ising1_readme.md

Part 1

(1) Set up a 2D lattice in the program, i.e., an indexed 2D matrix where each matrix site is indexed by i and j, where both variables run from 0 to 19 (or 1 to 20, depending on the convention in your program of choice), and each matrix site indexed by ij contains either “+1” or “-1”.

(2) Initialize the matrix by assigning +1 and -1 values at random to each matrix site. This will require the use of a random number generator. You will also want to have this part set up so that you can also start your simulation in the all-up (all +1) or all-down (all -1) states too.

(3) Calculate the initial energy for your system, using the traditional Ising Model energy. Make H and J both variables you can change easily. Implement 2D periodic boundary conditions, so that row 0 interacts with row 19 as well as with row 1, and the same for column 0 and column 19.

(4) Start the sampling of various states via a loop. Make the loop of variable length – 10,000 steps through the loop is a good starting place (but making it only 10 steps is more appropriate as you are writing and debugging the code). Each time you execute the loop, your program should:

a. Choose 1 lattice site at random.

b. Calculate its interactions with its nearest neighbors and the field, i.e. the portion of the energy that is influenced by this single spin.

c. Flip it. (i.e. if it is -1, make it +1, if it is +1 make it -1. Can you think of a computationally efficient way to do this transformation?)

d. Recalculate its interaction with its nearest neighbors and the field.

e. Use what you calculated in part (b) and (d) to calculate $\Delta E_{Ising}$ for this “spin flip.” [NOTE – there is no need to calculate the TOTAL energy again here, just calculating what has changed due to the flip of the single, randomly-chosen spin, is enough and far more computationally efficient. To check that this is working correctly, though, you may want to add a step within the loop that does recalculate the entire energy – just make it so that you only do a few times while you are looping. For instance, if you go through the loop 10,000 times, then maybe check your constantly-updated energy against the full energy calculation every 1,000 times through.)

f. If the spin flip reduced the energy, accept the change and update the energy for the entire system (by adding the $\Delta E_{Ising}$ to the total energy you had before the flip). If it increased the energy, reject the change, flip the spin back, and keep the total energy you had before flipping the spin.

g. Print out a snapshot of the system (not every timestep, just a subset of times during the simulation.) How to do this? You can simply print out an array of 0s and 1s or 0s and _s, etc...

h. Take any other measurements (leave as a section you can fill in with some tasks later). Be sure to take this measurement whether or not you accepted the spin flip.

i. Start the loop again. Once this is running, check it in several ways to make sure it is doing what you want it to be doing. Then print out some images of a series of microstates that start from completely random and go towards some order. Choose appropriate J and T values to make this happen. Try a few different options for the value of H. You may need to vary the number of times you go through the loop to make sure it is reaching a good endpoint.

Turn in a report that includes your code, your chosen values of J and H (along with a justification), 3-4 images for each series you investigated – include these in professional figure-style format with captions. (I’m expecting about 3-4 images for each series). Then explain your observations in professional language. The code and the figures will take up several pages, but, in total, the written portion should not exceed 1 page.

In Part 2, you will implement a more advanced spin flip “acceptance criteria,” and make more quantitative measurements of the system. In Part 3, you will implement a biasing potential to help us observe rare states of the system