Regular and Chaotic Dynamics

, Volume 13, Issue 3, pp 141–154 | Cite as

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

  • V. V. Kozlov
Research Articles


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.

Key words

Hamiltonian system sympathetic oscillators weak convergence thermostat 

MSC2000 numbers

37A60 60K35 70H05 82B30 40A9 


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Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.V. A. Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

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