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

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/ξμ)(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.

Revised version (20/5/2010) accepted for publication in EPJB

References in corpus (10)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• Kinetic and hydrodynamic models of chemotactic aggregation
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Jeans type analysis of chemotactic collapse
• Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions
• Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions
• Critical chemotactic collapse
• Newtonian gravity in d dimensions

Cited by in corpus (10)

• Solution of the Fokker-Planck equation with a logarithmic potential
• First-passage and first-exit times of a Bessel-like stochastic process
• Fluctuations of time averages for Langevin dynamics in a binding force field
• Scaling Green-Kubo relation and application to three aging systems
• Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas
• Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials
• Exact analytical solution of the collapse of self-gravitating Brownian particles and bacterial populations at zero temperature
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations
• Stochastic Averaging of The Einstein Vacuum Equations on a Toroidal Manifold with Randomly Perturbed Radial Moduli: Stability Criteria and Induced 'Cosmological Constant' Terms