Communications in Mathematical Physics

, Volume 358, Issue 3, pp 995–1006 | Cite as

Infinite Ergodic Index of the Ehrenfest Wind-Tree Model

  • Alba Málaga Sabogal
  • Serge Eugene Troubetzkoy


The set of all possible configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology is a compact metric space. For a typical configuration we show that the wind-tree dynamics has infinite ergodic index in almost every direction. In particular some ergodic theorems can be applied to show that if we start with a large number of initially parallel particles their directions decorrelate as the dynamics evolve, answering the question posed by the Ehrenfests.


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  1. AvHu.
    Avila, A., Hubert, P.: Recurrence for the wind-tree model. Annales de l’Institut Henri Poincaré: Analyse non linéaireGoogle Scholar
  2. BaKhMaPl.
    Bachurin P., Khanin K., Marklof J., Plakhov A.: Perfect retroreflectors and billiard dynamics. J. Mod. Dyn. 5, 33–48 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. BiRo.
    Bianca C., Rondoni L.: The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?. Chaos 19, 013121 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. De.
    Delecroix V.: Divergent trajectories in the periodic wind-tree model. J. Mod. Dyn. 7, 1–29 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. DeHuLe.
    Delecroix V., Hubert P., Lelièvre S.: Diffusion for the periodic wind-tree model. Ann. Sci. ENS 47, 1085–1110 (2014)MathSciNetzbMATHGoogle Scholar
  6. DeZo.
    Delecroix, V., Zorich, A.: Cries and whispers in wind-tree forests. arXiv:1502.06405 (2015)
  7. DeCoVB.
    Dettmann, C.P., Cohen, E.G.D., Van Beijeren, H.: Statistical 379 mechanics: microscopic chaos from brownian motion? Nature 401, 875. (1999)
  8. EhEh.
    Ehrenfest, P., Ehrenfest, T.: Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (in German) (The conceptual foundations of the statistical approach in mechanics, trans. Moravicsik, M.J.) pp. 10–13. Cornell University Press, Itacha (1959)Google Scholar
  9. FrHu.
    Fra¸czek, K., Hubert, P.: Recurrence and non-ergodicity in generalized wind-tree models. arXiv:1506.05884 (2015)
  10. FrUl.
    Fra¸czek K., Ulcigrai C.: Non-ergodic \({\mathbb{Z}}\) -periodic billiards and infinite translation surfaces. Invent. Math. 197, 241–298 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. Ga.
    Gallavotti G.: Divergences and the approach to equilibrium in the Lorentz and the wind-tree models. Phys. Rev. 185, 308–322 (1969)ADSCrossRefGoogle Scholar
  12. GaBoGe.
    Gallavotti G., Bonetto F., Gentile G.: Aspects of Ergodic Qualitative and Statistical Theory of Motion. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  13. HaWe.
    Hardy J., Weber J.: Diffusion in a periodic wind-tree model. J. Math. Phys. 21, 1802–1808 (1980)ADSMathSciNetCrossRefGoogle Scholar
  14. HaCo.
    Hauge E.H., Cohen E.G.D.: Normal and Abnormal Diffusion in Ehrenfest’s Wind-Tree Model. J. Math. Phys. 10, 397–414 (1969)ADSCrossRefGoogle Scholar
  15. HaCo1.
    Hauge E.H., Cohen E.G.D.: Normal and Abnormal Diffusion in Ehrenfest’s Wind-Tree Model. Phys. Lett. A 25, 78–79 (1967)ADSCrossRefGoogle Scholar
  16. HuLeTr.
    Hubert, P., Pascal, Lelièvre, S., Troubetzkoy, S.: The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion, J. Reine Angew. Math. 656, 223–244 (2011)Google Scholar
  17. KeMaSm.
    Kerckhoff S., Masur H., Smillie J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math. (2) 124(2), 293–311 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. MSTr1.
    Málaga Sabogal A., Troubetzkoy S.: Minimality of the Ehrenfest wind-tree model. J. Mod. Dyn. 10, 209–228 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. MSTr2.
    Málaga Sabogal A., Troubetzkoy S.: Ergodicity of the Ehrenfest wind-tree model. C. R. Math. 354, 1032–1036 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. MSTr3.
    Málaga Sabogal A., Troubetzkoy S.: Weakly-mixing polygonal billiards. Bull. Lond. Math. Soc. 49, 141–147 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Tr1.
    Troubetzkoy S.: Typical recurrence for the Ehrenfest wind-tree model. J. Stat. Phys. 141, 60–67 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. VBHa.
    Van Beyeren H., Hauge E.H.: Abnormal diffusion in Ehrenfest’s wind-tree model. Phys. Lett. A 39, 397–398 (1972)ADSCrossRefGoogle Scholar
  23. WoLa.
    Wood W., Lado F.: Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest’s wind-tree model. J. Comp. Phys. 7, 528–546 (1971)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.InriaParisFrance
  2. 2.Aix Marseille Univ, CNRS, Centrale MarseilleMarseilleFrance
  3. 3.Marseille CEDEX 9France

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