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 firstname.lastname@example.org
Revised: 20 March 2007
Published online: 5 July 2007
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 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.
PACS: 05.45.-a – Nonlinear dynamics and chaos / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 05.20.-y – Classical statistical mechanics / 04.40.-b – Self-gravitating systems; continuous media and classical fields in curved spacetime
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2007