Benois O., Esposito R., Marra R.
Equilibrium Fluctuations for Lattice Gases
(104K, TeX)
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 $\xi^\e(t)=\exp
\{-\ve^{-1}t E\}\z^\ve$, 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.