Showing papers in "Journal of Statistical Physics in 2008"

A Fredholm Determinant Representation in ASEP

Abstract: In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice ℤ. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers ℤ+ and consider the distribution function for the mth particle from the left. In Tracy and Widom (Commun. Math. Phys. 279:815–844, 2008) an infinite series of multiple integrals was derived for the distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.

Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior

Abstract: This paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results.

Quenched Invariance Principles for Random Walks with Random Conductances

Abstract: We prove an almost sure invariance principle for a random walker among i.i.d. conductances in ℤ d , d≥2. We assume conductances are bounded from above but we do not require that they are bounded from below.

On Steady Distributions of Kinetic Models of Conservative Economies

Abstract: We analyze the large-time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years. As specific examples, we study models with fixed saving propensity introduced by Chakraborti and Chakrabarti (Eur. Phys. J. B 17:167–170, 2000), as well as models involving both exchange between agents and speculative trading as considered by Cordier et al. (J. Stat. Phys. 120:253–277, 2005) We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail. In particular, we prove the absence of Pareto tails in pointwise conservative models, like the one in (Eur. Phys. J. B 17:167–170, 2000), while models with speculative trades introduced in (J. Stat. Phys. 120:253–277, 2005) develop fat tails if the market is “risky enough”. The results are derived by a Fourier-based technique first developed for the Maxwell-Boltzmann equation (Gabetta et al. in J. Stat. Phys. 81:901–934, 1995; Bisi et al. in J. Stat. Phys. 118(1–2):301–331, 2005; Pareschi and Toscani in J. Stat. Phys. 124(2–4):747–779, 2006) and from a recursive relation which allows to calculate arbitrary moments of the stationary state.

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

Abstract: We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.

The mean field ising model trough interpolating techniques

Abstract: Aim of this paper is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the self-consistency relation, the phase transition, the critical behavior and the self-averaging properties. At the end a formulation of a Parisi-like theory is tried and discussed.

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

Abstract: We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ3 for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6π ν j where j is the current density of the particles, and of a friction term 6π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius e=1/N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure $\frac{1}{N}\sum_{1\le k\le N}\delta_{x_{k},v_{k}}$ , where x k is the center of the k-th ball and v k its instantaneous velocity. This can be seen as a generalization of Allaire’s result in [Arch. Ration. Mech. Anal. 113:209–259, [1991]] who considered the case of periodically distributed x k s with v k =0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays.

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

Abstract: A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N strongly correlated random variables for all values of N (and not just for large N).

Universality for the Distance in Finite Variance Random Graphs

Abstract: We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical Erdos-Renyi graph). In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like log(nu) N, where the nu depends on the capacities and N denotes the size of the graph. In addition, the random fluctuations around this asymptotic mean log(nu) N are shown to be tight. We also consider the case where the capacities are independent copies of a positive random Lambda with P (Lambda>x) 3, againg resulting in graphs where the degrees have finite variance. The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of van der Hofstad et al.

Long Range Correlations and Phase Transitions in Non-equilibrium Diffusive Systems

Abstract: We obtain explicit expressions for the long range correlations in the ABC model and in diffusive models conditioned to produce an atypical current of particles. In both cases, the two-point correlation functions allow one to detect the occurrence of a phase transition as they become singular when the system approaches the transition.

Fluctuations of Quantum Currents and Unravelings of Master Equations

Abstract: The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the so-called quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems. In contrast with the approaches found in the literature, we do not assume that the system is continuously monitored. Instead, our starting point is a relatively realistic unitary dynamics of the full system

Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig relations

Abstract: We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Bootstrap Percolation on Homogeneous Trees Has 2 Phase Transitions

Abstract: We study the threshold θ bootstrap percolation model on the homogeneous tree with degree b+1, 2≤θ≤b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call pf, such that a) for p>pf, the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p pc, then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p

Information and Entropy in Quantum Brownian Motion : Thermodynamic Entropy versus von Neumann Entropy ()

Abstract: We compare the thermodynamic entropy of a quantum Brownian oscillator derived from the partition function of the subsystem with the von Neumann entropy of its reduced density matrix. At low temperatures we find deviations between these two entropies which are due to the fact that the Brownian particle and its environment are entangled. We give an explanation for these findings and point out that these deviations become important in cases where statements about the information capacity of the subsystem are associated with thermodynamic properties, as it is the case for the Landauer principle.

From ballistic to diffusive behavior in periodic potentials

Abstract: The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.

Sums of hermitian squares and the BMV conjecture

Abstract: We show that all the coefficients of the polynomial $$\mathop{\mathrm{tr}}((A+tB)^{m})\in\mathbb{R}[t]$$ are nonnegative whenever m≤13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for m≤7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.

Metastability for reversible probabilistic cellular automata with self-interaction

Abstract: The problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the Freidlin–Wentzel regime, namely, finite volume, small magnetic field, and small temperature. The model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. Our strategy is based on recent powerful approaches, not needing a complete description of the fixed points of the dynamics, but relying on few model dependent results. We compute the exit time, in the sense of logarithmic equivalence, and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state. We need to supply two model dependent inputs: (1) the communication energy, that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one; (2) a recurrence property stating that for any configuration different from the metastable state there exists a path, starting from such a configuration and reaching a lower energy state, such that its maximal energy is lower than the communication energy.

Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk

Abstract: We consider self-avoiding walk, percolation and the Ising model with long and flnite range. By means of the lace expansion we prove mean-fleld behavior for these models if d > 2(fi ^ 2) for self-avoiding walk and the Ising model, and d > 3(fi ^ 2) for percolation, where d denotes the dimension and fi the power-law decay exponent of the coupling function. We provide a simplifled analysis of the lace expansion based on the trigonometric approach in Borgs et al. (13).

Motility States of Molecular Motors Engaged in a Stochastic Tug-of-War

Abstract: Intracellular transport is mediated by molecular motors that pull cargos along cytoskeletal filaments. Many cargos move bidirectionally and are transported by two teams of motors which move into opposite directions along the filament. We have recently introduced a stochastic tug-of-war model for this situation. This model describes the motion of the cargo as a Markov process on a two-dimensional state space defined by the numbers of active plus and active minus motors. In spite of its simplicity, this tug-of-war model leads to a complex dependence of the cargo motility on the motor parameters. We present new numerical results for the dependence on the number of involved motors. In addition, we derive a simple and intuitive sharp maxima approximation, from which one obtains the cargo motility state from only four simple inequalities. This approach provides a fast and reliable method to determine the cargo motility.

Random Close Packing of Granular Matter

Abstract: We propose an interpretation of the random close packing of granular materials as a phase transition, and discuss the possibility of experimental verification.

Weakly Non-Ergodic Statistical Physics

Abstract: For weakly non ergodic systems, the probability density function of a time average observable $\overline{{\mathcal{O}}}$ is $f_{\alpha}(\overline{{\mathcal{O}}})=-{1\over \pi}\lim_{\epsilon\to 0}\mbox{Im}{\sum_{j=1}^{L}p^{\mathrm{eq}}_{j}(\overline{{\mathcal{O}}}-{\mathcal{O}}_{j}+i\epsilon)^{\alpha -1}\over \sum_{j=1}^{L}p^{\mathrm{eq}}_{j}(\overline{{\mathcal{O}}}-{\mathcal{O}}_{j}+i\epsilon)^{\alpha}}$ where ${\mathcal{O}}_{j}$ is the value of the observable when the system is in state j=1,…L. p eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p eq is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x 2〉∼t α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered $\lim_{\alpha \to 1}f_{\alpha}(\overline{{\mathcal{O}}})=\delta (\overline{{\mathcal{O}}}-\langle {\mathcal{O}}\rangle )$ . We briefly discuss possible physical applications in single particle experiments.

The Airy1 Process is not the Limit of the Largest Eigenvalue in GOE Matrix Diffusion

Abstract: Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion.

Chemomechanical coupling of molecular motors: Thermodynamics, network representations, and balance conditions

Abstract: Molecular motors are considered that convert the chemical energy released from the hydrolysis of adenosine triphosphate (ATP) into mechanical work. Such a motor represents a small system that is coupled to a heat reservoir, a work reservoir, and particle reservoirs for ATP, adenosine diphosphate (ADP), and inorganic phosphate (P). The discrete state space of the motor is defined in terms of the chemical composition of its catalytic domains. Each motor state represents an ensemble of molecular conformations that are thermally equilibrated. The motor states together with the possible transitions between neighboring states define a network representation of the motor. The motor dynamics is described by a continuous-time Markov process (or master equation) on this network. The consistency between thermodynamics and network dynamics implies (i) local and nonlocal balance conditions for the transition rates of the motor and (ii) an underlying landscape of internal energies for the motor states. The local balance conditions can be interpreted in terms of constrained equilibria between neighboring motor states; the nonlocal balance conditions pinpoint chemical and/or mechanical nonequilibrium.

On the Jarzynski relation for dissipative quantumdynamics

Abstract: In this note, we will discuss how to compactly express the Jarzynski identity for an open quantum system with dissipative dynamics. In quantum dynamics we must avoid explicitly measuring the work directly, which is tantamount to continuously monitoring the state of the system, and instead measure the heat ?ow from the environment. These measurements can be concisely represented with Hermitian map superoperators, which provide a convenient and compact representations of correlation functions and sequential measurements of quantum systems.

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

Abstract: We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules.

Random Motions at Finite Speed in Higher Dimensions

Abstract: We present a general method of studying the transport process \(\bold X(t)\) , t≥0, in the Euclidean space ℝm, m≥2, based on the analysis of the integral transforms of its distributions. We show that the joint characteristic functions of \(\bold X(t)\) are connected with each other by a convolution-type recurrent relation. This enables us to prove that the characteristic function (Fourier transform) of \(\bold X(t)\) in any dimension m≥2 satisfies a convolution-type Volterra integral equation of second kind. We give its solution and obtain the characteristic function of \(\bold X(t)\) in terms of the multiple convolutions of the kernel of the equation with itself. An explicit form of the Laplace transform of the characteristic function in any dimension is given. The complete solution of the problem of finding the initial conditions for the governing partial differential equations, is given.

Energy Spectrum of Bound-Spinons in the Quantum Ising Spin-Chain Ferromagnet

Abstract: We study the excitation energy spectrum in the S=1/2 ferromagnetic Ising spin chain with the easy axis z in a magnetic field h={h x ,0,h z }. According to Wu and McCoy’s scenario of weak confinement, the fermionic spinon excitations (kinks), being free at h z =0 in the ordered phase, are coupled into bosonic bound states at arbitrary small h z >0. We calculate the energy spectrum of such excitations in the leading order in small h z , using different perturbative methods developed for the similar problem in the Ising Field Theory.

On the Problem of Dynamical Localization in the Nonlinear Schrödinger Equation with a Random Potential

Abstract: We prove a dynamical localization in the nonlinear Schrodinger equation with a random potential for times of the order of O(β −2 ) ,w hereβ is the strength of the nonlinear- ity.

Slow Rates of Mixing for Dynamical Systems with Hyperbolic Structures

Abstract: We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable return time. We show that the decay of correlations of the SRB measure associated to that hyperbolic structure is related to the tail of the recurrence times. We also give sufficient conditions for the validity of the Central Limit Theorem. This extends previous results by Young in (Ann. Math. 147: 585–650, [1998]; Israel J. Math. 110: 153–188, [1999]).

Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats.

Abstract: We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary states as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling.