Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions
✍ Scribed by Pierre-Henri Chavanis
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 325 KB
- Volume
- 384
- Category
- Article
- ISSN
- 0378-4371
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✦ Synopsis
We show that the critical mass M c ¼ 8p of bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature T c ¼ 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 ¼ 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, twodimensional vortices.