Bálint Tóth

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

Abstract:  We prove an invariance principle for a 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 $\rho$, in the limit $r \to 0$, $\rho \to \infty$, $r^{2}\rho \to 1$, up to time scales of order $T = o(r^{-2} |\log r|^{-2})$. This represents the first significant progress towards solving this problem in mathematically rigorous 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 fight process, and probabilistic and geometric controls on the efficiency of this coupling.


Joint work with Christopher Lutsko. Commun. Math. Phys. (2020).