Critical dynamics of self-gravitating Langevin particles and bacterial populations

Abstract

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 [Chavanis & Sire, PRE, 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 $n_3=d/(d-2)$ (where $d\ge 2$ is the dimension of space), there exists a critical temperature ${\Theta }_c$ (for a given mass) or a critical mass $M_c$ (for a given temperature). For $\Theta >{\Theta }_c$ or $M<M_c$ the system tends to an incomplete polytrope confined by the box (in a bounded domain) or evaporates (in an unbounded domain). For $\Theta <{\Theta }_c$ or $M>M_c$ the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction $M_c$ of the total mass surrounded by a halo. This study extends the critical dynamics of the ordinary Smoluchowski-Poisson system and Keller-Segel model in $d=2$ corresponding to isothermal configurations with $n_3\to +\infty$. 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.

References in corpus (8)

• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Logotropic distributions
• White dwarf stars in D dimensions
• Dynamical stability of collisionless stellar systems and barotropic stars: the nonlinear Antonov first law
• Kinetic and hydrodynamic models of chemotactic aggregation
• Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot
• Critical mass of bacterial populations in a generalized Keller-Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars

Cited by in corpus (1)

• Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models