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

Minimality of the Ehrenfest wind-tree model

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.
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, ().

[2]

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

[3]

C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?,, Chaos, 19 (2009). 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. 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. 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.

[7]

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

[8]

P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik,, Encykl. d. Math. Wissensch. IV 2 II, (1912), 10.

[9]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces,, Invent. Math., 197 (2014), 241. 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. doi: 10.1103/PhysRev.185.308.

[11]

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties,, Bull. AMS, 50 (1944), 915. 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. 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.

[14]

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

[15]

P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Discrete Contin. Dyn. Syst., 33 (2013), 4341. 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. doi: 10.1515/CRELLE.2011.052.

[17]

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

[18]

M. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25. 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. doi: 10.1090/S0894-0347-05-00490-X.

[21]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, Handbook of dynamical systems, 1 (2002), 1015. 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.

[23]

S. Troubetzkoy, Approximation and billiards,, Dynamical systems and Diophantine approximation, (2009), 173.

[24]

S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model,, J. Stat. Phys., 141 (2010), 60. 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. doi: 10.1007/BF01388890.

[26]

Y. Vorobets, Periodic geodesics on translation surfaces,, Algebraic and topological dynamics, (2005), 205. 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. 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. 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, ().

[2]

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

[3]

C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?,, Chaos, 19 (2009). 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. 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. 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.

[7]

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

[8]

P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik,, Encykl. d. Math. Wissensch. IV 2 II, (1912), 10.

[9]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces,, Invent. Math., 197 (2014), 241. 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. doi: 10.1103/PhysRev.185.308.

[11]

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties,, Bull. AMS, 50 (1944), 915. 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. 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.

[14]

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

[15]

P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Discrete Contin. Dyn. Syst., 33 (2013), 4341. 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. doi: 10.1515/CRELLE.2011.052.

[17]

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

[18]

M. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25. 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. doi: 10.1090/S0894-0347-05-00490-X.

[21]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, Handbook of dynamical systems, 1 (2002), 1015. 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.

[23]

S. Troubetzkoy, Approximation and billiards,, Dynamical systems and Diophantine approximation, (2009), 173.

[24]

S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model,, J. Stat. Phys., 141 (2010), 60. 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. doi: 10.1007/BF01388890.

[26]

Y. Vorobets, Periodic geodesics on translation surfaces,, Algebraic and topological dynamics, (2005), 205. 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. 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. doi: 10.1016/0021-9991(71)90109-4.

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