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)+1z(x-1) = z(x-1)-1

• If the slope at any one point is higher than some critical value zc, a piece of sand fallsdown:

z(x) = z(x) -2z(x±1) = z(x±1)+1

BTW Model in 1-D

BTW Model in 1-D

• The pile is stable if z(x) ≤ zc for all x, so there are zcN

stable configurations.

• If sand is added randomly, the system will reach the minimally stable state (z(x) = zc 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/fb for b≈1.• This differs from random “white noise”, which is given by

1/f0 and that of a random (Brownian) walk, which is given by 1/f2. 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).