Introduction to Percolation

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Introduction to Percolation
?????

• basic concept and something else

Seung-Woo Son Complex System and Statistical
Physics Lab.
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Index

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

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What is Percolation?
-_- ?
Giant cluster






The number and properties of clusters ?
?????
Percolation
- First discussed by Hammersley in 1957
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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
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Threshold concentration
P 0.6
P 0.5
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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

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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
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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.
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Bond percolation site percolation

• Site percolation is dealt more frequently, even
  though bond percolation historically came first.
• Site-bond percolation(?)

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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

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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 ?
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Exact solution
Its very simple example.
The correlation length is proportional to a
typical cluster diameter.
Unfortunately the higher dimension, the
more complicated.
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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
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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
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Bethe lattice ( Cayley tree )
http//mathworld.wolfram.com/CayleyTree.html
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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.
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Power law behavior near Pc
briefly!
1st moment of cluster size
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Power law behavior near Pc
briefly!!
2nd moment of cluster size
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Scaling relation
Near p pc
Exact results on a Bethe Lattice ( Cayley tree )
These are the results in the limit of d ? ? !!
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Exponents
Universality !!
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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
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Small cell renormalization
?3??3 triangular lattice
Recursion relation
Fixed point
2?2 square lattice bond percolation (?)
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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
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Universality Class
All exponents of the continuum percolation
models with short- range interactions were found
to be the same as for the lattice percolation.
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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

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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
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Dynamics ?