Discontinuous Condensation Transition and Nonequivalence of Ensembles in a Zero-Range Process

Abstract

We study a zero-range process where the jump rates do not only depend on the local particle configuration, but also on the size of the system. Rigorous results on the equivalence of ensembles are presented, characterizing the occurrence of a condensation transition. In contrast to previous results, the phase transition is discontinuous and the system exhibits ergodicity breaking and metastable phases. This leads to a richer phase diagram, including nonequivalence of ensembles in certain phase regions. The paper is motivated by results from granular clustering, where these features have been observed experimentally.

Zero-range processes with multiple condensates: statics and dynamics

“…We refer to choice (3) as algebraic hopping rates. We note that the hopping rates (3) depend on the system size; another ZRP with size-dependent hopping rates that induce condensation has been studied in [14].…”

Emergent motion of condensates in mass-transport models

“…A primary role in these studies was played by the zero-range process (ZRP), an exactly solvable model in which particles hop between sites with rates that depend only on the number of particles in the departure site [9][10][11]. Extensions and variations of the ZRP have been used to study the emergence of multiple condensates [12], first-order condensation transitions [13,14], and the effect of interactions [15] and disorder [16] on condensation. Moreover, one-dimensional phase separation transitions in exclusion processes and other driven diffusive systems can quite generally be understood by a mapping on ZRPs [17].…”

Emergence of particle clusters in a one-dimensional model: connection to condensation processes

“…The limiting process that we took in order to arrive at this situation was somewhat unusual, but similar methods have been used in zero-range processes [3,17]. Our analysis of this model further accentuates the rich phenomenology that is accessible even in deceptively simple interacting particle systems.…”

“…These are the gaps that contribute to (19). If we consider the empirical density µ(x) defined in (17), the fact that the hydrodynamic limit consists of clusters separated by large gaps means that µ does not converge to any smooth profile µ 0 . Rather, assuming that a hydrodynamic description exists, we should think that µ, which is a sum of N Dirac delta functions, should converge (as N → ∞) to some µ 0 (x) = 1 n j M j δ(x −X j ) where M j is the mass of the jth cluster andX j is its position.…”