Permutation cycles in the Bose-Einstein condensation of a trapped ideal gas

We consider Bose-Einstein condensation for non-interacting particles trappe d in a harmonic potential by considering the length of permutation cycles a rising from wave function symmetry. This approach had been considered previ ously by Matsubara and Feynman for a homogeneous gas in a box with periodic boundary conditions. For the ideal gas in a harmonic potential, one can tr eat the problem nearly exactly by analytical means. One clearly sees that t he noncondensate is made up of permutation loops that are of length less th an or equal to N-1/3, and that the phase transition consists of the sudden growth of longer permutation cycles. The condensate is seen to consist of c ycles of all possible lengths with nearly equal likelihood. (C) 2000 Elsevi er Science B.V. All rights reserved.