Strong asymmetric limit of the quasi-potential of the boundary driven weakly asymmetric exclusion process

TL;DR

This paper investigates the behavior of the quasi-potential in a boundary-driven weakly asymmetric exclusion process, revealing its strong asymmetric limit aligns with known functionals for the asymmetric case, thus connecting weak and strong asymmetry regimes.

Contribution

It establishes the strong asymmetric limit of the quasi-potential for the boundary-driven weakly asymmetric exclusion process, linking it to previously known functionals for the asymmetric exclusion process.

Findings

Quasi-potential expressed via a boundary value problem for weak asymmetry.

Recovery of the Derrida-Lebowitz-Speer functional in the strong asymmetric limit.

Connection between weak and strong asymmetry regimes in exclusion processes.

Abstract

We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. In the case in which the bulk asymmetry is in the same direction as the drift due to the boundary reservoirs, we prove that the quasi-potential can be expressed in terms of the solution to a one-dimensional boundary value problem which has been introduced by Enaud and Derrida \cite{de}. We consider the strong asymmetric limit of the quasi-potential and recover the functional derived by Derrida, Lebowitz, and Speer \cite{DLS3} for the asymmetric exclusion process.