  We study the mean-field theory, and the properties of fluctuations, in an out
of equilibrium Bose-Fermi system, across the transition to a quantum condensed
phase. The system is driven out of equilibrium by coupling to multiple baths,
which are not in equilibrium with each other, and thus drive a flux of
particles through the system. We derive the self-consistency condition for an
uniform condensed steady state. This condition can be compared both to the
laser rate equation and to the Gross-Pitaevskii equation of an equilibrium
condensate. We study fluctuations about the steady state, and discuss how the
multiple baths interact to set the system's distribution function. In the
condensed system, there is a soft phase (Bogoliubov, Goldstone) mode, diffusive
at small momenta due to the presence of pump and decay, and we discuss how one
may determine the field-field correlation functions properly including such
soft phase modes. In the infinite system, the correlation functions differ both
from the laser and from an equilibrium condensate; we discuss how in a finite
system, the laser limit may be recovered.
