Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions |
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Authors: | P H Chavanis |
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Institution: | (1) Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France |
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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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Keywords: | PACS" target="_blank">PACS 05 45 -a Nonlinear dynamics and chaos 05 40 -a Fluctuation phenomena random processes noise and Brownian motion 05 20 -y Classical statistical mechanics 04 40 -b Self-gravitating systems continuous media and classical fields in curved spacetime |
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