33 research outputs found
From urn models to zero-range processes: statics and dynamics
The aim of these lecture notes is a description of the statics and dynamics
of zero-range processes and related models. After revisiting some conceptual
aspects of the subject, emphasis is then put on the study of the class of
zero-range processes for which a condensation transition arises.Comment: Lecture notes for the Luxembourg Summer School 200
Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
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.Comment: 54 pages, 9 figures, review articl