We address the thermodynamics ( equilibrium density profiles , phase diagram , instability analysis… ) and the collapse of a self-gravitating gas of Brownian particles in D dimensions , in both canonical and microcanonical ensembles .
In the canonical ensemble , we derive the analytic form of the density scaling profile which decays as f ( x ) \sim x ^ { - \alpha } , with \alpha = 2 .
In the microcanonical ensemble , we show that f decays as f ( x ) \sim x ^ { - \alpha _ { max } } , where \alpha _ { max } is a non-trivial 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 .