  Based on the Lindblad master equation approach we obtain a detailed
microscopic model of photons in a dye-filled cavity, which features
condensation of light. To this end we generalise a recent non-equilibrium
approach of Kirton and Keeling such that the dye-mediated contribution to the
photon-photon interaction in the light condensate is accessible due to an
interplay of coherent and dissipative dynamics. We describe the steady-state
properties of the system by analysing the resulting equations of motion of both
photonic and matter degrees of freedom. In particular, we discuss the existence
of two limiting cases for steady states: photon Bose-Einstein condensate and
laser-like. In the former case, we determine the corresponding dimensionless
photon-photon interaction strength by relying on realistic experimental data
and find a good agreement with previous theoretical estimates. Furthermore, we
investigate how the dimensionless interaction strength depends on the
respective system parameters.

Quantum Gases;Computational Physics;Quantum Physics