On a Two-Temperature Problem for Wave Equation

Abstract

Consider the wave equation with constant or variable coefficients in $\R^3$. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_3\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the random solution at a time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$ that means central limit theorem for the wave equation. The proof is based on the Bernstein `room-corridor' argument. The application to the case of the Gibbs measures $μ_\pm=g_\pm$ 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 and equals to $-C(0,0,T_+ -T_-)$ with $C>0$ for the convolution with a nontrivial test function.