Abstract:
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.
Keywords:the harmonic crystal coupled to a scalar field, Cauchy problem,
random initial data, space-time statistical solutions, weak convergence of measures.
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