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 Regul. Chaotic Dyn., 2008, Volume 13, Issue 3, Pages 141–154 (Mi rcd566)

Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat

V. V. Kozlov

V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

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 mean 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

DOI: https://doi.org/10.1134/S1560354708030015

Bibliographic databases:

MSC: 37A60, 60K35, 70H05, 82B30, 40A9
Accepted:28.10.2007
Language:

Citation: V. V. Kozlov, “Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat”, Regul. Chaotic Dyn., 13:3 (2008), 141–154

Citation in format AMSBIB
\Bibitem{Koz08} \by V.~V.~Kozlov \paper Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat \jour Regul. Chaotic Dyn. \yr 2008 \vol 13 \issue 3 \pages 141--154 \mathnet{http://mi.mathnet.ru/rcd566} \crossref{https://doi.org/10.1134/S1560354708030015} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2415369} \zmath{https://zbmath.org/?q=an:1229.82091} 

• http://mi.mathnet.ru/eng/rcd566
• http://mi.mathnet.ru/eng/rcd/v13/i3/p141

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This publication is cited in the following articles:
1. Andrea Carati, Luigi Galgani, Alberto Maiocchi, Fabrizio Gangemi, Roberto Gangemi, “The FPU Problem as a Statistical-mechanical Counterpart of the KAM Problem, and Its Relevance for the Foundations of Physics”, Regul. Chaotic Dyn., 23:6 (2018), 704–719
2. A. L. Kuzemsky, “Nonequilibrium statistical operator method and generalized kinetic equations”, Theoret. and Math. Phys., 194:1 (2018), 30–56
3. Carati A. Galgani L. Gangemi F. Gangemi R., “Relaxation Times and Ergodic Properties in a Realistic Ionic-Crystal Model, and the Modern Form of the Fpu Problem”, Physica A, 532 (2019), 121911