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DY: Fachverband Dynamik und Statistische Physik
DY 24: Poster I
DY 24.35: Poster
Mittwoch, 28. März 2007, 16:00–18:00, Poster D
Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity — •Thimo Rohlf1 and Constantino Tsallis2 — 1Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM87501, USA — 2Centro Brasileiro de Pesquisas Fisicas, Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
We study the dynamics of elementary 1D cellular automata (CA), where the 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),... [1]. We assume that the weight of this memory decays proportionally to τ−α, with α ≥ 0. 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, 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 sensitive change at α ≃ 1. We focuse on the class II rules 61 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. 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.
[1] Rohlf, T. and Tsallis, C., preprint: cond-mat/0604459