On Brownian motion, Boltzmann’s equation, and the Fokker-Planck equation

In order to describe Brownian motion rigorously, Boltzmann’s integral equation must be used. The Fokker-Planck type of equation is only an approximation to the Boltzmann equation and its domain of validity is worth examining. A treatment of the Brownian motion in velocity space of a particle with known initial velocity based on Boltzmann’s integral equation is given. The integral equation, which employs a suitable scattering kernel, is solved and its solution compared with that of the corresponding Fokker-Planck equation. It is seen that when $M/m$, the mass ratio of the particles involved, is sufficiently high and the dispersion of the velocity distribution sufficiently great, the Fokker-Planck equation is an excellent description. Even when the dispersion is small, the first and second moments of the Fokker-Planck solution are reliable. The higher moments, however, are then in considerable error—an error which becomes negligible as the dispersion increases.