Petrenko S. Conditions for existence of Gibbs measures for the infinite systems of interaction particles

The thesis is devoted to the research of conditions for the interaction potentials when at least one Gibbs measure will exist. The paper gives an overview of the basic notions of equilibrium statistical mechanics, connected with the construction of Gibbs measures in limited volume and space of infinite configurations. The sufficient conditions of superstability for two-body potential have been obtained. The conditions for two-body infinite range interaction potential which can provide uniform boundedness of correlation functions family have been obtained. Such conditions could give the possibility to take new proof of the existence and uniqueness of Gibbs measure when the values of activity are low. The method of cluster expansions in density of configurations for the potentials with infinite range proposed by O. L. Rebenko has been modified. For the first time the conditions on the family of -body infinite range potentials which ensure uniform by volume finiteness of correlation functions and allow to prove the set of Gibbs measures to be not empty have been obtained. For the systems with many-body interaction by the example of pressure thermodynamic function the method of quasicontinuous approximation has been described. It has been proved that for the arbitrary positive values of parameters and the approximating correlation function coincides with the value of the correlation function of an infinite system when the parameter of approximation tends to zero.