Dynamics of Fluctuating Bose-Einstein Condensates

We present a generalized Gross-Pitaevskii equation that describes also the dissipative dynamics of a trapped partially Bose condensed gas. It takes the form of a complex nonlinear Schr"odinger equation with noise. We consider an approximation to this Langevin field equation that preserves the correct equilibrium for both the condensed and the noncondensed parts of the gas. We then use this formalism to describe the reversible formation of a one-dimensional Bose condensate, and compare with recent experiments. In addition, we determine the frequencies and the damping of collective modes in this case.