Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions

Abstract

We show that the critical mass M_c=8πof bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature T_c=GMm/4k_B of self-gravitating Brownian particles in two-dimensional gravity. We obtain these critical values by using the Virial theorem or by considering stationary solutions of the Keller-Segel model and Smoluchowski-Poisson system. We also consider the case of one dimensional systems and develop the connection with the Burgers equation. Finally, we discuss the evolution of the system as a function of M or T in bounded and unbounded domains in dimensions d=1, 2 and 3 and show the specificities of each dimension. This paper aims to point out the numerous analogies between bacterial populations, self-gravitating Brownian particles and, occasionally, two-dimensional vortices.

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References in corpus (2)

• Logotropic distributions
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot

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• Self-gravitating clusters of Fermi-Dirac gas with planar, cylindrical, or spherical symmetry: evolution of density profiles with temperature
• Self-gravitating clusters of Bose-Einstein gas with planar, cylindrical, or spherical symmetry: gaseous density profiles and onset of condensation