A Correction to the Hydrodynamic Limit of Boundary Driven Weakly Asymmetric Exclusion Processes in a Quasi-Static Time Scale

We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval [ 0 , N ] ∩ Z in the quasi-static time scale N 2 ϵ N - 1 , where 1 ≪ ϵ N - 1 ≪ 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 ϵ N .