Abstract
We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1/2 and nott 1/2 as in the classical case. We find the covariance matrix of the limit distribution.
Similar content being viewed by others
References
H. A. Lorentz, The motion of electrons in metallic bodies,Proc. Amst. Acad. 7:438, 585, 604 (1905).
G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models,Phys. Rev. 185:308–314 (1969).
E. H. Hauge, What one can learn from Lorentz models, inLecture Notes in Physics, Vol. 31 (1974), pp. 337–367.
Sh. Goldstein, J. Lebowitz, and M. Aizenman, Ergodic properties of infinite systems, inLecture Notes in Physics 38:112–143 (1975).
G. Gallavotti, Lectures on the billiard, inLecture Notes in Physics, Vol. 38 (1975), pp. 236–295.
Ya. G. Sinai, Ergodic properties of the Lorentz gas,Funkts. Anal. Ego Prilozh. 13:46–59 (1979).
L. A. Bunimovich and Ya. G. Sinai, Markov partitions for dispersed billiards,Commun. Math. Phys. 78:247–280 (1980).
L. A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers,Commun. Math. Phys. 78:479–497 (1981).
L. A. Bunimovich, Decay of correlations in dynamical systems with chaotic behavior,Zh. Eksp. Teor. Fiz. 89:1452–1471 (1985) [Sov. Phys. JETP 62:842–852 (1985).
A. Krámli and D. Szász, Central limit theorem for the Lorentz process via perturbation theory,Commun. Math. Phys. 91:519–528 (1983).
G. Casati, G. Comparin, and I. Guarneri, Decay of correlations in certain hyperbolic systems,Phys. Rev. A 26:717–719 (1982).
J. Machta, Power law decay of correlations in a billiard problem,J. Stat. Phys. 32:555–564 (1983).
J. Machta and R. Zwanzig, Diffusion in a periodic Lorentz gas,Phys. Rev. Lett. 50:1959–1962 (1983).
B. Friedman, Y. Oono, and I. Kubo, Universal behavior of Sinai billiard systems in the small-scatterer limit,Phys. Rev. Lett. 52:709–712 (1984).
B. Friedman and R. F. Martin, Jr., Decay of the velocity autocorrelation function for the periodic Lorentz gas,Phys. Lett. 105A:23–26 (1984).
J.-P. Bouchaud and P. Le Doussal, Numerical study of a D-dimensional periodic Lorentz gas with universal properties,J. Stat. Phys. 41:225–248 (1985).
P. R. Baldwin, Soft billiard systems,Physica D 29:321–342 (1988).
B. Friedman and R. F. Martin, Jr., Behavior of the velocity autocorrelation function for the periodic Lorentz gas,Physica D 30:219–227 (1988).
P. M. Bleher, Statistical properties of a particle moving in a periodic scattering billiard, inStochastic Methods in Experimental Sciences (Proc. 1989 COSMEX Meeting, Sklarska Poreba, Poland 8–14 Sept. 1989), W. Kasprzak and A. Weron, eds. (World Scientific, Singapore, 1990), pp. 43–58.
Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards,Usp. Math. Nauk 25:141–192 (1970).
I. Kubo, Billiard problem,Sem. Prob. 37 (1972).
G. Gallavotti and D. Ornstein, Billiards and Bernoulli schemes,Commun. Math. Phys. 38:83–101 (1974).
L. A. Bunimovich and Ya. G. Sinai, On a fundamental theorem in the theory of dispersing billiards,Math. USSR-Sb. 90:407–423 (1973).
L. A. Bunimovich, Ya. G. Sinai, and N. I., Tchernov, Markov partitions for two-dimensional hyperbolic billiards,Uspekhi Math. Nauk 45:97–134 (1990).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. (Wiley, New York, 1971).
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
K. M. Efimov, The central limit theorem for the Lorentz gas and martingales,Stochastics 14:1–10 (1984).
A. Zacherl, T. Geisel, J. Nierwetberg, and G. Radons, Power spectra for anomalous diffusion in the extended billiard,Phys. Lett. 114A:317–321 (1986).
L. A. Bunimovich, Ph.d. Dissertation, Moscow (1985).
G. Keller, Diplomarbeit. Universitaet Erlangen (1977).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bleher, P.M. Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J Stat Phys 66, 315–373 (1992). https://doi.org/10.1007/BF01060071
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01060071