COLLOQUIUM DI MATEMATICA

Balint Toth
(University of Bristol, UK and Renyi Institute, Budapest)

Titolo: Invariance principle for the random Lorentz gas beyond the
[Boltzmann-Grad / Gallavotti-Spohn] limit

Mercoledi' 23 Maggio 2018 ORE 16:00

Dipartimento di Matematica e Fisica 
Universita' degli Studi Roma Tre
Aula F, primo piano, edificio Aule - Largo San Leonardo Murialdo,1

Abstract
Let hard ball scatterers of radius $r$ be placed in $mathbb R^d$,
centred at the points of a Poisson point process of intensity $rho$.
The volume fraction $r^d rho$ is assumed to be sufficiently low so
that with positive probability the origin is not trapped in a finite
domain fully surrounded by scatterers. The Lorentz process is the
trajectory of a point-like particle starting from the origin with
randomly oriented unit velocity subject to elastic collisions with the
fixed (infinite mass) scatterers. The question of diffusive scaling
limit of this process is a major open problem in classical statistical
physics. 
Gallavotti (1969) and Spohn (1978) proved that under the so-called
Boltzmann-Grad limit, when $r to 0$, $rho to infty$ so that
$r^{d-1}rho to 1$ and the time scale is fixed, the Lorentz process
(described informally above) converges to a Markovian random flight
process, with independent exponentially distributed free flight times
and Markovian scatterings. It is essentially straightforward to see
that taking a second diffusive scaling limit (after the
Gallavotti-Spohn limit) yields invariance principle. 
I will present new results going beyond the [Boltzmann-Grad /
Gallavotti-Spohn] limit, in $d=3$: Letting $r to 0$, $rho to infty$ so
that $r^{d-1}rho to 1$ (as in B-G) and simultaneously rescaling time
by $T sim r^{-2+epsilon}$ we prove invariance principle (under
diffusive scaling) for the Lorentz trajectory. Note that the B-G limit
and diffusive scaling are done simultaneously and not in sequel. The
proof is essentially based on control of the effect of re-collisions
by probabilistic coupling arguments. The main arguments are valid in
$d=3$ but not in $d=2$. 
Joint work with Chris Lutsko (Bristol)