Self Organized Criticality Benjamin Good March 21, 2008

What is SOC? Self Organized Criticality (SOC) is a concept first put forth by Bak, Tang, and Wiesenfeld (all theoretical physicists) in their famous 1988 paper. SOC models have been applied to earthquakes, evolution, neuron- processes, quantum gravity, and many other areas. It centers around two key concepts: Criticality: a concept from statistical physics characterized by a lack of a characteristic time/length scale, fractal behavior, and power laws. Self Organized: this critical state arises naturally, regardless of initial conditions, rather than requiring exactly tuned parameters.

Characteristics of Criticality Divergence of the correlation length ξ introduced in Yeomans’ book. Certain observables (e.g. distribution of patch sizes) obey power laws Universality – extremely different systems display the same behavior regardless of their dynamical rules. System is often sensitive to small perturbations. However, criticality is usually obtained by finely tuning a parameter (e.g. temperature for phase transitions), so they would be unlikely to naturally arise.

The BTW Model: Sand Piles The original SOC model was based on sand piles. Sand piles can reach a “critical state” where the addition of just one more piece of sand can trigger an avalanche of any size. The concept of universality ensures that studying this model can tell us much about other processes, despite the silly nature of the particular example.

There are N points on a line, and the height of sand at a point x is given by h(x). The slope, z(x), at a point x is then given by z(x) = h(x) – h(x)+1 To add a piece of sand at a point x, we take: z(x) = z(x)+1 z(x-1) = z(x-1)-1 If the slope at any one point is higher than some critical value z c, a piece of sand falls down: z(x) = z(x) -2 z(x±1) = z(x±1)+1 BTW Model in 1-D

The pile is stable if z(x) ≤ z c for all x, so there are z c N stable configurations. If sand is added randomly, the system will reach the minimally stable state (z(x) = z c for all x). If an extra piece of sand is added, it just falls all the way down the pile and off the edge. Thus, no interesting behavior in the 1-Dimensional case. SOC is not present.

BTW Model in 2-D To see interesting behavior, the model must have D ≥ 2. In two dimensions, the height is given by h(x,y) and the slope by z(x,y) = 2h(x,y)-h(x+1)-h(x,y+1) Our rules new 2-D rules become: Adding: z(x,y) = z(x,y)+2 z(x-1,y)=z(x-1,y)-1 z(x,y-1)=z(x,y-1)-1 Falling: z(x,y) = z(x,y) – 4 z(x ± 1,y) = z(x ± 1, y)+1 z(x,y ± 1) = z(x, y ± 1)+1

BTW Model in 2-D Will the system evolve towards the minimally stable state again? NO! Because each point is connected to more than one other point, a small perturbation amplifies and travels throughout the entire pile. Thus, the minimally stable state is unstable with respect to small fluctuations and cannot be an “attractor” of the dynamics.

BTW Model in 2-D As the system evolves, more and more “more than critically stable” patches arise and will impede the motion of the perturbation. Thus, the pile seems to be in a critical state, and since it arose on its own, it is a self organized critical state. Now we start looking for power laws.

Power Laws and Avalanche Sizes Critical states are slightly perturbed and the resulting avalanches are measured. We then form a distribution of avalanche sizes D(s) and try to fit it to a power law.

1/f Noise The critical sandpile also displays a phenomenon called 1/f noise. This means that its power spectrum, defined by follows the form S(f) = 1/f b for b≈1. This differs from random “white noise”, which is given by 1/f 0 and that of a random (Brownian) walk, which is given by 1/f 2. 1/f noise is usually defined as anything with 0<b<2

Applications to evolution Many researchers (including Bak himself) have been inspired to apply SOC concepts to evolution. This was prompted by several studies that discovered power law- like behavior in extinctions, evolutionary activity, etc. SOC models are appealing in this context, because they offer a natural explanation for how these phenomena arose (they self organized), whereas many existing models yield the desired power laws only if certain parameters are tuned. However, much of the justification for SOC models is based only on the fact that power laws are observed.

Are power laws enough evidence for SOC? Newman, in a 1995 paper on the evidence for SOC in evolution, asks whether power laws are actually present in the data and whether this is enough to imply SOC in evolutionary processes. Results: although the evidence for power laws is good, additional non-SOC mechanisms could account for them (e.g. environmental stresses).