Critical properties of a trapped interacting Bose gas

We develop a practical theoretical formalism for studying the critical properties of a trapped Bose-Einstein condensate using the projected Gross-Pitaevskii equation. We show that this approach allows us to investigate the behavior of the correlation length, condensate mode, and its number fluctuations about the critical point. Motivated by recent experiments [Donner , Science 315, 1556 (2007)], we calculate the critical exponent for the correlation length of the trapped system. We observe finite-size effects in our results when the correlation length becomes comparable to the Ginzburg length. We extend the Binder cumulant to the trapped system and discuss an experimental method for measuring number fluctuations.