A correction to the hydrodynamic limit of boundary driven exclusion processes in a super-diffusive time scale

TL;DR

This paper corrects the hydrodynamic limit description of a boundary driven exclusion process in a super-diffusive time scale, accounting for boundary conditions and external fields evolving smoothly.

Contribution

It provides a refined derivation of the density evolution equation for a weakly asymmetric exclusion process in a specific super-diffusive regime.

Findings

Derived an equation for density evolution up to order epsilon_N.

Analyzed the effects of boundary conditions and external fields.

Extended previous models to include super-diffusive time scales.

Abstract

We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval $[0,N]\cap Z$ in the super-diffusive time scale $N^2 \epsilon^{-1}_N$, where $1\ll \epsilon^{-1}_N \ll N^{1/4}$. We assume that the external field and the chemical potentials, which fix the density at the boundaries, evolve smoothly in the macroscopic time scale. We derive an equation which describes the evolution of the density up to the order $\epsilon_N$.