Hydrodynamic limit for a particle system with degenerate rates

Gonçalves, Patrícia; Landim, Cláudio; Toninelli, Cristina

doi:10.1214/09-aihp210

Cited by 50 publications

(110 citation statements)

References 10 publications

(26 reference statements)

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“…Secondly, the Gibbs measures (grand canonical measures) of our process are not product, while they are Bernoulli product measures in [9]. In particular, when the density is close to 1 2 , these measures exhibit spatial correlations, and adapting the entropy method for these non-product grand canonical measures requires significant extra technical work.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hydrodynamic limit for a facilitated exclusion process

Blondel¹,

Erignoux²,

Sasada³

et al. 2020

Ann. Inst. H. Poincaré Probab. Statist.

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We study the hydrodynamic limit for a periodic 1-dimensional exclusion process with a dynamical constraint, which prevents a particle at site x from jumping to sitex ± 1 unless site x ∓ 1 is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than 1 2 , or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density 1 2 , the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

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Section: Introductionmentioning

confidence: 99%

“…As the nature of microscopic dynamics is different from [9], the macroscopic equation is also different from the PME, and we obtain the following macroscopic equation…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamic limit for a facilitated exclusion process

Blondel¹,

Erignoux²,

Sasada³

et al. 2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…Furthermore for the same models it is established that the self-diffusion coefficient of the tagged particle is strictly positive. Moreover the hydrodynamic limit has been succesfully analyzed for a special class of gradient type in [13].…”

Section: Definition 11 a Model Is Said To Be Non-cooperative If Itsmentioning

confidence: 99%

Kinetically Constrained Lattice Gases

Cancrini

Martinelli

Roberto³

et al. 2010

Commun. Math. Phys.

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ABSTRACT. Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. Their distintictive signature is the following: i) reversibility w.r.t. product i.i.d. Bernoulli measure at any particle density and ii) vanishing of the exchange rate across any edge unless the particle configuration around the edge satisfies a proper constraint besides hard core. Because of degeneracy of the exchange rates the models can show anomalous time decay in the relaxation process w.r.t. the usual high temperature lattice gas models particularily in the so called cooperative case, when the vacancies have to collectively cooperate in order for the particles to move through the systems. Here we focus on the Kob-Andersen (KA) model, a cooperative example widely analyzed in the physics literature, both in a finite box with particle reservoirs at the boundary and on the infinite lattice. In two dimensions (but our techniques extend to any dimension) we prove a diffusive scaling O(L 2 ) (apart from logarithmic corrections) of the relaxation time in a finite box of linear size L. We then use the above result to prove a diffusive decay 1/t (again apart from logarithmic corrections) of the spin-spin time autocorrelation function at any particle density, a result that has been sometimes questioned on the basis of numerical simulations. The techniques that we devise, based on a novel combination of renormalization and comparison with a long-range Glauber type constrained model, are robust enough to easily cover other choices of the kinetic constraints.

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“…Microscopic derivations of the heat and porous medium equations have already been obtained, see [5], [7], [9] and [12], for instance. Here we provide a derivation of the hydrodynamic limit for a fast diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Lattice model for fast diffusion equation

Hernández

Jara

Valentim

2020

Stochastic Processes and their Applications

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We obtain a fast diffusion equation (FDE) as scaling limit of a sequence of zero-range process with symmetric unit rate. Fast diffusion effect comes from the fact that the diffusion coefficient goes to infinity as the density goes to zero. Therefore, in order to capture the behaviour for an arbitrary small density of particles, we consider a proper rescaling of a model with a typically high number of particles per site. Furthermore, we obtain some results on the convergence for the method of lines for FDE.2010 Mathematics Subject Classification. 60H30, 60J60, 82C22 .

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Hydrodynamic limit for a particle system with degenerate rates

Cited by 50 publications

References 10 publications

Hydrodynamic limit for a facilitated exclusion process

Hydrodynamic limit for a facilitated exclusion process

Kinetically Constrained Lattice Gases

Lattice model for fast diffusion equation

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