Partial equivalence of statistical ensembles and kinetic energy

TL;DR

This paper investigates how kinetic energy influences the partial equivalence of statistical ensembles, showing that it can remove partial equivalence and induce phase transitions in the microcanonical ensemble, with solutions for the mean-field spherical model.

Contribution

It demonstrates that kinetic energy significantly affects ensemble equivalence and phase transition behavior, providing a microcanonical solution for the mean-field spherical model with kinetic energy.

Findings

Kinetic energy removes partial ensemble equivalence.

Nonanalytic points indicate phase transitions in the canonical ensemble.

Microcanonical solutions are provided for finite and infinite systems.

Abstract

The phenomenon of partial equivalence of statistical ensembles is illustrated by discussing two examples, the mean-field XY and the mean-field spherical model. The configurational parts of these systems exhibit partial equivalence of the microcanonical and the canonical ensemble. Furthermore, the configurational microcanonical entropy is a smooth function, whereas a nonanalytic point of the configurational free energy indicates the presence of a phase transition in the canonical ensemble. In the presence of a standard kinetic energy contribution, partial equivalence is removed and a nonanalyticity arises also microcanonically. Hence in contrast to the common belief, kinetic energy, even though a quadratic form in the momenta, has a non-trivial effect on the thermodynamic behaviour. As a by-product we present the microcanonical solution of the mean-field spherical model with kinetic energy for finite and infinite system sizes.