Cellular Automata (Q2R and Creutz)

Abstract

In the simulations of the previous chapter we kept the temperature constant, and thus the energy fluctuated in the “canonical ensemble”. Now we deal with “micro-canonical” alternatives which keep the energy constant and thus have a fluctuating temperature. This method gives us an opportunity to introduce deterministic cellular automata. Those are lattices where each site k carries an integer variable S _k , called a spin, which has a limited number of values it can take. We let it be either -1 or +1 (or 0 and 1). The time t proceeds in steps of one and thus is also an integer. For time t + 1 the spin at site k gets a value determined uniquely from the spin values of its neighbors at time t, and in some models also from its own orientation S _k (t) at time t. For example, on an L*L square lattice with nearest neighbors k - 1, k + 1, k - L and k + L of site k (see Introduction), deterministic cellular automata are defined by the rule

$$ {S_k}(t + 1) = {f_k}[{S_{{k - 1}}}(t),{S_{{k + 1}}}(t),{S_{{k - L}}}(t),{S_{{k + L}}}(t),{S_{{k + L}}}(t),{S_k}(t)] $$

.