Entropy production inequalities for the Kac Walk

Eric A. Carlen; Maria C. Carvalho; Amit Einav

doi:10.3934/krm.2018012

Article Contents

2018, Volume 11, Issue 2: 219-238. Doi: 10.3934/krm.2018012

This issue Previous Article Hypocoercive estimates on foliations and velocity spherical Brownian motion Next Article Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation

Entropy production inequalities for the Kac Walk

1.
Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854-8019, USA
2.
CMAF-CIO, University of Lisbon, P 1749-016 Lisbon, Portugal
3.
Departments of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

Received: April 2017

Published: April 2018

¹Work partially supported by U.S. National Science Foundation grant DMS 1501007. ²Work partially supported by supported by Fundação para a Ciência e Tecnologia (PTDC/MAT/100983/2008, PEst-OE/MAT/UI0209/2013, UID/MAT/04561/2013). ³Work supported by EPSRC grant EP/L002302/1

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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.

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Mathematics Subject Classification: Primary: 81V99, 82B10, 94A17.

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