A difference in temperature leads to a net flow of heat energy.
The same equation describes the random motion of particles.
If is either the density of “beads” or the probability of finding a bead at a particular place (if the movement of the beads is independent, then these two situations are interchangeable), then once again the diffusion equation describes how the bead density changes over time.
This is why the idea of “heat beads” is a useful intuition to use; the same math that describes the random motion of particles also describes how heat spreads through materials.
In one of his terribly clever 1905 papers, Einstein described how the random motion of individual atoms gives rise to diffusion. Adding up the probabilities from every possible starting position is the sort of thing integrals were made for: So far this is standard probability fare.
Call that something “k” and you’ve got the diffusion equation, .