On convergence to equilibrium for one-dimensional chain of harmonic oscillators in the half-line

The initial-boundary value problem for an infinite one-dimensional chain of harmonic oscillators on the half-line is considered. The large time asymptotic behavior of solutions is studied. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution $\mu_t$ of the random solution at time moments $t\in\mathbb{R}$. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$. We find stationary states in which there is a non-zero energy current at origin.