In Jaouadi et al. [Phys. Rev. A 83, 023616 (2011)] the authors derive an analytical finite-size expansion for the Bose-Einstein condensation critical temperature of an ideal Bose gas in a generic power-law trap. In the case of a harmonic trap, this expansion adds higher-order terms to the well-known first-order correction. We point out a delicate point in connection to these results, showing that the claims of Jaouadi et al. should be treated with caution. In particular, for a harmonic trap, the given expansion yields results that, depending on what is considered to be the critical temperature of the finite system, do not generally improve on the established first-order correction. For some nonharmonic traps, the results differ at first order from other results in the literature.
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number
N of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature
Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of
x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of
Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with
104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.
The onset of Bose-Einstein condensation in a sample of N atoms confined to harmonic traps is critically examined. The monotonic growth of the condensate fraction f and the nonmonotic variation of the specific heat C of the sample are studied in fine detail. While no temperature is truly critical, our analysis does provide a strong theoretical basis for recognizing a temperature
T∗, that has been identified and advocated by previous authors, as the transition temperature of the system. The specific-heat maximum
Tm (which happens to be another important ``marker'' in this transition), the value of the specific heat at
T=Tm, and the ``jump'' in the specific heat of the system when the number N becomes very large are all explicitly evaluated. Wherever comparison is possible, our analytical results are found to be in perfect agreement with the ones obtained numerically by Kirsten and Toms [Phys. Rev. A 54, 4188 (1996)] and by Haugerud, Haugest, and Ravndal [Phys. Lett. A 225, 18 (1997)].