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

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

2004-02-08

arxiv.org/abs/cond-mat/0402227v1

Phys. Rev. E, 70, 026115 (2004)

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

10.1103/PhysRevE.70.026115

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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