Equilibrium Fluctuations for Lattice Gases

Abstract

The authors in a previous paper proved the hydrodynamic incompressible limit in $d\ge 3$ for a thermal lattice gas, namely a law of large numbers for the density, velocity field and energy. In this paper the equilibrium fluctuations for this model are studied and a central limit theorem is proved for a suitable modification of the vector fluctuation field $\z(t)$, whose components are the density, velocity and energy fluctuations fields. We consider a modified fluctuation field $ξ^\e(t)=\exp \{-\ve^{-1}t E\}\z^\ve$, where $E$ is the linearized Euler operator around the equilibrium and prove that $ξ^\e(t)$ converges to a vector generalized Ornstein-Uhlenbeck process $\xi (t)$, which is formally solution of the stochastic differential equation $d\xi (t)=N\xi (t)dt+Bd{W}_t$, with $B{B}^{\ast }=-2NC$, where $C$ is the compressibility matrix, $N$ is a matrix whose entries are second order differential operators and $B$ is a mean zero Gaussian field. The relation $-2NC=B{B}^{\ast }$ is the fluctuation-dissipation relation.

32 pages