## Kinetic theory for the low-density Lorentz gas

### With Jens Marklof (Bristol)

# Kinetic theory for the low-density Lorentz gas

The Lorentz gas is one of the simplest and most widely-studied models

for particle transport in matter. It describes a cloud of

non-interacting gas particles in an infinitely extended array of

identical spherical scatterers, whose radii are small compared to their

mean separation. The model was introduced by Lorentz in 1905 who,

following the pioneering ideas of Maxwell and Boltzmann, postulated that

its macroscopic transport properties should be governed by a linear

Boltzmann equation. A rigorous derivation of the linear Boltzmann

equation from the underlying particle dynamics was given, for random

scatterer configurations, in three seminal papers by Gallavotti, Spohn

and Boldrighini-Bunimovich-Sinai. The objective of this lecture is to

develop an approach for a large class of deterministic scatterer

configurations, including various types of quasicrystals. We prove the

convergence of the particle dynamics to transport processes that are in

general (depending on the scatterer configuration) not described by the

linear Boltzmann equation. This was previously understood only in the

case of the periodic Lorentz gas through work of Caglioti-Golse and

Marklof-Strombergsson. Our results extend beyond the classical Lorentz

gas with hard sphere scatterers, and in particular hold for general

classes of spherically symmetric finite-range potentials. We employ a

rescaling technique that randomises the point configuration given by the

scatterers’ centers. The limiting transport process is then expressed in

terms of a point process that arises as the limit of the randomised

point configuration under a certain volume-preserving one-parameter

linear group action.

Joint work with Andreas Strombergsson (Uppsala)

- Speaker: Jens Marklof (Bristol)
- Tuesday 03 December 2019, 14:00–15:00
- Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
- Series: Probability; organiser: Perla Sousi.