Eur. Phys. J. B 57, 391-409 (2007)
https://doi.org/10.1140/epjb/e2007-00187-2

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

Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Corresponding author: ^a chavanis@irsamc.ups-tlse.fr

Received: 6 December 2006
Revised: 20 March 2007
Published online: 5 July 2007

Abstract

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.