Exact analytical solution of the collapse of self-gravitating Brownian particles and bacterial populations at zero temperature

Abstract

We provide an exact analytical solution of the collapse dynamics of self-gravitating Brownian particles and bacterial populations at zero temperature. These systems are described by the Smoluchowski-Poisson system or Keller-Segel model in which the diffusion term is neglected. As a result, the dynamics is purely deterministic. A cold system undergoes a gravitational collapse leading to a finite time singularity: the central density increases and becomes infinite in a finite time t_coll. The evolution continues in the post collapse regime. A Dirac peak emerges, grows and finally captures all the mass in a finite time t_end, while the central density excluding the Dirac peak progressively decreases. Close to the collapse time, the pre and post collapse evolution is self-similar. Interestingly, if one starts from a parabolic density profile, one obtains an exact analytical solution that describes the whole collapse dynamics, from the initial time to the end, and accounts for non self-similar corrections that were neglected in previous works. Our results have possible application in different areas including astrophysics, chemotaxis, colloids and nanoscience.