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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.