Zero-range process in random environment

TL;DR

This paper surveys recent research on one-dimensional attractive zero range processes in a random environment with site disorder, focusing on invariant measures, convergence conditions, hydrodynamic limits, and local equilibrium results.

Contribution

It introduces new conditions for convergence to maximal invariant measures and develops a hydrodynamic theory for non-equilibrium profiles in disordered environments.

Findings

Necessary and sufficient conditions for distributional convergence.

Hydrodynamic limits for non-equilibrium density profiles.

Strong local equilibrium results.

Abstract

We survey our recent articles dealing with one dimensional attractive zero range processes moving under site disorder. We suppose that the underlying random walks are biased to the right and so hyperbolic scaling is expected. Under the conditions of our model the process admits a maximal invariant measure. The initial focus of the project was to find conditions on the initial law to entail convergence in distribution to this maximal distribution, when it has a finite density. Somewhat surprisingly, necessary and sufficient conditions were found. In this part hydrody-namic results were employed chiefly as a tool to show distributional convergence but subsequently we developed a theory for hydrodynamic limits treating profiles possessing densities that did not admit corresponding equilibria. Finally we derived strong local equilibrium results.