Condensation in the zero range process: stationary and dynamical properties

TL;DR

This paper investigates the zero range process, revealing a phase transition where excess particles cluster at a single site, and analyzes the dynamics of this clustering through rigorous proofs and simulations.

Contribution

It provides a rigorous proof of phase transition and clustering behavior in the zero range process with attractive interactions, including analysis of symmetric and asymmetric dynamics.

Findings

Excess particles condense at a single site at high density.

Clustering dynamics differ between symmetric and asymmetric jump rates.

An effective master equation describes late-stage clustering behavior.

Abstract

The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites.