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MiS Preprint
26/2000
On convergence to statistic equilibrium in two-temperature problem for wave equations with mixing
Tatiana V. Dudnikova, Alexander Komech and Herbert Spohn
Abstract
The wave equation in the whole space $R^3$ is considered. The initial datum is a random function with finite mean density of the energy which also fits the mixing condition of Ibragimov-Linnik-Rosenblatt type. The random function converges to different space-homogeneous processes as $x_3 \to \pm \infty$, with the distributions $\mu _\pm$. We study the distribution $\mu_i$ of the random solution at the moments $t\in R$. The main result is the convergence of $\mu_i$ to an equilibrium Gaussian translation-invariant measure as $t \to \infty$. The application to the case of the Gibbs measures $\mu_\pm =g_\pm$ with two different temperatures $T_\pm$ is given. Limiting mean energy current density formally is $-\infty \cdot (0,0,T_+ - T_- )$ for the Gibbs measures, and it is finite $-C(0,0,T_+ -T_- )$ with C>0 for the convolution with a nontrivial test function.
