Chapter 7: Mechanisms in Programs and Nature

Section 7: Origins of Discreteness

Rate equations

In standard chemical kinetics one assumes that molecules are uniformly distributed in space, so that the rates for particular reactions are proportional to the products of the densities of the molecules that react in them. Conditions for equilibrium where rates balance thus tend to be polynomial equations for densities—with discontinuous jumps in solutions sometimes occurring as parameters are changed. Analogous equations arise in probabilistic approximations to systems like cellular automata, as on page 953. But here—as well as in fast chemical reactions—correlations in spatial arrangements of elements tend to be important, invalidating simple probabilistic approaches. (For the cellular automaton on page 339 the simple condition for equilibrium is p==p^2 (3 - 2p), which correctly implies that 0, 1/2 and 1 are possible equilibrium densities.)

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