# Self-Organized Criticality and Cellular Automata

Later version available View entry history

**DOI:**https://doi.org/10.1007/978-3-642-27737-5_474-6

## Definition of the Subject

Self-organized criticality is a concept invoked to explain the frequent occurrence of fractal structures and multiscale phenomena in nature. In contrast with the ideas of chaos, here simple common features appear in systems with many degrees of freedom. For modeling this phenomenon, cellular automata provide an elegant class of dynamical systems which are easily simulated numerically.

## Introduction

Cellular automata provide a fascinating class of dynamical systems based on very simple rules of evolution yet capable of displaying highly complex behavior. These include simplified models for many phenomena seen in nature. Among other things, they provide insight into self-organized criticality, wherein dissipative systems naturally drive themselves to a critical state with important phenomena occurring over a wide range of length and time scales.

This entry begins with an overview of self-organized criticality. This is followed by a discussion of a few examples of...

## Keywords

Cellular Automaton Sandpile Model Relaxation Operator Successive Time Step Live Neighbor## Bibliography

## Primary Literature

- Anderson R et al (1989) Disks, balls, and walls: analysis of a combinatorial game. Am Math Mon 96:981; Björner A, Lovasz L, Shor P (1991) Chip-firing games on graphs. Eur J Comb 12:283; Eriksson K (1996) Chip-firing games on mutating graphs. SIAM J Discret Math 9:11822. Dhar D (1990) Phys Rev Lett 64:1613Google Scholar
- Bak P, Creutz M (1994) Fractals and self-organized criticality. In: Bunde A, Havlin S (eds) Fractals in science. Springer, Berlin, pp 26–47Google Scholar
- Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality. Phys Rev Lett 59:381; (1988) Phys Rev A 38:3645Google Scholar
- Bak P, Chen K, Creutz M (1989) Self-organized criticality in the ‘Game of Life’. Nature 342:780CrossRefADSGoogle Scholar
- Bennett C, Bourzutschy M (1991) ‘Life’ not critical?. Nature 350:468CrossRefADSGoogle Scholar
- Berlekamp E, Conway J, Guy R (1982) Winning ways for your mathematical plays, vol 2. Academic, New York. ISBN:0-12-091152-3, chapter 25Google Scholar
- Bogosian B (1993) Lattice gas hydrodynamics. Nucl Phys B Proc Suppl 30:204CrossRefADSGoogle Scholar
- Christensen K (1992) Self-organization in models of sandpiles, earthquakes, and fireflies. PhD thesis, University of AarhusGoogle Scholar
- Clar S, Drossel B, Schwabl F (1996) Exact results for the one-dimensional self-organized critical forest-fire model. Phys J Condens Matter 8:6803CrossRefADSGoogle Scholar
- Creutz M (1991) Abelian sandpiles. Comput Phys 5:198CrossRefADSGoogle Scholar
- Creutz M (1992) Abelian sandpiles. Nucl Phys B Proc Suppl 26:252CrossRefADSGoogle Scholar
- Creutz M (1997) Cellular automata and self organized criticality. In: Bhanot G, Chen S, Seiden P (eds) Some new directions in science on computers. World Scientific, Singapore, pp 147–169CrossRefGoogle Scholar
- Dhar D (1990) Self-organized critical state of sandpile automaton models. Phys Rev Lett 64:1613MathSciNetCrossRefADSMATHGoogle Scholar
- Dhar D, Majumdar SN (1990) Abelian sandpile model on the Bethe lattice. Phys J A 23:4333MathSciNetCrossRefGoogle Scholar
- Dhar D, Ramaswamy R (1989) Exactly solved model of self-organized critical phenomena. Phys Rev Lett 63:1659MathSciNetCrossRefADSGoogle Scholar
- Frette V et al (1996) Avalanche dynamics in a pile of rice. Nature 379:49CrossRefADSGoogle Scholar
- Gardner M (1983) Wheels, life, and other mathematical amusements. W.H. Freeman, New YorkMATHGoogle Scholar
- Goles E, Margenstern M (1996) Sand pile as a universal computer. Int J Mod Phys C 7:113MathSciNetCrossRefADSMATHGoogle Scholar
- Levy M, Solomon S, Ram G (1996) Dynamical explanation for the emergence of power law in a stock market model. Int J Mod Phys C 7:65CrossRefADSGoogle Scholar
- Majumdar SN, Dhar D (1992) Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model. Physica A 185:129CrossRefADSGoogle Scholar
- Nagel K, Paczuski M (1995) Emergent traffic jams. Phys Rev E 51:2909CrossRefADSGoogle Scholar
- Paczuski M, Maslov S, Bak P (1996) Avalanche dynamics in evolution, growth, and depinning models. Phys Rev E 53:414CrossRefADSGoogle Scholar
- Press W, Teukolsky S, Vetterling W, Flannery B (1988) Numerical recipes in C. Cambridge University Press, CambridgeMATHGoogle Scholar
- The latest version of the xtoys package is available at http://thy.phy.bnl.gov/www/xtoys/xtoys.html
- Toffoli T, Margolus N (1987) Cellular automata machines. MIT Press, CambridgeGoogle Scholar
- Wikipedia (2007) Conway’s game of life. http://enwikipedia.org/wiki/Conwav’s life. Accessed 6 Apr 2007
- Wikipedia (2007) Garden of Eden pattern. http://en.wikipedia.org/wiki/Garden_of_Eden_pattem. Accessed 6 Apr 2007
- Wolfram S (1986) Theory and applications of cellular automata. World Scientific, SingaporeMATHGoogle Scholar

## Books and Reviews

- Bak P (1996) How nature works: the science of self-organised criticality. Springer, BerlinCrossRefGoogle Scholar
- Gore A (1992) Earth in the balance: ecology and the human spirit. Plume, BostonGoogle Scholar
- Jensen HJ (1998) Self-organized criticality: emergent complex behavior in physical and biological systems. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- Toffoli T, Margolus N (1987) Cellular automata machines: a new environment for modeling. MIT Press, CambridgeGoogle Scholar
- Wolfram S (1994) Cellular automata and complexity: collected papers. Westview Press, BoulderMATHGoogle Scholar