The Galton board: Limit theorems and recurrence
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- by N. Chernov and D. Dolgopyat;
- J. Amer. Math. Soc. 22 (2009), 821-858
- DOI: https://doi.org/10.1090/S0894-0347-08-00626-7
- Published electronically: October 21, 2008
Abstract:
We study a particle moving in $\mathbb {R}^2$ under a constant (external) force and bouncing off a periodic array of convex domains (scatterers); the latter must satisfy a standard ‘finite horizon’ condition to prevent ‘ballistic’ (collision-free) motion. This model is known to physicists as the Galton board (it is also identical to a periodic Lorentz gas). Previous heuristic and experimental studies have suggested that the particle’s speed $v(t)$ should grow as $t^{1/3}$ and its coordinate $x(t)$ as $t^{2/3}$. We prove these conjectures rigorously; we also find limit distributions for the rescaled velocity $t^{-1/3} v(t)$ and position $t^{-2/3} x(t)$. In addition, quite unexpectedly, we discover that the particle’s motion is recurrent. That means that a ball dropped on an idealized Galton board will roll down, but from time to time it should bounce all the way back up (with probability one).References
- Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
- L. A. Bunimovich and Ya. G. Sinaĭ, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/81), no. 4, 479–497. MR 606459
- L. A. Bunimovich, Ya. G. Sinaĭ, and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Uspekhi Mat. Nauk 46 (1991), no. 4(280), 43–92, 192 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 4, 47–106. MR 1138952, DOI 10.1070/RM1991v046n04ABEH002827 CS Chepelianskii A. D. and Shepelyansky D. L. Dynamical Turbulent Flow on the Galton Board with Friction, Phys. Rev. Lett. 87 (2001) paper 034101 (4 pages).
- N. Chernov, Decay of correlations and dispersing billiards, J. Statist. Phys. 94 (1999), no. 3-4, 513–556. MR 1675363, DOI 10.1023/A:1004581304939
- N. I. Chernov, Sinai billiards under small external forces, Ann. Henri Poincaré 2 (2001), no. 2, 197–236. MR 1832968, DOI 10.1007/PL00001034 CD1 Chernov N. and Dolgopyat D. Brownian Motion I, to appear in Memoirs AMS.
- Nikolai Chernov and Dmitry Dolgopyat, Hyperbolic billiards and statistical physics, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1679–1704. MR 2275665
- N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinaĭ, Steady-state electrical conduction in the periodic Lorentz gas, Comm. Math. Phys. 154 (1993), no. 3, 569–601. MR 1224092 CELSp 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.
- Nikolai Chernov and Roberto Markarian, Chaotic billiards, Mathematical Surveys and Monographs, vol. 127, American Mathematical Society, Providence, RI, 2006. MR 2229799, DOI 10.1090/surv/127 DM Dettmann C. P. and Morriss G. P., Crisis in the periodic Lorentz gas, Phys. Rev. E. 54 (1996), 4782–4790.
- Dmitry Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004), no. 2, 389–449. MR 2031432, DOI 10.1007/s00222-003-0324-5
- Dmitry Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1637–1689. MR 2034323, DOI 10.1090/S0002-9947-03-03335-X
- Dmitry Dolgopyat, Averaging and invariant measures, Mosc. Math. J. 5 (2005), no. 3, 537–576, 742 (English, with English and Russian summaries). MR 2241812, DOI 10.17323/1609-4514-2005-5-3-537-576
- Dmitry Dolgopyat, Bouncing balls in non-linear potentials, Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 165–182. MR 2410953, DOI 10.3934/dcds.2008.22.165 D5 Dolgopyat D. Fermi acceleration, Cont. Math. 469 (2008), 149–166.
- Dmitry Dolgopyat, Domokos Szász, and Tamás Varjú, Recurrence properties of planar Lorentz process, Duke Math. J. 142 (2008), no. 2, 241–281. MR 2401621, DOI 10.1215/00127094-2008-006 DSV2 Dolgopyat D., Szasz D. and Varju T. Limit theorems for perturbed planar Lorentz process, preprint.
- Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153 G Galton F., Natural Inheritance, MacMillan, 1989 (facsimile available at www.galton.org). F Fermi E. On the origin of cosmic radiation, Phys. Rev. 75 (1949), 1169–1174.
- V. V. Kozlov and M. Yu. Mitrofanova, Galton board, Regul. Chaotic Dyn. 8 (2003), no. 4, 431–439. MR 2023046, DOI 10.1070/RD2003v008n04ABEH000255 KR Krapivsky P. and Redner S., Slowly divergent drift in the field-driven Lorentz gas, Phys. Rev. E 56 (1997), 3822.
- J. E. Littlewood, On the problem of $n$ bodies, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), 143–151. MR 54375 L05 Lorentz H. A., The motion of electrons in metallic bodies, Proc. Amst. Acad. 7 (1905), 438–453.
- Arthur Lue and Howard Brenner, Phase flow and statistical structure of Galton-board systems, Phys. Rev. E (3) 47 (1993), no. 5, 3128–3144. MR 1377898, DOI 10.1103/PhysRevE.47.3128
- Bill Moran, William G. Hoover, and Stronzo Bestiale, Diffusion in a periodic Lorentz gas, J. Statist. Phys. 48 (1987), no. 3-4, 709–726. MR 914903, DOI 10.1007/BF01019693
- Bernt Øksendal, Stochastic differential equations, 6th ed., Universitext, Springer-Verlag, Berlin, 2003. An introduction with applications. MR 2001996, DOI 10.1007/978-3-642-14394-6 PW Piasecki J. and Wajnryb E., Long-time behavior of the Lorentz electron gas in a constant, uniform electric field, J. Stat. Phys. 21 (1979), 549–559.
- K. Ravishankar and L. Triolo, Diffusive limit of the Lorentz model with a uniform field starting from the Markov approximation, Markov Process. Related Fields 5 (1999), no. 4, 385–421. MR 1734241
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- Ja. G. Sinaĭ, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk 25 (1970), no. 2(152), 141–192 (Russian). MR 274721 YK Yamada T., Kawasaki K. Nonlinear effects in the shear viscosity of a critical mixture, Prog. Theor. Phys. 38 (1967), 1031–1051.
- Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
- G. M. Zaslavsky, Chaos in dynamic systems, Harwood Academic Publishers, Chur, 1985. Translated from the Russian by V. I. Kisin. MR 780371
Bibliographic Information
- N. Chernov
- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
- D. Dolgopyat
- Affiliation: Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
- MR Author ID: 357840
- Received by editor(s): December 12, 2007
- Published electronically: October 21, 2008
- © Copyright 2008 by the authors
- Journal: J. Amer. Math. Soc. 22 (2009), 821-858
- MSC (2000): Primary 37D50
- DOI: https://doi.org/10.1090/S0894-0347-08-00626-7
- MathSciNet review: 2505302