Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps

TL;DR

This paper studies a symmetric exclusion process with long jumps connected to reservoirs, showing how different reservoir interaction rates lead to various reaction-diffusion equations with specific boundary conditions.

Contribution

It introduces a model with long jumps and variable reservoir interaction rates, deriving the resulting reaction-diffusion equations and boundary conditions based on the rate parameter.

Findings

Different reservoir rates lead to distinct boundary conditions in the limiting PDEs.

The model captures both slow and fast reservoir dynamics in a unified framework.

Reaction-diffusion equations describe the macroscopic behavior depending on the parameter .

Abstract

We consider an exclusion process with long jumps in the box $\Lambda\_N=\{1, \ldots,N-1\}$, for $N \ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The jump rate is described by a transition probability $p(\cdot)$ which is symmetric, with infinite support but with finite variance. The reservoirs add or remove particles with rate proportional to $\kappa N^{-\theta}$, where $\kappa>0$ and $\theta \in\mathbb R$. If $\theta>0$ (resp. $\theta<0$) the reservoirs add and fastly remove (resp. slowly remove) particles in the bulk. According to the value of $\theta$ we prove that the time evolution of the spatial density of particles is described by some reaction-diffusion equations with various boundary conditions.