Small deviations from local equilibrium for a process which exhibits hydrodynamical behavior. I

Abstract

The symmetric simple exclusion process where infinitely many particles move randomly on ℤ, jump with equal probability on nearest-neighbor sites, and interact by simple exclusion is considered. It is known that the only extremal invariant measures are Bernoulli, that each measure, in a suitable class, after a “macroscopic” time is locally described, at a zero-order approximation, by a Bernoulli measure with parameter depending on macroscopic space and time, and that the so-defined equilibrium profile satisfies the heat equation. Small deviations from local equilibrium in the hydrodynamical limit are investigated. It is proven, under suitable assumptions, that at first order the state is Gibbs with one- and two-body potentials whose strength depends only on macroscopic space and time and on the equilibrium profile. More precisely, the one-body potential is linear (on the microscopic positions of the particles) and proportional to the macroscopic space gradient of the equilibrium parameter at that time, so that Fourier law holds. The two-body potential varies on a macroscopic scale and does not depend on the microscopic positions of the particles; it is given by the value of the covariance of the Gaussian “macroscopic density fluctuation field.”