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
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
Table of Contents
Introduction
Phase Transitions and Critical Behavior
Self Organized Critical Behavior
Forest-Fire Model
Summary
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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)
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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.
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
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 .
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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)
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
The Sandpile Model
[4]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
The Sandpile Model - Results
[4]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
Analogy to Ising-Model
Ising Model Sandpile Model
Temperature T Slope ΘMagnetization M Spontaneous flow jExternal Field H Input flow jin
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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.
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
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 .
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
Forest-Fire Model - Results
[4]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
Forest-Fire Model - Results
[4]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
Summary
Pattern Formation
[1] [1]
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
Forest-Fire Model
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
Self OrganizedCriticality
Jonas J. Funke
Introduction
Phase Transitionsand CriticalBehavior
Self OrganizedCritical Behavior
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.
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