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6/11/2010 Introduction to percolations
Introduction to Percolation Theory
Danica Stojiljkovic
6/11/2010 Introduction to percolations11
6/11/2010 Introduction to percolations
System in concern
•Discrete system in d dimensions•Lattices
– 1D: array– 2D: square, triangular, honeycomb– 3D: cubic, bcc, fcc, diamond– dD: hypercubic– Bethe lattice (Cayley tree)
6/11/2010 Introduction to percolations22
6/11/2010 Introduction to percolations
System in concern
6/11/2010 Introduction to percolations33
6/11/2010 Introduction to percolations
System in concern
•Each lattice site is occupied randomly and independently with probability p•Nearest neighbors: sites with one side in common•Cluster: a group of neighboring sites
6/11/2010 Introduction to percolations44
6/11/2010 Introduction to percolations
Site and Bond Percolation
•A “site” can be a field or a node of a lattice•Bond percolation: bond is present with probability x•Site-bond percolation: continuous transition between site and bond percolation
6/11/2010 Introduction to percolations55
6/11/2010 Introduction to percolations
Percolation thresholds
6/11/2010 Introduction to percolations66
• At some occupation probability pc a spanning (infinite) cluster will appear for the first time
• Percolation thresholds are different for different systems
20 x 20 site lattice
6/11/2010 Introduction to percolations
Percolation thresholds
6/11/2010 Introduction to percolations77
6/11/2010 Introduction to percolations
Cluster numbers
•Number of cluster with s sites per lattice sites: ns •For 1D lattice:
•In general:
•t – perimeter
•gst –number of lattice animals with size s and perimeter t
2)1( ppn ss −=
∑ −=t
tssts ppgn )1(
6/11/2010 Introduction to percolations88
6/11/2010 Introduction to percolations
Average cluster size
•Probability that random site belongs to a cluster of size s is ws=sns/∑sns
•Average size of a cluster that we hit if we point to and arbitrary occupied site that is not a part of an infinite cluster
S = ∑wss = ∑s2ns/∑sns
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6/11/2010 Introduction to percolations
Average cluster size
•For 1D: S = (1+p)/(1p)
~ | pcp |1
•For Bethe lattice: S ~ | pcp |1
•In general S ~ | pcp |γ •compressibility, susceptibility
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6/11/2010 Introduction to percolations
Cluster numbers at pc
•If ns at pc decays exponentially, than S would be finite, therefore:
ns(pc) ~ sτ
•We calculate for Bethe ns(p)/ ns(pc) ~ exp (Cs)
whereC ~ | pc – p |1/σ
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6/11/2010 Introduction to percolations
Scaling assumption
•Generalizing Bethe result: ns(p) ~ sτ exp(|ppc|1/ σ s)
(1D result does not fit)
•We can derive all other critical exponents from τ and σ•Scaling assumption
ns(p) ~ sτ F(|ppc|1/ σ s)
or ns(p) ~ sτ ((ppc)sΦ σ)
6/11/2010 Introduction to percolations1212
6/11/2010 Introduction to percolations
Universality
• (z) Φ depends only on dimensionality, not on lattice structure• τ and σ, as well as function (z) Φare universal•Plotting ns(p)sτ versus (ppc)s σ would fall on the same line
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6/11/2010 Introduction to percolations
Strength of the infinite network
•P – probability of an arbitrary site belonging to the infinite network
P + ∑sns = p•Applicable only at p > pc•For Bethe lattice P ~ (p – pc)•In general
P ~ |p – pc|β•Spontaneous magnetization, difference between vapor and liquid density
6/11/2010 Introduction to percolations1414
6/11/2010 Introduction to percolations
Correlation function
•g(r) -Probability that a site at distance r from an occupied site belongs to the same cluster•In 1D array:
g(r) = pr = exp(r/ξ) ξ = 1/ln(p) = 1/(pcp)
• ξ Correlation length
6/11/2010 Introduction to percolations1515
6/11/2010 Introduction to percolations
Radius of gyration
•Average of the square radii
•r0 – center of mass
•Relation of gyration radii and average site distance
6/11/2010 Introduction to percolations1616
∑=
−=
s
i
is s
rrR
1
202
∑−
=ij
jis s
rrR 2
2
22
6/11/2010 Introduction to percolations
Correlation length
•Average distance between two sites belonging to the same cluster:
•Near the percolation threshold ξ ~ |p – pc| ν
6/11/2010 Introduction to percolations1717
∑∑∑∑
=
=
r s
s ss
r
r
ns
nsR
rg
rgr
2
22
22
2
)(
)(ξ
6/11/2010 Introduction to percolations
Fractal dimension
•At p = pc :M ~ LD
where M is mass of the largest cluster in a box with sides L•D<d and it is called fractal dimension•For gyration radii: Rs ~ s1/D
6/11/2010 Introduction to percolations1818
6/11/2010 Introduction to percolations
Crossover phenomena
•For finite lattices whose linear dimension L <<ξ , cluster behaves as fractals M ~ L D•ξ acts like a measuring stick, and in its absence all the relevant functions becomes power laws•For L >ξ, M ~ (L/ ξ)d ∙ ξD ~ L d
6/11/2010 Introduction to percolations1919
6/11/2010 Introduction to percolations
References
•Introduction to Percolation Theory, 2nd revised edition,1993by Dietrich Stauffer and Amnon Aharony
6/11/2010 Introduction to percolations2020
6/11/2010 Introduction to percolations
To be continued…
•Exact solution for 1D and Bethe lattice•Scaling assumption•Deriving relations between critical coefficients •Numerical methods•Renormalization group
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