Virial coefficients from unified statistical thermodynamics of quantum gases trapped under generic power law potential in $d$ dimension and the equivalence of trapped quantum gases

TL;DR

This paper derives virial coefficients for quantum gases in various dimensions and potentials, revealing an equivalence between Bose and Fermi gases under harmonic trapping in one dimension and linking 2D free gases to 1D trapped gases.

Contribution

It systematically derives virial coefficients for trapped quantum gases and uncovers novel equivalences between Bose and Fermi gases in specific trapping conditions.

Findings

Virial coefficients of bosons and fermions become equal in 1D harmonic traps (except second coefficient)

Virial coefficients of 2D free gases match those of 1D harmonically trapped gases

Established the equivalence of Bose and Fermi gases in certain trapping scenarios

Abstract

From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases which are trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in $d=3$ and $d=2$. But more importantly we found that, the virial coefficients of bosons and fermions become equal (except the the second virial coefficient, where the sign is different) when we trap the gases under harmonic potential in $d=1$. This result suggests the equivalence between Bose and Fermi gases which is already established for $d=1$ by M M Faruk (J Stat Phys, DOI 10.1007/s10955-015-1344-4). Surprisingly our investigation also shows that the virial coefficients of two dimensional free quantum gases are identical to the virial coefficients of one dimensional harmonically trapped quantum gases.