Large deviation functional of the weakly asymmetric exclusion process

We obtain the large deviation functional of a density profile for theasymmetric exclusion process of L sites with open boundary conditionswhen the asymmetry scales like 1/L. We recover as limitingcases the expressions derived recently for the symmetric (SSEP) and theasymmetric (ASEP) cases. In the ASEP limit, the non linear differentialequation one needs to solve can be analysed by a method which resemblesthe WKB method