Why a Naive CA Approach Does Not Work

Suppose we wanted to model an Ising model in the most straightforward possible manner using a CA approach. For argument s sake, let s consider modeling a basic Ising model without any external fields. Using binary site variables 7 = (1 — 5) to make any possible connection to conventional CA transparent, our task is to come up with a CA rule / , 

From this equation we see that a single spin flip a —) change AH equal to 

We can now take one of two approaches (1) construct a probabilistic CA along lines with the Metropolis Monte Carlo algorithm outlined above (see section 7.1.3.1), or (2) define a deterministic but reversible rule consistent with the microcanonical prescription. As we shall immediately see, however, neither approach yields the expected results. 

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