Exact canonical occupation numbers in a Fermi gas with finite level spacing and a q-analog of Fermi-Dirac distribution

TL;DR

This paper derives exact formulas for occupation numbers in a finite-level Fermi gas, revealing deviations from standard Fermi-Dirac statistics and introducing a q-analog distribution related to special functions.

Contribution

It provides a new recurrence relation and explicit formulas for occupation numbers in a finite-level Fermi gas using generating functions and q-polynomials.

Findings

Exact occupation numbers expressed in terms of q-polynomials.

Deviations from Fermi-Dirac distribution linked to chemical potential gaps.

Close-form expressions involving Rogers-Ramanujan functions.

Abstract

We consider equilibrium level occupation numbers in a Fermi gas with a fixed number of particles, n, and finite level spacing. Using the method of generating functions and the cumulant expansion we derive a recurrence relation for canonical partition function and an explicit formula for occupation numbers in terms of single-particle partition function at n different temperatures. We apply this result to a model with equidistant non-degenerate spectrum and obtain close-form expressions in terms of q-polynomials and Rogers-Ramanujan partial theta function. Deviations from the standard Fermi-Dirac distribution can be interpreted in terms of a gap in the chemical potential between the particle and the hole excitations with additional correlations at temperatures comparable to the level spacing.