The first part of this review deals with the single-phase approach to the statistical theory of phase transitions. This approach is based on the assumption that a first-order phase transition is due to the loss of stability of the parent phase. We demonstrate that it is practically impossible to find the coordinates of the transition points using this criterion in the framework of the global Gibbs theory which describes the state of the entire macroscopic system. On the basis of the Ornstein-Zernike equation we formulate a local approach that analyzes the state of matter inside the correlation sphere of radius R_c \approx 10 Å. This approach is proved to be as rigorous as the Gibbs theory. In the context of the local approach we formulate a criterion that allows finding the transition points without calculating the chemical potential and the pressure of the second conjugate phase. In the second part of the review we consider second-order phase transitions (critical phenomena). The Kadanoff-Wilson theory of critical phenomena is analyzed, based on the global Gibbs approach. Again we use the Ornstein-Zernike equation to formulate a local theory of critical phenomena. With regard to experimentally established quantities this theory yields precisely the same results as the Kadanoff-Wilson theory; secondly, the local approach allows the prediction of many previously unknown details of critical phenomena, and thirdly, the local approach paves the way for constructing a unified theory of liquids that will describe the behavior of matter not only in the regular domain of the phase diagram, but also at the critical point and in its vicinity.

PACS: 05.70.−a, 64.60.−i, 64.70.−p, 81.60.-s (all) DOI: 10.1070/PU1999v042n06ABEH000543 URL: https://ufn.ru/en/articles/1999/6/b/ 000081542900002 Citation: Martynov G A "The problem of phase transitions in statistical mechanics" Phys. Usp. 42 517–543 (1999) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Оригинал: Мартынов Г А «Проблемы фазовых переходов в статистической механике» УФН 169 595–624 (1999); DOI: 10.3367/UFNr.0169.199906b.0595