Preprint 25/2000

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
Pages: 30
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 tex2html_wrap_inline8 with an arbitrary odd tex2html_wrap_inline10 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 tex2html_wrap_inline12 of the random solution at the moment tex2html_wrap_inline14. The main result is the convergence of tex2html_wrap_inline12 to some Gaussian measure as tex2html_wrap_inline18. 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.

24.11.2021, 02:11