Phase transitions and configuration space topology

Equilibrium phase transitions may be defined as nonanalytic points of thermodynamic functions, e. g., of the canonical free energy. Given a certain physical system, it is of interest to understand which properties of the system account for the presence, respectively the absence, of a phase transition, and an understanding of these properties may lead to a deeper understanding of the physical phenomenon. One possible approach of this issue, reviewed and discussed in the present article, is the study of topology changes in configuration space which, remarkably, are found to be related to equilibrium phase transitions in classical statistical mechanical systems. For the study of configuration space topology, one considers the subsets Mv, consisting of all points from configuration space with a potential energy per particle equal to or less than a given v. For finite systems, topology changes of Mv are intimately related to nonanalytic points of the microcanonical entropy (which, as a surprise to many, do exist). In the thermodynamic limit, a more complex relation between nonanalytic points of thermodynamic functions (i. e., phase transitions) and topology changes is observed. For some class of short-range systems, a topology change of the Mv at v = vt was proved to be necessary, but not sufficient, for a phase transition to take place at a potential energy vt. In contrast, phase transitions in systems with long-range interactions or in systems with non-confining potentials need not be accompanied by such a topology change. Instead, for such systems the nonanalytic point in a thermodynamic function is found to have some maximization procedure at its origin. These results may foster our insight into the mechanisms which lead to the occurrence of a phase transition, and thus may help to explore the origin of this physical phenomenon.