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Accueil du site > Publications > Publications 2004 > 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

Pierre-Henri Chavanis and Clément Sire

par Clément Sire - 11 avril 2006

We determine an asymptotic expression of the blow-up time t_{\rm coll} for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_{\rm 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.

Preprint cond-mat/0402227