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

Thimo Rohlf and Constantino Tsallis

Physica A: Statistical Mechanics and its Applications, 2007, vol. 379, issue 2, 465-470

Abstract: 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)∝t1/(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.

Keywords: Cellular automata; Nonextensive statistical mechanics; Long-range memory (search for similar items in EconPapers)
Date: 2007
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:379:y:2007:i:2:p:465-470

DOI: 10.1016/j.physa.2007.02.015

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