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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
25/2000

On convergence to statistic equilibrium in wave equations with mixing

Tatiana V. Dudnikova, Alexander Komech and Nikita E. Ratanov

Abstract

The wave equation with constant or variable coefficients in the whole space $\mathbb{R}^n$ with an arbitrary odd $n \geq 3$ is considered. The initial datum is a translation-invariant random function with zero expectation and finite mean density of the energy, which also fits the mixing condition of Ibragimov-Linnik-Rosenblatt type. We study the distribution $\mu_t$ of the random solution at the moment $t \in \mathbb{R}$. The main result is the convergence of $\mu_t$ to some Gaussian measure as $t \to \infty$. This is the central limit theorem for linear wave equations. For the case of constant coefficients the proof is based on a new analysis of Kirchhoff's and Herglotz-Petrovskii's integral representations of the solution and on S.N.Bernstein's "room-corridors" method. The case of variable coefficients is reduced to constant coefficients. For this purpose the scattering theory for infinite energy solutions is constructed. The relation to Gibbs measures is discussed. The investigation is inspired by the problems of the mathematical foundation of the statistical physics.

Received:
31.03.00
Published:
31.03.00

Related publications

inJournal
2002 Repository Open Access
T. V. Dudnikova, Alexander Komech, N. E. Ratanov and Y. M. Suhov

On convergence to equilibrium distribution. II. The wave equation in odd dimensions, with mixing

In: Journal of statistical physics, 108 (2002) 5-6, pp. 1219-1253