Classifying Minimum Energy States for Interacting Particles: Regular Simplices

Abstract

Densities of particles on $${{{\textbf{R}}}^n}$$ which interact pairwise through an attractive-repulsive power-law potential $$W_{\alpha ,\beta }(x) = |x|^\alpha /\alpha -|x|^\beta /\beta $$ have often been used to explain patterns produced by biological and physical systems. In the mildly repulsive regime $$\alpha > \beta \ge 2$$ with $$n \ge 2$$ , we show there exists a decreasing homeomorphism $$\alpha _{\Delta ^n}$$ 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 $$\alpha \ge \alpha _{\Delta ^n}(\beta )$$ . Moreover this minimum is uniquely attained up to rigid motions when $$\alpha > \alpha _{\Delta ^n}(\beta )$$ . We estimate $$\alpha _{\Delta ^n}(\beta )$$ 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 $$(\alpha ,\beta )=(4,2)$$ 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 $$W_{\alpha ,\beta }$$ and corresponding energies in the case where $$n=1$$ , and (ii) for the attractive-repulsive potentials $$D_\alpha (x) = |x|^\alpha (\alpha \log |x|-1)$$ that arise in the limit $$\beta \nearrow \alpha $$ .