Journal
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
Volume 387, Issue 8-9, Pages 1999-2009Publisher
ELSEVIER
DOI: 10.1016/j.physa.2007.10.075
Keywords
chemotaxis; generalized thermodynamics; Nonlinear mean field Fokker-Planck equations; self-gravitating Brownian particles
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We point out a remarkable analogy between the limiting mass of relativistic white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial populations in a generalized Keller-Segel model of chemotaxis [P.H. Chavanis, C. Sire, Phys. Rev. E 69 (2004) 016116]. This model is based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations similar to gaseous polytropes in astrophysics. For the critical index n(3) = d/(d - 2) (where d >= 2 is the dimension of space), the theory of polytropes leads to a unique value of the mass M-c that we interpret as a limiting mass. In d = 3, we find Mc = 202.8956 . . . and in d = 2, we recover the well-known result M-c = 8 pi (in suitable units). For M < M-c, the system evaporates (in an infinite domain) or tends to an equilibrium state (for box-confined configurations). For M > M-c, the system collapses and forms a Dirac peak containing a mass M-c surrounded by a halo. This paper exposes the model and shows, by simple considerations, the origin of the critical mass. A detailed description of the critical dynamics of the generalized Keller-Segel model will be given in a forthcoming paper. (c) 2007 Elsevier B.V. All rights reserved.
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