Unified statistical thermodynamics of quantum gases trapped under generic power law potential in $d$ dimension and equivalence in $d=1$

TL;DR

This paper develops a unified thermodynamic framework for quantum gases in arbitrary power law traps across dimensions, revealing an equivalence in one dimension and discussing the impact of external potentials on known free gas relations.

Contribution

It introduces a unified description of trapped Bose and Fermi gases in arbitrary dimensions and explores their thermodynamic equivalences, especially in one dimension.

Findings

Unified thermodynamic description for quantum gases in power law traps.

Established equivalence between Bose and Fermi gases in 1D.

External potentials disrupt known free gas equivalences in 2D.

Abstract

A unified description for the Bose and Fermi gases trapped in an external generic power law potential $U=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}$ is presented using the grandpotential of the system in $d$ dimensional space. The thermodynamic quantities of the quantum gases are derived from the grand potential. An equivalence between the trapped Bose and Fermi gases is constructed in one dimension ($d=1$) using the Landen relation. It is also found that the established equivalence between the ideal free Bose and Fermi gases in $d=2$ (M. H. Lee, Phys. Rev. E 55, 1518 (1997)) is lost when external potential is applied.