Percolation on a 2D Square Lattice and Cluster Distributions

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Percolation on a 2D Square Lattice and Cluster
Distributions

• Kalin Arsov
• Second Year Undergraduate Student
• University of Sofia, Faculty of Physics
• Adviser Prof. Dr. Ana Proykova
• University of Sofia, Department of Atomic Physics

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CONTENTS

• What is Percolation? Applications
• Types of Percolation
• Size and dimensional effects
• Clusters and their distributions
• Hoshen-Kopelman labeling algorithm
• Results
• Acknowledgements

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What is Percolation? Passage of a substance
through a medium

• Every day examples
• Coffee making with a coffee percolator
• Infiltration of gas through gas masks
• Mathematical theory

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Some Applications of Percolation Theory

• Physical Applications
• Flow of liquid in a porous medium
• Conductor/insulator transition in composite
  materials
• Polymer gelation, vulcanization
• Non-Physical Applications
• Social models
• Forest fires
• Biological evolution
• Spread of diseases in a population

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Types of Percolation

• Depending on the relevant entities
• site percolation
• bond percolation
• Depending on the lattice type we consider
  percolation on
• a square lattice
• a triangular lattice
• a honeycomb lattice
• a bow-tie lattice

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Types of Percolation

• Site percolation
• The connectivity is defined for squares sharing
  sites (the substance passes through squares
  sharing sites)
• Bond percolation
• the substance passes through adjacent bonds

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Main Lattice Types
square lattice
triangular lattice
honeycomb lattice
bow-tie lattice
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Size and Dimensional Effects

• Influence of the linear size of the system
• increase of the spanning probability with the
  system size
• Influence of the dimensionality
• smaller spanning probabilities for higher
  dimensions

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Some Percolation Thresholds
Lattice pc (site percolation) pc (bond percolation)
cubic (body-centered) 0.246 0.1803
cubic (face-centered) 0.198 0.119
cubic (simple) 0.3116 0.2488
diamond 0.43 0.388
honeycomb 0.6962 0.65271
4-hypercubic 0.197 0.1601
5-hypercubic 0.141 0.1182
6-hypercubic 0.107 0.0942
7-hypercubic 0.089 0.0787
square 0.592746 0.50000
triangular 0.50000 0.34729
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Clusters and Their Distributions

• A cluster is a group of two or more neighboring
  sites (bonds) sharing a side (vertex)
• Cluster-size distribution ns (ns total number of
  s-clusters divided by L2, L the linear size of
  the system)
• Near the critical point ns? s-t, t
  187/912.05(5)
• If
• then

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Clusters and Their Distributions

• Mean cluster size S(p) is the mean size of the
  cluster (without the percolating cluster, if it
  exists) to which a randomly chosen occupied site
  belongs
• or

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Hoshen-Kopelman Labeling Algorithm

• Developed in 1976 by Hoshen and Kopelman
• Advantages
• simple
• fast
• no need of huge data files (the lattice is
  created on the fly)
• uses less memory than other algorithms
• gives us the clusters sizes as a secondary
  effect!!!

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Hoshen-Kopelman Algorithm

• Clever ideas one line, instead of a matrix, is
  kept cluster labels are divided into
• good a positive number, denoting the size of
  the cluster
• and
• bad negative, denoting the opposite to the
  good cluster
• label they are connected to.

N(1) 2 N(2) 3 N(3) 1
N(1) 11 N(2) -1 N(3) -2
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Results Cluster-Size Distribution ns
t 2.065 Excellent agreement with the
theoretically predicted t 2.055
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Results Spanning Probability W(p)

• Spanning probability W(p) is the ratio
• No_of_percolated_systems
• all_systems

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Results Mean Cluster Size S(p)
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Acknowledgements

• Ministry of Education and Science Grant for
  Stimulation of Research at the Universities, 2003
• The members of the Monte Carlo group
• My parents

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Monte Carlo Group Members

• Prof. Dr. Ana Proykova, Group leader
• M.Sc. Stoyan Pisov, Ass. Prof.
• M.Sc. Evgenia P. Daykova, Ph.D. Student
• B.Sc. Histo Iliev, Ph.D. Student
• Mr. Kalin Arsov, Undergraduate Student
• M.Sc. Ivan P. Daykov, Ph.D. Student (Cornell
  USA/UoS)

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More Information

• http//cluster.phys.uni-sofia.bg8080/kalin/