Bose-Einstein condensation of a finite number of particles confined to harmonic traps

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 T-m (which happens to be another important "marker'' in this transition), the value of the specific heat at T = T-m, 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)]. [S1050-2947(98)10008-2].