introduction to percolation : basic concept and something else seung-woo son complex system and...
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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 ?
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