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Show HN: 2D Coulomb Gas Simulator
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.
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