Nonequilibrium phase transition in a non-integrable zero-range process

Godrèche, Claude

doi:10.1088/0305-4470/39/29/003

Cited by 7 publications

(6 citation statements)

References 12 publications

(25 reference statements)

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“…The existence of a threshold density ρ 0 at which the background density has a discontinuous jump (see (5.7)) is reminiscent of what occurs in the model studied in [21], namely a ZRP with two species of particles, and with rates such that the stationary-state measure does not have a product form. When the densities ρ (1) and ρ (2) of the two species are equal, the behaviour of the system is qualitatively the same as that of the canonical ZRP (with one species).…”

Section: Discussionmentioning

confidence: 89%

Structure of the stationary state of the asymmetric target process

Luck¹,

Godrèche²

2007

J. Stat. Mech.

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Abstract. We introduce a novel migration process, the target process. This process is dual to the zero-range process (ZRP) in the sense that, while for the ZRP the rate of transfer of a particle only depends on the occupation of the departure site, it only depends on the occupation of the arrival site for the target process. More precisely, duality associates to a given ZRP a unique target process, and vice-versa. If the dynamics is symmetric, i.e., in the absence of a bias, both processes have the same stationary-state product measure. In this work we focus our interest on the situation where the latter measure exhibits a continuous condensation transition at some finite critical density ρ c , irrespective of the dimensionality. The novelty comes from the case of asymmetric dynamics, where the target process has a nontrivial fluctuating stationary state, whose characteristics depend on the dimensionality. In one dimension, the system remains homogeneous at any finite density. An alternating scenario however prevails in the high-density regime: typical configurations consist of long alternating sequences of highly occupied and less occupied sites. The local density of the latter is equal to ρ c and their occupation distribution is critical. In dimension two and above, the asymmetric target process exhibits a phase transition at a threshold density ρ 0 much larger than ρ c . The system is homogeneous at any density below ρ 0 , whereas for higher densities it exhibits an extended condensate elongated along the direction of the mean current, on top of a critical background with density ρ c .

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Section: Discussionmentioning

confidence: 89%

Structure of the stationary state of the asymmetric target process

Luck¹,

Godrèche²

2007

J. Stat. Mech.

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“…So far a discontinuous transition in a zero-range process has only been observed heuristically in a two-species system where the stationary state is not known [15]. The above features only concern the stationary measure, and for systems without L-dependence they have been shown rigorously in a general context [16].…”

Section: Differences To Previous Resultsmentioning

confidence: 99%

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

Großkinsky

Schütz

2008

J Stat Phys

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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.

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“…On the other hand, simple rates may lead to zero-range processes for which the stationary distribution is unknown and not of product form. For such models, a recent study revealed the possibility of a discontinuous condensation transition [31].…”

Section: Generic Examplesmentioning

confidence: 99%

Equivalence of ensembles for two-species zero-range invariant measures

Großkinsky

2008

Stochastic Processes and their Applications

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We study the equivalence of ensembles for stationary measures of interacting particle systems with two conserved quantities and unbounded local state space. The main motivation is a condensation transition in the zero-range process which has recently attracted attention. Establishing the equivalence of ensembles via convergence in specific relative entropy, we derive the phase diagram for the condensation transition, which can be understood in terms of the domain of grand-canonical measures. Of particular interest, also from a mathematical point of view, are the convergence properties of the Gibbs free energy on the boundary of that domain, involving large deviations and multivariate local limit theorems of subexponential distributions.

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Nonequilibrium phase transition in a non-integrable zero-range process

Cited by 7 publications

References 12 publications

Structure of the stationary state of the asymmetric target process

Structure of the stationary state of the asymmetric target process

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

Equivalence of ensembles for two-species zero-range invariant measures

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