  We study the kinetics of polariton condensation accounting for the
condensation process as well as the energy relaxation of condensed polaritons
due to their scattering with phonons and excitons. By assuming a Boltzmann
kinetic description of the scattering process, we show that intra-condensate
relaxation can be accounted for by an additional time-dependent term in the
Gross-Pitaevskii equation. As an example, we apply the formalism to the
experimental results recently obtained in polariton microwires [E. Wertz, et
al., Nature Phys. 6, 860 (2010)]. In the presence of a local non-resonant
optical pump, a dynamic balance between spatially dependent relaxation and
particle loss develops and excites a series of modes, roughly equally spaced in
energy. Upon comparison, excellent agreement is found with the experimental
data.
