  We show how to adapt the ideas of local energy and momentum conservation in
order to derive modifications to the Gross-Pitaevskii equation which can be
used phenomenologically to describe irreversible effects in a Bose-Einstein
condensate. Our approach involves the derivation of a simplified quantum
kinetic theory, in which all processes are treated locally. It is shown that
this kinetic theory can then be transformed into a number of phase-space
representations, of which the Wigner function description, although
approximate, is shown to be the most advantageous. In this description, the
quantum kinetic master equation takes the form of a GPE with noise and damping
added according to a well-defined prescription--an equation we call the
stochastic GPE. From this, a very simplified description we call the
phenomenological growth equation can be derived. We use this equation to study
i) the nucleation and growth of vortex lattices, and ii) nonlinear losses in a
hydrogen condensate, which it is shown can lead to a curious instability
phenomenon.
