On the Convergence to a Statistical Equilibrium for the Dirac Equation

Abstract

We consider the Dirac equation in $\R^3$ with constant coefficients and study the distribution $\mu_t$ of the random solution at time $t\in\R$. It is assumed that the initial measure $\mu_0$ has zero mean, a translation-invariant covariance, and finite mean charge density. We also assume that $\mu_0$ satisfies a mixing condition of Rosenblatt- or Ibragimov-Linnik-type. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$. The proof uses the study of long time asymptotics of the solution and S.N. Bernstein's ``room-corridor'' method.