Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state $sigma_i(t) in {0,1}$ of a cell $i$ does not only depend on the states in its local neighborhood at time $t-1$, but also on the memory of its own past states $sigma_i(t-2), sigma_i(t-3),...,sigma_i(t-tau),...$. We assume that the weight of this memory decays proportionally to $tau^{-alpha}$, with $alpha ge 0$ (the limit $alpha to infty$ corresponds to the usual CA). Since the memory function is summable for $alpha>1$ and nonsummable for $0le alpha le 1$, we expect pronounced %qualitative and quantitative changes of the dynamical behavior near $alpha=1$. This is precisely what our simulations exhibit, particularly for the time evolution of the Hamming distance $H$ of initially close trajectories. We typically expect the asymptotic behavior $H(t) propto t^{1/(1-q)}$, where $q$ is the entropic index associated with nonextensive statistical mechanics. In all cases, the function $q(alpha)$ exhibits a sensible change at $alpha simeq 1$. We focus on the class II rules 61, 99 and 111. For rule 61, $q = 0$ for $0 le alpha le alpha_c simeq 1.3$, and $q<0$ for $alpha> alpha_c$, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size $N$ indicate that the range of the power-law regime for $H(t)$ typically diverges $propto N^z$ with $0 le z le 1$. Similar studies have been carried out for other rules, e.g., the famous universal computer rule 110.