Entropy and chaos in the Kac model

Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani

Research output: Contribution to journalArticle

44 Scopus citations


We investigate the behavior in N of the N-particle entropy func-tional 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 prop-agation 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.

Original languageEnglish (US)
Pages (from-to)85-122
Number of pages38
JournalKinetic and Related Models
Issue number1
StatePublished - Mar 1 2010

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation


  • Entropy
  • Propagation of chaos

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    Carlen, E. A., Carvalho, M. C., Le Roux, J., Loss, M., & Villani, C. (2010). Entropy and chaos in the Kac model. Kinetic and Related Models, 3(1), 85-122. https://doi.org/10.3934/krm.2010.3.85