  We outline the general features of the conventional mean-field theory for the
description of Bose-Einstein condensates at near zero temperatures. This
approach, based on a phenomenological model, appears to give excellent
agreement with experimental data. We argue, however, that such an approach is
not rigorous and cannot contain the full effect of collisional dynamics due to
the presence of the mean-field. We thus discuss an alternative microscopic
approach and explain, within our new formalism, the physical origin of these
effects. Furthermore, we discuss the potential formulation of a consistent
finite-temperature mean-field theory, which we claim necessiates an analysis
beyond the conventional treatment.

Statistical Mechanics