title: 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 postcollapse 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 postcollapse evolutions are 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.
authors.affiliation: Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université Paul Sabatier, F-31062 Toulouse, France.
authors.forename: Pierre-Henri
authors.initials: PH
authors.lastname: Chavanis
authors.forename: Clément
authors.initials: C
authors.lastname: Sire
mesh_terms: D000465:Algorithms; D001419:Bacteria / Q000378:metabolism; D018407:Bacterial Physiological Phenomena; D001703:Biophysics / Q000379:methods*; D003198:Computer Simulation; D004058:Diffusion; D015233:Models, Statistical; D009038:Motion; D010316:Particle Size; D016012:Poisson Distribution; D013269:Stochastic Processes; D013696:Temperature