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

TL;DR

This paper extends a two-scale approach to demonstrate that microscopic variables in a spin system follow a local Gibbs state, leading to the convergence of microscopic entropy to hydrodynamic entropy.

Contribution

It introduces a novel application of the two-scale approach to establish local Gibbs behavior in a hydrodynamic limit setting.

Findings

Microscopic variables follow a local Gibbs distribution.

Microscopic entropy converges to hydrodynamic entropy.

The approach applies to Kawasaki dynamics.

Abstract

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