Brownian Motion of Stars, Dust, and Invisible Matter

Abstract

Treating the motion of a dust particle suspended in a liquid as a random walk, Einstein in 1905 derived an equation describing the diffusion of the particle's probability distribution in configuration space. Fokker and Planck extended this work to describe the velocity distribution of the particles. Their equation and its solutions have been applied to many problems in nature starting with the motion of Brownian particles in a liquid. Chandrasekhar derived the Fokker-Planck equation for stars and showed that long-range gravitational encounters provide a drag force, dynamical friction, which is important in the evolution of star clusters and the formation of galaxies. In certain circumstances, Fokker-Planck evolution also describes the evolution of dark (invisible) matter in the universe. In the early universe, the thermal decoupling of weakly interacting massive particles from the plasma of relativistic leptons and photons is governed by Fokker-Planck evolution. The resulting dissipation imprints a minimum length scale for cosmic density fluctuations. Still later, these density fluctuations produce stochastic gravitational forces on the dark matter as it begins to cluster under gravity. The latter example provides an exact derivation of the Fokker-Planck equation without the usual assumption of Markovian dynamics.