Level density of a Fermi gas and integer partitions: a Gumbel-like finite-size correction

TL;DR

This paper studies the finite-size effects on the level density of a Fermi gas, revealing Gumbel-like corrections at higher temperatures and connecting these findings to number theory and extreme value statistics.

Contribution

It introduces a finite-size correction to the Fermi gas level density, linking statistical physics with number theory and extreme value theory.

Findings

Finite-size corrections lead to Gumbel-like contributions.

Connections established between Fermi gas level density and number partition problems.

Differences highlighted between Fermi and Bose gases at finite sizes.

Abstract

We investigate the many-body level density of gas of non-interacting fermions. We determine its behavior as a function of the temperature and the number of particles. As the temperature increases, and beyond the usual Sommerfeld expansion that describes the degenerate gas behavior, corrections due to a finite number of particles lead to Gumbel-like contributions. We discuss connections with the partition problem in number theory, extreme value statistics as well as differences with respect to the Bose gas.