Nonequilibrium statistical mechanics of the zero-range process and related models

By M. Evans, T. Hanney. 2005

500 citations recorded by the source corpus.

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; 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0bda4b2ffb122c59513d437cab08dc62b9627cd8

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In Condensed Matter Physics

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Citations in the corpus

Random walks and diffusion on networks (2016, background): “t, and their very close relatives. We only very rarely mention some of the numerous other types of random walks, which include correlated RWs [69], self-avoiding RWs [4, 70, 71], zero-range processes [72], multiplicative random processes [73, 74], adaptive RWs (including reinforced RWs [75]), branch”
Metastability of reversible condensed zero range processes on a finite set (2009, result): “We refer to the recent review by Evans and Hanney [4].”
Extreme value statistics of correlated random variables: A pedagogical review (2019, background): “At long times, the system reaches a stationary state where the joint distribution of the occupation numbers at different sites is given by PZRP(`1, · · · , `N |L) = 1 ZN (L) N∏ i=1 f(`i) δ∑N i=1 `i,L , (55) where f(`) = 1/ ∏` k=1 U(k) and ZN (L) is the normalization constant.”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “11 Zero-range processes were introduced by Spitzer [122] and have been extensively studied in the physics literature, see [44].”
Critical phenomena in complex networks (2007, background): “Particularly, z = 2 for a d > 2-regular lattice (Evans and Hanney, 2005). In a finite network, the condensate at a given vertex exists a finite time τm(N). After “melting” at this vertex, the condensate appears at another vertex, then—at another one, and so on. For a homogeneous network, e.g., for a”
Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport (2011, unknown): “Much progress has emerged, especially in the last two decades (see [332] for a review).”
Nonequilibrium steady states of matrix-product form: a solver's guide (2007, background): “tions for factorisation in a broad class of models which includes the ZRP have been established [11]. Zero-range processes and various generalisations have been reviewed recently in a companion paper [12] and so we do not discuss them further here. Matrix product steady state A matrix product steady”
Dragon-kings: Mechanisms, statistical methods and empirical evidence (2012, background): “…one have an average number equal to that for zero potential and one box has a finite fraction of the total number of particles, while the distribution of particles per box is a power law with exponent β, the inverse temperature (Bialas et al., 1999; Evans and Hanney, 2005; Evans et al., 2006).”
Stochastic models of intracellular transport (2013, background): “…2007; Lipowsky and Klumpp, 2005; Reimann, 2002), reaction kinetics (Hanggi et al., 1990), confined diffusion (Burada et al., 2009), random intermittent search processes (Benichou et al., 2011b), and exclusion processes (Chou et al., 2011; Evans and Hanney, 2005; Schadschneider et al., 2010).”