J. D. Faires, and A. M. Burden, Numerical
+Analysis (Cengage Learning, 2015).
+[10] E. Barkai, E. Aghion, and D. Kessler, Phys. Rev. X 4,
+021036 (2014).
+[9] J. W. Pitman, Adv. Appl. Probab. 7, 511 (1975).
+[3] A. J. F. Siegert, Phys. Rev. 81, 617 (1951).
+[8] U. Seifert, Annu. Rev. Condens. Matter Phys. 10, 171
+(2019).
+[12] G. Cowan, Statistical Data Analysis (Oxford University
+Press, 1998).
+[212] G. Teschl, Ordinary Differential Equations and Dynami-
+cal Systems (American Mathematical Society, 2012).
+[213] C. W. Gardiner, Handbook of Stochastic Methods for
+Physics, Chemistry and the Natural Sciences, 3rd ed.,
+
+9
+Springer Series in Synergetics, Vol. 13 (Springer-Verlag,
+Berlin, 2004).
+[214] S. C. Ibanez, Concentration inequalities for Markov jump
+processes (2022).
+
+1
+Supplementary Material for:
+Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime
+Rick Bebon and Aljaˇz Godec
+Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, Am Faßberg 11, 37077 G¨ottingen
+In this Supplementary Material (SM) we present additional background and details of the calculations, auxiliary
+results, numerical methods, and mathematical proofs of the claims made in the Letter. The sections are organized in
+the order as they appear in the Letter.
+CONTENTS
+References
+5
+S1. Spectral representation and preparatory Lemmas
+2
+A. Spectral representation
+2
+B. Lemma 1: All weights are non-negative for equilibrium initial conditions
+3
+C. Lemma 2: Sum of positive weights is bounded from above
+3
+S2. Extreme value bounds and comparison with Cram´er-Chernoff bounds
+4
+A. Extreme value bounds
+4
+B. Comparison of Cram´er-Chernoff vs Extreme value Bounds
+4
+S3. Complete proof of concentration inequalities and their asymptotics
+5
+A. Theorem 1: Cram´er-Chernoff bound for the right tail τ ≥ ⟨τ⟩
+5
+B. Theorem 2: Cram´er-Chernoff bound for the left tail ⟨τ⟩ < τ
+7
+C. Behavior of upper bounds U±
+n (t) for large sample sizes
+9
+D. Proof of bounds on C and model-free concentration inequalities
+9
+S4. Model systems and details on numerical methods
+10
+A. Continuous-time discrete-state Markov jump process
+10
+1. Transitions rates of the 8-state toy protein model
+11
+2. Transitions rates of the calmodulin protein model
+11
+B. Spatially confined Brownian molecular search process
+11
+C. Statistics of first-passage times ⟨τ⟩ and the sample mean τ n
+12
+S5. Uncertainty quantification with confidence intervals
+13
+References
+15
+
+2
+S1.
+SPECTRAL REPRESENTATION AND PREPARATORY LEMMAS
+In this section we provide additional background on the spectral analysis of first-passage problems and some auxiliary
+Lemmas. In particular, we prove that for equilibrium initial conditions all spectral first-passage weights wk(˜peq) are
+non-negative and that general initial conditions p0(x) the sum of positive spectral weights is always bounded.
+A.
+Spectral representation
+First, we recall some general results using the spectral representation of first-passage processes (for more details on
+see e.g. [1, 2]). As stated in the Letter, we consider time-homogeneous Markov processes xt on a continuous or discrete
+state-space Ω with (forward) generator ˆL corresponding to a Markov rate-matrix or an effectively one-dimensional
+Fokker-Planck operator. Let the transition probability density to find xt at x at time t given that it evolved from x0
+be pt(x|x0) ≡ eˆLtδx0(x) where δx0(x) denotes the Dirac or Kronecker delta for continuous and discrete state-spaces,
+respectively. We assume the process to be ergodic limt→∞ pt(x|x0) = peq(x), where peq(x) ≡ e−ϕ(x) denotes the
+equilibrium probability density and ϕ(x) the corresponding generalized potential in units of thermal energy kBT. We
+assume that ˆL obeys detailed balance, such that it is self-adjoint in the left eigenspace with respect to a scalar product
+weighted by e−ϕ(x) and the operator eϕ(x)/2 ˆLe−ϕ(x)/2 is self-adjoint with respect to a flat measure.
+We assume that ˆL is either (i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, or that (iii) Ω is infinite but ϕ(x)
+is sufficiently confining (precisely, we require that ϕ(x) satisfies the Poincar´e inequality, i.e. lim|x|→∞(|∇ϕ(x)|2/2 −
+∇2ϕ(x)) = ∞.). Each of the conditions (i)-(iii) ensures that the eigenvalue spectrum of ˆL is discrete. The relaxation
+eigenvalue problem (for the inner product (·|·) defined with respect to a flat Lebesgue measure) reads −ˆLΨR
+k (x) =
+νkΨR
+k (x) with ΨL
+k(x) = ΨR
+k (x)eϕ(x), ν0 = 0 and νk≥1 > 0.
+The first-passage time to a target a for xt=0 drawn from a density p0(x) is defined as τ = inft[ t |xt = a, p0(x0)].
+We will use ⟨·⟩ to denote an average over all first-passage paths {xt′}0≤t′≤τ, i.e. those that hit a only once. The
+first-passage time density to a, ℘a(t|x0) = ⟨δ(t − τ[{xt′}])⟩ to reach the absorbing target at x = a, starting initially
+from x0, has the general spectral representation
+℘a(t|x0) =
+�
+k≥1
+wk(x0)µke−µkt,
+(S1)
+where µk is the k-th first-passage rate and wk(x0) its corresponding first-passage weight. In similar fashion the survival
+probability is expressed as
+Sa(t|x0) ≡
+� ∞
+t
+℘a(t′|x0)dt′ =
+�
+k≥1
+wk(x0)e−µkt.
+(S2)
+We note that in contrast to the relaxation eigenvalues νk, the first-passage rates µk = µk(a) depend in the location of
+the absorbing target. Moreover, for any target location a the interlacing theorem holds [1, 2] :
+νk−1 ≤ µk(a) ≤ νk
+∀k, a
+(S3)
+where equality occurs iff wk(x0) = 0, i.e. for a where ΨR
+k (a) = 0.
+Laplace transforming the spectral expansion of the first-passage time density (S1)—according to ˜f(s) ≡
+�
+e−stf(t) dt
+with f being a generic function locally integrable on t ∈ [0, ∞)—yields
+˜℘a(s) =
+�
+k≥1
+wk(x0)µk
+s + µk
+.
+(S4)
+The first-passage weights are then obtained by using the residue theorem to invert the Laplace transformed renewal
+theorem [1–3]
+wk(x0) = ˜p(a, −µk|x0)
+µk ˙˜p(a, −µk|a) =
+�
+l≥0(1 − νl/µk)−1ΨR
+l (a)ΨL
+l (x0)
+�
+l≥0(1 − νl/µk)−2ΨR
+l (a)ΨL
+l (a) < ∞,
+(S5)
+where ˙˜p(a, s|a) = ∂s˜p(a, s|a) is taken at s = −µk and {νl, ΨR
+l , ΨL
+l } are the corresponding relaxation eigenmodes [1, 2].
+The weights satisfy �
+k≥1 wk(x0) = 1 and the first non-zero weight is strictly positive w1(x0) > 0. Moreover, the
+relaxation eigenvalues ν0 = 0 and all νk>0 ≥ 0 are real as a result of detailed balance.
+
+3
+B.
+Lemma 1: All weights are non-negative for equilibrium initial conditions
+In the Letter we focus on equilibrium initial conditions, that is we assume that x0 is drawn from the invariant
+measure, peq(x0), which in the particular case of diffusion processes is assumed to have a reflecting boundary at a
+(i.e. we focus on the one-sided first-passage process). We further introduce the non-negative modified spectral weights
+¯wk(x0) ≡ wk(x0)θ(sgn[wk(x0)]) and now prove that for a normalized equilibrium probability density of initial conditions
+p0(x0) that excludes the target—i.e. ˜peq(x0) ≡ peq(x0)[1 − δa(x0)]/(1|peq(x0)[1 − δa(x0)]) where δa(x0) is the Dirac
+measure (note that (1|˜peq) = 1)—all weights wk are rendered non-negative. We thus have ¯wk(˜peq) = wk(˜peq) ≥ 0, ∀k.
+Namely, because ΨL
+l (a) = eβU(a)ΨR
+l (a) we have ΨR
+l (a)ΨL
+l (a) ≥ 0, ∀l, and from bi-orthogonality (ΨL
+l |peq) = δl,0 it
+follows that
+˜wk ≡ (wk|˜peq) = ˜peq(a)
+1 − �
+l≥0(1 − νl/µk)−1ΨR
+l (a)ΨL
+l (a)
+�
+l≥0(1 − νl/µk)−2ΨR
+l (a)ΨL
+l (a)
+=
+˜peq(a)
+�
+l≥0(1 − νl/µk)−2ΨR
+l (a)ΨL
+l (a) ≥ 0
+(S6)
+because by definition µk > 0, ∀k ≥ 1 denotes the zeros of ˜p(a, s|a), i.e. ˜p(a, −µk|a) = �
+l≥0(νl − µk)−1ΨR
+l (a)ΨL
+l (a) =
+µ−1
+k
+�
+l≥0(1 − νl/µk)−1ΨR
+l (a)ΨL
+l (a) = 0 which completes the proof of the Lemma.
+C.
+Lemma 2: Sum of positive weights is bounded from above
+For the sake of completeness we here additionally present results for general initial conditions p0(x0). Recall from
+the Letter that we require some additional conditions on ϕ(x) or Ω in this more general setting.
+In particular, we assume that ϕ(x) is sufficiently confining to assure a “nice” asymptotic growth of eigenvalues,
+limk→∞ νk = bkβ with β > 1/2 and 0 < b < ∞. The latter condition is automatically satisfied when Ω is finite, since
+regular Sturm-Liouville problems display Weyl asymptotics with β = 2 [4]. The condition is in fact satisfied by most
+physically relevant processes with discrete spectra, incl. the (Sturm-Liouville irregular) Ornstein-Uhlenbeck or Rayleigh
+process [5] with β = 1. This implies, by the interlacing theorem (S3) that b(k − 1)β ≤ µk ≤ bkβ and therefore there
+exists a real constant C ∈ (0, ∞) such that limk→∞ µk diverges as Ckβ.
+Recall further that the m-th moment of τ is given by ⟨τ m⟩ = m! �
+k≥1 wk(p0)/µm
+k . By construction we obtain
+2 �
+k≥1 ¯wk(p0)/µ2
+k ≡ ⟨¯τ 2
+p0⟩ ≥ 2 �
+k≥1 wk(p0)/µ2
+k ≡ ⟨τ 2
+p0⟩, where equality holds when p0 = ˜peq (since in this case all
+wk ≥ 0, i.e., ¯wk(˜peq) = wk(˜peq) as discussed before).
+Moreover, because we only consider Markov jump processes on finite state-spaces as well as processes for which
+limk→∞ µk = Ckα with 0 < C < ∞ and α > 1/2 (this includes confined Markov jump processes on infinite state-spaces
+and all regular Sturm-Liouville problems) convergence is ensured, i.e. 2 �
+k≥1 ¯wk(p0)/µ2+n
+k
+< ∞, ∀n ≥ 0.
+To prove this consider wmax ≡ maxk≥k∗ ¯wk(p0) such that wmax/µ2+n
+k
+≥ wk(p0)/µ2+n
+k
+, ∀k. Let the smallest k for
+which the asymptotic scaling holds be k∗ then we may split the summation as �
+k≥1 = �k∗−1
+k=1 + �
+k≥k∗ such that
+�
+k≥1
+¯wk(p0)
+µ2+n
+k
+≤
+k∗−1
+�
+k=1
+¯wk(p0)
+µ2+n
+k
++
+�
+k≥k∗
+wmax
+µ2+n
+k
+.
+Because the first term is nominally finite we only need to prove convergence of the second sum, which we do by
+means of the integral test. We define a function f(k) ≡ wmax/µ2+n
+k
+that is monotonically decaying in k. This
+implies f(x) ≤ f(k), ∀x ∈ [k, ∞) and f(x) ≥ f(k), ∀x ∈ [k∗, k].
+We then have for every integer k ≥ k∗ that
+� k+1
+k
+f(x)dx ≤
+� k+1
+k
+f(k)dx = f(k) and conversely, for every integer k ≥ k∗+1 that
+� k
+k−1 f(x)dx ≥
+� k
+k−1 f(k)dx = f(k).
+We now sum over all k ≥ k∗ to obtain, using µk = Ckα∀k ≥ k∗
+� ∞
+k∗
+wmax
+(Cxα)(2+n) dx ≤
+�
+k
+wmax
+µ2+n
+k
+≤
+wmax
+(Ckα∗ )2+n +
+� ∞
+k∗
+wmax
+(Cxα)2+n dx →
+wmaxC−(2+n)k1−α(2+n)
+∗
+α(2 + n) − 1
+≤
+�
+k
+wmax
+µ2+n
+k
+≤ wmax(Ckα
+∗ )−(2+n) + wmaxC−(2+n)k1−α(2+n)
+∗
+α(2 + n) − 1
+< ∞
+where the last integral converges because 1−α(2+n) < 0, ∀n ≥ 0, which in turn proves convergence of �
+k≥1 ¯wk(p0)/µ2
+k.
+
+4
+S2.
+EXTREME VALUE BOUNDS AND COMPARISON WITH CRAM´ER-CHERNOFF BOUNDS
+In the Letter we derive lower bounds L±
+n (t) on the deviation probability P(τ n − ⟨τ⟩ ≥ t) and P(⟨τ⟩ − τ n ≥ t)
+by utilizing extremal events, i.e., we consider the maximal and minimal first-passage time in a sample of n ≥ 1
+i.i.d. realizations. In this section we derive analogous upper bounds building on the same ideas.
+A.
+Extreme value bounds
+Recall that for the reversible Markov dynamics considered the equilibrium survival probability Sa(t|˜peq) ≡ Sa(t) in
+its spectral representation (S2) obeys
+w1e−µ1t ≤ Sa(t) ≤ e−µ1t.
+(S7)
+For the upper bound we use µk ≤ µk+1 and that �
+k>0 wk = 1 are normalized, whereas the lower bound follows since
+wk ≥ 0, ∀k, as we consider equilibrium initial conditions throughout. Moreover, from extreme value theory it follows
+P(τ min
+n
+≥ t) = Sa(t)n
+⇔
+P(τ min
+n
+≤ t) = 1 − Sa(t)n,
+P(τ max
+n
+≤ t) = (1 − Sa(t))n
+⇔
+P(τ max
+n
+≥ t) = 1 − (1 − Sa(t))n ,
+(S8)
+where we introduce τ max
+n
+≡ maxi∈[1,n] τi and τ min
+n
+≡ mini∈[1,n] τi, respectively. Clearly, since τ min
+n
+≤ τ n ≤ τ max
+n
+we
+can write P(τ min
+n
+≥ t) ≤ P(τ n ≥ t) ≤ P(τ max
+n
+≥ t) and analogously P(τ min
+n
+≤ t) ≥ P(τ n ≤ t) ≥ P(τ max
+n
+≤ t). Using
+Eq. (S8) in combination with Eq. (S7) we directly arrive at the lower bounds L±
+n (t) (see Eq. (4) in the Letter)
+P(τ n ≥ ⟨τ⟩ + t) ≥ P(τ min
+n
+≥ ⟨τ⟩ + t) = Sa(t + ⟨τ⟩)n ≥
+�
+w1e−µ1(⟨τ⟩+t)�n
+(S9)
+P(τ n ≤ ⟨τ⟩ − t) ≥ P(τ max
+n
+≤ ⟨τ⟩ − t) = (1 − Sa(⟨τ⟩ − t))n ≥
+�
+1 − e−µ1(⟨τ⟩−t)�n
+.
+(S10)
+Introduced considerations are, however, not restricted to only lower bounds such that we can further leverage bounds
+on the equilibrium survival probability (S7) to analogously obtain corresponding upper bounds as
+P(τ n ≥ ⟨τ⟩ + t) ≤ P(τ max
+n
+≥ t) = 1 − (1 − Sa(⟨τ⟩ + t))n ≤ 1 −
+�
+1 − eµ1(⟨τ⟩+t)�n
+,
+P(τ n ≤ ⟨τ⟩ − t) ≤ P(τ min
+n
+≤ t) = 1 − Sa(t)n ≤ 1 −
+�
+w1e−µ1(⟨τ⟩−t)�n
+.
+(S11)
+As we will illustrate next, the upper bounds (S11) are much weaker than those derived with the Cram´er-Chernoff
+approach (Eq. (7) in the Letter) and require more information about the dynamics.
+B.
+Comparison of Cram´er-Chernoff vs Extreme value Bounds
+In this section we directly compare the concentration-based upper bounds U±
+n (t) (see Eq. (7) in the Letter) that
+are obtained with the Cram´er-Chernoff approach, with the upper bounds (S11) which are based on extreme value
+considerations in analogy to the lower bounds L±
+n (t). Similar to Fig. 2e-h of the Letter we now exemplify and compare
+both upper bounds in Fig. S1 for the model systems shown in Fig. 1b-d.
+In Fig. S1a-d we equivalently express re-scaled deviation probabilities P1/n(sgn(t)δτ n ≥ |t|) in a single panel, i.e.,
+for the left tail we formally let t → −t such that t as shown now has support in [−⟨τ⟩, ∞) and sgn(x) = ±1 for
+±x > 0 and sgn(0) = 0 denotes the signum function. Empirical deviation probabilities (symbols) as a function of t are
+computed from statistics obtained by sampling τ n for different fixed n values. Extreme value lower bounds L±
+n (t) (S10)
+(or Eq. (4)) for both tails are depicted in red. Here we now focus on comparing the upper bounds. Concentration
+inequalities U±
+n (t; C) (Eq.(7)) are again depicted as black lines whereas the corresponding extreme value upper bounds
+are represented as dashed/dotted lines where the respective coloring indicates the number of realizations n. Note, that
+the concentration bounds (and the lower bounds) collapse onto a single master curve due to the employed scaling P1/n,
+whereas the extreme value upper bounds do not due to their different functional form (compare Eq. (S11)). Evidently,
+while for n = 1 the extreme value bounds remains close to the actual deviation probability, already for n = 3 they
+become considerably less tight and overshoot heavily for all considered models. Moreover, extreme value upper bounds
+become increasingly weak (even trivial at times) as n increases, therefore highlighting that Cram´er-Chernoff-type
+bounds are vastly more suitable.
+
+5
+Motivated by the discussion above we next want to gain more quantitative insights for which sample sizes n the
+Cram´er-Chernoff approach becomes more favorable. For this purpose we introduce a quality factor Q ∈ [0, ∞) that is
+informally defined as
+Q ≡
+Extreme value upper bound
+Cram´er-Chernoff-type upper bound.
+(S12)
+A value Q > 1 therefore indicates that the Cram´er-Chernoff bound is tighter and Q < 1 suggests that the extreme
+value bound should be favored, respectively. In Fig. S1e-h we illustrate the quality factor Q as a function of sample
+size n for different fixed dimensionless deviation values µ1t (star symbols in Fig. S1a-d). Remarkably for all model
+systems considered—which span a large range of possible C values—the Cram´er-Chernoff approach is already superior
+even in the small-sample regime n ≲ 4. Moreover, we can further study the particular n∗, for which one would reach
+Q = 1, as a function of some desired deviation µ1t relative to the longest time scale 1/µ1. Note, that again for the
+left tail we let t → −t (see discussion above). As depicted in Fig. S1i-l for our model systems, n∗ (blue) generally is
+found to be well below n = 8, i.e., even for most small sample sizes the derived Cram´er-Chernoff-type bounds can be
+considered to be the better choice, especially when considering large µ1t (i.e. large deviations).
+Lastly, one could ask the question why the extreme value upper bound is so “weak” when n increases even just
+slightly. To answer this question we recall that—since we are interested in deviations of the sample mean τ n around
+⟨τ⟩—we bound the sample mean with the minimal and maximal first-passage time according to τ min
+n
+≤ τ n ≤ τ max
+n
+which is further used, in combination with bounds on the survival probability (S7), to derive corresponding upper
+bounds (S11). Clearly, as n increases we expect this bound to become increasingly loose as by larger sample sizes we
+increase the chances of sampling rare first-passage times, i.e., maximal and minimal first-passage time that strongly
+deviate from the (sample) mean—this also explain why bounds (S10) and (S11) are only particularly tight for n = 1
+as here τ min
+n
+= τ 1 = τ max
+n
+. In contrast, the Cram´er-Chernoff method requires a much more delicate mathematical
+analysis involving bounds of the moment generating function. The Cram´er-Chernoff-type bound has the additional
+advantage that it can be further used to universally bound deviation probabilities where no specific information about
+the underlying system is required (see Eq. (11) in the Letter). Moreover, even the version of Cram´er-Chernoff bounds
+U±
+n (t; C) that require input of one system-dependent constant C still require less information about the dynamics since
+extreme value upper bounds (S11) partly also require knowledge about the first-passage weight w1 and ⟨τ⟩ itself.
+S3.
+COMPLETE PROOF OF CONCENTRATION INEQUALITIES AND THEIR ASYMPTOTICS
+In this section we provide various additional details on the upper bounds U±
+n (t; C) (Eq. (7) of the Letter). In
+particular, we prove the required bounds on the cumulant generating function, compute their corresponding Cram´er
+transform, and give further information about the large-sample limit n → ∞, as well as the model-free version of the
+bounds.
+A.
+Theorem 1: Cram´er-Chernoff bound for the right tail τ ≥ ⟨τ⟩
+We begin with the right tail, i.e. upwards deviations such that τ ≥ ⟨τ⟩, and start by proving a bound for the
+moment generating function of the deviation of the first-passage time τ from the mean ⟨τ⟩. Using the spectral
+representation (S1) and the inequality x ≤ ex−1, ∀x ∈ R, we find
+⟨eλ(τ−⟨τ⟩)⟩ = e−λ⟨τ⟩ �
+k>0
+wk
+1 − λ/µk
+≤ exp
+�
+−λ⟨τ⟩ +
+�
+k>0
+wk
+1 − λ/µk
+− 1
+�
+(S13)
+for all λ < µk. Moreover, for |λ| < µ1 we may further expand the sum �
+k>0
+wk
+1−λ/µk = �
+m≥0 λm �
+k>0 wk/µm
+k using
+the geometric series. Recall that the moments are given by ⟨τ m⟩ = m! �
+k>0 wk/µm
+k , such that we obtain
+⟨eλ(τ−⟨τ⟩)⟩ ≤ exp
+�
+��
+m≥2
+λm �
+k>0
+wk
+µm
+k
+�
+� = exp
+�
+λ2 ⟨τ 2⟩
+2
++
+�
+m>2
+λm �
+k>0
+wk
+µm
+k
+�
+.
+(S14)
+
+6
+FIG. S1. Comparison between Cram´er-Chernoff-type upper bounds U±
+n (t; C) and extreme value upper bounds for a spatially
+confined Brownian search process in dimensions (a,e,i) d = 1 and (b,f,j) d = 3, and discrete-state Markov jump processes for
+(c,d,k) the inferred model of calmodulin and (d,h,l) a 8-state toy protein. (a-d) Scaled probabilities P1/n(sgn(t)δτ n ≥ |t|) that
+the sample mean τ n inferred from n ≥ 1 realizations deviations from ⟨τ⟩ by more than t in either direction. Right tail areas
+are shown for t > 0 and left for t < 0, respectively. Cram´er-Chernoff upper bounds U±
+n (t; C) as black and extreme value upper
+bounds as dashed lines, respectively. Corresponding lower bounds L±
+n (t) are depicted as red lines and symbols denoted scaled
+empirical deviation probabilities obtained from the statistics of τ n for different n. (e-h) Quality factor Q as a function of n for
+different fixed relative deviations µ1t (see star symbols (a-d)). (i-l) Sample size n∗ (blue) for which both upper bounds are
+equal, i.e., Q = 1, as a function of re-scaled deviations.
+Since µ1 ≤ µk>1 and all first-passage weights wk are positive (due to equilibrium initial conditions) we find
+⟨eλ(τ−⟨τ⟩)⟩ ≤ exp
+�
+λ2 ⟨τ 2⟩
+2
++
+�
+m>2
+λm �
+k>0
+wk
+µm
+k
+�
+≤ exp
+�
+λ2 ⟨τ 2⟩
+2
++
+�
+m>2
+λm
+µm−2
+1
+�
+k>0
+wk
+µ2
+k
+�
+≤ exp
+�
+λ2 ⟨τ 2⟩
+2
+�
+1 +
+λ
+µ1 − λ
+��
+= exp
+�
+λ2
+⟨τ 2⟩/2
+(1 − λ/µ1)
+�
+.
+Introducing ψδτ(λ) ≡ ln⟨eλδτ⟩, with δτ = τ − ⟨τ⟩ for the right tail, we immediately identify the upper bound
+ψδτ(λ) ≤ λ2
+2
+⟨τ 2⟩
+1 − λ/µ1
+=
+˜λ2
+2
+C
+1 − ˜λ ≡ φδτ(˜λ; C)
+τ ≥ ⟨τ⟩,
+(S15)
+which concludes the derivation of the upper expression in Eq. (6) of the Letter. Note that we further have introduced
+the dimensionless quantities ˜t ≡ µ1t, C = µ2
+1⟨τ 2⟩, and ˜λ = λ/µ1 in the last step. In the case of general initial conditions
+p0(x0) ̸= peq(x0) we must simply replace µ2
+1⟨τ 2⟩ → C from Lemma 2.
+Next, we find the optimizing value of ˜λ, i.e., we compute the Cram´er transform of Eq. (S15) defined as
+φ∗
+δτ(˜t; C) ≡
+sup
+˜λ∈[0,1)
+[˜λ˜t − φδτ(˜λ; C)] =
+sup
+˜λ∈[0,1)
+�
+˜λ˜t −
+˜λ2
+2
+C
+1 − ˜λ
+�
+.
+(S16)
+
+7
+φδτ(˜λ; C) is differentiable, non-negative, convex, and increasing on
+˜
+lambda ∈ [0, 1), which implies that Eq. (S16) can be
+obtained by differentiation of ˜λ˜t − φδτ(˜λ; C) with respect to ˜λ, hence φ∗
+δτ(˜t; C) = ˜λ†˜t − φδτ(˜λ†; C) where the optimum ˜λ†
+solves φ′
+δτ(˜λ†; C) = t. Accordingly, we find the supremum to be attained at ˜λ†(˜t) = 1 − 1/
+�
+1 + 2˜t/C. For convenience
+we further introduce the auxiliary function h+(u) ≡ 1 + u − √1 + 2u such that we finally arrive at
+φ∗
+δτ(˜t; C) = Ch+(˜t/C) = Ch+(µ1t/C),
+0 ≤ t ≤ ⟨τ⟩.
+(S17)
+By using Chernoff’s inequality we subsequently obtain the upper bound P(δτ n ≥ t) ≤ e−nφ∗
+δτ (t;C) ≡ U+
+n (t; C) for
+0 ≤ t ≤ ∞ which completes the proof of Theorem 1 and thus the first announced inequality (7) in the Letter.
+0
+0.005
+0.010 (a)
+˜t = 0.1
+˜λ˜t
+φ∗
+δτ (˜t; 2)
+φ∗
+δτ (˜t; C)
+φδτ (˜λ; C)
+φδτ (˜λ; 2)
+0
+0.005
+0.010 (b)
+0
+0.005
+0.010
+(c)
+0
+0.01
+0.02 (d)
+0
+0.05
+0.10
+˜λ
+0
+0.002
+0.004 (e)
+˜λ†(˜t; C)
+˜λ†(˜t; 2)
+φ∗
+δτ (˜t; C)
+φ∗
+δτ (˜t; 2)
+˜λ˜t − φδτ (˜λ; C)
+˜λ˜t − φδτ (˜λ; 2)
+0
+0.05
+0.10
+˜λ
+0
+0.001
+0.002
+0.003 (f)
+0
+0.05
+0.10
+˜λ
+0
+0.002
+0.004 (g)
+0
+0.1
+0.2
+˜λ
+0
+0.004
+0.008 (h)
+FIG. S2. Illustration of the Cram´er-Chernoff bounding method for the right tail with ˜t = 0.1 and parameters for spatially
+confined Brownian search process in dimensions d = 1 (a,e) or d = 3 (b,f), and discrete-state Markov jump processes for
+the model of calmodulin (c,d) and a 8-state toy protein (d,h). Top row depicts bounds of the cumulant generating function
+φδτ(˜λ; C) (black) and φδτ(˜λ; 2) (yellow) as a function of ˜λ, respectively. Bottom row shows the differences ˜λ˜t − φδτ(˜λ; C) (red)
+and ˜λ˜t − φδτ(˜λ; 2) (green) as a function of ˜λ, respectively (see also top row with ˜λ˜t in blue). The corresponding suprema are
+obtained at ˜λ†(˜t; C) and ˜λ†(˜t; 2) (dotted lines) and define the Cram´er transforms φ∗
+δτ(˜t; C) and φ∗
+δτ(˜t; 2) (compare top row). For
+all considered models we demonstrate φδτ(˜λ; C) ≤ φδτ(˜λ; 2) and φ∗
+δτ(˜t; C) ≥ φ∗
+δτ(˜t; 2) as derived in the maintext. Note for the
+panels (b,f) we have φδτ(˜λ; C) ⪅ φδτ(˜λ; 2) and φ∗
+δτ(˜t; C) ⪆ φ∗
+δτ(˜t; 2) since C = 1.99 ≈ 2.
+B.
+Theorem 2: Cram´er-Chernoff bound for the left tail ⟨τ⟩ < τ
+Next we turn to the left tail, τ < ⟨τ⟩, where the corresponding moment generating function analogously reads
+⟨eλ(⟨τ⟩−τ)⟩ = eλ⟨τ⟩ �
+k>0
+wk
+1 + λ/µk
+≤ exp
+�
+λ⟨τ⟩ +
+�
+k>0
+wk
+1 + λ/µk
+− 1
+�
+for λ < µk. Using equivalent arguments as for the right tail above we may further write
+eλ(⟨τ⟩−τ)⟩ ≤ exp
+�
+��
+m≥2
+(−λ)m �
+k>0
+wk
+µm
+k
+�
+�
+= exp
+�
+λ2 ⟨τ 2⟩
+2
++
+�
+m>2
+(−λ)m �
+k>0
+wk
+µm
+k
+�
+≤ exp
+�
+λ2 ⟨τ 2⟩
+2
++
+�
+m>0
+λ2m
+µ2m−2
+1
+�
+k>0
+wk
+µ2
+k
+�
+= exp
+�
+λ2
+⟨τ 2⟩/2
+1 − (λ/µ1)2
+�
+.
+(S18)
+Recall the definition of the cumulant generating function, ψδτ(λ) ≡ ln⟨eλδτ⟩, such that Eq. (S18) directly yields
+ψδτ(λ) ≤ λ2
+2
+⟨τ 2⟩
+1 − (λ/µ1)2 =
+˜λ2
+2
+C
+1 − ˜λ2 ≡ φδτ(˜λ; C)
+(S19)
+
+8
+which completes the derivation of the lower expression in Eq. (6) of the Letter. Note that for the left tail we
+have δτ = ⟨τ⟩ − τ and we again let ˜t ≡ µ1t, ˜λ ≡ λ/µ1, and C ≡ µ2
+1⟨τ 2⟩. In the case of general initial conditions
+p0(x0) ̸= peq(x0) we must simply replace µ2
+1⟨τ 2⟩ → C from Lemma 2.
+Analogous to the right tail we next compute the Cram´er transform of Eq. (S19), i.e.,
+φ∗
+δτ(˜t; C) ≡
+sup
+˜λ∈[0,1)
+[˜λ˜t − φδτ(˜λ; C)] =
+sup
+˜λ∈[0,1)
+�
+˜λ˜t −
+˜λ2
+2
+C
+1 − ˜λ2
+�
+,
+(S20)
+where we find the optimal value ˜λ†(˜t; C) to be determined by the transcendental quartic, ˜λ†(˜t) : (1− ˜λ2)2 −C˜t˜λ = 0 with
+C˜t ≡ C/˜t, which we solve according to the method of Descartes. First, we re-arrange the quartic as ˜λ4−2˜λ2−C˜t˜λ+1 = 0
+and make the factorization ansatz
+(˜λ2 − y˜t˜λ2 + w˜t)(˜λ2 + y˜t˜λ2 + z˜t) = 0
+w˜t + z˜t − y2
+˜t = −2
+y˜t(w˜t − z˜t) = −C˜t
+z˜tw˜t = 1.
+(S21)
+The system of equations (S21) is solved by
+w˜t(y˜t) = (y2
+˜t − 2 − C˜t/y˜t)/2,
+z˜t(y˜t) = (y2
+˜t − 2 + C˜t/y˜t)/2,
+(S22)
+where y2
+˜t ≡ Y˜t is the solution of the cubic Y 3
+˜t − 4Y 2
+˜t − C2
+˜t = 0. Moreover, since the discriminant D is strictly
+negative, i.e. D = −28C2
+˜t − 33C4
+˜t < 0, the qubic has only one real root.
+The corresponding depressed qubic
+reads ˜t3 − 24/3˜t − (27/33 + C2
+˜t ) = 0 with ˜tY˜t − 4/3.
+Let p = −24/3 < 0 and q = −(27/33 + C2
+˜t ) < 0 then
+22p3 + 33q2 = −212/33 + 33(27/33 + C2
+˜t )2 > 0 for any ˜t ≥ 0. We can express the unique real root as
+y2
+˜t = 4
+3
+�
+1 + 2 cosh
+�1
+3arcosh
+�
+1 + 33C2
+˜t
+27
+���
+(S23)
+and y = ±
+�
+y2 with y2 from Eq. (S23) can now be plugged into Eqs. (S22) to obtain w˜t(y) and z˜t(y) that are required
+to solve the pair of quadratic equations (S21). The four roots of the transcendental quartic are hence given by
+˜λ1(˜t) = y˜t
+2
+�
+1 +
+�
+1 − 4w˜t(y˜t)/y2
+˜t
+�
+,
+˜λ2(˜t) = y˜t
+2
+�
+1 −
+�
+1 − 4w˜t(y˜t)/y2
+˜t
+�
+,
+˜λ3(˜t) = −y˜t
+2
+�
+1 −
+�
+1 − 4z˜t(y˜t)/y2
+˜t
+�
+,
+˜λ4(˜t) = −y˜t
+2
+�
+1 +
+�
+1 − 4z˜t(y˜t)/y2
+˜t
+�
+.
+(S24)
+Moreover, we find w˜t(y˜t)/y2
+˜t = (1 − 2/y2
+˜t − C˜t/y3
+˜t )/2 and z˜t(y˜t)/y2
+˜t = (1 − 2/y2
+˜t + C˜t/y3
+˜t )/2. Since y˜t > 0 while
+˜λ ∈ [0, 1), ˜λ2, ˜λ3 in Eq. (S24) are excluded automatically (note also that the square root in ˜λ2, ˜λ3 becomes complex for
+˜t → ∞). We also have lim˜t→∞ y2
+˜t = 4 and lim˜t→∞ w˜t(y˜t) = 1 such that lim˜t→∞ ˜λ1 = ˜λ2 = 1. Conversely, we find that
+lim˜t→0 ˜t2/3y2
+˜t = C2/3 = lim˜t→0 ˜t2/3C˜t/y˜t such that lim˜t→0 w˜t(y˜t) = −1 while lim˜t→0 w˜t(y˜t)y2
+˜t = −C2/3 = 0. Therefore,
+lim˜t→0 ˜λ1 = y˜t → ∞ whereas lim˜t→0 ˜λ2(˜t) = y˜t × 0/2 ↘ 0. We recall that ˜λ ∈ [0, 1) which therefore excludes ˜λ1(˜t)
+and identifies ˜λ†(˜t) = ˜λ2(˜t) as the supremum. Finally, we introduce the auxiliary functions
+g(u) ≡
+2
+√
+3
+�
+1 + 2 cosh
+�1
+3arcosh
+�
+1 +
+33
+27u2
+���1/2
+and
+Λ(u) ≡ 1
+2
+�
+g(u) −
+�
+4 + 2/g(u)u − g(u)2
+�
+,
+(S25)
+as well as
+h−(u) ≡ Λ(u)u − 1
+2
+Λ(u)2
+1 − Λ(u)2
+(S26)
+which allows us to obtain and write the Cram´er transform as
+φ∗
+δτ(˜t; C) = Ch−(˜t/C) = Ch−(µ1t/C),
+0 ≤ t ≤ ⟨τ⟩.
+(S27)
+In the last step we use Chernoff’s inequality to obtain the bound P(δτ n ≥ t) ≤ e−nφ∗
+δτ (t;C) ≡ U−
+n (t; C) for 0 ≤ t ≤ ⟨τ⟩
+which completes the proof of Theorem 2 and hence the derivation of the lower expression in Eq. (7) of the Letter.
+
+9
+0
+0.005
+0.010 (a)
+˜t = 0.1
+˜λ˜t
+φ∗
+δτ (˜t; 2)
+φ∗
+δτ (˜t; C)
+φδτ (˜λ; C)
+φδτ (˜λ; 2)
+0
+0.005
+0.010 (b)
+0
+0.005
+0.010
+(c)
+0
+0.01
+0.02 (d)
+0
+0.05
+0.10
+˜λ
+0
+0.002
+0.004 (e)
+˜λ†(˜t; C)
+˜λ†(˜t; 2)
+φ∗
+δτ (˜t; C)
+φ∗
+δτ (˜t; 2)
+˜λ˜t − φδτ (˜λ; C)
+˜λ˜t − φδτ (˜λ; 2)
+0
+0.05
+0.10
+˜λ
+0
+0.001
+0.002
+0.003 (f)
+0
+0.05
+0.10
+˜λ
+0
+0.002
+0.004 (g)
+0
+0.1
+0.2
+˜λ
+0
+0.004
+0.008 (h)
+FIG. S3. Illustration of the Cram´er-Chernoff bounding method for the left tail with ˜t = 0.1 and parameters for spatially confined
+Brownian search process in dimensions d = 1 (a,e) or d = 3 (b,f), and discrete-state Markov jump processes for the model of
+calmodulin (c,d) and a 8-state toy protein (d,h). Top row depicts bounds of the cumulant generating function φδτ(˜λ; C) (black)
+and φδτ(˜λ; 2) (yellow) as a function of ˜λ, respectively. Bottom row shows the differences ˜λ˜t − φδτ(˜λ; C) (red) and ˜λ˜t − φδτ(˜λ; 2)
+(green) as a function of ˜λ, respectively (see also top row with ˜λ˜t in blue). The corresponding suprema are obtained at ˜λ†(˜t; C)
+and ˜λ†(˜t; 2) (dotted lines) and define the Cram´er transforms φ∗
+δτ(˜t; C) and φ∗
+δτ(˜t; 2) (compare top row). For all considered models
+we demonstrate φδτ(˜λ; C) ≤ φδτ(˜λ; 2) and φ∗
+δτ(˜t; C) ≥ φ∗
+δτ(˜t; 2) as derived in the maintext. Note for the panels (b,f) we have
+φδτ(˜λ; C) ⪅ φδτ(˜λ; 2) and φ∗
+δτ(˜t; C) ⪆ φ∗
+δτ(˜t; 2) since C = 1.99 ≈ 2.
+C.
+Behavior of upper bounds U±
+n (t) for large sample sizes
+Here, we present some further remarks about the limit of large sample sizes. Asymptotically as n → ∞, U±
+n (t) is
+substantial only for ˜t/C ≪ 1. For the right tail bound h+(u) we immediately find that for u ≪ 1 we can Taylor expand
+√1 + 2u = 1 + u − u2/2 + O(u3). Consequently we directly obtain h+(u) = −u2/2 + O(u3), i.e., the upper tail is
+sub-Gaussian for small deviations and will converge to a Gaussian as n → ∞. For the left tail we furthermore have
+arcosh(1 + x) = ln(1 + x +
+�
+x(x + 2)) and thus limx→∞ arcosh(1 + x) = ln(2x) − 1/(2x)2. As a result it follows that
+1
+3 limu→0 arcosh(1 + 33/27u2) ≃ 1
+3 ln(33/26u2) = ln(3/4u2/3) − u4212/37 and thus
+lim
+u→0 g(u) ≃
+2
+√
+3{1 + 2 cosh ln(3/4u2/3)}1/2 =
+2
+√
+3[1 + 3/4u2/3]1/2
+(S28)
+= u−1/3[1 + 4u2/3/3]1/2 = u−1/3[1 + 2u2/3/3 + O(u4/3)] = u−1/3 + 2
+3u1/3 + O(u).
+(S29)
+A lengthy but straightforward calculation subsequently reveals that limu→0 Λ(u) = u − O(u3) such that
+lim
+u→0
+Λ(u)2
+1 − Λ(u)2 ≃
+u2
+1 − u2 = u2 + O(u4).
+(S30)
+We therefore have that limu→0 h−(u) = u2/2 − O(u4), i.e., both tails are sub-Gaussian for ˜t/C ≪ 1 with C ≡ µ2
+1⟨τ 2⟩.
+D.
+Proof of bounds on C and model-free concentration inequalities
+Notably, system details only enter the Cram´er transforms (S17) and (S27) (and consequently upper bounds on the
+deviation probability due to Chernoff’s inequality) in the form of a system-specific constant C ≡ µ2
+1⟨τ 2⟩. Note that here
+we only allow for equilibrium initial conditions. Recalling that the moments of the first-passage time τ are expressed
+as ⟨τ m⟩ = m! �
+k>0 wk/µm
+k allows us to write
+0 ≤ 2w1
+µ2
+1
+≤ ⟨τ 2⟩ = 2
+�
+k>0
+wk
+µ2
+k
+≤ 2
+�
+k>0
+wk
+µ2
+1
+= 2
+µ2
+1
+(S31)
+
+10
+for equilibrium initial conditions where we have used that wk are non-negative, normalized, and µ1 ≤ µk>1. Conse-
+quently, by Eq. (S31), we immediately find that the system-constant itself is bounded 0 ≤ 2w1 ≤ C ≤ 2. Note that
+analogous considerations can be used to more generally obtain 0 ≤ m!w1 ≤ µm
+1 ⟨τ m⟩ ≤ m! for the m-th moment.
+The fact that C ∈ (0, 2] can now be further leveraged to arrive at the model-free bounds (Eq. (11) in the Letter)
+which require no information about the underlying system. Recall the upper bounds of the cumulant generating
+function φδτ(˜λ; C) and their corresponding Cram´er transform φ∗
+δτ(˜t; C), i.e.,
+φδτ(˜λ; C) =
+�
+�
+�
+�
+�
+�
+�
+˜λ2
+2
+C
+1 − ˜λ
+τ ≥ ⟨τ⟩
+˜λ2
+2
+C
+1 − ˜λ2
+τ < ⟨τ⟩,
+and
+φ∗
+δτ(˜t; C) =
+�
+Ch+(˜t/C)
+τ ≥ ⟨τ⟩
+Ch−(˜t/C)
+τ < ⟨τ⟩.
+(S32)
+Since φδτ(˜λ; C) is monotonically increasing in C it follows that φδτ(˜λ; C) ≤ φδτ(˜λ; 2), ∀˜λ ∈ [0, 1) (see Figs S2 and S3 top
+row). By definition of φ∗
+δτ(˜t; C) ≡ sup˜λ∈[0,1)(˜λ˜t−φδτ(˜λ; C)) this bound in turn implies that φ∗
+δτ(˜t; C) ≥ φ∗
+δτ(˜t; 2) (compare
+Figs. S2 and S3 bottom row). With Chernoff’s inequality we moreover arrive at P(δτ n ≥ t) ≤ e−nφ∗
+δτ (t;C) ≤ e−nφ∗
+δτ (t;2)
+and hence U±
+n (t; C) ≤ U±
+n (t; 2) which completes the derivation of Eq. (11) in the Letter.
+S4.
+MODEL SYSTEMS AND DETAILS ON NUMERICAL METHODS
+In the Letter we exemplify our results by considering a Brownian molecular search process in dimensions d = 1 and
+d = 3, as well as discrete-state Markov-jump models of protein folding for a 8-state toy protein and the experimentally
+inferred model of calmodulin (compare Fig. 1b-d). In this section we present further details on the model systems and
+their numerical treatment.
+A.
+Continuous-time discrete-state Markov jump process
+As illustrative discrete-state continuous-time Markov-jump models of protein folding we consider a simple 8-state
+toy protein [2, 6] and further use the experimentally inferred folding network of the cellular calcium sensor protein
+calmodulin [7]. Since we consider equilibrium initial conditions, proteins start from an initial state drawn from the
+equilibrium density ˜peq(x)—note that the tilde denotes that the absorbing target is excluded—from which they search
+the native state a (here a = (1, 1, 1) for the 8-state model and a = F1234 for calmodulin; cf. Fig. 1b-d). Arrows in the
+networks denote possible transitions, e.g. a transition from state i to state j that occurs with the corresponding rate
+Lji. We consider reversible dynamics, i.e., the resulting transition matrix ˆL of the relaxation process satisfies detailed
+balance peq,j/peq,i = Lji/Lij = exp(Fi − Fj) and transitions rates are connected to the free energy of the states Fi [8].
+We recall that the first-passage time density ℘a(t) can be evaluated by using the spectral representation (S1). To this
+end we set up the modified transition matrix, adopting in this section the Dirac bra-ket notation, ˆLa = ˆL−|a⟩⟨a| where
+|a⟩ ≡ (0, . . . , 0, 1, 0, . . .)⊺ defines a vector with all entries zero expect at the a-th position of the absorbing state where
+it equals one. This effectively removes all transitions that correspond to jumps leaving the absorbing state a. Next, we
+carry out an eigendecomposition of ˆLa and determine the eigenvalues µk, right eigenvectors |φR⟩, and left eigenvectors
+⟨φL|. We subsequently use obtained eigenmodes to compute the first-passage weights wk(x0) = −⟨a|φR
+k ⟩⟨φL
+k|x0⟩
+(see [1, 2]), and recall that µk and wk determine the moments according to ⟨τ m⟩ = m! �
+k>0 wk/µm
+k . Corresponding
+relevant parameters of the Markov jump models are listed in Tab. I. Next we give further details on how corresponding
+transition rates are constructed.
+TABLE I. Parameters for the Markov jump models for the 8-state toy protein and the inferred model of calmodulin. Listed are
+values for the first-passage eigenvalues µk, first-passage weights wk, and the first ⟨τ⟩ and second moment ⟨τ 2⟩.
+Model
+µ1
+w1
+µ2
+w2
+µ3
+w3
+µ4
+w4
+µ5
+w5
+µ6
+w6
+µ7
+w7
+⟨τ⟩
+⟨τ 2⟩
+Toy protein 0.976 0.337 6.148 0.009 1.551
+0.583
+4.203
+0.001
+4.396
+0.0001 6.233 0.060 12.834 0.010 0.385 0.713
+Calmodulin 0.469 0.651 3.763 0.349 19.097 9.98E-5 143.749 2.42E-9 1581.629 1.52E-6
+–
+–
+–
+–
+1.479 5.958
+
+11
+1.
+Transitions rates of the 8-state toy protein model
+For the 8-state toy protein model we randomly generate a free energy level Fi for each state i ∈ {1, 2, 3, 4, 5, 6, 7, 8}
+with Fi uniformly distributed within the interval 0 ≤ Fi ≤ 10. Transition rates that satisfy detailed balance are then
+obtained using the ansatz
+ki→j ≡ Lji = exp(∆Fi/2)
+and
+kj→i ≡ Lij = exp(−∆Fi/2),
+(S33)
+where ∆Fi ≡ Fi − Fj and thus ln(Lji/Lij) = ∆Fi = Fi − Fj. Obtained individual transition rates are listed in Tab. II.
+TABLE II. Transition rates for the 8-state toy protein model obtained via the ansatz described in the main text.
+transition rate ki→j transition rate ki→j transition rate ki→j transition rate ki→j
+1 → 2
+1.878
+2 → 5
+2.648
+3 → 7
+4.549
+5 → 8
+0.106
+2 → 1
+5.327
+5 → 2
+3.421
+7 → 3
+1.00994
+8 → 5
+124.477
+1 → 3
+0.00463
+2 → 6
+0.527
+4 → 6
+0.358
+6 → 8
+0.712
+3 → 1
+0.507
+6 → 2
+36.0577
+6 → 4
+36.457
+8 → 6
+15.794
+1 → 4
+0.326
+3 → 5
+1.109
+4 → 7
+0.523
+7 → 8
+0.322
+4 → 1
+0.623
+5 → 3
+0.0371
+7 → 4
+6.670
+8 → 7
+56.998
+2.
+Transitions rates of the calmodulin protein model
+In the experimental setup a constant external force f, a so-called pretension, is applied to the calmodulin protein
+via optical tweezers. Folding and unfolding processes are observed at different pretensions ranging from 6pN to 13 pN
+and corresponding force-dependent transition rates ki→j(f) = Lji(f) between two conformational states i and j are
+measured. Note that i, j ∈ {Unfold, F12, F123, F23, F34, F1234} and we further map states according to Unfold ↔ 1,
+F12 ↔ 2, F123 ↔ 3, F23 ↔ 4, F34 ↔ 5, and F1234 ↔ 6 for convenience. For our purposes we choose, without loss of
+generality, a pretension of f = 9 pN and obtain the corresponding measured transitions rates from Fig. S8 in the
+Supplementary Material of [7]. Clearly, experimental transitions rates are accompanied with measurement uncertainties
+which is reflected in slight “deviations” from a mathematically precise definition of detailed balance. To mitigate this
+issue, and to ensure that transition rates precisely obey detailed balance ki→jpeq,i = kj→ipeq,j, we further have to
+slightly adjust the rates.
+First, we compute the invariant density peq from the experimental rates and obtain a corresponding free energy
+level Fi = − ln(peq,i). Next, we use the ansatz (S33), i.e., Lji = Ai exp(∆Fi/2) and Lij = Ai exp(−∆Fi/2) where we
+introduce a constant Ai. Finally, Ai’s are chosen such that resulting transition rates fall within experimental error
+bars in Ref. [7]. Obtained transition rates are listed in Table III.
+TABLE III. Transition rates of the Markov jump model for the calmodulin protein. Rates are extracted from the Supplemental
+Material of Ref. [7] and modified such that they obey detailed balance precisely according to the maintext.
+transition rate ki→j transition rate ki→j transition rate ki→j
+1 → 2
+5.997
+1 → 4
+13.439
+1 → 5
+15.330
+2 → 1
+0.774
+4 → 1
+127.968
+5 → 1
+0.121
+5 → 6
+3.749
+2 → 3
+1514.820
+2 → 6
+13.441
+6 → 5
+13.326
+3 → 2
+53.0661
+6 → 2
+2.922
+B.
+Spatially confined Brownian molecular search process
+We also test our theory for Markov processes on a continuous state-space. More precisely, we consider the spatially
+confined diffusive search of a Brownian particle in a d-dimensional unit sphere with a reflecting boundary at R = 1 and
+a perfectly absorbing spherical target of radius 0 < a < 1, here a = 0.1, in the center (compare Fig. 1b). The closest
+distance of the particle to the surface of the absorbing sphere at time t is a confined Bessel process (see e.g. [2, 9, 10])
+which time evolution obeys the Itˆo equation
+dxt = (d − 1)x−1
+t dt +
+√
+2dWt,
+(S34)
+
+12
+where dWt is the increment of a Wiener process (i.e. Gaussian white noise) with ⟨dWt⟩ = 0 and ⟨dWtdWt′⟩ = δ(t−t′)dt,
+and we have set, without loss of generality, D = 1. The general case with any 0 < D < �� and a sphere of radius R is
+covered by expressing time in units of R2/D.
+For d = 1 Eq. (S34) reduces to a 1 dimensional Brownian motion which has the equilibrium first-passage weights
+weq
+k = 2
+π2
+1 − sin [(k − 1)π]
+(k − 1/2)2
+(S35)
+and matching first-passage eigenvalues are obtained as µk = π2(k − 1/2)2. Moreover, for d = 3 the first-passage time
+probability density of the Bessel process can be evaluated exactly and has the equilibrium weights
+weq
+k = 2
+µk
+3a2
+1 − a3
+tan[(1 − a)√µk] +
+1
+õk
+(1 − a) tan[(1 − a)√µk] −
+a
+õk
+,
+(S36)
+with the first-passage eigenvalues µk being the solutions of the transcendental equation √µk = tan([1 − a]√µk) that
+can be solved analytically using Newton’s series [2]. Relevant parameters for the spatially confined Brownian search
+process with a = 0.1 are listed in Tab. IV.
+TABLE IV. Parameters for the spatially confined Brownian molecular search process in dimensions 1 and 3. Listed are values
+for the first 5 first-passage eigenvalues µk, first-passage weights wk, and the first ⟨τ⟩ and second moment ⟨τ 2⟩, respectively a.
+Model
+µ1
+w1
+µ2
+w2
+µ3
+w3
+µ4
+w4
+µ5
+w5
+⟨τ⟩
+⟨τ 2⟩
+1D Brownian motion 2.467 0.811 22.207 0.0901 61.685
+0.0324
+120.903
+0.0165
+199.859 0.01001 0.333 0.267
+3D Bessel process
+0.363 0.994 25.174 0.00277 73.926 9.163E-4 147.037 4.573E-4 244.516 2.742E-4 2.739 1.509
+a For the numerical evaluation of ⟨τ⟩ and ⟨τ 2⟩ as listed we truncate the sum after M = 1000 terms.
+C.
+Statistics of first-passage times ⟨τ⟩ and the sample mean τ n
+Here we provide some further details on the sampling method used to obtain the statistics of (i) the first-passage
+time τ and (ii) the sample-mean τ n ≡ �
+i τi/n at some fixed value n for our considered models.
+We recall that after determining the first-passage eigenvalues µk and first-passage weights wk, the first-passage time
+density ℘a(t) (S1) and survival probability Sa(t) (S2) are fully characterized. To now sample the random variable τ,
+i.e. individual realizations of the first-passage process, we employ the so-called inversion sampling method [11]. This
+method allows us to generate independent samples of τ from ℘a(t) given its cumulative distribution function (CDF)
+which is directly related to the survival probability according to 1 − Sa(t). Note that for discrete-state dynamics the
+number of states M is finite, i.e. k = 1, . . . , M and therefore Eq. (S2) (and hence the CDF) is a finite sum. In contrast,
+for continuous-state dynamics we formally have M = ∞, meaning that sums are here not finite. For the following
+numerical evaluation of the spatially confined Brownian search process we therefore truncate the sum after M = 1000
+terms. The first-passage time densities ℘a(t) obtained via inversion sampling (symbols) for all considered models are
+shown in Fig. S4a-d and corroborated by the corresponding analytical result (S1) (dashed black line).
+For Fig. 2a-d in the Letter empirical probabilities that τ n − ⟨τ⟩ lies within a desired range of ± 10% of the
+longest first-passage time scale µ−1
+1 , P(µ1[τ n − ⟨τ⟩] ∈ [−0.1, 0.1]), are computed using statistics of the sample
+mean τ n by fixing n, i.e., the number of individual realizations the average is taken over. In particular, we have
+n ∈ {1, 2, 3, 5, 10, 20, 30, 40, 50, 75, 100, 150, 200, 300, 400, 500}. Subsequently, for each individual fixed n the sample
+mean τ n itself is sampled a total of N = 106 times. That is, we first draw n first-passage times τ, compute τ n by
+averaging over the drawn n realizations, and finally repeat this step N = 106 times to obtain statistics of τ n for all n
+values introduced above. Probability densities of the sample mean are shown in Fig. S4e-h for n ∈ {3, 5, 10, 20} and all
+model systems. Corresponding true mean first-passage times ⟨τ⟩ are highlighted in grey.
+In Fig. 2e-h of the Letter the probabilities to deviate more than t in either direction, P(±[τ n − ⟨τ⟩] ≥ t), are
+computed from analogous statistics of the sample mean τ n. Since we also consider empirical probabilities for rare
+events with large deviations (i.e. large µ1t) we however require substantially more statistics of τ n. To this end we now
+have N = 107 for n ∈ {1, 3} and N = 1011 for n ∈ {5, 10, 20}. In addition it should be further noted that we re-scale
+obtained probabilities according to P1/n. To compute an empirical deviation probability where e.g. P1/20 = 0.1 one
+would be thus required to sample rare events that occur with a probability of ≃ 10−20.
+
+13
+In Fig. 3a of the Letter each data point corresponds to the relative error µ1(τ n − ⟨τ⟩) (note that µ1 and ⟨τ⟩ are
+different for each model) where the sample mean τ n is again obtained by first fixing n and then sampling n first-passage
+times τ according to the inversion sampling method and subsequently taking the average.
+10−9
+10−5
+10−1
+t
+10−2
+101
+104
+℘a(t)
+(a)
+10−5
+10−1
+t
+10−2
+100
+102 (b)
+10−4
+10−2
+100
+t
+10−2
+10−1
+100
+101
+(c)
+10−5
+10−2
+t
+100
+102
+(d)
+0
+0.5
+1
+τ n
+0
+1
+2
+3
+4
+5
+p(τ n)
+⟨τ⟩
+(e)
+n = 3
+n = 5
+n = 10
+n = 20
+0
+2
+4
+6
+τ n
+0
+0.2
+0.4
+0.6
+0.8 (f)
+0
+2
+4
+τ n
+0
+0.5
+1 (g)
+0
+0.5
+1
+τ n
+0
+1
+2
+3
+4
+5 (h)
+FIG. S4. Inversion sampling of first-passage statistics for a spatially confined Brownian search process in dimensions (a,e) d = 1
+and (b,f) d = 3, and discrete-state Markov jump processes for (c,d) the inferred model of calmodulin and (d,h) a 8-state toy
+protein. (a-d) First-passage time density ℘a(t) obtained using inversion sampling (symbols) and analytical result as black dashed
+lines. (e-h) Empirical probability density of the sample mean τ n for different n values. True mean first-passage times ⟨τ⟩ are
+shown in grey.
+S5.
+UNCERTAINTY QUANTIFICATION WITH CONFIDENCE INTERVALS
+In this section we extend the discussion and present some further details on the confidence intervals introduced in
+the Letter. Our derived upper bounds U±
+n (t) can be applied to construct non-asymptotic performance guarantees such
+as confidence intervals. In particular, they can be employed to bound the probability that δτ n ≡ τ n − ⟨τ⟩ is found to
+be in some interval [−t−
+α−, t+
+α+], i.e.,
+P(δτ n ∈ [−t−
+α−, t+
+α+]) = P(−t−
+α− ≤ δτ n ≤ t+
+α+)
+= P(δτ n ≥ −t−
+α− ∩ δτ n ≤ t+
+α+)
+≥ 1 − P(δτ n ≤ −t−
+α−) − P(δτ n ≥ t+
+α+)
+≥ 1 − U−
+n (t−
+α−)
+�
+��
+�
+≡α−
+− U+
+n (t+
+α+)
+�
+��
+�
+≡α+
+.
+(S37)
+In passing from the second to the third line we have applied Boole’s second inequality, and from the third to forth
+line we use bounds (7) of the Letter. In the last line we additionally introduced acceptable right and left tail error
+probabilities α±. The implicit interval [−t−
+α−, t+
+α+] therefore defines a confidence interval at a confidence level of 1 − α
+with α ≡ α+ + α−, and α+ + α− < 1. In general the choice of the confidence interval for a fixed probability 1 − α is
+not unique. Some common options in the literature (see e.g. [12, 13]), all having the same confidence level, are listed
+below.
+• One common choice are so-called central intervals (blue lines in Fig. S5) which correspond to equal tail probabilities
+α+ = α− = α/2 for the complementary intervals [−⟨τ⟩, −t−
+α−] and [t+
+α+, ∞). Notably, we remark that central
+confidence intervals do not generally imply that t+
+α+ and t−
+α− are equidistant from another, i.e., t+
+α+ ̸= t−
+α−.
+
+14
+• As an alternative one could likewise choose t+
+α+ = t−
+α− ≡ ∆t/2, which subsequently leads to the symmetric
+interval [−∆t/2, ∆t/2] with total length ∆t (see red lines in Fig. S5) . Analogously, a symmetric interval does
+not necessarily imply that the corresponding tail probabilities are equal, i.e., in general α+ ̸= α−.
+• Both considerations above lead to two-sided intervals. However, another possible choice includes the fully
+asymmetric intervals [−⟨τ⟩, t+
+α+] and [−t−
+α−, ∞), i.e., one-sided intervals with a corresponding confidence level
+1 − α+ (for the upper limit t+
+α+) and 1 − α− (for the lower limit t−
+α−), respectively,
+P(±δτ n ≤ t±
+α±) ≥ 1 − α±.
+(S38)
+0
+0.5
+1
+µ1t+
+α+
+0
+0.5
+1
+µ1t−
+α−
+n = 15
+(a)
+0.5
+0.7
+0.9
+central int.
+symm. int.
+0
+0.5
+1
+µ1t+
+α+
+n = 20
+(b)
+0.5
+0.7
+0.9
+0
+0.5
+1
+µ1t+
+α+
+n = 30
+(c)
+0.5
+0.7
+0.9
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1 − α
+FIG. S5.
+Contour plot of different choices of possible two-sided confidence intervals [−µ1t−
+α−, µ1t+
+α+] for a fixed confidence level
+α and (a) n = 15, (b) n = 20, (c) n = 30. Contour lines for α ∈ {0.1, 0.3, 0.5} are depicted in white. Specific choices of central
+and symmetric are shown in blue and red, respectively, and we let C = 1 for all panels.
+Confidence intervals are practically useful as they answer questions such as e.g.:
+How many realizations are required to achieve a desired accuracy with a specified probability?
+Or: For a given number of realizations a desired accuracy is achieved with at least what probability?
+In the case of symmetric confidence intervals t+
+α+ = t−
+α− (see Fig. S5 red lines) the interval endpoints are implicitly
+defined via the last line of Eq. (S37) which is easily solved using standard root-finding procedures like the bi-section
+method [14]. The same holds true for other interval choices, however, when specifying the error probabilities α±
+directly—as done for e.g. two-sided central intervals (α± = α/2) or one-sided intervals—it suffices to solve Eq. (S38) with
+the respective α±. Hereby, the lower confidence limit t−
+α− is again easily obtained using standard root-finding methods.
+Notably, the upper confidence limit t+
+α+ can now be solved analytically. To show this we consider U+
+n (t+
+α+; C) = α+,
+i.e., we identify the t+
+α+ that solves
+0 = −nCh+(µ1t+
+α+/C) − ln(α+).
+(S39)
+The roots are identified as
+t1 = −ln (α+)
+µ1n
+−
+√
+2
+�
+− ln (α+)
+µ1
+�
+n/C
+and
+t2 = −ln (α+)
+µ1n
++
+√
+2
+�
+− ln (α+)
+µ1
+�
+n/C
+,
+(S40)
+and we identify t+
+α+ = t2 as the relevant solution. Having obtained an explicit expression for t+
+α+ further allows us to
+re-insert it into the left-hand side of Eq. (S38), i.e., we find that with a probability of at least 1 − α+
+δτ n ≤ −ln (α+)
+µ1n
++
+√
+2
+�
+− ln (α+)
+µ1
+�
+n/C
+.
+(S41)
+The required number of realizations n∗ to ensure with a probability of at least 1 − α that δτ n is found within some
+interval [−t−
+α−, t+
+α+] (e.g. symmetric interval in Fig. 3b) is analogous identified according to Eq. (14) in the Letter
+Un∗(t+
+α+; C) + Un∗(t−
+α−; C) = α,
+(S42)
+
+15
+which once again is readily solved via e.g. the bisection method. Moreover, in the case of one-sided intervals one
+immediately finds the corresponding analytical expression
+n∗ ≥ −
+ln(α±)
+Ch±(µ1t/C),
+(S43)
+where n∗ denotes the required number to ensure that ±δτ n ≤ t with at least 1 − α±.
+∗ [email protected]
+[1] D. Hartich and A. Godec, New J. Phys. 20, 112002 (2018).
+[2] D. Hartich and A. Godec, J. Phys. A: Math. Theor. 52, 244001 (2019).
+[3] A. J. F. Siegert, Phys. Rev. 81, 617 (1951).
+[4] G. Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Society, 2012).
+[5] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd ed., Springer Series
+in Synergetics, Vol. 13 (Springer-Verlag, Berlin, 2004).
+[6] G. R. Bowman, V. S. Pande, and F. No´e, An Introduction to Markov State Models and their Application to Long Timescale
+Molecular Simulation, Vol. 797 (Springer Science & Business Media, 2013).
+[7] J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt, and M. Rief, Science 334, 512 (2011).
+[8] U. Seifert, Annu. Rev. Condens. Matter Phys. 10, 171 (2019).
+[9] J. W. Pitman, Adv. Appl. Probab. 7, 511 (1975).
+[10] E. Barkai, E. Aghion, and D. Kessler, Phys. Rev. X 4, 021036 (2014).
+[11] L. Devroye, Non-Uniform Random Variate Generation (Springer New York, 1986).
+[12] G. Cowan, Statistical Data Analysis (Oxford University Press, 1998).
+[13] L. Lista, Statistical Methods for Data Analysis in Particle Physics (Springer International Publishing, 2017).
+[14] R. L. Burden, J. D. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015).
+

cNE5T4oBgHgl3EQffQ_r/content/tmp_files/2301.05626v1.pdf.txt

@@ -0,0 +1,1078 @@
+Channel Measurement for Holographic MIMO:
+Benefits and Challenges of Spatial Oversampling
+Tengjiao Wang∗, Yongxi Liu†, Ming Zhang†, Wei E. I. Sha‡, Cen Ling∗, Chao Li∗, Shaobo Wang∗
+∗Wireless Network RAN Research Department, Huawei Technologies CO., Ltd, Shanghai, China
+†School of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi, China
+‡College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China
+Emails: ∗{wangtengjiao6, lingcen, lichao18, shaobo.wang}@huawei.com,
+†[email protected], [email protected], ‡ [email protected]
+Abstract—In this paper, the channel of an indoor holographic
+multiple-input multiple-output (MIMO) system is measured. It
+is demonstrated through experiments for the first time that
+the spatial oversampling of holographic MIMO systems is able
+to increase the capacity of a wireless communication system
+significantly. However, the antenna efficiency is the most crucial
+challenge preventing us from getting the capacity improvement.
+An extended EM-compliant channel model is also proposed
+for holographic MIMO systems, which is able to take the
+non-isotropic characteristics of the propagation environment,
+the antenna pattern distortion, the antenna efficiency, and the
+polarization characteristics into consideration.
+Index Terms—Holographic MIMO, massive MIMO, spatial
+oversampling, channel measurement, electromagnetic informa-
+tion theory.
+I. INTRODUCTION
+In order to increase the spectral efficiency and energy
+efficiency for the future 5G-Advanced and 6G wireless com-
+munication systems, the concept of holographic multiple-input
+multiple-output (MIMO) is proposed recently [1]. By inte-
+grating an infinite number of antennas into a limited surface,
+holographic MIMO is expected to fully exploit the propagation
+characteristics offered by the electromagnetic channel and ap-
+proach the fundamental performance limit [2]. For holographic
+MIMO systems, both the accurate channel modeling and the
+realistic performance evaluation are key problems.
+In the literature, many efforts have been devoted to accu-
+rately model the channel of holographic MIMO systems [3]–
+[7]. In [3], a Fourier plane-wave series expansion-based chan-
+nel model is proposed for holographic MIMO systems. Based
+on the Fourier spectral representation, it provides a physically
+meaningful model capturing the propagation characteristics of
+the electromagnetic (EM) wave. In [4], the authors extend the
+Fourier plane-wave channel model to a multi-user scenario.
+In [5], the non-isotropic scattering environment is further con-
+sidered by using the von Mises-Fisher (VMF) distributions [8].
+Then in our previous works [6], [7], an EM-compliant channel
+model is proposed. By combining the VMF distributions [8]
+and the 3GPP TR 38.901 channel model [9], a realistic angular
+power spectrum is modeled. The non-ideal factors caused by
+mutual coupling at the transceivers [10], including the antenna
+pattern distortion and the decrease of antenna efficiency are
+also accounted for. However, in the state-of-the-art channel
+models [3]–[7], the polarization of EM wave is not taken into
+consideration. The polarization is an inherent characteristic of
+EM waves and will have significant impact on the performance
+of holographic MIMO systems.
+Another key problem for holographic MIMO is the accurate
+performance evaluation. In [5], the ergodic capacity of a
+single-user holographic MIMO system is analyzed, which
+shows the capacity improvement of holographic MIMO over
+conventional MIMO. In [11], the capacity of a single-user
+holographic MIMO system is investigated from a continuous
+point of view, the capacity enhancement is also demonstrated.
+In our previous work [6], both the single-user and the multi-
+user downlink channel capacities of holographic MIMO sys-
+tems are investigated. However, the performance evaluations
+in the state-of-the-art researches [5], [6], [11] are all numerical
+simulations based on theoretical models. No experiments have
+been done to verify the accuracy of the performance evalua-
+tions.
+In this paper, we try to solve the above two problems
+for holographic MIMO systems. Firstly, an extended EM-
+compliant channel model is proposed based on the channel
+model in our previous work [6]. Secondly, the channel of holo-
+graphic MIMO is measured through real-world experiments,
+and the performance is evaluated based on the measurement
+results. The contributions of this paper can be summarized as
+follows:
+• An extended EM-compliant channel model is proposed
+for holographic MIMO systems. In the extended model,
+not only the non-isotropic characteristics of the propaga-
+tion environment, the antenna pattern distortion, the an-
+tenna efficiency, but also the polarization of the antennas
+and the propagation environment can be modeled.
+• Based on the extended channel model, the real-world
+channel for holographic MIMO systems is measured for
+the first time in an indoor environment. An experiment
+is carefully designed, in which an electrically controlled
+virtual dense array is used to realize arbitrary element
+spacings of holographic MIMO.
+• The channel capacity of holographic MIMO systems is
+evaluated according to the measurement results. It is
+arXiv:2301.05626v1  [cs.IT]  13 Jan 2023
+
+demonstrated that the spatial oversampling of holographic
+MIMO is able to provide a two to three times capacity
+enhancement, without considering the antenna efficiency
+loss. The antenna efficiency is the most crucial challenge
+for holographic MIMO.
+The rest of this paper is organized as follows. In Section II,
+the proposed extended EM-compliant channel model for holo-
+graphic MIMO is explained in details. Then, the setup for the
+channel measurement is given in Section III. The measurement
+results and the corresponding performance evaluations are
+provided in Section IV. Finally, this paper is concluded in
+Section V.
+II. EXTENDED EM-COMPLIANT CHANNEL MODEL
+A. EM-Compliant Channel Model
+In this subsection, the EM-compliant channel model for
+holographic MIMO in our previous work [6] is briefly in-
+troduced. The central frequency and wavelength are denoted
+by fc and λ. Planar antenna arrays with size {Lx
+R, Ly
+R} and
+{Lx
+S, Ly
+S} are equipped at the receiver and the transmitter,
+respectively. The numbers of antenna elements are denoted
+by NR and NS. The spacings between antenna elements are
+denoted by {∆x
+R, ∆y
+R} and {∆x
+S, ∆y
+S}. The coordinates of the
+antenna elements are represented by rq = (rx
+q , ry
+q, rz
+q), q =
+1, 2, · · · , NR and sp = (sx
+p, sy
+p, sz
+p), p = 1, 2, · · · , NS, respec-
+tively.
+According to [6], the channel matrix H ∈ CNR×NS can be
+expressed as
+H = ΓRΨRHaΨH
+S ΓS,
+(1)
+where Ha ∈ CnR×nS denotes the wavenumber-domain chan-
+nel matrix, which has nR × nS elements. Here, nR = |ER|
+and nS = |ES| are the cardinalities of the sets ER and
+ES, with ER =
+�
+(lx, ly) ∈ Z2 :
+�
+lxλ
+Lx
+R
+�2
++
+�
+lyλ
+Ly
+R
+�2
+≤ 1
+�
+and
+ES =
+�
+(mx, my) ∈ Z2 :
+�
+mxλ
+Lx
+S
+�2
++
+�
+myλ
+Ly
+S
+�2
+≤ 1
+�
+. Each el-
+ement [Ha]β,α of Ha is a random Fourier coefficient fol-
+lowing the complex Gaussian distribution CN(0, σ2
+β,α), β =
+1, 2, · · · , nR, α = 1, 2, · · · , nS. The variance can be further
+given by
+σ2
+β,α =
+��
+ΩR(lx
+β,ly
+β)
+A2(θR, φR) sin θRdθRdφR×
+��
+ΩS(mxα,my
+α)
+A2(θS, φS) sin θSdθSdφS,
+(2)
+where A2(θR, φR) and A2(θS, φS) denote the angular power
+spectrum at the receiver and the transmitter, respectively. The
+angular power spectrum can be further modeled by a mixture
+of VMF distributions [8]
+A2
+R(θR, φR) =
+Nc
+�
+i=1
+wR,ipR,i(θR, φR),
+(3)
+and
+A2
+S(θS, φS) =
+Nc
+�
+i=1
+wS,ipS,i(θS, φS),
+(4)
+where Nc denotes the number of clusters of the scatters in the
+propagation environment. wR,i and wS,i denote the normal-
+ization factor with �Nc
+i
+wR,i = �Nc
+i
+wS,i = 1. pR,i(θR, φR)
+and pS,i(θS, φS) denote the probability functions of the VMF
+distribution, which can be further expressed as [8]
+pR,i(θR, φR) =
+αR,i
+4πsinh(αR,i)×
+eαR,i(sin θR sin ¯θR,i cos(φR− ¯φR,i)+cos θR cos ¯θR,i),
+(5)
+and
+pS,i(θS, φS) =
+αS,i
+4πsinh(αS,i)×
+eαS,i(sin θS sin ¯θS,i cos(φS− ¯φS,i)+cos θS cos ¯θS,i),
+(6)
+where {¯φR,i, ¯θR,i} and {¯φS,i, ¯θS,i} denote the elevation and
+azimuth angles of the i-th cluster at the receiver and the
+transmitter. αS,i and αR,i denote the concentration parameters
+for the i-th cluster. These angles can be derived from the 3GPP
+TR 38.901 channel model [9] according to the relationship
+defined in [6].
+In (1), ΨR ∈ CNR×nR and ΨS ∈ CNS×nS denote the
+modified Fourier harmonics, which take the antenna pattern
+distortion into consideration. Each element of ΨR and ΨS
+can be further derived as
+[ΨR]q,β =
+1
+√NR
+e
+j
+�
+2πlx
+β
+Lx
+R rx
+q +
+2πly
+β
+Ly
+R
+ry
+q +γR(lx
+β,ly
+β)rz
+q
+�
+× FR,q
+�
+ˆθR(lx
+β, ly
+β), ˆφR(lx
+β, ly
+β)
+�
+,
+(7)
+and
+[ΨS]p,α =
+1
+√NS
+e
+j
+�
+2πmx
+α
+Lx
+S
+sx
+p+ 2πmy
+α
+Ly
+S
+sy
+p+γS(mx
+α,my
+α)sz
+p
+�
+× FS,p
+�
+ˆθS(mx
+α, my
+α), ˆφS(mx
+α, my
+α)
+�
+,
+(8)
+where
+γR(lx, ly)
+=
+�
+( 2π
+λ )2 − ( 2πlx
+Lx
+R )2 − ( 2πly
+Ly
+R )2
+and
+γS(mx, my)
+=
+�
+( 2π
+λ )2 − ( 2πmx
+Lx
+S )2 − ( 2πmy
+Ly
+S )2.
+FR,q (θR, φR) and FS,p (θS, φS) represent the embedded
+element directivity pattern of the q-th antenna at the receiver
+and the p-th antenna at the transmitter. The corresponding
+elevation and azimuth angles {ˆφR(lx, ly), ˆθR(lx, ly)} and
+{ˆφS(mx, my), ˆθS(mx, my)} for the Fourier harmonics (lx, ly)
+and (mx, my) can be calculated by a transformation from the
+wavenumber domain to the angular domain [6].
+In the end, ΓR ∈ RNR×NR and ΓS ∈ RNS×NS are diagonal
+matrices representing the efficiency of the antenna element
+at the receiver and the transmitter, respectively. More details
+of the channel model can be found in [6]. However, the
+polarization characteristics of the antenna and the environment
+are not taken into consideration.
+B. Extended Channel Model with Polarization
+In this subsection, we extend the EM-compliant channel
+model to account for the polarization characteristics of the
+antennas and the propagation environment.
+
+From (1), the channel between the p-th antenna at the trans-
+mitter and the q-th antenna at the receiver can be expressed
+as
+[H]q,p =
+nR
+�
+β=1
+nS
+�
+α=1
+ηR,q[ΨR]q,β[Ha]β,α[ΨS]∗
+p,αηS,p,
+(9)
+where ηR,q and ηS,p denote the diagonal elements of ΓR and
+ΓS, representing the antenna efficiency of the corresponding
+antenna. According to [10], the polarization pattern of an EM
+wave can be decomposed into two components orthogonal to
+the propagation direction, i.e., the vertical polarization and the
+horizontal polarization. Therefore, the channel considering the
+polarization characteristics can be written as
+[H]pol
+q,p =
+nR
+�
+β=1
+nS
+�
+α=1
+ηR,q[Ψθ
+R]q,β[Hθθ
+a ]β,α[Ψθ
+S]∗
+p,αηS,p
++
+nR
+�
+β=1
+nS
+�
+α=1
+ηR,q[Ψθ
+R]q,β[Hθφ
+a ]β,α[Ψφ
+S]∗
+p,αηS,p
++
+nR
+�
+β=1
+nS
+�
+α=1
+ηR,q[Ψφ
+R]q,β[Hφθ
+a ]β,α[Ψθ
+S]∗
+p,αηS,p
++
+nR
+�
+β=1
+nS
+�
+α=1
+ηR,q[Ψφ
+R]q,β[Hφφ
+a ]β,α[Ψφ
+S]∗
+p,αηS,p,
+(10)
+where Ψθ
+R
+∈ CNR×nR and Ψθ
+S
+∈ CNS×nS denote the
+modified Fourier harmonics with the horizontal polarization,
+while Ψφ
+R ∈ CNR×nR and Ψφ
+S ∈ CNS×nS denote the modified
+Fourier harmonics with vertical polarization. Hθθ
+a ∈ CnR×nS,
+Hφφ
+a
+∈ CnR×nS, Hθφ
+a
+∈ CnR×nS, and Hφθ
+a
+∈ CnR×nS
+denote the co-polarization and cross-polarization wavenumber-
+domain channel matrices. The details of these parameters are
+explained in the following paragraphs.
+Polarization of Antennas: Firstly, the polarization charac-
+teristics of the antennas are modeled. The polarization of a
+specific antenna can be described by its antenna pattern [10].
+Therefore, we further modify the Fourier harmonics to take the
+polarization of the antennas into consideration. The modified
+Fourier harmonics with polarization at the receiver can be
+expressed as
+[Ψθ
+R]q,β =
+1
+√NR
+e
+j
+�
+2πlx
+β
+Lx
+R rx
+q +
+2πly
+β
+Ly
+R
+ry
+q +γR(lx
+β,ly
+β)rz
+q
+�
+× F θ
+R,q
+�
+ˆθR(lx
+β, ly
+β), ˆφR(lx
+β, ly
+β)
+�
+,
+(11)
+and
+[Ψφ
+R]q,β =
+1
+√NR
+e
+j
+�
+2πlx
+β
+Lx
+R rx
+q +
+2πly
+β
+Ly
+R
+ry
+q +γR(lx
+β,ly
+β)rz
+q
+�
+× F φ
+R,q
+�
+ˆθR(lx
+β, ly
+β), ˆφR(lx
+β, ly
+β)
+�
+,
+(12)
+where F θ
+R,q (θR, φR) and F φ
+R,q (θR, φR) denote the embedded
+element directivity patterns in the horizontal and the vertical
+polarization. Similarly, the modified Fourier harmonics at the
+transmitter can be expressed as
+[Ψθ
+S]p,α =
+1
+√NS
+e
+j
+�
+2πmx
+α
+Lx
+S
+sx
+p+ 2πmy
+α
+Ly
+S
+sy
+p+γS(mx
+α,my
+α)sz
+p
+�
+× F θ
+S,p
+�
+ˆθS(mx
+α, my
+α), ˆφS(mx
+α, my
+α)
+�
+,
+(13)
+and
+[Ψφ
+S]p,α =
+1
+√NS
+e
+j
+�
+2πmx
+α
+Lx
+S
+sx
+p+ 2πmy
+α
+Ly
+S
+sy
+p+γS(mx
+α,my
+α)sz
+p
+�
+× F φ
+S,p
+�
+ˆθS(mx
+α, my
+α), ˆφS(mx
+α, my
+α)
+�
+,
+(14)
+where F θ
+S,p (θS, φS) and F φ
+S,p (θS, φS) denote the embedded
+element directivity patterns in the horizontal and the vertical
+polarization for the p-th antenna at the transmitter.
+Polarization of Propagation Environment: Secondly, the
+polarization characteristics of the propagation environment is
+modeled. Similar to [9], we involve random phase shifts and
+cross polarization power ratios (XPR) to model the polariza-
+tion characteristics of the propagation environment. For the
+co-polarization wavenumber-domain channels, a phase shift
+is added to account for the polarization distortion by the
+propagation environment, which can be expressed as
+[Hθθ
+a ]β,α = [Ha]β,α × ejΦθθ
+β,α,
+(15)
+and
+[Hφφ
+a ]β,α = [Ha]β,α × ejΦφφ
+β,α.
+(16)
+Otherwise, the cross-polarization wavenumber-domain chan-
+nels can be expressed as
+[Hθφ
+a ]β,α = [Ha]β,α × ejΦθφ
+β,α ×
+�
+κ−1
+β,α,
+(17)
+and
+[Hφθ
+a ]β,α = [Ha]β,α × ejΦφθ
+β,α ×
+�
+κ−1
+β,α,
+(18)
+where Φθθ
+β,α, Φφφ
+β,α, Φφθ
+β,α, and Φθφ
+β,α are random phase shifts
+following the uniform distribution within [−π, π]. κβ,α de-
+notes the XPR of the propagation environment, which follows
+the log-normal distribution κβ,α = 10Xβ,α/10 with Xβ,α ∼
+N(µXPR, σ2
+XPR).
+Matrix Formulation: Finally, we transform the item-wise
+channel model into a matrix form. The item-wise channel
+model in (10) can be further expressed as
+Hpol =ΓRΨθ
+RHθθ
+a Ψθ
+S
+HΓS + ΓRΨθ
+RHθφ
+a Ψφ
+S
+HΓS
++ ΓRΨφ
+RHφθ
+a Ψθ
+S
+HΓS + ΓRΨφ
+RHφφ
+a Ψφ
+S
+HΓS
+=ΓR
+�
+Ψθ
+R, Ψφ
+R
+� �Hθθ
+a
+Hθφ
+a
+Hφθ
+a
+Hφφ
+a
+� �
+Ψθ
+S, Ψφ
+S
+��H ΓS.
+(19)
+Therefore, the final channel matrix can be derived as
+Hpol = ΓRΨpol
+R Hpol
+a Ψpol
+S
+HΓS,
+(20)
+where Ψpol
+R
+= [Ψθ
+R, Ψφ
+R], Ψpol
+S
+= [Ψθ
+S, Ψφ
+S], and Hpol
+a
+=
+�Hθθ
+a
+Hθφ
+a
+Hφθ
+a
+Hφφ
+a
+�
+.
+
+Network Analyzer
+Computer
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11 12 13 14 15 16
+17
+33
+49
+65
+81
+97
+113
+129
+145
+161
+177
+193
+209
+225
+241
+120 121
+128
+1
+2
+3
+4
+13
+16
+Receiver
+Transmitter
+Virtual 
+Antenna 
+Array
+Position
+Control
+Data
+Collect
+Antenna 
+Array
+Channel
+Signal 
+Generate
+Signal 
+Detect
+ൗ
+𝜆 2
+ൗ
+𝜆 2
+ൗ
+𝜆 4
+ൗ
+𝜆 8
+9
+5
+6
+7
+8
+10
+11
+12
+13
+14
+15
+16
+(120,6)
+(121,6)
+(113,6)
+(128,6)
+Fig. 1. Schematic diagram of the measurement equipment.
+As a result, not only the non-isotropic characteristics of the
+propagation environment, the antenna pattern distortion, the
+antenna efficiency, but also the polarization characteristics of
+the antennas and the propagation environment are all taken
+into consideration in the extended channel model.
+III. MEASUREMENT SETUP
+In order to evaluate the extended EM-compliant channel
+model and implement a realistic performance evaluation for
+holographic MIMO systems, an experiment is conducted to
+measure the real-world channel of holographic MIMO sys-
+tems. To the best of our knowledge, it is the first attempt to
+measure the channel of a holographic MIMO system.
+A schematic diagram of the measurement equipment is
+shown in Fig. 1. In the experiment, the dense antenna array of
+holographic MIMO is realized by a virtual antenna array. A
+discone antenna is used to achieve an omnidirectional pattern
+and the position of it is controlled by an electrical machine.
+Through computer programming, the antenna can be moved
+to different positions to construct a virtual dense array with
+arbitrary element spacings. In the experiment, the virtual an-
+tenna array is equipped at the receiver to realize a holographic
+MIMO array with spacing ∆x
+R = ∆y
+R ∈ {λ/8, λ/4, λ/2}. At
+the transmitter, a conventional antenna array with NS = 16
+antennas and spacings ∆x
+S = ∆y
+S = λ/2 is used. It is
+composed of patch antennas whose half power beam width
+is 70◦. A calibrated network analyzer is used to measure the
+channel and a computer is utilized to collect the measurement
+results. The center frequency is fc = 4.7 GHz and the
+bandwidth is 200 MHz from 4.6 GHz to 4.8 GHz with 1023
+samples.
+The experiment is performed in an indoor environment
+where the line-of-sight path is blocked by a metal object. Many
+scatters are present to create a rich scattering environment. The
+schematic diagram of the measurement environment is shown
+in Fig. 2. We consider two scenarios. In the first scenario, the
+virtual receive array plane is perpendicular to the transmitter,
+Receiver
+(Scenario2)
+Receiver
+(Scenario1)
+Transmitter
+Work Bench
+Storage Rack
+Medal Object
+1.55m
+5.65m
+1.70m
+2.18m
+2.00m
+Fig. 2. Schematic diagram of the measurement environment.
+while in the second scenario, the virtual receive array plane is
+parallel to the transmit array plane.
+IV. MEASUREMENT RESULTS AND EVALUATIONS
+In this section, we use the measurement results to evaluate
+the performance of an indoor holographic MIMO system.
+The measurement results are shown in Section IV-A and
+corresponding performance evaluations are provided in Sec-
+tion IV-B. Because the dense array of holographic MIMO
+is implemented virtually, the antenna efficiency loss is not
+accounted for in the measurement. Finally, the performance
+evaluations with antenna efficiency loss are provided in Sec-
+tion IV-C.
+A. Channel Measurement Results
+After measurement, a channel matrix Hpol with size NR ×
+NS can be derived. Here we plot several channel measurement
+results to show the correlation of antennas at different position.
+We use the pair (q, p) to represent the q-th antenna in the
+virtual receive array and the p-th antenna in the transmit array.
+The channel responses corresponding to (120, 6) and
+(121, 6) transceiver pairs are shown in Fig. 3a. In these two
+pairs, the transmit antennas are the same and the receive
+antennas are adjacent, we can observe that the channel re-
+sponses are quite similar. Instead, if we choose transceiver
+pairs whose receiver elements are not adjacent, e.g., (113, 6)
+and (128, 6), the results are shown in Fig. 3b. Since their
+receiver elements are separated by 2λ, we can find that the
+red line differs from the blue line in the spectrum, showing
+a low correlation compared with the results in Fig. 3a. The
+difference between these two figures shows the effect of spatial
+coherence. In an array, adjacent elements are more likely
+to sense the channel inside the same cluster, and thus the
+correlation of their channel responses are stronger.
+B. Performance Evaluation without Antenna Efficiency Loss
+Once the channel matrix Hpol is obtained, we can evaluate
+the channel capacity based on the measurement results. The
+variation of channel capacity with different element spacings
+in both scenarios are shown in Fig 4. In these figures, the
+antenna spacings are ∆x
+R = ∆y
+R ∈ {λ/8, λ/4, λ/2} and the
+corresponding numbers of antennas are NR ∈ {256, 64, 16}
+at the receiver. The signal to noise ratio (SNR) is set to 0 dB.
+
+4.6
+4.62
+4.64
+4.66
+4.68
+4.7
+4.72
+4.74
+4.76
+4.78
+4.8
+Frequency [GHz]
+-100
+-95
+-90
+-85
+-80
+-75
+-70
+-65
+-60
+S parrameter [dB]
+(a)
+4.6
+4.62
+4.64
+4.66
+4.68
+4.7
+4.72
+4.74
+4.76
+4.78
+4.8
+Frequency [GHz]
+-100
+-95
+-90
+-85
+-80
+-75
+-70
+-65
+-60
+S parrameter [dB]
+(b)
+Fig. 3.
+Channel response between q-th antenna at the receiver and p-th
+antenna at the transmitter in Scenario 2. (a) q = 120, 121, p = 6; (b)
+q = 113, 128, p = 6.
+Both the water filling and the equal power allocation strategies
+are adopted to evaluate the channel capacity performance. The
+blue lines correspond to the channel capacity, while the red
+lines correspond to the relative capacity with respect to the
+case with ∆x
+R = ∆y
+R = λ/2 spacing.
+From the results in Fig. 4, we can see that the spatial
+oversampling of holographic MIMO is able to increase the
+channel capacity. Using equal power allocation strategy, a four
+times spatial oversampling with ∆x
+R = ∆y
+R = λ/4 can offer
+about 120% capacity gain, and a 16 times oversampling with
+∆x
+R = ∆y
+R = λ/8 provides more than 300% capacity gain.
+While using the water filling strategy, the corresponding ca-
+pacity gains are about 80% and 200%. Therefore, the capacity
+enhancement capability of holographic MIMO stated in the
+previous research works [5], [6], [11] is verified by practical
+experiment. It is worth noting that the antenna efficiency
+loss at the receiver is not taken into consideration in the
+measurement because the dense array is realized virtually,
+Antenna spacing
+Capacity [bit/s/Hz]
+Water filling
+Equal power
+Water filling
+Equal power
+(a)
+Antenna spacing
+Capacity [bit/s/Hz]
+Water filling
+Equal power
+Water filling
+Equal power
+(b)
+Fig. 4. Channel capacity and relative capacity with different antenna spacings.
+(a) Scenario 1; (b) Scenario 2.
+which means ηR,q = 1. In the next subsection, the antenna
+efficiency loss is further considered in analyzing the capacity
+of a holographic MIMO system.
+C. Performance Evaluation with Antenna Efficiency Loss
+In a practical dense antenna array with small element
+spacing, the efficiency of the antenna elements will decrease
+because of the mutual coupling among them. In [12], a
+relationship between the antenna efficiency and the element
+spacing is established for a dense array, which is called
+Hannan’s element efficiency. According to [12], for a practical
+dense array at the receiver, the efficiency of the antenna
+element can be estimated as
+ηR,q ≈ π∆x
+R∆y
+R
+λ2
+,
+(21)
+which means that the element efficiency is proportional to the
+area allocated to the element. It implies that when the spacing
+of antenna element is small (∆x
+R∆y
+R < λ2/π), the element
+
+Antenna spacing
+Capacity [bit/s/Hz]
+Water filling
+Equal power
+Water filling
+Equal power
+(a)
+Antenna spacing
+Capacity [bit/s/Hz]
+Water filling
+Equal power
+Water filling
+Equal power
+(b)
+Fig. 5. Channel capacity and relative capacity with antenna efficiency loss. (a) Scenario 1; (b) Scenario 2.
+efficiency cannot reach 1, and it will decrease as the antenna
+elements are placed closer.
+Using the efficiency estimation in (21), we modify the chan-
+nel measurement results and evaluate the channel capacity of
+holographic MIMO systems. The results are shown in Fig. 5.
+It can be seen that in both scenarios, the channel capacities
+will not keep increasing with more antenna elements and
+smaller element spacings. Using the equal power allocation
+strategy, a 16 times oversampling with ∆x
+R = ∆y
+R = λ/8
+can only provide 4% capacity gain. While using the water
+filling strategy, the channel capacities even slightly decrease.
+The reason behind this is that the array gain and multiplexing
+gain by deploying more antenna elements are reduced by the
+decrease of the antenna efficiency.
+From the above analyses, we can find that although the
+channel correlation increases with smaller antenna spacings,
+the spatial oversampling of holographic MIMO is able to
+offer an obvious capacity enhancement. However, the antenna
+efficiency loss due to mutual coupling will greatly decrease the
+capacity gain, which is one of the most important challenges
+for a practical holographic MIMO system. Therefore, design-
+ing a dense antenna array with element efficiency above the
+Hannan’s efficiency scaling law will be the promising ways to
+exploit the benefit of spatial oversampling for the holographic
+MIMO systems.
+V. CONCLUSION
+In this paper, an extended EM-compliant channel model
+is proposed for holographic MIMO systems, which takes the
+non-isotropic characteristics of the propagation environment,
+the antenna pattern distortion, the antenna efficiency, and the
+polarization into account. An experiment is also conducted to
+measure the channel of an indoor holographic MIMO system
+through virtual antenna arrays. It is demonstrated through
+experiments for the first time that the spatial oversampling
+of holographic MIMO is able to increase the capacity of a
+wireless communication system significantly. However, the
+antenna efficiency is the most crucial challenge preventing us
+from getting the capacity improvement.
+REFERENCES
+[1] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen,
+R. Zhang, M. D. Renzo, and M. Debbah, “Holographic MIMO surfaces
+for 6G wireless networks: Opportunities, challenges, and trends,” IEEE
+Wireless Commun., vol. 27, no. 5, pp. 118–125, Oct. 2020.
+[2] D. Dardari and N. Decarli, “Holographic communication using intelli-
+gent surfaces,” IEEE Commun. Mag., vol. 59, no. 6, pp. 35–41, Jun.
+2021.
+[3] A. Pizzo, T. L. Marzetta, and L. Sanguinetti, “Spatially-stationary
+model for holographic MIMO small-scale fading,” IEEE J. Sel. Areas
+Commun., vol. 38, no. 9, pp. 1964–1979, Sep. 2020.
+[4] L. Wei, C. Huang, G. Alexandropoulus, W. E. I. Sha, Z. Zhang,
+M. Debbah, and C. Yuen, “Multi-user holographic MIMO surfaces:
+Channel modeling and spectral efficiency analysis,” IEEE J. Sel. Topics
+Signal Process., vol. 16, no. 5, pp. 1112–1124, Aug. 2022.
+[5] A. Pizzo, L. Sanguinetti, and T. L. Marzetta, “Fourier plane-wave
+series expansion for holographic MIMO communications,” IEEE Trans.
+Wireless Commun., vol. 21, no. 9, pp. 6890–6905, Sep. 2022.
+[6] T. Wang, W. Han, Z. Zhong, J. Pang, G. Zhou, S. Wang, and
+Q. Li, “Electromagnetic-compliant channel modeling and performance
+evaluation for holographic MIMO,” in IEEE Global Communications
+Conference (GLOBECOM), Rio de Janeiro, Brazil, Dec. 2022.
+[7] Y. Liu, M. Zhang, and T. Wang, “Effect of antenna pattern on electro-
+magnetic MIMO communication,” in IEEE International Conference on
+Communication Technology (ICCT), Nanjing, China, Nov. 2022.
+[8] K. Mammasis, R. W. Stewart, and J. S. Thompson, “Spatial fading
+correlation model using mixtures of von Mises Fisher distributions,”
+IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 2046–2055, Apr. 2009.
+[9] 3GPP TR 38.901, “Study on channel model for frequencies from 0.5 to
+100 GHz,” 2020.
+[10] C. A. Balanis, Antenna Theory: Analysis and Design.
+New York: John
+Wiley & Sons Ltd, 2015.
+[11] Z. Zhang and L. Dai, “Continuous-aperture MIMO for electromagnetic
+information theory,” arXiv, 2021. [Online]. Available: https://arxiv.org/
+abs/2111.08630
+[12] P. Hannan, “The element-gain paradox for a phased-array antenna,”
+IEEE Trans. Antenna Propag., vol. 12, no. 4, pp. 423–433, Jul. 1964.
+

cNE5T4oBgHgl3EQffQ_r/content/tmp_files/load_file.txt

@@ -0,0 +1,418 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf,len=417
+page_content='Channel Measurement for Holographic MIMO:  Benefits and Challenges of Spatial Oversampling  Tengjiao Wang∗, Yongxi Liu†, Ming Zhang†, Wei E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Sha‡, Cen Ling∗, Chao Li∗, Shaobo Wang∗  ∗Wireless Network RAN Research Department, Huawei Technologies CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', Ltd, Shanghai, China  †School of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi, China  ‡College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China  Emails: ∗{wangtengjiao6, lingcen, lichao18, shaobo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='wang}@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='com,  †liuyongxi@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='cn, ming20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='zhang@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='cn, ‡ weisha@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='cn  Abstract—In this paper, the channel of an indoor holographic  multiple-input multiple-output (MIMO) system is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' It  is demonstrated through experiments for the first time that  the spatial oversampling of holographic MIMO systems is able  to increase the capacity of a wireless communication system  significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' However, the antenna efficiency is the most crucial  challenge preventing us from getting the capacity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  An extended EM-compliant channel model is also proposed  for holographic MIMO systems, which is able to take the  non-isotropic characteristics of the propagation environment,  the antenna pattern distortion, the antenna efficiency, and the  polarization characteristics into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Index Terms—Holographic MIMO, massive MIMO, spatial  oversampling, channel measurement, electromagnetic informa-  tion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' INTRODUCTION  In order to increase the spectral efficiency and energy  efficiency for the future 5G-Advanced and 6G wireless com-  munication systems, the concept of holographic multiple-input  multiple-output (MIMO) is proposed recently [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' By inte-  grating an infinite number of antennas into a limited surface,  holographic MIMO is expected to fully exploit the propagation  characteristics offered by the electromagnetic channel and ap-  proach the fundamental performance limit [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' For holographic  MIMO systems, both the accurate channel modeling and the  realistic performance evaluation are key problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  In the literature, many efforts have been devoted to accu-  rately model the channel of holographic MIMO systems [3]–  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In [3], a Fourier plane-wave series expansion-based chan-  nel model is proposed for holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Based  on the Fourier spectral representation, it provides a physically  meaningful model capturing the propagation characteristics of  the electromagnetic (EM) wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In [4], the authors extend the  Fourier plane-wave channel model to a multi-user scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  In [5], the non-isotropic scattering environment is further con-  sidered by using the von Mises-Fisher (VMF) distributions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Then in our previous works [6], [7], an EM-compliant channel  model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' By combining the VMF distributions [8]  and the 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='901 channel model [9], a realistic angular  power spectrum is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The non-ideal factors caused by  mutual coupling at the transceivers [10], including the antenna  pattern distortion and the decrease of antenna efficiency are  also accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' However, in the state-of-the-art channel  models [3]–[7], the polarization of EM wave is not taken into  consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The polarization is an inherent characteristic of  EM waves and will have significant impact on the performance  of holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Another key problem for holographic MIMO is the accurate  performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In [5], the ergodic capacity of a  single-user holographic MIMO system is analyzed, which  shows the capacity improvement of holographic MIMO over  conventional MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In [11], the capacity of a single-user  holographic MIMO system is investigated from a continuous  point of view, the capacity enhancement is also demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  In our previous work [6], both the single-user and the multi-  user downlink channel capacities of holographic MIMO sys-  tems are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' However, the performance evaluations  in the state-of-the-art researches [5], [6], [11] are all numerical  simulations based on theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' No experiments have  been done to verify the accuracy of the performance evalua-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  In this paper, we try to solve the above two problems  for holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Firstly, an extended EM-  compliant channel model is proposed based on the channel  model in our previous work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Secondly, the channel of holo-  graphic MIMO is measured through real-world experiments,  and the performance is evaluated based on the measurement  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The contributions of this paper can be summarized as  follows:  An extended EM-compliant channel model is proposed  for holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In the extended model,  not only the non-isotropic characteristics of the propaga-  tion environment, the antenna pattern distortion, the an-  tenna efficiency, but also the polarization of the antennas  and the propagation environment can be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Based on the extended channel model, the real-world  channel for holographic MIMO systems is measured for  the first time in an indoor environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' An experiment  is carefully designed, in which an electrically controlled  virtual dense array is used to realize arbitrary element  spacings of holographic MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  The channel capacity of holographic MIMO systems is  evaluated according to the measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' It is  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='05626v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='IT]  13 Jan 2023  demonstrated that the spatial oversampling of holographic  MIMO is able to provide a two to three times capacity  enhancement, without considering the antenna efficiency  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The antenna efficiency is the most crucial challenge  for holographic MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In Section II,  the proposed extended EM-compliant channel model for holo-  graphic MIMO is explained in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Then, the setup for the  channel measurement is given in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The measurement  results and the corresponding performance evaluations are  provided in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Finally, this paper is concluded in  Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' EXTENDED EM-COMPLIANT CHANNEL MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' EM-Compliant Channel Model  In this subsection, the EM-compliant channel model for  holographic MIMO in our previous work [6] is briefly in-  troduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The central frequency and wavelength are denoted  by fc and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Planar antenna arrays with size {Lx  R, Ly  R} and  {Lx  S, Ly  S} are equipped at the receiver and the transmitter,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The numbers of antenna elements are denoted  by NR and NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The spacings between antenna elements are  denoted by {∆x  R, ∆y  R} and {∆x  S, ∆y  S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The coordinates of the  antenna elements are represented by rq = (rx  q , ry  q, rz  q), q =  1, 2, · · · , NR and sp = (sx  p, sy  p, sz  p), p = 1, 2, · · · , NS, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  According to [6], the channel matrix H ∈ CNR×NS can be  expressed as  H = ΓRΨRHaΨH  S ΓS,  (1)  where Ha ∈ CnR×nS denotes the wavenumber-domain chan-  nel matrix, which has nR × nS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Here, nR = |ER|  and nS = |ES| are the cardinalities of the sets ER and  ES, with ER =  �  (lx, ly) ∈ Z2 :  �  lxλ  Lx  R  �2  +  �  lyλ  Ly  R  �2  ≤ 1  �  and  ES =  �  (mx, my) ∈ Z2 :  �  mxλ  Lx  S  �2  +  �  myλ  Ly  S  �2  ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Each el-  ement [Ha]β,α of Ha is a random Fourier coefficient fol-  lowing the complex Gaussian distribution CN(0, σ2  β,α), β =  1, 2, · · · , nR, α = 1, 2, · · · , nS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The variance can be further  given by  σ2  β,α =  ��  ΩR(lx  β,ly  β)  A2(θR, φR) sin θRdθRdφR×  ��  ΩS(mxα,my  α)  A2(θS, φS) sin θSdθSdφS,  (2)  where A2(θR, φR) and A2(θS, φS) denote the angular power  spectrum at the receiver and the transmitter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The  angular power spectrum can be further modeled by a mixture  of VMF distributions [8]  A2  R(θR, φR) =  Nc  �  i=1  wR,ipR,i(θR, φR),  (3)  and  A2  S(θS, φS) =  Nc  �  i=1  wS,ipS,i(θS, φS),  (4)  where Nc denotes the number of clusters of the scatters in the  propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' wR,i and wS,i denote the normal-  ization factor with �Nc  i  wR,i = �Nc  i  wS,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' pR,i(θR, φR)  and pS,i(θS, φS) denote the probability functions of the VMF  distribution, which can be further expressed as [8]  pR,i(θR, φR) =  αR,i  4πsinh(αR,i)×  eαR,i(sin θR sin ¯θR,i cos(φR− ¯φR,i)+cos θR cos ¯θR,i),  (5)  and  pS,i(θS, φS) =  αS,i  4πsinh(αS,i)×  eαS,i(sin θS sin ¯θS,i cos(φS− ¯φS,i)+cos θS cos ¯θS,i),  (6)  where {¯φR,i, ¯θR,i} and {¯φS,i, ¯θS,i} denote the elevation and  azimuth angles of the i-th cluster at the receiver and the  transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' αS,i and αR,i denote the concentration parameters  for the i-th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' These angles can be derived from the 3GPP  TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='901 channel model [9] according to the relationship  defined in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  In (1), ΨR ∈ CNR×nR and ΨS ∈ CNS×nS denote the  modified Fourier harmonics, which take the antenna pattern  distortion into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Each element of ΨR and ΨS  can be further derived as  [ΨR]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β =  1  √NR  e  j  �  2πlx  β  Lx  R rx  q +  2πly  β  Ly  R  ry  q +γR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='ly  β)rz  q  �  × FR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q  �  ˆθR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly  β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ˆφR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly  β)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (7)  and  [ΨS]p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α =  1  √NS  e  j  �  2πmx  α  Lx  S  sx  p+ 2πmy  α  Ly  S  sy  p+γS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='my  α)sz  p  �  × FS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p  �  ˆθS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my  α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ˆφS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my  α)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (8)  where  γR(lx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly)  =  �  ( 2π  λ )2 − ( 2πlx  Lx  R )2 − ( 2πly  Ly  R )2  and  γS(mx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my)  =  �  ( 2π  λ )2 − ( 2πmx  Lx  S )2 − ( 2πmy  Ly  S )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  FR,q (θR, φR) and FS,p (θS, φS) represent the embedded  element directivity pattern of the q-th antenna at the receiver  and the p-th antenna at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The corresponding  elevation and azimuth angles {ˆφR(lx, ly), ˆθR(lx, ly)} and  {ˆφS(mx, my), ˆθS(mx, my)} for the Fourier harmonics (lx, ly)  and (mx, my) can be calculated by a transformation from the  wavenumber domain to the angular domain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  In the end, ΓR ∈ RNR×NR and ΓS ∈ RNS×NS are diagonal  matrices representing the efficiency of the antenna element  at the receiver and the transmitter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' More details  of the channel model can be found in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' However, the  polarization characteristics of the antenna and the environment  are not taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Extended Channel Model with Polarization  In this subsection, we extend the EM-compliant channel  model to account for the polarization characteristics of the  antennas and the propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  From (1), the channel between the p-th antenna at the trans-  mitter and the q-th antenna at the receiver can be expressed  as  [H]q,p =  nR  �  β=1  nS  �  α=1  ηR,q[ΨR]q,β[Ha]β,α[ΨS]∗  p,αηS,p,  (9)  where ηR,q and ηS,p denote the diagonal elements of ΓR and  ΓS, representing the antenna efficiency of the corresponding  antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' According to [10], the polarization pattern of an EM  wave can be decomposed into two components orthogonal to  the propagation direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', the vertical polarization and the  horizontal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' the channel considering the  polarization characteristics can be written as  [H]pol  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p =  nR  �  β=1  nS  �  α=1  ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q[Ψθ  R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β[Hθθ  a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α[Ψθ  S]∗  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p  +  nR  �  β=1  nS  �  α=1  ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q[Ψθ  R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β[Hθφ  a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α[Ψφ  S]∗  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p  +  nR  �  β=1  nS  �  α=1  ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q[Ψφ  R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β[Hφθ  a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α[Ψθ  S]∗  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p  +  nR  �  β=1  nS  �  α=1  ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q[Ψφ  R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β[Hφφ  a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α[Ψφ  S]∗  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (10)  where Ψθ  R  ∈ CNR×nR and Ψθ  S  ∈ CNS×nS denote the  modified Fourier harmonics with the horizontal polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  while Ψφ  R ∈ CNR×nR and Ψφ  S ∈ CNS×nS denote the modified  Fourier harmonics with vertical polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Hθθ  a ∈ CnR×nS,  Hφφ  a  ∈ CnR×nS, Hθφ  a  ∈ CnR×nS, and Hφθ  a  ∈ CnR×nS  denote the co-polarization and cross-polarization wavenumber-  domain channel matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The details of these parameters are  explained in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Polarization of Antennas: Firstly, the polarization charac-  teristics of the antennas are modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The polarization of a  specific antenna can be described by its antenna pattern [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Therefore, we further modify the Fourier harmonics to take the  polarization of the antennas into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The modified  Fourier harmonics with polarization at the receiver can be  expressed as  [Ψθ  R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β =  1  √NR  e  j  �  2πlx  β  Lx  R rx  q +  2πly  β  Ly  R  ry  q +γR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='ly  β)rz  q  �  × F θ  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q  �  ˆθR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly  β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ˆφR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly  β)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (11)  and  [Ψφ  R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='β =  1  √NR  e  j  �  2πlx  β  Lx  R rx  q +  2πly  β  Ly  R  ry  q +γR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='ly  β)rz  q  �  × F φ  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q  �  ˆθR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly  β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ˆφR(lx  β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ly  β)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (12)  where F θ  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q (θR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' φR) and F φ  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='q (θR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' φR) denote the embedded  element directivity patterns in the horizontal and the vertical  polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' the modified Fourier harmonics at the  transmitter can be expressed as  [Ψθ  S]p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α =  1  √NS  e  j  �  2πmx  α  Lx  S  sx  p+ 2πmy  α  Ly  S  sy  p+γS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='my  α)sz  p  �  × F θ  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p  �  ˆθS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my  α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ˆφS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my  α)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (13)  and  [Ψφ  S]p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='α =  1  √NS  e  j  �  2πmx  α  Lx  S  sx  p+ 2πmy  α  Ly  S  sy  p+γS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='my  α)sz  p  �  × F φ  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p  �  ˆθS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my  α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' ˆφS(mx  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' my  α)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (14)  where F θ  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p (θS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' φS) and F φ  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='p (θS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' φS) denote the embedded  element directivity patterns in the horizontal and the vertical  polarization for the p-th antenna at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Polarization of Propagation Environment: Secondly, the  polarization characteristics of the propagation environment is  modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Similar to [9], we involve random phase shifts and  cross polarization power ratios (XPR) to model the polariza-  tion characteristics of the propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' For the  co-polarization wavenumber-domain channels, a phase shift  is added to account for the polarization distortion by the  propagation environment, which can be expressed as  [Hθθ  a ]β,α = [Ha]β,α × ejΦθθ  β,α,  (15)  and  [Hφφ  a ]β,α = [Ha]β,α × ejΦφφ  β,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (16)  Otherwise, the cross-polarization wavenumber-domain chan-  nels can be expressed as  [Hθφ  a ]β,α = [Ha]β,α × ejΦθφ  β,α ×  �  κ−1  β,α,  (17)  and  [Hφθ  a ]β,α = [Ha]β,α × ejΦφθ  β,α ×  �  κ−1  β,α,  (18)  where Φθθ  β,α, Φφφ  β,α, Φφθ  β,α, and Φθφ  β,α are random phase shifts  following the uniform distribution within [−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' κβ,α de-  notes the XPR of the propagation environment, which follows  the log-normal distribution κβ,α = 10Xβ,α/10 with Xβ,α ∼  N(µXPR, σ2  XPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Matrix Formulation: Finally, we transform the item-wise  channel model into a matrix form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The item-wise channel  model in (10) can be further expressed as  Hpol =ΓRΨθ  RHθθ  a Ψθ  S  HΓS + ΓRΨθ  RHθφ  a Ψφ  S  HΓS  + ΓRΨφ  RHφθ  a Ψθ  S  HΓS + ΓRΨφ  RHφφ  a Ψφ  S  HΓS  =ΓR  �  Ψθ  R, Ψφ  R  � �Hθθ  a  Hθφ  a  Hφθ  a  Hφφ  a  � �  Ψθ  S, Ψφ  S  �H ΓS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (19)  Therefore, the final channel matrix can be derived as  Hpol = ΓRΨpol  R Hpol  a Ψpol  S  HΓS,  (20)  where Ψpol  R  = [Ψθ  R, Ψφ  R], Ψpol  S  = [Ψθ  S, Ψφ  S], and Hpol  a  =  �Hθθ  a  Hθφ  a  Hφθ  a  Hφφ  a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Network Analyzer  Computer  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16  17  33  49  65  81  97  113  129  145  161  177  193  209  225  241  120 121  128  1  2  3  4  13  16  Receiver  Transmitter  Virtual  Antenna  Array  Position  Control  Data  Collect  Antenna  Array  Channel  Signal  Generate  Signal  Detect  ൗ  𝜆 2  ൗ  𝜆 2  ൗ  𝜆 4  ൗ  𝜆 8  9  5  6  7  8  10  11  12  13  14  15  16  (120,6)  (121,6)  (113,6)  (128,6)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Schematic diagram of the measurement equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  As a result, not only the non-isotropic characteristics of the  propagation environment, the antenna pattern distortion, the  antenna efficiency, but also the polarization characteristics of  the antennas and the propagation environment are all taken  into consideration in the extended channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' MEASUREMENT SETUP  In order to evaluate the extended EM-compliant channel  model and implement a realistic performance evaluation for  holographic MIMO systems, an experiment is conducted to  measure the real-world channel of holographic MIMO sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' To the best of our knowledge, it is the first attempt to  measure the channel of a holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  A schematic diagram of the measurement equipment is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In the experiment, the dense antenna array of  holographic MIMO is realized by a virtual antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' A  discone antenna is used to achieve an omnidirectional pattern  and the position of it is controlled by an electrical machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Through computer programming, the antenna can be moved  to different positions to construct a virtual dense array with  arbitrary element spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In the experiment, the virtual an-  tenna array is equipped at the receiver to realize a holographic  MIMO array with spacing ∆x  R = ∆y  R ∈ {λ/8, λ/4, λ/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' At  the transmitter, a conventional antenna array with NS = 16  antennas and spacings ∆x  S = ∆y  S = λ/2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' It is  composed of patch antennas whose half power beam width  is 70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' A calibrated network analyzer is used to measure the  channel and a computer is utilized to collect the measurement  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The center frequency is fc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='7 GHz and the  bandwidth is 200 MHz from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='6 GHz to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='8 GHz with 1023  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  The experiment is performed in an indoor environment  where the line-of-sight path is blocked by a metal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Many  scatters are present to create a rich scattering environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The  schematic diagram of the measurement environment is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' We consider two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In the first scenario, the  virtual receive array plane is perpendicular to the transmitter,  Receiver  (Scenario2)  Receiver  (Scenario1)  Transmitter  Work Bench  Storage Rack  Medal Object  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='55m  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='65m  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='70m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='18m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='00m  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Schematic diagram of the measurement environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  while in the second scenario, the virtual receive array plane is  parallel to the transmit array plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' MEASUREMENT RESULTS AND EVALUATIONS  In this section, we use the measurement results to evaluate  the performance of an indoor holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  The measurement results are shown in Section IV-A and  corresponding performance evaluations are provided in Sec-  tion IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Because the dense array of holographic MIMO  is implemented virtually, the antenna efficiency loss is not  accounted for in the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Finally, the performance  evaluations with antenna efficiency loss are provided in Sec-  tion IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Channel Measurement Results  After measurement, a channel matrix Hpol with size NR ×  NS can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Here we plot several channel measurement  results to show the correlation of antennas at different position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  We use the pair (q, p) to represent the q-th antenna in the  virtual receive array and the p-th antenna in the transmit array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  The channel responses corresponding to (120, 6) and  (121, 6) transceiver pairs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In these two  pairs, the transmit antennas are the same and the receive  antennas are adjacent, we can observe that the channel re-  sponses are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Instead, if we choose transceiver  pairs whose receiver elements are not adjacent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', (113, 6)  and (128, 6), the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Since their  receiver elements are separated by 2λ, we can find that the  red line differs from the blue line in the spectrum, showing  a low correlation compared with the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The  difference between these two figures shows the effect of spatial  coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In an array, adjacent elements are more likely  to sense the channel inside the same cluster, and thus the  correlation of their channel responses are stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Performance Evaluation without Antenna Efficiency Loss  Once the channel matrix Hpol is obtained, we can evaluate  the channel capacity based on the measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The  variation of channel capacity with different element spacings  in both scenarios are shown in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In these figures, the  antenna spacings are ∆x  R = ∆y  R ∈ {λ/8, λ/4, λ/2} and the  corresponding numbers of antennas are NR ∈ {256, 64, 16}  at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The signal to noise ratio (SNR) is set to 0 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='62  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='64  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='66  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='68  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='72  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='76  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='78  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='8  Frequency [GHz]  100  95  90  85  80  75  70  65  60  S parrameter [dB]  (a)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='62  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='64  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='66  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='68  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='72  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='76  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='78  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='8  Frequency [GHz]  100  95  90  85  80  75  70  65  60  S parrameter [dB]  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Channel response between q-th antenna at the receiver and p-th  antenna at the transmitter in Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' (a) q = 120, 121, p = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' (b)  q = 113, 128, p = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Both the water filling and the equal power allocation strategies  are adopted to evaluate the channel capacity performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The  blue lines correspond to the channel capacity, while the red  lines correspond to the relative capacity with respect to the  case with ∆x  R = ∆y  R = λ/2 spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  From the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 4, we can see that the spatial  oversampling of holographic MIMO is able to increase the  channel capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Using equal power allocation strategy, a four  times spatial oversampling with ∆x  R = ∆y  R = λ/4 can offer  about 120% capacity gain, and a 16 times oversampling with  ∆x  R = ∆y  R = λ/8 provides more than 300% capacity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  While using the water filling strategy, the corresponding ca-  pacity gains are about 80% and 200%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Therefore, the capacity  enhancement capability of holographic MIMO stated in the  previous research works [5], [6], [11] is verified by practical  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' It is worth noting that the antenna efficiency  loss at the receiver is not taken into consideration in the  measurement because the dense array is realized virtually,  Antenna spacing  Capacity [bit/s/Hz]  Water filling  Equal power  Water filling  Equal power  (a)  Antenna spacing  Capacity [bit/s/Hz]  Water filling  Equal power  Water filling  Equal power  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Channel capacity and relative capacity with different antenna spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  (a) Scenario 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' (b) Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  which means ηR,q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In the next subsection, the antenna  efficiency loss is further considered in analyzing the capacity  of a holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Performance Evaluation with Antenna Efficiency Loss  In a practical dense antenna array with small element  spacing, the efficiency of the antenna elements will decrease  because of the mutual coupling among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' In [12], a  relationship between the antenna efficiency and the element  spacing is established for a dense array, which is called  Hannan’s element efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' According to [12], for a practical  dense array at the receiver, the efficiency of the antenna  element can be estimated as  ηR,q ≈ π∆x  R∆y  R  λ2  ,  (21)  which means that the element efficiency is proportional to the  area allocated to the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' It implies that when the spacing  of antenna element is small (∆x  R∆y  R < λ2/π), the element  Antenna spacing  Capacity [bit/s/Hz]  Water filling  Equal power  Water filling  Equal power  (a)  Antenna spacing  Capacity [bit/s/Hz]  Water filling  Equal power  Water filling  Equal power  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Channel capacity and relative capacity with antenna efficiency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' (a) Scenario 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' (b) Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  efficiency cannot reach 1, and it will decrease as the antenna  elements are placed closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Using the efficiency estimation in (21), we modify the chan-  nel measurement results and evaluate the channel capacity of  holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  It can be seen that in both scenarios, the channel capacities  will not keep increasing with more antenna elements and  smaller element spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Using the equal power allocation  strategy, a 16 times oversampling with ∆x  R = ∆y  R = λ/8  can only provide 4% capacity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' While using the water  filling strategy, the channel capacities even slightly decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  The reason behind this is that the array gain and multiplexing  gain by deploying more antenna elements are reduced by the  decrease of the antenna efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  From the above analyses, we can find that although the  channel correlation increases with smaller antenna spacings,  the spatial oversampling of holographic MIMO is able to  offer an obvious capacity enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' However, the antenna  efficiency loss due to mutual coupling will greatly decrease the  capacity gain, which is one of the most important challenges  for a practical holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Therefore, design-  ing a dense antenna array with element efficiency above the  Hannan’s efficiency scaling law will be the promising ways to  exploit the benefit of spatial oversampling for the holographic  MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' CONCLUSION  In this paper, an extended EM-compliant channel model  is proposed for holographic MIMO systems, which takes the  non-isotropic characteristics of the propagation environment,  the antenna pattern distortion, the antenna efficiency, and the  polarization into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' An experiment is also conducted to  measure the channel of an indoor holographic MIMO system  through virtual antenna arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' It is demonstrated through  experiments for the first time that the spatial oversampling  of holographic MIMO is able to increase the capacity of a  wireless communication system significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' However, the  antenna efficiency is the most crucial challenge preventing us  from getting the capacity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  REFERENCES  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Hu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Alexandropoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zappone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Yuen,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Renzo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Debbah, “Holographic MIMO surfaces  for 6G wireless networks: Opportunities, challenges, and trends,” IEEE  Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 118–125, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Dardari and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Decarli, “Holographic communication using intelli-  gent surfaces,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 35–41, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Pizzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Marzetta, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Sanguinetti, “Spatially-stationary  model for holographic MIMO small-scale fading,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Areas  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 1964–1979, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Wei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Alexandropoulus, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Sha, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zhang,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Debbah, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Yuen, “Multi-user holographic MIMO surfaces:  Channel modeling and spectral efficiency analysis,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Topics  Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 1112–1124, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Pizzo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Sanguinetti, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Marzetta, “Fourier plane-wave  series expansion for holographic MIMO communications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 6890–6905, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zhong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Pang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Wang, and  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Li, “Electromagnetic-compliant channel modeling and performance  evaluation for holographic MIMO,” in IEEE Global Communications  Conference (GLOBECOM), Rio de Janeiro, Brazil, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zhang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Wang, “Effect of antenna pattern on electro-  magnetic MIMO communication,” in IEEE International Conference on  Communication Technology (ICCT), Nanjing, China, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Mammasis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Stewart, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Thompson, “Spatial fading  correlation model using mixtures of von Mises Fisher distributions,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2046–2055, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [9] 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='901, “Study on channel model for frequencies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='5 to  100 GHz,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Balanis, Antenna Theory: Analysis and Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  New York: John  Wiley & Sons Ltd, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='  [11] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Zhang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Dai, “Continuous-aperture MIMO for electromagnetic  information theory,” arXiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='org/  abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content='08630  [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Hannan, “The element-gain paradox for a phased-array antenna,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' Antenna Propag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 423–433, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}
+page_content=' 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'}

dtE3T4oBgHgl3EQfGwnT/content/tmp_files/2301.04318v1.pdf.txt

@@ -0,0 +1,1832 @@
+Beyond Graph Convolutional Network: An Interpretable Regularizer-centered
+Optimization Framework
+Shiping Wang1,2, Zhihao Wu1,2, Yuhong Chen1,2, Yong Chen3*
+1 College of Computer and Data Science, Fuzhou University
+2 Fujian Provincial Key Laboratory of Network Computing and Intelligent Information Processing, Fuzhou University
+3 School of Computer Science, Beijing University of Posts and Telecommunications
+[email protected], [email protected], [email protected], [email protected].
+Abstract
+Graph convolutional networks (GCNs) have been attracting
+widespread attentions due to their encouraging performance
+and powerful generalizations. However, few work provide a
+general view to interpret various GCNs and guide GCNs’ de-
+signs. In this paper, by revisiting the original GCN, we in-
+duce an interpretable regularizer-centerd optimization frame-
+work, in which by building appropriate regularizers we can
+interpret most GCNs, such as APPNP, JKNet, DAGNN, and
+GNN-LF/HF. Further, under the proposed framework, we de-
+vise a dual-regularizer graph convolutional network (dubbed
+tsGCN) to capture topological and semantic structures from
+graph data. Since the derived learning rule for tsGCN con-
+tains an inverse of a large matrix and thus is time-consuming,
+we leverage the Woodbury matrix identity and low-rank ap-
+proximation tricks to successfully decrease the high computa-
+tional complexity of computing infinite-order graph convolu-
+tions. Extensive experiments on eight public datasets demon-
+strate that tsGCN achieves superior performance against quite
+a few state-of-the-art competitors w.r.t. classification tasks.
+Introduction
+Owing to the powerful ability to aggregate neighborhood in-
+formation, Graph Convolutional Network (GCN) has been
+successfully applied to diverse domains, such as computer
+vision [1, 2, 3], recommender systems [4, 5], privacy pre-
+serving [6], and traffic forecasting [7, 8]. Rooted in a series
+of theoretical foundations, GCN extends convolution opera-
+tions to the non-Euclidean spaces and effectively propagates
+label signals, and therefore its variants have been extensively
+employed for a variety of graph-related tasks, including clas-
+sification [9, 10], clustering [11, 12] and link prediction
+[13, 14]. In a nutshell, GCN generates the graph embedding
+with the well-established graph convolutional layers gath-
+ering semantics from neighbors according to the network
+topology, which are revealed to be the most critical com-
+ponent.
+Although GCN has behaved well in many machine learn-
+ing tasks, lots of studies have pointed out its certain draw-
+backs and made efforts for further improvements. Bo et al.
+[15] indicated that the propagation mechanism could be con-
+sidered as a special form of low-pass filter, and presented a
+*Corresponding author
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+GCN with an adaptive frequency. Zhang et al. [16] argued
+that most GCN-based methods ignored the global informa-
+tion and proposed SHNE, which leveraged the structure and
+feature similarity to capture latent semantics. Wang et al.
+[17] revealed that the original GCN aggregated informa-
+tion from node neighbors inadequately, and then developed
+a multi-channel GCN by utilizing feature-based semantic
+graph. In spite of the performance boosts of these GCN
+variants, they didn’t establish a generalized framework, i.e.,
+these approaches understood and enhanced GCN from cer-
+tain and non-generalizable perspectives, thereby they are ex-
+ceedingly difficult to be further developed, and with limited
+interpretability.
+Consequently, it is expected to construct a unified frame-
+work for various GCNs with better interpretability; however,
+it is a pity that this kind of work is still in shortage. Zhao et
+al. [18] linked GCN and Graph-regularized PCA (GPCA),
+and then proposed a multi-layer network by stacking the
+GPCA layers. Zhu et al. [19] attempted to interpret exist-
+ing GCN-based methods with a unified optimization frame-
+work, under which they devised an adjustable graph filter
+for a new GCN variant. Yang et al. [20] designed a family of
+graph convolutional layers inspired by the updating rules of
+two typical iterative algorithms. Although these efforts have
+contributed to better understanding of GCNs, they only ex-
+plained GCNs in partial aspects, promoting the expectation
+of a more comprehensive analysis of GCNs.
+To tackle the aforementioned issues, this paper induces an
+interpretable regularizer-centered optimization framework,
+which provides a novel perspective to digest various GCNs,
+i.e., this framework captures the common essential proper-
+ties of existing state-of-the-art GCN variants and could de-
+fines them just by devising different regularizers. Moreover,
+in light of the analyses on current representative GCNs, we
+find that most of the existing approaches only consider cap-
+turing the topological regularization, while the feature-based
+semantic structure is underutilized, and hence this motivates
+us to design a dual-regularizer graph convolutional network
+(called tsGCN) within the regularizer-centered optimization
+framework for the fullest explorations of both structures
+and semantics from graph data. Due to the high computa-
+tional complexity of performing infinite-order graph con-
+volutions, the unified framework provides a straightforward
+way employing truncated polynomials to approximate the
+arXiv:2301.04318v1  [cs.LG]  11 Jan 2023
+
+graph Laplacian, similar to the truncated Chebyshev poly-
+nomials by vanilla GCN, restricting the message passing of
+a single graph convolution to the first-order neighborhood.
+The main contributions of this paper can be summarized
+as the following three aspects:
+• Propose a regularizer-centered constrained optimization
+framework, which interprets various existing GCNs with
+specific regularizers.
+• Establish a new dual-regularizer graph convolutional net-
+work (tsGCN), which exploits topological and semantic
+structures of the given data; and develop an efficient algo-
+rithm to reduce the computational complexity of solving
+infinite-order graph convolutions.
+• Conduct a series of experiments to show that tsGCN per-
+forms much better than many SOTA GCNs, and also con-
+sumes much less time than the newly GNN-HF/LF.
+Related Work
+Graph Convolutional Networks
+The original GCN was first introduced by Kipf et al. [21],
+who generalized the convolution operations from the Eu-
+clidean domain to the non-Euclidean domain. SGC [22] as-
+sumed that the nonlinear transform of GCN was not that
+significant, and then devised a simplified GCN by remov-
+ing the nonlinear activation functions and collapsing the
+weight matrices. PPNP [23] employed the relationship be-
+tween PageRank and GCN for the improvement on the prop-
+agation mechanism of GCN, and an iterative version called
+APPNP was further proposed to reduce the high compu-
+tational complexity. Attempting to adaptively learn the in-
+fluence radii for each node and task, JKNet [24] combined
+various aggregations at the last layer and was able to learn
+representations of different orders for graph substructures.
+GNN-LF and GNN-HF [19] considered the low-pass and the
+high-pass filter as the convolution kernels to improve GCN’s
+expressive power, respectively. AdaGCN [25] leveraged Ad-
+aboost strategy for the enhancement of GCN, allowing in-
+formation to be shared between layers. To sum up, a main
+characteristic of these methods is exploring GCN from the
+perspectives of redesigning information aggregation strate-
+gies or modifying graph convolutions, and few work try to
+construct a unified framework to interpret various GCNs and
+reveal the underlying common principles.
+Further Insights on GCNs
+Quite a few studies have been devoted to explore the mech-
+anisms of GCN for deeper insights. Li et al. [26] indicated
+that the convolutional operation of GCN was a special form
+of Laplacian smoothing, attributed to which GCN suffered
+from the so-called over-smoothing problem. Specifically, the
+performance of GCN will decrease as the number of layers
+increases, which has been validated by many other studies.
+However, Liu et al. [27] held a different opinion that the en-
+tanglement of two steps in GCN damages the performance
+of the deep GCN, where the two steps were explained as
+propagation and transformation. Based on this view, they de-
+coupled the two operations and further presented a deeper
+GCN. Zhu et al. [19] also decomposed the convolution op-
+eration of GCN into two separate stages, called aggregation
+and transformation, and focused on the aggregation process,
+formulating an optimization objective to interpret it. Yang et
+al. [28] explored network topology refinement, leveraging a
+topology optimization process for the explanation. Oono et
+al. [29] analyzed the forward propagation of GCN and in-
+terpreted it with a specific dynamical system, allowing GCN
+to be related to the underlying topological structures. Over-
+all, these studies have contributed to the interpretability of
+GCNs, and also let researchers better understand GCNs. In
+this paper, we build a unified optimization framework from a
+novel view of graph regularizers to interpret and understand
+GCNs.
+Mathematical Notations
+For the convenience of formal descriptions, derivations, and
+analyses, necessary notations are narrated as below. A graph
+is denoted as G = (V, E, A), where V marks the vertex set
+with |V| = N (N is the total number of nodes in graph G),
+E marks the edge set, and A = [Aij]N×N marks an affinity
+matrix of which Aij measures the similarity between the i-
+th and the j-th node. In addition, D = [Dij]N×N represents
+the degree matrix of G with Dii = �N
+j=1 Aij, and then the
+normalized symmetrical graph Laplacian of G is computed
+as �L = I − �A with �A = D− 1
+2 AD− 1
+2 .
+Revisiting Graph Convolutional Network
+For a graph G = (V, E, A), the svd of its graph Laplacian is
+L = UΛU⊤, where U ∈ RN×N is comprised of orthonor-
+mal eigenvectors and Λ = diag(λ1, · · · , λN) is a diagonal
+matrix with λi denoting the i-th eigenvalue and λi ≥ λj
+(i = 1, · · · , N). Essentially, this decomposition induces a
+Fourier transform on the graph domain, where eigenvectors
+correspond to Fourier components and eigenvalues represent
+frequencies of the graph. For an input signal x ∈ RN defined
+on the graph G, the corresponding graph Fourier transform
+of x is �x = U⊤x, and its inverse transform is derived as
+x = U�x. Consequently, the graph convolution between the
+signal x and the filter g ∈ RN is
+g ∗ x = U(�g ⊙ �x) = U((U⊤g) ⊙ (U⊤x)),
+(1)
+where ⊙ is the Hadamard product between two vectors. Par-
+ticularly, denoting gΘ = diag(Θ) := U⊤g parameterized
+by Θ ∈ RN, the graph convolution between x and g can be
+rewritten as
+g ∗ x = U(�g ⊙ �x) = UgΘU⊤x,
+(2)
+where Θ is regarded as the filter coefficients to be optimized.
+Especially, Θ is assumed to be the polynomials of the spec-
+trums of the graph Laplacian [30], i.e.,
+Θ = Θ(Λ) =
+K
+�
+i=1
+ΘiΛi,
+(3)
+where K is the order of Chebyshev polynomials. By fixing
+K = 2, the graph convolutional network (GCN) [21] takes
+an effective form
+g ∗ x = θ(I + L)x,
+(4)
+
+Methods
+Propagation Rules
+Regularizer L(H(l); G)
+Projective Set
+GCN
+H(l) = σ
+�
+�AH(l−1)Θ(l)�
+Tr
+�
+H(l)⊤�LH(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex
+SGC
+H(l) = σ
+�
+�AH(l−1)Θ(l)�
+Tr
+�
+H(l)⊤�LH(l)�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+APPNP
+H(l) = σ
+�
+(1 − α) �AH(l−1) + αH(0)�
+Tr
+�
+1
+1−αH(l)⊤ �A−1(H(l) − 2αH(0))
+�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+JKNet
+H(l) = σ
+��K
+k=1 αk �AkH(l−1)Θ(l)�
+Tr
+�
+H(l)⊤ �A−1(I + β�L)H(l)�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+DAGNN
+H(l) = σ
+��K
+k=0 αk �AkH(l−1)�
+Tr
+�
+H(l)⊤(I + β�L)H(l)�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+GNN-HF
+H(l) = σ
+�
+(I + α�L)−1(I + β�L)H(l−1)Θ(l)�
+Tr
+�
+H(l)⊤(I + β�L)−1(I + α�L)H(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex.
+GNN-LF
+H(l) = σ
+�
+(I + α �A)−1(I + β �A)H(l−1)Θ(l)�
+Tr
+�
+H(l)⊤(I + β �A)−1(I + α �A)H(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex
+tsGCN
+H(l) = σ
+�
+(I + α�LG + β�LX )−1H(l−1)Θ(l)�
+Tr
+�
+H(l)⊤(I + α�LG + β�LX )H(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex
+Table 1: Different regularizers can derive different GCN variants under the regularizer-centered optimization framework.
+where Θ = [θ] is a parameter to be optimized. When ex-
+tending single channel signal x and filter θ to multi-channel
+H(l) ∈ RN×dl and Θ(l) ∈ Rdl×fl, the GCN is converted to
+H(l) = σ( �AH(l−1)Θ(l)),
+(5)
+where �A is a normalized version of I + �A, σ(·) is an acti-
+vation function, and H(l) ∈ RN×dl is the output of the l-th
+layer with H(0) = X being the input feature matrix.
+An Interpretable Regularizer-centered
+Optimization Framework for GCNs
+Given the input H(l−1) of the (l)-th layer, GCN can compute
+the output H(l) by minimizing
+L = −Tr(H(l)⊤H(l−1)Θ(l)) + 1
+2Tr(H(l)⊤�LH(l))
+(6)
+s.t. H(l) ≥ 0,
+where
+1
+2Tr(H(l)⊤�LH(l)) =
+1
+4
+�N
+j=1
+�N
+i=1 Aij|| h(l)
+i
+√Dii −
+h(l)
+j
+√
+Djj ||2
+2 with H(l) = [h(l)
+1 ; · · · ; h(l)
+N ]; it is a normalized reg-
+ularizer to preserve the pairwise similarity of any two nodes
+in the given graph. Besides, the −Tr(H(l)⊤H(l−1)Θ(l)) is
+actually a fitting loss term bewteen H(l) and H(l−1)Θ(l),
+i.e., ||H(l)−H(l−1)Θ(l)||2
+F with H(l−1) and Θ(l) fixed when
+optimizing H(l). Note that the square term ||H(l)||2
+F is a L2-
+regularized smoother, which can be ignored or absorbed in
+the second graph regularizer Tr(H(l)⊤�LH(l)).
+Taking derivative of L with respect to H(l) and setting it
+to zero, we obtain H(l+) as
+H(l+) = (I − �A)−1H(l−1)Θ(l);
+(7)
+and then there yields
+H(l) = σ
+�
+H(l+)�
+,
+(8)
+when the nonnegative constraints H(l) ≥ 0 are further con-
+sidered. Notice that σ(·) is the ReLU(·) activation function.
+Here, if the matix inverse (I− ˜A)−1 = �∞
+i=0 ˜Ai is approxi-
+mated by the first-order expansion, i.e., (I− ˜A)−1 ≈ I+ ˜A,
+then Eq. (8) will lead to the updating rule (5) of GCN.
+Usually, the activation functions in GCN are ReLU(·) and
+Softmax(·), which could be converted to different projec-
+tion optimizations. Concretely, the ReLU(·) activation func-
+tion is equivalent to project a point x onto the non-negative
+plane S+ = {s ∈ Rd|s ≥ 0}, i.e.,
+ReLU(x) = arg min
+y∈S+
+−x⊤y + 1
+2||y||2
+2.
+(9)
+By the way, we denote S = {s ∈ Rd}, which corresponds
+to an identity activation function. In terms of the Softmax(·)
+activation function, it can be regarded as projecting x onto
+the set Ssimplex = {s ∈ Rd|1⊤s = 1, s ≥ 0}, i.e.,
+Softmax(x) = arg min
+y∈Ssimplex
+−x⊤y + y⊤ log(y),
+(10)
+where y⊤ log(y) = �d
+i=1 yi log(yi) is the negative entropy
+of y [31]. In fact, with respect to other activation functions,
+they can also be equivalent to project a point onto some fea-
+sible set with some metric.
+Up to present, we have actually utilized a constrained op-
+timization problem to interpret GCN, including information
+propagations (i.e., Eq. (7)) and the nonlinear activation func-
+tions (i.e., ReLU(·) and Softmax(·)).
+The above analyses can not only explain the vanilla GCN,
+but also stimulate a regularizer-centered optimization frame-
+work that can further unify various GCNs. By extending the
+optimization (6), a more general framework is written as
+L = −Tr(H(l)⊤H(l−1)Θ(l)) + 1
+2L(H(l); G)
+(11)
+s.t. H(l) ∈ {S+ or S}, l ∈ [L − 1], H(L) ∈ Ssimplex.
+Under this framework, different regularizers could derive
+different GCNs, for example,
+
+Algorithm 1: Topological and Semantic Regularized GCN
+Require: Graph data G = (V, E, A), labels y, number of
+layers L, and hyperparameters {α, β, r}.
+Ensure: Predicted label set {y∗
+i }N
+i=n+1.
+1: Initialize model parameters {H(l), Θ(l)}L
+l=1;
+2: Compute the joint graph Laplacian α�LG + β�LX and its
+low-rank factorization WV⊤;
+3: Substitute the matrix inverse (I + α�LG + β�LX )−1 with
+I − W(I + V⊤W)−1V⊤;
+4: while not convergent do
+5:
+Calculate hidden layers {H(l)}L
+l=1 by Eq. (14);
+6:
+Update weights: Θ(l+1) ← Θ(l) − η
+∂L
+∂Θ(l) ;
+7: end while
+8: return The predicted labels: y∗
+i = arg maxj H(L)
+ij .
+• If L(H(l); G) = Tr
+�
+H(l)⊤(I + µ�L)−1(I + λ�L)H(l)�
+with λ = β +
+1
+α − 1, µ = β, and �L = I − �A,
+then
+it
+induces
+the
+updating
+rule
+H(l)
+=
+σ
+�
+(I + α �A)−1(I + β �A)H(l−1)Θ(l)�
+,
+which
+cor-
+responds to GNN-HF [19].
+• If L(H(l); G) = Tr
+�
+H(l)⊤(I + µ �A)−1(I + λ �A)H(l)�
+with λ
+=
+−αβ+2α−1
+αβ−α+1
+and µ
+=
+1
+β − 1, then
+it
+gives
+rise
+to
+the
+updating
+rule
+H(l)
+=
+σ
+�
+(I + α �A)−1(I + β �A)H(l−1)Θ(l)�
+,
+which
+cor-
+responds to GNN-LF [19].
+For more cases, their results are summarized in Table 4, and
+the derivation details can refer to those of the original GCN
+(from Eq. (7) to Eq. (10)) and the supplementary.
+Remarks. The work [19] is most similar to our work with
+the same research idea: they both want to propose a unified
+framework to interpret the current GCNs and guide the de-
+sign of new GCN variants; however, they are realized in dif-
+ferent ways. To be specific, (1) Zhu et al. [19] develop an
+optimization framework to explain different GCNs’ prop-
+agation processes; whereas we propose a constrained op-
+timization framework not only to interpret various GCNs’
+propagation processes, but also explain the nonlinear activa-
+tion layers; (2) [19] unifies various GCNs via devising vari-
+ous fitting items which are essentially constructed by limited
+graph filters; while our work derives different GCNs through
+designing different regularizers. To sum up, our work inter-
+prets the whole (not partial) GCNs with regularizer-centered
+constrained optimizations.
+tsGCN: Topological and Semantic Regularized
+Graph Convolutional Network
+One finding from most existing GCNs is that they often ig-
+nored feature-based semantic structures, which can weaken
+the representation learning abilities of graph networks, then
+Table 2: Dataset statistics.
+Datasets
+#Nodes
+#Edges
+#Classes
+#Features
+#Train/Val/Test
+Cora
+2,708
+5,429
+7
+1,433
+140/500/1,000
+Citeseer
+3,327
+4,732
+6
+3,703
+120/500/1,000
+Pubmed
+19,717
+44,338
+3
+500
+60/500/1,000
+ACM
+3,025
+13,128
+3
+1,870
+60/500/1,000
+BlogCatalog
+5,196
+171,743
+6
+8,189
+120/500/1,000
+CoraFull
+19,793
+65,311
+70
+8,710
+1,400/500/1,000
+Flickr
+7,575
+239,738
+9
+12,047
+180/500/1,000
+UAI
+3,067
+28,311
+19
+4,973
+367/500/,1000
+we focus on two regularizers, i.e.,
+L1(H(l); G) = 1
+2Tr
+�
+{H(l)}⊤(1
+2I + α�LG)H(l)
+�
+,
+(12)
+L2(H(l); X) = 1
+2Tr
+�
+{H(l)}⊤(1
+2I + β�LX )H(l)
+�
+,
+(13)
+where �LG is a graph Laplacian from the given adjacency ma-
+trix (e.g., �LG = �L), and �LX is a graph Laplacian calculated
+from the pairwise similarity of any two graph nodes. Hence,
+we devise a dual-regularizer, i.e., L(H(l)) = L1(H(l); G) +
+L2(H(l); X), and if it is under the optimization framework
+(19), then there yields the following updating rule
+H(l) = σ
+�
+(I + α�LG + β�LX )−1H(l−1)Θ(l)�
+.
+(14)
+Since this method seeks to preserve both the topological and
+semantic structures for more accurate presentations, we call
+it tsGCN (i.e., Topological and Semantic regularized GCN).
+Notably, the computational complexity of (I + α�LG +
+β�LX )−1 is O(N 3), which tends to be unaffordable in prac-
+tical applications. To this end, a low-rank approximation is
+operated, i.e., α�LG +β�LX ≈ WV⊤, where W, V ∈ RN×r
+with r ≪ N. This leads to the Woodbury matrix identity:
+(I + WV⊤)−1 = I − W(I + V⊤W)−1V⊤,
+(15)
+of which the computational complexity costs O(N 2).
+Given that the optimal M∗ of the following problem
+min
+M∈RN×N: rank(M)=r ||M − (α�LG + β�LX )||2
+F
+(16)
+is attained at the r-truncated singular value decomposition
+of α�LG + β�LX , i.e., M∗ = UΣU⊤, where Σ ∈ Rr×r is a
+diagonal matrix containing the r largest singular values. An
+optimal {W∗, V∗} to α�LG + β�LX ≈ WV⊤ can be given
+by an analytic form of W∗ = V∗ = UΣ
+1
+2 .
+To obtain the optimum {W∗, V∗}, the iterative algorithm
+[32] with O(N 2) is leveraged as
+Z(t+1) ← (α�LG + β�LX )U(t),
+(17)
+{U(t+1), R(t+1)} ← QR(Z(t+1)),
+(18)
+where QR(·) denotes the QR-decomposition. Note that this
+algorithm can converge to the r largest eigenvalues R(t+1)
+and its corresponding eigenvectors Z(t+1) when the iterative
+number t is large enough. Finally, there will be W∗ = V∗ =
+U(t+1)[R(t+1)]
+1
+2 .
+Gathering all analyses mentioned above, the procedure for
+tsGCN is summarized in Algorithm 1.
+
+Table 3: Accuracy and F1-score (mean% and standard deviation%) of all methods, where the best results are in red and the
+second-best are in blue. Note that GraphSAGE fails to work on the ACM dataset, and thus its results are marked with “—”.
+Metrics
+Methods / Datasets
+Cora
+Citeseer
+Pubmed
+ACM
+BlogCatalog
+CoraFull
+Flickr
+UAI
+Chebyshev
+76.2 (0.7)
+69.3 (0.4)
+74.0 (0.8)
+82.8 (1.4)
+68.3 (1.6)
+57.2 (1.1)
+38.5 (1.6)
+49.7 (0.4)
+GraphSAGE
+76.7 (0.6)
+64.4 (0.9)
+75.5 (0.2)
+—
+57.8 (0.7)
+59.9 (0.7)
+32.7 (1.0)
+41.7 (1.4)
+GAT
+79.1 (0.8)
+68.3 (0.5)
+78.4 (0.3)
+84.6 (0.5)
+67.1 (1.7)
+62.4 (0.4)
+40.4 (0.9)
+49.7 (3.0)
+GCN
+80.6 (1.4)
+69.1 (1.5)
+77.6 (1.3)
+88.8 (0.5)
+84.2 (0.6)
+62.8 (0.4)
+51.0 (1.2)
+58.5 (1.1)
+SGC
+79.3 (1.0)
+66.4 (1.7)
+76.8 (2.0)
+80.8 (2.7)
+81.3 (0.2)
+62.9 (2.2)
+51.0 (0.1)
+56.5 (3.5)
+APPNP
+78.0 (0.1)
+65.8 (0.2)
+78.0 (0.0)
+88.2 (0.0)
+87.7 (0.3)
+63.1 (0.5)
+57.5 (0.2)
+62.3 (1.2)
+JKNet
+83.1 (0.1)
+72.3 (0.1)
+80.1 (0.2)
+82.3 (0.6)
+75.7 (0.1)
+62.6 (0.0)
+54.0 (0.3)
+45.6 (0.5)
+DAGNN
+81.9 (0.7)
+70.0 (1.1)
+80.6 (0.7)
+87.4 (0.9)
+84.6 (1.9)
+65.6 (0.3)
+54.6 (5.9)
+46.7 (12.4)
+GNN-LF
+81.1 (0.5)
+72.3 (0.9)
+80.0 (0.4)
+90.8 (0.5)
+86.7 (0.6)
+63.5 (0.9)
+56.6 (0.6)
+36.6 (19.8)
+GNN-HF
+80.7 (0.2)
+68.8 (1.3)
+77.7 (0.2)
+91.2 (0.5)
+84.5 (0.4)
+63.0 (0.7)
+60.7 (0.4)
+54.8 (1.4)
+tsGCN (inv)
+80.3 (0.3)
+73.3 (0.4)
+78.4 (0.3)
+85.1 (1.6)
+87.8 (6.3)
+67.0 (0.9)
+53.3 (12.6)
+64.2 (1.8)
+ACC
+tsGCN
+82.0 (0.3)
+73.1 (0.4)
+82.4 (0.1)
+92.8 (0.3)
+92.3 (0.5)
+67.9 (0.9)
+79.1 (3.0)
+67.9 (0.6)
+Chebyshev
+76.3 (0.7)
+65.4 (0.8)
+73.9 (0.7)
+82.5 (1.4)
+64.3 (1.6)
+40.0 (0.5)
+38.4 (1.5)
+39.1 (0.2)
+GraphSAGE
+76.7 (0.5)
+60.7 (0.5)
+74.7 (0.2)
+—
+54.7 (0.6)
+51.9 (0.6)
+31.0 (1.1)
+35.3 (1.0)
+GAT
+77.1 (0.7)
+64.6 (0.5)
+78.2 (0.2)
+84.8 (0.5)
+66.3 (1.9)
+46.4 (0.4)
+38.1 (1.1)
+40.8 (1.3)
+GCN
+79.4 (1.4)
+65.2 (2.4)
+77.2 (1.4)
+88.9 (0.5)
+82.4 (0.5)
+52.8 (0.8)
+50.0 (1.7)
+45.0 (1.1)
+SGC
+77.7 (0.9)
+61.5 (1.7)
+76.5 (2.3)
+81.1 (2.6)
+80.7 (0.3)
+53.2 (2.1)
+44.2 (0.2)
+46.7 (1.7)
+APPNP
+77.6 (0.1)
+63.2 (0.2)
+77.7 (0.0)
+88.3 (0.0)
+85.7 (0.3)
+48.2 (0.7)
+56.9 (0.2)
+48.6 (1.6)
+JKNet
+82.3 (0.3)
+67.8 (0.1)
+79.3 (0.3)
+82.2 (0.6)
+75.0 (0.1)
+51.3 (0.1)
+51.1 (0.5)
+31.7 (1.5)
+DAGNN
+80.0 (0.7)
+65.7 (0.7)
+80.7 (0.7)
+87.5 (0.9)
+83.8 (2.4)
+53.0 (0.9)
+55.5 (6.7)
+39.3 (11.2)
+GNN-LF
+79.1 (0.7)
+66.7 (0.4)
+80.2 (0.5)
+90.9 (0.5)
+85.9 (0.6)
+50.5 (1.9)
+54.3 (1.0)
+29.7 (15.1)
+GNN-HF
+78.6 (0.3)
+64.3 (1.7)
+78.1 (0.2)
+91.3 (0.5)
+83.8 (0.4)
+49.0 (1.1)
+58.6 (0.6)
+44.9 (0.8)
+tsGCN (inv)
+78.5 (0.3)
+69.6 (0.4)
+78.7 (0.3)
+85.1 (1.5)
+85.2 (7.1)
+57.2 (1.1)
+52.9 (15.8)
+48.5 (0.8)
+F1
+tsGCN
+80.5 (0.5)
+69.0 (0.3)
+82.4 (0.1)
+92.8 (0.4)
+90.1 (0.6)
+58.7 (0.7)
+79.3 (2.9)
+50.1 (0.1)
+Experiment
+This section will show tsGCN’s effectiveness and efficiency
+via comprehensive experiments.
+Datasets
+Cora, Citeseer and Pubmed are citation networks, and Cora-
+Full is a larger version of Cora; ACM is a paper network, and
+BlogCatalog and Flickr are social networks; UAI has been
+utilized for community detection. The detailed statistics of
+the above eight public datasets are concluded in Table 2.
+Compared Methods
+Two types of methods are employed here for comparisons.
+Chebyshev [33], GraphSAGE [34] and GAT [35] are clas-
+sical graph neural networks. GCN, SGC [22], APPNP [23],
+JKNet [24], DAGNN [27], GNN-LF and GNN-HF [19] are
+selected as state-of-the-art GCN variants.
+Parameter Setups
+For all experiments, we randomly split samples into a small
+set of 20 labeled samples per class for training, a set of 500
+samples for validating and a set of 1, 000 samples for testing.
+In terms of the ten baseline methods, all their configurations
+are set as the default in their original papers. With respect to
+tsGCN, following the vanilla GCN, the learning rate, weight
+decay and the size of hidden units are set to 1 × 10−2, 5 ×
+10−4 and 32, respectively. The hyperparameters α and β are
+selected in {0.1, 0.2, . . . , 1.0} for different datasets, and r is
+chosen in {⌊ d
+211 ⌋, ⌊ d
+210 ⌋, . . . , ⌊ d
+23 ⌋}, where d is the feature
+dimension of the original data.
+Semi-supervised Classification
+Performance Comparisons. The semi-supervised classifi-
+cation task is conducted on selected datasets, whose results
+are recorded in Table 3. Specifically, we compare our tsGCN
+with the ten baseline methods in terms of both accuracy and
+F1-score, marking the best and second-best results on each
+dataset. Note that tsGCN (inv) denotes tsGCN without the
+low-rank approximation, which directly calculates the ma-
+trix inverse in Eq. (14). From Table 3, we have the following
+observations:
+• tsGCN achieves the best performances on most datasets,
+and is only slightly inferior to the JKNet method on the
+smallest Cora dataset.
+• tsGCN yields higher scores than JKNet and APPNP, es-
+pecially on Pubmed, CoraFull, BlogCatalog, and Flickr,
+where the first two are relatively large datasets and the
+latter two have dense edges. tsGCN even outperforms the
+second-best approach GNN-HF by about 20% on Flickr.
+It is worth mentioning that tsGCN utilizes high-order
+information by the infinite-order graph convolution, and
+JKNet and APPNP also develop different techniques for the
+same goal. Hence, the advantage of tsGCN over APPNP and
+JKNet implies that the infinite-order graph convolution im-
+plemented by the low-rank approximation not only requires
+less computational complexity, but also effectively captures
+high-order neighborhood information and filters significant
+noises.
+Runtime Comparisons. This section collects the train-
+ing time (i.e., runtime) of all methods on two selected large
+datasets, i.e., Pubmed and CoraFull, as exhibited in Fig. 1(a):
+the first three columns correspond to classical GNNs, while
+
+(a) Runtime
+(b) Classification Accuracy
+Figure 1: (a) All methods’ runtime on two large datasets. (b) The classification accuracy of tsGCN w.r.t. (α, β) on all datasets.
+(a) Cora
+(b) Citeseer
+(c) Pubmed
+(d) ACM
+(e) BlogCatalog
+(f) CoraFull
+(g) Flickr
+(h) UAI
+Figure 2: The classification accuracy of tsGCN w.r.t. hyperparameters α and β on all datasets.
+the rest are GCNs. From Fig. 1(a), we find that tsGCN takes
+much less runtime than Chebyshev, GAT, and GraphSAGE;
+however, it performs moderately well among the state-of-
+the-art GCN variants. Specifically, tsGCN is (1) inferior to
+SGC, JKNet, and DAGNN; (2) well-matched with the orig-
+inal GCN; (3) but advantageous over the recently proposed
+GNN-LF and GNN-HF.
+Parameter Sensitivity Analysis
+Fig. 1(b) curves the accuracy of tsGCN w.r.t. various ranks
+by fixing other parameters α and β. Considering that differ-
+ent datasets hold different distributions, their optimal ranks
+to the optimization (16) are also different. For example, in
+regard to the curves on BlogCatalog and ACM, their accu-
+racy first go up to a high value and then keep steady, which
+indicates that when rank r = ⌊d/512⌋, the low-rank approx-
+imation is effective and efficient enough. When it comes to
+the curve on Pubmed, the trend of its performance mono-
+tonically decreases as rank r becomes bigger, which implies
+that a very low-rank (i.e., r = ⌊d/2048⌋) approximation is
+sufficient enough to preserve abundant information for good
+results. However, with respect to the other curves such as on
+Flickr and Cora, the y-axis’ scores generally rise to a peak
+first and then fall continuously as the rank r increases. For
+these cases, the optimal ranks differ at their peaks.
+Fig. 2 plots the accuracy of tsGCN w.r.t. (α, β) by fixing
+the optimal ranks. On Cora, Citeseer, Pubmed, BlogCatalog,
+and CoraFull, tsGCN performs well with large α and small
+β; while, on ACM, Flickr, and UAI, tsGCN generates high
+accuracy when these two parameters are both large.
+For detailed settings of these hyperparameters, please ref-
+erence the codes and datasets to be released on Github.
+Ablation Study
+The results of GCN, tsGCN-s, tsGCN-t, tsGCN (inv), and ts-
+GCN are plotted in Fig. 3 (notice that tsGCN-s and tsGCN-t
+are with semantic and topological regularizer, respectively),
+telling us:
+• The performance is unsatisfactory when the two regular-
+izers are adopted alone, while tsGCN can always effec-
+tively fuse the two to better capture underlying structures.
+• tsGCN (inv) is even worse than single-regularizer model
+on some datasets, indicating that the infinite-order graph
+
+Accuracy (%)
+Accuracy (%)
+F1-score (%)
+F1-score (%)
+Figure 3: Accuracy and F1-score of tsGCN and its variants on all datasets.
+(a) Chebyshev
+(b) GraphSAGE
+(c) GAT
+(d) GCN
+(e) SGC
+(f) APPNP
+(g) JKNet
+(h) DAGNN
+(i) GNN-LF
+(j) GNN-HF
+(k) tsGCN (inv)
+(l) tsGCN
+Figure 4: Different methods’ t-SNE visualizations on BlogCatalog, where each color corresponds to one class.
+convolutions implemented by the matrix inverse can pull-
+in instability to the model.
+• Compared to GCN, tsGCN (inv) performs comparable or
+even worse, whereas tsGCN shows substantial improve-
+ments on all datasets, which indicates that the low-rank
+approximation enhances the robustness of infinite-order
+graph convolutions.
+Data Visualization
+Fig. 4 exhibits the graph representations learned by different
+methods on BlogCatalog. As can be seen clearly, the results
+of the three classical graph neural networks, i.e., Chebyshev,
+GraphSAGE and GAT, are unsatisfactory; while for the other
+competitors, there are:
+• Both tsGCN (inv) and tsGCN are better than other GCNs,
+which indicates that the dual-regularizer can extract more
+accurate inter-relationships from the topological and se-
+mantic structures.
+• Comparing the embeddings learned by tsGCN with those
+of tsGCN (inv), classes in the former sub-figure are more
+clearly recognized and the within-clusters are more com-
+pact, which testifies the effectiveness of the low-rank ap-
+proximation.
+In a nutshell, the embeddings of the proposed model show
+the best inter-class separation and intra-class aggregation.
+Conclusion
+By revisiting GCN, this paper puts forward an interpretable
+regularizer-centered optimization framework, in which the
+connections between existing GCNs and diverse regularizers
+are revealed. It’s worth mentioning that this framework pro-
+vides a new perspective to interpret existing work and guide
+new GCNs just by designing new graph regularizers. Im-
+pressed by the significant effectiveness of the feature based
+semantic graph, we further combine it with nodes’ topolog-
+ical structures, and develop a novel dual-regularizer graph
+convolutional network, called tsGCN. Since the analytical
+updating rule of tsGCN contains a time-consuming matrix
+inverse, we devise an efficient algorithm with low-rank ap-
+proximation tricks. Experiments on node classification tasks
+demonstrate that tsGCN performs much better than quite a
+few state-of-the-art competitors, and also exhibit that tsGCN
+runs much faster than the very recently proposed GCN vari-
+ants, e.g., GNN-HF and GNN-LF.
+Acknowledgments
+This work is in part supported by the National Natu-
+ral Science Foundation of China (Grant Nos. U21A20472
+and 62276065), the Natural Science Foundation of Fujian
+Province (Grant No. 2020J01130193).
+Supplementary
+In this supplementary, we mainly present specific details to
+link various GCNs with various graph regularizers under the
+regularizer-centered optimization framework. Besides, more
+experimental settings and results are provided to further en-
+rich the main paper.
+The Framework Review
+An interpretable regularizer-centered constrained optimiaza-
+tion framework is induced as
+arg min
+H(l) L = −Tr(H(l)⊤H(l−1)Θ(l))
+�
+��
+�
+fitting
++ 1
+2L(H(l); G)
+�
+��
+�
+regularization
+(19)
+
+s.t. H(l) ∈ {S+ or S}, l ∈ [L − 1], H(L) ∈ Ssimplex,
+with the aim to unify various GCNs in an interpretable way,
+and also to guide the design of new GCN variants. Note that
+the first term in optimization (19) is equivalent to the fitting
+loss between the forward propagation H(l−1)Θ(l) and the
+output H(l), while the second term is the priors-based graph
+regularizer. Besides, S, S+ and Ssimplex are separately de-
+fined to be
+S = {s ∈ Rd},
+(20)
+S+ = {s ∈ Rd|s ≥ 0},
+(21)
+and
+Ssimplex = {s ∈ Rd|1⊤s = 1, s ≥ 0}.
+(22)
+The above three sets correspond to the Identity(·), Relu(·),
+and Softmax(·) activation functions frequently used in graph
+convolutional networks, respectively.
+It’s claimed that by designing different regularizers, this
+framework can give birth to different GCN methods. In the
+following, we will give specific details about how they could
+be derived from optimization (19).
+Link Various GCNs with Various Regularizers
+Theorem 1. The updating rule of the vanilla GCN [21]
+H(l) = σ
+�
+�AH(l−1)Θ(l)�
+, l ∈ [L],
+(23)
+is equivalent to solving the following optimization
+H(l) = arg min
+H∈S(l) J (l)
+(24)
+s.t. S(l) ∈ {S or S+ or Ssimplex},
+where
+J (l) = −Tr
+�
+H⊤H(l−1)Θ(l)�
++ 1
+2Tr
+�
+H⊤�LH
+�
+.
+(25)
+Proof. Taking derivative of J (l) w.r.t. H, we obtain
+∂J (l)
+∂H
+= −H(l−1)Θ(l) + �LH;
+(26)
+if it ( ∂J (l)
+∂H ) is further set to zero, then there yields
+H∗ = (I − �A)−1H(l−1)Θ(l).
+(27)
+By projecting H∗ onto S(l), we could arrive at
+H(l) = σ(H∗).
+(28)
+Notably, (I − �A)−1 = �∞
+i=0 �Ai; and when its first-order
+approximation is leveraged, i.e., (I − �A)−1 ≈ I + ˜A = �A,
+Eq. (28) gives birth to the updating rule (23).
+The above analyses reveal that when the regularizer is de-
+signed to 1
+2Tr
+�
+H⊤�LH
+�
+, the framework (19) could gener-
+ate the vanilla GCN [21].
+Theorem 2. Given H(0) = f MLP
+Θ
+(X) and α ∈ [0, 1), the
+updating rule of APPNP [23]
+H(l) = σ
+�
+(1 − α) �AH(l−1) + αH(0)�
+, l ∈ [L],
+(29)
+is equivalent to solving the following optimization
+H(l) = arg min
+H∈S(l) J (l),
+(30)
+s.t. H(l) = S, l ∈ [L − 1], H(L) = Ssimplex,
+where
+J (l) = −Tr
+�
+H⊤H(l−1)Θ(l)�
++ 1
+2Tr
+�
+1
+1 − αH⊤ �A−1(H − 2αH(0))
+�
+.
+(31)
+Proof. Taking the derivative of J (l) w.r.t. H and setting it
+to zero, we come to
+∂J (l)
+∂H
+= −H(l−1) +
+1
+1 − α
+�A−1(H − αH(0)) = 0, (32)
+which leads to
+H∗ = (1 − α) �AH(l−1) + αH(0).
+(33)
+By projecting H∗ onto S(l), we could achieve
+H(l) = σ
+�
+(1 − α) �AH(l−1) + αH(0)�
+, l ∈ [L],
+(34)
+which completes the proof.
+The above analyses reveal that when the regularizer is de-
+vised to 1
+2Tr
+�
+1
+1−αH⊤ �A−1(H − 2αH(0))
+�
+, the framework
+(19) could give birth to APPNP [23].
+Theorem 3. The updating rule of JKNet [24]
+H(l) = σ
+� K
+�
+k=1
+αk �AkH(l−1)Θ(l)
+�
+, l ∈ [L],
+(35)
+is equivalent to solving the following optimization
+H(l) = arg min
+H∈S(l) J (l)
+(36)
+s.t. H(l) = S, l ∈ [L − 1], H(L) = Ssimplex,
+where
+J (l) = −Tr
+�
+H⊤H(l−1)Θ(l)�
++ 1
+2Tr
+�
+H⊤ �A−1(I + β�L)H
+�
+.
+(37)
+Proof. Taking the derivative of J (l) w.r.t. H and setting it
+to zero, we have
+∂J (l)
+∂H
+= −H(l−1)Θ(l) + �A−1(I + β�L)H = 0,
+(38)
+which leads to
+H∗ =
+1
+β + 1
+�
+I −
+β
+β + 1
+�A
+�−1
+�AH(l−1)Θ(l).
+(39)
+
+Methods
+Propagation Rules
+Regularizer L(H(l); G)
+Projective Set
+GCN
+H(l) = σ
+�
+�AH(l−1)Θ(l)�
+Tr
+�
+H(l)⊤�LH(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex
+SGC
+H(l) = σ
+�
+�AH(l−1)Θ(l)�
+Tr
+�
+H(l)⊤�LH(l)�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+APPNP
+H(l) = σ
+�
+(1 − α) �AH(l−1) + αH(0)�
+Tr
+�
+1
+1−αH(l)⊤ �A−1(H(l) − 2αH(0))
+�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+JKNet
+H(l) = σ
+��K
+k=1 αk �AkH(l−1)Θ(l)�
+Tr
+�
+H(l)⊤ �A−1(I + β�L)H(l)�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+DAGNN
+H(L) = σ
+��K
+k=0 αk �AkH(0)�
+Tr
+�
+H(l)⊤(I + β�L)H(l)�
+�
+S(l) = S, l ∈ [L−1],
+S(L) = Ssimplex
+GNN-HF
+H(l) = σ
+�
+(I + α�L)−1(I + β�L)H(l−1)Θ(l)�
+Tr
+�
+H(l)⊤(I + β�L)−1(I + α�L)H(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex.
+GNN-LF
+H(l) = σ
+�
+(I + α �A)−1(I + β �A)H(l−1)Θ(l)�
+Tr
+�
+H(l)⊤(I + β �A)−1(I + α �A)H(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex
+tsGCN
+H(l) = σ
+�
+(I + α�LG + β�LX )−1H(l−1)Θ(l)�
+Tr
+�
+H(l)⊤(I + α�LG + β�LX )H(l)�
+�
+S(l) = S+, l ∈ [L−1],
+S(L) = Ssimplex
+Table 4: Different regularizers can derive different GCN variants under the regularizer-centered optimization framework.
+It is noted that the spectral radius of
+β
+β+1 �A is smaller than
+one, indicating
+�
+I −
+β
+β + 1
+�A
+�−1
+=
+∞
+�
+k=0
+�
+β
+β + 1
+�A
+�k
+.
+(40)
+If its (K−1)-order approximation is employed, then there
+goes
+�
+I −
+β
+β + 1
+�A
+�−1
+≈
+K−1
+�
+k=0
+βk
+(β + 1)k �Ak,
+(41)
+which suggests that H∗ can be approximated by
+H∗ =
+K
+�
+k=1
+βk−1
+(β + 1)k �AkH(l−1)Θ(l).
+(42)
+If denote αk =
+βk−1
+(β+1)k (k ∈ [K]), then {αk}∞
+k=1 is a set
+of parameters with �∞
+k=1 αk =
+1
+β+1
+1
+1−
+β
+β+1 = 1.
+By projecting H∗ onto S(l), we can realize
+H(l) = σ
+� K
+�
+k=1
+αk �AkH(l−1)Θ(l)
+�
+, l ∈ [L],
+(43)
+which completes the proof.
+The above analyses reveal that when the regularizer is
+devised to 1
+2Tr
+�
+H⊤ �A−1(I + β�L)H
+�
+, the framework (19)
+can produce JKNet [24].
+Theorem 4. Given H(0) = f MLP
+Θ
+(X) and a trainable pro-
+jection vector α ∈ RK+1, the updating rule of DAGNN [27]
+H(l) = σ
+� K
+�
+k=0
+αk �AkH(0)
+�
+,
+(44)
+is equivalent to solving the following optimization
+H(l) = arg min
+H∈S(l) J (l)
+(45)
+s.t. H(l) = S, l ∈ [L − 1], H(L) = Ssimplex,
+where
+J (l) = −Tr
+�
+H⊤H(l−1)Θ(l)�
++ 1
+2Tr
+�
+H⊤(I + β�L)H
+�
+.
+(46)
+Proof. Taking the derivative of J (l) w.r.t. H and setting it
+to zero, we can harvest
+∂J (l)
+∂H
+= −H(0) + (I + β�L)H = 0.
+(47)
+Similar to the proof of Theorem 3, the K-order approxi-
+mation of
+�
+I + β�L
+�−1
+=
+1
+β+1
+�
+I −
+β
+β+1 �A
+�−1
+is utilized,
+and then we obtain
+H∗ =
+K
+�
+k=0
+αk �AkH(0).
+(48)
+By projecting H∗ onto S(l), we can arrive at
+H(l) = σ
+� K
+�
+k=0
+αk �AkH(0)
+�
+, l ∈ [L],
+(49)
+which completes the proof.
+The above analyses reveal that when the regularizer is
+devised to 1
+2Tr
+�
+H⊤(I + β�L)H
+�
+, the framework (19) can
+produce DAGNN [27].
+Theorem 5. The updating rule of GNN-HF [19]
+H(l) = σ((β + 1
+α)I
++ (1 − β − 1
+α) �A−1(I + β�L)H(l−1)Θ(l)),
+(50)
+
+is equivalent to solving the following optimization
+H(l) = arg min
+H∈S(l) J (l)
+(51)
+s.t. H(l) = S+, l ∈ [L − 1], H(L) = Ssimplex,
+where
+J (l) = −Tr
+�
+H⊤H(l−1)Θ(l)�
++ 1
+2Tr
+�
+H⊤(I + µ�L)−1(I + λ�L)H
+�
+(52)
+with λ = β + 1
+α − 1 and µ = β.
+Proof. Taking the derivative of J (l) w.r.t. H and setting it
+to zero, we own
+∂J (l)
+∂H
+= −H(l−1)Θ(l)+(I+µ�L)−1(I+λ�L)H = 0. (53)
+which yields
+H∗ = (I + λ�L)−1(I + µ�L)H(l−1)Θ(l).
+(54)
+Substituting λ = β + 1
+α − 1 and µ = β into Eq. (54), we
+obtain
+(I + λ�L)−1 =
+�
+(1 + λ)I − λ �A
+�−1
+=
+�
+(β + 1
+α)I + (1 − β − 1
+α) �A
+�−1
+.
+(55)
+By projecting H∗ onto S(l), we can ahieve
+H(l) = σ (H∗) , l ∈ [L],
+(56)
+which completes the proof.
+The above analyses reveal that when the regularizer is de-
+vised to 1
+2Tr
+�
+H⊤(I + µ�L)−1(I + λ�L)H
+�
+, the framework
+(19) can produce GNN-HF [19].
+Theorem 6. The updating rule of GNN-LF [19]
+H(l) = σ((βI + (1 − β) �A
++ ( 1
+α − 1)�L)−1(βI + (1 − β) �A)H(l−1)),
+(57)
+is equivalent to solving the following optimization
+H(l) = arg min
+H∈S(l) J (l)
+(58)
+s.t. H(l) = S+, l ∈ [L − 1], H(L) = Ssimplex,
+where
+J(l) = −Tr
+�
+H⊤H(l−1)Θ(l)�
++ 1
+2Tr
+�
+H⊤(I + µ �A)−1(I + λ �A)H
+�
+(59)
+with λ = −αβ+2α−1
+αβ−α+1
+and µ = 1
+β − 1.
+Proof. Taking the derivative of J (l) w.r.t. H and setting it
+to zero, we can get
+∂J (l)
+∂H
+= −H(l−1)Θ(l)+(I+µ �A)−1(I+λ �A)H = 0, (60)
+which leads to
+H∗ = (I + λ �A)−1(I + µ �A)H(l−1)Θ(l).
+(61)
+Absorbing the scale
+αβ
+αβ−α+1 into the to-be-learnt variable
+Θ(l), and substituting λ = −αβ+2α−1
+αβ−α+1
+and µ = 1
+β − 1 into
+Eq. (61), we can harvest
+αβ
+αβ − α + 1(I + λ �A)−1(I + µ �A)
+=
+αβ
+αβ − α + 1(I + −αβ + 2α − 1
+αβ − α + 1
+�A)−1(I + ( 1
+β − 1) �A)
+=
+�
+(1 − 1
+β + 1
+αβ )I + (−1 + 2
+β − 1
+αβ ) �A
+�−1
+(I + ( 1
+β − 1) �A)
+=
+�
+(β − 1 + 1
+α)I + (−β + 2 − 1
+α) �A
+�−1
+(βI + (1 − β) �A).
+(62)
+By projecting H∗ onto S(l), we can attain
+H(l) = σ (H∗) , l ∈ [L],
+(63)
+For notation consistency, we denote λ and µ as α and β
+in Table 4, completing the proof.
+The above analyses reveal that when the regularizer is set
+to 1
+2Tr
+�
+H⊤(I + µ �A)−1(I + λ �A)H
+�
+, the framework (19)
+can generate GNN-LF [19].
+More Experimental Settings and Results
+In this part, we provide more experimental settings and re-
+sults for tsGCN, including hyperparameter settings, t-SNE
+visualizations of various methods, and the parameter sensi-
+tivity analysis of tsGCN w.r.t. F1-score.
+Hyperparameter Settings. The detailed values of several
+hyperpaerameters are recorded in Table 5, which can be used
+to reproduce the reported experimental results. And the code
+is also provided as a supplementary file.
+More visualizations. We draw the t-SNE of embeddings
+generated by all methods on all datasets from Fig. 5 to
+Fig. 11, from which we have the following observations:
+• The results are generally matched with the quantitative
+performance, i.e., tsGCN achieves better results than the
+others on most datasets.
+• Embeddings generated by tsGCN achieve better inter-
+class separation alongside intra-class clustering than
+those generated by tsGCN (inv), even when their quanti-
+tative performance is comparable.
+Parameter Sensitivity. It can be seen that the F1-scores of
+tsGCN w.r.t. (α, β) hold the similar trends with the classifi-
+cation accuracies of tsGCN.
+
+Table 5: Specific (α, β, r) and other parameters of tsGCN on all datasets.
+Datasets/Parameters
+α
+β
+r
+Learning rate
+Weight decay
+Hidden units
+Cora
+1.0
+0.2
+⌊d/16⌋
+1 × 10−2
+5 × 10−4
+32
+Citeseer
+1.0
+0.4
+⌊d/16⌋
+1 × 10−2
+5 × 10−4
+32
+Pubmed
+1.0
+0.3
+⌊d/2048⌋
+1 × 10−2
+5 × 10−4
+32
+ACM
+1.0
+0.9
+⌊d/64⌋
+1 × 10−2
+5 × 10−4
+32
+BlogCatalog
+1.0
+0.5
+⌊d/64⌋
+1 × 10−2
+5 × 10−4
+32
+CoraFull
+1.0
+0.1
+⌊d/8⌋
+1 × 10−2
+5 × 10−4
+32
+Flickr
+1.0
+1.0
+⌊d/64⌋
+1 × 10−2
+5 × 10−4
+32
+UAI
+1.0
+1.0
+⌊d/16⌋
+1 × 10−2
+5 × 10−4
+32
+Figure 5: Different methods’ t-SNE visualizations on Cora, where each color corresponds to one class.
+Figure 6: Different methods’ t-SNE visualizations on Citeseer, where each color corresponds to one class.
+
+Figure 7: Different methods’ t-SNE visualizations on Pubmed, where each color corresponds to one class.
+Figure 8: Different methods’ t-SNE visualizations on ACM, where each color corresponds to one class. Note that GraphSAGE
+fails to run on ACM.
+Figure 9: Different methods’ t-SNE visualizations on CoraFull, where each color corresponds to one class.
+
+Figure 10: Different methods’ t-SNE visualizations on Flickr, where each color corresponds to one class.
+Figure 11: Different methods’ t-SNE visualizations on UAI, where each color corresponds to one class.
+(a) Cora
+(e) BlogCatalog
+(b) Citeseer
+(f) CoraFull
+(c) Pubmed
+(g) Flickr
+(d) ACM
+(h) UAI
+Figure 12: The classification F1-scores of tsGCN w.r.t. different hyperparameters α and β on all datasets.
+
+References
+[1] Z. Chen, X. Wei, P. Wang, Y. Guo, Multi-label im-
+age recognition with graph convolutional networks, in:
+CVPR, 2019, pp. 5177–5186.
+[2] W. Nie, Y. Zhao, A. Liu, Z. Gao, Y. Su, Multi-graph
+convolutional network for unsupervised 3d shape re-
+trieval, in: MM, 2020, pp. 3395–3403.
+[3] Y. Wang, M. Cao, Z. Fan, S. Peng, Learning to detect
+3d facial landmarks via heatmap regression with graph
+convolutional network, in: AAAI, 2022, pp. 2595–
+2603.
+[4] F. Xu, J. Lian, Z. Han, Y. Li, Y. Xu, X. Xie, Relation-
+aware graph convolutional networks for agent-initiated
+social e-commerce recommendation, in: CIKM, 2019,
+pp. 529–538.
+[5] Y. Chen, L. Huang, C. Wang, J. Lai, Hybrid-order
+gated graph neural network for session-based recom-
+mendation, IEEE Transactions on Industrial Informat-
+ics 18 (3) (2022) 1458–1467.
+[6] H. Hu, L. Cheng, J. P. Vap, M. Borowczak, Learn-
+ing privacy-preserving graph convolutional network
+with partially observed sensitive attributes, in: WWW,
+2022, pp. 3552–3561.
+[7] B. Yu, H. Yin, Z. Zhu, Spatio-temporal graph convolu-
+tional networks: A deep learning framework for traffic
+forecasting, in: IJCAI, 2018, pp. 3634–3640.
+[8] W. Chen, L. Chen, Y. Xie, W. Cao, Y. Gao, X. Feng,
+Multi-range attentive bicomponent graph convolu-
+tional network for traffic forecasting, in: AAAI, 2020,
+pp. 3529–3536.
+[9] Y. Zhang, S. Pal, M. Coates, D. ¨Ustebay, Bayesian
+graph
+convolutional
+neural
+networks
+for
+semi-
+supervised
+classification,
+in:
+AAAI,
+2019,
+pp.
+5829–5836.
+[10] M. Yang, Y. Shen, R. Li, H. Qi, Q. Zhang, B. Yin, A
+new perspective on the effects of spectrum in graph
+neural networks, in: ICML, 2022, pp. 25261–25279.
+[11] S. Fan, X. Wang, C. Shi, E. Lu, K. Lin, B. Wang,
+One2multi graph autoencoder for multi-view graph
+clustering, in: WWW, 2020, pp. 3070–3076.
+[12] H. Zhu, P. Koniusz, Simple spectral graph convolution,
+in: ICLR, 2021, pp. 1–11.
+[13] H. Chen, H. Yin, X. Sun, T. Chen, B. Gabrys, K. Mu-
+sial, Multi-level graph convolutional networks for
+cross-platform anchor link prediction, in: KDD, 2020,
+pp. 1503–1511.
+[14] N. Halliwell, Evaluating explanations of relational
+graph convolutional network link predictions on
+knowledge graphs, in: AAAI, 2022, pp. 12880–12881.
+[15] D. Bo, X. Wang, C. Shi, H. Shen, Beyond low-
+frequency information in graph convolutional net-
+works, in: AAAI, 2021, pp. 3950–3957.
+[16] Z. Zhang, C. Chen, Y. Chang, W. Hu, X. Xing, Y. Zhou,
+Z. Zheng, Shne: Semantics and homophily preserv-
+ing network embedding, IEEE Transactions on Neural
+Networks and Learning Systems (2021) 1–12.
+[17] X. Wang, M. Zhu, D. Bo, P. Cui, C. Shi, J. Pei, Am-
+gcn: Adaptive multi-channel graph convolutional net-
+works, in: KDD, 2020, pp. 1243–1253.
+[18] L. Zhao, L. Akoglu, Connecting graph convolutional
+networks and graph-regularized pca, arXiv preprint
+arXiv:2006.12294 (2020).
+[19] M. Zhu, X. Wang, C. Shi, H. Ji, P. Cui, Interpreting and
+unifying graph neural networks with an optimization
+framework, in: WWW, 2021, pp. 1215–1226.
+[20] Y. Yang, T. Liu, Y. Wang, J. Zhou, Q. Gan, Z. Wei,
+Z. Zhang, Z. Huang, D. Wipf, Graph neural networks
+inspired by classical iterative algorithms, in: ICML,
+2021, pp. 11773–11783.
+[21] T. N. Kipf, M. Welling, Semi-supervised classification
+with graph convolutional networks, in: ICLR, 2017,
+pp. 1–13.
+[22] F. Wu, A. H. S. Jr., T. Zhang, C. Fifty, T. Yu,
+K. Q. Weinberger, Simplifying graph convolutional
+networks, in: ICML, 2019, pp. 6861–6871.
+[23] J. Klicpera, A. Bojchevski, S. G¨unnemann, Predict
+then propagate: Graph neural networks meet person-
+alized pagerank, in: ICLR, 2019, pp. 1–15.
+[24] K.
+Xu,
+C.
+Li,
+Y.
+Tian,
+T.
+Sonobe,
+K.
+ichi
+Kawarabayashi, S. Jegelka, Representation learning on
+graphs with jumping knowledge networks, in: ICML,
+2018, pp. 5449–5458.
+[25] K. Sun, Z. Zhu, Z. Lin, Adagcn: Adaboosting graph
+convolutional networks into deep models, in: ICLR,
+2021, pp. 1–15.
+[26] Q. Li, Z. Han, X. Wu, Deeper insights into graph con-
+volutional networks for semi-supervised learning, in:
+AAAI, 2018, pp. 3538–3545.
+[27] M. Liu, H. Gao, S. Ji, Towards deeper graph neural
+networks, in: KDD, 2020, pp. 338–348.
+[28] L. Yang, Z. Kang, X. Cao, D. Jin, B. Yang, Y. Guo,
+Topology optimization based graph convolutional net-
+work, in: AAAI, 2019, pp. 4054–4061.
+[29] K. Oono, T. Suzuki, Graph neural networks exponen-
+tially lose expressive power for node classification, in:
+ICLR, 2020, pp. 1–8.
+[30] D. K. Hammond, P. Vandergheynst, R. Gribonval,
+Wavelets on graphs via spectral graph theory, Applied
+and Computational Harmonic Analysis 30 (2011) 129–
+150.
+[31] B. Amos, Differentiable optimization-based modeling
+for machine learning, Ph.D. thesis, Carnegie Mellon
+University (2019).
+[32] J. Sun, Z. Xu, Neural diffusion distance for image seg-
+mentation, in: NeurIPS, 2019, pp. 1441–1451.
+[33] M. Defferrard, X. Bresson, P. Vandergheynst, Convo-
+lutional neural networks on graphs with fast localized
+spectral filtering, in: NeurIPS, 2016, pp. 1–9.
+[34] W. L. Hamilton, Z. Ying, J. Leskovec, Inductive repre-
+sentation learning on large graphs, in: NeurIPS, 2017,
+pp. 1024–1034.
+
+[35] P. Velickovic, G. Cucurull, A. Casanova, A. Romero,
+P. Li`o, Y. Bengio, Graph attention networks, in: ICLR,
+2018, pp. 1–12.
+

gNFKT4oBgHgl3EQftS6b/content/tmp_files/2301.11886v1.pdf.txt

@@ -0,0 +1,1600 @@
+Machine Learning Over Heuristic:
+a Learned Cache Eviction Framework with Minimal Overhead
+Dongsheng Yang1, Daniel S. Berger2, Kai Li1, and Wyatt Lloyd2
+1Princeton University
+2Microsoft Research
+Abstract
+Recent work shows the effectiveness of Machine Learning
+(ML) to reduce cache miss ratios by making better eviction de-
+cisions than heuristics. However, state-of-the-art ML caches
+require many predictions to make an eviction decision, mak-
+ing them impractical for high-throughput caching systems.
+This paper introduces Machine learning At the Tail (MAT),
+a framework to build efficient ML-based caching systems by
+integrating an ML module with a traditional cache system
+based on a heuristic algorithm. MAT treats the heuristic al-
+gorithm as a “filter” to receive high-quality samples to train
+an ML model and likely candidate objects for evictions. We
+evaluate MAT on 8 production workloads, spanning storage,
+in-memory caching, and CDNs. The simulation experiments
+show MAT reduces the number of costly ML predictions-per-
+eviction from 63 to 2, while achieving comparable miss ratios
+to the state-of-the-art ML cache system. We compare a MAT
+prototype system with an LRU-based caching system in the
+same setting and show that achieve similar request rates.
+1
+Introduction
+Software caching systems are ubiquitous in modern comput-
+ing infrastructure. Examples of large-scale use cases include
+include content delivery networks (CDNs), in-memory caches,
+and storage systems. CDNs protect expensive and scarce In-
+ternet backbone bandwidth and are expected to serve 72%
+of Internet traffic by 2022 [16]. In-memory caches protect
+computationally expensive services are extensively used in
+the data centers of Facebook [32] and Twitter [42]. Storage
+caches reduce the data movement of large objects in the net-
+work and an essential part of cloud services [22].
+Caching systems seek to minimize their miss ratio, i.e.,
+the fraction of requests not served by the cache. The lower
+the miss ratios, the lower the load on backend servers and
+Internet traffic (for CDNs). To decide which objects to keep
+in the cache, current caching systems [3, 6, 12, 32] rely on
+heuristic algorithms, such as Least Recently Used (LRU), and
+First In First out (FIFO), and Least Frequently Used (LFU).
+Recent work [8, 13, 35, 38, 40] shows that machine learning
+based eviction algorithms (ML-based caching systems) signif-
+icantly outperform these heuristics by using a history of past
+access patterns to predict future access patterns. These accu-
+rate predictions reduce miss ratios by up to 25% compared to
+heuristic caches [35].
+Bringing ML-based caching systems from research to pro-
+duction faces a key challenge due to their computational over-
+head and hardware cost. In particular, ML-based caching sys-
+tems are not yet applicable in systems with high throughput
+demands [10, 23] or when CPU resources are scarce due to
+being coloated with other applications [12].
+The overhead of ML-based caches is significantly higher
+than heuristic caching systems for two reasons. First, ML-
+based caching systems need to update the model online fre-
+quently to retrain with more recent access patterns. For ex-
+ample, a state-of-the-art ML-based caching system for CDNs,
+LRB [35], uses all cache requests to generate training entries,
+which leads to a large training data volume and a slow training
+process.
+Second, ML-based caches require running many predic-
+tions to find an object to evict. For example, LRB samples 64
+eviction candidates randomly within the cache to run predic-
+tions. Running 64 predictions per eviction can be slow and
+expensive especially in bursty production systems that can
+face pressure to evict hundreds of thousands of evictions in a
+second due to burst arrivals.
+While the overheads of ML-based caches are known, it is
+less known which of their decisions are actually required to
+improve miss ratios compared to heuristics. An answer to
+this question can guide applying costly ML predictions only
+where they are needed. In fact, when comparing the eviction
+decisions of heuristics to an offline optimal algorithm, we find
+that they evict most of the objects that the optimal algorithm
+evicts, but they sometimes evict objects they should keep. This
+leads to our main insight that heuristic algorithms can serve
+as good filters. The ML algorithm will only run predictions
+on the objects evicted by the heuristic algorithm, instead of
+1
+arXiv:2301.11886v1  [cs.OS]  27 Jan 2023
+
+all objects in the cache. It can dramatically reduce the number
+of predictions without affecting miss ratios.
+We propose an efficient ML-based caching framework, Ma-
+chine learning At the Tail (MAT), which builds on the insight
+that we can effectively pair a heuristic with an ML predic-
+tor. We define the tail as evictions of the heuristic algorithm,
+e.g., the least recently used items in LRU. MAT feeds the tail
+objects into a novel ML predictor. This ML predictor then de-
+cides which objects to keep in the cache and which objects to
+truly evict. This allows MAT to identify good objects to evict
+using only a few predictions because the heuristic’s tail is a
+small subset of objects in the cache. Similarly, it allows MAT
+to focus its training on this small subset of objects. In turn,
+this means MAT does far less computation per eviction than
+prior ML-based caching system. But, because the heuristic’s
+tail contains nearly all the objects that should be evicted, MAT
+can achieve the same miss ratio as state-of-the-art ML-based
+caching systems.
+An additional challenge in many caches is handling scenar-
+ios where computation power becomes scarce during certain
+time intervals such as request load spiking or an increase in
+higher-priority work that is collocated with the cache [12].
+MAT’s design robustly handles these scarce computation sce-
+narios by falling back to the heuristic algorithm when the ML
+model cannot keep up with the request load.
+To compare MAT with a variety of algorithms including
+LRB, the state-of-the-art ML cache, We have also imple-
+mented it in a cache simulator and run experiments with 8
+production workloads from CDNs, in-memory caches, and
+storage caches. Our results show that while achieving compa-
+rable miss ratios, MAT dramatically reduces the overhead for
+ML: it reduces the average number of predictions per evic-
+tion by 31 times (from 63 to 2) and the average prediction
+overhead per eviction including metadata feature building
+overhead by 21 times (from 300us to 9.3us).
+We have implemented MAT in Cachelib [12], which is an
+open-source cache system developed by Facebook. We com-
+pare its performance with the Cachelib instance with LRU al-
+gorithm. Our end-to-end evaluation shows that MAT achieves
+similar request rates to the LRU-based caching system.
+Section 2 elaborates on the motivation of our work. Sec-
+tion 3 details the observation that leads to our design. In
+Section 4 we present the design of our framework MAT. Sec-
+tion 5 describes how MAT is implemented in the simulator
+and in the Cachelib prototype. Section 6 presents an evalu-
+ation of MAT. We cover related work in Section 7 and we
+conclude in Section 8.
+2
+Background and Motivation
+This section covers background on offline caching algorithm,
+heuristic caching algorithms, and ML caching algorithms.
+We will use examples from each group to study the decision
+quality of heuristic algorithms in Section 3.
+2.1
+Optimal Offline Algorithm
+In 1966, Belady proposed the MIN algorithm that evicts a data
+object whose next access occurs furthest in the future [11].
+This algorithm provably optimal for caching equal-sized data
+objects [31] such as cacheline or video chunks. Since knowl-
+edge about future accesses is not typically available in an
+online setting, we call Belady’s MIN algorithm an offline
+optimal or oracle algorithm.
+2.2
+Heuristic Caching Algorithms
+Figure 1: Heuristic cache algorithms that maintain the
+rank of objects in a priority queue.
+The most common class of caching algorithms used in
+production systems are based on heuristics. A heuristic is
+designed to decide which object to evict from a cache to
+admit an object that is currently not cached. Most heuristic
+algorithms form an explicit or implicit ranking of objects in
+the cache. If the cache request is a miss, it inserts the requested
+object into the ranking and evicts the object with the lowest
+rank. If the cache request is a hit, it re-ranks the requested
+object in the cache. Figure 1 shows the typical structure of a
+heuristic caching algorithm.
+A well-known example is Least-Recently-Used (LRU),
+which maintains a queue to implicitly rank objects by their
+most recent access times. LRU inserts the most recently ac-
+cessed object at the head of the queue and evicts the one at
+the tail of the queue.
+The main advantage of heuristic caching algorithms is their
+simplicity and efficiency. For example, LRU can be imple-
+mented with a doubly-linked list as its priority queue, and a
+hash table to speedup the lookup operation. This implemen-
+tation of LRU is the default algorithm in many production
+systems such as Cachelib [12].
+The main drawback of heuristic caching algorithms is that
+they work well for certain workloads or access patterns while
+working poorly for others [35].
+2.3
+ML-Based Caching Algorithms
+Machine learning is changing how caches are designed. An
+ML-based cache trains a model with past access patterns and
+then uses the model to predict which objects in the cache
+2
+
+Object index
+Lookup
+Reinsert
+Requests
+Head 
+Tail
+Insert
+Evict
+Priority queueshould be evicted. Recent studies [13, 35, 41, 44] show that
+ML-based approaches can adapt to different workloads dy-
+namically and can reduce wide area network traffic by around
+20% compared to the state-of-the-art heuristic algorithms.
+Two Key Challenges.
+There are two key challenges with
+ML-based caching systems. The first is the overhead for train-
+ing ML models. Adapting to recent access patterns requires
+training and updating the model frequently. This overhead
+can be significant in space and time as hardware accelerators
+are usually not equipped on caching servers.
+The second is the overhead for making eviction decisions.
+To mimic the optimal offline oracle in a straightforward way,
+the ML-based caching system needs to predict the next access
+times of all objects in the cache and evicts the one with the
+furthest time in the future. The prediction overhead would
+be prohibitively high for large caches. Therefore, ML-based
+caching systems for software caching have to decide which
+subset of objects to run predictions on.
+Figure 2: The state-of-the-art ML-based caching system,
+Learning Relaxed Belady (LRB).
+ML-Based Caching by Sampling.
+LRB [35] is the state-
+of-the-art ML-based caching system and it is based on sam-
+pling for both training and for eviction selection. Figure 2
+shows how it trains the ML model and uses the ML model to
+make evictions.
+LRB overcomes the first challenge by using a Gradient
+Boosted Decision Tree (GBDT), a relatively simple ML ap-
+proach. It trains and updates the GBDT model online with
+a relatively small training dataset (about 128K objects) ran-
+domly sampled from a window of recent request history. The
+training overhead is small enough to update the model every
+few seconds.
+It overcomes the second challenge by relaxing the eviction
+criteria of the Belady’s MIN algorithm. Instead of finding
+the object in the cache whose next access time occurs the
+furthest in the future, it picks any object whose predicted next
+access time is far enough. With this approximation, LRB ap-
+proach runs predictions on only k randomly sampled objects
+in the cache, where k is set to 64. When k = 64, with a high
+probability, LRB can find at least one object whose predicted
+next access time is close to the largest next access time in the
+cache.
+LRB achieved better miss ratios than 14 state-of-the-art
+heuristic caching algorithms over various cache sizes with 6
+production CDN workloads. Since LRB is designed for CDN
+workloads whose objects are quite large and requests rates
+are low, it can afford some computational overhead. However,
+we find that LRB requires more than two orders of magni-
+tude more CPU resources than heuristic algorithms. We seek
+to reduce this overhead to enable us to deploy ML-based
+algorithms in high-throughput environments, application-
+embedded environments, or low-power environments such
+as the Internet edge.
+Insertion-time ML caching algorithms.
+Another cate-
+gory of ML caching algorithms runs a prediction on each
+object as its request arrives [13]. It maintains a data structure
+that remembers the ranking of the predicted scores of all ob-
+jects in the cache and chooses the lowest ranked object for an
+eviction.
+The prediction overhead of this method is significant for
+two main reasons. First, the number of requests can be larger
+than the number of evictions by an order of magnitude, de-
+pending on cache miss ratios. Second, after updating the ML
+model with new training data, the previous predictions are not
+consistent with the new model. It needs to rerun predictions
+of all objects in the cache to make the ranking up-to-date and
+consistent [44]. If the frequency of retraining the model is
+high, the total cost for predictions can be extremely high.
+ML-based caching systems make better eviction decisions
+and thus they can save cost on storage media and network
+bandwidth. However, ML-based caching systems require a lot
+more computation than heuristic algorithms. In high through-
+put caching systems, the available computation power is not
+enough for the ML-based caching systems to keep up with the
+line speed. In other cases, it is also highly preferred to reduce
+the computational overhead of ML-based caching systems
+to save up power budget for other applications such as edge
+computing. Thus, our goal is to have as good eviction deci-
+sions as the state-of-the-art ML-based caching systems, while
+have orders of magnitude lower computational overhead.
+3
+Heuristic Algorithms as Filters
+The key idea in this paper is to use a heuristic caching al-
+gorithm as a filter in front of an ML-based caching system
+to reduce the predictions per eviction and the samples for
+training an ML model. The question is how good heuristic
+algorithms are as such filters.
+3
+
+Label
+Requests
+Training
+Candidates
+LRB Metadata
+Sample
+Randomly
+Object index
+Model
+Sample
+Randomly-
+Eviction
+Candidates
+Choose one to evictLRU
+FIFO
+LFUDA
+LRUK
+Belady’s MIN
+TTA<T
+TTA>T
+TTA<T
+TTA>T
+TTA<T
+TTA>T
+TTA<T
+TTA>T
+TTA<T
+TTA>T
+CDN1
+55%
+90%
+65%
+90%
+47%
+87%
+363%
+97%
+0%
+100%
+CDN2
+47%
+95%
+57%
+95%
+36%
+96%
+487%
+92%
+0%
+100%
+CDN3
+42%
+94%
+67%
+91%
+23%
+95%
+41%
+96%
+0%
+100%
+Wikipedia
+129%
+94%
+196%
+87%
+86%
+95%
+89%
+93%
+0%
+100%
+Table 1: Fractions of evicted objects whose TTA < T and TTA > T by 5 caching algorithms (LRU, FIFO, LFUDA, LRUK
+and Belady’s MIN) with 4 workloads (CDN1, CDN2, CDN3 and Wikipedia). All fractions are normalized to the total
+number of objects evicted by Belady’s MIN.
+To answer this question, we compare the distribution of
+evicted objects by a heuristic algorithm to Belady’s MIN
+(optimal offline) algorithm. A good filter should pass over
+most good eviction candidates that Belady’s MIN evicts, even
+at the cost of passing over some bad candidates. We will first
+look at LRU algorithm as a filter and then look at several other
+heuristic algorithms.
+Compare LRU to Belady’s MIN
+Figure 3: Time-To-next-Access (TTA) distribution of the
+evicted objects by LRU and Belady’s MIN algorithms.
+The result is collected from the Wikipedia trace with
+256GB cache size.
+Figure 3 compares the distributions of evicted objects by
+LRU and Belady’s MIN for Wikipedia workload with a cache
+size of 256 GB. The figure groups evicted objects according to
+the log of their Time-To-next-Accesses (TTAs) from the time
+when they are evicted by a given algorithm. The time is the
+logical time, which means it is a counter that increments on
+each request by 1. The figure plots the percentage of evictions
+in each group normalized by the total number of evictions of
+the Belady’s MIN algorithm.
+The threshold T in the figure separates good eviction de-
+cisions from bad ones. All evicted objects by Belady’s MIN
+have their TTAs > T (on the right hand side of the threshold),
+none have TTAs < T (on the left hand side of threshold).
+We have two observations about the distributions of LRU.
+First, the total number of evicted objects whose TTAs > T by
+LRU is close to that by Belady’s MIN. This means that LRU
+evicts most of objects that Belady’s MIN does. In other words,
+LRU rarely keeps objects it should evict. This indicates that in
+most cases, LRU does not filter out most of the good eviction
+candidates.
+Second, the number of objects evicted by LRU on the left
+hand side of the threshold is similar to that on the right hand
+side. In other words, although LRU frequently evicts objects
+it should keep, one of every two evictions is a good decision
+on average for this workload. This means that using LRU as a
+filter, we can reduce the number of predictions from 64 to 2!
+Compare Other Heuristics to Belady’s MIN.
+Table 1 shows the distributions of good and bad eviction
+decisions with four workloads by LRU, FIFO, LFUDA, LRUK
+and Belady’s MIN algorithms. As Belady’s MIN is an optimal
+offline algorithm, 100% of its evicted objects have TTA > T.
+The main observation is that all heuristic algorithms in the
+table can serve as filters well. They can evict 87-97% of the
+objects (TTA > T) Belady’s MIN evicts.
+Among these heuristic algorithms as filters, LRU and
+LFUDA are better overall. LRU evicts 90%, 95%, 94%, and
+94% of the objects that Belady’s MIN evicts with CD1, CDN2,
+DDN3 and Wiki workloads respectively. LFUDA evicts 87%,
+96%, 95%, and 95% of those that Belady’s MIN evicts respec-
+tively. LFUDA has the smallest fractions (23-86%) of evicted
+objects whose TTA < T.
+LRUK achieves the highest coverage (92-97%) of the ob-
+jects that Belady’s MIN evicts, but it has large fractions of bad
+eviction decisions with CDN1 and CDN2 worklaods (363%
+and 487% respectively).
+FIFO can also be a good filter but it is slightly worse for
+Wiki workload, evicting 87% of the objects that Belady’s
+MIN evicts.
+Filtering Training Samples
+Using a heuristic caching algorithm as a filter can deliver bet-
+ter training samples than random sampling. Our key insight is
+that since the ML-based caching algorithm takes objects from
+the tail of the heuristic algorithm as eviction candidates, the
+ML model needs to learn only from such historical candidates
+to make better predictions.
+4
+
+Threshold T
+Fraction of evicted objects
+50%
+Should not evict 
+Should evict
+40%
+LRU
+■Belady
+30%
+20%
+10%
+0%
+¥1112　131415　16　17　18　1920 ≥21
+≤10
+log(Time-To-next-Access)Figure 4: The architecture and the data flow of the MAT system. The ML module makes predictions on objects removed
+from the tail of the heuristic cache and decides to insert the objects back or evict the objects.
+4
+Machine Learning at the Tail
+This section describes our approach called Machine Learning
+at the Tail (MAT). Section 4.1 describes the two components
+of MAT, which are the heuristic cache system and the ML
+module, and their interfaces. Section 4.2 describes and dis-
+cusses how MAT uses a dynamic threshold to decide which
+object should be evicted. Section 4.3 introduces an imple-
+mentation of the machine learning method of MAT. Finally,
+Section 4.3 describes how training data are generated in MAT.
+4.1
+Architecture and Algorithm
+Figure 4 shows the architecture of MAT, consisting of two
+main modules: a heuristic cache and an ML module.
+Heuristic cache.
+It is a traditional cache using a priority-
+queue based heuristic algorithm. It needs to provide two calls
+to interface with the ML module:
+• RemoveFromTail(): removes an object from the the tail
+of the priority queue(s) of the heuristic algorithm and
+return the object to the ML model.
+• Insert(x, rank): insert object x back to a priority queue
+of the heuristic algorithm. The ML model can inform
+the heuristic algorithm to insert to a specific position of
+the queue by providing the rank.
+In the case that the heuristic algorithm uses multiple queues,
+such as Segmented LRU, 2Q, and TinyLFU, the two proce-
+dures need to pick one of the queues.
+The MAT framework is general since most heuristic al-
+gorithms use priority queue(s). We have implemented and
+experimented MAT with LRU, 2Q [25] and TinyLFU [19]
+algorithms. The two calls are simple to implement.
+ML Module.
+In the MAT design, it consists of a training
+pipeline and a prediction (or inference) pipeline. The predic-
+tion pipeline implements Evict() which returns an object for
+eviction.
+The main data structure in the training pipeline is a training
+dataset, which is the recent historical candidates from the
+candidate queue. The training thread uses the dataset to train
+a model and update the current model with the newly trained
+model.
+Two kinds of threads are used in the prediction pipeline:
+ML threads and eviction thread, connected by two queues:
+candidate queue and eviction queue.
+An ML thread removes a batch of candidate objects from
+the candidate queue, predicts the time-to-next-access (TTA)
+of each object in a way similar to that of LRB [35]. If the
+TTA is greater than threshold T, the object will be put on the
+eviction queue.
+The eviction thread is responsible for removing ob-
+jects from the tail of the heuristic algorithm (by calling
+RemoveFromTail()) and putting them in the candidate queue,
+and insert them back into the priority queue(s) (by calling
+Insert()).
+When the cache system needs to evict an object from the
+cache, it will call Evict() which will remove an object from
+the eviction queue and return it as the object for eviction. If
+the eviction queue is empty, Evict() will remove an object
+at the tail of priority queue(s) of the heuristic algorithm. In
+either case, the cache system will evict the returned object
+from the cache and also removes it from its related priority
+queue.
+The main advantage to allow the cache system to go ahead
+when the eviction queue is empty is that the cache system can
+run at a speed (or throughput) similar to the heuristic cache
+system with a ML module. In this case, eviction decisions
+can fall back to the heuristic algorithm. The cache system
+continues functioning well when the ML thread is slow or
+even fails.
+5
+
+Requests
+Training
+Training
+data
+thread
+Cache
+Heuristic cache
+Candidate queue
+algorithm gueue
+RemoveFromTail()
+Eviction
+ML
+Model
+thread
+thread
+Insert(), Evict()
+Eviction queue
+Heuristic cache system
+ML module4.2
+Eviction Decision
+Algorithm 1: MAT Eviction
+Input: The expected number of predictions per
+eviction k; The TTA threshold T;
+r := 1;
+while r ≤ L do
+obj := Heuristic.RemoveFromTail();
+TTA := MLModel.Predict(obj);
+if TTA ≥ T then
+break;
+else
+r += 1;
+Heuristic.Insert(obj, TTA);
+end
+end
+if r > k then
+T *= (1-δ);
+end
+if r < k then
+T *= (1+δ);
+end
+return obj;
+As shown in Algorithm 1, the ML thread evaluates eviction
+candidates one at a time by predicting its TTA and compares
+it with a threshold T. If the TTA of object x is ≥ T, object x
+will be put on the eviction queue.
+If the number of iterations reaches a limit L, it means none
+of the L candidates satisfies TTA ≥ T. In this case, we will
+choose the one with the largest TTA for eviction, putting it
+on the eviction queue. Our system uses L = 10 to bound the
+maximal cost for an eviction decision.
+The threshold T is dynamically adjusted to achieve a target
+average number of predictions per eviction, denoted as k. If it
+takes fewer than k iterations to find an object whose TTA ≥ T,
+we will increase the threshold slightly T = (1+δ)T. If it takes
+more than k iterations, we will decrease the threshold slightly
+T = (1−δ)T.
+The rationale to adjust the threshold T dynamically is to
+tolerate variations of the workloads as request distributions
+change over time. We find that there is no optimal constant
+value of T for an entire workload. Our system uses δ = 1e−4
+as the default.
+What happen to the objects inserted back to the priority
+queue? These are the objects that the heuristic algorithm
+would like to evict, but the ML model disagrees. When an
+object is inserted back into the priority queue, it may take a
+while for it to reach the tail of the queue again. By then, its
+metadata (e.g., the time since last access) has changed. The
+next time it becomes a candidate, the ML model might choose
+to evict it.
+40
+60
+80
+100
+Number of Requests (Million)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+TTA Threshold T
+1e8
+Wikipedia
+(a) Optimal threshold
+2
+4
+8
+16
+32
+64
+# predictions per eviction (k)
+0
+10
+20
+30
+Miss ratio reduction of 
+ Threshold over Top-1 (%)
+CDN1
+CDN2
+CDN3
+Wikipedia
+(b) Threshold vs. Top-1
+Figure 5: Why using dynamic threshold to select object
+to evict. (a) The best TTA threshold is continuously changing.
+(b) The dynamic threshold method reduces up to 30% misses
+of the Top-1 method.
+Why does MAT uses threshold T to make eviction deci-
+sions? An alternative is to choose the object with the Top-1
+TTA among k eviction candidates from the heuristic algo-
+rithm. As shown in Figure 5, there are two advantages of us-
+ing threshold T over the Top-1 method. First, in the case that
+the heuristic algorithm continuously sends good candidates
+with TTA ≥ T, each will be selected by our threshold method,
+whereas the Top-1 method will miss k −1 good candidates.
+Second, in the situation that the heuristic algorithm contin-
+uously sends bad candidates with TTA (< T), our threshold
+method choose the one with the largest TTA among L can-
+didates, whereas the Top-1 method chooses the the best one
+among k candidates. Since k ≪ L, the Top-1 method may
+choose many bad candidates.
+4.3
+Machine Learning Methods
+MAT is a general framework that can run with any supervised
+ML module. Here, we will introduce one ML method as an
+example. While this is our implementation of MAT, many
+other implementations can also work well.
+Machine learning models.
+The cache request stream can
+be viewed as time-series data. To predict the TTA of an object,
+we are interested in the past access pattern of the object and
+also in the context of other objects. The access pattern of this
+object can be viewed as low dimension tabular data, while the
+context is sequence data.
+We have experimented with several simple and efficient
+ML approaches including Linear Regression (LR), Gradi-
+ent Boosted Decision Trees (GBDT), Multi-layer Perceptron
+(MLP), and Recursive Neural Network (RNN). By default,
+MAT uses the GDBT model as it has a better trade-off be-
+tween accuracy and computational overhead.
+Metadata of input objects
+The ML module in the MAT-
+based caching system needs to learn from the past access
+patterns to perform predictions in order to make the eviction
+6
+
+decisions to minimize cache miss ratios. We call such data
+the metadata for ML and they vary depending on the choice
+of ML models.
+To study MAT, we use metadata similar to those in
+LRB [35]. The metadata keeps track of three clusters of fea-
+tures for each object:
+• Deltai is the interval between the ith and the i+1th most
+recent accesses (e.g., delta1 is the interval between the
+most recent and second most recent accesses).
+• Exponential Decayed Counter (EDC) is a counter that is
+incremented on each access and is halved after a certain
+time. EDCi is halved after each 2i requests in the cache.
+• Each object also has static features that are not related
+to access, such as object size, object class, etc.
+By default, we maintain 32 deltas, 10 EDCs and 2 static fea-
+tures and the size of the metadata for each object is at most
+192 Bytes. Objects with fewer past accesses will take less
+space.
+4.4
+Training Dataset
+In the MAT framework, a training dataset is a set of recent
+historical eviction candidates, as opposed to all objects.
+The training data generation involves a tagging phase
+and a labelling phase. When an object is selected as
+an eviction candidate by the heuristic algorithm (calling
+RemoveFromTail()), it is tagged as eligible for training.
+When an object is requested, if it is tagged, the tag is re-
+set and MAT calculates the true label of TTA with regard to
+its last access. Then the object features and the label are in-
+serted into the training batch. Once the training batch reaches
+a predefined batch size (e.g., 1 million ), it is used to retrain a
+ML model online and then replaces the current model.
+This method uses the heuristic algorithm to filter out ob-
+jects that are not heuristically determined eviction candidates.
+The intuition is that they are not relevant to predictions, so
+excluding them will not affect the learning of the ML model.
+In fact, it reduces the noise in the training data and can im-
+prove the accuracy of the ML model. Depending on the miss
+ratio, the tagged objects can usually be 10% to 50% of all the
+objects, so the amount of the training data can be reduced by
+50% to 90%.
+4.5
+Time-To-next-Access Prediction
+The main goal of using the ML model is to predict the Time-
+To-next-Access (TTA) of a given object. The inference opera-
+tion with the ML model outputs the predicted distance to the
+next access, which is defined as the difference between the
+timestamps of the last access and the next access of an object.
+TTA is calculated as this predicted distance minus the time
+passed since the last access.
+In the case that the predicted distance to the next access is
+shorter than the time passed since the last access, we use the
+time passed since the last access minus the predicted distance
+to the next access as an estimation of TTA.
+The intuition is that when the time passed since the last
+access is only slightly larger than the predicted distance to
+the next access, we still have confidence in the ML prediction
+and we estimate the TTA to be small.
+However, if the object still does not come and the time
+past since last access becomes much larger than the predicted
+distance to the next access, we want to stop keeping this object
+in the cache.
+5
+Implementations
+5.1
+Optimizations
+Batched predictions.
+The basic MAT makes one eviction
+decision at a time. To take advantage of modern processors,
+MAT can exploit the data parallelism for eviction decisions.
+This approach runs parallel predictions on B objects before
+the tail of the heuristic algorithm. The predicted TTAs are
+recorded in the metadata. When the ML model receives an
+eviction candidate from the heuristic algorithm, it can directly
+use the recorded TTA for making an eviction decision. If there
+is no recorded TTA, it will initiate a new parallel prediction
+task.
+The batch size B has influence on the parallelism and the
+miss ratio. If B is too small, the parallelism is not fully realized.
+If B is too large, the prediction results are stall and the miss
+ratio will be hurt. In our design, we use B = 64.
+5.2
+Prototype
+We have implemented a MAT prototype in Cachelib [12],
+which is an open-source C++ caching library. Our implemen-
+tation adds about 1,000 lines of code, with about 900 for MAT
+itself and about 100 lines for integrating it into Cachelib. We
+use LightGBM [27] to implement Gradient Boosted Decision
+Trees. The LightGBM model has 32 trees and each tree has
+no more than 32 leaves. The bagging frequency is 5 and the
+bagging fraction is 0.8. The learning rate is 0.1.
+5.3
+Simulators
+To compare MAT to LRB, we also integrated MAT-LRU into
+LRB’s simulation framework [2], The simulator measures
+the miss ratios of caching algorithms by replaying all cache
+requests in the traces. It only maintains metadata of objects
+and does not allocate physical space for the objects.
+The advantages are that it can simulate cache sizes much
+larger than the memory on the simulation machine and the sys-
+tem is always bottlenecked on the caching algorithm so that
+we can measure the running time of the caching algorithms.
+7
+
+Trace
+Type
+Object Size
+Number of Requests
+Requested Bytes
+Default
+Cache Size
+Mean
+Max
+Total
+Unique
+Total
+Unique
+CDN1
+CDN
+2 MB
+2 MB
+300 M
+31 M
+585 TB
+60 TB
+4 TB
+CDN2
+CDN
+2 MB
+2 MB
+220 M
+19 M
+430 TB
+38 TB
+4 TB
+CDN3
+CDN
+451 KB
+1 GB
+200 M
+22 M
+72 TB
+9.5 TB
+4 TB
+Wikipedia [7]
+CDN
+116 KB
+1.3 GB
+200 M
+15 M
+7.9 TB
+1.7 TB
+256 GB
+Memcachier [4]
+In-memory
+4.6 KB
+1 MB
+500 M
+9 M
+1 TB
+40 GB
+1 GB
+InMem
+In-memory
+337 B
+400 KB
+500 M
+62 M
+159 GB
+19 GB
+8 GB
+IBM merged [1]
+Storage
+3.1 M
+4 MB
+500 M
+30 M
+1832 TB
+89 TB
+16 TB
+Microsoft [5]
+Storage
+445 KB
+6 MB
+200 M
+48 M
+5.1 TB
+2 TB
+512 GB
+Table 2: Overview of the traces used for evaluation.
+We mainly use the simulator to perform an apple-to-apple
+comparison between MAT and LRB. LRB is only available in
+the simulator because it is non-trivial to implement a bug-free
+LRB in the Cachelib prototype. The results in Section 6.2 and
+6.3 are collected from the simulator.
+6
+Evaluation
+Our evaluation answers the following questions:
+• How many predictions does MAT need for each eviction
+decision to achieve comparable miss ratios to SOTA?
+• What is the software overhead of MAT?
+• What performance can MAT prototype system achieve?
+• Is MAT sensitive to heuristic algorithm choices?
+• How well can MAT tolerate slow ML predictions?
+In the following, we will first describe our experimental setup,
+implementations, and experimental results to answer these
+questions.
+6.1
+Experimental Setup
+Two hardware settings are used in our experiments. All simu-
+lation experiments are run on servers in Cloudlab [18], each
+with two 2 GHz Intel E5-2683v3 CPUs (14 physical cores)
+and 256 GB of RAM.
+Prototype experiments are run on two servers in Microsoft
+Azure cloud, each with a 2.4 GHz AMD EPYC 7763 CPU (48
+cores) and 378 GB of RAM. The two servers are connected
+to 40 Gbps local area network.
+Workloads.
+We use 4 CDN traces and 4 other workloads
+in our experiments. Table 2 shows the characteristics of these
+workloads. In addition,
+• CDN1, CDN2 are collected from different caching servers
+(located in different regions) of same anonymous service
+provider.
+• CDN3 is collected from the caching server of another video
+service provider.
+• Wikipedia trace is from wikipedia servers.
+• Memcachier is from a in-memory application cache.
+• InMem is an anonymous trace collected from an in-memory
+key value store of social media company.
+• IBM merged is a combined workload of 99 traces from IBM
+object store, which is a cloud storage service. We merge the
+traces based on request timestamps.
+• Microsoft is a storage trace from Microsoft.
+Warmup.
+The first 50 million requests of each trace are
+used as a warm-up period. The byte miss ratios and through-
+put are measured after the warmup period.
+We measure 3 aspects of algorithms.
+• Byte Miss Ratio: This metric is the total size of the objects
+that are not present in the cache when requested divided by
+the total size of the objects of all requests. This metric is an
+indicator of the network traffic volume, which is the main
+optimization goal for CDN caching.
+• Request Processing Rate: This metric measures the number
+of requests processed by the caching system per second. It
+is greatly influenced by the object size in the workload.
+• Software Overhead: Software overhead refers to the extra
+computational overhead introduced by the ML algorithms.
+We measure the normalized running time of each part of
+the machine learning pipeline, including trainings, predic-
+tions, and building features. The running time is normalized
+by the number of evictions. In addition, we measure the
+number of predictions and the number of training entries
+incurred by the ML based caching algorithm for each evic-
+tion.
+8
+
+256
+512
+1024
+2048
+4096
+Cache Size (MB)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Byte Miss Ratio
+MIN*
+LRB(64)
+LRU
+MAT(2)
+MAT(3)
+MAT(4)
+1
+2
+4
+8
+16
+Cache Size (TB)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Byte Miss Ratio
+(a) CDN1
+1
+2
+4
+8
+16
+Cache Size (TB)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Byte Miss Ratio
+(b) CDN2
+1
+2
+4
+8
+16
+Cache Size (TB)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Byte Miss Ratio
+(c) CDN3
+64
+128
+256
+512
+1024
+Cache Size (GB)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Byte Miss Ratio
+(d) Wikipedia
+256
+512
+1024
+2048
+4096
+Cache Size (MB)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Byte Miss Ratio
+(e) Memcachier
+2
+4
+8
+16
+32
+Cache Size (GB)
+0.00
+0.05
+0.10
+0.15
+Byte Miss Ratio
+(f) InMem
+4
+8
+16
+32
+64
+Cache Size (TB)
+0.0
+0.2
+0.4
+0.6
+Byte Miss Ratio
+(g) IBM merged
+128
+256
+512
+1024
+2048
+Cache Size (GB)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Byte Miss Ratio
+(h) Microsoft
+Figure 6: Miss ratio comparisons in the simulator. The algorithm names show the number of predictions per eviction. MAT
+has better or similar miss ratios compared to LRB while MAT has an order of magnitude fewer number of predictions.
+6.2
+Predictions per Eviction
+To answer the first question, we would like to experimen-
+tally evaluate how many predictions MAT needs to make
+an eviction decision, while achieving similar miss ratios to
+the LRB [35] with various workloads. We conduct our ex-
+periments using the simulator with 8 workloads as shown in
+Figure 6. Our experiments compare 3 MAT cases (2, 3, and 4
+predictions per eviction) with LRB which runs 64 predictions
+per eviction (default).
+The results show that MAT reduces the number of predic-
+tions by 31 times compared to LRB without degrading the
+miss ratios. MAT with 2, 3, and 4 predictions per eviction
+have similar miss ratios to LRB with 64 predictions.
+The differences among the 4 cases are small. Figure 6a-
+6d are the results on the 4 CDN traces and the miss ratios of
+MAT(2), MAT(3), and MAT(4) are almost identical. MAT(2)’s
+miss ratios are slightly better than LRB(64)’s miss ratios on
+CDN1 and Wikipedia, while LRB(64) is slightly better on
+CDN1 and CDN2. Figure 6e, 6f are for in-memory traces.
+Compared to LRB(64), MAT(2) has 1% and 5% average rela-
+tive reductions in the miss ratios on Memcachier and InMem,
+respectively. Figure 6g, 6h show the results on storage traces.
+MAT(2) has in average 1% relatively lower miss ratio than
+LRB(64) on IBM merged. LRB(64) has in average 2% rela-
+tively lower miss ratio than MAT(2) on Microsoft workload.
+All ML algorithms achieve significant improvements over
+the LRU approach. For instance, on CDN1 with a 2 TB cache
+MAT has a 18% miss ratio compared to LRU’s 21%, which
+would reduce wide-area traffic by (21%-18%)/21%=14%.
+When the cache sizes are large, the differences among them
+diminish.
+However, there is still a significant gap between these algo-
+rithms and the the optimal offline algorithm. As MAT frame-
+work can reduce the number of predictions to 2, it allows
+the community to explore more sophisticated ML models to
+reduce this gap.
+6.3
+Software Overhead
+To see how much overhead MAT can reduce compared to
+LRB with similar ML modules, we run both in the same
+simulation environment with 256 GB cache size.
+Table 3 shows the average prediction overhead per eviction.
+The average prediction overhead including the feature build-
+ing time per eviction is reduced by reduction is 32X (from
+300us to 9.3us).
+LRB
+MAT
+Reduction
+Number of predictions
+63.2
+2.0
+32 times
+Prediction time (us)
+240
+6.4
+38 times
+Feature building time (us)
+60
+2.9
+21 times
+Total time (us)
+300
+9.3
+32 times
+Table 3: Average overhead per eviction (256 GB cache
+size, Wikipedia workload).
+9
+
+128
+256
+512
+1024
+2048
+Cache Size (GB)
+0.2
+0.3
+0.4
+Object Miss Ratio
+LRU
+MAT-LRU
+2Q
+MAT-2Q
+Tiny
+MAT-Tiny
+128
+256
+512
+1024
+2048
+Cache Size (GB)
+0.2
+0.3
+0.4
+Byte Miss Ratio
+(a) CDN1
+128
+256
+512
+1024
+2048
+Cache Size (GB)
+0.20
+0.25
+0.30
+0.35
+Byte Miss Ratio
+(b) CDN2
+128
+256
+512
+1024
+2048
+Cache Size (GB)
+0.2
+0.3
+0.4
+Byte Miss Ratio
+(c) CDN3
+8
+16
+32
+64
+128
+256
+Cache Size (GB)
+0.4
+0.5
+0.6
+0.7
+0.8
+Byte Miss Ratio
+(d) Wikipedia
+Figure 7: The byte miss ratios of MAT with LRU, 2Q, and TinyLFU as the base algorithms. MAT in average reduce the
+byte miss ratio of LRU, 2Q, and TinyLFU by relatively 12%, 12%, and 10% respectively.
+MAT reduces the average number of predictions by 31X
+(from 63.2 to 2) while reducing the average feature building
+time by 21X (from 60us to 2.9us). Feature building refers to
+converting the object metadata to the feature matrix which
+serves as the input of the ML model, for both training and
+predictions.
+Table 4 shows the average overheads per eviction of LRB
+and MAT. The average training time reduction of MAT is
+9.5X (from 160us to 16.9us).
+LRU
+MAT
+Reduction
+Number of training samples
+8.6
+0.93
+9 times
+Training time (us)
+100
+14
+7 times
+Table 4: Average training overhead per eviction ( 256 GB
+cache size, Wikipedia workload).
+The average number of training samples per eviction is
+reduced by 9.2X (from 8.6 to 0.93). The average training time
+overhead is reduced by 7X (from 100us to 14us).
+In summary, the total overhead of MAT’s ML module is
+23.3 us per eviction, 17X reduction over LRB.
+6.4
+Prototype Performance
+To evaluate the performance of MAT prototype, we compare
+its performance with Cachelib with LRU algorithm. We are
+interested in the request processing rate of MAT prototype
+compared to that of LRU-based caching system. Since both
+MAT and LRU are implemented in Cachelib. We conduct
+experiments in the same setting.
+To run such experiments, we extended the Cachebench
+module in the Cachelib library to support running experi-
+ments over a network. We implement a Cachebench client
+instance, a Cachebench server instance, and an Nginx server
+instance. The Cachebench client instance reads requests from
+a trace file and sends them through HTTP to the Nginx server
+using CURL. The Nginx server accepts requests and forwards
+them to the Cachebench server instance using Fastcgi. The
+Cachebench server runs either prototype which executes the
+requests and returns the results using the Fastcgi API to the
+Nginx server. The Nginx server then forwards the requests
+back to the Cachebench client.
+Trace
+Cache Size
+MAT
+LRU
+Wikipedia
+32 GB
+23787 req/s
+24465 req/s
+128 GB
+25117 req/s
+25577 req/s
+512 GB
+30258 req/s
+31181 req/s
+Memcachier
+Infinite
+52883 req/s
+51380 req/s
+Table 5: Request rates of MAT and LRU prototypes with
+Wikipedia and Memcachier workloads.
+The main result is that MAT prototype achieves similar
+request rates as LRU prototype. Table 5 shows the request
+rates of MAT and LRU with Wikipedia and Memcacheir work-
+loads.
+For Wikipedia workload whose average object size is
+116KB, MAT achieves 23,878, 25,117, and 30,258 re-
+quests/sec with cache sizes of 32GB, 128GB, and 512GB
+respectively. It is 2.8%, 1.8%, and 3.0% slower than LRU.
+The these results include the warmup period. Without the
+warmup, we expect MAT will achieve higher request rates
+than above since its miss ratio is substantially lower than
+LRU.
+The reason for using Memcachier workload is to test maxi-
+mal request rates as its average object size is relatively small
+(4.6KB). MAT achieves 52,883 requests/sec whereas LRU
+achieves 51,380 requests/sec. In this case, MAT is 2.9% faster.
+6.5
+Heuristic Algorithm Choices
+To understand the effects of different heuristic algorithm
+choices, we compare three heuristic algorithms (LRU, 2Q
+and TinyLFU) in Cachelib with their corresponding MAT im-
+plementations (MAT-LRU, MAT-2Q and MAT-TinyLFU) as
+10
+
+shown in Figure 7. We conduct experiments with 4 workloads:
+CDN1, CDN2, CDN3 and Wikipedia.
+The experiments show two results. First, MAT is not sensi-
+tive to heuristic algorithm choices. MAT-LRU, MAT-2Q and
+MAT-TinyLFU achieve similar miss ratios with all 4 work-
+loads. MAT-TinyLFU achieves slightly worse miss ratios than
+MAT-LRU and MAT-2Q with CDN2 workload.
+Second, MAT can correct the eviction mistakes by its
+heuristic algorithm. Dramatic examples are TinyLFU with
+CDN1 and CDN2. In both cases, TinyLFU has much higher
+miss ratios than LRU and 2Q. However, MAT-TinyLFU can
+achieve miss ratios similar to those of MAT-LRU and MAT-
+2Q. The ML model in MAT can effectively make good evic-
+tion decisions, adapting to different request patterns.
+6.6
+Tolerance to Slow ML Predictions
+256
+1024
+Cache Size (MB)
+0.00
+0.05
+0.10
+0.15
+0.20
+Reduction in Miss Ratio (%)
+MAT(0us)
+MAT(1us)
+MAT(10us)
+MAT(100us)
+MAT(1ms)
+LRU
+Figure 8: MAT with slow Predictions. When the ML model
+is slower, the miss ratio degenerates gracefully.
+MAT has the ability to fall back to its heuristic algorithm
+when its ML threads cannot keep up with the rate of evictions.
+To answer the question how well MAT can tolerate slow ML
+predictions, we conduct experiments on CDN1 by stalling
+each prediction for 1 us, 10 us, 100 us, and 1 ms. Note that a
+prediction itself takes 3 us on average.
+Figure 8 shows that the miss ratio reduction of MAT us-
+ing LRU as the baseline. MAT can always deliver better or
+equal miss ratio to LRU and MAT degrades gracefully when
+the ML model stalls. When the stall is 10 us, which means
+the ML model runs at 25% of its full speed, the miss ratio
+reduction is about 40% of the full speed ML model. This
+makes MAT particularly practical for deployment because it
+can run elastically with any available amount of computation
+resources.
+7
+Related Work
+This section discusses related work on learning-based cache
+algorithms and heuristic cache algorithms. We also review
+systems targeting other aspects of CDN cache system.
+Learning-based cache replacement
+Existing works on ap-
+plying machine learning cache algorithms can be categorized
+into supervised learning and reinforcement learning. Most
+supervised learning approaches use regression. LRB [35] is
+the most similar approach to MAT. As we have seen in Sec-
+tion 6, LRB processing time overhead is 17× higher than
+MAT. LFO [13] uses boosting tress to predict and imitate the
+admission policy of OPT. It requires complex offline training,
+and thus is not practical to adapt quickly to workload changes.
+LFO also performed poorly in prior experiments [35]. Similar
+to regression on a single value, [44] learns the next access
+distribution, and uses it to compute a utility. However, to get
+the distribution training data, it is also limited to offline train-
+ing. Different from regression, Parrot [29] takes a ranking
+approach. Instead of predicting the time to next access, it di-
+rectly learns to rank objects. This approach is more end-to-end
+but the computation overhead is much higher than regression.
+Another line of works [17, 28, 38] apply reinforcement
+learning to cache algorithms. Instead of predicting the time
+to next access, the eviction policy is directly modeled and op-
+timized. Unfortunately, feedback loops in production systems
+would be in the tens of millions of steps, which exceeds the
+capability of current reinforcement learning frameworks. Con-
+sequently, the performance of reinforcement-learning-based
+approaches is worse than supervised learning.
+MAT is the first learning-based algorithm that can provide
+both a low miss ratio and high throughput. This is because its
+design provides efficient online training and inference.
+Heuristic-based cache replacement
+Over the past five
+decades, many cache algorithms based on heuristics have
+been proposed. Prominent examples include LRU, FIFO,
+SLRU [26], 2Q [25], LIRS [24], ARC [33], LeCaR [37], and
+LHD [10]. MAT is agnostic to most base heuristic algorithms
+and can use any priority-queue based heuristic as its base
+algorithm.
+Learning-augmented
+cache
+replacement
+Several
+learning-augmented replacement algorithms have been pro-
+posed to combine heuristic with learning-based algorithms.
+To the best of our knowledge, previous works [9, 30, 34, 39]
+focus on theoretical analysis on the competitive ratio of
+learning-augmented cache, or only evaluate the cache miss
+ratio regardless of the overhead [15]. MAT is the first
+learning-augmented cache that achieves a low byte miss
+ratio and high throughput on production cache software and
+workloads.
+Cache admission policies
+Multiple systems focus on learn-
+ing smart cache admission policies while relying on exist-
+ing cache replacement heuristics such as TinyLFU [19, 20],
+AdaptSize [14], Flashield [21], and CacheLib [12]. While
+there is no public implementation of Flashield, our evaluation
+11
+
+of TinyLFU, AdaptSize, and CacheLib shows that admission
+policies are not sufficient to achieve state-of-the-art miss ra-
+tios. LRB outperforms both systems’ miss ratio, and MAT
+achieves similar miss ratios to LRB while significantly im-
+proving throughput and reducing overhead.
+High-throughput
+in-memory
+caching
+systems
+MemC3 [23] and LHD [10] show how to significantly
+increase the throughput of in-memory caching systems based
+on replacement heuristics. The throughput challenges faced
+by a ML-based caching systems are different from the setting
+in MemC3. Some of MemC3’s techniques, such as faster
+hashing and fast approximations for a heuristic like LRU,
+are complementary to MAT. In fact, MemC3’s replacement
+heuristic can be used to create candidates for MAT’s eviction
+filter. LHD’s primary design based on sampling is similar to
+LRB. MAT effectively overcomes the challenges of sampling
+which requires too many evaluations and calls to a prediction
+model. SegCache [43] is designed for small objects with
+TTLs. It groups objects with similar TTLs together to reduce
+memory fragmentation and thus improves the hit ratios.
+CDN cache systems
+Many works optimize CDN cache sys-
+tems from other aspects. RIPQ [36] co-locates small writes to
+reduce SSD write amplification. AViC [8] designs the eviction
+algorithm based on the video chunk sequential accessed at a
+constant speed and leverages the properties of video delivery
+to optimize the hit ratio. The design of MAT is flexible for
+general cache and can be applied together with these systems.
+8
+Conclusion
+MAT is proposed as a general framework for building an
+efficient ML-based caching system by adding an ML module
+to an existing cache system based on a heuristic algorithm.
+The key idea behind MAT is to treat a heuristic algorithm
+as a filter for the ML module. Most heuristic algorithms can
+serve as good filters, as we demonstrate that they evict most
+objects an optimal algorithm evicts while evicting some ob-
+jects they should keep. The role of ML module is to correct
+these mistakes.
+We show several simulation and prototyping results. First,
+it reduces the average number of predictions per eviction
+to 2, a 31 times reduction compared to the state-of-the-art
+ML-based caching system while achieving comparable miss
+ratios. Second, MAT is not sensitive to the choice of heuristic
+algorithms. Third, MAT can fall back to the heuristic algo-
+rithm which allows it to run efficiently, tolerating slow ML
+inferences or lack of computing power.
+The ML module used in our implementations is Gradient
+Boosted Decision Tree (GBDT) due to its simplicity and effi-
+ciency. Other ML methods that are less efficient may further
+reduce miss ratios. Since MAT framework dramatically re-
+duces the ML overhead to merely 2 predictions per eviction,
+it enables us to explore some sophisticated ML approaches.
+References
+[1] Ibm object storage.
+http://iotta.snia.org/traces/
+key-value/36305.
+[2] Lrb github repository.
+https://github.com/sunnyszy/
+lrb.
+[3] memcached - a distributed memory object caching system.
+http://memcached.org/, .
+[4] Memcachier. https://www.memcachier.com/, .
+[5] Microsoft.
+http://iotta.snia.org/traces/block-io/
+388.
+[6] redis. https://redis.io/.
+[7] Wikipedia
+trace.
+http://lrb.cs.princeton.edu/
+wiki2019.tr.tar.gz.
+[8] Z. Akhtar, Y. Li, R. Govindan, E. Halepovic, S. Hao, Y. Liu, and
+S. Sen. Avic: a cache for adaptive bitrate video. In Proceedings
+of the 15th International Conference on Emerging Networking
+Experiments And Technologies, pages 305–317, 2019.
+[9] A. Antoniadis, C. Coester, M. Elias, A. Polak, and B. Simon.
+Online metric algorithms with untrusted predictions. In In-
+ternational Conference on Machine Learning, pages 345–355.
+PMLR, 2020.
+[10] N. Beckmann, H. Chen, and A. Cidon. LHD: Improving cache
+hit rate by maximizing hit density. In USENIX NSDI, pages
+389–403, 2018.
+[11] L. A. Belady. A study of replacement algorithms for a virtual-
+storage computer. IBM Systems journal, 5(2):78–101, 1966.
+[12] B. Berg, D. S. Berger, S. McAllister, I. Grosof, S. Gunasekar,
+J. Lu, M. Uhlar, J. Carrig, N. Beckmann, M. Harchol-Balter,
+et al. The cachelib caching engine: Design and experiences
+at scale. In 14th {USENIX} Symposium on Operating Sys-
+tems Design and Implementation ({OSDI} 20), pages 753–768,
+2020.
+[13] D. S. Berger. Towards lightweight and robust machine learning
+for cdn caching. In ACM HotNets, pages 134–140, 2018.
+[14] D. S. Berger, R. Sitaraman, and M. Harchol-Balter. Adapt-
+size: Orchestrating the hot object memory cache in a content
+delivery network. In USENIX NSDI, pages 483–498, 2017.
+[15] J. Chł˛edowski, A. Polak, B. Szabucki, and K. T. ˙Zołna. Robust
+learning-augmented caching: An experimental study. In Inter-
+national Conference on Machine Learning, pages 1920–1930.
+PMLR, 2021.
+[16] CISCO.
+Cisco visual networking index: Forecast and
+trends
+2022, February
+2019.
+Available
+at https:
+//www.cisco.com/c/en/us/solutions/collateral/
+service-provider/visual-networking-index-vni/
+white-paper-c11-741490.pdf, accessed 09/18/19.
+[17] R. Costa and J. Pazos. Mlcache: A multi-armed bandit policy
+for an operating system page cache. Technical report, Technical
+Report. University of British Columbia, 2017.
+[18] D. Duplyakin, R. Ricci, A. Maricq, G. Wong, J. Duerig, E. Eide,
+12
+
+L. Stoller, M. Hibler, D. Johnson, K. Webb, A. Akella, K. Wang,
+G. Ricart, L. Landweber, C. Elliott, M. Zink, E. Cecchet, S. Kar,
+and P. Mishra. The design and operation of CloudLab. In Pro-
+ceedings of the USENIX Annual Technical Conference (ATC),
+pages 1–14, July 2019. URL https://www.flux.utah.edu/
+paper/duplyakin-atc19.
+[19] G. Einziger and R. Friedman. Tinylfu: A highly efficient cache
+admission policy. In IEEE Euromicro PDP, pages 146–153,
+2014.
+[20] G. Einziger, O. Eytan, R. Friedman, and B. Manes. Adaptive
+software cache management. In ACM Middleware, pages 94–
+106, 2018.
+[21] A. Eisenman, A. Cidon, E. Pergament, O. Haimovich,
+R. Stutsman, M. Alizadeh, and S. Katti. Flashield: a hybrid
+key-value cache that controls flash write amplification. In
+USENIX NSDI, pages 65–78, 2019.
+[22] O. Eytan, D. Harnik, E. Ofer, R. Friedman, and R. Kat. It’s
+time to revisit {LRU} vs.{FIFO}. In 12th USENIX Workshop
+on Hot Topics in Storage and File Systems (HotStorage 20),
+2020.
+[23] B. Fan, D. G. Andersen, and M. Kaminsky. MemC3: Compact
+and concurrent memcache with dumber caching and smarter
+hashing. In USENIX NSDI, pages 371–384, 2013.
+[24] S. Jiang and X. Zhang. LIRS: an efficient low inter-reference
+recency set replacement policy to improve buffer cache perfor-
+mance. ACM SIGMETRICS, 30(1):31–42, 2002.
+[25] T. Johnson and D. Shasha. 2Q: A low overhead high perfor-
+mance buffer management replacement algorithm. In VLDB,
+pages 439–450, 1994.
+[26] R. Karedla, J. S. Love, and B. G. Wherry. Caching strategies
+to improve disk system performance. IEEE Computer, 27(3):
+38–46, 1994.
+[27] G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye,
+and T.-Y. Liu. Lightgbm: A highly efficient gradient boosting
+decision tree. In Advances in Neural Information Processing
+Systems, pages 3146–3154, 2017.
+[28] V. Kirilin, A. Sundarrajan, S. Gorinsky, and R. K. Sitaraman.
+Rl-cache: Learning-based cache admission for content delivery.
+IEEE Journal on Selected Areas in Communications, 38(10):
+2372–2385, 2020.
+[29] E. Liu, M. Hashemi, K. Swersky, P. Ranganathan, and J. Ahn.
+An imitation learning approach for cache replacement. In
+International Conference on Machine Learning, pages 6237–
+6247. PMLR, 2020.
+[30] T. Lykouris and S. Vassilvtiskii. Competitive caching with ma-
+chine learned advice. In International Conference on Machine
+Learning, pages 3296–3305. PMLR, 2018.
+[31] R. L. Mattson, J. Gecsei, D. R. Slutz, and I. L. Traiger. Eval-
+uation techniques for storage hierarchies. In IBM Systems
+journal, volume 9, pages 78–117, 1970.
+[32] S. McAllister, B. Berg, J. Tutuncu-Macias, J. Yang, S. Gu-
+nasekar, J. Lu, D. S. Berger, N. Beckmann, and G. R. Ganger.
+Kangaroo: Caching billions of tiny objects on flash. In Pro-
+ceedings of the ACM SIGOPS 28th Symposium on Operating
+Systems Principles, pages 243–262, 2021.
+[33] N. Megiddo and D. S. Modha. ARC: A self-tuning, low over-
+head replacement cache. In USENIX FAST, volume 3, pages
+115–130, 2003.
+[34] D. Rohatgi. Near-optimal bounds for online caching with
+machine learned advice. In Proceedings of the Fourteenth
+Annual ACM-SIAM Symposium on Discrete Algorithms, pages
+1834–1845. SIAM, 2020.
+[35] Z. Song, D. S. Berger, K. Li, A. Shaikh, W. Lloyd, S. Ghorbani,
+C. Kim, A. Akella, A. Krishnamurthy, E. Witchel, et al. Learn-
+ing relaxed belady for content distribution network caching.
+In 17th {USENIX} Symposium on Networked Systems Design
+and Implementation ({NSDI} 20), pages 529–544, 2020.
+[36] L. Tang, Q. Huang, W. Lloyd, S. Kumar, and K. Li. RIPQ:
+advanced photo caching on flash for facebook. In USENIX
+FAST, pages 373–386, 2015.
+[37] G. Vietri, L. V. Rodriguez, W. A. Martinez, S. Lyons, J. Liu,
+R. Rangaswami, M. Zhao, and G. Narasimhan. Driving cache
+replacement with ML-based LeCaR. In USENIX HotStorage,
+2018.
+[38] H. Wang, H. He, M. Alizadeh, and H. Mao. Learning caching
+policies with subsampling. In NeurIPS Machine Learning for
+Systems Workshop, 2019.
+[39] A. Wei. Better and simpler learning-augmented online caching.
+arXiv preprint arXiv:2005.13716, 2020.
+[40] G. Yan and J. Li. Rl-bélády: A unified learning framework for
+content caching. In Proceedings of the 28th ACM International
+Conference on Multimedia, pages 1009–1017, 2020.
+[41] G. Yan, J. Li, and D. Towsley. Learning from optimal caching
+for content delivery. In Proceedings of the 17th International
+Conference on emerging Networking EXperiments and Tech-
+nologies, pages 344–358, 2021.
+[42] J. Yang, Y. Yue, and K. Rashmi. A large scale analysis of
+hundreds of in-memory cache clusters at twitter.
+In 14th
+{USENIX} Symposium on Operating Systems Design and Im-
+plementation ({OSDI} 20), pages 191–208, 2020.
+[43] J. Yang, Y. Yue, and R. Vinayak. Segcache: a memory-efficient
+and scalable in-memory key-value cache for small objects. In
+18th USENIX Symposium on Networked Systems Design and
+Implementation (NSDI 21), pages 503–518, 2021.
+[44] G. Zhou and M. Maas. Learning on distributed traces for data
+center storage systems. Proceedings of Machine Learning and
+Systems, 3, 2021.
+13
+

htE3T4oBgHgl3EQfgQqz/content/tmp_files/2301.04560v1.pdf.txt

@@ -0,0 +1,2135 @@
+Model reduction for nonlinearizable dynamics via
+delay-embedded spectral submanifolds
+Joar Ax˚as1, George Haller1∗
+1Institute for Mechanical Systems, ETH Z¨urich,
+Leonhardstrasse 21, 8092 Zurich, Switzerland
+12.01.2023
+Abstract
+Delay embedding is a commonly employed technique in a wide range
+of data-driven model reduction methods for dynamical systems, including
+the Dynamic mode decomposition (DMD), the Hankel alternative view
+of the Koopman decomposition (HAVOK), nearest-neighbor predictions
+and the reduction to spectral submanifolds (SSMs). In developing these
+applications, multiple authors have observed that delay embedding ap-
+pears to separate the data into modes, whose orientations depend only
+on the spectrum of the sampled system. In this work, we make this ob-
+servation precise by proving that the eigenvectors of the delay-embedded
+linearized system at a fixed point are determined solely by the correspond-
+ing eigenvalues, even for multi-dimensional observables. This implies that
+the tangent space of a delay-embedded invariant manifold can be pre-
+dicted a priori using an estimate of the eigenvalues. We apply our results
+to three datasets to identify multimodal SSMs and analyse their nonlin-
+ear modal interactions.
+While SSMs are the focus of our study, these
+results generalize to any delay-embedded invariant manifold tangent to a
+set of eigenvectors at a fixed point. Therefore, we expect this theory to
+be applicable to a number of data-driven model reduction methods.
+1
+Background
+Much recent effort in nonlinear dynamics has focused on data-driven model
+reduction methods. Such algorithms return a simplified model of the system
+dynamics based on sampled trajectories from experiments or simulations. Com-
+monly pursued objectives for developing these methods include dimensionality
+reduction, sparsity, and interpretability. Prevalent methods include the proper
+orthogonal decomposition (POD) [1, 2] and the dynamic mode decomposition
+(DMD) [3, 4, 5], which fit models to data under various linearity assumptions.
+∗Corresponding author. E-mail: [email protected]
+1
+arXiv:2301.04560v1  [math.DS]  11 Jan 2023
+
+Linear models cannot, however, capture characteristically nonlinear (or non-
+linearizable) phenomena. Such phenomena include the coexistence of, and the
+transition between, isolated and compact stationary states, such as fixed points,
+limit cycles, and invariant tori [6]. To address this shortcoming, the Sparse iden-
+tification of nonlinear dynamics (SINDy) algorithm fits a sparse nonlinear model
+to training data using a library of nonlinear functions [7]. However, the choice of
+this library depends on the user [8] and the coordinate system used. Addition-
+ally, the size of the library scales up quickly with the problem dimensionality [9].
+While neural networks can pattern-match nonlinear phenomena [10, 11, 12], the
+models they return are often difficult to interpret and generalize poorly outside
+the range of training data [13].
+In the last few years, spectral submanifolds (SSMs) have appeared as an
+alternative for model reduction in intrinsically nonlinear systems. An SSM is
+the unique smoothest invariant manifold tangent to a nonresonant spectral sub-
+space emanating from a fixed point [14] or a periodic or quasiperiodic orbit
+[15]. Therefore, an attracting SSM is the ideal candidate for a low-dimensional
+model of a nonlinear system [16, 17]. Concepts related to SSMs include nonlinear
+normal modes (NNMs) defined either as sets of periodic motions in conserva-
+tive systems [18, 19, 20] or invariant manifolds [21, 22] and invariant spectral
+foliations [23]. Here, we will apply SSMs, as they are unique, exist under well-
+defined conditions in dissipative systems, can have arbitrary dimensions, and
+can include internally resonant modes.
+After the computation of an SSM, we can project either equations or data
+onto it to reduce the system to a high-fidelity model. Automated model re-
+duction to SSMs from equations [24] can successfully predict responses to small
+harmonic forcing [25, 26, 27] and bifurcations of those responses [28, 29], and has
+also been extended to constrained mechanical systems [30]. Recently, Ref. [31]
+developed a data-driven method which identifies the SSM geometry and its re-
+duced dynamics to trajectories in an observable space [32]. This approach also
+transforms the SSM-reduced dynamics to a normal form, which describes the
+dynamics as sparsely as possible while maintaining essential nonlinearities [33].
+SSM-based model reduction has since been applied to both numerical and ex-
+perimental datasets in fluid and structural dynamics [34, 35] and control [36].
+Ref. [37] showed how to improve the computational efficiency of data-driven
+SSM identification through a simplified formulation of the algorithm.
+Delay embedding is the method of reconstructing invariant sets by viewing
+a select number of measurements separated by a timelag as independent ob-
+servables. This method is routinely used to aid data-driven model identification
+in nonlinear dynamical systems. Examples of model reduction methods based
+on delay embedding include the extended dynamic mode decomposition (DMD)
+[38, 39], the Hankel alternative view of the Koopman decomposition (HAVOK)
+[40], the eigensystem realization algorithm (ERA) [41], closure modeling [42],
+and nearest-neighbor prediction [43, 44].
+In addition, delay embedding has
+been extensively employed in SSM-based model reduction from data [31, 34].
+For SSMs, a closer understanding of the delay embedding map improves fits to
+data and produces more accurate reduced-order models [37]. This has motivated
+2
+
+our present study on how invariant manifolds can be efficiently and accurately
+reconstructed in delay coordinates.
+The main driver behind the introduction of delay embedding as a tool in dy-
+namical systems was the discovery that it could reconstruct strange attractors
+from scalar measurements of chaotic systems [45, 46]. Floris Takens’s celebrated
+embedding theorem [47] and its later extension [48] show that, in principle, de-
+lay embedding recovers invariant sets from the full state space in a suitable
+observable space under generic assumptions. In practice, however, the choice of
+the timelag and embedding dimension is critical to obtain robust models [49,
+50, 51]. The many methods for choosing delay parameters for chaotic attractor
+reconstruction include minimization of the mutual information between subse-
+quent samples [52], minimization of false nearest neighbors [53, 54], and a Monte
+Carlo decision tree search formulation [55].
+Recent work has also explored the geometric structure of delay embedded
+invariant sets in an effort to improve model order reduction. For periodic data,
+singular value decomposition (SVD) on the delay-embedded snapshot matrix
+has been shown to converge to a Fourier analysis [56]. The number of delays
+required to recover such periodic orbits equals the number of coefficients of
+the Fourier spectrum [57]. Fitting a linear map between subsequent snapshots
+of such a delay-embedded periodic orbit produces a companion matrix, whose
+eigenvectors are given by the inverse Vandermonde matrix [58, 59].
+Furthermore, connections to convolutional coordinates [60] and the Frenet-
+Serret frame [61] have been made, and an interpretation of SVD modes in delay
+coordinates as principal component trajectories has been proposed [62]. For
+the special case of an observed signal composed of oscillating sinusoidal func-
+tions, the observable space contains invariant spaces determined by the signal
+frequencies [37].
+Recently, it was shown that subsequent components of the
+DMD modes of delay-embedded linear systems are related by a multiplication
+of the corresponding eigenvalue [63].
+In this work, we explore the local dynamics close to a fixed point of a nonlin-
+ear delay-embedded system. We show that the linear part of the delay-embedded
+dynamics depends solely on the corresponding eigenvalues, and not on the ob-
+servable function and the full state space eigenvectors. In particular, the eigen-
+vectors in the observable space are given by the columns of the Vandermonde
+matrix of the exponential of the eigenvalues multiplied by the timelag. Unlike
+available previous work, we do not attempt a linearization of the nonlinear dy-
+namics, nor do we restrict our attention to periodic orbits. Instead, our results
+imply that the nonlinear delay-embedded system has an SSM whose tangent
+space coincides with the column space of this Vandermonde matrix. We exploit
+this structure to aid the data-driven identification of SSMs in three mechanical
+examples. We believe that these results enhance the understanding of delay
+embedding in reduced-order modeling and also reveal new opportunities for
+SSM-based model reduction.
+The structure of this paper is the following. First, Sect. 2 briefly introduces
+SSM theory and summarizes a method for fast SSM-based data-driven modeling.
+Sect. 3 outlines a new theory for delay-embedding tangent spaces of invariant
+3
+
+manifolds and discusses their application to SSM-based model reduction. In
+Sect. 4, we use these results to identify SSMs in examples of a 2-degree-of-
+freedom oscillator, simulations of multimodal vibrations in a von K´arm´an beam,
+and experiments of complex behavior in a sloshing tank. In Sect. 5, we draw
+conclusions from these examples and discuss possible further extensions of our
+theory.
+Finally, Appendix A contains the proofs of the results presented in
+Sect. 3.
+2
+Model reduction to spectral submanifolds
+Here, we outline previous results on rigorous model order reduction to SSMs in
+smooth nonlinear systems. We also summarize fastSSM, the algorithm we use
+here to identify SSMs from data.
+2.1
+Spectral submanifold theory
+Consider a nonlinear, autonomous dynamical system of class Cl, l ∈ {N+, ∞, a},
+where a denotes analyticity, in the form
+˙x = Ax + g(x),
+x ∈ Rn,
+g ∼ O(|x|2),
+g : Rn → Rn.
+(1)
+Let us denote the flow map of the system by F t(x0) := x(t, x0), with x(t, x0)
+denoting the trajectory of (1) starting from x0 at time 0.
+We assume that
+A ∈ Rn×n is diagonalizable and that the real parts of its eigenvalues are either
+all strictly negative or all strictly positive. We take d eigenvectors of A and
+denote their span by E, i.e., a d-dimensional spectral subspace of Rn. In this
+step, we often choose the d slowest eigendirections.
+Provided that the d eigenvalues corresponding to E are non-resonant with the
+remaining n−d eigenvalues of A, the nonlinear system has a unique smoothest,
+invariant manifold M tangent to E at the origin, i.e., T0M = E [14]. Following
+[16], we call M a spectral submanifold (SSM). In case of a resonance between
+E and the rest of the spectrum of A, the d-dimensional SSM does not exist in
+general, and we must then include the resonant modal subspace in E to obtain
+a higher-dimensional SSM. If all eigenvalues of A are stable, the slowest SSM
+attracts nearby trajectories, which makes it suitable for model order reduction.
+The open-source numerical package SSMTool computes SSMs from arbitrary
+finite-dimensional nonlinear systems [24, 27]. More recently, the SSMLearn pack-
+age was developed to find SSMs in data from nonlinear dynamical systems [31,
+34].
+Here, we will apply the simplified data-driven SSM algorithm fastSSM
+introduced by Ref. [37].
+2.2
+Fast data-driven model order reduction to spectral
+submanifolds
+The objective of dynamics-based machine learning is to reconstruct SSMs from
+data, and then use SSM-reduced models for predictions of the full system re-
+4
+
+sponse [31]. Here, we use fastSSM [64] to identify the SSM from snaphots of
+trajectories in an observable space. The procedure consists of two steps: man-
+ifold geometry detection and normal form computation. The summary below
+follows Ref. [37], to which we refer for further details.
+Whereas that refer-
+ence differentiates between the algorithm for cubic polynomial approximations
+of two-dimensional SSMs and its extension to arbitrary order and dimension,
+here, we will simply refer to both algorithms as fastSSM.
+The SSM is parametrized in the graph style, that is, we construct M as a
+graph over the spectral subspace E. The data consists of snapshots y(ti) ∈ Rp in
+a p-dimensional observable space. For each trajectory we construct the snapshot
+matrix Y ∈ Rp×N from N snapshots as
+Y =
+�
+�
+|
+|
+|
+y(t1)
+y(t2)
+. . .
+y(tN)
+|
+|
+|
+�
+�
+(2)
+Let T ∈ Rp×d be a matrix whose columns approximately span the SSM
+tangent space. In fastSSM, the standard procedure is to obtain T through SVD
+on the snapshot matrix Y . However, T can also be prescribed if the tangent
+space is known a priori. Denoting by (·)† the Moore-Penrose pseudoinverse, we
+project each snapshot yi onto this subspace to obtain d-dimensional reduced
+coordinates ξ as
+ξ = T †y.
+(3)
+We write Ξ ∈ Cd×N for the projection of the snapshot matrix onto the tangent
+space.
+Next, we seek to approximate the embedding of M as the graph of a multi-
+variate polynomial of order m from the data:
+y(ξ) = Mξ1:m,
+M
+= [M1, M2, . . . , Mm],
+Mi ∈ Rp×di,
+(4)
+where di is the number of d-variate monomials at order i and the superscript
+in (·)1:l denotes a vector of all monomials from order 1 up to l. We obtain the
+manifold parametrization coefficients M ∈ Rp×d1:m by a polynomial regression,
+which yields the solution
+M = Y (Ξ1:m)†.
+(5)
+The reduced dynamics are approximated by another O(r) polynomial re-
+gression, with a coefficient matrix G ∈ Cd×d1:r, in the form
+˙ξ ≈ Gξ1:r,
+G = ˙Ξ(Ξ1:r)†.
+(6)
+Finally, we compute the normal form [33] of the SSM-reduced dynamics up
+to order h. This amounts to a near-identity polynomial transformation with
+coefficients H ∈ Cd×d1:h from the new coordinates ζ ∈ Cd such that
+ξ = Hζ1:h = ζ + H2:hζ2:h,
+˙ζ = Nζ1:h = Λζ + N2:hζ2:h.
+(7)
+5
+
+The normal form and the reduced dynamics are conjugate dynamical systems.
+Therefore, we substitute (7) into (6) to obtain
+Dζ(Hζ1:h)Nζ1:h = G(Hζ1:h)1:r.
+(8)
+The matrices H and N are computed by solving (8) recursively at increasing
+orders with SSMTool [27].
+This procedure requires that the training data lies sufficiently close to the
+SSM, which can be achieved by removing initial transients from the input signal,
+as identified by a spectral analysis on the training data [34]. Since the SSM built
+over the slowest d modes is unique and attracting, this method ensures relevant
+training data.
+3
+Delay-embedding the tangent spaces of invari-
+ant manifolds
+Here, we show how tangent spaces of invariant manifolds at a fixed point can be
+analytically recovered when the observable space arises from delay embedding
+of a signal. We also describe how the recovered tangent spaces facilitate the
+reconstruction of spectral submanifolds in such observable spaces.
+3.1
+Theoretical results
+For the dynamical system (1), we define a scalar observable µ(x(t)), where
+µ : Rn → R is a differentiable function that returns a measured feature of
+system (1), such as a displacement coordinate. In order to reconstruct features
+of the full phase space from the observable, we use delay embedding. We stack p
+consecutive measurements separated by a timelag τ > 0 to create an observable
+space of dimension p. This yields a trajectory in the form y(t) = S(x(t)) ∈ Rp,
+where we define the sampling map
+S : Rn → Rp,
+x �→
+�
+������
+µ(x)
+µ(F τ(x))
+µ(F 2τ(x))
+...
+µ(F (p−1)τ(x))
+�
+������
+.
+(9)
+An important question is how invariant sets of system (1) in Rn are re-
+produced in the observable space Rp. In particular, when the full state space
+trajectory x(t) resides on a d-dimensional invariant manifold M, will y(t) also
+do so? Takens’s embedding theorem gives an affirmative answer. It states that
+if µ is generic and no small integer multiple of τ coincides with the period of
+any possible periodic orbit of (1) lying in M, then for
+p ≥ 2d + 1,
+(10)
+6
+
+the manifold M will have a diffeomorphic copy
+˜
+M in Rp via the mapping
+(9) [47]. Whereas Takens’s theorem was formulated only for scalar observable
+functions, this result has since been extended to multi-dimensional µ as long as
+the total observable space dimension exceeds 2d [65].
+Both the nonlinear geometry and dynamics of M and the observable function
+influence the geometry of ˜
+M. It is therefore difficult to predict its geometry for
+a general flow map. Around the fixed point q = S(0) ∈ Rp, however, the O(1)
+expansion of ˜
+M, i.e., its tangent space Tq ˜
+M, can be directly determined, as we
+will show next. Note that since the flow map is the identity at the origin, q lies
+on the diagonal in the observable space, with each of its identical components
+given by µ(0).
+We start by rewriting (1) in modal coordinates:
+˙z = f(z) = Λz + E−1g(Ez),
+(11)
+where E = [e1, . . . , en] contains the eigenvectors of A and Λ = diag(λ1, . . . , λn)
+the corresponding eigenvalues, which we assume to be distinct. We define modal
+coordinates z ∈ Cn by letting z = E−1x. Whereas the observable function is
+defined as a function of x, it is notationally convenient to define it as a function
+of z, as µ(x) = µ(Ez).
+Let M be a d-dimensional invariant manifold of (1) intersecting the origin
+0 ∈ Rn, where it is tangent to a set of d eigenvectors e1, e2, . . . , ed of A with
+corresponding eigenvalues λ1, . . . , λd. We define the Vandermonde matrix V ∈
+Cp×d of the d eigenvalues governing the linearized dynamics on M as Vjk =
+eλkjτ, i.e.,
+V =
+�
+������
+1
+1
+. . .
+1
+eλ1τ
+eλ2τ
+. . .
+eλdτ
+e2λ1τ
+e2λ2τ
+. . .
+e2λdτ
+...
+...
+...
+...
+e(p−1)λ1τ
+e(p−1)λ2τ
+. . .
+e(p−1)λdτ
+�
+������
+.
+(12)
+Theorem 1. Under the assumptions of a generic observable function µ : Rn →
+R and distinct eigenvalues λ1 ̸= . . . ̸= λd, the tangent space of the observable
+manifold
+˜
+M at the fixed point can be written
+Tq ˜
+M = range V .
+(13)
+Proof. See Appendix A.1.
+This result is illustrated in Fig. 1. Note that the observable function must
+have full rank, as spelled out in the following remark.
+Remark 1. For (13) to hold, we must have
+∂µ
+∂zk |0 ̸= 0 ∀k ∈ {1, . . . , d}, which
+defines the genericity of µ. If the gradient of the observable function is orthog-
+onal to any of the eigenvectors e1, . . . , ed, the sampling map S will not be an
+embedding of M.
+7
+
+Manifold in full phase space
+Manifold in delay embedding space
+�
+����
+1
+eλ1τ
+e2λ1τ
+...
+�
+����
+�
+����
+1
+eλ2τ
+e2λ2τ
+...
+�
+����
+Sampling map
+S : Rn → Rp
+Tangent space
+embedding DS(T0M)
+x1
+x2
+x3,...,n
+y1
+y2
+y3,...,p
+M
+T0M
+˜
+M
+Tq ˜
+M
+Figure 1: Delay embedding of the tangent space T0M of an invariant manifold
+M. The full state space manifold M (left) has a diffeomorphic copy
+˜
+M in the
+observable space (right) by Takens’s theorem. The shape of the reconstructed
+manifold
+˜
+M depends on the flow map, but its tangent space, Tq ˜
+M, is directly
+given by the eigenvalues at the fixed point, independent of the geometry of M
+and the observable function µ.
+This should be kept in mind particularly when dealing with symmetries of
+engineering structures, as we will show in our examples below.
+Theorem 2. The columns of V are eigenvectors of the linearized delay-embedded
+system at the fixed point. Indeed, the dynamics in the observable space can be
+written
+˙y = V ΛV †(y − q) + o(|y − q|)
+(14)
+Proof. See Appendix A.2.
+Corollary 1. In the observable space Rp, the timelag τ and the eigenvalues
+λk fully determine the tangent space and the linear part of the dynamics. In
+particular, the linear dynamics are independent of both the full eigenvectors and
+the observable function.
+In the following, we will demonstrate how this structure can be exploited for
+parametrizing spectral submanifolds from data, when the corresponding eigen-
+values are approximately known.
+Finally, when the observable function is multi-dimensional, the tangent space
+is influenced by the relative dependency of each component µℓ of the observable
+function on each modal coordinate zk.
+Theorem 3. For a multidimensional observable µ : Rn → Rq with components
+8
+
+µ1, . . . , µq, the tangent space Tq ˜
+M ⊂ Rpq can be expressed as
+Tq ˜
+M = range
+�
+����
+V diag
+�
+∂µ1
+∂z
+���
+0
+�
+...
+V diag
+�
+∂µq
+∂z
+���
+0
+�
+�
+���� .
+(15)
+Proof. See Appendix A.3.
+When the observable function is a set of displacements, the linearized multi-
+dimensional observable function
+∂µ
+∂z
+���
+0 corresponds to the mode shapes of the
+system in terms of those displacements.
+Therefore, if the mode shapes and
+eigenvalues of the observed system are known, we can directly compute the
+tangent space of ˜
+M. In the special case of a scalar observable, the tangent space
+is independent of the observable function and we do not need any information
+about the mode shapes.
+3.2
+Delay-embedded spectral submanifold reconstruction
+These theoretical results can be exploited as a constraint to aid SSM identi-
+fication from data. In the case of a scalar signal and with the eigenvalues of
+interest λ1, . . . , λd approximately known, we select the matrix representation of
+the tangent space T appearing in (3) as the Vandermonde matrix (12), i.e.,
+T := V .
+(16)
+We have seen that the gradient of a multi-dimensional observable func-
+tion µ enters the expression for the tangent space (15).
+While an expres-
+sion for this gradient is typically not available in experiments, mode shapes
+ˆE = [ˆe1, . . . , ˆed] ∈ Rq×d are often known from theory or obtained experimen-
+tally. Here, each mode shape ˆek ∈ Rq describes how the eigenvector ek ∈ Rn is
+observed. Specifically, they are related by
+ˆek = ck
+∂µ
+∂x
+����
+0
+ek,
+(17)
+where ck ∈ C is a nonzero constant that only rescales the eigenvectors. We
+select T as the columnwise Kronecker product of the Vandermonde matrix and
+the observable mode shapes, i.e.,
+T :=
+�
+��
+V diag (ˆe1)
+...
+V diag (ˆed)
+�
+�� .
+(18)
+A sketch of the geometry is shown in Fig. 2. In the case of unknown mode
+shapes, it may be possible to first project low-amplitude data onto the delay-
+embedded eigenvectors and then extract the observable mode shapes via SVD,
+although we do not explore this idea further in this work.
+9
+
+q
+µ1(x(t + τ))
+µ1(x(t))
+µ2(x(t))
+V
+ˆE
+T
+˜
+M ⊂ Rpq
+Figure 2: The tangent space Tq ˜
+M of the delay-embedded manifold
+˜
+M for a
+q-dimensional observable function µ is the range of the matrix T , defined as the
+columnwise Kronecker product of the Vandermonde matrix V and the mode
+shapes ˆE in terms of the observable.
+Prescribing T and projecting the data onto its columns yields modal reduced
+coordinates. This diagonalization of the system simplifies the learning of the
+geometry and the reduced dynamics of the SSM via the algorithm outlined in
+Sect. 2.2.
+Choosing proper delay-embedding parameters to reconstruct nonlinear sys-
+tems can be a challenge. For the linear part of the system, however, our results
+suggest picking the timelag τ and embedding dimensionality p so as to obtain
+numerically favorable reduced coordinates along the SSM. We ideally want the
+columns of the Vandermonde matrix (12) to be orthogonal in order to maxi-
+mize the signal-to-noise ratio in each of the observed modes. To this end, we
+formulate a minimization problem,
+(κ⋆, p⋆) = argmin
+κ,p∈N+
+��V (κ∆t, p)⊤V (κ∆t, p) − I
+��
+F ,
+(19)
+in which the columns of V are normalized and ∥·∥F denotes the Frobenius norm.
+Since the timelag is an integer multiple of the sampling timestep, τ = κ∆t, (19)
+defines an optimization over a set of discrete variables which can be solved
+simply by brute force.
+Bearing in mind the nonlinear part of the system, however, an optimal choice
+of delay parameters is not as straightforward. Increasing the timelag and em-
+bedding dimension tends to curve the SSM, requiring higher orders of approx-
+10
+
+imation and in extreme cases folding the manifold, so that it can no longer be
+parametrized as a graph. Taking into account the nonlinear part of the SSM
+reconstruction, therefore, we typically want the total delay embedding to be as
+small as possible. While solving (19) gives some guidance, a suitable choice of τ
+and p will also depend on the nonlinearity of the system in the data range and
+the amount of signal noise.
+4
+Applications
+We now apply our method to three datasets: two from simulations and one from
+experiments. The eigenvalues in these examples are known either from theory
+or simulations. We infer the delay-embedded tangent space accordingly before
+parametrizing the SSM. The examples include an oscillator chain, a clamped-
+clamped beam and tank sloshing.
+4.1
+Two-degree-of-freedom oscillator with nonlinear springs
+As our first example, we consider an oscillator chain of two masses, both at-
+tached with linear springs to each other and to the ground. In addition, the
+spring connecting the left mass to the ground has a quadratic softening non-
+linearity and the spring connecting the masses is of cubic hardening type. The
+masses and linear spring stiffnesses are set to 1, the softening parameter is −2
+and the hardening parameter is 1. Each of the springs also has a linear damp-
+ing coefficient of 0.03. Fig. 3a shows the configuration. The sampling time is
+∆t = 0.1 s.
+m
+x1(t)
+k, κ
+c
+m
+x2(t)
+k, γ
+c
+k
+c
+(a)
+-0.5
+0.5
+0
+-0.5
+x4
+0.5
+0
+0
+x1
+x3
+0.5 -0.5
+M
+Data
+Re E1
+Im E1
+T0M
+(b)
+0.5
+-0.5
+0.5
+0
+0
+x1(t)
+x1(t + 30"t)
+0.5
+0
+x1(t + 15"t)
+-0.5
+-0.5
+~
+M
+Data
+Re V1
+Im V1
+T0 ~
+M
+(c)
+Figure 3: (a) Setup for the two-degree-of-freedom oscillator example with two
+nonlinear springs. (b) The slow 2D SSM (gray) in the full state space, along
+with its tangent space (red). (c) The delay-embedded SSM in the observable
+space.
+We compute an initial condition on the slow 2D SSM for the single training
+trajectory using SSMTool. Our observable function is the first mass displace-
+ment, µ(x) = x1. The trajectory in the full phase space and the SSM are shown
+in Fig. 3b. The first two eigenvectors span the tangent space of the SSM.
+11
+
+Next, we delay embed the trajectory with a timelag τ = 15∆t and embedding
+dimension p = 5, and seek the 2D SSM in this observable space using fastSSM.
+We obtain reduced coordinates by projection of the trajectory data onto the
+columns of the Vandermonde matrix V as predicted by our theory.
+Fig. 3c
+shows the SSM in the first three coordinates of the observable space. Indeed,
+the tangent space of this observable space is identical to the column space of
+V .
+0.5
+-0.5
+0.5
+0
+0
+x2(t)
+x2(t + 30"t)
+0.5
+0
+x2(t + 15"t)
+-0.5
+-0.5
+~
+M
+Data
+Re V1
+Im V1
+T0 ~
+M
+(a)
+0.5
+-0.5
+0
+0
+x3(t) + x4(t)
+x3(t + 30"t) + x4(t + 30"t)
+0.5
+0.5
+x3(t + 15"t) + x4(t + 15"t)
+0
+-0.5
+-0.5
+~
+M
+Data
+Re V1
+Im V1
+T0 ~
+M
+(b)
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+!x1(t + 30"t) + x2(t + 30"t)
+-0.2
+0.1
+-0.1
+!x1(t) + x2(t)
+0
+!x1(t + 15"t) + x2(t + 15"t)
+0
+-0.1
+0.1
+-0.2
+Data
+(c)
+Figure 4: (a,b) Changing the observable function leads to different SSM geome-
+tries, but the tangent space remains the same as in Fig. 3c. (c) A nongeneric
+observable function however, observing only the second mode, does not embed
+the manifold.
+Corollary 1 predicts that this tangent space will be independent of the ob-
+servable function, provided that it is generic. To illustrate this, we plot the
+delay-embedded SSM for various observable functions, µ(x) = x1 (Fig. 3c),
+µ(x) = x2 (Fig. 4a), and µ(x) = ˙x1 + ˙x2 (Fig. 4b). These different observable
+functions clearly produce different SSM geometries, but the eigenvectors and
+tangent spaces of the manifolds all agree.
+One exception is when we observe the distance between the masses, µ(x) =
+x2 − x1 (Fig. 4c). In this case, the delay-embedded trajectory no longer lies on
+an invariant manifold, as is evident by the nonsmooth cusp in the data at the
+origin. The reason is that this observable is non-generic precisely in the sense
+of our theory; it coincides with the mode shape of the second, fast mode of the
+full system. This means that the observable function acts orthogonally to the
+slow SSM at the fixed point and thus the delay mapping is not an embedding,
+by Remark 1.
+Next, we pick µ(x) = x2 and use fastSSM to approximate the cubic order
+reduced dynamics on the SSM from the data.
+Computing the normal form
+yields
+� ˙ρ1
+˙θ1
+�
+=
+� −0.0014 ρ13 − 0.0148 ρ1
+1.0025 − 0.0919 ρ12
+�
+.
+(20)
+The trajectory projected onto the columns of V is shown in Fig. 5a. Integrating
+the obtained normal form and mapping back to the observable space yields a
+good reconstruction of the training data.
+Finally, following Theorem 3, we demonstrate how to determine the tangent
+12
+
+-0.1
+0
+0.1
+Re(V yy)
+-0.1
+0
+0.1
+Im(V yy)
+(a)
+0
+500
+1000
+1500
+2000
+time
+-0.2
+0
+0.2
+0.4
+x2(t)
+Simulation
+Prediction
+(b)
+0.4
+0.2
+-0.2
+0
+x1(t)
+0
+0.5
+x2(t + 15"t)
+0.2
+-0.2
+0.4
+x1(t + 15"t)
+0
+-0.4
+-0.5
+~
+M
+Data
+Re V1
+Im V1
+T0 ~
+M
+(c)
+Figure 5: (a) Projection of the data onto the delay-embedded tangent space
+predicted by our theory. (b) fastSSM predicts a model that successfully recon-
+structs the decay of the trajectory. (c) A view of the SSM in a delay-embedded
+space from a multi-dimensional observable, with the tangent space predicted by
+our theory.
+space of the SSM at the fixed point when the observable is a vector. When
+we choose µ(x) = [x1, x2]⊤, unlike for a scalar observable function, the tangent
+space orientation is influenced not only by the eigenvalues, but also by the shape
+of the first mode. This first mode shape corresponds to the masses moving in
+unison, i.e.
+ˆe1 = ˆe2 =
+� 1
+1
+�
+.
+(21)
+Then, by (18), we obtain vectors spanning the tangent space as the columns of
+the matrix
+T =
+� V diag(ˆe1)
+V diag(ˆe2)
+�
+=
+� V
+V
+�
+,
+(22)
+where V is the Vandermonde matrix (12). A view of the SSM and its tangent
+space in this 10-dimensional observable space is shown in Fig. 5c.
+The relation of this mode shape to the observable function is given by (17).
+In particular, we can compute the derivative of the observable function with
+respect to the modal coordinates as
+�
+∂µ1
+∂z1 (0)
+∂µ1
+∂z2 (0)
+∂µ2
+∂z1 (0)
+∂µ2
+∂z2 (0)
+�
+=
+� c1
+c2
+c1
+c2
+�
+,
+(23)
+where c1, c2 ∈ C are nonzero constants depending on the scaling of the eigen-
+vectors.
+For simplicity, in (21) we chose c1 = c2 = 1, such that T is the
+Vandermonde matrix vertically stacked twice.
+4.2
+6D SSM in a nonlinear finite-element model of a beam
+We train an SSM-reduced model with data from numerical simulations of a
+finite-element (FE) representation of a clamped-clamped von K´arm´an nonlinear
+beam [66]. This example was previously studied in Refs. [31, 37], which identified
+13
+
+the slowest 2D SSM in the delay-embedded observable space, predicted the
+forced response and analyzed the radius of convergence of the analytical normal
+form.
+Here, thanks to our results on the tangent spaces of delay-embedded
+SSMs, we can extend the analysis to the six-dimensional SSM emanating from
+the three slowest modes of the linear part of the system.
+Each node in the FE model has three degrees of freedom: axial deformation
+u, transverse deflection w, and rotation w′. The von K´arm´an axial strain is
+given by
+ϵ11 = u′(x) + 1
+2 (w′(x))2 − zw′′(x).
+(24)
+The axial stress is given by
+σ = Eϵ11 + c˙ϵ11,
+(25)
+where E = 70 GPa denotes the Young’s modulus and c = 1.0 × 106 Pa · s the
+material rate of viscous damping. Based on a convergence analysis, we set the
+number of elements to 12, resulting in a 33-degree of freedom mechanical system,
+i.e., a 66-dimensional phase space. We set the beam length to 1000 mm, width
+50 mm, and thickness 20 mm. The sampling time is ∆t = 0.0955 s.
+w(t)
+w(t)
+w(t)
+Figure 6: von K´arm´an beam: schematic first, second, and third mode shapes.
+The scalar observable must be generic in the sense that it must have contri-
+butions from all modes of interest. For instance, the midpoint displacement
+is not excited by the second mode. Instead, we choose the shown transverse
+displacement at 1/4 of the beam length.
+By Remark 1, the observable function µ must have significant contributions
+from all modes zk that we wish to model. For example, the midpoint displace-
+ment chosen as observable function in Refs. [31, 37] was sufficient to model the
+2D SSM, but cannot be employed for higher-dimensional SSMs. This is be-
+cause the antisymmetric shape of the second mode has zero displacement at the
+14
+
+midpoint (see Fig. Figure 6), i.e.
+∂µ
+∂z3
+(0) = ∂µ
+∂z4
+(0) = 0.
+(26)
+Instead, we choose the transverse displacement of the beam at one fourth of the
+total length, µ = w(l/4), as this degree of freedom has nonzero contributions
+from all three mode shapes.
+For our data-driven modeling objectives, we need training data containing
+the first three modes. To generate initial conditions for such trajectories, we
+use linear combinations of the mode shapes of the system computed from its
+linear part. Since the SSM is normally attracting, these trajectories will quickly
+approach it and we can use them to train our reduced-order model. With this
+method, we produce three trajectories close to the 6D SSM with different ini-
+tial conditions, of which we use two as training data and one as test data. For
+validation purposes, we also pick the individual mode shapes as initial condi-
+tions and use as test data. The individual modal contributions in these initial
+conditions were chosen as follows:
+Initial
+Mode
+Type
+condition
+1
+2
+3
+1
+1
+0
+0
+Test
+2
+0
+1
+0
+Test
+3
+0
+0
+1
+Test
+4
+0.8
+-0.8
+0.8
+Train
+5
+-0.1
+0.8
+0.8
+Train
+6
+-0.6
+-0.2
+-0.8
+Test
+We choose the delay embedding parameters guided by the observations in
+Sect. 3.2.
+Setting κ = 1 such that the timelag τ = ∆t and the embedding
+dimension to p = 50 gives a local optimum of the function (19) with the com-
+puted eigenvalues, while still keeping the maximal delay κp moderate to prevent
+folding of the embedding.
+Fig. 7a shows the delay embedding of the single-mode trajectories 1-3, cor-
+responding to the first three modes, in three of the 50 delay coordinates. These
+trajectories visualize the orientations of the corresponding eigenspaces in the
+observable space. Indeed, minimization of (19) corresponds to making these
+planes orthogonal, simplifying their identification.
+Fig. 7b similarly displays the delay embedding of the first training trajectory
+along with a visualization of the columns of the Vandermonde matrix as vectors.
+Our delay theory predicts that projection of the data onto these vectors yields
+modal coordinates, as shown in Fig. 7c. This space will serve as the reduced
+coordinates of the SSM.
+After projection onto these eigenvectors, we approximate the geometry of the
+6D SSM with a 3rd order polynomial. For the reduced dynamics in fastSSM, we
+also use a 3rd order approximation. We compute the normal form of this reduced
+dynamics up to 7th order and obtain our model for the reduced dynamics. The
+15
+
+w(t + 16"t)
+w(t)
+w(t + 8"t)
+Mode 1
+Mode 2
+Mode 3 (#2)
+(a)
+w(t + 16"t)
+w(t)
+w(t + 8"t)
+Traj. 4
+Re V1
+Re V3
+Re V5
+(b)
+(V yy)2
+(V yy)3
+(V yy)1
+Projection of traj. 4 onto V
+(c)
+Figure 7: (a) The trajectories with single modal contributions visualize the
+modal subspaces in the delay-embedded space. The third mode data has been
+scaled by a factor 2 to increase visibility. (b) The same delay-embedded view
+of the first training trajectory, along with the delay-embedded eigenvectors. (c)
+After projection of this trajectory onto the eigenvectors, the modal structure
+becomes clear.
+terms up to third order in polar form are found by fastSSM to be of the form
+�
+�
+�
+�
+�
+�
+�
+�
+˙ρ1
+ρ1 ˙θ1
+˙ρ2
+ρ2 ˙θ2
+˙ρ3
+ρ3 ˙θ3
+�
+�
+�
+�
+�
+�
+�
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+0.3058 ρ13 + 2.088ρ1 ρ22 − 3.091ρ1
+102.0 ρ13 + 82.70 ρ22ρ1 + 657.2ρ1
+−2.705 ρ12ρ2 + 1.723 ρ23 − 23.72ρ2
+95.64 ρ12ρ2 + 115.6 ρ23 + 1812ρ2
+−8.968ρ3 ρ12 − 13.27ρ3 ρ22 − 88.47ρ3
+115.9 ρ12ρ3 + 85.04 ρ22ρ3 + 3558ρ3
+�
+�
+�
+�
+�
+�
+�
+�
++ O(5).
+(27)
+We transform the initial conditions from the observable space to the normal
+form and integrate our model to predict signal decay. This produces a normal-
+ized mean trajectory error (as defined in [31]) of 2.2 % on the test data. Some
+of the predictions are shown in Fig. 8.
+0
+0.05
+0.1
+0.15
+0.2
+time [s]
+-5
+0
+5
+u [m]
+#10-3
+Training data
+Reconstruction
+(a)
+0
+0.05
+0.1
+0.15
+0.2
+time [s]
+-5
+0
+5
+u [m]
+#10-3
+Test data
+Reconstruction
+(b)
+0
+0
+1
+0.5
+;3
+;2
+;1
+0.5
+0.5
+1
+0
+1
+Traj. 1
+Traj. 2
+Traj. 3
+Traj. 4
+Traj. 5
+Traj. 6
+(c)
+Figure 8: (a,b) Predictions from fastSSM for the decaying trajectories 5 and 6
+(c) Phase portrait of the trajectories after transformation to the normal form.
+We also visualize our reduced-order model by plotting the instantaneous
+frequency and damping as predicted by the normal form (27) for varying ampli-
+tudes of mode 1 and 2. For instance, our model predicts hardening of the first
+16
+
+mode with respect to both the first and the second modal amplitudes (Fig. 9a),
+a decrease in the instantaneous damping of mode 1 with respect to mode 2
+(Fig. 9b), and independence of the third instantaneous frequency with respect
+to itself (Fig. 9c). The predictions for each of the trajectories are included for
+reference.
+650
+1
+700
+1
+_31
+750
+;2
+0.5
+;1
+800
+0.5
+0
+0
+(a)
+-3
+1
+-2
+1
+_;1=;1
+;2
+0.5
+-1
+;1
+0.5
+0
+0
+(b)
+3550
+1
+3600
+1
+_33
+;3
+0.5
+3650
+;1
+0.5
+0
+0
+(c)
+Figure 9: Visualization of the normal form (27) with the trajectories for (a)
+instantaneous frequency and (b) damping of mode 1, as well as (c) frequency of
+mode 3.
+4.3
+Multimodal sloshing of water in a tank
+4
+B. B¨auerlein and K. Avila
+Figure 2. Sketch of the experiment. A motor (a) drives an eccentric disk which converts the
+rotary motion of the motor via a pushing rod (b) into a quasi-harmonic horizontal oscillation of
+the platform. A positioning sensor (c) directly records the motion of the platform on which
+the tank (d), two high speed cameras (e) and an USB-camera (f) are mounted. For the
+PIV measurements a light sheet (g) is provided by a laser passing through a cylinder lens
+(implemented in the stationary laser guiding arm).
+dynamics. We find that neither the exact surface shape, nor the frequency spectrum
+are useful to determine the nonlinear resonance maxima. The key indicator is the
+phase-lag between driving and response. We systematically investigate the role of initial
+conditions, characterise the sloshing amplitude with the motion of the liquid’s centre
+of mass and directly measure the damping coefficient. The results obtained with our
+approach are compared to common approaches used in the literature. The paper is
+structured as follows. In the next section, we describe the experimental methods and in
+§3 the quantitative characterisation of the sloshing phenomena. In §4 and §5, the Duffing
+and multimodal model of sloshing are respectively described and briefly compared to
+our measured data. Detailed measurements of large-amplitude sloshing are presented
+in §6 with focus on the nonlinear dynamics of the system, including multiplicity and
+competition of several flow states. The experimental response curves obtained for several
+amplitudes are presented and compared to the Duffing and multimodal model in §7. An
+assessment of the strengths and weakness of these models in capturing the experimentally
+measured response is given in §8 before the conclusion in §9.
+2. Methods
+Our experiments were performed in a rectangular container subjected to harmonic
+horizontal excitation. As illustrated in figure 1, the flow is quasi-two-dimensional. Slosh-
+ing waves reaching from a quasi-planar surface, up to run-up at the tank walls and
+wave-breaking were investigated. A distinct feature of the sloshing waves in an oscillated
+(or pitched) tank is their asymmetric shape leading to an oscillation of the liquid’s
+centre of mass (shown as a red dot in figure 1). Many fundamental studies consider
+sloshing in wavemaker tanks (Taylor 1953; Fultz 1962; Chester 1968a). A key difference
+between oscillated and wavemaker tanks is that in the latter the primary resonant mode
+is symmetric and the liquid’s centre of mass is steady in the lateral direction.
+2.1. Experimental setup
+A sketch of our experimental setup is shown in figure 2. The tank (width w = 500 mm,
+depth l = 50 mm) is mounted on a platform and filled with water at room temperature
+(a)
+0
+100
+200
+300
+400
+500
+x [mm]
+Mode 1
+Mode 2
+Mode 3
+Mode 4
+(b)
+Figure 10: (a) Experimental setup for tank sloshing (adapted from [67]) (b) The
+first four sloshing mode shapes.
+For our final example, we apply our results to sloshing experiments. Sloshing
+models have a wide range of industrial applications, including fluid container
+interaction with ship motion [68], road transportation of fluids [69], damping
+devices in towers [70], and fuel tank design in spacecraft [71, 72].
+A tank
+partially filled with water exhibits several nonlinear phenomena under horizontal
+harmonic excitation [73]. On the one hand, intensified fluid motion can alter
+the instantaneous damping and frequency of the first sloshing mode [74]. On
+the other hand, increasing the amplitude further activates several nonlinearly
+coupled modes of the system and gives rise to a range of different wave motions
+[75, 76].
+17
+
+Our training data comes from experiments described in Ref. [67] with a rect-
+angular tank of width w = 500 mm and thickness 50 mm, partially filled with
+water up to a height of h = 400 mm. The tank was attached to a horizon-
+tally moving platform harmonically excited by a motor at different frequencies.
+Then, once the system had reached a steady state, the motor was turned off.
+Depending on the forcing frequency, this periodic response exhibited planar,
+wave-breaking, or three-periodic motion. The three-periodic forced state was
+characterized by an increase in the response amplitude every third forcing cycle,
+while the wave-breaking response was defined as overturning of the water close
+to the walls [67]. A camera detected the surface profile h with the sampling
+time ∆t = 0.01 s. Figure 10a displays the experimental setup.
+While previous work successfully captured the dynamics of the main sloshing
+mode using a 2D SSM for the center of mass signal [31] and the full surface
+profile [37], here, we model the decay from a multimodal state by identifying
+a 6D SSM, corresponding to the nonlinear extension of the three dominant
+oscillatory modes. We train on three decaying measurements: Trajectory 1 and
+2 start at a three-periodic state, and Trajectory 3 starts at a wave-breaking
+state.
+The observable vector µ is the surface profile measured at 1 771 points along
+the tank width. Since this function is multi-dimensional, in order to apply our
+theory on delay-embedded tangent spaces, we need an estimate of the eigen-
+values and linear mode shapes in our observable. The eigenfrequencies can be
+computed from potential theory [74] as
+ωk =
+�
+gπ
+w k tanh
+�
+πk h
+w
+�
+,
+(28)
+which scales approximately with the square root of the mode number k for
+our configuration.
+The first five eigenfrequencies are [7.80, 11.1, 13.6, 15.7,
+17.6] rad/s, with an approximate 1:2 resonance between frequencies 1 and 4.
+The mode shapes by the same theory are
+ˆek = cos(kx/w),
+x ∈ [0, w],
+(29)
+shown in Fig. 10b. For the tangent space, in principle, we also need the lin-
+ear damping of each mode. In practice, this real part of the eigenvalues has
+very little influence on V for limited delay embedding and we pick the values
+[−0.05, −0.07, −0.08, −0.09, −0.1] based on previous fits of the first mode and
+the assumption of increasing damping with the mode number.
+Based on (19), we delay-embed the data with timelag τ = 5∆t and dimension
+p = 47.
+A projection of the delay-embedded data onto the eigenvectors T
+predicted by our theory appears to yield modal coordinates, as indicated in
+Fig. 11a.
+Consequently, the norm of these projections can be used as a heuristic mea-
+sure of the modal content in the signal. This procedure should be used with
+caution, since it does not take manifold curvature into account, but it can be
+18
+
+-200
+-2000
+0
+2000
+(T yy)3
+(T yy)1
+0
+(T yy)2
+0
+200
+2000
+-2000
+Projection of traj. 2 onto T
+(a)
+0
+10
+20
+30
+40
+time [s]
+0
+1000
+2000
+3000
+Projected amplitude
+Mode 1
+Mode 2
+Mode 3
+Mode 4
+Mode 5
+(b)
+0
+5
+10
+15
+time [s]
+0
+100
+200
+300
+Projected amplitude
+Mode 1
+Mode 2
+Mode 3
+Mode 4
+Mode 5
+(c)
+Figure 11: (a) Projecting one of the trajectories onto the tangent space vec-
+tors unveils the modal structure.
+(b) By projecting the trajectory onto the
+eigenvectors and taking the absolute value, we can estimate the relative modal
+contributions in the signal (c) A zoomed-in view of (b) indicates that modes 1,
+2, and 4 dominate.
+employed to provide an initial guess for the SSM dimension. In Fig. 11b, we
+plot the absolute value of the projection onto each modal subspace of T over
+time for Trajectory 2. This plot shows that the first mode dominates, while the
+zoomed-in view (Fig. 11c) indicates that the second and fourth modes appear to
+be the most prevalent of the higher modes throughout the decay. The third and
+fifth mode are present at first but quickly die out. All amplitudes are decaying
+except for the fourth mode, which instead initially grows. Based on this analy-
+sis, we will identify a 6D SSM emanating from the spectral subspace of modes
+1, 2, and 4. This choice also takes SSM theory into account, by which the 1:2
+resonance requires that the modal subspace of the fourth mode is included in
+the spectral subspace of the SSM. We choose to start our training data after
+1.2 s, as the third and fifth modal amplitudes are small thereafter and we expect
+the trajectory to lie sufficiently close to the SSM.
+With an SSM parametrization order m = 4, reduced dynamics order r = 3,
+and normal form order h = 3, we compute the SSM geomety and dynamics and
+integrate our reduced-order model to predict the decay from the various flow
+states. This yields a normalized mean trajectory error (NMTE) [31] of 2.6 %.
+fastSSM successfully detects and accounts for the internal resonance by adding
+phase-dependent terms to the computed normal form, which reads
+˙ρ1
+ρ1 = −0.056 − 0.0069 sin(ψ − 0.26)ρ4 − 0.0015ρ2
+4 − 0.039ρ2
+2 + 0.023ρ2
+1
+˙θ1 = 7.78 + 0.0069 cos(ψ − 0.26)ρ4 + 0.040ρ2
+4 + 0.016ρ2
+2 − 0.82ρ2
+1
+˙ρ2
+ρ2 = −0.13 + 0.15ρ2
+4 − 0.89ρ2
+2 + 0.37ρ2
+1
+˙θ2 = 11.4 + 0.57ρ2
+4 − 0.0085ρ2
+2 − 2.2ρ2
+1
+˙ρ4
+ρ4 = −0.30 − 0.29ρ2
+4 + 0.67ρ2
+2 − 0.27 sin(ψ + 1.4)ρ2
+1 + 1.2ρ2
+1
+˙θ4 = 15.9 − 0.085ρ2
+4 + 1.2ρ2
+2 + 0.27 cos(ψ + 1.4)ρ2
+1 − 2.0ρ2
+1
+(30)
+where ψ = θ4 − 2θ1 and the subscripts denote the corresponding mode number.
+Looking at the linear part, we see that the eigenfrequencies are well captured.
+19
+
+Good agreement between the experimentally measured surface profile eleva-
+tion at the tank’s leftmost point and the delay-embedded SSM-reduced predic-
+tion is shown for the first period-three initial state in Fig. 12a and the wave-
+breaking state in Fig. 12b. Further, our 6D reduced model can accurately predict
+the full surface profile decay, with snapshots shown in Figure 13.
+0
+5
+10
+15
+20
+time
+-100
+0
+100
+200
+hx=0 [mm]
+Original
+Reconstruction
+(a)
+0
+10
+20
+30
+40
+time
+-100
+-50
+0
+50
+100
+150
+hx=0 [mm]
+Original
+Reconstruction
+(b)
+0
+0.2
+0.4
+0.6
+0
+0.8
+;4
+;1
+0.5
+;2
+0.6
+0.4
+0.2
+0
+Traj. 1
+Traj. 2
+Traj. 3
+(c)
+Figure 12: The prediction on the 6D SSM for the decay of (a) Trajectory 1 and
+(b) Trajectory 3. (c) Phase portrait of the amplitudes of the normal form coor-
+dinates on the SSM for each of the trajectories shows the modal contributions
+and development for different intial flow states.
+We project the training trajectories onto the SSM and transform them to
+the normal form in polar coordinates. The development of the amplitudes in the
+normal form are shown in Fig. 12c for each trajectory. This plot suggests that (i)
+the wave-breaking motion (Traj. 3) does not seem to have any significant content
+of the second mode, (ii) the amplitude of the fourth mode indeed increases after
+motor detachment, (iii) there is a small oscillation in these signals not captured
+by our model, which may be due to noise, insufficient separation of the modal
+subspaces, a mode outside our model, or some other phenomenon.
+0
+100
+200
+300
+400
+500
+x [mm]
+-100
+0
+100
+200
+Elevation h [mm]
+t = 1:2 s
+Experiment
+Simulation
+0
+100
+200
+300
+400
+500
+x [mm]
+t = 1:77 s
+0
+100
+200
+300
+400
+500
+x [mm]
+t = 14:49 s
+Figure 13: The experimentally measured surface profile decay agrees with our
+6D SSM model prediction for Trajectory 2.
+We note that the combined higher modal content in the signal is small -
+only about 10 % with respect to the first mode. This is because the data is
+decaying from steady states induced by forcing near the first eigenfrequency.
+Due to their symmetric shape, isolated forcing of the second and fourth modes
+is not possible with horizontal harmonic excitation. Nevertheless, we are able
+20
+
+to capture these smaller oscillations on the SSM. The key technology allowing
+this enhancement is the enforcement of the delay-embedded tangent space in
+our SSM reconstruction, based on the theoretical eigenfrequencies and mode
+shapes.
+Due to the small activation of the higher modes, our model is expected to be
+sensitive to noise. It is, however, stable with respect to changes in starting time,
+manifold order, and delay parameters. A more robust model can be obtained by
+decreasing the manifold dimension to 4, neglecting the relatively minor influence
+of the fourth mode, resulting in an average NMTE of 3.9 %. Here, since our
+objective was modal analysis of different flow states, we chose the more detailed
+6D model.
+5
+Conclusions
+We have shown that for a scalar observation of an invariant manifold tangent
+to a spectral subspace at a fixed point, the delay-embedded reconstruction of
+the tangent space is dependent only on the corresponding eigenvalues of the full
+system linearized at that point. In particular, we have proven that the columns
+of a Vandermonde matrix, given by repeated multiplication of the exponential
+of the eigenvalues times the timelag, are eigenvectors for the linearized system
+in the observable space. Therefore, the Vandermonde matrix diagonalizes the
+linear part of the delay-embedded dynamics. We have also shown that when sev-
+eral quantities are measured and delay-embedded simultaneously, the tangent
+space can be expressed given the Vandermonde matrix and the mode shapes
+expressed in the observable function components. These results hold for any
+invariant manifold tangent to a modal subspace with distinct eigenvalues, in-
+cluding, e.g., classic stable manifolds. Here, our focus was the application of
+this result to spectral submanifolds of hyperbolic fixed points.
+In an attempt to exploit this uncovered structure, we have shown that
+for data-driven SSM model reduction, when the eigenvalues are approximately
+known, we can analytically predict the tangent space of the embedded SSM
+a priori to achieve local modal decomposition and aid the reconstruction of the
+nonlinear reduced dynamics. We have found that even for small activation of
+higher modes, this trick helps modeling complex multimodal nonlinear dynam-
+ics on an SSM, which in turn allows for analysis of modal energy interchange
+and instantaneous frequencies.
+Our theory assumes a generic observable function, which we describe in
+more detail in our first and second example, and distinct eigenvalues. While
+the second assumption is a generic one in a mathematical sense, it is not always
+satisfied for engineering structures with symmetry. Vibrations in a square plate
+is an example where our theory would fail, as it has repeated eigenvalues. Using
+a vector-valued observable may help in differentiating between the modes in
+such a case. Further, while technically covered by the theory, possible practical
+difficulties related to the conditioning of the Vandermonde matrix include highly
+different or very similar eigenvalues, or eigenvalues of different stability type.
+21
+
+In our third example with data from experiments, we also devised a new
+heuristic scheme for using delay embedding to study modal contents in a signal.
+With this method, that served as an initial guess, we projected the data onto
+the respective prescribed modal subspaces, thereby implicitly assuming that the
+SSM of each mode is nearly flat. An interesting development of this idea would
+be its use as a filter, which could remove or keep certain frequencies in a signal.
+Another idea would be to use the tangent space condition as a verification or
+for iterative adjustment of the linear fit of the reduced dynamics. Finally, in
+analogy with a Fourier analysis, it would be possible to estimate both instanta-
+neous frequency and damping of a signal by singular value decomposition of the
+trajectory in delay coordinates followed by analysis of the Vandermonde matrix
+columns. This ties in with several other observations made in the literature; for
+example, for a linear system, these columns agree with the recently proposed
+notion of principal component trajectories [62]. Overall, we believe that our
+findings shed more light on delay-embedding invariant manifolds and selecting
+delay parameters in particular. For that reason, we expect these results to be
+of use for a wide range of data-driven methods.
+Acknowledgements
+We are grateful to Kerstin Avila and Bastian B¨auerlein (U. Bremen) for sharing
+their experimental surface profile data from Ref. [67] with us.
+A
+Appendix
+A.1
+Proof of Theorem 1
+Let M be a d-dimensional invariant manifold of (1) containing the origin of Rn.
+The tangent space of M at the origin can be written T0M = span {ek}k∈K,
+where K ⊂ {1, . . . , n} is an index set labeling the d eigenvectors ek spanning
+the spectral subspace from which the manifold is emanating. For example, for
+a stable manifold, K = {k : Re λk < 0}. We also assume that the eigenvalues
+in question are distinct, i ̸= k ⇔ λi ̸= λk, for i, k ∈ K.
+To simplify the notation, we transform the full state space to modal coor-
+dinates (11). We rewrite the observable function on the system x ∈ Rn as an
+observable on the modal coordinate system z ∈ Cn as ˆµ(z) = µ(Ez). We de-
+note by Φt = E ◦F t ◦E−1 the flow in Cn. Consider the sampling map in modal
+coordinates
+ˆS =
+�
+������
+ˆµ
+ˆµ ◦ Φτ
+ˆµ ◦ Φ2τ
+...
+ˆµ ◦ Φ(p−1)τ
+�
+������
+: Cn → Rp,
+y = ˆS(z).
+(31)
+22
+
+Under the conditions of Takens’s embedding theorem, the delay embedding
+map Ψ = ˆS|M : M →
+˜
+M is a smooth embedding with a smooth inverse
+Ψ−1 : ˜
+M → M, and Ψ(0) = q.
+In order to derive the tangent space Tq ˜
+M, we compute the derivative of the
+embedding at 0:
+DΨ(0) =
+�
+����
+Dˆµ(0)
+Dˆµ(Φτ(0)) ◦ DΦτ(0)
+...
+Dˆµ(Φ(p−1)τ(0)) ◦ DΦ(p−1)τ(0)
+�
+���� =
+�
+����
+Dˆµ(0)
+Dˆµ(0) ◦ eΛτ
+...
+Dˆµ(0) ◦ eΛ(p−1)τ
+�
+���� .
+(32)
+Now note that the jth component expressed in modal coordinates is
+Dˆµ(0) ◦ eΛjτ(z) =
+�
+k∈K
+∂ˆµ
+∂zk
+����
+0
+eλkjτzk,
+j ∈ {1, . . . , p}.
+(33)
+We define the Vandermonde matrix V of the d eigenvalues {λk}k∈K governing
+the linearized dynamics on M as Vjk = eλkjτ. We conclude that the tangent
+space of the observable manifold at the fixed point in modal coordinates can be
+written as
+Tq ˜
+M = {DΨ(0)z, z ∈ Cn} = range
+�
+V diag
+� ∂ˆµ
+∂z
+����
+0
+��
+= range V ,
+(34)
+where the diagonal matrix acts only as a rescaling of each component of z.
+Therefore, one matrix representation of the tangent space of the manifold in
+the observable space is V , which is independent both of the matrix E of full
+system eigenvectors and the observable function µ.
+Note that we must have
+∂ ˆµ
+∂zk |0 ̸= 0
+∀k ∈ K, which defines the genericity of
+µ. In practice, this implies that the linearized observable function must contain
+contributions from all modal coordinates that we wish to model. In addition,
+note that for the embedding of the tangent space itself, i.e., the linear system,
+p = d suffices.
+A.2
+Proof of Theorem 2
+The flow on
+˜
+M is
+˜Φt = Ψ ◦ Φt ◦ Ψ−1.
+(35)
+We compute the ODE on
+˜
+M, ˙y = ˜f(y), as
+˜f = d
+dt
+˜Φt = DΨ ◦ f ◦ Ψ−1,
+(36)
+where f is given by (11). The derivative of ˜f at the fixed point is, therefore,
+D ˜f(q) = DΨ(0) ◦ Λ ◦ DΨ−1(q) = V diag
+� ∂ˆµ
+∂z
+����
+0
+�
+Λ diag
+� ∂ˆµ
+∂z
+����
+0
+�−1
+V †
+= V ΛV †,
+(37)
+23
+
+where we used the commutative property of multiplication of diagonal matri-
+ces, which eliminates the linearized observable function terms, and the fact
+that DΨ−1(q) = diag
+�
+∂ ˆµ
+∂z
+���
+0
+�−1
+V † is well-defined. To see this, note that the
+derivative of the delay embedding map composed with its inverse
+DΨ(0) ◦ DΨ−1(q) = V diag
+� ∂ˆµ
+∂z
+����
+0
+�
+diag
+� ∂ˆµ
+∂z
+����
+0
+�−1
+V † = V V †,
+(38)
+maps all points in T0M to themselves, since V has full rank under the assump-
+tion of distinct eigenvalues {λk}k∈K.
+Taylor-expanding the ODE on
+˜
+M in the observable space yields
+˙y = D ˜f(q)(y − q) + o(|y − q|) = V ΛV †(y − q) + o(|y − q|).
+(39)
+Therefore, under the assumptions of a generic observable function and distinct
+eigenvalues, the tangent space Tq ˜
+M and the linearized dynamics D ˜f(q) in
+the observable space Rp are both fully determined by the timelag τ and the
+eigenvalues λk, k ∈ K.
+In the special case that M = Rn, if p ≥ 2n + 1, the entire phase space can
+be reconstructed. For a linear system, p = n suffices, and the delay embedding
+reduces to a linear operator.
+A.3
+Proof of Theorem 3
+For a vector-valued observable, µ : Rn → Rq, the delay embedding map reads
+Ψ =
+�
+��
+Ψµ1
+...
+Ψµq
+�
+�� : M → ˜
+M ⊂ Rpq,
+DΨ(0) =
+�
+��
+DΨµ1(0)
+...
+DΨµq(0)
+�
+�� ,
+(40)
+where Ψµℓ is the delay embedding map corresponding to the ℓth component of
+the observable function µ. The derivatives are given by
+DΨµℓ(0) = V diag
+� ∂ˆµℓ
+∂z
+����
+0
+�
+.
+(41)
+In this case, the tangent space is not independent of the observable function.
+Instead, it is affected by the relative dependency of each component µℓ of the ob-
+servable function on each modal coordinate zk. The tangent space can, however,
+be expressed as
+Tq ˜
+M = range
+�
+����
+V diag
+�
+∂ ˆµ1
+∂z
+���
+0
+�
+...
+V diag
+�
+∂ ˆµq
+∂z
+���
+0
+�
+�
+���� .
+(42)
+24
+
+References
+[1]
+J. L. Lumley. “The Structure of Inhomogeneous Turbulent Flows”. Atmo-
+spheric Turbulence and Radio Wave Propagation (1967), pp. 166–177.
+[2]
+J. Awrejcewicz, V. A. Krys’ko, and A. F. Vakakis. “Order Reduction
+by Proper Orthogonal Decomposition (POD) Analysis”. Nonlinear Dy-
+namics of Continuous Elastic Systems. Springer, Berlin, Heidelberg, 2004,
+pp. 279–320.
+[3]
+P. Schmid. “Dynamic mode decomposition of numerical and experimental
+data”. J. Fluid Mech. 656 (2010), pp. 5–28.
+[4]
+J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor. Dynamic
+Mode Decomposition. Philadelphia, PA: SIAM, 2016.
+[5]
+P. J. Schmid. “Dynamic Mode Decomposition and Its Variants”. Annual
+Review of Fluid Mechanics 54.1 (2022), pp. 225–254. doi: 10 . 1146 /
+annurev-fluid-030121-015835.
+[6]
+J. Page and R. Kerswell. “Koopman mode expansions between simple
+invariant solutions”. J. Fluid Mech. 879 (2019), pp. 1–27.
+[7]
+S. L. Brunton, J. L. Proctor, and J. N. Kutz. “Discovering governing equa-
+tions from data by sparse identification of nonlinear dynamical systems”.
+Proceedings of the National Academy of Sciences 113.15 (2016), pp. 3932–
+3937.
+[8]
+D. Bertsimas and W. Gurnee. Learning Sparse Nonlinear Dynamics via
+Mixed-Integer Optimization. 2022. doi: 10.48550/ARXIV.2206.00176.
+url: https://arxiv.org/abs/2206.00176.
+[9]
+J. N. Kutz and S. L. Brunton. “Parsimony as the ultimate regularizer
+for physics-informed machine learning”. Nonlinear Dynamics 107.3 (Jan.
+2022), pp. 1801–1817. doi: 10.1007/s11071-021-07118-3. url: https:
+//doi.org/10.1007/s11071-021-07118-3.
+[10]
+S. Chen and S. A. Billings. “Neural networks for nonlinear dynamic system
+modelling and identification”. International Journal of Control 56.2 (Aug.
+1992), pp. 319–346. doi: 10 . 1080 / 00207179208934317. url: https :
+//doi.org/10.1080/00207179208934317.
+[11]
+T. Daniel, F. Casenave, N. Akkari, and D. Ryckelynck. “Model order re-
+duction assisted by deep neural networks (ROM-net)”. Adv. Model. and
+Simul. in Eng. Sci. 7 (2020), p. 105786.
+[12]
+L. Salmela, N. Tsipinakis, A. Foi, C. Billet, J. M. Dudley, and G. Genty.
+“Predicting ultrafast nonlinear dynamics in fibre optics with a recurrent
+neural network”. Nature Machine Intelligence 3.4 (Feb. 2021), pp. 344–
+354. doi: 10.1038/s42256-021-00297-z. url: https://doi.org/10.
+1038/s42256-021-00297-z.
+25
+
+[13]
+J.-C. Loiseau, S. L. Brunton, and B. R. Noack. “From the POD-Galerkin
+method to sparse manifold models”. Model Order Reduction, Volume 3:
+Applications. Ed. by P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza,
+W. Schilders, and L. M. Silveira. De Gruyter, Berlin, 2020, pp. 279–320.
+[14]
+Cabr´e, E. Fontich, and R. de la Llave. “The parameterization method for
+invariant manifolds I: Manifolds associated to non-resonant subspaces”.
+Indiana Univ. Math. J. 52.2 (2003), pp. 283–328.
+[15]
+A. Haro and R. de la Llave. “A parameterization method for the compu-
+tation of invariant tori and their whiskers in quasi-periodic maps: rigorous
+results”. J. Differential Eqs. 228.2 (2006), pp. 530–579.
+[16]
+G. Haller and S. Ponsioen. “Nonlinear normal modes and spectral sub-
+manifolds: existence, uniqueness and use in model reduction”. Nonlinear
+Dyn. 86.3 (2016), pp. 1493–1534.
+[17]
+S. Ponsioen, T. Pedergnana, and G. Haller. “Automated computation of
+autonomous spectral submanifolds for nonlinear modal analysis”. J. Sound
+and Vibration 420 (2018), pp. 269–295.
+[18]
+R. M. Rosenberg. “The Normal Modes of Nonlinear n-Degree-of-Freedom
+Systems”. Journal of Applied Mechanics 29.1 (Mar. 1962), pp. 7–14. doi:
+10.1115/1.3636501. url: https://doi.org/10.1115/1.3636501.
+[19]
+A. Vakakis. “Non-linear normal modes (NNMs) and their applications in
+vibration theory: an overview”. Mechanical Systems and Signal Processing
+11.1 (Jan. 1997), pp. 3–22. doi: 10.1006/mssp.1996.9999. url: https:
+//doi.org/10.1006/mssp.1996.9999.
+[20]
+G. Kerschen, M. Peeters, J. Golinval, and A. Vakakis. “Nonlinear nor-
+mal modes, Part I: A useful framework for the structural dynamicist”.
+Mechanical Systems and Signal Processing 23.1 (Jan. 2009), pp. 170–194.
+doi: 10.1016/j.ymssp.2008.04.002. url: https://doi.org/10.1016/
+j.ymssp.2008.04.002.
+[21]
+S. Shaw and C. Pierre. “Non-linear normal modes and invariant mani-
+folds”. Journal of Sound and Vibration 150.1 (Oct. 1991), pp. 170–173.
+doi: 10.1016/0022-460x(91)90412-d. url: https://doi.org/10.
+1016/0022-460x(91)90412-d.
+[22]
+S. Shaw and C. Pierre. “Normal modes for non-linear vibratory systems”.
+J. Sound and Vibration 164.1 (1993), pp. 85–124.
+[23]
+R. Szalai. “Invariant spectral foliations with applications to model order
+reduction and synthesis”. Nonlinear Dyn. 101 (2020), pp. 2645–2669.
+[24]
+S. Jain, T. Thurner, M. Li, and G. Haller. SSMTool: Computation of
+invariant manifolds and their reduced dynamics in high-dimensional me-
+chanics problems. 2021. doi: 10 . 5281 / zenodo . 4614201. url: http :
+//www.georgehaller.com.
+[25]
+S. Ponsioen, T. Pedergnana, and G. Haller. “Analytic prediction of iso-
+lated forced response curves from spectral submanifolds”. Nonlinear Dyn.
+98 (2019), pp. 2755–2773.
+26
+
+[26]
+S. Ponsioen, S. Jain, and G. Haller. “Model reduction to spectral sub-
+manifolds and forced-response calculation in high-dimensional mechanical
+systems”. J. Sound and Vibration 488 (2020), p. 115640.
+[27]
+S. Jain and G. Haller. “How to compute invariant manifolds and their
+reduced dynamics in high-dimensional finite element models?” Nonlinear
+Dynamics 107.2 (Oct. 2021), pp. 1417–1450. doi: 10.1007/s11071-021-
+06957-4. url: https://doi.org/10.1007/s11071-021-06957-4.
+[28]
+M. Li, S. Jain, and G. Haller. “Nonlinear analysis of forced mechanical
+systems with internal resonance using spectral submanifolds – Part I: Pe-
+riodic response and forced response curve”. Nonlinear Dynamics (2022).
+[29]
+M. Li and G. Haller. “Nonlinear analysis of forced mechanical systems
+with internal resonance using spectral submanifolds – Part II: Bifurcation
+and quasi-periodic response”. Nonlinear Dynamics (2022). doi: https:
+//doi.org/10.1007/s11071-022-07476-6.
+[30]
+M. Li, S. Jain, and G. Haller. Model reduction for constrained mechanical
+systems via spectral submanifolds. 2022. doi: 10.48550/ARXIV.2208.
+03119. url: https://arxiv.org/abs/2208.03119.
+[31]
+M. Cenedese, J. Ax˚as, B. B¨auerlein, K. Avila, and G. Haller. “Data-driven
+modeling and prediction of non-linearizable dynamics via spectral sub-
+manifolds”. Nat. Commun. 13.1 (Feb. 2022). doi: 10.1038/s41467-022-
+28518-y. url: https://doi.org/10.1038/s41467-022-28518-y.
+[32]
+M. Cenedese, J. Ax˚as, and G. Haller. SSMLearn. 2021. url: http://www.
+georgehaller.com.
+[33]
+J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Sys-
+tems and Bifircation of Vector Fields. Springer, New York, 1983.
+[34]
+M. Cenedese, J. Ax˚as, H. Yang, M. Eriten, and G. Haller. “Data-driven
+nonlinear model reduction to spectral submanifolds in mechanical sys-
+tems”. Philosophical Transactions of the Royal Society A: Mathematical,
+Physical and Engineering Sciences 380.2229 (June 2022). doi: 10.1098/
+rsta.2021.0194. url: https://doi.org/10.1098/rsta.2021.0194.
+[35]
+B. Kasz´as, M. Cenedese, and G. Haller. “Dynamics-based machine learn-
+ing of transitions in Couette flow”. Physical Review Fluids 7 (8 2022),
+p. L082402.
+[36]
+J. I. Alora, M. Cenedese, E. Schmerling, G. Haller, and M. Pavone. Data-
+Driven Spectral Submanifold Reduction for Nonlinear Optimal Control of
+High-Dimensional Robots. 2022. doi: 10.48550/ARXIV.2209.05712. url:
+https://arxiv.org/abs/2209.05712.
+[37]
+J. Ax˚as, M. Cenedese, and G. Haller. “Fast data-driven model reduction
+for nonlinear dynamical systems”. Nonlinear Dyn. (2022).
+[38]
+M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. “A Data–Driven
+Approximation of the Koopman Operator: Extending Dynamic Mode De-
+composition”. J Nonlinear Sci. 9 (2015), pp. 1307–1346.
+27
+
+[39]
+D. Dylewsky, D. Barajas-Solano, T. Ma, A. M. Tartakovsky, and J. N.
+Kutz. “Stochastically Forced Ensemble Dynamic Mode Decomposition
+for Forecasting and Analysis of Near-Periodic Systems”. IEEE Access 10
+(2022), pp. 33440–33448. doi: 10.1109/ACCESS.2022.3161438.
+[40]
+S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz.
+“Chaos as an intermittently forced linear system”. Nat. Commun. 8.1
+(May 2017). doi: 10.1038/s41467-017-00030-8. url: https://doi.
+org/10.1038/s41467-017-00030-8.
+[41]
+J.-N. Juang and R. S. Pappa. “An eigensystem realization algorithm for
+modal parameter identification and model reduction”. Journal of Guid-
+ance, Control, and Dynamics 8.5 (Sept. 1985), pp. 620–627. doi: 10.
+2514/3.20031. url: https://doi.org/10.2514/3.20031.
+[42]
+S. Pan and K. Duraisamy. “Data-Driven Discovery of Closure Models”.
+SIAM Journal on Applied Dynamical Systems 17.4 (Jan. 2018), pp. 2381–
+2413. doi: 10.1137/18m1177263. url: https://doi.org/10.1137/
+18m1177263.
+[43]
+A. Pikovsky. “Noise filtering in the discrete time dynamical systems”. Sov.
+J. Commun. Technol. Electron 31.5 (1986), pp. 911–914.
+[44]
+G. Sugihara and R. M. May. “Nonlinear forecasting as a way of distin-
+guishing chaos from measurement error in time series”. Nature 344.6268
+(Apr. 1990), pp. 734–741. doi: 10.1038/344734a0. url: https://doi.
+org/10.1038/344734a0.
+[45]
+J. P. Crutchfield. “Prediction and stability in classical mechanics”. Senior
+thesis in physics and mathematics, University of California, Santa Cruz.
+1979.
+[46]
+N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. “Geometry
+from a Time Series”. Phys. Rev. Lett. 45 (9 1980), pp. 712–716. doi:
+10.1103/PhysRevLett.45.712. url: https://link.aps.org/doi/10.
+1103/PhysRevLett.45.712.
+[47]
+F. Takens. “Detecting strange attractors in turbulence”. Dynamical Sys-
+tems and Turbulence, Warwick 1980. Ed. by D. Rand and L. Young.
+Springer Berlin Heidelberg, 1981, pp. 366–381.
+[48]
+T. Sauer, J. Yorke, and M. Casdagli. “Embedology”. J. Stat. Phys. 65
+(1991), pp. 579–616.
+[49]
+D. Broomhead and G. P. King. “Extracting qualitative dynamics from ex-
+perimental data”. Physica D: Nonlinear Phenomena 20.2 (1986), pp. 217–
+236. issn: 0167-2789. doi: https://doi.org/10.1016/0167-2789(86)
+90031-X. url: https://www.sciencedirect.com/science/article/
+pii/016727898690031X.
+28
+
+[50]
+M. Casdagli, S. Eubank, J. Farmer, and J. Gibson. “State space recon-
+struction in the presence of noise”. Physica D: Nonlinear Phenomena 51.1
+(1991), pp. 52–98. issn: 0167-2789. doi: https://doi.org/10.1016/
+0167 - 2789(91 ) 90222 - U. url: https : / / www . sciencedirect . com /
+science/article/pii/016727899190222U.
+[51]
+H. L. Yap and C. Rozell. “Stable Takens’ Embedding for Linear Dynamical
+Systems”. Vol. 59. Dec. 2010, pp. 2948–2953. doi: 10.1109/TSP.2011.
+2160629.
+[52]
+A. M. Fraser and H. L. Swinney. “Independent coordinates for strange
+attractors from mutual information”. Phys. Rev. A 33 (2 Feb. 1986),
+pp. 1134–1140. doi: 10.1103/PhysRevA.33.1134. url: https://link.
+aps.org/doi/10.1103/PhysRevA.33.1134.
+[53]
+M. B. Kennel, R. Brown, and H. D. I. Abarbanel. “Determining embed-
+ding dimension for phase-space reconstruction using a geometrical con-
+struction”. Phys. Rev. A 45 (6 Mar. 1992), pp. 3403–3411. doi: 10.1103/
+PhysRevA . 45 . 3403. url: https : / / link . aps . org / doi / 10 . 1103 /
+PhysRevA.45.3403.
+[54]
+H. D. I. Abarbanel and M. B. Kennel. “Local false nearest neighbors
+and dynamical dimensions from observed chaotic data”. Phys. Rev. E 47
+(5 May 1993), pp. 3057–3068. doi: 10.1103/PhysRevE.47.3057. url:
+https://link.aps.org/doi/10.1103/PhysRevE.47.3057.
+[55]
+K. H. Kraemer, M. Gelbrecht, I. Pavithran, R. I. Sujith, and N. Marwan.
+“Optimal state space reconstruction via Monte Carlo decision tree search”.
+Nonlinear Dynamics 108.2 (Mar. 2022), pp. 1525–1545. doi: 10.1007/
+s11071-022-07280-2. url: https://doi.org/10.1007/s11071-022-
+07280-2.
+[56]
+E. Bozzo, R. Carniel, and D. Fasino. “Relationship between Singular Spec-
+trum Analysis and Fourier analysis: Theory and application to the mon-
+itoring of volcanic activity”. Computers and Mathematics with Applica-
+tions 60.3 (Aug. 2010), pp. 812–820. doi: 10.1016/j.camwa.2010.05.
+028. url: https://doi.org/10.1016/j.camwa.2010.05.028.
+[57]
+S. Pan and K. Duraisamy. “On the structure of time-delay embedding
+in linear models of non-linear dynamical systems”. Chaos: An Interdis-
+ciplinary Journal of Nonlinear Science 30.7 (July 2020), p. 073135. doi:
+10.1063/5.0010886. url: https://doi.org/10.1063/5.0010886.
+[58]
+C. W. Rowley, I. Mezi´c, S. Bagheri, P. Schlachter, and D. Henningson.
+“Spectral analysis of nonlinear flows”. J. Fluid Mech. 641 (2009), pp. 115–
+127.
+[59]
+Z. Drmaˇc, I. Mezi´c, and R. Mohr. “Data Driven Koopman Spectral Anal-
+ysis in Vandermonde–Cauchy Form via the DFT: Numerical Method and
+Theoretical Insights”. SIAM Journal on Scientific Computing 41.5 (2019),
+A3118–A3151. doi: 10.1137/18M1227688. url: https://doi.org/10.
+1137/18M1227688.
+29
+
+[60]
+M. Kamb, E. Kaiser, S. L. Brunton, and J. N. Kutz. Time-Delay Ob-
+servables for Koopman: Theory and Applications. 2018. doi: 10.48550/
+ARXIV.1810.01479. url: https://arxiv.org/abs/1810.01479.
+[61]
+S. M. Hirsh, S. M. Ichinaga, S. L. Brunton, J. N. Kutz, and B. W. Brunton.
+“Structured time-delay models for dynamical systems with connections to
+Frenet–Serret frame”. Proceedings of the Royal Society A: Mathematical,
+Physical and Engineering Sciences 477.2254 (Oct. 2021). doi: 10.1098/
+rspa.2021.0097. url: https://doi.org/10.1098/rspa.2021.0097.
+[62]
+D. Dylewsky, E. Kaiser, S. L. Brunton, and J. N. Kutz. “Principal com-
+ponent trajectories for modeling spectrally continuous dynamics as forced
+linear systems”. Phys. Rev. E 105 (1 Jan. 2022), p. 015312. doi: 10.1103/
+PhysRevE.105.015312. url: https://link.aps.org/doi/10.1103/
+PhysRevE.105.015312.
+[63]
+E. Bronstein, A. Wiegner, D. Shilo, and R. Talmon. “The spatiotemporal
+coupling in delay-coordinates dynamic mode decomposition”. Chaos: An
+Interdisciplinary Journal of Nonlinear Science 32.12 (Dec. 2022), p. 123127.
+doi: 10.1063/5.0123101. url: https://doi.org/10.1063/5.0123101.
+[64]
+J. Ax˚as and G. Haller. fastSSM: Algorithm for fast computation of spec-
+tral submanifolds from data. 2022. url: https://github.com/haller-
+group/SSMLearn/tree/main/fastSSM.
+[65]
+E. R. Deyle and G. Sugihara. “Generalized Theorems for Nonlinear State
+Space Reconstruction”. PLoS ONE 6.3 (2011). url: https://doi.org/
+10.1371/journal.pone.0018295.
+[66]
+S. Jain, P. Tiso, and G. Haller. “Exact nonlinear model reduction for a
+von K´arm´an beam: slow-fast decomposition and spectral submanifolds”.
+Journal of Sound and Vibration 423 (2018), pp. 195–211.
+[67]
+B. B¨auerlein and K. Avila. “Phase lag predicts nonlinear response maxima
+in liquid-sloshing experiments”. J. Fluid Mech. 925 (2021), A22. doi: 10.
+1017/jfm.2021.576.
+[68]
+S. Mitra, L. V. Hai, L. Jing, and B. C. Khoo. “A fully coupled ship motion
+and sloshing analysis in various container geometries”. Journal of Marine
+Science and Technology 17.2 (Feb. 2012), pp. 139–153. doi: 10.1007/
+s00773-012-0157-2. url: https://doi.org/10.1007/s00773-012-
+0157-2.
+[69]
+F. Fleissner, A. Lehnart, and P. Eberhard. “Dynamic simulation of slosh-
+ing fluid and granular cargo in transport vehicles”. Vehicle System Dy-
+namics 48.1 (Jan. 2010), pp. 3–15. doi: 10.1080/00423110903042717.
+url: https://doi.org/10.1080/00423110903042717.
+[70]
+J. Hickey, B. Broderick, B. Fitzgerald, and H. Moore. “Mitigation of
+wind induced accelerations in tall modular buildings”. Structures 37 (Mar.
+2022), pp. 576–587. doi: 10.1016/j.istruc.2022.01.037. url: https:
+//doi.org/10.1016/j.istruc.2022.01.037.
+30
+
+[71]
+F. Dodge. The New ”Dynamic Behavior of Liquids in Moving Contain-
+ers”. Southwest Research Inst., 2000. url: https://books.google.ch/
+books?id=RltitwAACAAJ.
+[72]
+H. Abramson, ed. The dynamic behavior of liquids in moving contain-
+ers: with applications to space vehicle technology / edited by H. Norman
+Abramson. NASA SP-106. Washington, D.C: Scientific, Technical Infor-
+mation Division, National Aeronautics, and Space Administration, 1966.
+[73]
+G. Taylor. “An experimental study of standing waves”. Proceedings of the
+Royal Society of London. Series A. Mathematical and Physical Sciences
+218.1132 (1953), pp. 44–59.
+[74]
+O. Faltinsen and A. Timokha. Sloshing. Cambridge University Press, 2009.
+isbn: 9780521881111. url: https : / / books . google . ch / books ? id =
+81qkPwAACAAJ.
+[75]
+G. S. Narimanov. “Movement of a tank partly filled by a fluid: the taking
+into account of non-smallness of amplitude”. Prikl. Mat. Mekh. 21 (1957).
+in Russian, pp. 513–524.
+[76]
+O. M. Faltinsen, O. F. Rognebakke, I. A. Lukovsky, and A. N. Timokha.
+“Multidimensional modal analysis of nonlinear sloshing in a rectangu-
+lar tank with finite water depth”. Journal of Fluid Mechanics 407 (Mar.
+2000), pp. 201–234. doi: 10 . 1017 / s0022112099007569. url: https :
+//doi.org/10.1017/s0022112099007569.
+31
+