Quasi-static limit for the asymmetric simple exclusion

TL;DR

This paper analyzes the quasi-static evolution of a one-dimensional asymmetric simple exclusion process with time-dependent boundary rates, showing that the macroscopic density profile follows a stationary Burgers equation with boundary conditions evolving slowly over time.

Contribution

It establishes the quasi-static limit for the asymmetric simple exclusion process with boundary rates changing on a slow time scale, extending understanding of boundary-driven particle systems.

Findings

The system evolves quasi-statically at the time scale N^{1+a}.

The macroscopic density profile is given by the entropy solution of the stationary Burgers equation.

Different boundary rate types (Liggett and reversible) are analyzed within the framework.

Abstract

We study the one-dimensional asymmetric simple exclusion process on the lattice $\{1, \dots,N\}$ with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale $N^{-a}$ with $a>0$. We prove that at the time scale $N^{1+a}$ the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy--entropy flux pairs and a coupling argument.