Optional Project 2

Optional Project 2: Ising Model Simulation (3 possible quiz grades)

-Write a computer program that simulates the Ising Model (2D) using the Metropolis method described at http://ace.acadiau.ca/science/phys/ising/. (quiz grade 1)

-Try your code on 10/10 2D matrix then on a 100/100 matrix for various temperatures 0, 0.1, 0.2, 1 (find the critical temperature). (quiz grade 1)

-Modify the program for simulation of a 3D model using the same approach and a 10x10x10 matrix for the same temperatures. Find the critical temperature and compare this with the 2D critical temperature. (quiz grade 2)

-Modify the program for a 1-d simulation of a polymer chain (polyethylene) using the 9 rotational isomeric states from pentane to generate the lowest energy chain conformation for a chain of length N. Find the critical temperature for a planar zig-zag conformation. This is the simulated crystallization temperature. (quiz grade 3)

Possible approach:

The Metropolis Algorithm: (Copied Directly From http://ace.acadiau.ca/science/phys/ising/)

The Ising Model employs the Metropolis algorithm in order to show that the overall energy of the lattice is converging. The Metropolis algorithm involves flipping a random lattice point, and determining if the net energy decreases. If the energy does decrease because of the random flip, then the new state is allowed, and the simulation continues. If the new lattice configuration causes the overall energy to increase, then a random number between 0 and 1 is generated. If the exponential of the temperature and change in energy is less than this randomly generated number, then the state is allowed, and the simulation continues. If this exponential is less than the randomly generated value, the flip is not allowed, the flipped lattice point is returned to its previous state, a different random lattice point is chosen and the simulation continues.

Here is the Metropolis Algorithm in step form:
1. Flip the spin of a random lattice point, creating a new state n'.
2. Compute DE = E_n' - E_n.
3. If DE < 0, then the new state is allowed and the algorithm repeats.
   If DE > 0, a random number x between [0,1] is chosen.
          If e^-bDE > x, then the new state is allowed and the algorithm repeats.
          If e^-bDE < x, then the state is rejected, the flipped lattice point is returned to its original state, and the algorithm repeats.

Also See: http://bartok.ucsc.edu/peter/java/ising/ising.html

Figure 1 showing results using the above routine for the Ising Metropolis simulation. Random start, T = 1, T = 0.2, T = 0.1, T = 0. 5000 steps, 50x50 grid.