Regensburg 2004 – scientific programme
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SYFT: Fat-Tail Distributions - Applications from Physics to Finance
SYFT 2: Fat-Tail Distributions - Applications from Physics to Finance
SYFT 2.3: Fachvortrag
Thursday, March 11, 2004, 12:30–12:45, H1
The Sierpinski signal: 1/fα noise in a simple 1D automaton model of pattern formation — •Jens Christian Claussen1, Jan Nagler2, and Heinz Georg Schuster1 — 1Theoretische Physik, Universität Kiel — 2Theoretische Physik, Universität Bremen
Cellular automata have been considered widely as a model for complexity and self-organized criticality: In classic models (Game of Life, Bak-Sneppen, BTW, OFC), power law spectra and avalanche distributions have been observed and compared with neural oscillators, extinction dynamics, and earthquakes [1]. An even more simplified model, yet exhibiting complex patterns, is the Sierpinski automaton xi(t+1)= (xi+1(t)+xi−1(t)) mod 2. Sierpinski patterns appear quite generically in 1D reaction-diffusion pattern formation and have been observed in catalytic reactions and in molluscs (Olivia porphyria).
In [2] we investigate the spectral properties of the time dependence of total activity X(t)=∑i xi(t) of the Sierpinski automaton with initial condition of one single seed. The spectrum is derived analytically and shows a power-law decay with exponent 1.15. The relation to the paradigmatic Thue-Morse sequence is discussed. While in our model the strict lattice topology is crucial for the analytic result and seems to prevent a direct mapping on economic agent networks, cellular (and stochastic) automata may serve as candidates for modelling of fat-tailed observables resulting from an underlying complex spatio-temporal dynamics.
[1] P. Bak, How nature works; H. J. Jensen, Self-organized criticality.
[2] J. C. Claussen, J. Nagler, and H. G. Schuster, cond-mat/0308277.