The loop-gas approach to Bose-Einstein condensation for trapped particles

We examine Bose-Einstein condensation (BEC) for particles trapped in a harm onic potential by considering it as a transition in the length of permutati on cycles that arise from wave-function symmetry. This "loop-gas" approach was originally developed by Feynman in his path-integral study of BEC for a homogeneous gas in a box. For the harmonic oscillator potential it is poss ible to treat the ideal gas exactly so that one can easily see how standard approximations become more accurate in the thermodynamic limit (TDL). One clearly sees that the condensate is made up of very long permutation loops whose length fluctuates ever more widely as the number of particles increas es. In the TDL, the Wentzel-Kramers-Brillouin approximation, equivalent to the standard approach to BEG, becomes precise for the noncondensate; howeve r, this approximation neglects completely the long cycles that make up the condensate. We examine the exact form for the density matrix for the system and show how it describes the condensate and behaves in the TDL. (C) 2000 American Association of Physics teachers.