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From the unified statistical thermodynamics of quantum gases, the virialcoefficients of ideal Bose and Fermi gases which are trapped under genericpower law potential are derived systematically. From the general result ofvirial coefficients, one can produce the known results in $d=3$ and $d=2$. Butmore importantly we found that, the virial coefficients of bosons and fermionsbecome equal (except the the second virial coefficient, where the sign isdifferent) when we trap the gases under harmonic potential in $d=1$. Thisresult suggests the equivalence between Bose and Fermi gases which is alreadyestablished for $d=1$ by M M Faruk (J Stat Phys, DOI10.1007/s10955-015-1344-4). Surprisingly our investigation also shows that thevirial coefficients of two dimensional free quantum gases are identical to thevirial coefficients of one dimensional harmonically trapped quantum gases.

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