Applications of Monte Carlo Methods to Statistical Physics

1. Applications of Monte Carlo Methods to Statistical Physics Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at http://www.utdallas.edu/~ahoward/montecarlo

2. What is a Monte Carlo Method? An Introduction

3. Motivation: An Example • Consider calculation of an integral: • How can we calculate this? • Midpoint Method • Trapezoid Method • But these have problems…

4. Why do the standard methods fail? 1 Dimensional Integral 2 Dimensional Integral

5. Solving the Problem using Randomness • To prevent the so-called “curse of dimensionality,” we can randomly sample our space instead. • Example: Calculating π.

6. Calculating π

7. Calculating π

8. An Important Point • There is not “one” Monte Carlo (MC) method! • MC simulations do not come in a well defined equation or package. • The MC method can better be thought of as a process or systematic approach.

9. Percolation An example of Monte Carlo Methods in Action

10. Percolation What is Percolation?

11. What is Percolation? • Percolation describes the flow of a fluid through a porous material. • This is in contrast to diffusion, which is the spread of particulates through a fluid. Image from Wikimedia Commons

12. How Can We Model Percolation? • To model percolation (in 2D), we represent the material by an n x n “lattice” of points, called nodes,

13. How Can We Model Percolation? • Connected by line segments called bonds. • POROSITY (pō•ros′i•ty): The ratio of the volume of a material’s pores to that of its solid content. Webster’s New Universal Unabridged Dictionary

14. How Can We Model Percolation? • Then we go through and randomly assign the property of open of closed to each line segment. Let us say the probabilty a particular line is open is p.

15. How Can We Model Percolation? • And we see how many “paths” from top to bottom we can trace using only “open” line segments.

16. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms:

17. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Straightforward “Brute Force” Method

18. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Straightforward “Brute Force” Method Problems with this approach: • Far too many computations: • First, we have to trace all possible paths from one node on the surface.

19. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Straightforward “Brute Force” Method Problems with this approach: • Far too many computations: • Then we have to repeat for every one of the nodes.

20. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Straightforward “Brute Force” Method Problems with this approach: • Far too many computations: • In order to use the MC method, we need many, many “runs" with the same probability, so we must repeat the whole process a number of times with the same value of p.

21. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Straightforward “Brute Force” Method Problems with this approach: • Far too many computations: • Finally, in order to get the percolation P (p) as a function of p, we must repeat all of this many times for different values of p.

22. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Straightforward “Brute Force” Method Problems with this approach: • Net result: this method is far too inefficient to work in practice.

23. Determining the Number of Paths • How do we count the number of paths which “span” the matrix? • There are a number of algorithms: • Hoshen-Kopelman Algorithm

24. The Hoshen-Kopelman Algorithm • First improvement is that we transform from a matrix of the bonds:

25. The Hoshen-Kopelman Algorithm • To one of the nodes. • Each node is given the property of open or closed, as before, and we consider percolation to occur between two open nodes.

26. The Hoshen-Kopelman Algorithm • Thus, our problem is reduced to finding the proportion of “clusters” of open nodes which are large enough that they span from the top edge to the bottom edge.

27. The Hoshen-Kopelman Algorithm • The Hoshen-Kopelman Algorithm (HKA) essentially labels clusters of adjoining elements of a matrix which have the same value

28. The Hoshen-Kopelman Algorithm • Specifically, HKA transforms a matrix of data to a matrix of labels, with a different label used for each cluster of adjoining elements of the data matrix which have the same value.

29. The Hoshen-Kopelman Algorithm 1 and 0 (essentially true and false) denote open and closed nodes, respectively. 2 1 4 3

30. Operation of HKA • Unfortunately, due to time constraints, we will not be able to discuss the specifics of HKA here. • However, it is discussed in our paper, available on WebCT, and on the internet with this presentation.

31. Results of HKA applied to a Model System • Consider the following example: • Grid is a 500 x 500 2D matrix • Generate 5,000 matrices for each value of p. • Calculate P(p) for values of p spaced a distance 0.05 apart. • One obtains the following graph.

32. Key Points: • Percolation Threshold • Phase Transition • Appropriate Limiting Behavior Pc ≈ 0.6 Results of HKA applied to a Model System n

33. (Number of clusters of size larger than 1) Results of HKA applied to a Model System

35. Ising Model What is the Ising Model? -Simplified model for magnetic systems -Only two possible directions for spin -There are interactive forces between spins, but only neighbors

36. Ising Model A few equations for us to recall

37. The Monte Carlo Approach Ising Model

38. Ising Model

39. Ising Model -Divide system into a lattice structure -Set initial conditions spin direction and H -Flip spin direction and calculate new energy (E*)

40. Ising Model

41. Ising Model -If ∆E<0 we retain it -If ∆E>0 we perform the following -Choose a random number between (0,1] -Calculate the probability (P) of the system attaining this state -If P>random number spin flip retained -If P<random number spin not flipped

42. Ising Model

43. Summary - The key ingredient in a Monte Carlo method is random numbers. - In both Ising Model and percolation, Monte Carlo method is a valuable tool.

44. This presentation is available at http://www.utdallas.edu/~ahoward/montecarlo