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This work is a follow-up on [GOVW]. In that previous work a two-scaleapproach was used to prove the logarithmic Sobolev inequality for a system ofspins with fixed mean whose potential is a bounded perturbation of a Gaussian,and to derive an abstract theorem for the convergence to the hydrodynamiclimit. This strategy was then successfully applied to Kawasaki dynamics. Herewe shall use again this two-scale approach to show that the microscopicvariable in such a model behaves according to a local Gibbs state. As aconsequence, we shall prove the convergence of the microscopic entropy to thehydrodynamic entropy.

A two-scale approach to the hydrodynamic limit, part II: local Gibbs behavior