Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity
✍ Scribed by Thimo Rohlf; Constantino Tsallis
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 732 KB
- Volume
- 379
- Category
- Article
- ISSN
- 0378-4371
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✦ Synopsis
We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state s i ðtÞ 2 f0; 1g of a cell i does not only depend on the states in its local neighborhood at time t À 1, but also on the memory of its own past states s i ðt À 2Þ; s i ðt À 3Þ; . . . ; s i ðt À tÞ; . . . . We assume that the weight of this memory decays proportionally to t Àa , with aX0 (the limit a ! 1 corresponds to the usual CA). Since the memory function is summable for a41 and nonsummable for 0pap1, we expect pronounced changes of the dynamical behavior near a ¼ 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 HðtÞ / t 1=ð1ÀqÞ , where q is the entropic index associated with nonextensive statistical mechanics. In all cases, the function qðaÞ exhibits a sensible change at a ' 1. We focus on the class II rules 61, 99 and 111. For rule 61, q ¼ 0 for 0papa c ' 1:3, and qo0 for a4a 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 HðtÞ typically diverges / N z with 0pzp1.