Chapter 9: Fundamental Physics

Section 4: Conserved Quantities and Continuum Phenomena

Non-standard diffusion

To get ordinary diffusion behavior of the kind that occurs in gases—and is described by the diffusion equation—it is in effect necessary to have perfect uncorrelated randomness, with no structure that persists too long. But for example in the rule (a) picture on page 463 there is in effect a block of solid that persists in the middle—so that no ordinary diffusion behavior is seen. In rule (c) there is considerable apparent randomness, but it turns out that there are also fluctuations that last too long to yield ordinary diffusion. And thus for example whenever there is a structure containing s identical cells (as on page 462), this typically takes about s2 steps to decay away. The result is that on page 464 the limiting form of the average behavior does not end up being an ordinary Gaussian.

From Stephen Wolfram: A New Kind of Science [citation]