The two-scale approach to hydrodynamic limits for non-reversible dynamics

In \cite{GOVW09}, a new method to study hydrodynamic limits, called the two-scale approach, was developed for reversible dynamics. In this work, we generalize this method to a family of non-reversible dynamics. As an application, we obtain quantitative rates of convergence to the hydrodynamic limit for a weakly asymmetric version of the Ginzburg\tire Landau model endowed with Kawasaki dynamics. These results also imply local Gibbs behaviour, following a method of \cite{Fat13}.