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.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.Call that something “k” and you’ve got the diffusion equation, .
You can actually derive useful math from it, which is a damn sight better than most science metaphors (E.g., “space is like a rubber sheet” is not useful for actual astrophysicists).The diffusion of heat is governed, not surprisingly, by the “diffusion equation”.The same equation describes the random motion of particles.Vi ber deg nok en gang om å holde indentiteten deres hemmelig! : I’m having a debate with my wife that I think you can help us resolve. It has an electric heater, which we set to keep the pool water at 85 degrees Fahrenheit. My wife says we should turn the heater off while we’re away to save energy. The one and only gain from leaving a pool heater on is that it will be warm when you get in.But if the pool is hotter, then more “beads” will randomly roll out than randomly roll in.