Project 6

Self-Organized Criticality

Introduction

Sandpile cellular automata were introduced almost twenty years ago as a paradigmatic example of self-organized criticality (SOC) [1,2], the tendency of slowly driven dissipative systems to display a scale free avalanche response. Such ideas have had an enormous impact in different fields, ranging from magnetic systems, superconductors and mechanics, to geophysics and plasma physics, including in particular the magnetosphere.

The influence also extends beyond physics, to for example biology, human (heart) physiology and cognitive processes or neuroscience. The reason for this success lies in the wide variety of non-equilibrium systems displaying an avalanche response to an external driving. One of the primary aims of SOC was, originally [1], to explain the wide occurrence of $1/f^{\alpha }$ noise in natural phenomena, through a direct relation between avalanche scaling and spectral properties.

The local rules for the so-called Bak-Tang-Wiesenfeld (BTW) cellular automaton update the integer field $z(i,j)$ defined on a square lattice synchronously according to:

$\mathrm {If} \ z(i,j)>4\ \mathrm {then}$

$z(i,j)\rightarrow z(i,j)-4$
$z(i\pm 1,j)\rightarrow z(i\pm 1,j)+1$
$z(i,j\pm 1)\rightarrow z(i,j\pm 1)+1$

The criticality of this state, obtained without fine tuning of parameters (such as the tuning of temperature in the case of conventional thermal critical phenomena), is manifested as the absence of characteristic spatial and temporal scales of these avalanches. In less fancy language: large avalanches occur rather often (there is no exponential decay of avalanche sizes, which would result in a characteristic avalanche size), and there is a variety of power laws without cutoffs in various properties of the system. The paradigm model for this type of behavior is the celebrated sandpile cellular automaton also know as the Bak-Tang-Wiesenfeld (BTW) model.

References

• [1] P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A 38, 364 (1988). [PDF]

• [2] S. K. Grumbacher, K. M. McEwen, D. A. Halverson, D. T. Jacobs, and J. Lindner, Self‐organized criticality: An experiment with sandpiles, Am. J. Phys. 61, 329 (1993). [PDF]

• [3] L. Laurson, M. J. Alava, and S. Zapperi, Power spectra of self-organized critical

sandpiles, J. Stat. Mech. L11001, (2005).