Thermodynamics of self-gravitating systems.
We study the thermodynamics and the collapse of a self-gravitating gas of Brownian particles. We consider a high-friction limit in order to simplify the problem. This results in the Smoluchowski-Poisson system. Below a critical energy or below a critical temperature, there is no equilibrium state and the system develops a self-similar collapse leading to a finite time singularity. In the microcanonical ensemble, this corresponds to a "gravothermal catastrophe" and in the canonical ensemble to an "isothermal collapse." Self-similar solutions are investigated analytically and numerically.