Phase transitions and configuration space topology

Abstract

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 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 paper, 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 M_v, 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 M_v 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 M_v at v=v_t was proved to be necessary for a phase transition to take place at a potential energy v_t. 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 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.

22 pages, 6 figures

References in corpus (10)

• An Introduction to the Thermodynamic and Macrostate Levels of Nonequivalent Ensembles
• Phase transitions and topology changes in configuration space
• Partial equivalence of statistical ensembles and kinetic energy
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Application of large deviation theory to the mean-field phi^4-model
• Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials
• On the mean-field spherical model
• Topology, phase transitions and the spherical model
• Topological approach to phase transitions and inequivalence of statistical ensembles
• Topological conditions for discrete symmetry breaking and phase transitions

Cited by in corpus (3)

• Phase transitions induced by saddle points of vanishing curvature
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• On the Equilibrium Fluctuations of an Isolated System