Abstract
We discuss various experiments on the time decay of velocity autocorrelation functions in billiards. We perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by Friedman and Martin (FM): they do not seem compatible with the stretched exponentials believed, in spite of FM and more recently of Chernov, to describe the mixing. The analysis leads to several byproducts: we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timet m (ast m →∞), finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles.
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References
G. Benettin, Power law behaviour of Lyapunov exponents in some conservative dynamical systems,Physica D 13:211–213 (1984).
P. Bleher, Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon,J. Stat. Phys. 66:315–373 (1992).
J. P. Bouchaud and P. Le Doussal, Numerical study of aD-dimensional periodic Lorentz gas with universal properties,J. Stat. Phys. 41:225–248 (1985).
R. Bowen, Equilibrium and ergodic theory of Anosov diffeomorphisms, inLecture Notes in Mathematics, Vol. 470 (Springer-Verlag, Berlin, 1975); see also G. Gallavotti, Aspettidella teoria ergodica qualitativa e statistica del moto,Quaderni UMI (Pitagora, Bologna)21 (1980).
Bunimovitch and Ya. Sinai, Markov partitions for dispersed billiards,Commun. Math. Phys. 73:247–280 (1980); Statistical properties of Lorentz gas with periodic configuration of scatterers,Commun. Math. Phys. 78:479–497 (1981).
L. Bunimovitch, Ya. Sinai, and N. Chernov, Statistical properties of two dimensional hyperbolic billiards,Russ. Math. Surv. 45(3):105–152 (1990).
Chernov, Ergodic and statistical properties of piecewise liear hyperbolic automorphisms of the 2-torus,J. Stat. Phys. 69:111–134 (1992).
G. Casati, O. Comparin, and I. Guarneri, Decay of correlations in certain hyperbolic systems,Phys. Rev. A 26:717–719 (1982).
B. Friedman and R. Martin, Decay of the velocity autocorrelation function for the periodic Lorentz gas,Phys. Lett. 105A:23–26 (1984).
G. Gallavotti, Lectures on the billiards, inDynamical Systems, Theory and Applications, J. Moser, ed., (Springer-Verlag, Berlin, 1975), pp. 236–295.
G. Gallavotti and D. Ornstein, Billiards and Bernoulli schemes,Commun. Math. Phys. 38:83–101 (1974).
Laskar and P. Robutel, The chaotic obliquity of the planets,Nature 361:608–612 (1993).
Ya. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards,Russ. Math. Surv. 25:137–189 (1970).
Ya. Sinai, Markov partitions andC-diffeomorphisms,Funct. Anal Appl. 2(1):64–89 (1968).
Ya. Sinai, Construction of Markov partitions,Funct. Anal. Appl. 2(2):70–80 (1968).
Ya. Sinai, Gibbs measure in ergodic theory,Russ. Math. Surv. 166:21–69 1972.
S. Vaienti, Properties of the discontinuous sawthooth map,J. Stat. Phys. 67:251–269 (1992).
F. Vivaldi, G. Casati, and I. Guarneri, Origin of long time tails in strongly chaotic systems,Phys. Rev. Lett. 51:727–730 (1983).
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This paper is dedicated to Philippe Choquard on his 65th birthday
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Garrido, P.L., Gallavotti, G. Billiards correlation functions. J Stat Phys 76, 549–585 (1994). https://doi.org/10.1007/BF02188675
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DOI: https://doi.org/10.1007/BF02188675