Showing papers in "Journal of Statistical Mechanics: Theory and Experiment in 2015"

Variational and optimal control representations of conditioned and driven processes

Abstract: We have shown recently that a Markov process conditioned on rare events involving time-integrated random variables can be described in the long-time limit by an effective Markov process, called the driven process, which is given mathematically by a generalization of Doob's $h$-transform. We show here that this driven process can be represented in two other ways: first, as a process satisfying various variational principles involving large deviation functions and relative entropies and, second, as an optimal stochastic control process minimizing a cost function also related to large deviation functions. These interpretations of the driven process generalize and unify many previous results on maximum entropy approaches to nonequilibrium systems, spectral characterizations of positive operators, and control approaches to large deviation theory. They also lead, as briefly discussed, to new methods for analytically or numerically approximating large deviation functions.

Quenching the XXZ spin chain: Quench action approach versus generalized Gibbs ensemble

Abstract: Following our previous work [PRL 113 (2014) 09020] we present here a detailed comparison of the quench action approach and the predictions of the generalized Gibbs ensemble, with the result that while the quench action formalism correctly captures the steady state, the GGE does not give a correct description of local short-distance correlation functions. We extend our studies to include another initial state, the so-called q-dimer state. We present important details of our construction, including new results concerning exact overlaps for the dimer and q-dimer states, and we also give an exact solution of the quench-action-based overlap-TBA for the q-dimer. Furthermore, we extend our computations to include the xx spin correlations besides the zz correlations treated previously, and give a detailed discussion of the underlying reasons for the failure of the GGE, especially in the light of new developments.

Entanglement entropy and negativity of disjoint intervals in CFT: some numerical extrapolations

Abstract: The entanglement entropy and the logarithmic negativity can be computed in quantum field theory through a method based on the replica limit. Performing these analytic continuations in some cases is beyond our current knowledge, even for simple models. We employ a numerical method based on rational interpolations to extrapolate the entanglement entropy of two disjoint intervals for the conformal field theories given by the free compact boson and the Ising model. The case of three disjoint intervals is studied for the Ising model and the non compact free massless boson. For the latter model, the logarithmic negativity of two disjoint intervals has been also considered. Some of our findings have been checked against existing numerical results obtained from the corresponding lattice models.

Physics at the entangling surface

Abstract: To consider the entanglement between the spatial region A and its complement in a QFT, we need to assign a Hilbert space to the region. Usually, some boundary condition on ?A is implicitly chosen, but we argue that the choice of the boundary condition at ?A is physically meaningful and affects the subleading contributions to the entanglement R?nyi entropy. We investigate these issues in the context of 2d CFTs, and show that we can indeed read off the Cardy states of the c?=?1/2 minimal model from the entanglement entropy of the critical Ising chain.

Effective Langevin equations for constrained stochastic processes

Abstract: We propose a novel stochastic method to exactly generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time $t_{f}$. These paths are weighted with a probability given by the overdamped Langevin dynamics. We show how these paths can be exactly generated by a local stochastic differential equation. The method is illustrated on the generation of Brownian bridges, Brownian meanders, Brownian excursions and constrained Ornstein-Uehlenbeck processes. In addition, we show how to solve this equation in the case of a general force acting on the particle. As an example, we show how to generate constrained path joining the two minima of a double-well. Our method allows to generate statistically independent paths, and is computationally very efficient.

Global characteristics of all eigenstates of local many-body Hamiltonians: participation ratio and entanglement entropy

Abstract: In the spectrum of many-body quantum systems appearing in condensed matter physics, the low-energy eigenstates were the traditional focus of research. The interest in the statistical properties of the full eigenspectrum has grown more recently, in particular in the context of non-equilibrium questions. Wave functions of interacting lattice quantum systems can be characterized either by local observables or by global properties such as the participation ratio (PR) in a many-body basis or the entanglement between various partitions. We present a study of the PR and of the entanglement entropy (EE) between two roughly equal spatial partitions of the system, in all the eigenfunctions of local Hamiltonians. Motivated by the similarity of the PR and EE?both are generically larger in the bulk and smaller near the edges of the spectrum?we quantitatively analyze the correlation between them. We elucidate the effect of (proximity to) integrability, showing how low-entanglement and low-PR states appear also in the middle of the spectrum as one approaches integrable points. We also determine the precise scaling behaviour of the eigenstate-to-eigenstate fluctuations of the PR and EE with respect to system size and characterize the statistical distribution of these quantities near the middle of the spectrum.

An experimental study on four-directional intersecting pedestrian flows

Abstract: Intersecting pedestrian flows especially multi-directional ones are complicated in dynamics. People will face unavoidable head-on conflicts and obstruct each other. In this paper, controlled experiments of a four-directional intersecting pedestrian flow were conducted. Up to 364 university students took part in the experiments and their trajectories were extracted by a mean-shift algorithm. The global density–velocity relations in the cross area in different scenarios are compared. Moreover, local density–velocity and local density-flow relations in the cross area are investigated. In order to adapt the study of a fundamental diagram for four directional intersecting flows, a new coordinate system based on pedestrian motion is built. The results indicate that the coordinate system is suitable for the analysis of multi-directional flows. The local density–velocity relation seems consistent with previous results obtained from an actual high-density pedestrian flow. At high densities, the average local velocity in the cross area is a bit larger than a previous study. The reason may be due to the density difference between the cross area and the corridors, which can be observed in real life.

Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity

Abstract: We study generalized anomalous diffusion processes whose diffusion coefficient D(x, t) ~ D0|x|αtβ depends on both the position x of the test particle and the process time t. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalized diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalized diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.

Current fluctuations for anharmonic chains in thermal equilibrium

Abstract: We study the total current correlations for anharmonic chains in thermal equilibrium, putting forward predictions based on the second moment sum rule and on nonlinear fluctuating hydrodynamics. We compare with molecular dynamics simulations for hard collision models. For the first time we investigate the full statistics of time-integrated currents. Generically such a quantity has Gaussian statistics on a scale . But if the time integration has its endpoint at a moving sound peak, then the fluctuations are suppressed and only of order t1/3. The statistics is governed by the Baik–Rains distribution, known already from the fluctuating Burgers equation.

A consistent description of fluctuations requires negative temperatures

Abstract: We review two definitions of temperature in statistical mechanics, and , corresponding to two possible definitions of entropy, and , known as surface and volume entropy respectively. We limit our attention to a class of systems with bounded energy, and such that the second derivative of , with respect to energy, is always negative. The second condition holds in systems where the number N of degrees of freedom is sufficiently large (examples are shown where is sufficient) and without long-range interactions. We first discuss the basic role of , even when negative, as the parameter describing fluctuations of observables in a sub-system. Then, we focus on how can be measured dynamically, i.e. averaging over a single long experimental trajectory. The same approach cannot be used in a generic system for , since the equipartition theorem may be impaired by boundary effects due to the limited energy. These general results are substantiated by the numerical study of a Hamiltonian model of interacting rotators with bounded kinetic energy. The numerical results confirm that the kind of configurational order realized in the regions at small , or equivalently at small , depends on the sign of .

Entanglement over the rainbow

Abstract: In one dimension the area law for the entanglement entropy is violated maximally by the ground state of a strongly inhomogeneous spin chain, the so called concentric singlet phase (CSP), that looks like a rainbow connecting the two halves of the chain. In this paper we show that, in the weak inhomogeneity limit, the rainbow state is a thermo field double of a conformal field theory with a temperature proportional to the inhomogeneity parameter. This result suggests some relation of the CSP with black holes. Finally, we propose an extension of the model to higher dimensions.

Thermodynamic limits to information harvesting by sensory systems

Abstract: In view of the relation between information and thermodynamics we investigate how much information about an external protocol can be stored in the memory of a stochastic measurement device given an energy budget. We consider a layered device with a memory component storing information about the external environment by monitoring the history of a sensory part coupled to the environment. We derive an integral fluctuation theorem for the entropy production and a measure of the information accumulated in the memory device. Its most immediate consequence is that the amount of information is bounded by the average thermodynamic entropy produced by the process. At equilibrium no entropy is produced and therefore the memory device does not add any information about the environment to the sensory component. Consequently, if the system operates at equilibrium the addition of a memory component is superfluous. Such a device can be used to model the sensing process of a cell measuring the external concentration of a chemical compound and encoding the measurement in the amount of phosphorylated cytoplasmic proteins.

Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current

Abstract: We study the asymmetric zero-range process (ZRP) with L sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob h-transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the steady state of the conditioned dynamics, and, under generic conditions on the jump rates of the unconditioned ZRP, to an intriguing supercritical bulk region where condensates can grow. These results provide a microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly asymmetric case: it turns out that the predictions of MFT remain valid in the non-rigorous limit of finite asymmetry. In addition, the microscopic results yield the correct scaling factor for the asymmetry that MFT cannot predict.

Thermodynamics of accuracy in kinetic proofreading: dissipation and efficiency trade-offs

Abstract: The high accuracy exhibited by biological information transcription processes is due to kinetic proofreading, i.e. by a mechanism which reduces the error rate of the information-handling process by driving it out of equilibrium. We provide a consistent thermodynamic description of enzyme-assisted assembly processes involving competing substrates, in a master equation framework. We introduce and evaluate a measure of the efficiency based on rigorous non-equilibrium inequalities. The performance of several proofreading models are thus analyzed and the related time, dissipation and efficiency versus error trade-offs exhibited for different discrimination regimes. We finally introduce and analyze in the same framework a simple model which takes into account correlations between consecutive enzyme-assisted assembly steps. This work highlights the relevance of the distinction between energetic and kinetic discrimination regimes in enzyme-substrate interactions.

Density form factors of the 1D Bose gas for finite entropy states

Abstract: We consider the Lieb-Liniger model for a gas of bosonic delta-interacting particles. Using Algebraic Bethe Ansatz results we compute the thermodynamic limit of the form factors of the density operator between finite entropy eigenstates such as finite temperature states or generic non-equilibrium highly excited states. These form factors are crucial building blocks to obtain the thermodynamic exact dynamic correlation functions of such physically relevant states. As a proof of principle we compute an approximated dynamic structure factor by including only the simplest types of particle-hole excitations and show the agreement with known results.

Partial transpose of two disjoint blocks in XY spin chains

Abstract: We consider the partial transpose of the spin reduced density matrix of two disjoint blocks in spin chains admitting a representation in terms of free fermions, such as XY chains. We exploit the solution of the model in terms of Majorana fermions and show that such partial transpose in the spin variables is a linear combination of four Gaussian fermionic operators. This representation allows to explicitly construct and evaluate the integer moments of the partial transpose. We numerically study critical XX and Ising chains and we show that the asymptotic results for large blocks agree with conformal eld theory predictions if corrections to the scaling are properly taken into account.

Evaluating accuracy of community detection using the relative normalized mutual information

Abstract: The normalized mutual information (NMI) has been widely used to evaluate the accuracy of community detection algorithms. However in this article we show that the NMI is seriously affected by systematic errors due to finite size of networks, and may give a wrong estimate of performance of algorithms in some cases. We give a simple theory to the finite-size effect of NMI and test our theory numerically. Then we propose a new metric for the accuracy of community detection, namely the relative normalized mutual information (rNMI), which considers statistical significance of the NMI by comparing it with the expected NMI of random partitions. Our numerical experiments show that the rNMI overcomes the finite-size effect of the NMI.

Authorship recognition via fluctuation analysis of network topology and word intermittency

Abstract: Statistical methods have been widely employed in many practical natural language processing applications. More specifically, complex network concepts and methods from dynamical systems theory have been successfully applied to recognize stylistic patterns in written texts. Despite the large number of studies devoted to representing texts with physical models, only a few studies have assessed the relevance of attributes derived from the analysis of stylistic fluctuations. Because fluctuations represent a pivotal factor for characterizing a myriad of real systems, this study focused on the analysis of the properties of stylistic fluctuations in texts via topological analysis of complex networks and intermittency measurements. The results showed that different authors display distinct fluctuation patterns. In particular, it was found that it is possible to identify the authorship of books using the intermittency of specific words. Taken together, the results described here suggest that the patterns found in stylistic fluctuations could be used to analyze other related complex systems. Furthermore, the discovery of novel patterns related to textual stylistic fluctuations indicates that these patterns could be useful to improve the state of the art of many stylistic-based natural language processing tasks.

Approximate message-passing with spatially coupled structured operators, with applications to compressed sensing and sparse superposition codes

Abstract: We study the behavior of approximate message-passing (AMP), a solver for linear sparse estimation problems such as compressed sensing, when the i.i.d matrices—for which it has been specifically designed—are replaced by structured operators, such as Fourier and Hadamard ones. We show empirically that after proper randomization, the structure of the operators does not significantly affect the performances of the solver. Furthermore, for some specially designed spatially coupled operators, this allows a computationally fast and memory efficient reconstruction in compressed sensing up to the information-theoretical limit. We also show how this approach can be applied to sparse superposition codes, allowing the AMP decoder to perform at large rates for moderate block length.

Critical behavior in spherical and hyperbolic spaces

Abstract: We study the effects of curved background geometries on the critical behavior of scalar field theory. In particular we concentrate on two maximally symmetric spaces: d-dimensional spheres and hyperboloids. In the first part of the paper, by applying the Ginzburg criterion, we find that for a large correlation length the Gaussian approximation is valid on the hyperboloid for any dimension d 2, while it is not trustable on the sphere for any dimension. This is understood in terms of various notions of the effective dimension, such as the spectral and Hausdorff dimension. In the second part of the paper, we apply functional renormalization group methods to develop a different perspective on such phenomena and to deduce them from a renormalization group analysis. By making use of the local potential approximation, we discuss the consequences of having a fixed scale in the renormalization group equations. In particular, we show that in the case of spheres there is no true phase transition, as symmetry restoration always occurs at large scales. In the case of hyperboloids, the phase transition is still present, but as the only true fixed point is the Gaussian one, mean field exponents are valid also in dimensions lower than four.

Pre-relaxation in weakly interacting models

Abstract: We consider time evolution in models close to integrable points with hidden symmetries that generate infinitely many local conservation laws that do not commute with one another. The system is expected to (locally) relax to a thermal ensemble if integrability is broken, or to a so-called generalised Gibbs ensemble if unbroken. In some circumstances expectation values exhibit quasi-stationary behaviour long before their typical relaxation time. For integrability-breaking perturbations, these are also called pre-thermalisation plateaux, and emerge e.g. in the strong coupling limit of the Bose-Hubbard model. As a result of the hidden symmetries, quasi-stationarity appears also in integrable models, for example in the Ising limit of the XXZ model. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-$\frac{1}{2}$ chain with additional perturbations that break integrability. One of the most intriguing results of the analysis is the appearance of persistent oscillatory behaviour. To unravel its origin, we study in detail a toy model: the transverse-field Ising chain with an additional nonlocal interaction proportional to the square of the transverse spin per unit length [Phys. Rev. Lett. 111, 197203 (2013)]. Despite being nonlocal, this belongs to a class of models that emerge as intermediate steps of the mean-field mapping and shares many dynamical properties with the weakly interacting models under consideration.

Can dynamical synapses produce true self-organized criticality?

Abstract: Neuronal networks can present activity described by power-law distributed avalanches presumed to be a signature of a critical state. Here we study a random-neighbor network of excitable cellular automata coupled by dynamical synapses. The model exhibits a very similar to conservative self-organized criticality (SOC) models behavior even with dissipative bulk dynamics. This occurs because in the stationary regime the model is conservative on average, and, in the thermodynamic limit, the probability distribution for the global branching ratio converges to a delta-function centered at its critical value. So, this non-conservative model pertain to the same universality class of conservative SOC models and contrasts with other dynamical synapses models that present only self-organized quasi-criticality (SOqC). Analytical results show very good agreement with simulations of the model and enable us to study the emergence of SOC as a function of the parametric derivatives of the stationary branching ratio.

Fractional diffusion on circulant networks: emergence of a dynamical small world

Abstract: In this paper, we study fractional random walks on networks defined from the equivalent of the fractional diffusion equation in graphs. We explore this process analytically in circulant networks; in particular, interacting cycles and limit cases such as a ring and a complete graph. From the spectra and the eigenvectors of the Laplacian matrix, we deduce explicit results for different quantities that characterize this dynamical process. We obtain analytical expressions for the fractional transition matrix, the fractional degrees and the average probability of return of the random walker. Also, we discuss the Kemeny constant, which gives the average number of steps necessary to reach any site of the network. Throughout this work, we analyze the mechanisms behind fractional transport on circulant networks and how this long-range process dynamically induces the small-world property in different structures.

Dynamical properties of single-file diffusion

Abstract: We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagged particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory—the full single-time statistics, all multiple-time correlation functions, etc are merely special cases.

Multipartite information flow for multiple Maxwell demons

Abstract: The second law of thermodynamics dictates the fundamental limits to the amount of energy and information that can be exchanged between physical systems. In this work, we extend a thermodynamic formalism describing this flow of energy and information developed for a pair of bipartite systems to many multipartite systems. We identify a natural thermodynamic quantity that describes the information exchanged among these systems. We then introduce and discuss a refined version. Our results are illustrated with a model of two, competing Maxwell demons.

Singularities in large deviation functions

Abstract: Large deviation functions of configurations exhibit very different behaviors in and out of thermal equilibrium. In particular, they exhibit singularities in a broad range of non-equilibrium models, which are absent in equilibrium. These singularities were first identified in finite-dimensional systems in the weak-noise limit. Recent studies have shown that they are also present in driven diffusive systems with an infinite-dimensional configuration space. This short review describes singularities appearing in both types of systems under a unified framework, presenting a classification of singularities into two broad categories. The types of singularities which were identified for finite-dimensional cases are compared to those found in driven diffusive systems.

Conformism-driven phases of opinion formation on heterogeneous networks: the q-voter model case

Abstract: The q -voter model, a variant of the classic voter model, has been analyzed by several authors. While allowing us to study opinion dynamics, this model is also believed to be one of the most representative among the many defined in the wide field of sociophysics. Here, we investigate the consequences of conformity on the consensus reaching process, by numerically simulating a q -voter model with agents behaving either as conformists or nonconformists, embedded on heterogeneous network topologies (as small-world and scale-free). In fact, although it is already known that conformity enhances the reaching of consensus, the related process is often studied only on fully-connected networks, thus strongly limiting our full understanding of it. This paper represents a first step in the direction of analyzing more realistic social models, showing that different opinion formation phases, driven by the conformist agents density, are observable. As a result, we identify threshold values of the density of conformist agents, varying across different topologies and separating different regimes of our system, ranging from a disordered phase, where different opinions coexist, to a gradually more ordered phase, where consensus is eventually reached.

Many-body microhydrodynamics of colloidal particles with active boundary layers

Abstract: Colloidal particles with active boundary layers—regions surrounding the particles where non-equilibrium processes produce large velocity gradients—are common in many physical, chemical and biological contexts. The velocity or stress at the edge of the boundary layer determines the exterior fluid flow and, hence, the many-body interparticle hydrodynamic interaction. Here, we present a method to compute the many-body hydrodynamic interaction between N spherical active particles induced by their exterior microhydrodynamic flow. First, we use a boundary integral representation of the Stokes equation to eliminate bulk fluid degrees of freedom. Then, we expand the boundary velocities and tractions of the integral representation in an infinite-dimensional basis of tensorial spherical harmonics and, on enforcing boundary conditions in a weak sense on the surface of each particle, obtain a system of linear algebraic equations for the unknown expansion coefficients. The truncation of the infinite series, fixed by the degree of accuracy required, yields a finite linear system that can be solved accurately and efficiently by iterative methods. The solution linearly relates the unknown rigid body motion to the known values of the expansion coefficients, motivating the introduction of propulsion matrices. These matrices completely characterize hydrodynamic interactions in active suspensions just as mobility matrices completely characterize hydrodynamic interactions in passive suspensions. The reduction in the dimensionality of the problem, from a three-dimensional partial differential equation to a two-dimensional integral equation, allows for dynamic simulations of hundreds of thousands of active particles on multi-core computational architectures. In our simulation of 104 active colloidal particle in a harmonic trap, we find that the necessary and sufficient ingredients to obtain steady-state convective currents, the so-called 'self-assembled pump', are (a) one-body self-propulsion and (b) two-body rotation from the vorticity of the Stokeslet induced in the trap.

Correlations and symmetry of interactions influence collective dynamics of molecular motors

Abstract: Enzymatic molecules that actively support many cellular processes, including transport, cell division and cell motility, are known as motor proteins or molecular motors. Experimental studies indicate that they interact with each other and they frequently work together in large groups. To understand the mechanisms of collective behavior of motor proteins we study the effect of interactions in the transport of molecular motors along linear filaments. It is done by analyzing a recently introduced class of totally asymmetric exclusion processes that takes into account the intermolecular interactions via thermodynamically consistent approach. We develop a new theoretical method that allows us to compute analytically all dynamic properties of the system. Our analysis shows that correlations play important role in dynamics of interacting molecular motors. Surprisingly, we find that the correlations for repulsive interactions are weaker and more short-range than the correlations for the attractive interactions. In addition, it is shown that symmetry of interactions affect dynamic properties of molecular motors. The implications of these findings for motor proteins transport are discussed. Our theoretical predictions are tested by extensive Monte Carlo computer simulations.

Analysis of adaptive walks on NK fitness landscapes with different interaction schemes

Abstract: Fitness landscapes are genotype to fitness mappings commonly used in evolutionary biology and computer science which are closely related to spin glass models. In this paper, we study the NK model for fitness landscapes where the interaction scheme between genes can be explicitly defined. The focus is on how this scheme influences the overall shape of the landscape. Our main tool for the analysis are adaptive walks, an idealized dynamics by which the population moves uphill in fitness and terminates at a local fitness maximum. We use three different types of walks and investigate how their length (the number of steps required to reach a local peak) and height (the fitness at the endpoint of the walk) depend on the dimensionality and structure of the landscape. We find that the distribution of local maxima over the landscape is particularly sensitive to the choice of interaction pattern. Most quantities that we measure are simply correlated to the rank of the scheme, which is equal to the number of nonzero coefficients in the expansion of the fitness landscape in terms of Walsh functions.