Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

TL;DR

This paper proves the invariance principle for a 3D random Lorentz gas under simultaneous diffusive and Boltzmann-Grad limits, advancing the rigorous understanding of particle trajectories in nonequilibrium statistical physics.

Contribution

It introduces a probabilistic coupling approach to establish the invariance principle, extending previous results to longer time scales and different physical settings.

Findings

Proves the invariance principle for the 3D Lorentz gas under combined limits.

Establishes a longer validity time scale for diffusive approximation.

Uses probabilistic coupling to connect mechanical trajectories with Markovian processes.

Abstract

We prove the invariance principle for a \emph{random Lorentz-gas} particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius $r$, placed according to a Poisson point process of density $\varrho$, in the limit $\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order $T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969), Spohn (1978) and Boldrighini-Bunimovich-Sinai (1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski-Ryzhik (2006), respectively, Erd\H os-Salmhofer-Yau (2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2)The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation -- expressed in terms of the kinetic time scale -- is much longer and fully explicit.