Ising
The images below picture configurations of the Ising model. A very simple model of a magnetic material that nevertheless exhibits a surprisingly complex behavior.
In this model, we imagine that "atoms" are placed on a square lattice two-dimensional lattice. Each atom acts like a small magnet, or "spin", that can take only two configurations, up and down, symbolised in the pictures below by black and white. If two neighboring spins are both up or both down, the spins "repels" each other and the energy of the system is higher than when one is up and the other is down. The minimum energy state of the system is therefore when the spins are either all up or all down. This corresponds to a "magnetized" material.
To make the system more interesting, we introduce a non-zero "temperature". The effect of the temperature is to randomly switch some of the spins. The higher the temperature, the more the spins are randomly switched. If the temperature is very high, it is clear that a typical state of the system will not be one of the states of minimal energy, but rather a state in which the spins are randomly up or down. The material is demagnetised.
An interesting question is to determine how exactly the material changes from a magnetised state to a demagnetised one as a function of the temperature. It turns out that in the limit of a large system, there is a sharp transistion at a critical temperature. When the pattern formed by the spins is neither monotone nor random: they rather form a random fractal pattern in which arbitrarily large clusters of either spins coexist.
This what the pictures below illustrate.
For more information about the Ising model, check out this very good blog post by the author of the code I used to generate the pictures below.
First, this is what happens when the system start from a random state, and is left to evolve at a temperature below the critical one. We see that large clusters of spins form. If the system was left to evolve for long enough, a single cluster would eventually take over the whole picture.
Next this is what happens above the critical temperature. Although small cluster do form, they never get big. In the limit where the size of the system become infinite, these configurations are indistinguishable from the random starting configuration.
Finally, here is what happens at the critical temperature. The last image is the most interesting one. Here we see that although we have arbitrarily large clusters of spins, there is no region where they take over the opposite spins, resulting in a characteristic fractal pattern.
And here is a video showing the process. The structures build up in a time exponential in their size, so in the video below the time accelerate exponentially. This makes it a bit slow to start building small structures, but it's worth being patient!
