On the energy current for harmonic crystals

We consider a $d$-dimensional harmonic crystal, $d\ge 1$, and study the Cauchy problem with random initial data. We assume that the random initial function is close to different translation-invariant processes for large values of $x_1,\dots,x_k$ with some $k\in\{1,\dots,d\}$. The distribution $\mu_t$ of the solution at time $t\in\mathbb{R}$ is studied. We prove the convergence of correlation functions of the measures $\mu_t$ to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in mean) are obtained in the terms of the initial covariance. We give the application to the case of the Gibbs initial measures with different temperatures. In particular, we find stationary states in which there is a constant non-zero energy current flowing through the harmonic crystal. Furthermore, the weak convergence of $\mu_t$ to a limit measure is proved. We also study the initial boundary value problem for the harmonic crystal with zero boundary condition and obtain the similar results.