On the convergence to statistical equilibrium for harmonic crystals

We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n arbitrary, d,n ≥1, and study the distribution μ _t, of the solution at time t ∈ R. The initial measure μ ₀ has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt - resp. Ibragimov-Linnik type mixing condition. The main result is the convergence of μ _t, to a Gaussian measure as t → ∞. The proof is based on the long time asymptotics of the Green's function and on Bernstein's "room-corridors" method.