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 Tc at which the diffusion coefficient vanishes. For T < Tc, 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 Tc 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.
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Chavanis, P. Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions. Eur. Phys. J. B 57, 391–409 (2007). https://doi.org/10.1140/epjb/e2007-00187-2
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DOI: https://doi.org/10.1140/epjb/e2007-00187-2