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 μt of the solution at time t∈R. The initial measure μ0 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 μt 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.