On Justification of Gibbs Distribution

The paper develop a new approach to the justification of Gibbs canonical distribution for Hamiltonian systems with finite number of degrees of freedom. It uses the condition of nonintegrability of the ensemble of weak interacting Hamiltonian systems. 1. Gibbs distribution. We consider the probability distribution in the phase space of Hamiltonian system with the density ρ = ce− H kτ , (1) where H is a Hamiltonian, τ is an absolute temperature, k is the Boltzmann constant, c is a normalized factor. It plays the key role in the equilibrium statistical mechanics. Gibbs show in [1] that the averaging with respect to probability measure with density (1) give rise to the fundamental relations of equilibrium thermodynamics. To deduce the canonical Gibbs distribution one usually consider the ensemble of Hamiltonian systems with Hamiltonian function of the following form H = H0(P, Q) + εH1(P, Q), (2)