On the energy current for the harmonic crystals in the half-space

We consider the dynamics of the harmonic crystal in the half-space with zero boundary condition. We assume that the initial data are a random function. We prove the convergence of the distributions of the solutions to a limiting measure for large times. The formula for the limiting energy current density (in mean) is derived. The application to the case of the Gibbs initial measures is given. We find the stationary non-equilibrium states, in which there exists a non-zero heat flow passing through the points of the crystal.