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

Abstract

We consider Bose–Einstein condensation for non-interacting particles trapped in a harmonic potential by considering the length of permutation cycles arising from wave function symmetry. This approach had been considered previously 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 treat the problem nearly exactly by analytical means. One clearly sees that the noncondensate is made up of permutation loops that are of length ⩽ N 1/3, and that the phase transition consists of the sudden growth of longer permutation cycles. The condensate is seen to consist of cycles of all possible lengths with nearly equal likelihood.