10217

# 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.

### DETAILS

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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.