The problem of phase transitions in statistical mechanics

The first part of this review deals with the one-phase approach to the stat istical theory of phase transitions. This approach is based on the assumpti on that a phase transition of the first kind is due to the loss of stabilit y by the host phase. We demonstrate that it is practically impossible to fi nd the coordinates of the points of phase transition using this criterion i n the framework of the global Gibbs theory which describes the state of the entire macroscopic system. On the basis of Ornstein-Zernicke equation we f ormulate the local approach that analyzes the state of matter inside the co rrelation sphere of radius R-c approximate to 10 Angstrom. This approach is proved to be as rigorous as the Gibbs theory. In the context of the local approach we formulate the criterion that allows finding the points of phase transition without calculating the chemical potential and the pressure of the second concurrent phase. In the second part of the review we consider p hase transitions of the second kind (critical phenomena). Based on the glob al Gibbs approach, the Kadanov-Wilson theory of critical phenomena is analy zed. Again we use the Ornstein - Zernicke equation to formulate the local t heory of critical phenomena. With regard to experimentally observable quant ities this theory yields precisely the same results as the Kadanov-Wilson t heory; secondly, the local approach allows predicting many previously unkno wn details of critical phenomena, and thirdly, the local approach paves the way towards constructing a unified theory of liquids that will describe th e behavior of matter not only in the regular part of the phase diagram, but also at the critical point and in its neighborhood.