Invariant Measures for the Zero Range Process

Andjel, Enrique D.

doi:10.1214/aop/1176993765

Cited by 220 publications

(195 citation statements)

References 16 publications

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“…The model has three parameters: b, α, and the density ρ ≡ N/L, which is conserved by the dynamics. In the usual ZRP (the case of α = 1), the stationary distribution is known to factorize into a product of single site terms, and so can be exactly calculated [9,27,28]. This factorization property renders the ZRP quite special, as slight variations of the ZRP dynamics result in nonfactorizable models.…”

Section: Modelmentioning

confidence: 99%

Emergent motion of condensates in mass-transport models

2013

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We examine the effect of spatial correlations on the phenomenon of real-space condensation in driven masstransport systems. We suggest that in a broad class of models with a spatially correlated steady state, the condensate drifts with a nonvanishing velocity. We present a robust mechanism leading to this condensate drift. This is done within the framework of a generalized zero-range process (ZRP) in which, unlike the usual ZRP, the steady state is not a product measure. The validity of the mechanism in other mass-transport models is discussed.

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Section: Modelmentioning

confidence: 99%

Emergent motion of condensates in mass-transport models

2013

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“…The product measure µ N with geometric distribution is invariant for the zero range process (cf. Andjel (1982)). Motion by mean curvature for the sets (A σ N 2 t N ) corresponds then to the hydrodynamic limit for the zero range process.…”

Section: Resultsmentioning

confidence: 99%

The Initial Drift of a 2D Droplet at Zero Temperature

Cerf

Louhichi

2006

Probab. Theory Relat. Fields

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We consider the 2D stochastic Ising model evolving according to the Glauber dynamics at zero temperature. We compute the initial drift for droplets which are suitable approximations of smooth domains. A specific spatial average of the derivative at time 0 of the volume variation of a droplet close to a boundary point is equal to its curvature multiplied by a direction dependent coefficient. We compute the explicit value of this coefficient. The first version of this work dealt only with the deterministic initial condition; we are very grateful to Herbert Spohn for explaining to us the relevance of the randomness in the initial condition.

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“…the density fluctuations move with a speed v. Using the mapping (13) and retaining the quadratic term that carries information about the decay of the density fluctuations, we obtain the KPZ equation for the height profile in one (space) dimension [11,12]:…”

Section: Discussionmentioning

confidence: 99%

“…The class of models that we consider here are related to a zero range process whose stationary state is known exactly [13,14] or a misanthrope process for which the exact steady state measure can be found in certain cases [15,16]. In these models, a jamming transition appears as a condensation transition where a macroscopically large mass cluster forms on a single site at high densities but at low densities, the mass is distributed uniformly.…”

Section: Introductionmentioning

confidence: 99%

Critical dynamics of the jamming transition in one-dimensional nonequilibrium lattice-gas models

Priyanka

Jain

2016

Phys. Rev. E

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We consider several one-dimensional driven lattice-gas models that show a phase transition in the stationary state between a high-density fluid phase in which the typical length of a hole cluster is of order unity and a low-density jammed phase where a hole cluster of macroscopic length forms in front of a particle. Using a hydrodynamic equation for an interface growth model obtained from the driven lattice-gas models of interest here, we find that in the fluid phase, the roughness exponent and the dynamic exponent that, respectively, characterize the scaling of the saturation width and the relaxation time of the interface with the system size are given by the Kardar-ParisiZhang exponents. However, at the critical point, we show analytically that when the equal time density-density correlation function decays slower than inverse distance, the roughness exponent varies continuously with a parameter in the hop rates but it is one half otherwise. Using these results and numerical simulations for the density-density autocorrelation function, we further find that the dynamic exponent z = 3/2 in all cases.

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Invariant Measures for the Zero Range Process

Cited by 220 publications

References 16 publications

Emergent motion of condensates in mass-transport models

Emergent motion of condensates in mass-transport models

The Initial Drift of a 2D Droplet at Zero Temperature

Critical dynamics of the jamming transition in one-dimensional nonequilibrium lattice-gas models

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