1. Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at http://www.utdallas.edu/~ahoward/montecarlo

2. An Introduction

3.  Consider calculation of an integral:  How can we calculate this? ◦ Midpoint Method ◦ Trapezoid Method  But these have problems…

4. 1 Dimensional Integral2 Dimensional Integral

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

8.  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. An example of Monte Carlo Methods in Action

10. What is Percolation?

11.  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.  To model percolation (in 2D), we represent the material by an n x n “lattice” of points, called nodes,

13.  Connected by line segments called bonds.  POROSITY (pōros′ity): The ratio of the volume of a material’s pores to that of its solid content. Webster’s New Universal Unabridged Dictionary

14.  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.  And we see how many “paths” from top to bottom we can trace using only “open” line segments.

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

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

18.  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.  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.  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.  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.  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.  How do we count the number of paths which “span” the matrix?  There are a number of algorithms: ◦ Hoshen-Kopelman Algorithm

24.  First improvement is that we transform from a matrix of the bonds:

25.  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.  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 (HKA) essentially labels clusters of adjoining elements of a matrix which have the same value

28.  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. 0111 1010 0100 1101 1 and 0 (essentially true and false) denote open and closed nodes, respectively. 1 1 2 2 3 3 4 4 0111 2010 0300 3304

30.  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.  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. n Key Points: Percolation Threshold Phase Transition Appropriate Limiting Behavior Pc ≈ 0.6

33. (Number of clusters of size larger than 1)

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. Ising Model The Monte Carlo Approach

38. Ising Model

39. -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. -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