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 shown that the density of particles near the boundary follows an erfc distribution 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 thesis or this blog post.
Exact pairwise repulsion is O(n²); very large n may be slow.
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