Real-space condensation in a dilute Bose gas at low temperature

We show with a direct numerical analysis that a dilute Bose gas in an external potential-which is chosen for simplicity as a radial parabolic well-undergoes at a certain temperature T-c a phase transition to a state supporting a macroscopic fraction of particles at the origin of the phase space (r=0, p=0). Quantization of particle motion in a well wipes out the sharp transition but supports a distribution of a radial particle density rho (r) peaked at r=0 (a real-space condensate) as well as a phase-space Wigner distribution density W(r,p) peaked at r=0 and p=0 below a crossover temperature T-c(*) of order of T-c. A fixed-particle-number canonical ensemble, which is a combination of the fixed-N condensate part and the fixed-mu excitation part, is suggested to resolve the difficulty of large fluctuation of the particle number (deltaN similar toN) in the Bose-Einstein condensation problem treated within the orthodox grand canonical ensemble formalism. (C) 2001 American Institute of Physics.