Critical mass of bacterial populations in a generalized Keller–Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars

Abstract

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 n3=d/(d−2) (where d≥2 is the dimension of space), the theory of polytropes leads to a unique value of the mass Mc 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 Mc=8π (in suitable units). For M<Mc, the system evaporates (in an infinite domain) or tends to an equilibrium state (for box-confined configurations). For M>Mc, the system collapses and forms a Dirac peak containing a mass Mc 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.