  We develop a theory for the dynamics of the density matrix describing a
multimode polariton condensate. In such a condensate several single-particle
orbitals become highly occupied, due to stimulated scattering from reservoirs
of high-energy excitons. A generic few-parameter model for the system leads to
a Lindblad equation which includes saturable pumping, decay, and condensate
interactions. We show how this theory can be used to obtain the population
distributions, and the time-dependent first- and second-order coherence
functions, in such a multimode condensate. As a specific application, we
consider a polaritonic Josephson junction, formed from a double-well potential.
We obtain the population distributions, emission line shapes, and widths
(first-order coherence functions), and predict the dephasing time of the
Josephson oscillations.
