On a two-temperature problem for Klein–Gordon equation

We consider the Klein–Gordon equation in $\mathbf{R}^n$, $n\geq 2$, with constant or variable coefficients. The initial datum is a random function with a finite mean density of the energy and satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. We also assume that the random function is close to different space-homogeneous processes as $x_n\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution $\mu_t$ of the random solution at time $t\in\mathbf{R}$. The main result is the convergence of $\mu_t$ to a Gaussian translation-invariant measure as $t\to\infty$ that means the central limit theorem for the Klein–Gordon equation. The proof is based on the Bernstein “room-corridor” method and oscillatory integral estimates. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. It is proved that limit mean energy current density formally is $-\infty\cdot(0,\dots,0,T_+-T_-)$ for the Gibbs measures, and it is finite and equals $-C(0,\dots,0,T_+-T_-)$ with some positive constant $C>0$ for the smoothed solution. This corresponds to the second law of thermodynamics.