Thimo Rohlf, Constantino Tsallis

Paper #: 06-05-017

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state σi (t) ∈ {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 σi (t − 2), σi (t − 3), …, σi (t − τ), …. We assume that the weight of this memory decays proportionally to τ -α , with α > 0 (the limit α → ∞ corresponds to the usual CA). Since the memory function is summable for α > 1 and nonsummable for 0 ≤ α ≤ 1, we expect pronounced changes of the dynamical behavior near α = 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) ∝ t 1/(1-q) , where q is the entropic index associated with nonextensive statistical mechanics. In all cases, the function q(α) exhibits a sensible change at α ≈ 1. We focus on the class II rules 61, 99 and 111. For rule 61, q = 0 for 0 ≤ α ≤ αc ≈ 1.3, and q < 0 for α > αc , 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 ∝ Nz with 0 ≤ z ≤ 1. Similar studies have been carried out for other rules, e.g., the famous “universal computer” rule 110.

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