Abstract
We review known results and derive some new ones about the mean free path, Kolmogorov-Sinai entropy, and Lyapunov exponents for billiard-type dynamical systems. We focus on exact and asymptotic formulas for these quantities. The dynamical systems covered in this paper include the priodic Lorentz gas, the stadium and its modifications, and the gas of hard balls. Some open questions and numerical observations are discussed.
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References
P. R. Baldwin, The billiard algorithm and KS entropy,J. Phys. A 24: L941-L947 (1991).
G. Benettin, Power law behavior of Lyapunov exponents in some conservative dynamical systems,Phys. D 13: 211–220 (1984).
J.-P. Bouchaud and P. Le Doussal, Numerical study of ad-dimensional periodic Lorentz gas with universal properties,J. Statist. Phys. 41: 225–248 (1985).
L. A. Bunimovich, On billiards close to dispersing,Math. USSR Sbornik 23: 45–67 (1974).
L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards,Comm. Math. Phys. 65: 295–312 (1979).
L. A. Bunimovich,On absolutely focusing mirrors, Lect. Notes Math., 1514, Springer, New York, 1990.
S. Chapman and T. G. Cowling,The mathematical theory of nonuniform gases, Cambridge U. Press, 1970.
N. I. Chernov, A new proof of Sinai's formula for entropy of hyperbolic billiards. Its application to Lorentz gas and stadium,Funct. Anal. Appl. 25: 204–219 (1991).
N. I. Chernov, and R. Markarian, Entropy of nonuniformly hyperbolic plane billiards,Bol. Soc. Brasil Math. 23: 121–135 (1992).
N. I. Chernov and C. Haskell, Nonuniformly hyperbolic K-systems are Bernoulli,Ergodic Theory and Dynamical Systems 16: 19–44 (1996).
N. I. Chernov and J. L. Lebowitz, Stationary nonequilibrium states in boundary driven Hamiltonian systems: shear flow,J. Statist. Phys. 86: 953–990 (1997).
V. Donnay, Using intergrability to produce chaos: billiards with positive entropy,Comm. Math. Phys. 141: 225–257 (1991).
J. J. Erpenbeck and W. W. Wood, Molecular-dynamics calculations of the velocity-autocorrelation function. Methods, hard-disk results,Phys. Rev. A 26: 1648–1675 (1982).
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).
G. Gallavotti and D. Ornstein, Billiards and Bernoulli schemes,Comm. Math. Phys. 38: 83–101 (1974).
D. Gass, Enskog theory for a rigid disk fluid,J. Chem. Phys. 54: 1898–1902 (1971).
F. Golse, private communication.
A. Katok and J.-M. Strelcyn,Invariant manifolds, entropy and billiards; smooth maps with singularities, Lect. Notes Math., 1222, Springer, New York, 1986.
A. Katok, The groth rate for the number of singular and periodic orbits for a polygonal billiard,Comm. Math. Phys. 111: 151–160 (1987).
M. Lach-had, E. Blaisten-Barojas, and T. Sauer, Irregular scattering of particles confined to ring-bounded cavities, manuscript, 1996.
A. Latz, H. van Beijeren, and J. R. Dorfman, Lyapunov spectrum and the conjugate pairing rule for a thermostatted random Lorentz gas: kinetic theory, manuscript, 1996; C. Dellago and H. A. Posch, Lyapunov spectrum and the conjugate pairing rule for a thermostatted random Lorentz gas: numerical simulations, manuscript, 1996.
C. Liverani and M. Wojtkowski, Ergodicity in Hamiltonian systems,Dynamics Reported 4: 130–202 (1995).
R. Markarian, Billiards with Pesin region of measure one,Comm. Math. Phys. 118: 87–97 (1988).
G. Matheron,Random sets and integral geometry, J. Wiley & Sons, New York, 1975.
Ya. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,Russ. Math. Surv. 32: 55–114 (1977).
L. A. Santaló,Integral geometry and geometric probability, Addison-Wesley Publ. Co., Reading, Mass., 1976.
Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards,Russ. Math. Surv. 25: 137–189 (1970).
Ya. G. Sinai, Entropy per particle for the system of hard spheres, Harvard Univ. Preprint, 1978.
Ya. G. Sinai, Development of Krylov's ideas, Afterwards to N. S. Krylov,Works on the foundations of statistical physics, Princeton Univ. Press, 1979, pp. 239–281.
Ya. G. Sinai and N. I. Chernov, Entropy of a gas of hard spheres with respect to the group of space-time translations,Trudy Seminara Petrovskogo 8: 218–238 (1982) English translation in:Dynamical Systems, Ed. by Ya. Sinai, Adv. Series in Nonlinear Dynamics, Vol. 1.
Ya. G. Sinai and N. I. Chernov, Ergodic properties of some systems of 2-dimensional discs and 3-dimensional spheres,Russ. Math. Surv. 42: 181–207 (1987).
D. Szasz, Boltzmann's ergodic hypothesis, a conjecture for centuries?,Sudia Sci. Math. Hung. 31: 299–322 (1996).
F. Vivaldi, G. Casati, and I. Guarneri, Origin of long-time tails in strongly chaotic systems,Phys. Rev. Lett. 51: 727–730 (1983).
M. P. Wojtkowski, Invariant families of cones and Lyapunov exponents,Ergod. Th. Dynam. Sys. 5: 145–161 (1985).
M. P. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents,Commun. Math. Phys. 105: 391–414 (1986).
M. P. Wojtkowski, Measure theoretic entropy of the system of hard spheres,Ergod. Th. Dynam. Sys. 8: 133–153 (1988).
G. M. Zaslavski, Stochasticity in quantum mechanics,Phys. Rep. 80: 147–250 (1981).
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Chernov, N. Entropy, Lyapunov exponents, and mean free path for billiards. J Stat Phys 88, 1–29 (1997). https://doi.org/10.1007/BF02508462
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DOI: https://doi.org/10.1007/BF02508462