Convergence to stationary non-equilibrium states for Klein–Gordon equations

We consider Klein–Gordon equations in $\mathbb{R}^d$, $d\ge2$, with constant or variable coefficients and study the Cauchy problem with random initial data. We investigate the distribution $\mu_t$ of a random solution at moments of time $t\in\mathbb{R}$. We prove the convergence of correlation functions of the measure $\mu_t$ to a limit as $t\to\infty$. The explicit formulae for the limiting correlation functions and the energy current density (in mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of $\mu_t$ to a limiting measure as $t\to\infty$. We apply these results to the case when the initial random function has the Gibbs distribution with different temperatures in some infinite “parts” of the space. In this case, we find states in which the limiting energy current density does not vanish. Thus, for the model being studied, we construct a new class of stationary non-equilibrium states.

Klein–Gordon equations, Cauchy problem, random initial data, weak convergence of measures, Gibbs measures, energy current density, non-equilibrium state