Momentum distribution functions in ensembles: the inequivalence of microcannonical and canonical ensembles in a finite ultracold system

TL;DR

This paper demonstrates that in finite ultracold systems at low energies, the microcanonical and canonical ensembles produce different momentum distribution functions, highlighting ensemble inequivalence outside the thermodynamic limit.

Contribution

It reveals the inequivalence of microcanonical and canonical ensembles in finite ultracold systems, providing explicit calculations of the microcanonical MDF for fermions and bosons.

Findings

Microcanonical MDF deviates from Fermi-Dirac/Bose-Einstein distributions.

Ensemble inequivalence is significant at low energies in finite systems.

Finite density of states causes differences in momentum distributions.

Abstract

It is demonstrated that in many thermodynamic textbooks the equivalence of the different ensembles is achieved in the thermodynamic limit. In this present work we remark the inequivalence of microcannonical and canonical ensembles in a finite ultracold system at low energies. We calculate the microcanonical momentum distribution function (MDF) in a system of identical fermions (bosons). We find that, the microcanonical MDF deviates from the canonical one, which is the Fermi-Dirac (Bose-Einstein) function, in a finite system at low energies where the single-particle density of states and its inverse are finite.