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

TL;DR

This paper explores the analogies between bacterial populations, self-gravitating Brownian particles, and vortices, focusing on critical mass and temperature thresholds across different dimensions using theoretical models.

Contribution

It establishes a connection between critical mass in bacterial chemotaxis and critical temperature in self-gravitating particles, extending the analysis to various dimensions and related systems.

Findings

Critical mass M_c=8π for bacterial populations in 2D.

Critical temperature T_c=GMm/4k_B for self-gravitating Brownian particles.

Dimension-specific behaviors and analogies among different physical systems.

Abstract

We show that the critical mass M_c=8\pi of 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.