Elementary Cellular Automata

In this activity we are tasked to implement the elementary cellular automata. Results for the 1D cellular automata rules in the case of an initial array with a non-zero value at the center only:

Next, we impose the different ECA rules for the case of a random initial list. I used an array with 256 elements drawn from a uniform distribution. The results for the different rules implemented for 256 iterations are shown in the animation below:

We continue to let the system evolve for 256 more iterations and take only this second half of the iterations to remove the transient period. For each time step, we determine the Shannon entropy, $S$, and the kinetic energy $K$, defined as:

$$ S = -\frac{1}{S_{max}}\sum_{i=0}^{i = 8} p_i log p_i$$

$$K  = |K_{t+1}-K_{t}| ^2 $$

where $p_i$ is the probability of occurrence of the ith triple.

We plot the average Shannon entropy vs the average Kinetic energy to check for possible clustering of the different rules. In the figure below, three possible clusters were identified.

Plot of entropy versus energy. The encircled parts are possible clusters.