On the asymptotical normality for lattice Hamiltonian dynamics. Energy transport equation

We consider the lattice dynamics in the whole space (in the half-space) and study the Cauchy (respectively, mixed initial-boundary value) problem with random initial data. We prove the weak convergence of statistical solutions to a limit for large time. Further, we assume that the initial measure enforces slow spatial variation on the linear scale $1/\varepsilon$. We check that for times of order $1/\varepsilon$, the limit covariance changes in time and is governed by the energy transport equation.