Journal of Statistical Physics

, Volume 156, Issue 4, pp 647–667 | Cite as

The Kac Model Coupled to a Thermostat

  • Federico Bonetto
  • Michael Loss
  • Ranjini Vaidyanathan


In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature \(\beta \). The system admits the canonical distribution at inverse temperature \(\beta \) as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one particle marginal, in the large system limit, satisfies an effective Boltzmann-type equation.


Kac model Thermostat Approach to equilibrium Propagation of chaos 

Mathematics Subject Classifications

47A63 15A90 



The authors would like to thank Eric Carlen, Joel Lebowitz and Hagop Tossounian for many enlightening discussions. We thank Hagop Tossounian for helping in the proof of Proposition 15. We also thank the referee for various helpful comments and the extremely detailed report. This work was partially supported by U.S. National Science Foundation grant DMS 1301555.   2013 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.


  1. 1.
    Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley and Los Angeles (1956)Google Scholar
  2. 2.
    McKean Jr, H.P.: Speed of approach to equilibrium for kac’s caricature of a maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Lanford III, O.E.: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1–111. Lecture Notes in Physics, vol. 38. Springer, Berlin (1975)Google Scholar
  4. 4.
    Lanford III, O.E.: On a derivation of the Boltzmann equation. In: International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), pp. 117–137. Astérisque, No. 40. Societe Mathematique De France, Paris (1976).Google Scholar
  5. 5.
    Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: “Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum” [Comm. Math. Phys. 105(2), 1986, pp. 189–203; MR0849204 (88d:82061)] and “Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum” [ibid. 113 (1987), no. 1, 79–85; MR0918406 (89b:82052)] by Pulvirenti. Comm. Math. Phys. 121(1), 143–146 (1989). Accessed 08 May 2014
  6. 6.
    Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29(1), 288–304 (2001). doi: 10.1214/aop/1008956330
  7. 7.
    Carlen, E., Carvalho, M.C., Loss, M.: Many-body aspects of approach to equilibrium. In: Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), pp. Exp. No. XI, 12. University of Nantes, Nantes (2000)Google Scholar
  8. 8.
    Maslen, D.K.: The eigenvalues of Kac’s master equation. Math. Z. 243(2), 291–331 (2003). doi: 10.1007/s00209-002-0466-y
  9. 9.
    Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234(3), 455–490 (2003). doi: 10.1007/s00220-002-0777-1
  10. 10.
    Einav, A.: On Villani’s conjecture concerning entropy production for the Kac master equation. Kinet. Relat. Models 4(2), 479–497 (2011). doi: 10.3934/krm.2011.4.479
  11. 11.
    Fröhlich, J., Gang, Z.: Exponential convergence to the Maxwell distribution for some class of Boltzmann equations. Commun. Math. Phys. 314(2), 525–554 (2012). doi: 10.1007/s00220-012-1499-7
  12. 12.
    Bonetto, F., Chernov, N., Korepanov, A., Lebowitz, J.L.: Nonequilibrium stationary state of a current-carrying thermostated system. EPL 102, 15001 (2013)Google Scholar
  13. 13.
    Bonetto, F., Carlen, E., Esposito, R., Lebowitz, J., Marra, R.: Propagation of chaos for a thermostated kinetic model.
  14. 14.
    Mischler, S., Mouhot, C.: Kac’s program in kinetic theory. Invent. Math. 193(1), 1–147 (2013). doi: 10.1007/s00222-012-0422-3
  15. 15.
    Stam, A.J.: Some inequalities satisfied by the quantities of information of fisher and shannon. Inf. Control 2, 101–112 (1959)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gross, L.: Logarithmic sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)CrossRefGoogle Scholar
  17. 17.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985). doi: 10.1007/BFb0075847
  18. 18.
    Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. In: Dirichlet Forms (Varenna, 1992). Lecture Notes in Mathematics, vol. 1563, pp. 54–88. Springer, Berlin (1993). doi: 10.1007/BFb0074091
  19. 19.
    Toscani, G.: Entropy production and the rate of convergence to equilibrium for the Fokker–Planck equation. Quart. Appl. Math. 57(3), 521–541 (1999)MATHMathSciNetGoogle Scholar
  20. 20.
    Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Am. Math. Soc 55, 961–962 (1949)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Carlen, E.A., Geronimo, J.S., Loss, M.: Determination of the spectral gap in the Kac model for physical momentum and energy-conserving collisions. SIAM J. Math. Anal. 40(1), 327–364 (2008). doi: 10.1137/070695423

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Federico Bonetto
    • 1
  • Michael Loss
    • 1
  • Ranjini Vaidyanathan
    • 1
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations