Introduction to Percolation

1. 산돌광수체 Introduction to Percolation : basic concept and something else Seung-Woo Son Complex System and Statistical Physics Lab.

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

3. 사전적 의미(?) percolationn.1여과; 삼출, 삼투 2퍼컬레이션 ((퍼컬레이터로 커피 끓이기)) Cluster Square lattice What is Percolation? -_-; ? Giant cluster The number and properties of clusters ? 통계물리학 Percolation - First discussed by Hammersley in 1957

4. 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%

5. Threshold concentration ( ) = 0.5927 ( 2D square site ) Threshold concentration P = 0.6 P = 0.5

6. 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

7. L  L lattice Tree Burning tree Burned tree Empty hole 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

8. p < pc It will most probably hit a small cluster. bad investment ! 광수생각 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. 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 !! 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.

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

10. 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

11. 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 ? ^^;

12. 1D case the number of s-clusters per lattice site (normalized cluster number) : the probability that an arbitrary site is part of an s-cluster for one dimension. It’s trivial. average cluster size • correlation function (pair connectivity) - the probability that a site a distance r apart from an occupied site belongs to the same cluster. ex) correlation(connectivity) length It’s very simple example. Exact solution The correlation length is proportional to a typical cluster diameter. Unfortunately the higher dimension, the more complicated.

13. 1 2-dimension square lattice animals 2 3 4 5 exponentially increase ! http://mathworld.wolfram.com/Polyomino.html Animals in d Dimensions = fixed polyomino monomino domino It is nice exercise to find all 63 configurations for s=5. ^^; triomino tetromino pentomino For s=4, 19 possible configurations

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

15. 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. z = 3 Path graph star graph http://mathworld.wolfram.com/CayleyTree.html Bethe lattice ( Cayley tree )

16. Bethe lattice ( with z branch ) = 1 D chain = 1 square bond percolation = 1/2 triangular site percolation = 1/2 triangular bond percolation = honeycomb bond percolation = honeycomb site percolation  1/2 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.

17. : density of clusters of size s number of clusters of size s per lattice site For For : probability that any given site belongs to the infinite cluster briefly~! Power law behavior near Pc 1st moment of cluster size

18. : average cluster size S (Percolation susceptibility) Percolation specific heat Consider the Gibbs free energy as the singular part of the zeroth moment of cluster size distribution. zeroth moment of cluster size Percolation correlation length briefly~!! Power law behavior near Pc 2nd moment of cluster size

19. Scaling relation Near p = pc Exact results on a Bethe Lattice ( Cayley tree ) These are the results in the limit of d   !!

20. Exponents Universality !!

21. Spanning probability : bb cell 11 cell Fixed point : 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

22. Small cell renormalization 33 triangular lattice Recursion relation Fixed point 22 square lattice bond percolation (?)

23. Swiss cheese model Inverse Swiss cheese model bonding probability Continuum percolation Fully penetrable sphere model Equi-sized particles of diameter σ are distributed randomly in a system of side L σ. Penetrable concentric shell model Particles of diameter σ contain impenrable core of diameter λσ Randomly bonded percolation Adhesive sphere model

24. For overlapping disks : For interacting particles : Universality Class All exponents of the continuum percolation models with short- range interactions were found to be the same as for the lattice percolation.

25. 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

26. 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/

28. Finite size scaling

29. Dynamics ?