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

Pierre-Henri Chavanis

doi:10.1016/j.physa.2007.03.056

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arXiv · 0706.3603

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

Pierre-Henri Chavanis

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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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Pierre-Henri Chavanis. 2007-06-25. Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions. https://doi.org/10.1016/j.physa.2007.03.056

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