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CATEGORIES:Geometric Analysis and Partial Differential Equati
ons seminar
SUMMARY:Kinetic theory for the low-density Lorentz gas (Co
mmon session with Probability Seminar) - Jens Mark
lof\, University of Bristol
DTSTART;TZID=Europe/London:20191203T140000
DTEND;TZID=Europe/London:20191203T150000
UID:TALK130969AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/130969
DESCRIPTION: Joint work with Andreas Strombergsson (Uppsala)\n
\nThe Lorentz gas is one of the simplest and most
widely-studied models for particle transport in ma
tter. It describes a cloud of non-interacting gas
particles in an infinitely extended array of ident
ical spherical scatterers\, whose radii are small
compared to their mean separation. The model was i
ntroduced by Lorentz in 1905 who\, following the p
ioneering ideas of Maxwell and Boltzmann\, postula
ted that its macroscopic transport properties shou
ld be governed by a linear Boltzmann equation. A r
igorous derivation of the linear Boltzmann equatio
n from the underlying particle dynamics was given\
, for random scatterer configurations\, in three s
eminal papers by Gallavotti\, Spohn and Boldrighin
i-Bunimovich-Sinai. The objective of this lecture
is to develop an approach for a large class of det
erministic scatterer configurations\, including va
rious types of quasicrystals. We prove the converg
ence of the particle dynamics to transport process
es that are in general (depending on the scatterer
configuration) not described by the linear Boltzm
ann equation. This was previously understood only
in the case of the periodic Lorentz gas through wo
rk of Caglioti-Golse and Marklof-Strombergsson. Ou
r results extend beyond the classical Lorentz gas
with hard sphere scatterers\, and in particular ho
ld for general classes of spherically symmetric fi
nite-range potentials. We employ a rescaling techn
ique that randomises the point configuration given
by the scatterers' centers. The limiting transpor
t process is then expressed in terms of a point pr
ocess that arises as the limit of the randomised p
oint configuration under a certain volume-preservi
ng one-parameter linear group action.
LOCATION:CMS\, MR12
CONTACT:Jessica Guerand
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