Self Organized Criticality

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Self Organized Criticality

• Benjamin Good
• March 21, 2008

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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 Criticalit
  y 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.

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Characteristics of Criticality

• Divergence of the correlation length ? 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.

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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.

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BTW Model in 1-D

• 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 zc, a piece of sand
  fallsdown z(x) z(x) -2 z(x1)
  z(x1)1

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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.

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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(x1)-h(x,y1)
• Our rules new 2-D rules becomeAdding z(x,y)
  z(x,y)2 z(x-1,y)z(x-1,y)-1
  z(x,y-1)z(x,y-1)-1Falling 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

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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.

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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.

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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.

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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 b1.
• 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 0ltblt2

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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.

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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).