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Critical dynamics of self-gravitating Langevin particles and bacterial populations

Clément Sire and Pierre-Henri Chavanis

par Clément Sire - 24 avril 2008

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.