On Convergence to Equilibrium Distribution, I.¶The Klein-Gordon Equation with Mixing

Abstract

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.

On Two-Temperature Problem for Harmonic Crystals

“…Developing this approach, we have proved the convergence for the wave and Klein-Gordon equations with translation-invariant initial measures. (7,8,18) In ref. 9 we have extended the results to the wave equation with the two-temperature initial measures.…”

On the asymptotical normality of statistical solutions for wave equations coupled to a particle

“…It remains to prove the convergence E (exp{i W t φ 0 , Π(Z) }) ≡μ B t (Π(Z)) to a limit as t → ∞. In [6,7], we have proved the convergence ofμ B t (f ) to a limit for f ∈ D 0 ≡ [C ∞ 0 (R 3 )] 2 . However, Π(Z) ∈ D 0 in general.…”

On convergence to equilibrium distribution for wave equation in even dimensions

“…This paper can be considered as a continuation of our papers [5]- [8], [11] which concern the long time convergence to equilibrium distribution for the linear wave, Klein-Gordon and Schrödinger equations.…”

“…We develop this approach for hyperbolic PDEs. In [5]- [8], [10]- [11] the convergence to equilibrium distributions has been proved for the linear wave, Klein-Gordon and Schrödinger equations with potentials, for the harmonic crystal, and for the free Dirac equation. The initial distribution are translation invariant and satisfy the mixing condition of Rosenblatt or Ibragimov-Linnik type.…”