Topological approach to phase transitions and inequivalence of statistical ensembles

TL;DR

This paper explores how topological changes in the state space relate to phase transitions in systems where different statistical ensembles are not equivalent, using the spherical model as a case study.

Contribution

It demonstrates the connection between topology changes and phase transitions specifically in non-ensemble-equivalent systems, highlighting the microcanonical perspective.

Findings

Topology changes correlate with microcanonical phase transitions.

Ensemble inequivalence affects the topological interpretation.

Topological approach aligns with microcanonical quantities in the studied model.

Abstract

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.