On the non-equilibrium states of the crystal lattice

We consider the Cauchy problem for an infinite crystal lattice in $\mathbb{Z}^d$, $d\geqslant1$, with random initial data. We study the behavior of the distributions of the solutions as $t\to\infty$. The main goal is to find the limiting stationary non-equilibrium states in which there is a constant non-zero heat flux passing through the lattice.