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SUMMARY:Kac’s Model and Villani’s Conjecture - Amit Einav (DPMMS)
DTSTART:20120319T160000Z
DTEND:20120319T170000Z
UID:TALK36071@talks.cam.ac.uk
CONTACT:Jonathan Ben-Artzi
DESCRIPTION:In his 1956 paper\, Marc Kac constructed a linear model of N p
 articles\, interacting through binary collision\, from which a ‘baby’ 
 version of the Boltzmann equation arose for special types of families – 
 chaotic ones. Kac proceeded to notice that solutions to his ‘Master Equa
 tion’ converge to equilibrium and conjectured that the spectral gap of t
 he 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 was
 n’t enough to show the desired exponential rate of convergence to equili
 brium.\n\nA different approach was taken\, one that involved the entropy a
 nd its ‘spectral gap’ equivalent – the entropy production. In his 20
 03 paper\, Villani managed to give a lower 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 pr
 oof to a ‘1+ epsilon’ version of Villani’s conjecture\, showing that
  the entropy approach in the most general case isn’t as promising as we 
 hoped.
LOCATION:CMS\, MR12
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