Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain

Abstract

We derive the Virial theorem appropriate to the generalized Smoluchowski-Poisson system describing self-gravitating Brownian particles and bacterial populations (chemotaxis). We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the Virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature T_c in dimension d=2 and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity or chemotactic attraction. We also study the collapse at T=T_c and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for d>2, we show that the evaporation of the system is essentially a pure diffusion slightly slowed-down by self-gravity. We also study the linear dynamical stability of stationary solutions of the generalized Smoluchowski-Poisson system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the dynamical stability of the system. Finally, we propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles and biological populations and recover known models in some particular limits.

References in corpus (10)

• On the lifetime of metastable states in self-gravitating systems
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• Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski
• Hamiltonian and Brownian systems with long-range interactions
• Chapman-Enskog derivation of the generalized Smoluchowski equation
• Post-collapse dynamics of self-gravitating Brownian particles in D dimensions
• Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions
• Generalized Fokker-Planck equations and effective thermodynamics
• 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

Cited by in corpus (30)

• Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• An approximate treatment of gravitational collapse
• Logotropic distributions
• White dwarf stars in D dimensions
• Kinetic and hydrodynamic models of chemotactic aggregation
• Dynamics of the Bose-Einstein condensation: analogy with the collapse dynamics of a classical self-gravitating Brownian gas
• The Brownian Mean Field model
• Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data
• Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions
• Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions
• Critical chemotactic collapse
• Jeans type instability for a chemotactic model of cellular aggregation
• Self-gravitating Brownian particles in two dimensions: the case of N=2 particles
• Exact analytical solution of the collapse of self-gravitating Brownian particles and bacterial populations at zero temperature
• Two-dimensional Brownian vortices
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot
• Caloric curves of classical self-gravitating systems in general relativity
• Gravitational phase transitions with an exclusion constraint in position space
• Virial theorem for rotating self-gravitating Brownian particles and two-dimensional point vortices
• Logarithmic scaling of the collapse in the critical Keller-Segel equation
• Stability of infinite time blow up for the Patlak Keller Segel system
• Critical mass of bacterial populations in a generalized Keller-Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars
• Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data
• Existence and stability of infinite time blow-up in the Keller-Segel system
• Anisotropic long-range interaction investigated with cold atoms
• Universal structure of two and three dimensional self-gravitating systems in the quasi-equilibrium state