I implemented a discretization of a weird 2D heat equation "mod $2\pi$",
$$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$
where (WARNING: handwavy, I'm not sure I understand it) $\Delta^*=\nabla^*\cdot\nabla^*$ where $\nabla^*$ is the gradient reduced into the interval $[-\pi,\pi]$.
Starting with a 2D random noise array and using periodic boundary conditions, I calculated the time-evolution of the array $A(t)$ and plotted the evolution as a complex phase plot, ie, I plotted $e^{iA(t)}$ using an argument-to-hue colormap. The results were pretty fascinating:
https://www.youtube.com/watch?v=oMp7OLojuTw
Here's some of the still frames, in the event you can't access YouTube:
Of note is that the noise smooths out into pairs of "particle-antiparticle pairs", which gradually attract and annihilate each other, eventually resulting in a constant solution. The "particles" are point discontinuities with clockwise hue and locally look like the complex function $f(z)=z$, and the "antiparticles" have counterclockwise hue and locally look like $f(z)=\bar{z}$.
I have run a bunch of simulations, and it appears that every particle has a corresponding antiparticle, and so given enough time, all pairs annihilate, resulting in a constant solution. I have not yet observed any stable nonconstant solutions.
I have two questions:
Is there a name for this equation or an equation with similar qualitative behavior, and if so, has it been studied?
Is it possible to prove that there is a pairing between particles and antiparticles? On a related note, are there any nonconstant equilibrium solutions?
I suspect there might be a topological reason why there appears to be a 1-to-1 correspondence between particles and antiparticles, but I'm too dumb to figure it out. At some point I may compute the time-evolution on a Klein bottle and see if any interesting solutions arise.
Implementation Details
The discretization used was similar to the traditional explicit discretization of the heat equation,
$$A_{i,j}^{k+1}=A_{i,j}^{k} + \alpha\left(A_{i-1,j}^{k}+A_{i+1,j}^{k}+A_{i,j-1}^{k}+A_{i,j+1}^{k}-4A_{i,j}^{k}\right)$$
but used a series of modifications. First, I ended up increasing the "radius" of the difference, ie, I used the stencil
$$A_{i,j}^{k+1}=A_{i,j}^{k} + \alpha\left(A_{i-r,j}^{k}+A_{i+r,j}^{k}+A_{i,j-r}^{k}+A_{i,j+r}^{k}-4A_{i,j}^{k}\right)$$
where $r>1$ because using $r=1$ gave "uninteresting" results. To insert the "mod $2\pi$" behavior of $\nabla^*$, I converted the above to the following:
$$A_{i,j}^{k+1}=A_{i,j}^{k} + \alpha\left[\text{mod}\left(A_{i-r,j}^{k}-A_{i,j}^{k},2\pi,\pi\right) + \text{mod}\left(A_{i+r,j}^{k}-A_{i,j}^{k},2\pi,\pi\right) + \text{mod}\left(A_{i,j-r}^{k}-A_{i,j}^{k},2\pi,\pi\right) + \text{mod}\left(A_{i,j+r}^{k}-A_{i,j}^{k},2\pi,\pi\right)\right]$$
where $\text{mod}(a,b,c)$ is $a$ reduced into the interval $[-c,b-c]$. Finally, I made the above time-evolution "stochastic" by computing only one difference at a time using a point within radius $r$ of the center:
$$A_{i,j}^{k+1}=A_{i,j}^{k} + \alpha\text{mod}\left(A_{i-a,j-b}^{k}-A_{i,j}^{k},2\pi,\pi\right)$$
where $a,b$ are randomly-chosen integers satisfying $a^2+b^2\leq r^2$. This was my final discretization, with $r\approx 7$ and $\alpha\approx 0.1$.