Phase transitions and configuration space topology

Michael Kastner
Rev. Mod. Phys. 80, 167 – Published 2 January 2008

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, or the absence, of a phase transition, and an investigation of these properties may lead to a deeper understanding of the physical phenomenon. One possible way to approach this problem, reviewed and discussed in the present paper, is the study of topology changes in configuration space which 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. 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 proven 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 nonconfining 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.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure

    DOI:https://doi.org/10.1103/RevModPhys.80.167

    ©2008 American Physical Society

    Authors & Affiliations

    Michael Kastner*

    • Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

    • *michael.kastner@uni-bayreuth.de

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 80, Iss. 1 — January - March 2008

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required


    ×
    ×

    Images

    ×

    Sign up to receive regular email alerts from Reviews of Modern Physics

    Log In

    Cancel
    ×

    Search


    Article Lookup

    Paste a citation or DOI

    Enter a citation
    ×