Critical dynamics of self-gravitating Langevin particles and bacterial populations

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 [P. H. Chavanis and C. Sire, Phys. Rev. E 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>or=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 MM_{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. We study these regimes numerically and, when possible, analytically by looking for self-similar or pseudo-self-similar solutions. 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}-->+infinity . 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.