Electrostatic Patterns in the Interior of a Circular Region

N.G. Nicolis
Bridges Donostia: Mathematics, Music, Art, Architecture, Culture (2007)
Pages 169–176


We examine the formation of symmetric patterns of N equal charges in static equilibrium, confined in the interior of a circular region by a harmonic oscillator potential. We develop a simulation procedure for the determination of their equilibrium positions. Initially, the charges are assumed to occupy random positions on the plane. The time evolution of the system is determined with a numerical integration of the equations of motion of the N charges. By imposing a proper kinetic energy dissipation mechanism, equilibrium configurations are obtained. They correspond to the absolute minimum (ground state) or to a secondary higher minimum (excited states) of the multidimensional potential energy surface. Ground state configurations are presented for N=5-30, 45 and 230. They involve arrangements of charges at the vertices of regular polygons inscribed in concentric circles. The sequence of occupation indicates a shell effect, similar to the one encountered in the occupation of the electron orbits of the atoms. With increasing N, the rotational symmetry tends to disappear, competing with the one of an infinite triangular lattice. Changes in the symmetry of the ground to the excited state(s) are discussed in typical cases. Interesting patterns of a greater complexity are produced by groups of unequal charges in the same confinement.