Hydrodynamic limit for a facilitated exclusion process

TL;DR

This paper establishes that a constrained exclusion process with dynamical rules converges to a fast diffusion equation at the macroscopic level, revealing phase transition behavior based on initial density.

Contribution

It introduces a hydrodynamic limit for a facilitated exclusion process with degenerate rates, linking microscopic dynamics to a macroscopic fast diffusion equation.

Findings

System reaches ergodic component in subdiffusive time

Macroscopic density evolves according to a fast diffusion equation

Phase transition occurs at critical density 1/2

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 $x\pm1$ unless site $x\mp1$ 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.