Thermodynamic Equivalence of Certain Ideal Bose and Fermi Gases

TL;DR

This paper generalizes the thermodynamic equivalence between certain ideal Bose and Fermi gases, showing it applies broadly to systems with an energy-independent density of states and proposing a conjecture for discrete spectra.

Contribution

It extends known equivalences to a larger class of noninteracting quantum gases with specific density of states and introduces a conjecture for systems with discrete spectra.

Findings

Thermodynamic equivalence applies to systems with energy-independent density of states.

Equivalence observed in 2D and 1D systems is part of a broader class.

Conjecture: equivalence holds for discrete spectra when certain parameters are small.

Abstract

We show that the recently discovered thermodynamic equivalence between noninteracting Bose and Fermi gases in two dimensions, and between one-dimensional Bose and Fermi systems with linear dispersion, both in the grand-canonical ensemble, are special cases of a larger class of equivalences of noninteracting systems having an energy-independent single-particle density of states. We also conjecture that the same equivalence will hold in the grand-canonical ensemble for any noninteracting quantum gas with a discrete ladder-type spectrum whenever $\sigma \Delta / N k_{\rm B} T$ is small, where $N$ is the average particle number and $\sigma$ its standard deviation, $\Delta$ is the level spacing, $k_{\rm B}$ is Boltzmann's constant, and $T$ is the temperature.