Introduction to Percolation

: basic concept and something else

Seung-Woo Son

Complex System and Statistical Physics Lab.

산돌광수체

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?

percolation n. 1 여과 ; 삼출 , 삼투 2 퍼컬레이션 (( 퍼컬레이터로 커피 끓이기 ))

사전적 의미 (?)

-_-; ?

통계물리학

Square lattice

2Ζ

Cluster

Giant cluster

- First discussed by Hammersley in 1957Percolation

The number and properties of clusters ?

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

Threshold concentration ( ) = 0.5927 ( 2D square site )

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

Tree

Burning tree

Burned tree

Empty hole

L L lattice

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 andbottom neighbor within same sweep throughthe 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 FractalsPercolation 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 )

p < pc

It will most probably hit a small cluster.

They must take out rock samples from the well !!

bad investment ! 광수생각

5~10 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.But near pc

…M(L) L1.9 fractal dimension D = 1.9 is not equal to Euclidean dimension 2.

So…Average density decays as L-0.1. For 100km size, (106)-0.1 ~ 0.25

Remaining 75% can’t directly extract.

Bond percolation & site percolation

Site percolation is dealt more frequently, even though bond percolation historically came first.

Site-bond percolation(?)

Site percolation 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 thresholdsIn 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 solution1D case It’s very simple example.

2)1( ppn ss the number of s-clusters per lattice site (normalized cluster number)

sns : the probability that an arbitrary site is part of an s-cluster

)( cs

s pppsn 1cp for one dimension. It’s trivial.

)( )1(

)1(2

cs

s ppp

p

sn

snS

average cluster size

correlation function (pair connectivity)- the probability that a site a distance r apart from an occupied site belongs to thesame cluster. ex)

rprg )(

1)0( g

)(

1

ln

1 )exp()(

ppp

rrg

c

correlation(connectivity) length

Srgr

)(

The correlation length is proportional to a typical cluster diameter.

)( cppS Unfortunately the higher dimension, the more complicated.

Animals in d Dimensions1

2

3

4

5

2-dimension square lattice animals

For s=4, 19 possible configurations

It is nice exercise to find all 63 configurations for s=5.^^;

?sn

monominodomino

triomino

tetromino

pentomino

= fixed polyomino

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

exponentially increase !

PerimeterPerimeter – the number of empty neighbors of a cluster. ( t ) c.f. cluster surface

stg- the number of lattice animals (cluster configurations) with size s and perimeter t

t

tssts ppgn )1( It is difficult to sum over all possible perimeter t.

)1( re whe)( pqqgp

nqD

t

tsts

ss 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

s

a

a

a

a

1)1(

ss constsg

tsts gg

Bethe lattice ( Cayley tree )

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

Path graph star graph

z = 3

1

1

z

pc

A tree in which each non-leaf graph vertex has a constant number of branches n is called an n-Cayley tree. 2-Cayley trees are path graphs. The unique n-Cayley tree on nodes is the star graph.

Exact percolation threshold Pc

Bethe lattice ( with z branch ) =1

1

z1 D chain = 1

square bond percolation = 1/2

triangular site percolation = 1/2

triangular bond percolation = )18

sin(2

honeycomb bond percolation = )18

sin(21

honeycomb site percolation 1/2

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

: density of clusters of size s number of clusters of size s per lattice sitesn )( c

ss pppsn

cpp For

cpp For

)( pfcen css

sns

)( /1

cccs

s ppppcesn

P: probability that any given site belongs to the infinite cluster

s

s psnP

)( 0 cppP

cc ppppP

briefly~!

1st moment of cluster size

Power law behavior near Pc

briefly~!!

s

s

ss

ss

nssn

nsS 2

2

: average cluster size S (Percolation susceptibility)

)(

)( '

cc

cc

ppppC

ppppC

universal ; '

C

CR

2nd moment of cluster size

Percolation specific heat

zeroth moment of cluster size

Consider the Gibbs free energy as the singular part of the zeroth moment of cluster size distribution.

sing

)()(

ssS pnpG

200 c

s sss ppnnsM

Percolation correlation length cpp

Scaling relation

Near p = pc

2

3

21

2

dD

Exact results on a Bethe Lattice ( Cayley tree )

1 , 1 , 2

1 ,

2

5 ,

1

1

z

pc

These are the results in the limit of d !!

ExponentsUniversality !!

Small cell renormalizationRescale bb cell into 11 cell

b

b

Spanning probability :

bb cell

11 cell

)( pRb'p

Fixed point :*' ppp **)( ppRb

Recursion relation

Correlation length

bb cell :

11 cell :

co pp'' ~

) bξξ ( ~ 0'00 cpp

cc ppppb '

bb

pppp cc

log

log

log

)/()'(log1

cppat '

)(

)'(

dp

dp

pp

pp

c

c

1D case…bpp ' 1* p fixed point

Small cell renormalization33 triangular lattice

)1(3)(' 23 ppppRp b

Recursion relation

1 ,2

1 ,0* pFixed point

...3547.1)2/3ln(

3ln ,

2

3'

*

p

pp

p p

p

22 square lattice bond percolation (?)

(exact) 3

4 (exact),

2

1 cp

2345

322345

2252

)1(2)1(8)1(5'

pppp

pppppppp

8

13' and 2 ,

2

1

*

* pp

dp

dpbp

428.1

Continuum percolationFully penetrable sphere model

Equi-sized particles of diameter σ are distributed randomly in a system of side L σ.

Swiss cheese model

Inverse Swiss cheese model

Penetrable concentric shell model

Particles of diameter σ contain impenrable core of diameter λσ

Randomly bonded percolation/

0

2

)( reprp bonding probability

Adhesive sphere model

Universality Class

For overlapping disks :

For interacting particles :

All exponents of the continuum percolation models with short-range interactions were found tobe 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 (199

4) 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/

Finite size scaling

Dynamics ?