Self-gravitating Brownian particles in two dimensions: the case of N=2 particles

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 (r(2)) 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 (r(2)) = r(0)(2) + (4k(B)/xi mu) (T-T-*) t, where k(B)T = G(m1m2)/2 is a critical teperature, and an anomalous diffusion for large times (r(2)) alpha t(1-T)*/(T). 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 < T (c) = GMm/(4k(B) ). 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.