Statistics of non-interacting bosons and fermions in micro-canonical, canonical and grand-canonical ensembles: A survey

TL;DR

This survey reviews the statistical properties of non-interacting bosons and fermions across micro-canonical, canonical, and grand-canonical ensembles, emphasizing formulas for finite systems and their thermodynamic limits.

Contribution

It compiles and derives formulas for non-interacting quantum gases in various ensembles, including new formulas for micro-canonical systems and their relation to other ensembles.

Findings

Formulas for micro-canonical ensemble are reported for the first time.

Differences between ensembles diminish in the thermodynamic limit.

Formulas for canonical ensemble derived from micro-canonical expressions.

Abstract

The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, particularly by Khinchin in 1950. Others were obtained only recently, often motivated by experimental results relating to atomic confinement. A number of formulas are reported for the first time in the present paper. Formulas applicable to the case where the system may exchange energy but not particles with a reservoir (canonical ensemble) are derived from the micro-canonical ensemble expressions. The differences between the three ensembles tend to vanish in the so-called Thermodynamics limit, that is, when the number of particles and the volume go to infinity while the particle number density remains constant. But we are mostly interested in systems of moderate size, often referred to as being mesoscopic, where the grand-canonical formalism is not applicable. The mathematical results rest primarily on the enumeration of partitions of numbers.