On the Convergence to a Statistical Equilibrium for the Dirac Equation

Abstract

We consider the Dirac equation in ${\mathbb{R}}^3$ with constant coefficients and study the distribution ${\mu }_t$ of the random solution at time $t\in \mathbb{R}$. 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.

12 pages

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