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

TL;DR

This paper analytically investigates the dynamics and coalescence probability of two gravitationally interacting overdamped Brownian particles in two dimensions, revealing diffusion behaviors, a critical temperature, and condensate growth characteristics.

Contribution

It provides an exact analytical solution for the probability distribution, coalescence likelihood, and diffusion properties of two particles under gravitational interaction in 2D, linking to chemotaxis and vortex models.

Findings

Normal diffusion at small times with gravity-modified coefficient

Anomalous diffusion at large times with t^(1-T*/T) scaling

Growth of condensate follows algebraic or exponential saturation depending on domain

Abstract

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 some similarities with the stochastic motion of two point vortices in viscous hydrodynamics [Agullo & Verga, Phys. Rev. E, 63, 056304 (2001)]. We analytically obtain the density probability 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 variance of the distribution <r^2> and discuss the proper form of the virial theorem for this system. The reduced particle has a normal diffusion behaviour for small times with a gravity-modified diffusion coefficient <r^2>=r_0^2+(4k_B/\xi\mu)(T-T_*)t, where k_BT_{*}=Gm_1m_2/2 is a critical temperature, and an anomalous diffusion for large times <r^2>~t^(1-T_*/T). As a by-product, our solution also describes the growth of the Dirac peak (condensate) that forms in the post-collapse regime of the Smoluchowski-Poisson system (or Keller-Segel model) 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.