On the stationary non-equilibrium measures for the “field–crystal” system

In the paper, we consider the Cauchy problem for a Hamiltonian system consisting of a Klein–Gordon field and an infinite harmonic crystal. We assume that the initial data of the problem are a random function and study the convergence of the distributions of the solutions to a limiting measure for large times. Under the condition that the initial random function in the “left” and “right” parts of the space has the Gibbs distribution with different temperatures, we find the stationary states of the system in which the limiting energy current density does not vanish. Thus, for this system, a class of stationary non-equilibrium states is constructed.