On convergence to statistic equilibrium in wave equations with mixing
Tatiana V. Dudnikova, Alexander Komech, and Nikita E. Ratanov
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Submission date: 31. Mar. 2000
published in: Journal of statistical physics, 108 (2002) 5-6, p. 1219-1253
with the following different title: On convergence to equilibrium distribution. II. The wave equation in odd dimensions, with mixing
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The wave equation with constant or variable coefficients in the whole space with an arbitrary odd 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 of the random solution at the moment . The main result is the convergence of to some Gaussian measure as . 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.