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Kac’s Model and Villani’s Conjecture

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If you have a question about this talk, please contact Jonathan Ben-Artzi.

In his 1956 paper, Marc Kac constructed a linear model of N particles, 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 Equation’ converge to equilibrium 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 convergence to equilibrium.

A different approach was taken, one that involved the entropy and its ‘spectral gap’ equivalent – the entropy production. In his 2003 paper, Villani managed to give a lower bound to the entropy production and conjectured that it is indeed the right order in N.

In 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 the entropy approach in the most general case isn’t as promising as we hoped.

This talk is part of the Partial Differential Equations seminar series.

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