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Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions

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  • P. H. Chavanis

Abstract

We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. Our formula generalizes the usual Einstein relation for a free Brownian motion to the context of two-dimensional gravity. We show the existence of a critical temperature T c at which the diffusion coefficient vanishes. For T > T c , the diffusion coefficient is negative and the gas undergoes gravitational collapse. This leads to the formation of a Dirac peak concentrating the whole mass in a finite time. We also stress that the critical temperature T c is different from the collapse temperature T * at which the partition function diverges. These quantities differ by a factor 1-1/N where N is the number of particles in the system. We provide clear evidence of this difference by explicitly solving the case N=2. We also mention the analogy with the chemotactic aggregation of bacteria in biology, the formation of “atoms” in a two-dimensional (2D) plasma and the formation of dipoles or “supervortices” in 2D point vortex dynamics. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Suggested Citation

  • P. H. Chavanis, 2007. "Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 57(4), pages 391-409, June.
    • Handle: RePEc:spr:eurphb:v:57:y:2007:i:4:p:391-409
    DOI: 10.1140/epjb/e2007-00187-2
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    Cited by:

    1. Alberto Dinelli & Jérémy O’Byrne & Agnese Curatolo & Yongfeng Zhao & Peter Sollich & Julien Tailleur, 2023. "Non-reciprocity across scales in active mixtures," Nature Communications, Nature, vol. 14(1), pages 1-10, December.

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