Classifying Minimum Energy States for Interacting Particles: Spherical Shells

Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems and to predict or explain many of the patterns they display. Apart from rare values of the attractive and repulsive exponents $(\alpha,\beta)$, the energy minimizing configurations of particles are not explicitly known, although simulations and local stability considerations have led to conjectures with strong evidence over a much wider region of parameters. For dimension $n\ge 2$ and for a segment $\beta=2<\alpha<4$ on the mildly repulsive frontier we employ strict convexity to conclude that the energy is uniquely minimized ($d_\infty$-locally, up to translation) by a spherical shell. If $n=1$ and $\beta=2<\alpha-1$, we prove that the spherical shell is (i) the unique global energy minimizer and (ii) the unique $d_\infty$-local energy minimizer in the class of even, compactly supported probability measures. In a companion work, we show that, in the mildly repulsive range $\alpha>\beta\ge2$, a unimodal threshold $2<\alpha_{\Delta^n}(\beta) \le \max\{\beta,4\}$ exists such that equidistribution of particles over a unit diameter regular $n$-simplex minimizes the energy if and only if $\alpha \ge \alpha_{\Delta^n}(\beta)$ (and minimizes uniquely up to rigid motions if strict inequality holds). For $n\ge 2,$ the point $(\alpha,\beta)=(2,4)$ separates these regimes. At this point we show the minimizers all lie on a sphere and are precisely characterized by sharing all first and second moments with the spherical shell. Although the minimizers need not be asymptotically stable, our approach establishes $d_\alpha$-Lyapunov nonlinear stability of the associated ($d_2$-gradient) aggregation dynamics near the minimizer in both of these adjacent regimes---without reference to linearization. The $L^\alpha$-Kantorovich--Rubinstein--Wasserstein distance $d_\alpha$ which quantifies stability is chosen to match the attraction exponent.

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