**78**, 139-165 (2010)

https://doi.org/10.1140/epjb/e2010-90839-3

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

^{1}
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}
chavanis@irsamc.ups-tlse.fr

Received:
30
October
2009

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 *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} + (4*k*_{B}/*ξ**μ*)(*T*–)*t*, where
*k*_{B} = *Gm*_{1}*m*_{2}/2 is a critical temperature, and an anomalous
diffusion for large times *r*^{2} . 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*/(4*k*_{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.

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2010*