Brownian Motion in 2D and the Fokker-Planck Equation

We show the Brownian motion of an evolving assembly of particles and the corresponding probability density. The probability density is a solution of the Fokker–Planck equation, which here reduces to a drift-diffusion partial differential equation. The center of mass of the particle distribution moves with a constant drift velocity while anisotropic diffusion is determined by the principal values of the diffusion matrix, along the Cartesian axes.

Brownian motion of a particle is described by a stochastic differential equation , where the are particle positions in , is the drift velocity, is an matrix and represents an -dimensional normal Wiener process. The Fokker–Planck equation (also called forward Kolmogorov equation) describes the temporal evolution of the probability density :

, where .

If and are constant, the Fokker–Planck equation reduces to a drift-diffusion equation that can be solved analytically. The fundamental solutions are Gaussian distributions which drift and widen with time.

This Demonstration shows the Brownian motion of a number of independent particles in 2D superimposed on the solution of the Fokker–Planck equation. To simplify the controls, the principal axes of the matrix are always the horizontal-vertical axes of the screen.