On the convergence to statistical equilibrium for harmonic crystals

Abstract

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution ${\mu }_t$ of the solution at time $t\in \mathbb{R}$. The initial measure ${\mu }_0$ 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 ${\mu }_t$ to a Gaussian measure as $t\to \infty$. The proof is based on the long time asymptotics of the Green's function and on Bernstein's ``room-corridors'' method.

References in corpus (1)

• On Convergence to Equilibrium Distribution, I. The Klein - Gordon Equation with Mixing

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