Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions

Abstract. We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. We show the existence of a critical temperature Tc at which the diffusion coefficient vanishes. For T < Tc the diffusion coefficient is negative and the gas undergoes gravitational collapse. This leads to the formation of a Dirac peak concentrating the whole mass in a finite time. We also stress that the critical temperature Tc is different from the collapse temperature T∗ at which the partition function diverges. These quantities differ by a factor 1− 1/N where N is the number of particles in the system. We provide clear evidence of this difference by explicitly solving the case N = 2. We also mention the analogy with the chemotactic aggregation of bacteria in biology, the formation of “atoms” in a two-dimensional (2D) plasma and the formation of dipoles or supervortices in 2D point vortex dynamics.