分类相互作用粒子的最小能量状态：正则单形
Classifying minimum energy states for interacting particles: Regular simplices
摘要：
$\Rn$上的粒子通过吸引-排斥幂律势$W_{\al,\bt}(x) = |x|^\al/\al-|x|^\bt/\bt$成对相互作用，它们的密度经常被用来解释生物和物理系统产生的模式。在温和的排斥制度$\al> \bt \ge 2$与$n \ge 2$，我们发现有一个减少同胚$\al_{Δ^n}$从$[2,4]$本身，使得：均匀分布的粒子在一个规则的单位直径$n$ -单纯形的顶点最小化的势能当且仅当$\al\ge \al_{\De^n}(\bt)$。此外，当$\al > \al_{\De^n}(\bt)$。我们估计$\al_{\De^n}(\bt)$以上和以下，并确定其限制的尺寸变大。这些结果是由指数空间中的一个新的东北比较原理导出的。在这个过渡曲线的端点$(\al,\bt)=(4,2)$，我们的特点是所有极小显示他们躺在一个球体上，并分享所有的第一和第二时刻的球壳。这些陈述的适当修改的版本也建立（i）$W_{α,β}$和相应的能量的情况下，$n=1$，和（ii）的吸引-排斥势$D_\al(x) = |x|^\al(\al\log |x|-1)$，出现在极限$\bt \nearrow \al$。
英文摘要：
Densities of particles on $\Rn$ which interact pairwise through an attractive-repulsive power-law potential $W_{\al,\bt}(x) = |x|^\al/\al-|x|^\bt/\bt$ have often been used to explain patterns produced by biological and physical systems. In the mildly repulsive regime $\al> \bt \ge 2$ with $n \ge 2$, we show there exists a decreasing homeomorphism $\al_{Δ^n}$ 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 $\al\ge \al_{\De^n}(\bt)$. Moreover this minimum is uniquely attained up to rigid motions when $\al > \al_{\De^n}(\bt)$. We estimate $\al_{\De^n}(\bt)$ 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 $(\al,\bt)=(4,2)$ 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 $D_\al(x) = |x|^\al(\al\log |x|-1)$ that arise in the limit $\bt \nearrow \al$.
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