A simple stochastic model for the dynamics of condensation

Drouffe, J.M.; Godrèche, Claude; Camia, Federico

doi:10.1088/0305-4470/31/1/003

Cited by 60 publications

(135 citation statements)

References 11 publications

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“…Nonequilibrium properties of the zeta urn model have been studied recently [6], both at criticality and in the condensed phase, pursuing earlier investigations [9]. The most salient results are as follows.…”

Section: Nonequilibrium Critical Dynamicsmentioning

confidence: 94%

“…The acceptance rates W k,ℓ (t) depend on the thermal part of the rule. With the notation (2.4), the rates for the Metropolis rule [9],…”

Section: Dynamicsmentioning

confidence: 99%

“…Nonequilibrium properties of dynamical urn models such as the backgammon model [7,8,11,12] or the zeta urn model [9,6] have been extensively studied in recent years.…”

Section: Equilibrium Vs Nonequilibrium Dynamics: Two-time Quantitiesmentioning

confidence: 99%

“…Its dynamics has been subsequently investigated [9,6]. The zeta urn model belongs to the Monkey class, with mean-field heat-bath dynamics, defined by the energy function E(N i ) = ln(N i + 1).…”

Section: The Zeta Urn Modelmentioning

confidence: 99%

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Nonequilibrium dynamics of urn models

Godrèche

Luck

2002

J. Phys.: Condens. Matter

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Abstract. Dynamical urn models, such as the Ehrenfest model, have played an important role in the early days of statistical mechanics. Dynamical manyurn models generalize the former models in two respects: the number of urns is macroscopic, and thermal effects are included. These many-urn models are exactly solvable in the mean-field geometry. They allow analytical investigations of the characteristic features of nonequilibrium dynamics referred to as aging, including the scaling of correlation and response functions in the two-time plane and the violation of the fluctuation-dissipation theorem. This review paper contains a general presentation of these models, as well as a more detailed description of two dynamical urn models, the backgammon model and the zeta urn model. PrologueUrns containing balls, just as dice or playing cards, are ubiquitous in writings on probability theory, reminding us that this branch of mathematics owes its early developments to practical questions arising in playing games. Dynamical urn models, such as the Ehrenfest model, have played an important role in the elucidation of conceptual problems in statistical mechanics. More recently, dynamical many-urn models have been investigated in the mean-field geometry. These exactly solvable models exhibit characteristic features of nonequilibrium dynamics referred to as aging, such as the scaling of the correlation and response functions in the two-time plane and the violation of the fluctuation-dissipation theorem. This paper contains a didactic introduction to urn models (Section 1), a presentation of static and dynamical properties of many-urn models (Section 2), a reminder of the main characteristic features of aging (Section 3), and an overview of recent results on two dynamical urn models, namely the backgammon model (Section 4) and the zeta urn model (Section 5).

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Section: Nonequilibrium Critical Dynamicsmentioning

confidence: 94%

“…The acceptance rates W k,ℓ (t) depend on the thermal part of the rule. With the notation (2.4), the rates for the Metropolis rule [9],…”

Section: Dynamicsmentioning

confidence: 99%

“…Nonequilibrium properties of dynamical urn models such as the backgammon model [7,8,11,12] or the zeta urn model [9,6] have been extensively studied in recent years.…”

Section: Equilibrium Vs Nonequilibrium Dynamics: Two-time Quantitiesmentioning

confidence: 99%

Section: The Zeta Urn Modelmentioning

confidence: 99%

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Nonequilibrium dynamics of urn models

Godrèche

Luck

2002

J. Phys.: Condens. Matter

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“…In recent years many studies have been devoted to nonequilibrium statistical-mechanical models yielding condensation, such as zero-range processes (ZRP) [1,2,3,4,5,6,7], dynamical urn models [8,9,10,11,12], and mass transport models [13]. In all these models the condensate manifests itself by the macroscopic occupation of a single site by a finite fraction of the whole available mass.…”

Section: Introductionmentioning

confidence: 99%

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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From Urn Models to Zero-Range Processes: Statics and Dynamics

Godrèche

Ageing and the Glass Transition

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A simple stochastic model for the dynamics of condensation

Cited by 60 publications

References 11 publications

Nonequilibrium dynamics of urn models

Nonequilibrium dynamics of urn models

Structure of the stationary state of the asymmetric target process

From Urn Models to Zero-Range Processes: Statics and Dynamics

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