A heuristic wave equation parameterizing BEC dark matter halos with a quantum core and an isothermal atmosphere

The Gross-Pitaevskii-Poisson equations that govern the evolution of self-gravitating Bose-Einstein condensates, possibly representing dark matter halos, experience a process of gravitational cooling and violent relaxation. We propose a heuristic parametrization of this complicated process in the spirit of Lynden-Bell's theory of violent relaxation for collisionless stellar systems. We derive a generalized wave equation (that was introduced phenomenologically in [P.H. Chavanis, Eur. Phys. J. Plus {\bf 132}, 248 (2017)]) involving a logarithmic nonlinearity associated with an effective temperature $T_{\rm eff}$ and a damping term associated with a friction $\xi$. These terms can be obtained from a maximum entropy production principle and are linked by a form of Einstein relation expressing the fluctuation-dissipation theorem. The wave equation satisfies an $H$-theorem for the Lynden-Bell entropy and relaxes towards a stable equilibrium state which is a maximum of entropy at fixed mass and energy. This equilibrium state represents the most probable state of a Bose-Einstein condensate dark matter halo. It generically has a core-halo structure. The quantum core prevents gravitational collapse and may solve the core-cusp problem. The isothermal halo leads to flat rotation curves in agreement with the observations. These results are consistent with the phenomenology of dark matter halos. Furthermore, as shown in a previous paper [P.H. Chavanis, Phys. Rev. D {\bf 100}, 123506 (2019)], the maximization of entropy with respect to the core mass at fixed total mass and total energy determines a core mass--halo mass relation which agrees with the relation obtained in direct numerical simulations. We stress the importance of using a microcanonical description instead of a canonical one. We also explain how our formalism can be applied to the case of fermionic dark matter halos.