Macroscopic evolution of particle systems with random field Kac interactions*

Abstract

We consider a lattice gas interacting via a Kac interaction Jγ(|x−y|) of range γ−1, γ>0, x,y∊ℤd and under the influence of an external random field given by independent bounded random variables with a translation invariant distribution. We study the evolution of the system through a conservative dynamics, i.e. particles jump to nearest neighbour empty sites, with rates satisfying a detailed balance condition with respect to the equilibrium measure. We prove that rescaling space as γ−1 and time as γ−2, in the limit γ→0, for dimension d⩾3, the macroscopic density profile ρ satisfies, a.s. with respect to the random field, a nonlinear integral differential equation, with a diffusion matrix determined by the statistical properties of the external random field. The result holds for all values of the density, also in the presence of phase segregation, and the equation is in the form of the flux gradient for the energy functional.