Jonas J. Funke Introduction Self Organized CriticalityJonas J. Funke Introduction Phase Transitions...

Self OrganizedCriticality

Jonas J. Funke

Introduction

Phase Transitionsand CriticalBehavior

Self OrganizedCritical Behavior

Forest-Fire Model

Summary

Self Organized Criticality(SOC)

Jonas J. Funke

Technische Universitat MunchenSeminar zur Selbstorganisation in physikalischen Systemen:

Rhythmen, Muster und Chaos

01.12.2010


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Table of Contents

Introduction

Phase Transitions and Critical Behavior

Self Organized Critical Behavior

Forest-Fire Model

Summary


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Power-law scaling in nature

Many systems are found to exhibit power-law scaling of theirdistribution functions, i.e. the number of an event N(s)scales with the event size s as

N(s) ∼ s−α (1)

[4]


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Self-Organized Criticality

I Since systems close to a (second order) phase transitionshow similar power-law scaling P. Bak, C. Tang and K.Wiesenfeld had the following idea

I Idea: These systems operate always at the phasetransition (at the critical point). The systems drivethemselves toward this phase transition.

What is SOC?Self-Organized Criticality =Self Organization + Critical Phenomena


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Introduction



Forest-Fire Model

Summary

Phase Transitions

The Ising-Model:

I Spins Si = ±1 on a d-dimensional lattice

I Hamiltonian:

H = H∑i

Si + K∑<i,j>

SiSj (2)

I Coupling constant K, External field H (J and B in thpicture)

I Usually only nearest-neighbor (local) interactions

[3]


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Introduction



Forest-Fire Model

Summary

Ising-Model

[3]

I Order parameter:Magnetization M = 〈Si 〉

I H = 0:

T > Tc M = 0 G (r) ≈ e−r/ξ

rD−2+η

T = Tc M =? G (r) ∼ 1rD−2+η

T < Tc M 6= 0 G (r) ≈ e−r/ξ

rD−2+η

(G(r(i, j)) =⟨

(Si − M)(Sj − M)⟩

spatial correlation

function)

I Correlation length ξ:

ξ ∼(T − Tc

T

)−ν

(3)


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Introduction



Forest-Fire Model

Summary

Critical Phenomena

I What does the divergence of the correlation length ξ atT = Tc mean?

I System is strongly correlated, i.e. small (local)pertubations can cause events at all scales (even global)

I non-linear responseI no specific length scale

[3]

T > Tc spinflip → only local eventT = Tc spinflip → local-global eventT < Tc spinflip → only local event


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Divergence of physical quantities and scaleinvariance

I No specific length scale ⇒ scale invariance⇒ power-law distributions

I non-linear respond (CvdT = dQ, χTdH = dM)

[3]


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Introduction



Forest-Fire Model

Summary

Main Idea of SOC

We have learned so far:

I Critical behavior at phase transitions leads to power-lawscaling

I But exact tuning of external parameters is necessary(i.e. H = 0, T → Tc)

How can that occure in nature?Idea:

I System drives itself towards the critical point (attractor)

I Energy input will be the driving force

Let us start with a simple toy-model.


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Introduction



Forest-Fire Model

Summary

The Sandpile Model

Cellular automata toy-model to study SOC-systems.

[1]Consider a grid in 2-dimensions:

I Grains of sand are continuously added at radom sites ofthe grid

h(i , j)→ h(i , j) + 1 (4)

I If the local slope z exceeds a critical threshold zc thegrains are redistributed among the neighbors


Jonas J. Funke

Introduction



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Summary

The Sandpile Model in 1-dimension

[2]

Slope zi = hi − hi+1

Adding a grain hi → hi + 1:

zi−1 → zi−1 − 1 (can be neglected(5)

zi → zi + 1 (6)

Redistribution via hi → hi − 1 andhi+1 → hi+1 + 1:

zi−1 → zi−1 + 1 (7)

zi → zi − 2 (8)

zi+1 → zi+1 + 1 (9)

⇒ forget about h, just look at theslope z .


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Introduction



Forest-Fire Model

Summary

The Sandpile Model - Rules

I Grains of sand are added to a grid, causing the slope toincrease:

z(i , j)→ z(i , j) + 1 (10)

I If a threshold z > zc is reached the grains areredistributed:

z(i , j)→ z(i , j)− 4 (11)

z(i ± 1, j)→ z(i ± 1, j) + 1 (12)

z(i , j ± 1)→ z(i , j ± 1) + 1 (13)


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Introduction



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Summary

The Sandpile Model

[4]


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

The Sandpile Model - Results

[4]


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Introduction



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Summary

Analogy to Ising-Model

Ising Model Sandpile Model

Temperature T Slope ΘMagnetization M Spontaneous flow jExternal Field H Input flow jin


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Key features of SOC models

What needs a system to be a SOC system?

I Constant energy input

I Thresholds (ability to store energy)

I Local interactions

What characteristics will it show?

I Avalanches at all scales (size and time)

I Power-law frequency-size distributions

Remark: There is still no 100% clean definition of SOC.


Jonas J. Funke

Introduction



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Summary

Another example of SOC - Forest-Fire Model

Lattice: Each site can be in one of three states:

I Tree

I Empty

I Burning

Rules:

i Trees are randomly grown with a probability pat empty sites at each time step.

ii Trees, which are on fire will burn down at thenext time step.

iii At the next time step, the fire will spread to allnearest neighbors.

iv A tree which is not burning (and has noburning neighbors ), will catch fire with aprobability f .


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Forest-Fire Model - Characteristics

Does the Forest-Fire Model fulfill the ’requirements’ forSOC?

I Constant energy input: trees are grown with prob p X

I Threshold: Cluster of trees have to burn down’instantaneously’One therefore requires p → 0 while f

p → 0

I Local interactions X


Jonas J. Funke

Introduction



Forest-Fire Model

Summary

Forest-Fire Model - Results

[4]


Jonas J. Funke

Introduction



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Summary

Pattern Formation

[1] [1]


Jonas J. Funke

Introduction



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Summary

Summary

I SOC = Systems that drive themselves towards a criticalpoint ⇒ always operate at phase transition

I Consant energy input, thresholds (ability to storeenergy)

I Avalanches/burst of all length and time scales

I Power-law scaling of frequency-size distribution


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Introduction



Forest-Fire Model

Summary

References

Per Bak.How Nature Works: The Science of Self-OrganisedCriticality.Copernicus Press, New York, 1996.

Henrik Jeldtoft Jensen.Self-Organized Criticality: Emergent Complex Behaviorin Physical and Biological Systems.Camebridge University Press, Camebride, New York,1998.

Franz Schwabl.Statistical Mechanics.Springer, Berlin, Heidelberg, second edition, 2006.

Donald L Turcotte.Self-organized criticality.Reports on Progress in Physics, 62(10):1377, 1999.

Jonas J. Funke Introduction Self Organized CriticalityJonas J. Funke Introduction Phase Transitions...

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