Perturbation Theory in a Microcanonical Ensemble

TL;DR

This paper develops perturbation theory within the microcanonical ensemble, demonstrating its effectiveness for classical systems and revealing differences from the canonical ensemble in quantum cases.

Contribution

It introduces a method to perform perturbation theory directly in the microcanonical ensemble, extending its applicability to non-interacting and interacting systems.

Findings

Microcanonical perturbation theory matches canonical results for classical systems.

Constructs crossover functions for specific heat using microcanonical ensemble.

Identifies differences between ensembles for quantum particles in a box.

Abstract

The microcanonical ensemble is a natural starting point of statistical mechanics. However, when it comes to perturbation theory in statistical mechanics, traditionally only the canonical and grand canonical ensembles have been used. In this article we show how the microcanonical ensemble can be directly used to carry out perturbation theory for both non-interacting and interacting systems. We obtain the first non-trivial order answers for the specific heat of anharmonic oscillators and for the virial expansion in real gases. They are in exact agreement with the results obtained from the canonical ensemble. In addition, we show how crossover functions for the specific heat of anharmonic oscillators can be constructed using a microcanonical ensemble and also how the subsequent terms of the virial expansion can be obtained. However, we find that if we consider quantum free particles in a one-dimensional box of extension L, then the two ensembles give strikingly different answers for the first correction to the specific heat in the high temperature limit.