On two-temperature problem for harmonic crystals

We consider the dynamics of a harmonic crystal in d dimensions with n com-ponents, 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 x1,..., xd−1 and converges to different translation-invariant processes as xd Q±., with the distributions m±. We study the distribution mt of the solution at time t ¥ R. The main result is the convergence of mt to a Gaussian translation-invariant measure as tQ.. 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 m±=g ± with two different temperatures T ± is given. Limit-ing mean energy current density is −(0,..., 0, C(T+−T−)) with some positive constant C> 0 what corresponds to Second Law. KEY WORDS: Harmonic crystal; random initial data; mixing condition; Gaussian measures; covariance matrices; characteristic functional