On the convergence to statistical equilibrium for harmonic crystals

TL;DR

This paper proves that the distribution of a harmonic crystal's solution converges to a Gaussian measure over time, under certain initial conditions and mixing assumptions, using Green's function asymptotics and Bernstein's method.

Contribution

It establishes the convergence to statistical equilibrium for harmonic crystals with general dimensions and components under broad initial conditions.

Findings

Distribution converges to Gaussian as time approaches infinity.

Convergence holds for arbitrary dimensions and components.

Uses Green's function asymptotics and Bernstein's method for proof.

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 $t\in\R$. 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.