Space-time statistical solutions for the Hamiltonian field-crystal system

We consider the dynamics of a scalar field coupled to a harmonic crystal with $n$ components in dimension $d$, $d, n\geqslant1$. The dynamics of the system is translation-invariant with respect to the discrete subgroup $\mathbb{Z}^d$ of $\mathbb{R}^d$. We study the Cauchy problem with random initial data. We assume that the initial measure has a finite mean energy density and the initial correlation functions are translation invariant with respect to the subgroup $\mathbb{Z}^d$. We prove the convergence of space-time statistical solutions to a Gaussian measure.