Basis of Local Approach in Classical Statistical Mechanics
  An ensemble of classical subsystems interacting with surrounding particles has been considered. In general case, a phase volume of the subsystems ensemble was shown to be a function of time. The evolutional equations of the ensemble are obtained as well as the simplest solution of these equations representing the quasi-local distribution with the temperature pattern being assigned. Unlike the Gibbs's distribution, the energy of interaction with surrounding particles appears in the distribution function, which make possible both evolution in the equilibrium case and fluctuations in the non-equilibrium one. The expression for local entropy is obtained. The exact expressions for changing entropy and quantity of the heat given by the environment have been obtained. A two-particle distribution function for pair interaction system has been obtained with the use of local conditional distribution functions. Its formula is exact disregarding edge conditions.The derivation of hydrodynamic equations from Boltzmann equation has been analyzed. The hydrodynamic equations obtained from Boltzmann equation were shown to be equations for ideal liquid. Reasons for stochastic description in deterministic Hamilton's systems, conditions of applicability of Poincares recurrence theorem as well as the problem of irreversibility have been considered. 
S. R. Sharov