Topological approach to phase transitions and inequivalence of statistical ensembles

The relation between thermodynamic phase transitions in classical systems and topology changes in their state space is discussed for systems in which equivalence of statistical ensembles does not hold. As an example, the spherical model with mean field-type interactions is considered. Exact results for microcanonical and canonical quantities are compared with topological properties of a certain family of submanifolds of the state space. Due to the observed ensemble inequivalence, a close relation is expected to exist only between the topological approach and one of the statistical ensembles. It is found that the observed topology changes can be interpreted meaningfully when compared to microcanonical quantities. Phase transitions, like the boiling and evaporating of water at a certain temperature and pressure, are common phenomena both in everyday life and in almost any branch of physics. Loosely speaking, a phase transition brings about a sudden change of the macroscopic properties of a system while smoothly varying a parameter (the temperature or the pressure in the above example). For the description of equilibrium phase transitions within the framework of statistical mechanics, several so-called statistical ensembles or Gibbs ensembles, like the microcanonical or the canonical one, are at disposal, each corresponding to a different physical situation. For a large class of systems with sufficiently short ranged interactions, these different approaches lead to identical numerical values for the typical system observables of interest, after taking the thermodynamic limit of the number of particles in the system going to infinity [1]. In this situation one speaks of equivalence of ensembles. Then, instead of selecting the statistical ensemble according to the physical situation of interest, one can revert to the ensemble most convenient for the computation intended. For systems with long range interactions, however, equivalence of ensembles does not hold in general. Systems showing such an inequivalence of ensembles in the thermodynamic limit (among those gravitational systems and Bose-Einstein condensates) have attracted much research interest in the last years (see Ref. [2] e-mail: [email protected]