Classifying Minimum Energy States for Interacting Particles: Regular Simplices

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 $$n , we show there exists a decreasing homeomorphism _{\Delta ^n}$$ from [2, 4] itself such that: distributing uniformly over vertices regular unit diameter n-simplex minimizes energy if only \alpha ^n}(\beta )$$ . Moreover this minimum is uniquely attained up rigid motions when We estimate above below, identify its limit as dimension grows large. These results are derived new northeast comparison principle in space exponents. At endpoint $$(\alpha )=(4,2)$$ transition curve, characterize all minimizers showing they lie sphere share first second moments spherical shell. Suitably modified versions these statements also established (i) for }$$ corresponding energies case where $$n=1$$ (ii) potentials $$D_\alpha (x) (\alpha \log |x|-1)$$ that arise $$\beta \nearrow