Project #1: Exploring the Ising Model

The Ising model is one of the oldest and well know models for ordering in cooperative systems. A cooperative system is one where there are many individual and simple parts that interact strongly. The ordering and patterns that appear in a cooperative system arrise from the interaction between these simple parts, and can sometime be quite complex. Because these pattern self-organize this is called "emergence" because the behavior emerges out of the rules for interaction rather that being prespecified as in an external blueprint.

What is the Ising model about?

The original Ising model was designed to model magnetic spins. Think of the spins as little arrows that either point up or down. These arrows can be thought of as a little magnets with a positive and negative pole. The magnets can flip up or down at random. A real magnet is made of many such spins. One of the reasons the Ising model is so popular is that is can also me mapped on to a lot of other porblems: everyting from populations dynamics, to models of the brain. This is because it is very analogous to many types of cooperative systems. So we will start our studies of complex systems by looking at the Ising model.

How does the Ising model work?

Start with a grid like a chess board, and put a spin in every square. Each spin is initially either "up" or "down". Now choose one of the spins at random and make a descision wherether to change the spin, or leave it the same. The choice is made at random, but is biased in the following way: the spin "looks" at its neighbors, the 4 squares to the north, south, east, and west of it and take a vote. Let s = the total number of neighboring spins with the SAME direction. Then the probability the the spin will stay the same is P = 1/ [1+exp(-2s/T)], and the probability it will flip is 1-P. After updating the spin, choose another at random and repeat over and over again. Here the parameter T is known as the temprature. As T is increased the randomness is also increased.

What is it about?

You might prefert to think of the Ising model like popularity contrst where each spin is bias and tries to be line up with its neighbors to a certain extent. Initially there is no concensus, but at some point ordering can set in spontaniously.

The Ising Model Applet

Assingment #1 use the Ising model applet. You can load it by following the "Link to Simulators" link on the Physics 110 home page, or by going here. The graphical display is a grid with white or blue squares indicating the two posible spin states of each "spin". Blue and white are interchangeble but you can think of blue as s = +1 and white as S = -1. There are several controls you will make use of. There is a slider bar to change the temperature. There are start and stop buttons for running the simulations, and there are also three buttons you can use to initialize the starting distributions of blue and whitle spins.

Assingment #1

Use the Ising model applet to do the following: 1. Set the lattice size to 30, an the temperaure to the lowest value. You may have to hit return each time you enter a new lattice size. Randomize the initial condition by pressing the "Init Random" button. Now hit the start button to run the simulation. The program now computes the rules describe above, and automatically makes changes of the spin color. Describe what happens in a few sentences. 2. Run the previous simulation over from the beginning at least 5 times, and compare both the final result and the process of geting there for all the runs. Write a few sentences about what you discover about the similarities and differneces between the runs. Explain in you onw word what you think is happening. 3. Now vary the temperature parameter and decribe the effect. Give you best emphasis for the temperature of the phase transition: the point on the border between ordered and random final states. 4. (Hard extra credit problem) Mean field theory is a method for calculating the approximate location of the phase transition where we pretend that the total number of plus and minus spins seen by each spin is exactly the same and the mean number of plus and minus spin (per site). We define P as the probabiltiy that a spin is in the plus state. In this case 1-P is the probability to be in the minus state. Let the difference be M = 2P-1 be called the order parameter since M will vanish if there is complete randomness. The probability to stay the same is W=1/[1+exp(-2s/T)] as we stated above. There will be an equilibrium when the total numner of spin flipping from + to - equals the total number of spins flipping from - to +, or when W(1-P)=(1-W)P. Show that this is equivalent to the situation where M=2W-1. The mean spin is s = P-1/2. Show that the equilibrium condition is equivalent to 2/ [1+exp(-M/T)]-1=M. Draw the right and left hand sides of this function for various values of T. The two curves will intersect for value of M that are solution to the equation above. How many solution are there. In you own words, what does this mean for the equilibrium values of the order parameter. (Hint: what happens at T = 1/2?)