On Soccer Balls and Linearized Inverse Statistical Mechanics

The classical inverse statistical mechanics question involves inferring properties of pairwise interaction potentials from exhibited ground states. For patterns that concentrate near a sphere, the ground states can range from platonic solids for small numbers of particles to large systems of particles exhibiting very complex structures. In this setting, previous work (von Brecht et al., Math. Models Methods Appl. Sci. 22, 2012) allows us to infer that the linear instabilities of the pairwise potential accurately characterize the resulting nonlinear ground states. Potentials with a small number of spherical harmonic instabilities may produce very complex patterns as a result. This leads naturally to the linearized inverse statistical mechanics question: given a finite set of unstable modes, can we construct a potential that possesses precisely these linear instabilities? If so, this would allow for the design of potentials with arbitrarily intricate spherical symmetries in the ground state. In this paper, we solve our linearized inverse problem in full, and present a wide variety of designed ground states.