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)

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