Bolukh V. Behavior of correlational and thermodynamic functions of continuous statistical systems.

The thesis is dedicated to the investigation of infinite systems of the classical statistical mechanics in the framework of infinite dimensional Poisson analysis. In the second chapter of the thesis some significant formulas and theorems of the statistical mechanics were proved in the framework of infinite dimensional Poisson analysis. Using this method the theorem of exponential representation of some integrals with respect to the Lebesgue-Poisson measure has been proved. It gives the possibility to represent the large partition function in the exponential form and write down the pressure of the infinite system. The new approaches of derivation of Kirkwood-Salzburg equations and Boholyubov chain of equations for correlational functions have been suggested. The third chapter of the thesis is dedicated to the construction and investigation of the new expansions for thermodynamic functions (the Mayer expansion). The Bridges-Federbush method of the Mayer expansion for nonintegrable potentials was constructed and the convergences of these expansions were proved. A new form of the Mayer expansion for thermodynamic potentials of infinite systems is presented. In the fourth chapter in the framework of the cell gas model we have formulated and proved the theorem that free energy is an approximation of the correspondent value of continuous system; characteristics of the free energy have been investigated as the monotonous, infinite, convex function from the specific volume v. It is established that approximated free energy will converge to the free energy of the continuous system if the parameter of approximation a>0 for any values of an inverse temperature and volume per particle.