We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution $\mu_t$ of the solution at time $t\in\R$. The initial measure $\mu_0$ has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green's function and on Bernstein's ``room-corridors'' method

Consider the Klein-Gordon equation (KGE) in IR n , n ≥ 2, with constant or variable coefficients. We study the distribution µ t of the random solution at time t ∈ IR. We assume that the initial probability measure µ 0 has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that µ 0 satisfies a Rosenblattor Ibragimov-Linnik-type mixing condition. The main result is the convergence of µ t to a Gaussian probability measure as t → ∞ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's 'room-corridor' argument. The case of variable coefficients is treated by using an 'averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.

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 Q ± ., with the distributions m ± . We study the distribution m t of the solution at time t ¥ R. The main result is the convergence of m t to a Gaussian translation-invariant measure as t Q .. 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. Limiting mean energy current density is − (0,..., 0, C(T + − T − )) with some positive constant C > 0 what corresponds to Second Law.

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Consider the wave equation with constant or variable coefficients in IR 3 . The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt-or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as x 3 → ±∞ , with the distributions µ ± . We study the distribution µ t of the random solution at a time t ∈ IR . The main result is the convergence of µ t to a Gaussian translation-invariant measure as t → ∞ that means central limit theorem for the wave equation. The proof is based on the Bernstein '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 formally is −∞ · (0, 0, T + −T − ) for the Gibbs measures, and it is finite and equals to −C(0, 0, T + −T − ) with C > 0 for the convolution with a nontrivial test function.1 Supported partly by research grants of DFG (436 RUS 113/615/0-1) and of RFBR