Simulation
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What is this?
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 described 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 minimum-energy state is a Fekete configuration. For more on the background and context of these systems see my bachelor thesis below.
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Potential
Symmetric potentials
Lemniscate potentials
Number of particles
Exact pairwise repulsion is O(n²); very large n may be slow.
Rendering
Particle radius
Particle opacity
Lemniscate & inserted scaling
Critical scaling
Interpolation (Lemniscate)
Inserted particle settings
Tune the strength $c$ and location $(x, y)$ of logarithmic insertions $-c\,\log|z - p|$ to pin particles at points $p_a$ and $p_b$ inside the droplet.
$c_{a}$
$a_{x}$
$a_{y}$
$c_{b}$
$b_{x}$
$b_{y}$