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Introduction to Percolation

?????

- basic concept and something else

Seung-Woo Son Complex System and Statistical

Physics Lab.

Index

- Basic Concept of the Percolation
- Lattice and Lattice animals
- Bethe Lattice ( Cayley Tree )
- Percolation Threshold
- Cluster Numbers Exponents
- Small Cell Renormalization
- Continuum Percolation

What is Percolation?

-_- ?

Giant cluster

The number and properties of clusters ?

?????

Percolation

- First discussed by Hammersley in 1957

Other fun example

Let's consider a 2D network as shown in left

figure. The communication network, represented

by a very large square-lattice network of

interconnections, is attacked by a crazed

saboteur who, armed with wire cutters, proceeds

to cut the connecting links at random.

Q. What fraction of the links(or bonds) must be

cut in order to electrically isolate the two

boundary bars?

A. 50

Threshold concentration

P 0.6

P 0.5

Examples of percolation in real world

- Water molecule in a coffee percolator
- Oil in a porous rock ground water
- Forest fires
- Gelation of boiled egg hardening of cement
- Insulator - conductor transition

Forest fires

A green tree is ignited and becomes red if it

neighbors another red tree which at that time is

still burning. Thus a just-ignited tree ignites

its right and bottom neighbor within same sweep

through the lattice, its top and left neighbor

tree at the next sweep.

Average termination time for forest fires, as

simulated on a square lattice. The center curve

corresponds to the simplest case. p 0.5928

Oil fields and Fractals

Percolation can be used as an idealized simple

model for the distribution of oil or gas inside

porous rocks in oil reservoirs.

The average concentration of oil concentration of

oil in the rock is represented by the occupation

probability p. ( porosity )

They must take out rock samples from the well !!

510 cm diameter long rock logs sample ?

extrapolate to the reservoir scale.

M(L) - how many points within this frame belong

to the same cluster ? L2 Average density of

points P M(L)/L2 is independent of L.

Bond percolation site percolation

- Site percolation is dealt more frequently, even

though bond percolation historically came first. - Site-bond percolation(?)

Lattice dimension

- Square lattice, triangular lattice, honeycomb

lattice 2D - Simple cubic, body-centered cubic, face-centered

cubic, diamond lattice -3D - Hypercubic lattice higher than 3

Percolation thresholds

In finite systems as simulated on a computer one

does not have in general a sharply defined

threshold any effective threshold values

obtained numerically or experimentally need to be

extrapolated carefully to infinite system size.

Thermodynamic limit - physicist

Mathematically exact ?

Exact solution

Its very simple example.

The correlation length is proportional to a

typical cluster diameter.

Unfortunately the higher dimension, the

more complicated.

Animals in d Dimensions

http//mathworld.wolfram.com/Polyomino.html

fixed polyomino

monomino

domino

It is nice exercise to find all 63

configurations for s5.

triomino

tetromino

pentomino

For s4, 19 possible configurations

Perimeter

Perimeter the number of empty neighbors of a

cluster. ( t ) c.f. cluster surface

It is difficult to sum over all possible

perimeter t.

Perimeter polynomial

There seems to be no exact solution for general t

and s available at present.

Asymptotic result

The perimeter t, averaged over all animals with a

given size s, seems to be proportional to s for

s ? ?.

It is appropriate to classify different animals

of the same large size s by the ratio a t/s .

If a is smaller than (1-pc)/pc , then gst varies

as

Bethe lattice ( Cayley tree )

http//mathworld.wolfram.com/CayleyTree.html

Exact percolation threshold Pc

For square site percolation and 3D percolation,

no plausible guess for exact result.

Next will be more serious calculation -_-

It will border you

and me.

Power law behavior near Pc

briefly!

1st moment of cluster size

Power law behavior near Pc

briefly!!

2nd moment of cluster size

Scaling relation

Near p pc

Exact results on a Bethe Lattice ( Cayley tree )

These are the results in the limit of d ? ? !!

Exponents

Universality !!

Small cell renormalization

Rescale b?b cell into 1?1 cell

b

Recursion relation

b

Correlation length

b?b cell

1?1 cell

1D case

fixed point

Small cell renormalization

?3??3 triangular lattice

Recursion relation

Fixed point

2?2 square lattice bond percolation (?)

Continuum percolation

Fully penetrable sphere model

Equi-sized particles of diameter s are

distributed randomly in a system of side L s.

Penetrable concentric shell model

Particles of diameter s contain impenrable core

of diameter ?s

Randomly bonded percolation

Adhesive sphere model

Universality Class

All exponents of the continuum percolation

models with short- range interactions were found

to be the same as for the lattice percolation.

Summary

- Basic Concept of the Percolation
- Lattice and Lattice animals
- Bethe Lattice ( Cayley Tree )
- Percolation Threshold
- Cluster Numbers Exponents
- Small Cell Renormalization
- Continuum Percolation
- Dynamics

Reference

- Dietrich Stauffer and Amnon Aharony, Introduction

to Percolation Theory 2nd (1994) - Hoshen-Kopelman algorithm
- J. Hoshen and R. Kopelman, PRB 14, 3438 (1976)
- Review of the renomalization
- M. E. Fisher, Rev. Mod. Phys. 46, 597 (1974)
- S. K. Ma, Rev. Mod. Phys. 45, 589 (1973)
- M. E. Fisher, Lecture notes in Physics (1983)
- Renormalization for percolation
- P. J. Reynolds, Ph. D. Thesis (MIT)
- P. J. Reynolds, H. E. Stanley, and W. Klein,

Phys. Rev. B 21, 1223 (1980) - For continuum percolation models
- D. Y. Kim et al. PRB 35, 3661 (1987)
- I. Balberg, PRB 37, 2361 (1988)
- Lee and Torquato, PRA 41, 5338 (1990)
- http//www-personal.umich.edu/mejn/percolation/

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Finite size scaling

Dynamics ?