We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are attractive at short distances and repulsive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number N. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-N limit for locally integrable (that is, less singular) potentials.

翻译：我们考虑N粒子相互作用势能的最小化子，并简要回顾用于计算它们的数值方法。我们考察了短距离吸引、长距离排斥的简单对势，重点关注双幂次和形式的实例。所研究的幂次范围包括著名的Lennard-Jones势，但我们也关注集体行为模型中相关的奇异性较弱的势函数。我们报告了基于Wales及其合作者为化学问题开发的GMIN软件的计算结果。在所有案例中，该算法均能给出粒子数N较低时的最小化子候选解。对于类似Lennard-Jones的势函数这一结论已广为人知，但在集体行为所关注的势函数范围内尚未被充分认识。文献中该领域采用的标准最小化方法往往只能得到非最小化子的稳态解。我们通过二维数值模拟展示了最小化子的若干特性，包括晶格结构、Wulff形状，以及局部可积（即奇异性较弱）势函数在连续大N极限下的行为。