Diffusion and the diffusion equation are central topics in both Physics and Mathematics, and their ranges of applicability span from stellar dynamics to the diffusion of particles governed by Schroedinger's equation. The latter is, for a free particle, nothing but the diffusion equation in complex time!
Let us consider the one-dimensional diffusion equation. We study a large ensemble of particles performing Brownian motion along the \( x \)-axis. There is no interaction between the particles.
We define \( w(x,t)dx \) as the probability of finding a given number of particles in an interval of length \( dx \) in \( x\in [x, x+dx] \) at a time \( t \). This quantity is our probability distribution function (PDF).