Percolation on a 2D Square Lattice and Cluster Distributions

Kalin ArsovSecond Year Undergraduate StudentUniversity of Sofia, Faculty of Physics

Adviser: Prof. Dr. Ana ProykovaUniversity of Sofia, Department of Atomic Physics

CONTENTS

• What is Percolation? Applications

• Types of Percolation

• Size and dimensional effects

• Clusters and their distributions

• Hoshen-Kopelman labeling algorithm

• Results

• Acknowledgements

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

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

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

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

Main Lattice Types

square lattice triangular lattice

honeycomb lattice bow-tie lattice

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

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

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, =187/91≈2.05(5)

If

then

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

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!!!

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

Results: Cluster-Size Distribution ns

≈ 2.065: Excellent agreement with the theoretically predicted ≈ 2.055

Results: Spanning Probability W(p)• Spanning probability W(p) is the ratio:

No_of_percolated_systems all_systems

Results: Mean Cluster Size S(p)

Acknowledgements

• Ministry of Education and Science: Grant for Stimulation of Research at the Universities, 2003

• The members of the “Monte Carlo” group

• My parents

“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)

More Information

• http://cluster.phys.uni-sofia.bg:8080/kalin/