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Publication details

Document type
Journal articles

Document subtype
Full paper

Title
Entropy and chaos in the Kac model

Participants in the publication
Eric A. Carlen (Author)
RUTGERS UNIVERSITY
Maria C. Carvalho (Author)
Dep. Matemática
CMAF - Centro de Matemática e Aplicações Fundamentais 
Jonathan Le Roux (Author)
University of Tokyo
Michael Loss (Author)
Cédric Villani (Author)

Scope
International

Refereeing
Yes

Summary
We investigate the behavior in \\nN\\n of the \\nN\\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.

Date of Publication
2010-03-01

Where published
Kinetic & Related Models

Publication Identifiers
ISSN - 1937-5077

Publisher
American Institute of Mathematical Sciences (AIMS)

Volume
3
Number
1

Number of pages
37
Starting page
85
Last page
122

Document Identifiers
DOI - https://doi.org/10.3934/krm.2010.3.85
URL - http://dx.doi.org/10.3934/krm.2010.3.85

Keywords
Entropy propagation of chaos


Export

APA
Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani, (2010). Entropy and chaos in the Kac model. Kinetic & Related Models, 3, 85-122. ISSN 1937-5077. eISSN . http://dx.doi.org/10.3934/krm.2010.3.85

IEEE
Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani, "Entropy and chaos in the Kac model" in Kinetic & Related Models, vol. 3, pp. 85-122, 2010. 10.3934/krm.2010.3.85

BIBTEX
@article{44955, author = {Eric A. Carlen and Maria C. Carvalho and Jonathan Le Roux and Michael Loss and Cédric Villani}, title = {Entropy and chaos in the Kac model}, journal = {Kinetic & Related Models}, year = 2010, pages = {85-122}, volume = 3 }