  We derive the exact expression of the diffusion coefficient of a
self-gravitating Brownian gas in two dimensions. Our formula generalizes the
usual Einstein relation for a free Brownian motion to the context of
two-dimensional gravity. We show the existence of a critical temperature T_{c}
at which the diffusion coefficient vanishes. For T<T_{c} 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 T_{c} 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.
