On Two-Temperature Problem for Harmonic Crystals

We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n≥1. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The random function is translation-invariant in x ₁,...,x _d?1 and converges to different translation-invariant processes as x _d →±∞, with the distributions μ _±. We study the distribution μ _t of the solution at time $t in mathbb{B}$ . The main result is the convergence of μ _t to a Gaussian translation-invariant measure as t→∞. The proof is based on the long time asymptotics of the Green function and on Bernstein's “room-corridor” argument. The application to the case of the Gibbs measures μ _±=g _± with two different temperatures T _± is given. Limiting mean energy current density is ?(0,...,0,C(T ₊?T _?)) with some positive constant C>0 what corresponds to Second Law.