Exclusion process with slow boundary

TL;DR

This paper analyzes the hydrodynamic and hydrostatic behavior of the symmetric exclusion process with slow boundary conditions, revealing different boundary behaviors depending on the boundary rate parameter .

Contribution

It characterizes the boundary conditions of the heat equation for the exclusion process with slow boundary, depending on the parameter .

Findings

Three linear profiles for hydrostatic states based on .

Different boundary conditions (Dirichlet, Robin, Neumann) for the heat equation depending on .

Hydrodynamic limit described by the heat equation with boundary conditions varying with .

Abstract

We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with \emph{slow boundary}. The term \emph{slow boundary} means that particles can be born or die at the boundary sites, at a rate proportional to $N^{-\theta}$, where $\theta > 0$ and $N$ is the scaling parameter. In the bulk, the particles exchange rate is equal to $1$. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $\theta$; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on $ \theta $. If $\theta \in(0,1)$, we get Dirichlet boundary conditions, (which is the same behavior if $\theta=0$, see \cite{f}); if $\theta=1$, we get Robin boundary conditions; and, if $\theta\in(1,\infty)$, we get Neumann boundary conditions.