Percolation

Recommended page

http://qiita.com/ssh0/items/20077cb8005e974b3110

Python scripts for percolation simulation is given in this page.

Percolation (Chap.12 in the textbook)

We introduce several geometrical structures constructed by a random process. In this regard, however, we do not treat any dynamical processes but analyse the geometrical specification of constructed structures.

If a container is filled with metall balls shown in the following figure and a battery is connected, the container is an conductor, this is, electric current passes through the system. What parcentage of metal balls is needed for the container to become a conductor? Analysises show the change from the insulator phase to the conductor phase has the sharp shreshold. This is an example of a percolation phase transition.

(copied from Fig. 12.1 of the text¹)

A simpler model is site percolation. The critical density of this problem is analytically found to be 0.59.

(a copy of Fig.12.3 of the text¹)

• The program "PercolationApp.java" produces a sample structure for a selected system size and the density. Check various parameter values.
• That program is included in "Statistical and Termal Physics Applications" package, which can be obtained from http://www.compadre.org/OSP/document/ServeFile.cfm?ID=7308&DocID=481

Find the critical behavior by use of an excellent algorithm for finding clusters

The program "Cluster.java" is to classify every particles into the clusters using Newman-Ziff algorithm.

Try to run "ClusterApp", and answer the problem 12.7 in the textbook. At first, (a) can be skipped.

(a)

Read the code for class Clusters and explain how the Newman-Ziff alogorithm is implemented.

(b)

Collect data for P_\infty (p), the probablity that an occupied site belongs to the spanning cluster, S(p), the mean cluster size, and S_{\rm span}(p), the probability of a spanning cluster, Consider L=8, 32, 128,\ {\rm and}\ 256 and average over at least 100 configurations. How does the qualitative behavior of these quantities change with increasing L? Discuss the qualitative dependence of S_\infty and S(p), on p for the largest lattice that you can simulate in a reasonable time.

(c)

At what value of p is S_{\rm span}(p) \approx 0.5 for each value of L ? Call this value P_c(L) to L \rightarrow \infty . For example, try fitting your data for p_c(L) = p_c - cL^{-x}, where P_c,\ c and x are fitting parameters. Because you will likely have insufficient data to deterministic three parameters with reasonable accuracy, take x = 3/4 and plot p_c(L) versus L^{-3/4} . How sensitive is your result for the intercept p_c on the assumed value of x ?

Useful pages

Ising model

http://physics.weber.edu/schroeder/software/demos/IsingModel.html

• Critical behavior of Ising model

  C \propto |T - T_c|^{-\alpha},\quad M\propto (T_c - T)^\beta, \quad \chi \propto |T - T_c|^{-\gamma}

• Correlation function

  C(r) = \frac{1}{r^{d-2+\eta}} e^{-r/ \xi }