Each dot represents an electron experiencing pairwise Coulomb repulsion with every other electron while being confined by an external potential $Q$. The energy of a configuration $z_1, \dots, z_n$ is given by the 2D log-gas Hamiltonian $$H(z_1,\ldots,z_n) = -\sum_{i \neq j} \log\lvert z_i - z_j \rvert + n\sum_{j=1}^n Q(z_j).$$ The 2D Coulomb gas is interesting because this type of Hamiltonian shows up in many different places across mathematics / mathematical physics:
- Eigenvalues of a random matrix with Gaussian random entries
- Zeroes of a polynomial with Gaussian random coefficient
- Fractional quantum hall effect
- Hele-Shaw/Laplacian growth
- Vortices in superconductors
Consequently, there is a large body of research devoted to deducing properties of this family of systems. For example, in 2017 in was
shownthat the density of particles near the boundary follows an
erfcdistribution by means of a remarkably long proof. Of course, with this simulator we minimize the Hamiltonion, not sample from it in a temperature dependent way. We therefore approximate the minimum-energy state which is known as a
Fekete configuration.
For more on the background and context of these systems, I implore you to look into my
bachelor thesisor this
blog post.
Comments
You have to painfully force quit Safari by going through the motions of the gestures, wait for the phone to respond, and then quit Safari from the card task switcher.
Note 2: In this case, 2D means that the force is proportional to distance^-1 instead of the usual distance^-2. It's not a 3D word projected into a plane, it's a real 2D word that lives inside a plane. It would be nice to have an option to switch and compare. I have no intuition about the difference.
Note 3: Somewhere in the middle, it's possible to switch the external potencial from z^2 to z^4 to z^20, where z is the distance from the center.
z^2 "Ginibre" is a nice round surface, like a wok. Electrons get evenly distributed.
z^20 "Mittag-Leffler lambda=10" is flat in the center and then goes up quickly, like a saucepan. Electrons escape from each other, a few of them remain in the center but most get squashed against the circular wall.
z^4 "Mittag-Leffler lambda=2" Something in between
Thanks for playing around with the tool!
With z^20, the problem is that when you change the number of particles, the ones are distributed randomly and the ones near the corners have a huge gradient and probably overflow and the inifinites/nans are viral and kill all the other particles. The trick is to switch to z^2, change N wait a moment and then change to z^20. Perhaps you can clip some values or try some trick like in stiff equations.
In 3D, I expected a z^2 potencial with a 1/z^2 force to generate an uniform distribution, for something something Gauss. (It's just bad hand-waving, I didn't have anything close to a proof.) It's interesting that it is so easy.