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

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

2005-08-26

arxiv.org/abs/math-ph/0508053v1

Russ. J. Math. Physics, 12 (2005), no. 3, 301-325

Physics

Mathematical Physics

33 pages

Scientific paper

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 $\mu_t$ of the solution at time $t\in\R$. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$, where $\mu_\infty$ is translation-invariant with respect to the subgroup $\Z^d$.

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