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.

References in corpus (6)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations
• Dynamics of colloidal particles with capillary interactions
• Brownian particles with long and short range interactions
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot

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• Jeans instability and turbulent gravitational collapse of Bose-Einstein Condensate dark matter halos
• A heuristic wave equation parameterizing BEC dark matter halos with a quantum core and an isothermal atmosphere
• Caloric curves of classical self-gravitating systems in general relativity
• On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis
• Logarithmic scaling of the collapse in the critical Keller-Segel equation
• On blowup dynamics in the Keller-Segel model of chemotaxis