Comptes Rendus
no. 10
Dynamical systems/Mathematical physics
Ergodicity of the Ehrenfest wind–tree model

Presented by: Claire Voisin

Alba Málaga Sabogal1; Serge Troubetzkoy2
1 Département de mathématiques, Bâtiment 425, faculté des sciences d'Orsay, Université Paris-Sud 11, 91405 Orsay cedex, France
2 Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1032-1036.

We consider aperiodic wind–tree models, and show that, for a generic (in the sense of Baire) configuration, the wind–tree dynamics is ergodic in almost every direction.

Nous considérons une modèle apériodique de vent dans des arbres et nous montrons que, pour une configuration générique (dans le sens de Baire), la dynamique de vent–arbre est ergodique dans presque toutes les directions.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.08.008

Alba Málaga Sabogal 1; Serge Troubetzkoy 2

1 Département de mathématiques, Bâtiment 425, faculté des sciences d'Orsay, Université Paris-Sud 11, 91405 Orsay cedex, France
2 Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France 
@article{CRMATH_2016__354_10_1032_0,
     author = {Alba M\'alaga Sabogal and Serge Troubetzkoy},
     title = {Ergodicity of the {Ehrenfest} wind{\textendash}tree model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1032--1036},
     publisher = {Elsevier},
     volume = {354},
     number = {10},
     year = {2016},
     doi = {10.1016/j.crma.2016.08.008},
     language = {en},
}
TY  - JOUR
AU  - Alba Málaga Sabogal
AU  - Serge Troubetzkoy
TI  - Ergodicity of the Ehrenfest wind–tree model
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1032
EP  - 1036
VL  - 354
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2016.08.008
LA  - en
ID  - CRMATH_2016__354_10_1032_0
ER  - 
%0 Journal Article
%A Alba Málaga Sabogal
%A Serge Troubetzkoy
%T Ergodicity of the Ehrenfest wind–tree model
%J Comptes Rendus. Mathématique
%D 2016
%P 1032-1036
%V 354
%N 10
%I Elsevier
%R 10.1016/j.crma.2016.08.008
%G en
%F CRMATH_2016__354_10_1032_0
Alba Málaga Sabogal; Serge Troubetzkoy. Ergodicity of the Ehrenfest wind–tree model. Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1032-1036. doi : 10.1016/j.crma.2016.08.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.08.008/

[1] A. Avila, P. Hubert, Recurrence for the wind–tree model, Ann. Inst. Henri Poincaré, Anal. Non Linéaire.

[2] A.S. Besicovitch A problem on topological transformations of the plane. II, Math. Proc. Camb. Philos. Soc., Volume 47 (1951), pp. 38-45

[3] C. Bianca; L. Rondoni The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles?, Chaos, Volume 19 (2009)

[4] V. Delecroix Divergent trajectories in the periodic wind–tree model, J. Mod. Dyn., Volume 7 (2013), pp. 1-29

[5] V. Delecroix; P. Hubert; S. Lelièvre Diffusion for the periodic wind–tree model, Ann. Sci. ENS, Volume 47 (2014), pp. 1085-1110

[6] C.P. Dettmann; E.G.D. Cohen; H. van Beijeren Statistical mechanics: microscopic chaos from Brownian motion?, Nature, Volume 401 (1999), p. 875 | DOI

[7] P. Ehrenfest; T. Ehrenfest; P. Ehrenfest; T. Ehrenfest, Encykl. d. Math. Wissensch. IV2 II, Volume Heft 6, Cornell University Press, Itacha, NY, USA (1912), pp. 10-13 90 p. (in German) (English translation by M.J. Moravicsik)

[8] K. Frączek; C. Ulcigrai Non-ergodic Z-periodic billiards and infinite translation surfaces, Invent. Math., Volume 197 (2014), pp. 241-298

[9] G. Gallavotti Divergences and the approach to equilibrium in the Lorentz and the wind–tree models, Phys. Rev., Volume 185 (1969), pp. 308-322

[10] G. Gallavotti; F. Bonetto; G. Gentile Aspects of Ergodic Qualitative and Statistical Theory of Motion, Springer, 2004

[11] J. Hardy; J. Weber Diffusion in a periodic wind–tree model, J. Math. Phys., Volume 21 (1980), pp. 1802-1808

[12] E.H. Hauge; E.G.D. Cohen Normal and abnormal diffusion in Ehrenfest's wind–tree model, J. Math. Phys., Volume 10 (1969), pp. 397-414

[13] P. Hooper; P. Hubert; B. Weiss Dynamics on the infinite staircase, Discrete Contin. Dyn. Syst., Volume 33 (2013), pp. 4341-4347

[14] P. Hubert; B. Weiss Ergodicity for infinite periodic translation surfaces, Compos. Math., Volume 149 (2013), pp. 1364-1380

[15] P. Hubert; S. Lelièvre; S. Troubetzkoy The Ehrenfest wind–tree model: periodic directions, recurrence, diffusion, J. Reine Angew. Math., Volume 656 (2011), pp. 223-244

[16] A. Katok; A. Zemlyakov Topological transitivity of billiards in polygons, Math. Notes, Volume 18 (1975), pp. 760-764

[17] S. Kerckhoff; H. Masur; J. Smillie Ergodicity of billiard flows and quadratic differentials, Ann. Math. (2), Volume 124 (1986) no. 2, pp. 293-311

[18] A. Málaga Sabogal Étude d'une famille de transformations préservant la mesure de Z×T, Université Paris-11, Paris, 2014 (PhD thesis)

[19] A. Málaga Sabogal; S. Troubetzkoy Minimality of the Ehrenfest wind–tree model, J. Mod. Dyn., Volume 10 (2016), pp. 209-228

[20] H. Masur; S. Tabachnikov Rational billiards and flat structures, Handbook of Dynamical Systems, vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089

[21] D. Ralston; S. Troubetzkoy Ergodic infinite group extensions of geodesic flows on translation surfaces, J. Mod. Dyn., Volume 6 (2012), pp. 477-497

[22] S. Troubetzkoy Approximation and Billiards, Dynamical Systems and Diophantine Approximation, Semin. Congr., vol. 19, Soc. Math. France, Paris, 2009, pp. 173-185

[23] S. Troubetzkoy Typical recurrence for the Ehrenfest wind–tree model, J. Stat. Phys., Volume 141 (2010), pp. 60-67

[24] H. Van Beyeren; E.H. Hauge Abnormal diffusion in Ehrenfest's wind–tree model, Phys. Lett. A, Volume 39 (1972), pp. 397-398

[25] W. Wood; F. Lado Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest's wind–tree model, J. Comput. Phys., Volume 7 (1971), pp. 528-546

Cited by Sources:

Comments - Policy