The thesis is devoted to the development of a method for describing the first order phase transition in one-component systems in the framework of the grand canonical ensemble. The calculations were performed using a cell fluid model, which is generalized in the thesis to the case of continuous systems. The idea of the cell fluid model consists in the conditional partition of the total volume of the system into a certain number of fixed size cells. Depending on the density of the system, each of the cells contains a certain amount of particles, without imposing any conditions on their number. The particles within one cell consider to repel each other and particles from different cells – to attract each other. The Curie-Weiss interaction, which fails to be a function of distance between constituents, is used as the interaction potential. Based on the method of describing the phase behavior proposed in the thesis, for the first time an accurate calculation of the grand partition function of a continuous system with Curie-Weiss interaction is performed, the equation of state of such a system is obtained, the cascade of phase transitions from the phase of lower density to the phase of increasingly higher density is detected. The regularity of decrease in the value of the critical temperature with decrease in the value of attraction relative to repulsion is established.
The above results are used to describe the phase behavior of systems with distance dependent interaction. For this purpose, the grand partition function of a cell fluid model with the Morse interaction potential is being calculated. The Morse potential has a Fourier transform, which allows applying the method of collective variables. The peculiarity of calculating the grand partition function of the cell fluid model is the introduction of a kind of a reference system. The later contains only a part of the repulsive component of the interaction potential and also allows calculating the Jacobian of transition to collective variables. Two approaches to the study of the phase behavior of the model are developed. The first method is common and does not depend on the form of the interaction potential. However, it is more cumbersome, since one of its stages is intermediate integration during the calculation of the Jacobian of transition. The second method — direct — is applicable to a limited class of interaction potentials. It is used only when the effective interaction potential resulting from the formation of the reference system potential from the initial potential does not change the sign. This approach made it possible to obtain an accurate representation of the grand partition function of a system with Morse potential in the form of an infinite cumulant series. It is established that the values of cumulants are expressed through new special functions, which are rapidly convergent series.
A characteristic feature of using a grand canonical distribution is to obtain an equation for the bond of chemical potential and density. The thermodynamic potential of the model is obtained, based on which the main characteristics of the first order phase transition are calculated. The values of the critical point parameters were found, the equation of state was calculated over a wide range of density and temperature above and below the critical point, based on which the coexistence curve and spinodal were plotted.
The thermodynamic potential in the vicinity of the critical point is also calculated with allowance for fluctuations. The basic idea of calculating the thermodynamic potential near Tc is to include the contributions from short- and long-wave modes of order parameter fluctuations. Shortwave modes are characterized by the presence of renormalization group symmetry and are described by non-Gaussian measure density. In this case, the method of renormalization group is used. The direct method of calculating thermodynamic potential is developed in the thesis, which includes both types of modes of fluctuations (long-wave and shortwave) in the supercritical region. A nonlinear equation relating the density and the chemical potential is obtained and investigated. The coordinates of the critical point are obtained, the equation of state and the isothermal compressibility in the supercritical region are calculated. The existence of maxima on compressibility isotherms as functions of density is established. The projection of a curve corresponding to these maxima is plotted on the pressure-temperature plane. Near the critical temperature, such a curve is identified as the Widom line. The latter is a boundary between the gaseous and liquid structures of the supercritical fluid.
Based on the model proposed in the thesis, the theoretical method is developed, which makes it possible in frames of the single approach to describe the phase behavior of simple one-component systems both in wide range of density and temperature, and in the critical region.
