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

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.