Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models

TL;DR

This paper reviews the zero-range process, a model of interacting particles with applications in granular gases and network dynamics, focusing on condensation phenomena and various generalisations with factorised steady states.

Contribution

It provides a comprehensive review of recent advances in understanding condensation transitions and generalisations of the zero-range process with factorised steady states.

Findings

Analysis of condensation transitions in homogeneous and heterogeneous systems

Progress in understanding the dynamics of condensation

Extensions to models with multiple species and non-conservation

Abstract

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics, also we discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorised form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarise recent progress in understanding the dynamics of condensation. We then turn to several generalisations which also, under certain specified conditions, share the property of a factorised steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.