A theory of complex patterns arising from 2D particle interactions

Pairwise particle interactions arise in diverse physical systems ranging from insect swarms to selfassembly of nanoparticles. This letter develops a fundamental theory for the morphology of patterns in two dimensions which can range from ring and annular states to more complex spot patterns with N-fold symmetry. Many of these patterns have been observed in nature although a general theory has been lacking, in particular how small changes to the interaction potential can lead to large changes in self-organized state. Emergence of these patterns is explained by a stability analysis of a ring solution. This analysis leads to analytic formulae involving the interaction potential that provide detailed information about the structure of complex equilibria.