Brownian Motion

A simple illustration of Brownian motion, and how it relates to diffusion. Each dot represents a particle which, at each time step, moves either up, down, right or left by a fixed amount.

The main observation is that the particles tend to diffuse over time, similar to the way a drop of ink would spread on a sheet of paper. This shows how random motion leads to diffusion.

Diffusion can be characterized by looking at the average and the standard deviation of the distance to the center of the box $r$, and how it evolves through time $t$. The average should vanish $\langle r \rangle = 0$, while the standard deviation squared should grow linearly with time $\langle r^2 \rangle + \langle r \rangle^2 = \langle r^2 \rangle \propto t$. As expected, this is observed in this simple simulation.