Thermalization in Quantum Fluids of Light: A Convection-Diffusion Equation

We develop a microscopic theory for the dynamics of quantum fluids of light, deriving an effective kinetic equation in momentum space that takes the form of the convection-diffusion equation. In the particular case of two-dimensional systems with parabolic dispersion, it reduces to the Bateman–Burgers equation. The hydrodynamic analogy unifies nonlinear wave phenomena, such as shock wave formation and turbulence, with non-equilibrium Bose–Einstein condensation of photons and polaritons in optical cavities. We introduce the Reynolds number $(\mathit{\text{Re}})$ and demonstrate that the condensation threshold corresponds exactly to a critical Reynolds number of unity $(\mathit{\text{Re}}=1)$, beyond which $(\mathit{\text{Re}}>1)$ a shock-like front emerges in the momentum space, characterized by the Bose–Einstein distribution for the particle density in states with high momentum.