Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations

Abstract

We determine an asymptotic expression of the blow-up time t_coll for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta_c is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the asymptotic expression of the relaxation time close but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.

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• Brownian theory of 2D turbulence and generalized thermodynamics
• Self-gravitating Brownian systems and bacterial populations with two or more types of particles
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
• 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
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• Caloric curves of self-gravitating fermions in general relativity
• 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
• A Field-Theoretic Model for Chemotaxis in Run and Tumble Particles