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

2010/09/30 by Pierre-Henri Chavanis, Clément Sire
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Brownian dynamics #Brownian motion #Classical mechanics #Cosmology and Gravitation Theories #Diffusion #Dirac (video compression format) #Dynamics (music) #Exact solutions in general relativity #Gravitation #Gravitational collapse #Mathematical analysis #Mathematics #Physics #Poisson distribution #Quantum mechanics #Singularity #Statistical Mechanics and Entropy #Statistical physics #Time evolution #Zero (linguistics) #cond-mat.stat-mech #q-bio.QM

paper · pdf · doi:10.1103/physreve.83.031131

published as Phys. Rev. E 83, 031131 (2011)

openalex publication_date 2011/03/25 · arxiv created 2011/07/25 · arxiv updated 2013/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 tcoll. The evolution continues in the postcollapse regime. A Dirac peak emerges, grows, and finally captures all the mass in a finite time tend, 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.

Citations

• Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dimensional turbulence
• Statistical mechanics and dynamics of solvable models with long-range interactions
• The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids
• Model for chemotaxis
• Thermodynamics and collapse of self-gravitating Brownian particles inDdimensions