American Institute of Mathematical Sciences

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2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209

Minimality of the Ehrenfest wind-tree model

Alba Málaga Sabogal 1, and  Serge Troubetzkoy 1,

1. 

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France, France

Received  June 2015 Revised  March 2016 Published  June 2016

We consider aperiodic wind-tree models and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions and has a dense set of periodic points.
Keywords: minimality, periodic orbit., Wind-tree.
Mathematics Subject Classification: Primary: 37D50; Secondary: 37A40, 37B0.
Citation: Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209
References:
[1]

A. Avila and P. Hubert, Recurrence for the Wind-Tree Model, Annales de l'Institut Henri Poincaré - Analyse non linéaire, to appear.

[2]

A. S. Besicovitch, A problem on topological transformations of the plane. II., Proc. Cambridge Philos. Soc., 47 (1951), 38-45. doi: 10.1017/S0305004100026347.

[3]

C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?, Chaos, 19 (2009), 013121, 10pp. doi: 10.1063/1.3085954.

[4]

M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons, Trans. Am. Math. Soc., 350 (1998), 3523-3535. doi: 10.1090/S0002-9947-98-02089-3.

[5]

V. Delecroix, Divergent trajectories in the periodic wind-tree model, J. Mod. Dyn., 7 (2013), 1-29. doi: 10.3934/jmd.2013.7.1.

[6]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. ENS, 47 (2014), 1085-1110.

[7]

C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Statistical mechanics: Microscopic chaos from brownian motion?, Nature, 401 (1999), p875. doi: 10.1038/44759.

[8]

P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (in German, translated in:) The conceptual foundations of the statistical approach in mechanics, (trans. Moravicsik, M. J.), 10-13 Cornell University Press, Itacha NY (1959).

[9]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z.

[10]

G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models, Phys. Rev., 185 (1969), 308-322. doi: 10.1103/PhysRev.185.308.

[11]

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. AMS, 50 (1944), 915-919. doi: 10.1090/S0002-9904-1944-08262-1.

[12]

J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633.

[13]

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

[14]

P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, Compos. Math., 149 (2013), 1364-1380. doi: 10.1112/S0010437X12000887.

[15]

P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase, Discrete Contin. Dyn. Syst., 33 (2013), 4341-4347. doi: 10.3934/dcds.2013.33.4341.

[16]

P. Hubert, Pascal, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion, J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052.

[17]

A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 291-300.

[18]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[19]

A. Málaga Sabogal, Étude D'une Famille de Transformations Préservant la Mesure de $\mathbbZ \times \mathbbT$, Thèse Paris 11, 2014.

[20]

S. Marmi, P. Moussa and Y.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. AMS, 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X.

[21]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, North-Holland, Amsterdam, 1 (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.

[22]

D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces, J. Mod. Dyn., 6 (2012), 477-497.

[23]

S. Troubetzkoy, Approximation and billiards, Dynamical systems and Diophantine approximation, 173-185, Semin. Congr., 19, Soc. Math. France, Paris, 2009.

[24]

S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, J. Stat. Phys., 141 (2010), 60-67. doi: 10.1007/s10955-010-0026-5.

[25]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97 (1989), 553-583. doi: 10.1007/BF01388890.

[26]

Y. Vorobets, Periodic geodesics on translation surfaces, Algebraic and topological dynamics, 205-258, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005 doi: 10.1090/conm/385/07199.

[27]

H. Van Beyeren and E. H. Hauge, Abnormal diffusion in Ehrenfest's wind-tree model, Physics Letters A, 39 (1972), 397-398. doi: 10.1016/0375-9601(72)90112-0.

[28]

W. Wood and F. Lado, Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest's wind-tree model, J. Comp. Physics, 7 (1971), 528-546. doi: 10.1016/0021-9991(71)90109-4.

show all references

References:
[1]

A. Avila and P. Hubert, Recurrence for the Wind-Tree Model, Annales de l'Institut Henri Poincaré - Analyse non linéaire, to appear.

[2]

A. S. Besicovitch, A problem on topological transformations of the plane. II., Proc. Cambridge Philos. Soc., 47 (1951), 38-45. doi: 10.1017/S0305004100026347.

[3]

C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?, Chaos, 19 (2009), 013121, 10pp. doi: 10.1063/1.3085954.

[4]

M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons, Trans. Am. Math. Soc., 350 (1998), 3523-3535. doi: 10.1090/S0002-9947-98-02089-3.

[5]

V. Delecroix, Divergent trajectories in the periodic wind-tree model, J. Mod. Dyn., 7 (2013), 1-29. doi: 10.3934/jmd.2013.7.1.

[6]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. ENS, 47 (2014), 1085-1110.

[7]

C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Statistical mechanics: Microscopic chaos from brownian motion?, Nature, 401 (1999), p875. doi: 10.1038/44759.

[8]

P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (in German, translated in:) The conceptual foundations of the statistical approach in mechanics, (trans. Moravicsik, M. J.), 10-13 Cornell University Press, Itacha NY (1959).

[9]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z.

[10]

G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models, Phys. Rev., 185 (1969), 308-322. doi: 10.1103/PhysRev.185.308.

[11]

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. AMS, 50 (1944), 915-919. doi: 10.1090/S0002-9904-1944-08262-1.

[12]

J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633.

[13]

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

[14]

P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, Compos. Math., 149 (2013), 1364-1380. doi: 10.1112/S0010437X12000887.

[15]

P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase, Discrete Contin. Dyn. Syst., 33 (2013), 4341-4347. doi: 10.3934/dcds.2013.33.4341.

[16]

P. Hubert, Pascal, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion, J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052.

[17]

A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 291-300.

[18]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[19]

A. Málaga Sabogal, Étude D'une Famille de Transformations Préservant la Mesure de $\mathbbZ \times \mathbbT$, Thèse Paris 11, 2014.

[20]

S. Marmi, P. Moussa and Y.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. AMS, 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X.

[21]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, North-Holland, Amsterdam, 1 (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.

[22]

D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces, J. Mod. Dyn., 6 (2012), 477-497.

[23]

S. Troubetzkoy, Approximation and billiards, Dynamical systems and Diophantine approximation, 173-185, Semin. Congr., 19, Soc. Math. France, Paris, 2009.

[24]

S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, J. Stat. Phys., 141 (2010), 60-67. doi: 10.1007/s10955-010-0026-5.

[25]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97 (1989), 553-583. doi: 10.1007/BF01388890.

[26]

Y. Vorobets, Periodic geodesics on translation surfaces, Algebraic and topological dynamics, 205-258, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005 doi: 10.1090/conm/385/07199.

[27]

H. Van Beyeren and E. H. Hauge, Abnormal diffusion in Ehrenfest's wind-tree model, Physics Letters A, 39 (1972), 397-398. doi: 10.1016/0375-9601(72)90112-0.

[28]

W. Wood and F. Lado, Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest's wind-tree model, J. Comp. Physics, 7 (1971), 528-546. doi: 10.1016/0021-9991(71)90109-4.

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