For a long time, the walls in my apartment have been empty. So empty that looking at them was like staring into the void and having the void stare right back at you. My girlfriend and I recently solved this situation by making a custom piece of art!

Concretely, we painted a picture of the Ising model on a sheet of metal. This model from statistical mechanics is used to explain how local interactions between particles can give rise to order on large scales. The original application was the study of magnetization but many other systems can be studied with the same model or slight modifications thereof. For instance, consensus formation in opinion dynamics can also be considered.

Formally, let $n\geq 1$ denote a positive integer, set $[n]:= \{1,\ldots,n \}$, and fix a scalar $\beta \geq 0$. Then, the *two-dimensional Ising model* refers to the probability distribution over the collection of all maps $\sigma:[n]^2 \to \{\text{red},\text{blue} \}$ with $$\mathbb{P}(\sigma) = \frac{1}{Z} \exp\Bigl(-\beta \sum_{i\sim j}\mathbb{1}(\sigma_i \neq \sigma_j)\Bigr).$$ Here, the sum runs over all $i,j\in [n]^2$ which are nearest neighbors and $Z$ is a normalization constant. Intuitively, the term $\mathbb{1}(\sigma_i \neq \sigma_j)$ expresses that adjacent atoms tend to have the same magnetization.

It may be easiest to understand with a picture. Behold!

The historical significance of this model is that is displays a phase transition. This means that there is some critical value $\beta_c>0$ such that the limiting behavior of the model for $n\to \infty$ is extremely different depending on whether or not $\beta < \beta_c$. For $\beta < \beta_c$, chaos is dominant and the configurations have no apparent structure. For $\beta > \beta_c$, the local interactions are sufficiently strong to overcome the randomness and order may be observed in the form of large-scale structures. This phenomenon was highly relevant at the time of the model’s invention. Physicists back then did not fully appreciate that finite-particle systems can give rise to sharp transitions in behavior when the number of particles tends to infinity.

If you want to play around with the model a bit, a simulator has been implemented by Daniel V. Schroeder at this link.