American Institute of Mathematical Sciences

Advanced
March  2010, 3(1): 85-122. doi: 10.3934/krm.2010.3.85

Entropy and chaos in the Kac model

Eric A. Carlen 1, , Maria C. Carvalho 2, , Jonathan Le Roux 3, , Michael Loss 4, and  Cédric Villani 5,

1. 

Department of Mathematics, Hill Center, Rutgers University, Piscataway, NJ 08854, United States
2. 

Department of Mathematics and CMAF, University of Lisbon, 1649-003 Lisbon, Portugal
3. 

Department of Information Physics and Computing, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
4. 

School of Mathematics, Georgia Institute of Technology, Atlanta GA, 30332, United States
5. 

UMPA, ENS Lyon, University of Lisbon, 46 allée d’Italie, 69364 Lyon Cedex 07, France

Received  April 2009 Revised  November 2009 Published  January 2010

We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochasticmodel of Boltzmann dynamics, and its relation to the entropy function forsolutions of Kac's one dimensional nonlinear model Boltzmann equation. We proveresults 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 hereshow 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..
Mathematics Subject Classification: Primary: 76P05, 60G50; Secondary: 54C7.
Citation: Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic and Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
[1]

Samir Salem. A gradient flow approach of propagation of chaos. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5729-5754. doi: 10.3934/dcds.2020243
[2]

Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
[3]

Vladimír Špitalský. Entropy and exact Devaney chaos on totally regular continua. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3135-3152. doi: 10.3934/dcds.2013.33.3135
[4]

Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic and Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1
[5]

Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic and Related Models, 2019, 12 (2) : 269-302. doi: 10.3934/krm.2019012
[6]

Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic and Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013
[7]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[8]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[9]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[10]

Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191
[11]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[12]

Arsen R. Dzhanoev, Alexander Loskutov, Hongjun Cao, Miguel A.F. Sanjuán. A new mechanism of the chaos suppression. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 275-284. doi: 10.3934/dcdsb.2007.7.275
[13]

Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161
[14]

Y. Charles Li. Chaos phenotypes discovered in fluids. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1383-1398. doi: 10.3934/dcds.2010.26.1383
[15]

Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917
[16]

Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
[17]

J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653
[18]

Chang-Yeol Jung, Alex Mahalov. Wave propagation in random waveguides. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 147-159. doi: 10.3934/dcds.2010.28.147
[19]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933
[20]

Piotr Oprocha. Specification properties and dense distributional chaos. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 821-833. doi: 10.3934/dcds.2007.17.821

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (143)
  • HTML views (0)
  • Cited by (39)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy