  A stochastic Gross-Pitaevskii equation is derived for partially condensed
Bose gas systems subject to binary contact interactions. The theory we present
provides a classical-field theory suitable for describing dissipative dynamics
and phase transitions of spinor and multi-component Bose gas systems comprised
of an arbitrary number of distinct interacting Bose fields. A new class of
dissipative processes involving distinguishable particle interchange between
coherent and incoherent regions of phase-space is identified. The formalism and
its implications are illustrated for two-component mixtures and spin-1
Bose-Einstein condensates. For systems comprised of atoms of equal mass, with
thermal reservoirs that are close to equilibrium, the dissipation rates of the
theory are reduced to analytical expressions that may be readily evaluated. The
unified treatment of binary contact interactions presented here provides a
theory with broad relevance for quasi-equilibrium and far-from-equilibrium
Bose-Einstein condensates.
