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On the work of Rodriguez Hertz on rigidity in dynamics
Minimality of the Ehrenfest wind-tree model
1. | Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France, France |
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), 38-45.
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, ().
|
[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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