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CATEGORIES:Geometric Analysis and Partial Differential Equati
ons seminar
SUMMARY:Kac’s Model and Villani’s Conjecture - Amit Einav
(DPMMS)
DTSTART;TZID=Europe/London:20120319T160000
DTEND;TZID=Europe/London:20120319T170000
UID:TALK36071AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/36071
DESCRIPTION:In his 1956 paper\, Marc Kac constructed a linear
model of N particles\, interacting through binary
collision\, from which a ‘baby’ version of the Bol
tzmann equation arose for special types of familie
s – chaotic ones. Kac proceeded to notice that sol
utions to his ‘Master Equation’ converge to equili
brium and conjectured that the spectral gap of the
associated linear operator will be bounded below
independently in N. It took 44 years to prove this
conjecture and even when it was solved it wasn’t
enough to show the desired exponential rate of con
vergence to equilibrium.\n\nA different approach w
as taken\, one that involved the entropy and its ‘
spectral gap’ equivalent – the entropy production.
In his 2003 paper\, Villani managed to give a low
er bound to the entropy production and conjectured
that it is indeed the right order in N.\n\nIn our
talk we’ll review and go into more details about
the above topics and give a proof to a ‘1+ epsilon
’ version of Villani’s conjecture\, showing that t
he entropy approach in the most general case isn’t
as promising as we hoped.
LOCATION:CMS\, MR12
CONTACT:Jonathan Ben-Artzi
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