Canonical Gibbs Distribution and Thermodynamics of Mechanical Systems with a Finite Number of Degrees of Freedom

Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit n → ∞. In the presented work we develop another approach to these problems assuming that n is fixed and n ≥ 2. The ergodic hypothesis (which frequently is not valid due to known results of the KAM-theory) is substituted by a weaker assumption that the perturbed system does not have additional first integrals independent of the energy integral. The proof of nonintegrability of perturbed Hamiltonian systems is based on the Poincaré method. Moreover, we use the natural Gibbs assumption concerning a thermodynamic equilibrium of subsystems at vanishing interaction. The general results are applied to the system of the weakly connected pendula. The averaging with respect to the Gibbs measure allows to pass from usual dynamics of mechanical systems to the classical thermodynamic model.