We derive a theoretical model which describes Bose-Einstein condensation in an open driven-dissipative system. It includes external pumping of a thermal reservoir, finite life time of the condensed particles and energy relaxation. The coupling betwee
n the reservoir and the condensate is described with semi-classical Boltzmann rates. This results in a dissipative term in the Gross-Pitaevskii equation for the condensate, which is proportional to the energy of the elementary excitations of the system. We analyse the main properties of a condensate described by this hybrid Boltzmann Gross-Pitaevskii model, namely, dispersion of the elementary excitations, bogolon distribution function, first order coherence, dynamic and energetic stability, drag force created by a disorder potential. We find that the dispersion of the elementary excitations of a condensed state fulfils the Landau criterion of superfluidity. The condensate is dynamically and energetically stable as longs it moves at a velocity smaller than the speed of excitations. First order spatial coherence of the condensate is found to decay exponentially in 1D and with a power law in 2D, similarly with the case of conservative systems. The coherence lengths are found to be longer due to the finite life time of the condensate excitations. We compare these properties with the ones of a condensate described by the popular diffusive models in which the dissipative term is proportional to the local condensate density. In the latter, the dispersion of excitations is diffusive which as soon as the condensate is put into motion implies finite mechanical friction and can lead to an energetic instability.
