You then add the number of \(+1\)s and \(-1\)s together, which will give you the total magnetization. To take this a little further, imagine that you assign a value of \(+1\) to each up arrow and a value of \(-1\) to each down arrow. Above the critical temperature, you will find that, on average, 50% of the arrows will point up and the other 50% will point down, much like in this image: Now, if we “turn on” the temperature, some of these arrows will start to flip their direction from up to down. The subsections in How the Simulation Works explain some of the details behind why the arrows would prefer to all point in the same direction. A grid of all up arrows is called ferromagnetism and is a model of what happens with iron at lower temperatures: It may sound overly simplistic, but a reasonable model for a permanent magnet is as a grid of arrows that either point up or down. This is the context for the Ising Model: we want to model what happens to a permanent magnet as you increase the temperature and it starts to demagnetize. And, it turns out, the demagnetization process that’s induced by heating a material is quite complicated, but yet can be modeled in a straightforward way. For example, above 1418 degrees Fahrenheit (the critical temperature) iron is no longer magnetic. However, this type of magnetism cannot be sustained under all conditions. Permanent magnets are different from other kinds of magnets in that they generate a magnetic field via their internal structure. Iron is a classic and familiar example of a permanent magnet. There has been great utility in studying this seemingly simple model, and it continues to play an important role in simulated-based science. Furthermore, Ising-like models have been used outside of the field of physics to explain complex behaviorial phenomena, including rational herding, segregation, and how languages change over time. The Monte Carlo method for simulating the two-dimensional Ising model is a classic problem in statistical mechanics and magnetic modeling, and the simulation itself is intriguing to watch when converted into an animation. Solving the Ising model is a statistical problem, and so its results can be numerically calculated using the Monte Carlo method to run simulations. Only a specific version of the two-dimensional Ising Model can be solved exactly, and the three-dimensional version cannot be solved exactly in any form. It should also be noted that the Ising Model is a special case of the more general Heisenberg Model, which is a magnetic model that is still in use to this day. This is an example of short-range local interactions giving rise to extended long-range behavior, which can be a counter-intuitive and unexpected outcome.
The Ising Model is interesting due to the two- and three-dimensional versions exhibiting a phase transition at a critical temperature, above which the model no longer exhibits permanent magnetism. It is named after Ernst Ising, who solved the one-dimensional version exactly as part of his 1924 thesis. The Ising Model is a model of a permanent magnet.
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