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Density shift of Bose gas due to the Casimir effect and mean field potential

The shift of density of Bose gas due to the mean field potential (MFP) and the Casimir effect is systematically investigated in the $d$-dimensional configuration space from the point of thermodynamic consideration. We show that, for $d=3$, the shift of density arises completely due to the Casimir effect and, the MFP remains totally ineffective regardless of the state, condensate or non-condensate. But for dimension $d>3$ the MFP plays an active role in shifting the density of Bose gas along with the Casimir interaction. The sign of density shift becomes positive in the present case. So, the corresponding critical temperature shift would be negative, because these two shifts are related as $Δn_{c}$/$n_{c}$ $\approx$ -$ΔT_{c}$/$T_{c}$. It is important to note that, the MFP causes a shift of density for $d>3$ even when the Casimir effect is not there, and the sign of shift becomes negative then; consequently, the critical temperature shift would be positive with MFP alone in this particular situation.