Showing papers in "Journal of Statistical Mechanics: Theory and Experiment in 2005"
TL;DR: It is found that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes.
Abstract: We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.
2,630 citations
TL;DR: In this paper, the authors studied the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian.
Abstract: We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.
1,191 citations
TL;DR: It is demonstrated that one can (i) find modules in complex networks and (ii) classify nodes into universal roles according to their pattern of within- and between-module connections, which yields a 'cartographic representation' of complex networks.
Abstract: Integrative approaches to the study of complex systems demand that one knows the manner in which the parts comprising the system are connected. The structure of the complex network defining the interactions provides insight into the function and evolution of the components of the system. Unfortunately, the large size and intricacy of these networks implies that such insight is usually difficult to extract. Here, we propose a method that allows one to systematically extract and display information contained in complex networks. Specifically, we demonstrate that one can (i) find modules in complex networks and (ii) classify nodes into universal roles according to their pattern of within- and between-module connections. The method thus yields a 'cartographic representation' of complex networks.
665 citations
TL;DR: In this paper, a linear relation between the velocity and the inverse of the density of a single-file movement along a line has been analyzed, which can be regarded as the required length for one pedestrian to move.
Abstract: The empirical relation between density and velocity of pedestrian movement has not been completely analysed, particularly with regard to the 'microscopic' causes which determine the relation at medium and high densities. The simplest system for the investigation of this dependence is the normal movement of pedestrians along a line (single-file movement). This paper presents experimental results for this system under laboratory conditions and discusses the following observations: the data show a linear relation between the velocity and the inverse of the density, which can be regarded as the required length for one pedestrian to move. Furthermore, we compare the results for the single-file movement with literature data for the movement in a plane. This comparison shows an unexpected conformance between the fundamental diagrams, indicating that lateral interference has negligible influence on the velocity–density relation in the density domain 1 m−2 > ρ > 5 m−2. In addition, we test a procedure for automatic recording of pedestrian flow characteristics. We present preliminary results on the measurement range and accuracy of this method.
546 citations
TL;DR: In this paper, the main theoretical concepts and techniques in the field of mean-field spin glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student.
Abstract: In these notes the main theoretical concepts and techniques in the field of mean-field spin glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spin-glass model is analysed (the p-spin spherical model) by using three different approaches: thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition; dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the mode coupling approximation, and the dynamical transition; and finally complexity, covering the mean-field (Thouless–Anderson–Palmer) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid to the mutual consistency of the results obtained from the different methods.
399 citations
TL;DR: In this article, the authors present exact results on universal quantities derived from the local density matrix ρ, for a free massive Dirac field in two dimensions, which can be written exactly in terms of the solutions of non-linear differential equations of the Painleve V type.
Abstract: We present some exact results on universal quantities derived from the local density matrix ρ, for a free massive Dirac field in two dimensions. We first find trρn in a novel fashion, which involves the correlators of suitable operators in the sine–Gordon model. These, in turn, can be written exactly in terms of the solutions of non-linear differential equations of the Painleve V type. Equipped with the previous results, we find the leading terms for the entanglement entropy, both for short and long distances, and showing that in the intermediate regime it can be expanded in a series of multiple integrals. The previous results have been checked by direct numerical calculations on the lattice, finding perfect agreement. Finally, we comment on a possible generalization of the entanglement entropy c-theorem to the alpha-entropies.
373 citations
TL;DR: In this paper, it was shown that the non-linear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation, and the usual backward computation can be replaced by a forward diffusion process that can be computed by stochastic integration or by the evaluation of a path integral.
Abstract: This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of stochastic control problems the non-linear Hamilton–Jacobi–Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton–Jacobi equation to the Schrodinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process that can be computed by stochastic integration or by the evaluation of a path integral. It is shown how in the deterministic limit the Pontryagin minimum principle formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as Monte Carlo sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in a number of examples. Examples are given that show the qualitative difference between stochastic and deterministic control and the occurrence of symmetry breaking as a function of the noise.
339 citations
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TL;DR: In this article, the authors review the mode-coupling theory of the glass transition from several perspectives, and discuss recent advances in the applications of the theory and its applications.
Abstract: In this set of lecture notes we review the mode-coupling theory of the glass transition from several perspectives. First, we derive mode-coupling equations for the description of density fluctuations from microscopic considerations with the use the Mori–Zwanzig projection operator technique. We also derive schematic mode-coupling equations of a similar form from a field-theoretic perspective. We review the successes and failures of mode-coupling theory, and discuss recent advances in the applications of the theory.
308 citations
TL;DR: In this paper, the potential energy landscape thermodynamic formalism and its applications to the study of supercooled glass forming liquids are discussed. But the authors focus on the statistical properties of the landscape, i.e. the number and the distribution in energy of the local minima of the surface for bulk systems.
Abstract: These notes review the potential energy landscape thermodynamic formalism and some of its recent applications to the study of supercooled glass forming liquids. They also review the techniques which have been recently developed to quantify the statistical properties of the landscape, i.e. the number and the distribution in energy of the local minima of the surface for bulk systems. Ac ritical examination of the approximations involved in such a calculation and results for models of simple and molecular liquids are reported. Finally, these notes discuss how an equation of state, expressed only in terms of statistical properties of the landscape, can be derived and under which conditions such an equation of state can be generalized to describe out-of-equilibrium liquids.
229 citations
TL;DR: The presented results suggest that the interplay between weight dynamics and spatial constraints is a key ingredient in order to understand the formation of real-world weighted networks.
Abstract: Motivated by the empirical analysis of the air transportation system, we define a network model that includes geographical attributes along with topological and weight (traffic) properties. The introduction of geographical attributes is made by constraining the network in real space. Interestingly, the inclusion of geometrical features induces non-trivial correlations between the weights, the connectivity pattern and the actual spatial distances of vertices. The model also recovers the emergence of anomalous fluctuations in the betweenness-degree correlation function as first observed by Guimera and Amaral (2004 Eur. Phys. J. B 38 381). The presented results suggest that the interplay between weight dynamics and spatial constraints is a key ingredient in order to understand the formation of real-world weighted networks.
191 citations
TL;DR: In this paper, the Jordan-Wigner transformation was extended to dimensions higher than one by introducing auxiliary degrees of freedom, represented by Majorana fermions, which allowed the transformation to be applied to local spin problems on a lattice in any dimension.
Abstract: We show how to map local fermionic problems onto local spin problems on a lattice in any dimension. The main idea is to introduce auxiliary degrees of freedom, represented by Majorana fermions, which allow us to extend the Jordan–Wigner transformation to dimensions higher than one. We also discuss the implications of our results in the numerical investigation of fermionic systems.
TL;DR: In this article, the authors compute all dynamical spin-spin correlation functions for the spin-1/2/XXZ anisotropic Heisenberg model in the gapless antiferromagnetic regime at zero temperature, using numerical sums of exact determinant representations for form factors of spin operators.
Abstract: We compute all dynamical spin–spin correlation functions for the spin-1/2 XXZ anisotropic Heisenberg model in the gapless antiferromagnetic regime at zero temperature, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier transform our results back to real space, allowing us in particular to make a comparison with known exact formulae for equal-time correlation functions for small separations in zero field, and with predictions for the zero-field asymptotics from conformal field theory.
TL;DR: In this paper, the long-time limit of the integrated current distribution for the one-dimensional zero-range process with open boundaries is discussed, and it is shown that the current fluctuations become site dependent above some critical current.
Abstract: We discuss the long-time limit of the integrated current distribution for the one-dimensional zero-range process with open boundaries. We observe that the current fluctuations become site dependent above some critical current and argue that this is a precursor of the condensation transition which occurs in such models. Our considerations for the totally asymmetric zero-range process are complemented by a Bethe ansatz treatment for the equivalent exclusion process.
TL;DR: In this article, an extension of the fluctuation theorem to stochastic models that, in the limit of zero external drive, are not able to equilibrate with their environment was discussed.
Abstract: We discuss an extension of the fluctuation theorem to stochastic models that, in the limit of zero external drive, are not able to equilibrate with their environment, extending earlier results of Sellitto. We show that if the entropy production rate is suitably defined, its probability distribution function verifies the fluctuation relation with the ambient temperature replaced by a (frequency dependent) effective temperature. We derive modified Green–Kubo relations. We illustrate these results with the simple example of an oscillator coupled to a non-equilibrium bath driven by an external force. We discuss the relevance of our results for driven glasses and the diffusion of Brownian particles in out-of-equilibrium media and propose a concrete experimental strategy for measuring the low frequency value of the effective temperature using the fluctuations of the work done by an ac conservative field. We compare our results to related ones that appeared in the literature recently.
TL;DR: In this article, a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices is introduced. But unlike cluster variational methods, the new approach takes into account fluctuations on all length scales.
Abstract: We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.
TL;DR: In this paper, a one-parameter ranking method was proposed based on a network representation of college football schedules, which is mathematically well founded, gives results in general accord with received wisdom concerning the relative strengths of the teams, and is based upon intuitive principles, allowing it to be accepted readily by fans and experts alike.
Abstract: American college football faces a conflict created by the desire to stage national championship games between the best teams of a season when there is no conventional play-off system for deciding which those teams are. Instead, ranking of teams is based on their records of wins and losses during the season, but each team plays only a small fraction of eligible opponents, making the system underdetermined or contradictory or both. It is an interesting challenge to create a ranking system that at once is mathematically well founded, gives results in general accord with received wisdom concerning the relative strengths of the teams, and is based upon intuitive principles, allowing it to be accepted readily by fans and experts alike. Here we introduce a one-parameter ranking method that satisfies all of these requirements and is based on a network representation of college football schedules.
TL;DR: A modified belief propagation algorithm, with over-relaxed dynamics, that turns out to be generally more stable and faster than ordinary belief propagation on the random satisfiability problem.
Abstract: We propose a modified belief propagation algorithm, with over-relaxed dynamics. Such an algorithm turns out to be generally more stable and faster than ordinary belief propagation. We characterize the performance of the algorithm, employed as a tool for combinatorial optimization, on the random satisfiability problem. Moreover, we trace a connection with a recently proposed double-loop algorithm for minimizing Bethe and Kikuchi free energies.
TL;DR: In this paper, the XXZ open spin chain with general integrable boundary conditions is shown to possess a q-deformed analogue of the Onsager's algebra as fundamental non-Abelian symmetry which ensures the integrability of the model.
Abstract: The XXZ open spin chain with general integrable boundary conditions is shown to possess a q-deformed analogue of the Onsager's algebra as fundamental non-Abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an Abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.
TL;DR: In this paper, a generic continuum model is proposed to account for randomness in the local stress-strain relationships as well as for long-range internal stresses that arise from the ensuing plastic strain heterogeneities.
Abstract: On microscopic and mesoscopic scales, plastic flow of crystals is characterized by large intrinsic fluctuations. Deformation by crystallographic slip occurs in a sequence of intermittent bursts ('slip avalanches') with power-law size distribution. In the spatial domain, these avalanches produce characteristic deformation patterns in the form of slip lines and slip bands which exhibit long-range spatial correlations. We propose a generic continuum model which accounts for randomness in the local stress–strain relationships as well as for long-range internal stresses that arise from the ensuing plastic strain heterogeneities. The model parameters are related to the local dynamics and interactions of lattice dislocations. The model explains experimental observations on slip avalanches as well as the associated slip and surface pattern morphologies.
TL;DR: In this article, it was shown that the indecomposable part of the centralizer of the one-boundary Temperley-Lieb algebra leads to degeneracies in the three mentioned Hamiltonians.
Abstract: We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley–Lieb algebra. For a system of size L these representations are all of dimension 2L and, for generic points of the algebra, equivalent. However, at exceptional points they can possess different indecomposable structures. We study a centralizer of the one-boundary Temperley–Lieb algebra in the 'non-diagonal' spin- representation and find its eigenvalues and eigenvectors. In the exceptional cases this centralizer becomes indecomposable. We show how to get a truncated space of 'good' states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.
TL;DR: In this paper, the stability of the high-temperature fixed point of the loopy belief propagation (LBP) algorithm was analyzed for binary networks with pairwise interactions and sufficient conditions for convergence of LBP to a unique fixed point were given.
Abstract: We analyse the local stability of the high-temperature fixed point of the loopy belief propagation (LBP) algorithm and how this relates to the properties of the Bethe free energy which LBP tries to minimize. We focus on the case of binary networks with pairwise interactions. In particular, we state sufficient conditions for convergence of LBP to a unique fixed point and show that these are sharp for purely ferromagnetic interactions. In contrast, in the purely antiferromagnetic case, the undamped parallel LBP algorithm is suboptimal in the sense that the stability of the fixed point breaks down much earlier than for damped or sequential LBP; we observe that the onset of instability for the latter algorithms is related to the properties of the Bethe free energy. For spin-glass interactions, damping LBP only helps slightly. We estimate analytically the temperature at which the high-temperature LBP fixed point becomes unstable for random graphs with arbitrary degree distributions and random interactions.
TL;DR: In this article, the authors considered a random walk in one dimension, where the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function.
Abstract: We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[Mn] of the walk up to n steps behaves asymptotically for large n as , where ?2 is the variance of the step lengths. While the leading behaviour is universal and easy to derive, the leading correction term turns out to be a nontrivial constant ?. For the special case of uniform distribution over [?1,1], Coffmann et al recently computed ? = ?0.516?068... by exactly enumerating a lengthy double series. Here we present a closed exact formula for ? valid for arbitrary symmetric distributions. We also demonstrate how ? appears in the thermodynamic limit as the leading behaviour of the difference variable E[Mn]?E[|xn|] where xn is the position of the walker after n steps. An application of these results to the equilibrium thermodynamics of a Rouse polymer chain is pointed out. We also generalize our results to L?vy walks.
TL;DR: In this paper, the transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity, and the Bethe ansatz solution is derived for transfer matrix eigenvalues.
Abstract: The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe ansatz solution for the transfer matrix eigenvalues for the special cases where all but one of the boundary parameters are zero, and the bulk anisotropy parameter is iπ/3, iπ/5, ....
TL;DR: In this article, the authors derived the 1-, 2-, 3-, and 4-enumerations of alternating sign matrices (ASMs) using Hankel determinants and orthogonal polynomials.
Abstract: The six-vertex model with domain wall boundary conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be derived in a very simple and straightforward way. The derivation is based on the standard relationship between Hankel determinants and orthogonal polynomials. For the particular sets of parameters corresponding to 1-, 2-, and 3-enumerations of ASMs, the Hankel determinant can be naturally related to Continuous Hahn, Meixner–Pollaczek, and Continuous Dual Hahn polynomials, respectively. This observation allows for a unified and simplified treatment of ASM enumerations. In particular, along the lines of the proposed approach, we provide a complete solution to the long-standing problem of the refined 3-enumeration of ASMs.
TL;DR: In this paper, the authors present basic properties of dipolar stochastic Loewner evolutions (SLEs), a new version of SLEs in which the critical interfaces end randomly on an interval of the boundary of a planar domain.
Abstract: We present basic properties of dipolar stochastic Loewner evolutions, a new version of stochastic Loewner evolutions (SLEs) in which the critical interfaces end randomly on an interval of the boundary of a planar domain. We present a general argument explaining why correlation functions of models of statistical mechanics are expected to be martingales and we give a relation between dipolar SLEs and conformal field theories (CFTs). We compute SLE excursion and/or visiting probabilities, including the probability for a point to be on the left/right of the SLE trace or that for being inside the SLE hull. These functions, which turn out to be harmonic, have a simple CFT interpretation. We also present numerical simulations of the ferromagnetic Ising interface that confirm both the probabilistic approach and the CFT mapping.
TL;DR: The largest eigenvalue of the reduced density matrix for quantum chains is shown to have a simple physical interpretation and power-law behaviour in critical systems as discussed by the authors, which is verified numerically for XXZ spin chains.
Abstract: The largest eigenvalue of the reduced density matrix for quantum chains is shown to have a simple physical interpretation and power-law behaviour in critical systems. This is verified numerically for XXZ spin chains.
TL;DR: In this article, a simple approximate theory, that treats the correlations around the junction position in a mean-field fashion, is developed in order to calculate stationary particle currents, density profiles and a phase diagram.
Abstract: Totally asymmetric simple exclusion processes on lattices with junctions, where particles interact with hard core exclusion and move on parallel lattice branches that at the junction combine into a single lattice segment, are investigated. A simple approximate theory, that treats the correlations around the junction position in a mean-field fashion, is developed in order to calculate stationary particle currents, density profiles and a phase diagram. It is shown that there are three possible stationary phases depending on the state of each of the lattice branches. At first-order phase boundaries, where the density correlations are important, a modified phenomenological domain wall theory, that accounts for correlations, is introduced. Extensive Monte Carlo computer simulations are performed to investigate the system, and it is found that they are in excellent agreement with theoretical predictions. The application of the theoretical method for other inhomogeneous asymmetric simple exclusion processes is outlined.
TL;DR: In this paper, the phase diagram and critical behavior of the two-dimensional square-lattice fully frustrated XY model (FFXY) and two related models, a lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model, were studied.
Abstract: We study the phase diagram and critical behaviour of the two-dimensional square-lattice fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau–Ginzburg–Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model We present a finite-size-scaling analysis of the results of high-precision Monte Carlo simulations on L × L square lattices, up to L = O (103) In the FFXY model and in the other models, when the transitions are continuous, there are two very close but separate transitions There is an Ising chiral transition characterized by the onset of chiral long-range order while spins remain paramagnetic Then, as temperature decreases, the systems undergo a Kosterlitz–Thouless spin transition to a phase with quasi-long-range order The FFXY model and the other models, in a rather large parameter region, show a crossover behaviour at the chiral and spin transitions that is universal to some extent We conjecture that this universal behaviour is due to a multicritical point The numerical data suggest that the relevant multicritical point is a zero-temperature transition A possible candidate is the O(4) point that controls the low-temperature behaviour of the 4-vector model
TL;DR: In this article, the authors show that the probability distribution function in each of these four instances can be evaluated with a unique formula, and numerically analyse some consequences that emerge from these four choices.
Abstract: In spite of its undeniable success, there are still open questions regarding Tsallis's non-extensive statistical formalism, whose founding stone was laid in 1988 in Journal of Statistical Physics. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes' MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) the 1988 Tsallis original, (ii) the Curado–Tsallis version, (iii) the Tsallis–Mendes–Plastino version, and (iv) the Tsallis–Mendes–Plastino version, but using centred operators as constraints. The 1988 version was promptly abandoned and replaced, mostly with versions (ii) and (iii). We will here (a) show that the 1988 version is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyse some consequences that emerge from these four choices.
TL;DR: In this paper, it was shown that the super Tonks-Girardeau gas-like state corresponds to a highly excited Bethe state in the integrable interacting Bose gas for which the bosons acquire hard-core behaviour.
Abstract: We provide evidence in support of a recent proposal by Astrakharchik et al for the existence of a super Tonks–Girardeau gas-like state in the attractive interaction regime of quasi-one-dimensional Bose gases. We show that the super Tonks–Girardeau gas-like state corresponds to a highly excited Bethe state in the integrable interacting Bose gas for which the bosons acquire hard-core behaviour. The gas-like state properties vary smoothly throughout a wide range from strong repulsion to strong attraction. There is an additional stable gas-like phase in this regime in which the bosons form two-body bound states behaving like hard-core bosons.