I'm not a physicist so my question may be really old hat, but whatever.
I can think of two situations in which one ends up with a diffusion equation but in which the underlying physics is quite different.
First, the flow of heat in a solid. Here there is a continuous 'heat flows down a temperature gradient' picture that is mathematically equivalent to a picture in which individual particles follow Brownian motions. Physically, the former is just a sort of averaged version of the latter - some accounting short cuts - while the latter is some way closer to rea...
Sorry I asked that wrong. I don't mean heat flow in the first case, there are no diffusing particles there. Say concentration of tracer in fluid suspension or something.