M R Evans and T Hanney 2005 J. Phys. A: Math. Gen. 38 R195 doi:10.1088/0305-4470/38/19/R01
M R Evans and T Hanney
Show affiliationsWe 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; we also 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 factorized 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 summarize recent progress in understanding the dynamics of condensation. We then turn to several generalizations which also, under certain specified conditions, share the property of a factorized 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.
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.70.Fh Phase transitions: general studies
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
82C22 Interacting particle systems (See also 60K35)
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. (See also 60G50)
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
82C26 Dynamic and nonequilibrium phase transitions (general)
Issue 19 (13 May 2005)
Received 18 January 2005, in final form 3 March 2005
Published 25 April 2005
M R Evans and T Hanney 2005 J. Phys. A: Math. Gen. 38 R195
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