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

Abstract

We show with a direct numerical analysis that a dilute Bose gas in an external potential - which is choosen for simplicity as a radial parabolic well - undergoes at certain temperature Tc a phase transition to a state supporting macroscopic fraction of particles at the origin of the phase space (r=0,p=0). Quantization of particle motion in a well wipes out sharp transition but supports a distribution of radial particle density ρ(r) peacked at r=0 (a real-space condensate) as well as the phase-space Wigner distribution density W(r, p) peaked at r=0 and p=0 below the crossover temperature Tc* of order of Tc. Fixed-particle-number canonical ensemble which is a combination of the fixed-μ condensate part and the fixed-m excitation part is suggested to resolve the difficulty of large fluctuation of the particle number (δN~N) in the Bose-Einstein condensation problem treated within the orthodox grand canonical ensemble formalism.