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Entropy production inequalities for the Kac Walk

Accepted version
Peer-reviewed

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Authors

Carlen, EA 
Carvalho, MC 
Einav, A 

Abstract

Mark Kac introduced what is now called 'the Kac Walk' with the aim of investigating the spatially homogeneous Boltzmann equation by probabilistic means. Much recent work, discussed below, on Kac's program has run in the other direction: using recent results on the Boltzmann equation, or its one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results for the Kac Walk. Here we investigate new functional inequalities for the Kac Walk pertaining to entropy production, and introduce a new form of chaoticity'. We then show how these entropy production inequalities imply entropy production inequalities for the Kac-Boltzmann equation. This results validate Kac's program for proving results on the non-linear Boltzmann equation via analysis of the Kac Walk, and they constitute a partial solution to the Almost' Cercignani Conjecture on the sphere.

Description

Keywords

math-ph, math-ph, math.MP, 60J25, 82C41, 82C41}

Journal Title

Kinetic and Related Models

Conference Name

Journal ISSN

1937-5077

Volume Title

Publisher

American Institute of Mathematical Sciences
Sponsorship
U.S. National Science Foundation grants DMS 1201354, PHY 0965859 and PHY-1265118; EPSRC grant EP/L002302/1.