2008/04/30 by Clément Sire, Clement Sire, Pierre-Henri Chavanis · 1 citation
Biochemistry, Genetics and Molecular Biology · Environmental Science · Mathematics · Physics and Astronomy · #Classical mechanics #Critical mass (sociodynamics) #Domain (mathematical analysis) #Ecosystem dynamics and resilience #Langevin dynamics #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematical physics #Mathematics #Physics #Polytrope #Polytropic process #Statistical Mechanics and Entropy #Statistical physics #Tsallis statistics #cond-mat.stat-mech #q-bio.PE #q-bio.QM
paper · pdf · doi:10.1103/physreve.78.061111
published as Phys. Rev. E, 78, 061111 (2008)
openalex publication_date 2008/12/11 · arxiv created 2009/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the critical dynamics of the generalized Smoluchowski-Poisson system (for self-gravitating Langevin particles) or generalized Keller-Segel model (for the chemotaxis of bacterial populations). These models [P. H. Chavanis and C. Sire, Phys. Rev. E 69, 016116 (2004)] are based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations with index n similar to polytropic stars in astrophysics. At the critical index n3=d(d-2) (where d>or=2 is the dimension of space), there exists a critical temperature Thetac (for a given mass) or a critical mass Mc (for a given temperature). For Theta>Thetac or M<Mc the system tends to an incomplete polytrope confined by the box (in a bounded domain) or evaporates (in an unbounded domain). For Theta<Thetac or M>Mc the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction Mc of the total mass surrounded by a halo. We study these regimes numerically and, when possible, analytically by looking for self-similar or pseudo-self-similar solutions. This study extends the critical dynamics of the ordinary Smoluchowski-Poisson system and Keller-Segel model in d=2 corresponding to isothermal configurations with n3-->+infinity . We also stress the analogy between the limiting mass of white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial populations in the generalized Keller-Segel model of chemotaxis.