Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle

TL;DR

This paper studies stochastic lattice gases with degenerate exchange rates, proving spectral gap and entropy decay properties, and analyzing tagged particle behavior, revealing diffusive limits in higher dimensions.

Contribution

It introduces models with degenerate rates, establishes spectral gap growth, links entropy decay to porous media equations, and characterizes tagged particle diffusion in infinite volume.

Findings

Spectral gap and logarithmic Sobolev constant grow as ^2.

Exponential decay of macroscopic entropy via scaling limits.

Tagged particle converges to Brownian motion in high dimensions.

Abstract

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 $\ell$ 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 $\ell^2$. It is also shown how one can obtain, via a scaling limit from the logarithmic Sobolev inequality, the exponential decay of a macroscopic entropy associated to 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.