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
NN0 as well as the two-point correlation function $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for cubes of arbitrary sizes. Numerical results are given for two specific sizes,
lL=40and100, where
L is the edge length of the enclosure and
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.
We investigate systems of interacting bosonic particles confined within slablike boxes of size ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ with $Z\ensuremath{\ll}L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature
Tc, and below
Tc where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at
TBKT<Tc depending on the thickness
Z). The low-temperature phase below
TBKT shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\ensuremath{\gtrsim}{T}_{c}$ to a quasi-2D critical behavior for $T\ensuremath{\lesssim}{T}_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ lattices with $Z\ensuremath{\ll}L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.