On convergence to equilibrium distribution for Dirac equation

TL;DR

This paper proves that solutions to the Dirac equation with a potential in three-dimensional space converge in distribution to a Gaussian measure over time, under certain initial conditions and mixing assumptions.

Contribution

It establishes the long-term convergence of the distribution of Dirac equation solutions to a Gaussian measure, extending understanding of quantum dynamics with random initial data.

Findings

Distribution of solutions converges to Gaussian measure over time.

Convergence occurs in the projection onto the continuous spectral space.

Results hold under mixing conditions and finite charge density.

Abstract

We consider the Dirac equation in $\R^3$ with a potential, and study the distribution $\mu_t$ of the random solution at time $t\in\R$. The initial measure $\mu_0$ has zero mean, a translation-invariant covariance, and a 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 long time convergence of projection of $\mu_t$ onto the continuous spectral space. The limiting measure is Gaussian.