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

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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.

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      Journal
      2010-09-15
      2011-07-25
      1009.2884
      10.1103/PhysRevE.83.031131

      http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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      Phys. Rev. E 83, 031131 (2011)
      cond-mat.stat-mech q-bio.QM

      Condensed matter, Quantitative & Systems biology

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