Finite Geometries and Ideal Bose Gases

It is shown that the lack of condensation in partially finite geometries is not dependent on the use of periodic boundary conditions or the symmetry-breaking technique for a class of independent-particle Bose-gas models. Condensation in an ideal Bose gas in a finite geometry is studied for one-, two-, and three-dimensional systems. Explicit expressions for the chemical potential in terms of the number of particles, the temperature, and the dimensions of the system are obtained by using a contour-integration technique to express sums as integrals plus corrections. Even though there is no true condensation, the range of the reduced density matrix for an ideal Bose gas in a thick slab ($L_1\times L_2\times L_3$ with $L_1$, $L_2\to \infty$ and $\infty >L_3\gg {\lambda }_T=\mathrm{the}\mathrm{thermal}\mathrm{wavelength}$) is found to go from order $\frac{{\lambda }_T{T}_{\mathrm{cB}}}{(T-T_{\mathrm{cB}})}$ to order $L_3\exp \left\{\frac{1}{2}\sqrt{\pi }{\zeta }_{\frac{3}{2}}\right(\frac{L_3}{{\lambda }_T}\left)\right[{\left(\frac{T_{\mathrm{cB}}}{T}\right)}^{\frac{3}{2}}-1\left]\right\}$ as the temperature is lowered through the bulk critical temperature $T_{\mathrm{cB}}·({\zeta }_{\chi } \mathrm{is}\mathrm{the}\mathrm{Riemann} \zeta \mathrm{function}.)$ The range exhibits a spatial asymmetry. The onset of this extremely long-range order is found to be reflected in PVT diagrams and the specific heat in a manner very similar to that in bulk condensation. It is conjectured that in calculations for infinite systems the existence of a long- but finite-range order may be a more relevant criterion for superfluidity and superconductivity than the usual criterion of an infinitely long-range order.