Equilibrium Fluctuations for Lattice Gases

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 $\xi^\e(t)=\exp \{-\ve^{-1}t E\}\z^\ve$, where $E$ is the linearized Euler operator around the equilibrium and prove that $\xi^\e(t)$ converges to a vector generalized Ornstein-Uhlenbeck process $\xi(t)$, which is formally solution of the stochastic differential equation $d \xi(t)=N\xi(t)dt+ B dW_t$, with $ BB^*=-2 NC$, 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 $-2NC=BB^*$ is the fluctuation-dissipation relation.