Abstract
We consider a physical Ehrenfests’ Wind–Tree model where a moving particle is a hard ball rather than (mathematical) point particle. We demonstrate that a physical periodic Wind–Tree model is dynamically richer than a physical or mathematical periodic Lorentz gas. Namely, the physical Wind–Tree model may have diffusive behavior as the Lorentz gas does, but it has more superdiffusive regimes than the Lorentz gas. The new superdiffusive regime where the diffusion coefficient \(D(t)\sim (\ln t)^2\) of dynamics seems to be never observed before in any model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avila, A., Hubert, P.: Recurrence for the wind-tree model. Ann. Inst. Henri Poincare Anal. non lineaire. https://doi.org/10.1016/j.anihpc.2017.11.006
Bianca, C., Rondoni, L.: The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles? Chaos 19, 013121 (2009)
Bleher, P.M.: Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Stat. Phys. 66, 315–373 (1992)
Bunimovich, L.A.: Physical versus mathematical billiards: from regular dynamics to chaos and back. Chaos 29, 091105 (2019)
Bunimovich, L.A., Sinai, Y.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1981)
Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47–106 (1991)
Cox, C., Feres, R., Zhang, H.K.: Stability of periodic orbits in no-slip billiards. Nonlinearity 31, 4443–4471 (2018)
Delecroix, V.: Divergent trajectories in the periodic wind–tree model. J. Mod. Dyn. 7, 1–29 (2013)
Delecroix, V., Hubert, P., Lelievre, S.: Diffusion for the periodic wind–tree model. Ann. Sci. 47, 1085–1110 (2014)
Dettmann, C.P., Cohen, E.G.D., Van Beijeren, H.: Statistical mechanics: microscopic chaos from Brownian motion? Nature 401, 875–875 (1999)
Ehrenfest, P., Ehrenfest, T.: Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, Encykl. d. Math. Wissensch, IV2 II Heft (6), 90 (1912) (in German)
Ehrenfest, P., Ehrenfest, T.: The Conceptual Foundations of the Statistical Approach in Mechanics, pp. 10-13. Cornell University Press, Ithaca (1959) (English translation)
Fraczek, K., Ulcigrai, C.: Non-ergodic Z-periodic billiards and infinite translation surfaces. Invent. Math. 197, 241–298 (2014)
Gallavotti, G.: Divergences and the approach to equilibrium in the Lorentz and the Wind–Tree Models. Phys. Rev. 185, 308–322 (1969)
Hardy, J., Weber, J.: Diffusion in a periodic wind–tree model. J. Math. Phys. 21, 1802–1808 (1980)
Hauge, E.H., Cohen, E.G.D.: Normal and abnormal diffusion in Ehrenfests’ Wind–Tree Model. J. Math. Phys. 10, 397–414 (1969)
Hooper, P., Hubert, P., Weiss, B.: Dynamics on the infinite staircase. Discret. Contin. Dyn. Syst. 33, 4341–4347 (2013)
Hubert, P., Weiss, B.: Ergodicity for infinite periodic translation surfaces. Compos. Math. 149, 1364–1380 (2013)
Hubert, P., Lelie’vre, S., Troubetzkoy, S.: The Ehrenfest wind–tree model: periodic directions, recurrence, diffusion. J. Reine Angew. Math. 656, 223–244 (2011)
Ralston, D., Troubetzkoy, S.: Ergodic infinite group extensions of geodesic flows on translation surfaces. J. Mod. Dyn. 6, 477–497 (2012)
Sabogal, A.M., Troubetzkoy, S.: Minimality of the Ehrenfest wind–tree model. J. Mod. Dyn. 10, 209–228 (2016)
Sabogal, A.M., Troubetzkoy, S.: Ergodicity of the Ehrenfest wind–tree model. C. R. Math. 354(10), 1032–1036 (2016)
Sinai, Y.G.: Dynamical systems with elastic reflections. Russ. Math. Surv. 25, 137–189 (1970)
Van Beyeren, H., Hauge, E.H.: Abnormal diffusion in Ehrenfest’s wind–tree model. Phys. Lett. A 39(5), 397–398 (1972)
Wood, W., Lado, F.: Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest’s wind–tree model. J. Comput. Phys. 7(3), 528–546 (1971)
Acknowledgements
We thank R. Feres for useful suggestions and for pointing to the paper [7]. This work was partially supported by the NSF Grant DMS-1600568.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Michael Aizenman.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
To Joel with love and admiration.
Rights and permissions
About this article
Cite this article
Attarchi, H., Bolding, M. & Bunimovich, L.A. Ehrenfests’ Wind–Tree Model is Dynamically Richer than the Lorentz Gas. J Stat Phys 180, 440–458 (2020). https://doi.org/10.1007/s10955-019-02460-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-019-02460-8