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

We show that the critical mass Mc = 8π of bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature Tc = GMm/4kB 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.