Invariance principle for the periodic Lorentz gas in the Boltzmann-Grad limit

TL;DR

This paper proves that in the low-density limit, the particle trajectories in a multidimensional periodic Lorentz gas converge to Brownian motion, extending previous results from a central limit theorem to a functional form.

Contribution

It extends prior work by establishing a functional central limit theorem for the Lorentz gas in the Boltzmann-Grad limit, showing convergence to Brownian motion.

Findings

Particle displacement satisfies a functional central limit theorem

Trajectory converges to Brownian motion in the limit

Generalizes previous superdiffusive results

Abstract

In earlier work we showed that the particle displacement for the multidimensional periodic Lorentz gas, in the limit of low scatterer density (Boltzmann-Grad limit), satisfies a central limit theorem with superdiffusive scaling. The present paper extends this result to a functional central limit theorem, i.e., the weak convergence of the particle trajectory to Brownian motion.