Ronald Westra

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• Ronald Westra
• Dep. Mathematics
• Faculty of Humanity and Sciences
• Universiteit Maastricht

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• Lecture 6
• Self-Organisation
• and
• Collective Phenomenae

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Percolation revisited
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Percolation (Definition from Wikipedia)In
chemistry and materials science, percolation
concerns the movement and filtering of fluids
through porous materials.
During the last three decades, percolation
theory, an extensive mathematical model of
percolation, has brought new understanding and
techniques to a broad range of topics in physics
and materials science.
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Percolation

• Given the probability p of an occupied site
• What is the size M(p) of the largest connected
  cluster?
• Clearly, M(p) grows with p

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Relation with fractal shapes

• p the probability of a site being occupied
• (1-p) the probability of a site being unoccupied
• At a certain threshold value pc, the size of the
  largest connected cluster becomes very large.
• pc is called the percolation threshold
• The largest connected cluster has a fractal shape

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Duration of a fire

• How long does a fire last (depending on p and L)?
• Graph shows duration as a function of p
• L 20, 100, 500 (from bottom to top graph)
• pc ? 0.5928

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Phase transitions

• The sudden transition from no percolation to
  percolation at the percolation threshold is a
  phase transition
• Compare the boiling of water at the critical
  temperature Tc 1000 Celsius.
• For L ?? (infinitely large forest) the cluster
  size becomes infinite too

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• Probability that a randomly-selected tree is
  reached by fire as a function of p
• The larger L, the more pronounced the transition
  from 0 to 1

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L ? ?

• What does it mean and why is it important?
• It means that percolation extends over an
  infinite distance
• At the transition point, systems exhibit
  universal (predictable) behaviour
• At the transition point, the system becomes scale
  invariant (? fractal)

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Intuitive understanding

• What is the relation between fractal shape and
  critical behaviour?
• Self-similarity across scales
• E.g., zooming in on the coast of England
• Google-earth
• Basis for the renormalization approach

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Renormalization

• At the phase (or percolation) transition, the
  system is scale invariant
• Scale invariance implies that the system exhibits
  identical behaviour at all spatial scales
• Zooming out renormalization

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Zooming out

• A new lattice is formed by grouping three sites
  of the original lattice into one site
• Occupation of the new site is determined by the
  majority rule
• If most are occupied, new site is occupied too.
  The new site is unoccupied otherwise.

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zooming out
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Mathematical derivation(for the triangular
lattice)

• p is the percolation value of the original
  lattice
• p is the percolation value of the new (zoomed
  out) lattice
• The probability for an occupied site on the
  original is p
• The probability for an unoccupied site on the
  original lattice (1-p)
• The probability for an occupied site on the new
  lattice is
• The probability that all three original sites are
  occupied p?p?p p3
• The probability that one of the three original
  sites is unoccupied (1-p)p2which can occur in
  three ways
• Hence, p p3 3(1-p)p2

please take a nap if you dont like this, or if
you dont (want) to understand this
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Mathematical derivation

• p p3 3(1-p)p2
• We are interested in the case p p
• i.e., no change in occupation probability when
  zooming out
• p p3 3(1-p)p2 ? p3 3(1-p)p2-p 0
• p3 3p2 (1-p) -p -2p(p-0.5)(p-1)
• Which gives zero for p 0, p 1, and p 0.5
• These three values of p correspond to the fixed
  points of the recursive relation p p3
  3(1-p)p2

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Graphical illustration
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Mathematical derivation(for the square lattice)

• p is the percolation value of the original
  lattice
• p is the percolation value of the new (zoomed
  out) lattice
• The probability for an occupied site on the
  original is p
• The probability for an unoccupied site on the
  original lattice (1-p)
• The probability for an occupied site on the new
  lattice is
• The probability that all four original sites are
  occupied p?p ?p?p p4
• The probability that one of the four original
  sites is unoccupied (1-p)p3which can occur in
  four ways
• Hence, p p4 4(1-p)p3

please take a nap if you dont like this, or if
you dont (want) to understand this
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Phase transition

• At a critical value of T, Tc, the behaviour of
  the systems changes (it becomes critical)
• The critical behaviour is characteristic for the
  system (and other systems)

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Phase transitions micro vs macro

• Phase transitions in different systems often
  possess the same characteristics. This is known
  as universality.
• Universality is a prediction of renormalization
  theory
• Renormalization theory states that the properties
  of a phase transition depend only on a small
  number of features, such as dimensionality and
  symmetry, and is insensitive to the underlying
  microscopic properties of the system.

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Universality

• Universality is the observation that there are
  macroscopic properties for a large class of
  systems that are independent of the microscopic
  dynamical details of the system.
• Systems that display universality tend to be
  chaotic and often have a large number of
  interacting parts

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Applications

• Percolation
• Spread of diseases
• Spread of rumours

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Conclusions

• Phase transitions are universal
• They occur in many systems
• Economics
• Foraging in ants
• Movements of people
• Growth patterns
• Birth of solar systems

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End of revisit of Percolation
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Self-Organization
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Overview

• Definition of self-organization
• Natural patterns
• Flocking
• Boids
• Period doubling in clapping
• Mexican Waves

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Self-organization

• Self-organization is a process where the
  organization of a system spontaneously increases,
  i.e., without this increase being controlled by
  the environment or an encompassing or otherwise
  external system.

Principia Cybernetica Webhttp//pespmc1.vub.ac.be
/SELFORG.html
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Natural patterns
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Metal leaves produced during the
electrochemical deposition of ZnSO4
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Colonial cooperative self-organization
Paenibacillus vortex
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Colonial cooperative self-organization
Paenibacillus dendritiformis
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Spontaneous clustering

• Brownian bugs are random walkers
• Two simple principles
• Birth always occurs near to a bug
• Death may occur anywhere

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time
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Self-organization in the brain

• Orientation sensitivity

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Self-organization in the brain

• Direction tuning

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Ant Self-Organization
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Flocking
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Steering behaviour for Autonomous Characters
www.red3d.com/cwr/steer
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Lion King sequence

• Simulation stampede simulated using boid
  techniques

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Self-organization of clapping
Global noise intensity
Local noise intensity
Correlation parameter
Average noise intensity
Clapping period
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Period doubling

• Transition from asynchronous to rhythmic
  clapping skip every second beat
• Yields a clapping mode with a double period

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Mexican Wave
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Mexican-Wave model
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Mexican-Wave demo
http//angel.elte.hu/wave
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Lane formation and other demos
http//www.helbing.org/
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Conclusions on self-organisation

• Self-organizing processes underlie patterns in
  nature, society, and culture
• What is the relation between individual behavior
  and collective behavior?
• Not always obvious, e.g., termite behaviour
• Nature evolves group behaviors that improve
  fitness
• Society evolves group behaviors that improve
  social acceptance

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• The End