Show HN: 2D Coulomb Gas Simulator

Source: Hacker News

Hamiltonian

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 simulator minimizes this Hamiltonian, approximating the minimum‑energy state known as a Fekete configuration.

Applications

The 2D Coulomb gas appears in many areas of mathematics and mathematical physics, including:

• Eigenvalues of a random matrix with Gaussian random entries
• Zeros of a polynomial with Gaussian random coefficients
• Fractional quantum Hall effect
• Hele‑Shaw / Laplacian growth
• Vortices in superconductors

References

• In 2017 it was shown that the density of particles near the boundary follows an erfc distribution, via a remarkably long proof.
• For more background and context, see the author’s bachelor thesis and this blog post.

Performance Note

Exact pairwise repulsion is (O(n^{2})); very large (n) may be slow.