!Converted with LaTeX2HTML 95.1 (Fri Jan 20 1995) by Nikos Drakos (email@example.com), CBLU, University of Leeds >
There are two approaches to this computation in the literature. One is called mean field theory and begins by assigning probabilities to each of the k states of the automaton, and then calculates the probabilities in the next generation on the basis of the usual combinatorial rules of probability, assuming that the probabilities for each of the cells in a neighborhood are independent. It is then possible to solve for a set of self consistent probabilities for each state. The results of such calculations are generally plausible but do deviate significantly from empirical observations. The suspect element in the calculation is the assumption of independence.
To see how this works, let us once again recall the transitions defining Rule 22:
Five neighborhoods evolve into zeroes, three into ones; thus one might predict 37.5% ones would be found each generation on the basis of the number of ancestral neighborhoods. This is a better estimate than saying that 50% of the cells ought to be ones because there are only two different values they can have, but we have no reason to believe that all neighborhoods are equally likely either.
Taking the probability of finding a one as p, its coprobability as q, we could estimate for the probability of finding a one in the following generation, based on the makeup of the three neighborhoods that evolve to one. Mean field theory takes the fixed point of this estimate as the equilibrium density of ones for this rule. The self-consistent values for p are 0 and or approximately 42%.
A slightly more detailed approach to the same information would be to set up an evolution matrix, in which the probabilities of each of the cell values are components of a vector, while the elements of the matrix describe the probabilities that one value of the cell evolves into another.
The eigenvalues of this column stochastic matrix are A matrix of column eigenvectors is
Self-consistency is judged as before, with the same equation for
but now additional information about the rate of decay
of disequilibrium is available. Disequilibrium is very long lived for
the self-consistent value p=0, and vanishes for