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

TL;DR

This paper draws an analogy between the critical mass of bacterial populations in a generalized Keller-Segel model and the Chandrasekhar limit of white dwarf stars, revealing a universal aspect of collapse phenomena across disciplines.

Contribution

It introduces a generalized Keller-Segel model based on Tsallis statistics and derives a critical mass analogous to astrophysical limits, providing new insights into chemotactic collapse.

Findings

Critical mass in 3D is approximately 203 units.

In 2D, the critical mass is 8 pi.

Below critical mass, the system reaches equilibrium or evaporates.

Abstract

We point out a remarkable analogy between the limiting mass of white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial populations in a generalized Keller-Segel model of chemotaxis [Chavanis & Sire, PRE, 69, 016116 (2004)]. 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 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 M_c=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.