Thermodynamics and collapse of self-gravitating Brownian particles inDdimensions

We address the thermodynamics and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. We study the equilibrium density profile and phase diagram of isothermal spheres and, for 2<D<10, determine the onset of instability in the series of equilibria. We also study the dynamics of self-gravitating Brownian particles in a high friction limit leading to the Smoluchowski-Poisson system. Self-similar solutions describing the collapse are investigated analytically and numerically. In the canonical ensemble (fixed temperature), we derive the analytic form of the density scaling profile which decays as f(x) approximately x(-alpha), with alpha=2. In the microcanonical ensemble (fixed energy), we show that f decays as f(x) approximately x(-alpha(max)), where alpha(max) is a nontrivial exponent. We derive exact expansions for alpha(max) and f in the limit of large D. Finally, we solve the problem in D=2, which displays rather rich and peculiar features with, in particular, the formation of a Dirac peak in the density profile.