Chapter 9. Self-Organized Criticality

Sand Piles

In 1987, Bak, Tang, and Wiesenfeld published a paper in Physical Review Letters, “Self-organized criticality: An explanation of noise.” You can download it from

The title takes some explaining. A system is critical if it is in transition between two phases; for example, water at its freezing point is a critical system.

A variety of critical systems demonstrate common behaviors:

  • Long-tailed distributions of some physical quantities: for example, in freezing water, the distribution of crystal sizes is characterized by a power law.

  • Fractal geometries: freezing water tends to form fractal patterns—the canonical example is a snowflake. Fractals are characterized by self-similarity; that is, parts of the pattern resemble scaled copies of the whole.

  • Variations in time that exhibit pink noise: what we call “noise” is a time series with many frequency components. In white noise, all of the components have equal power. In pink noise, low-frequency components have more power than high-frequency components. Specifically, the power at frequency f is proportional to . Visible light with this power spectrum looks pink, hence the name.

Critical systems are usually unstable. For example, to keep water in a partially ...

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