Classifying Minimum Energy States for Interacting Particles: Regular Simplices

Densities of particles on R n which interact pairwise through an attractive-repulsive power-law potential W α , β ( x ) = | x | α / α - | x | β / β have often been used to explain patterns produced by biological and physical systems. In the mildly repulsive regime α > β ≥ 2 with n ≥ 2 , we show there exists a decreasing homeomorphism α Δ n from [2, 4] to itself such that: distributing the particles uniformly over the vertices of a regular unit diameter n -simplex minimizes the potential energy if and only if α ≥ α Δ n ( β ) . Moreover this minimum is uniquely attained up to rigid motions when α > α Δ n ( β ) . We estimate α Δ n ( β ) above and below, and identify its limit as the dimension grows large. These results are derived from a new northeast comparison principle in the space of exponents. At the endpoint ( α , β ) = ( 4 , 2 ) of this transition curve, we characterize all minimizers by showing they lie on a sphere and share all first and second moments with the spherical shell. Suitably modified versions of these statements are also established (i) for W α , β and corresponding energies in the case where n = 1 , and (ii) for the attractive-repulsive potentials D α ( x ) = | x | α ( α log | x | - 1 ) that arise in the limit β ↗ α .