We calculate certain features of Bose–Einstein condensation in the ideal gas by using recurrence relations for the partition function. The grand canonical ensemble gives inaccurate results for certain properties of the condensate that are accurately provided by the canonical ensemble. Calculations in the latter can be made tractable for finite systems by means of the recurrence relations. The ideal one-dimensional harmonic Bose gas provides a particularly simple and pedagogically useful model for which detailed results are easily derived. An analysis of the Bose system via permutation cycles yields insight into the physical meaning of the recurrence relations.
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