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 exactly what your intuition should say about heat: if a place is hotter than the area around it, it will cool off.This is a very long-winded way of saying “think of heat as randomly moving particles, because the math is the same”.If the temperature of the pool and the air above it are equal, then the amount of heat that flows out of the pool is equal to the amount that flows in.But if the pool is hotter, then more “beads” will randomly roll out than randomly roll in.

The diffusion of heat is governed, not surprisingly, by the “diffusion equation”.Einstein’s cute trick was to say “Listen, I don’t know what ϕ() is a probability distribution (and the sum of probabilities over all possibilities is 1).So, can be written: To make the jump from discrete time steps to continuous time, we just let the time step, ), but it’s something, so is something too.It’s all well and good to talk about how heat beads randomly walk around inside of a material, but if that material isn’t uniform or has an edge, then suddenly the math gets remarkably nasty.Fortunately, if all you’re worried about is whether or not you should leave your heater on, then you’re probably not sweating the calculus.

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