Condensate fluctuations in finite Bose-Einstein condensates at finite temperature

A Langevin equation for the complex amplitude of a single-mode Bose-Einstein condensate is derived. The equation is first formulated phenomenologically, defining three transport parameters. It is then also derived microscopically. Expressions for the transport parameters in the form of Green-Kubo formulas are thereby derived and evaluated for simple trap geometries, a cubic box with cyclic boundary conditions and an isotropic parabolic trap. The number fluctuations in the condensate, their correlation time, and the temperature-dependent collapse-time of the order parameter as well as its phase-diffusion coefficient are calculated.