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

TL;DR

This paper derives the exact diffusion coefficient for a self-gravitating Brownian gas in two dimensions, revealing a critical temperature where diffusion ceases and gravitational collapse occurs, with implications across physics and biology.

Contribution

It provides the first exact expression for the diffusion coefficient in 2D self-gravitating Brownian systems, including the critical temperature and its relation to collapse phenomena.

Findings

Diffusion coefficient vanishes at critical temperature T_c

For T<T_c, the system undergoes gravitational collapse

Explicit solution for N=2 confirms the theoretical predictions

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.