austin howard & chris wohlgamuth april 28, 2009 this presentation is available at...
TRANSCRIPT
![Page 1: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/1.jpg)
Applications of Monte Carlo Methods to
Statistical PhysicsAustin Howard & Chris Wohlgamuth
April 28, 2009
This presentation is available at http://www.utdallas.edu/~ahoward/montecarlo
![Page 2: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/2.jpg)
What is a Monte Carlo Method?
An Introduction
![Page 3: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/3.jpg)
Consider calculation of an integral:
How can we calculate this?◦ Midpoint Method◦ Trapezoid Method
But these have problems…
Motivation: An Example
![Page 4: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/4.jpg)
Why do the standard methods fail?
1 Dimensional Integral 2 Dimensional Integral
![Page 5: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/5.jpg)
To prevent the so-called “curse of dimensionality,” we can randomly sample our space instead.
Example: Calculating π.
Solving the Problem using Randomness
![Page 6: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/6.jpg)
Calculating π
![Page 7: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/7.jpg)
Calculating π
![Page 8: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/8.jpg)
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.
An Important Point
![Page 9: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/9.jpg)
PercolationAn example of Monte Carlo Methods in Action
![Page 10: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/10.jpg)
PercolationWhat is Percolation?
![Page 11: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/11.jpg)
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.
What is Percolation?
Image from Wikimedia Commons
![Page 12: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/12.jpg)
To model percolation (in 2D), we represent the material by an n x n “lattice” of points, called nodes,
How Can We Model Percolation?
![Page 13: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/13.jpg)
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.
How Can We Model Percolation?
Webster’s New Universal Unabridged Dictionary
![Page 14: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/14.jpg)
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.
How Can We Model Percolation?
![Page 15: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/15.jpg)
And we see how many “paths” from top to bottom we can trace using only “open” line segments.
How Can We Model Percolation?
![Page 16: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/16.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:
Determining the Number of Paths
![Page 17: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/17.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Straightforward
“Brute Force” Method
Determining the Number of Paths
![Page 18: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/18.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Straightforward
“Brute Force” Method
Determining the Number of Paths
Problems with this approach:
Far too many computations:◦ First, we have to
trace all possible paths from one node on the surface.
![Page 19: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/19.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Straightforward
“Brute Force” Method
Determining the Number of Paths
Problems with this approach:
Far too many computations:◦ Then we have to
repeat for every one of the nodes.
![Page 20: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/20.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Straightforward
“Brute Force” Method
Determining the Number of Paths
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.
![Page 21: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/21.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Straightforward
“Brute Force” Method
Determining the Number of Paths
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.
![Page 22: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/22.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Straightforward
“Brute Force” Method
Determining the Number of Paths
Problems with this approach:
Net result: this method is far too inefficient to work in practice.
![Page 23: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/23.jpg)
How do we count the number of paths which “span” the matrix?
There are a number of algorithms:◦ Hoshen-Kopelman
Algorithm
Determining the Number of Paths
![Page 24: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/24.jpg)
First improvement is that we transform from a matrix of the bonds:
The Hoshen-Kopelman Algorithm
![Page 25: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/25.jpg)
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.
The Hoshen-Kopelman Algorithm
![Page 26: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/26.jpg)
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.
The Hoshen-Kopelman Algorithm
![Page 27: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/27.jpg)
The Hoshen-Kopelman Algorithm (HKA) essentially labels clusters of adjoining elements of a matrix which have the same value
The Hoshen-Kopelman Algorithm
![Page 28: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/28.jpg)
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.
The Hoshen-Kopelman Algorithm
![Page 29: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/29.jpg)
The Hoshen-Kopelman Algorithm
0 1 1 1
1 0 1 0
0 1 0 0
1 1 0 1
1 and 0 (essentially true and false) denote open and closed nodes, respectively.
12
3
4
0 1 1 1
2 0 1 0
0 3 0 0
3 3 0 4
![Page 30: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/30.jpg)
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.
Operation of HKA
![Page 31: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/31.jpg)
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.
Results of HKA applied to a Model System
![Page 32: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/32.jpg)
Results of HKA applied to a Model System
n
Key Points:• Percolation Threshold• Phase Transition• Appropriate Limiting
Behavior
Pc ≈ 0.6
![Page 33: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/33.jpg)
Results of HKA applied to a Model System
(Number of clusters of size larger than 1)
![Page 34: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/34.jpg)
![Page 35: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/35.jpg)
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
![Page 36: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/36.jpg)
Ising Model
A few equations for us to recall
E J i j
i, j
, i, j 1
P e E /T
![Page 37: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/37.jpg)
Ising ModelThe Monte Carlo Approach
![Page 38: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/38.jpg)
Ising Model
![Page 39: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/39.jpg)
Ising Model
-Divide system into a lattice structure
-Set initial conditions spin direction and H
-Flip spin direction and calculate new energy (E*)
![Page 40: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/40.jpg)
Ising Model
![Page 41: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/41.jpg)
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
![Page 42: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/42.jpg)
Ising Model
![Page 43: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/43.jpg)
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.
![Page 44: Austin Howard & Chris Wohlgamuth April 28, 2009 This presentation is available at ahoward/montecarlo](https://reader033.vdocuments.net/reader033/viewer/2022051401/5697bf7d1a28abf838c84d0f/html5/thumbnails/44.jpg)
This presentation is available at http://www.utdallas.edu/~ahoward/montecarlo