Description
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).)