Notes

Chapter 9: Fundamental Physics

Section 4: Conserved Quantities and Continuum Phenomena


Diffusion equation

In an appropriate limit the density distribution for cellular automaton (d) appears to satisfy the usual diffusion equation D[f[x, t], t] == c D[f[x, t], x, x] discussed on page 163. The solution to this equation with an impulse initial condition is Exp[-x^2/t], and with a block from -a to a it is (Erf[(a-x)/Sqrt[t]]+Erf[(a+x)/Sqrt[t]])/a.

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