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CATEGORIES:Probability
SUMMARY: Kinetic theory for the low-density Lorentz gas -
Jens Marklof (Bristol)
DTSTART;TZID=Europe/London:20191203T140000
DTEND;TZID=Europe/London:20191203T150000
UID:TALK131485AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/131485
DESCRIPTION:The Lorentz gas is one of the simplest and most wi
dely-studied models\nfor particle transport in mat
ter. It describes a cloud of\nnon-interacting gas
particles in an infinitely extended array of\niden
tical spherical scatterers\, whose radii are small
compared to their\nmean separation. The model was
introduced by Lorentz in 1905 who\,\nfollowing th
e pioneering ideas of Maxwell and Boltzmann\, post
ulated that\nits macroscopic transport properties
should be governed by a linear\nBoltzmann equation
. A rigorous derivation of the linear Boltzmann\ne
quation from the underlying particle dynamics was
given\, for random\nscatterer configurations\, in
three seminal papers by Gallavotti\, Spohn\nand Bo
ldrighini-Bunimovich-Sinai. The objective of this
lecture is to\ndevelop an approach for a large cla
ss of deterministic scatterer\nconfigurations\, in
cluding various types of quasicrystals. We prove t
he\nconvergence of the particle dynamics to transp
ort processes that are in\ngeneral (depending on t
he scatterer configuration) not described by the\n
linear Boltzmann equation. This was previously und
erstood only in the\ncase of the periodic Lorentz
gas through work of Caglioti-Golse and\nMarklof-St
rombergsson. Our results extend beyond the classic
al Lorentz\ngas with hard sphere scatterers\, and
in particular hold for general\nclasses of spheric
ally symmetric finite-range potentials. We employ
a\nrescaling technique that randomises the point c
onfiguration given by the\nscatterers' centers. Th
e limiting transport process is then expressed in\
nterms of a point process that arises as the limit
of the randomised\npoint configuration under a ce
rtain volume-preserving one-parameter\nlinear grou
p action.\nJoint work with Andreas Strombergsson (
Uppsala)\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Perla Sousi
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