On a Two‐Temperature Problem for the Klein–Gordon Equation

Abstract

We consider the Klein-Gordon equation in R n , n 2, with constant or variable coefficients. The initial datum is a random function with a finite mean density of the energy and satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. We also assume that the random function is close to different space-homogeneous processes as xn →± ∞, with the distributions μ±. We study the distribution μt of the random solution at time t ∈ R. The main result is the convergence of μt to a Gaussian translation-invariant measure as t →∞ that means the central limit theorem for the Klein-Gordon equation. The proof is based on the Bernstein room-corridor method and oscillatory integral estimates. The application to the case of the Gibbs measures μ± = g± with two different temperatures T± is given. It is proved that limit mean energy current density formally is −∞ · (0 ,..., 0 ,T + − T−) for the Gibbs measures, and it is finite and equals −C(0 ,..., 0 ,T + − T−) with some positive constant C> 0 for the smoothed solution. This corresponds to the second law of thermodynamics.