Self-gravitating Brownian particles in two dimensions: the case of N = 2 particles
Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, 31062 Toulouse, France
2 Dipartimento di Fisica `E. Fermi' (CNISM Unità di Pisa), CNR-INFM, Università di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
Corresponding author: a firstname.lastname@example.org
Revised: 24 May 2010
Published online: 20 October 2010
We study the motion of N = 2 overdamped Brownian particles in gravitational interaction in a space of dimension d = 2. This is equivalent to the simplified motion of two biological entities interacting via chemotaxis when time delay and degradation of the chemical are ignored. This problem also bears similarities with the stochastic motion of two point vortices in viscous hydrodynamics [O. Agullo, A. Verga, Phys. Rev. E 63, 056304 (2001)]. We analytically obtain the probability density of finding the particles at a distance r from each other at time t. We also determine the probability that the particles have coalesced and formed a Dirac peak at time t (i.e. the probability that the reduced particle has reached r = 0 at time t). Finally, we investigate the mean square separation r2 and discuss the proper form of the virial theorem for this system. The reduced particle has a normal diffusion behavior for small times with a gravity-modified diffusion coefficient r2 = r02 + (4kB/ξμ)(T–)t, where kB = Gm1m2/2 is a critical temperature, and an anomalous diffusion for large times r2 . As a by-product, our solution also describes the growth of the Dirac peak (condensate) that forms at large time in the post collapse regime of the Smoluchowski-Poisson system (or Keller-Segel model in biology) for T < Tc = GMm/(4kB). We find that the saturation of the mass of the condensate to the total mass is algebraic in an infinite domain and exponential in a bounded domain. Finally, we provide the general form of the virial theorem for Brownian particles with power law interactions.
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2010