how nature works

the science of self-organized criticalityPer BakPresented by Adam Murray

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Today’s DiscussionI) Introduction

1. Theory of self-organized criticalityComplexity and CriticalitySelf-Organized Criticality

2. Examples reinforcing the theorySandpilesEarthquakesGame of LifeEconomics

II) Observations & ConclusionsIII) Questions / Comments

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The nature of how nature worksA personable authorDigestible formatDual narratives

SOC theory and practiceThe author’s opinions and criticisms of modern science

Our emphasis and interest is on SOC

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Power Laws for DummiesA power law comes in the form:

N(s) = s-t

The author illustrates his results in the form of a log-log graph of the power rule shown above:

Log N(s) = -t log s(Straight lines with slope –t)

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Physics is simple - Nature is complexPhysics has simple laws, while nature is complexComplex behaviour in nature reflects the tendency of large systems with many components to evolve into a critical state

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Self-organized and criticalThe out-of-balance critical state leads to avalanches of all sizesNo outside help to get to critical stateMost changes are a result of catastrophic events, not a smooth pathLarge catastrophic events occur for the same reasons small ones do

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Criticality in a simple sandbox

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SOC walking a fine lineSystems in equilibrium do not show signs of SOCChaos theory cannot explain complexityWalking with careful balance between equilibrium and chaosPerhaps SOC is an underlying principle that can be the law for complex behaviour

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General Empirical ObservationsGutenberg-Richter power lawsFractals1/f signalsZipf’s law

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Regularity of earthquakes

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Regularity of Biological Extinctions

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Biological Extinctions follow a Power Law

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Fractals off the coast of Norway

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Zipf’s lawZipf made interesting observations regarding population distribution in cities, and word distribution in textHe observed how many cities in the world had more than a given number of inhabitants

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City Rankings in 1920

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Observed Values: Zipf10th most frequently used word: 2653 times20th: 1311 times20 000th: only once

logarithmic plot of rank vs. frequency is a straight line

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Spherical cows and toy models

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Reading a quotation from Mr. Per Bak

Reading a quotation

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Examples from the textCoupled PendulumsSandpilesEarthquakesGame of LifeEconomicsAnd some of the topics are in the book that we do not focus too much on today: Mass extinction, solar flares and black holes, the human brain, traffic jams, and evolutionary biology

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Coupled Pendulum Experiment

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Avalanche Diagram for Pendulums

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Better analogy: SandpileDifferent nomenclature, same mathThe critical state must be robust to modificationsExperimental representation is identical to coupled pendulumsAn example illustrates the simplicity of the model

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Toppling Avalanches in Action

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Oversimplification of modelsReal grains have different sizes & shapesInstability does not just happen at surfaceSo what makes the model acceptable?

Essential physicsDetachment from details! (I.e. we are not interested in sand)

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Why are the results significant?No organization is suggested, but in reality the pile has organized in a highly orchestrated, susceptible state.Plotting the log # of avalanches of a given magnitude vs. that magnitude reveals the GR power lawFractal geometry for the profile of sandpileSandpile dynamics obey Zipf’s law

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On real sandpiles, and landscape formationTesting on real sand is tedious

Sandpiles must be large to test power law behaviourWe do not have the patience nature has, nor the space nature has.

Not everything in this world is SOC!Long-grain rice approach seems most indicative of SOC behaviour

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Some Experiment Layouts (1)

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Some Experiment Layouts (2)

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Some Experiment Layouts (3)

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Earthquake ModelingThe model is mathematically identical to CP & sandpiles. If you study one, you study them allTectonic plate motion == tilting sandpileToppling grains == ruptures leading to other rupturesThe physical model is quite similar to CP

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Earthquake Experiment Display

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There are greater implicationsThe GR Power law is the fingerprint that the crust of the earth has self-organized to the critical stateBecause of robustness, the criticality does not depend on choice of modelThe crust of the Earth has organized itself into a critical state

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The game of life: complexity is criticalityThe game of life is a toy model of the formation of organized, complex, societiesComplex phenomena can be generated from simple local rules0’s and 1’s across a grid, at each time step all numbers are updated by a simple rule

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Game of Life Avalanches

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Economic SystemsIn equilibrium systems, everything adds up nicely and linearlyTraditional economics is not realisticReal economics is like sandA histogram of (1960) monthly variations of cotton prices displays a “Levy Distribution”, which has a power law for large eventsSo Per Bak builds a toy model…

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Economic Model Pre-Action

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Economic Model in Action

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Observations & ConclusionsComplexity is a consequence of criticalityReal-life operates at a critical point between order and chaosSOC is a theoretical foundation for catastrophism, and explaining complexityLarge fluctuations cannot be prevented by local manipulationsAny small behaviour in the critical state eventually affects everything in the systemHarder Conclusions would have been nice, such as the application of the said theory

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ReferencesPer Bak, How Nature Works (Springer, New York, 1996).

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Questions / Comments?Thank you for your attention!