Question

Implement the ising program on your favourite computer, using your favourite programming language. Run it for various lattice sizes and temperatures and observe the results. In particular:

(a) Run the program with a 20 x 20 lattice at T = 10, 5, 4, 3, and 2.5, for at least 100 iterations per dipole per run. At each temperature make a rough estimate of the size of the largest clusters.

(b) Repeat part (a) for a 40 x 40 lattice. Are the cluster sizes any different? Explain. (c) Run the program with a 20 x 20 lattice at T = 2, 1.5, and 1. Estimate the average magnetisation (as a percentage of total saturation) at each of these temperatures. Disregard runs in which the system gets stuck in a metastable state with two domains.

(d) Run the program with a 10x 10 lattice at T = 2.5. Watch it run for 100,000 iterations or so. Describe and explain the behaviour.

(e) Use successively larger lattices to estimate the typical cluster size at temperatures from 2.5 down to 2.27 (the critical temperature). The closer you are to the critical temperature, the larger a lattice you'll need and the longer the program will have to run. Quit when you realise that there are better ways to spend your time. Is it plausible that the cluster size goes to infinity as the temperature approaches the critical temperature?

Short answer

The pictures and codes are given.

Step-by-step solution

Given information and explanation

Implement the ising program on your favourite computer, using your favourite programming language. Run it for various lattice sizes and temperatures and observe the results.

A Python application is written, and this is how it looks:

Explanation

a)We can observe that when the temperature rises, the size of the domain shrinks.

b)We can see that the clusters are smaller for the larger lattice.

Explanation

c) We have the following scenarios for these three temperatures. The average magnetisation (number of black squares) is around 50% or higher, as can be shown.