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 (19)

• Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Thermodynamics of self-gravitating systems
• Phase transitions in self-gravitating systems. Self-gravitating fermions and hard spheres models
• Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions
• Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions
• Thermodynamics and Self-Gravitating Systems
• Chapman-Enskog derivation of the generalized Smoluchowski equation
• Post-collapse dynamics of self-gravitating Brownian particles in D dimensions
• Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain
• 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
• Self-gravitating Brownian systems and bacterial populations with two or more types of particles
• Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations
• 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 (16)

• Dynamics of a microorganism moving by chemotaxis in its own secretion
• Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles
• Inhomogeneous Tsallis distributions in the HMF model
• Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
• Critical chemotactic collapse
• Self-gravitating Brownian particles in two dimensions: the case of N=2 particles
• Phase transitions in self-gravitating systems and bacterial populations with a screened attractive potential
• On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis
• Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile
• Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion
• Logarithmic scaling of the collapse in the critical Keller-Segel equation
• Supernova implosion-explosion in the light of catastrophe theory
• Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data
• Blow-up dynamics of self-attracting diffusive particles driven by competing convexities
• On blowup dynamics in the Keller-Segel model of chemotaxis
• Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models