Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hydrodynamic limit for a facilitated exclusion process

Oriane Blondel, Clément Erignoux, Makiko Sasada, and Marielle Simon

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Abstract

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 site $x\pm 1$ unless site $x\mp 1$ 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 $\frac{1}{2}$, 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 $\frac{1}{2}$, 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.

Résumé

Nous étudions la limite hydrodynamique d’un système d’exclusion unidimensionnel avec une contrainte dynamique, qui empêche une particule en $x$ de sauter en $x\pm 1$ à moins que $x\mp 1$ soit occupé. Ce processus à taux de sauts dégénérés admet des états transients, qu’il finit par quitter pour atteindre une composante ergodique dans le cas où la densité initiale macroscopique est supérieure à $\frac{1}{2}$, ou un de ses états absorbants dans l’autre cas. Ce processus fait partie des gaz conservatifs sur réseau, qui ont été introduits dans la litérature physique comme systèmes présentant une transition de phase active-absorbante en présence d’un champ conservé. Nous montrons que pour des profils initiaux de densité suffisamment réguliers et strictement supérieurs à $\frac{1}{2}$, le profil de densité macroscopique évolue à l’échelle diffusive suivant une équation de diffusion rapide (FDE). La première étape de la preuve consiste à montrer que, typiquement, le système atteint une composante ergodique en temps sous-diffusif.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 1 (2020), 667-714.

Dates
Received: 23 July 2018
Revised: 5 February 2019
Accepted: 27 February 2019
First available in Project Euclid: 3 February 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1580720505

Digital Object Identifier
doi:10.1214/19-AIHP977

Mathematical Reviews number (MathSciNet)
MR4059004

Zentralblatt MATH identifier
07199321

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 35R35: Free boundary problems 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
Facilitated exclusion process Hydrodynamic limit Active-absorbing phase transition Conserved lattice gases Fast diffusion equation

Citation

Blondel, Oriane; Erignoux, Clément; Sasada, Makiko; Simon, Marielle. Hydrodynamic limit for a facilitated exclusion process. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 1, 667--714. doi:10.1214/19-AIHP977. https://projecteuclid.org/euclid.aihp/1580720505


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