Entropy and chaos in the Kac model
Eric A. Carlen Maria C. Carvalho Jonathan Le Roux Michael Loss Cédric Villani
We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved here show that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open and the upper bounds obtained here show that this problem is somewhat subtle.
keywords: Entropy propagation of chaos.
Improved interpolation inequalities on the sphere
Jean Dolbeault Maria J. Esteban Michał Kowalczyk Michael Loss
This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
keywords: Sobolev inequality Gagliardo-Nirenberg inequalities interpolation logarithmic Sobolev inequality heat equation hypercontractivity spectral decomposition.

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