Scalings limits for the exclusion process with a slow site

TL;DR

This paper investigates how a slow site affects the hydrodynamic limits and fluctuations in symmetric exclusion processes, revealing a phase transition depending on the slow site's jump rate scaling.

Contribution

It characterizes the hydrodynamic limits and fluctuations for exclusion processes with a slow site, identifying a phase transition driven by the slow site's jump rate scaling.

Findings

Hydrodynamic behavior follows the heat equation with periodic boundary conditions when g(n)=1+o(1).

Hydrodynamics follow the heat equation with Neumann boundary conditions when g(n)=α n^{-eta} with β>1.

A dynamical phase transition is established based on the slow site's jump rate scaling.

Abstract

We consider the symmetric simple exclusion processes with a slow site in the discrete torus with $n$ sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, \textit{the slow site}, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is given by a parameter $g(n)$. Two cases are treated, namely $g(n)=1+o(1)$, and $g(n)=\alpha n^{-\beta}$ with $\beta>1$, $\alpha>0$. In the former, both the hydrodynamic behavior and equilibrium fluctuations are driven by the heat equation (with periodic boundary conditions when in finite volume). In the latter, they are driven by the heat equation with Neumann boundary conditions. We therefore establish the existence of a dynamical phase transition. The critical behavior remains open.