Thermodynamic Equivalence of Certain Ideal Bose and Fermi Gases

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.