Exclusion Processes with Degenerate Rates: Convergence to Equilibrium and Tagged Particle

Stochastic lattice gases with degenerate rates, namely conservative particle systems where the exchange rates vanish for some configurations, have been introduced as simplified models for glassy dynamics. We introduce two particular models and consider them in a finite volume of size in contact with particle reservoirs at the boundary. We prove that, as for non-degenerate rates, the inverse of the spectral gap and the logarithmic Sobolev constant grow as ². It is also shown how one can obtain, via a scaling limit from the logarithmic Sobolev inequality, the exponential decay of a Lyapunov functional for a degenerate parabolic differential equation (porous media equation). We analyze finally the tagged particle displacement for the stationary process in infinite volume. In dimension larger than two we prove that, in the diffusive scaling limit, it converges to a Brownian motion with non-degenerate diffusion coefficient.     

Scaling Limits of a Tagged Particle in the Exclusion Process with Variable Diffusion Coefficient

We prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in ℤ with variable diffusion coefficient. The scaling limits are obtained from a similar result for the current through −1/2 for a zero-range process with bond disorder. For the CLT, we prove convergence to a fractional Brownian motion of Hurst exponent 1/4.     

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in the TASEP with the step initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of the TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.     

Effect of Detachments in Asymmetric Simple Exclusion Processes

Asymmetric simple exclusion processes are important for understanding low-dimensional multi-particle dynamic phenomena. The effect of irreversible detachments of particles on dynamics of asymmetric simple exclusion processes is studied using analytical and computer simulation techniques. In the simplest model, where particles can only detach from a single site in the bulk of the system, a theory is presented and used to calculate explicitly phase diagrams and particle density profiles. The complexity of the phase behavior is discussed in terms of a recent domain-wall theory for driven lattice systems. The theoretical results qualitatively and quantitatively agree with computer Monte Carlo simulations.     

Hydrodynamic Limit for Weakly Asymmetric Simple Exclusion Processes in Crystal Lattices

We investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices. We construct a suitable scaling limit by using a discrete harmonic map. As we shall observe, the quasi-linear parabolic equation in the limit is defined on a flat torus and depends on both the local structure of the crystal lattice and the discrete harmonic map. We formulate the local ergodic theorem on the crystal lattice by introducing the notion of local function bundle, which is a family of local functions on the configuration space. The ideas and methods are taken from the discrete geometric analysis to these problems. Results we obtain are extensions of ones by Kipnis, Olla and Varadhan to crystal lattices.     

Particle Model for the Reservoirs in the Simple Symmetric Exclusion Process

Journal of Statistical Physics - In this paper, we will study the long time behavior of the simple symmetric exclusion process in the “channel” $$varLambda _N=[1,N]cap mathbb {N}$$...     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Einstein Relation for a Class of Interface Models

 A class of SOS interface models which can be seen as simplified stochastic Ising model interfaces is studied. In the absence of an external field the long-time fluctuations of the interface are shown to behave as Brownian motion with diffusion coefficient given by a Green-Kubo formula. When a small external field h is applied, it is shown that the shape of the interface converges exponentially fast to a stationary distribution and the interface moves with an asymptotic velocity v(h). The mobility is shown to exist and to satisfy the Einstein relation: , where β is the inverse temperature. Received: 16 April 2002 / Accepted: 3 July 2002 Published online: 22 November 2002 RID="*" ID="*" Work partially supported by the N.S.F. through grants DMS-0071766 and DMS-0074152.     

Spectral Gap for Multi-species Exclusion Processes

We consider a multi-species generalization of the symmetric simple exclusion process in homogeneous and non-homogeneous hypercubes of Z ^d . In this model, the hyperplanes of configurations with given numbers of particles of each species are not necessarily irreducible. We give a sufficient condition of the dynamics to make them irreducible. In addition, assuming the irreducibility of them, we show some estimates of the spectral gap (the absolute value of the second largest eigenvalue of the generator), which plays an important role in the study of hydrodynamic limit.     

Einstein Relation for Nonequilibrium Steady States

The Einstein relation, relating the steady state fluctuation properties to the linear response to a perturbation, is considered for steady states of stochastic models with a finite state space. We show how an Einstein relation always holds if the steady state satisfies detailed balance. More generally, we consider nonequilibrium steady states where detailed balance does not hold and show how a generalisation of the Einstein relation may be derived in certain cases. In particular, for the asymmetric simple exclusion process and a driven diffusive dimer model, the external perturbation creates and annihilates particles thus breaking the particle conservation of the unperturbed model.     

On Tagged Particle Dynamics in Highly Confined Fluids

Heuristic approaches to the statistics of tagged particle motion in a one-dimensional hard point particle fluid are discussed. An exact expression is obtained for the finite N case with arbitrary single-particle interactionless dynamics. This is extended to the mean over tagged particles as N→∞, and a simple form presented in terms of elementary physical quantities. Extension to single-file flow under quasi-one-dimensional confinement is initiated.     

Tagged Particle Diffusion in One-Dimensional Gas with Hamiltonian Dynamics

We consider a one-dimensional gas of hard point particles in a finite box that are in thermal equilibrium and evolving under Hamiltonian dynamics. Tagged particle correlation functions of the middle particle are studied. For the special case where all particles have the same mass, we obtain analytic results for the velocity auto-correlation function in the short time diffusive regime and the long time approach to the saturation value when finite-size effects become relevant. In the case where the masses are unequal, numerical simulations indicate sub-diffusive behaviour with mean square displacement of the tagged particle growing as t/ln(t) with time t. Also various correlation functions, involving the velocity and position of the tagged particle, show damped oscillations at long times that are absent for the equal mass case.     

The Gradient Condition for One-Dimensional Symmetric Exclusion Processes

For every Gibbs measure on the one dimensional lattice Z with translation-invariant potential of finite range, an exchange rate for one-dimensional lattice gas which satisfy both the detailed balance condition relative to the Gibbs measure and the gradient condition is constructed.     

The Einstein Relation for Random Walks on Graphs

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs.     

On Viscosity and Fluctuation-Dissipation in Exclusion Processes

We consider the asymmetric simple exclusion process. We review some results in dimension d≥3 concerning the fluctuation-dissipation theorem and we prove regularity of viscosity coefficients.