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

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

cubic (body-centered) cubic (face-centered) cubic (simple) diamond honeycomb 4-hypercubic 5-hypercubic 6-hypercubic 7-hypercubic square triangular

p c

(site percolation)

0.246

0.198 0.3116 0.43 0.6962 0.197 0.141 0.107 0.089 0.592746

0.50000

p c

(bond percolation)

0.1803

0.119 0.2488 0.388

0.65271

0.1601 0.1182 0.0942 0.0787

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

n s

(

n s

: total number of

s

-clusters divided by

L 2 , L –

the linear size of the system )

Near the critical point: n s



s

-t , t

= 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

n s

t

≈ 2.065

: Excellent agreement with the theoretically predicted t

≈ 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/