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Gibbs ensembles, equidistribution of the energy of sympathetic oscillators and statistical models of thermostat. (English) Zbl 1229.82091
Summary: The paper develops an approach to the proof of the “zeroth” law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.

82B30 Statistical thermodynamics
37A60 Dynamical aspects of statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
70H05 Hamilton’s equations
Full Text: DOI
[1] Fermi, E., Pasta, J., and Ulam, S., Studies of Nonlinear Problems, Los Alamos Scientific Laboratory, Los Alamos, NM, 1955 (Reprinted in: Collected Works of Enrico Fermi, Vol. 2, University of Chicago Press, 1965). · Zbl 0353.70028
[2] Bogolubov, N.N., On Some Statistical Methods in Mathematical Physics, Kiev: Ukr. SSR Academy of Sciences, 1945.
[3] Szász, D., Ed., Hard Ball Systems and the Lorentz Gas, vol. 101 of Encycl. of Math. Sciencies, Berlin: Springer-Verlag, 2000. · Zbl 0953.00014
[4] Kozlov, V.V., Thermal Equilibrium in the Sense of Gibbs and Poincaré, Dokl. Math., 2002, vol. 65, no. 1, pp. 125–128. · Zbl 1143.82302
[5] Kozlov, V.V. and Treschev, D.V., Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems, Teoret. Mat. Fiz., 2003, vol. 136, no. 3, pp. 496–506 [Engl. transl.: Theoret. and Math. Phys., 2003, vol. 136, no. 3, pp. 1325–1335].
[6] Kozlov, V.V., Kinetics of Collisionless Continuous Medium, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 235–251. · Zbl 1006.82011
[7] Kozlov, V.V. and Treschev, D.V., Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian systems, Teoret. Mat. Fiz., 2003, vol. 134, no. 3, pp. 388–400 [Engl. transl.: Theoret. and Math. Phys., 2003, vol. 134, no. 3, pp. 339–350].
[8] Kozlov, V.V. and Treschev, D.V., On New Forms of the Ergodic Theorem, J. Dynam. Control Systems, 2003, vol. 9, no. 3, pp. 449–453. · Zbl 1021.37004
[9] Kozlov, V.V., Billiards, Invariant Measures, and Equilibrium Thermodynamics, II, Regul. Chaotic Dyn., 2004, vol. 9, no. 2, pp. 91–100. · Zbl 1078.82012
[10] Bogolubov, N.N., On Some Problems Connected with Validation of Statistical Mechanics, in: Bogolubov, N.N., Collected Works, Vol. VI, Moscow: Nauka, 2006, pp. 432–440.
[11] Kozlov, V.V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Berlin: Springer-Verlag, 1996. · Zbl 0921.58029
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