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On the Convergence to a Statistical Equilibrium in the Crystal Coupled to a Scalar Field

T. V. Dudnikova·A. I. Komech
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Abstract

We consider the dynamics of a field coupled to a harmonic crystal with $n$ components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics are translation-invariant with respect to the subgroup $\Z^d$ of $\R^d$. The initial data is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. Moreover, initial correlation functions are translation-invariant with respect to the discrete subgroup $\Z^d$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$, where $μ_\infty$ is translation-invariant with respect to the subgroup $\Z^d$.

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T. V. Dudnikova, A. I. Komech. 2005-08-26. On the Convergence to a Statistical Equilibrium in the Crystal Coupled to a Scalar Field. https://arxiv.org/abs/math-ph/0508053

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