Low-temperature behavior of Bose systems confined to restricted geometries: Growth of the condensate fraction and spatial correlations

Through an extensive use of the Poisson summation formula, we have elucidated the description of the phenomenon of Bose-Einstein condensation in finite systems in terms of a "collapse in the thermogeometric space." In particular, we have carried out an explicit evaluation of the temperature dependence of the thermogeometric parameter $y$ for a cubical enclosure under periodic boundary conditions, which in turn enables us to evaluate the temperature dependence of the condensate fraction $\frac{N_0}{N}$ as well as the two-point correlation function $\rho (\overrightarrow{\mathrm{r}},{\overrightarrow{\mathrm{r}}}^{\prime })$ for cubes of arbitrary sizes. Numerical results are given for two specific sizes, $\frac{L}{\overline{l}}=40\mathrm{} \mathrm{and} 100$, where $L$ is the edge length of the enclosure and $\overline{l}$ the mean interparticle distance in the system. In the appropriate limit, our results are in complete agreement with the Fisher-Barber scaling theory for finite-size effects. Application of the Poisson summation formula has also enabled us to extract information of direct physical interest about the growth of long-range order in general cuboidal geometries. The special case of the thin-film geometry has been studied in detail; the resulting formulas provide a considerable improvement over the ones obtained by previous workers.