The Beauty of Simplicity

Article
Zach Robertson

19 days ago by zachrobertson

If you have no interest in learning anything about physics I suggest you immediately click away from this post, for those of you still remaining I am going to explain a concept that I presented during my technical interview for my current job. It is an interesting display of how a simple model can produce amazing results.

The idea behind the Ising model was to represent a system of particles, in a lattice of N dimensions that resides in a thermodynamic system, with the least amount of interactions between particles as possible. To do this we can ignore all of the interactions between particles, except for those between nearest neighbor particles due to the spin of each particle (Spin refers to a quantum mechanical property of a particle and not the physical spin of a particle). These spins interact like two magnetic dipoles (I won’t get into why this is but if you want a way to conceptualize it, and have a good understanding of electromagnetism, think about a point charge spinning in a ring and what kind of magnetic field that produces) so we can describe the total energy E of a two dimensional lattice as :

Hamiltonian of the Ising model

were the sum over <i, j> means a sum over the nearest neighbors j for each point i in the lattice (in two dimensions the nearest neighbors would be the points directly above, below, left and right of the particle i). The second term can be ignored for now as it represents an external magnetic field and that will only complicate the problem and does not change whether a system will exhibit a phase transition or not.

Now that we have the total energy we also have the Hamiltonian H of the system, this is very important as it allows us to calculate quantities of interest, the main one being the Magnetization of the system as the temperature of the thermodynamic system is changed.

The reason magnetization was an important area of study is that the Bohr-Van Leeuwen theorem states that classical physics can not account for magnetism of any sort. This meant that scientist in the early nineteen-hundreds were looking for a quantum mechanical explanation for magnetism so the Ising model arose as a way to explore the mathematical consequences of quantum mechanical effects on a thermodynamic system. It was first solved by Ernst Ising in 1925 for the one dimensional case where he saw that there was no phase transition (this is not really a problem though as one dimensional systems do not exists in physical reality). In 1944 Lars Onsager used a transfer-matrix method approach to find an equation for the magnetization of the two dimensional system and saw that there was a phase transition of the magnetization (meaning that the spin interactions of particles could explain magnetization for two dimensional objects). The obvious next step is to try the transfer-matrix method on the three dimensional system, unfortunately no one has been able to find an exact solution for the magnetization of the three dimensional system with any method as of today.

We are not left hopeless though, we can use numerical methods to find an approximate representation of the magnetization of the three dimensional system. There are many approach to do this but the one I have used utilized a Monte Carlo method for numerical results. The basic idea being that we pick a random point from the system and have a set of criteria to determine whether we flip the spin of that point in the system (the spin can only be up(+1) or down(-1) ) or leave it alone, the criteria being based on the simulated “temperature” of the system. We do this random sampling a lot of times, until we are sure the system has had enough “time” to equilibrate and measure the magnetization of the system. Then we do it all over again for a wide range of temperatures and plot the results to see if we get a phase transition. For the three dimensional case the results of this numerical computation on a number of different sized lattices (all squares ) are shown below:

3D Magnetization Plot

This graph shows a plot of the magnetization of the lattice versus the simulated “temperature” of the system, and you can clearly see a phase transition. This means below a certain point there is a spontaneous bulk magnetization of the system.

While this might all sound like useless physics and boring computational analysis, there is an important take away from this. Physicists were attempting to break something down into its most basic form to see if it would yield any information about the system. While you would think that removing so much of the information of the system would completely destroy the properties of that system, the Isisng model shows that breaking down a problem into its smallest pieces and studying those pieces in great detail is a way to learn fundamental truths about the systems you are studying.

Thanks for reading
Zach

P.S. If you want to see a more technical write-up on the ising model as the well as the model I used to create the graph shown in the article, and some good 2D and 3D plots of the actual lattices, check out my github repo here