Equilibrium configurations of N equal charges on a sphere

The emergence of new levels of complexity that often accompanies the transition from few- to many-body systems is clearly illustrated by the progression of equilibrium states of N charges on the surface of a sphere as N increases. The characteristics of these electrostatic equilibrium states in the range 2<or=N<or=65 can be examined in detail by equilibrating 1000 randomized initial configurations for every value of N. As N increases, the equilibrium states undergo a succession of structural changes. For instance, a state with non-zero dipole moment appears at N=11, an enantiomeric or mirror image state appears at N=15, and a robust metastable state appears at N=16. For values of N exceeding 50, clusters of four or more metastable states with energies within 0.01% of each other are the dominant pattern. In analogy with some other complex systems, these energetically similar states have strikingly different asymmetric configurations.