Scale-Free Enumeration of Self-Avoiding Walks on Critical Percolation Clusters

Transcription

1 Scale-Free Enumeration of Self-Avoiding Walks on Critical Percolation Clusters Niklas Fricke and Wolfhard Janke Institut für Theoretische Physik, Universität Leipzig November 29, 2013

2 Self-avoiding walk (SAW) Universal scaling behavior Number of conformations: C N µ N N γ 1 Mean end-to-end distance: R 2 = Flory: ν = 3 2+D CN i=1 R 2 i C N N 2ν

3 Random site dilution x x x x Concentration of accessible sites p < 1 Quenched disorder averages [C N ] µ N N γ 1, [ R 2 ] N ν

4 Critical cluster "./components.dat" matrix Percolating cluster at p = p c D p c d f d l d Bf /

5 Methods Analytical: mean-field (Flory) approximations, real space renormalization, field theory 1 Monte Carlo: chain-growth methods, i.e., simple sampling, PERM 2 Exact enumeration 3 1 Meir and Harris (1989), v. Ferber et al. (2004) 2 Lee at. al. (1989), Blavatska and Janke (2008) 3 Lam (1990), Singh et al. (2009)

6 Methods Analytical: mean-field (Flory) approximations, real space renormalization, field theory 1 Monte Carlo: chain-growth methods, i.e., simple sampling, PERM 2 Exact enumeration 3 Problem: computation time µ N 1000 steps take > years! 1 Meir and Harris (1989), v. Ferber et al. (2004) 2 Lee at. al. (1989), Blavatska and Janke (2008) 3 Lam (1990), Singh et al. (2009)

7 Structure of the critical cluster "./chem.dat" matrix "./components.dat" matrix 1000 chemical shells around SAW origin Bi-connected components of region relevant for 1000 steps

8 Factorizing the enumeration Example: Two pieces A, B with only one connection C a b [N] = N C a x [i] C x b [N i] i=0

9 Hierarchical decomposition A B D C E F Exact enumeration of SAW configurations in polynomial time.

10 Time complexity 1000 N Time [s] N 2D: t N 2.14(1) (lower bound: N d l N 1.68 )

11 [ ] R 2 vs N D 3D 4D 5D 6D 7D N [ R 2 ] N (10 3 ) SAW steps on critical in 2-7 dimensions clusters for each dimension

12 The exponent ν D 3D 4D 6D 7D ln[ R 2 ] 2 ln N /N [ R 2 ] N 2ν ν = 1 d ln [ R 2 ] 2 dn D ν 0.777(5) 0.647(3) 0.581(5) 0.537(6) 0.513(8) 0.510(8)

13 Role of the backbone? Backbone of a 3D cluster

14 Backbone vs. full cluster 1e D full 2D backbone 3D full 3D backbone [ R 2 ] N The backbone seems to determine ν.

15 Backbone vs. full cluster D full 2D backbone 3D full 3D backbone ln[ R 2 ] 2 ln N /N The backbone seems to determine ν.

16 R 2 vs N single configurations 7e+05 6e+05 5e+05 [ R 2 ] 4e+05 3e+05 2e+05 1e+05 0e+00 0e+00 5e+03 1e+04 2e+04 2e+04 N Plateaus: One cluster region dominates average. Jumps: Endpoints are pushed out of dominating region.

17 R 2 vs N single configurations 1e+06 1e+05 N ν3d 1e+04 [ R 2 ] 1e+03 1e+02 1e+01 1e+00 1e-01 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05 N Strong fluctuations, but average slope is clearly there.

18 Current work: Self-attracting SAWs x x x x ɛ R 2 = CN i=1 e βe R 2 i CN i=1 e βe Is there a Θ-collapse transition? 4 If so, at what temperature and with which exponents? 4 see Roy et al. (1990), Barat et al. (1992), Blavatska and Janke (2009)

19 Θ-Transition? first results (D=3, N=10 3 ) CV / R T Nice collapse and coinciding cusp in C V for some configurations.

20 Θ-Transition? first results (D=3, N=10 3 ) CV / R T But at varying temperatures.

21 Θ-Transition? first results (D=3, N=10 3 ) CV / R T Quite different behaviors here.

22 Summary SAWs on critical percolation clusters: basic model for polymers in disordered media. Fractal nature of critical clusters can be exploited Exact enumeration of SAWs in polynomial time. Flory exponent determined for various dimensions. Numerically confirmed ν backbone = ν full cluster.

23 Summary SAWs on critical percolation clusters: basic model for polymers in disordered media. Fractal nature of critical clusters can be exploited Exact enumeration of SAWs in polynomial time. Flory exponent determined for various dimensions. Numerically confirmed ν backbone = ν full cluster. Thank you for the attention! Thanks to the Sassonian Research Group FOR877 and the graduate school BuildMoNA for funding and support!

24 ρ (ln Z 1000 ) ρ (ln Z 1000 ) N (378, 144) h ln Z 1000