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 repulsive at short distances and attractive 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：本文考虑N粒子相互作用势能的极小化子，简要回顾了用于计算这些极小化子的数值方法。我们关注短程排斥、长程吸引的简单对势，重点分析以两个幂次之和形式呈现的实例。考察的幂次范围涵盖经典的伦纳德-琼斯势，同时也涉及集体行为模型中相关度较低的非奇异势。基于Wales及其合作者开发的化学问题求解软件GMIN，我们报告了计算结果。对于所有算例，该算法在粒子数N相对较小时均能给出极小化子的良好候选解。这一结论对类似伦纳德-琼斯势的情形已为人熟知，但在集体行为所关注的幂次范围内尚属新发现。已有文献对该范围的极小化子采用标准优化流程求解，但此类方法更可能得到非极小化子的稳态构型。我们通过二维数值模拟，阐明了极小化子的若干特性，包括晶格结构、Wulff形状，以及局部可积（即较低奇异度）势在连续大N极限下的行为特征。