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Particle drift in self-similar billiards

Published online by Cambridge University Press:  01 April 2008

N. CHERNOV  and
D. DOLGOPYAT
N. CHERNOV
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA (email: [email protected])
D. DOLGOPYAT
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (email: [email protected])
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Abstract

We study a particle moving at unit speed in a channel made by connected self-similar billiard tables that grow in size by a factor r>1 from left to right (this model was recently introduced in the physics literature by Barra, Gilbert and Romo). Let q(T) denote the position of the particle at time T. Our main result is the existence of an asymptotic distribution of q(T)/T as and {ln T/ln r}→ρ for some 0≤ρ<1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 389 - 403
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Barra, F., Gilbert, T. and Romo, M.. Drift of particles in self-similar systems and its Liouvillian interpretation. Phys. Rev. E 73 (2006), 14 (paper 026211).CrossRefGoogle ScholarPubMed
[2]Barra, F. and Gilbert, T.. Steady-state conduction in self-similar billiards. Phys. Rev. Lett. 98 (2007), 4 (paper 130601).CrossRefGoogle ScholarPubMed
[3]Barra, F. and Gilbert, T.. Non-equilibrium Lorentz gas on a curved space. J. Stat. Mech. Theory Exp. (2007) paper L01003.CrossRefGoogle Scholar
[4]Barra, F., Chernov, N. and Gilbert, T.. Log-periodic drift oscillations in self-similar billiards. Nonlinearity 20 (2007), 2539–2549, available at http://www.math.uab.edu/chernov/papers/pubs.html.CrossRefGoogle Scholar
[5]Bleher, P. M.. Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Statist. Phys. 66 (1992), 315–373.CrossRefGoogle Scholar
[6]Bunimovich, L. A., Sinai, Ya. G. and Chernov, N. I.. Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys 46 (1991), 47–106.CrossRefGoogle Scholar
[7]Chernov, N.. Decay of correlations and dispersing billiards. J. Statist. Phys. 94 (1999), 513–556.CrossRefGoogle Scholar
[8]Chernov, N. I.. Sinai billiards under small external forces. Ann. Henri Poincaré 2 (2001), 197–236.CrossRefGoogle Scholar
[9]Chernov, N. and Dolgopyat, D.. Brownian motion—1. Mem. Amer. Math. Soc., to appear.Google Scholar
[10]Chernov, N. and Dolgopyat, D.. Hyperbolic billiards and statistical physics. Proceedings ICM-2006, Vol. II. European Mathematical Society, Zürich, pp. 1679–1704.Google Scholar
[11]Chernov, N. I., Eyink, G. L., Lebowitz, J. L. and Sinai, Ya. G.. Steady-state electrical conduction in the periodic Lorentz gas. Comm. Math. Phys. 154 (1993), 569–601.CrossRefGoogle Scholar
[12]Chernov, N. I., Eyink, G. L., Lebowitz, J. L. and Sinai, Ya. G.. Derivation of Ohm’s law in a deterministic mechanical model. Phys. Rev. Lett. 70 (1993), 2209–2212.CrossRefGoogle Scholar
[13]Chernov, N. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127). American Mathematical Society, Providence, RI, 2006.CrossRefGoogle Scholar
[14]Dettmann, C. P. and Morriss, G. P.. Hamiltonian formulation of the Gaussian isokinetic thermostat. Phys. Rev. E 54 (1996), 2495–2500.CrossRefGoogle ScholarPubMed
[15]Gaspard, P.. Chaos, Scattering and Statistical Mechanics. Cambridge University Press, Cambridge, 1999.Google Scholar
[16]Moran, B. and Hoover, W.. Diffusion in a periodic Lorentz gas. J. Statist. Phys. 48 (1987), 709–726.CrossRefGoogle Scholar
[17]Pesin, Ya. B.. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. & Dynam. Sys. 12 (1992), 123–151.CrossRefGoogle Scholar
[18]Szasz, D. and Varju, T.. Limit laws and recurrence for the planar Lorentz process with infinite horizon.J. Statist. Phys. to appear.Google Scholar
[19]Wojtkowski, M. P.. W-flows on Weyl manifolds and Gaussian thermostats. J. Math. Pures Appl. (9) 79 (2000), 953–974.CrossRefGoogle Scholar
[20]Wojtkowski, M. P.. Weyl manifolds and Gaussian thermostats. Proc. ICM-2002, Vol. III. Higher Education Press, Beijing, pp. 511–523.Google Scholar
[21]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585–650.CrossRefGoogle Scholar
[22]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar