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Nonequilibrium statistical mechanics of weakly ergodic systems. (English) Zbl 1471.82007
Summary: The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed. We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the equalization of the density and the temperature of a Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity of a billiard in a rectangular parallelepiped with a partition wall is established.
MSC:
82C03 Foundations of time-dependent statistical mechanics
82C23 Exactly solvable dynamic models in time-dependent statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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[1] Kac, M., Probability and Related Topics in Physical Sciences (1959), London: Interscience Publ., London · Zbl 0087.33003
[2] Uhlenbeck, G. E.; Ford, G. W., Lectures in Statistical Mechanics (1963), Providence, R.I.: AMS, Providence, R.I. · Zbl 0111.43802
[3] Kozlov, V. V., Thermal Equilibrium in the Sense of Gibbs and Poincaré, Dokl. Math., 65, 1, 125-128 (2002) · Zbl 1143.82302
[4] Kozlov, V. V.; Treshchev, D. V., Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems, Theoret. and Math. Phys., 134, 3, 339-350 (2003) · Zbl 1178.37050
[5] Kozlov, V. V., Thermal Equilibrium in the Sense of Gibbs and Poincaré (2002), Izhevsk: R&C Dynamics, Institute of Computer Science, Izhevsk · Zbl 1143.82302
[6] Kozlov, V. V., Gibbs Ensembles and Nonequilibrium Statistical Mechanics (2008), Izhevsk: R&C Dynamics, Institute of Computer Science, Izhevsk
[7] Poincaré, H., Réflexion sur la théorie cinétique des gaz, J. Phys. Théor. Appl., 4e sér., 5, 369-403 (1906) · JFM 37.0944.02
[8] Kozlov, V. V., Statistical Irreversibility of the Kac Reversible Circular Model, Regul. Chaotic Dyn., 16, 5, 536-549 (2011) · Zbl 1309.37015
[9] Kadomtsev, B. B., Landau Damping and Echo in a Plasma, Sov. Phys. Usp., 11, 3, 328-337 (1968)
[10] Maslov, V. P.; Fedoryuk, M. V., The Linear Theory of Landau Damping, Math. USSR-Sb., 55, 2, 437-465 (1986) · Zbl 0662.35035
[11] Mouhot, C.; Villani, C., On Landau Damping, Acta Math., 207, 29-201 (2011) · Zbl 1239.82017
[12] Kozlov, V. V., The Generalized Vlasov Kinetic Equation, Russian Math. Surveys, 63, 4, 691-726 (2008) · Zbl 1181.37006
[13] Ulam, S. M., On the Ergodic Behavior of Dynamical Systems, Analogies between Analogies: The Mathematical Reports of S. M. Ulam and His Los Alamos Collaborators, 155-162 (1990), Berkeley, Calif.: Univ. of California Press, Berkeley, Calif. · Zbl 0917.00003
[14] Kozlov, V. V., Statistical Properties of Billiards in Polytopes, Dokl. Math., 76, 2, 696-699 (2007) · Zbl 1152.82014
[15] Tabachnikov, S., Geometry and Billiards (2005), Providence, R.I.: AMS, Providence, R.I. · Zbl 1119.37001
[16] Lazutkin, V. F., The Existence of Caustics for a Billiard Problem in a Convex Domain, Math. USSR-Izv., 7, 1, 185-214 (1973) · Zbl 0277.52002
[17] Szász, D., Boltzmann’s Ergodic Hypothesis, a Conjecture for Centuries?, Hard Ball Systems and the Lorentz Gas, 421-448 (2000), Berlin: Springer, Berlin · Zbl 1026.82001
[18] Simányi, N., Conditional Proof of the Boltzmann - Sinai Ergodic Hypothesis, Invent. Math., 177, 2, 381-413 (2009) · Zbl 1178.37038
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