Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant \(d_\infty \) metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on \({{\mathbf {R}}}^n\) whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass \(1/(n+1)\) to each vertex of a (unit-diameter) regular simplex.

考虑一组通过吸引力-排斥势相互作用的粒子集合，该势作为功率定律之差给出并归一化，因此其唯一最小值出现在单位分离时。对于与轻度斥力和强吸引力相对应的一系列指数，我们表明，除了平移和旋转之外，通过将粒子均匀地分布在规则的三维单纯形的顶点（即，等边三角形中）上，可以唯一地获得最小能量配置。二维和常规四面体在三个）。如果假定吸引力不强，则表明这些配置至少是相关\（d_ \ infty \）中的局部能量最小化器衡量最佳运输的效率，其他所有无数不平衡配置都具有相同的支撑。我们推断出相变的存在。该证明部分基于简单的等径方差边界，该边界刻画了规则的单纯形。结果表明，在支持最多具有单位直径的\（{{\\ mathbf {R}}} ^ n \）的概率测度中，均值周围的方差被分配质量\（1 /（n +1）\）到（单位直径）规则单纯形的每个顶点。