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

Deviations from local equilibrium in a symmetric simple exclusion process are studied. Properties which mimic the hydrodynamical behavior of real physical systems are underlined. The only extremal invariant measures are the Bernoulli measures. Each measure in a suitable class, after a macroscopic time, at a zero order approximation, by a Bernoulli measure with parameter depending on macroscopic space and time. The so defined equilibrium profile satisfies the heat equation. It is proven that at first order, the state is the Gibbs phenomenon with one and two body potentials. Potential strength depends only on macroscopic space and time and on the equilibrium profile. 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 is not dependent on the microscopic positions of the particles, which is given by the value of the covariance of the Gaussian macroscopic density fluctuation field.