Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics]
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 Nelin. Dinam., 2007, Volume 3, Number 2, Pages 123–140 (Mi nd130)

Gibbs ensembles, equidistribution of the energy of sympathetic oscillators and statistical models of thermostat

V. V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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 average energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.

Keywords: Hamiltonian system, sympathetic oscillators, weak convergence, thermostat

Full text: PDF file (263 kB)
UDC: 530+517
MSC: 37A60, 60K35, 70H05, 82B30, 40A99

Citation: V. V. Kozlov, “Gibbs ensembles, equidistribution of the energy of sympathetic oscillators and statistical models of thermostat”, Nelin. Dinam., 3:2 (2007), 123–140

Citation in format AMSBIB
\Bibitem{Koz07} \by V.~V.~Kozlov \paper Gibbs ensembles, equidistribution of the energy of sympathetic oscillators and statistical models of thermostat \jour Nelin. Dinam. \yr 2007 \vol 3 \issue 2 \pages 123--140 \mathnet{http://mi.mathnet.ru/nd130} 

• http://mi.mathnet.ru/eng/nd130
• http://mi.mathnet.ru/eng/nd/v3/i2/p123

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This publication is cited in the following articles:
1. V. V. Kozlov, “The generalized Vlasov kinetic equation”, Russian Math. Surveys, 63:4 (2008), 691–726
2. A. V. Kargovsky, L. S. Bulushova, O. A. Chichigina, “Theorem on energy distribution over degrees of freedom for a quasistable symmetric anharmonic oscillator”, Theoret. and Math. Phys., 167:2 (2011), 636–644
3. A. P. Kuznetsov, S. P. Kuznetsov, L. V. Tyuryukina, I. R. Sataev, “Stsenarii Landau–Khopfa v ansamble vzaimodeistvuyuschikh ostsillyatorov”, Nelineinaya dinam., 8:5 (2012), 863–873
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