Critical Phenomena and Percolation Theory: II

Transcription

1 Critical Phenomena and Percolation Theory: II Kim Christensen Complexity & Networks Group Imperial College London Joint CRM-Imperial College School and Workshop Complex Systems Barcelona 8-13 April 2013

2 Outline 1 Percolation on the Bethe lattice 2 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1

3 A Bethe lattice is a tree where each site has z neighbours: l = 2 l = 3, generation number z = 3 l = 0 l = 1 sub-branch branch

4 Bethe lattice has no loops: Unique path between any sites i & j. Consider a percolating infinite cluster in the Bethe lattice. Perform a walk, where retracing of steps are forbidden. Each step has z 1 new sites (sub-branches). In average, p(z 1) sites occupied. Onset of percolation when p(z 1) = 1 p c = 1 z 1 = { 1 for z = 2;(d = 1) 1/2 for z = 3. p c decreases with increasing coordination number. Sensitive to lattice details. Non-universal quantity.

5 Assume p < p c. Let center site be occupied. to which this site belongs: χ(p) = contribution from center site + contribution from z branches = 1 + zb. B = contribution to average cluster size from a given branch = (1 p) 0+p (1+(z 1) B) parent site of branch is empty/occupied = p + p(z 1)B. Solve for B and insert in Eq. (1) above: χ(p) = 1 + p 1 p(z 1) = p c(1 + p) p c p with p c = 1 z 1 p c (1 + p c )(p c p) 1 for p p c (1)

6 z = 3 sub-branch branch

7 z = 3 sub-branch branch

8 z = 3 sub-branch branch

9 χ(p) p

10 Exponent characterizing the divergence is independent of coordination number. Universal χ(p) (p c p)

11 Probability center site belongs to percolating infinite cluster: P (p) = p (prob. at least one branch connects to percolating cluster) = p (1 - prob. none of z branches connect to percolating cluster) = p (1 Q z (p)). Q (p) = prob. a branch DOES NOT connect to percolating cluster = (1 p)+p Q z 1 (p) parent site of branch is empty/occupied For z = 3, solve quadratic equation for Q (p): { 1 for p pc Q (p) = 1 p p for p > p c.

12 z = 3 sub-branch branch

13 0 [ for p p c P (p) = ) ] 3 p 1 for p > p c. 1 ( 1 p p 0.8 P (p) p

14 P (p) = 10 0 { 0 for p p c 6 (p p c ) β for p p + c ;β = P (p) Exponent characterising the pick-up is independent of coordination number. Universal (p c p)

15 : Consider a cluster of size s. Define the perimeter t of cluster: t = no. of unoccupied nearest-neighbours of cluster. n(s, p) = g(s, t) (1 p) t p s. t=1 Clusters might not have unique geometry or orientation: g(s, t) = no. of different s-clusters with perimeter t. { 1 for t = 2 For d = 1 g(s, t) = 0 otherwise n(s, p) = (1 p)2 p s.

16 In Bethe lattice a unique relationship between s & t: t = 2 + s(z 2); For z = 3 we have t = 2 + s:

17 n(s, p) = t=1 g(s, t) (1 p)t p s = g(s, 2 + s) (1 p) 2+s p s. Trick to avoid enumerating g(s, 2 + s) by considering ratio: [ ] n(s, p) 1 p 2 [ ] (1 p ) p s n(s, p c ) = 1 p c (1 p c ) p c [ ] 1 p 2 ( [ ]) (1 p ) p = exp s ln 1 p c (1 p c ) p c [ ] 1 p 2 = exp ( s/s ξ ), 1 p c where we have defined the characteristic cluster size s ξ (p) = ln 1 [ (1 p ) p (1 p c) p c ] = 1 ln [ 1 4(p p c ) 2] 1 4 (p p c) 2 for p p c

18 sξ(p) p

19 Exponent characterising the divergence is independent of coordination number. Universal sξ(p) p c p

20 P (p) prob. site infinite cluster P s=1 sn(s, p) prob. site any finite cluster P (p) + P s=1 sn(s, p) = p p

21 The cluster no. density & the characteristic cluster size: [ ] 1 p 2 n(s, p) = n(s, p c ) exp ( s/s ξ ); 1 p c 1 s ξ (p) = [ ]. ln (1 p ) p (1 p c) p c sn(s, p) = p P (p) finite for all p, also at p = p c s=1 s 2 n(s, p) = χ(p) sn(s, p) s=1 s=1 for p p c ; diverges at p = p c

22 Ansatz: n(s, p c ) s τ for s 1: sn(s, p c ) = s 1 τ finite τ > 2 s=1 s 2 n(s, p c ) = s=1 s=1 s=1 s 2 τ infinite τ 3 The Bethe lattice has a cluster number density n(s, p) s 5/2 exp ( s/s ξ ) for s 1, p p c, s ξ (p) (p p c ) 2 for p p c,

23 At p = p c, n(s, p c ) s 5/2 for s 1 n(s, pc) s

24 For p p c : n(s, p) = n(s, p) { s τ decays rapidly for 1 s s ξ for s s ξ p = 0.35 p = 0.45 p = p = s

25 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 General scaling ansatz for cluster no. density: n(s, p) s τ G(s/s ξ ) for p p c, s 1, (2) s ξ (p) p p c 1/σ for p p c, Critical exponents: τ and σ. Scaling function G with dimensionless argument s/s ξ. The scaling ansatz Eq. (2) allows a data collapse because s τ n(s, p) G(s/s ξ ) for p p c, s 1. Plotting the transformed cluster no. density s τ n(s, p) vs. the re-scaled cluster size s/s ξ, all the data fall onto the graph of the scaling function G.

26 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Scaling function & data collapse for Bethe lattice: n(s, p) s 5/2 exp ( s/s ξ ) for p p c, s 1 (3) s ξ (p) (p c p) 2 for p p c The argument s/s ξ in the scaling fct is a re-scaled cluster size: G Bethe (s/s ξ ) = exp ( s/s ξ ), Eq. (3) allows a data collapse: s 5/2 n(s, p) = G Bethe (s/s ξ ). Plotting s 5/2 n(s, p) vs. s/s ξ the curves collapse onto the graph for the scaling function G Bethe. Another example of universality.

27 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Data collapse of cluster no. densities for perc. on Bethe lattice: n(s, p) p = 0.35 p = 0.45 p = p = s

28 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Data collapse of cluster no. densities for perc. on Bethe lattice: 10 0 s τ n(s, p) s

29 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Data collapse of cluster no. densities for perc. on Bethe lattice: 10 0 G Bethe (x) = exp ( x) s τ n(s, p) s

30 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Scaling function & data collapse for d = 1: n(s, p) = (p c p) 2 exp ( s/s ξ ) = s 2 [s(p c p)] 2 exp ( s/s ξ ) = s 2 (s/s ξ ) 2 exp ( s/s ξ ) for p p c = s 2 G 1d (s/s ξ ) for p p c (4) s ξ (p) = 1 ln p (p c p) 1 for p p c The argument s/s ξ in the scaling fct is a re-scaled cluster size: G 1d (s/s ξ ) = (s/s ξ ) 2 exp ( s/s ξ ), Eq. (4) allows a data collapse because s 2 n(s, p) = G 1d (s/s ξ ). Plotting s 2 n(s, p) vs. s/s ξ the curves collapse onto the graph for the scaling function G 1d. Another example of universality.

31 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Data collapse of the cluster no. densities for d = 1 percolation: 10 0 n(s, p) p = 0.4 p = p = 0.99 p = p = s

32 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Data collapse of the cluster no. densities for d = 1 percolation: s τ n(s, p) s

33 General scaling ansatz for cluster no. density Scaling function & data collapse for Bethe lattice Scaling function & data collapse for d = 1 Data collapse of the cluster no. densities for d = 1 percolation: 10 0 G 1d (x) = x 2 exp( x) 10-2 s τ n(s, p)