An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere

Determination of globally optimal arrangements of N pairwise-interacting particles is an important problem that occurs in a variety of biological, physical, and chemical applications. We propose a numerical procedure to compute putative optimal configurations. The procedure is able to handle a wide class of pairwise potentials. Locally and globally minimal arrangements of particles on the unit sphere, interacting via the Coulombic, logarithmic, and inverse square law, are computed as samples. We present new results for the logarithmic potential consisting of 45 new local minima for N <= 65 and two new global minima (N = 19, 46), as well as results for the inverse square law potential which has not previously been studied. We provide comprehensive tables of all minima found, and exclude all saddle points. The algorithm can perform computations exceeding N = 100 with reasonable execution times.求解N个两两相互作用粒子的全局最优排布，是一类广泛出现于生物、物理与化学领域的重要科学问题。本文提出一种数值方法，用于计算推定最优粒子构型。该方法可适配广泛类别的两两相互作用势（pairwise potentials）。我们针对通过库仑相互作用（Coulombic）、对数相互作用（logarithmic）以及平方反比律（inverse square law）进行作用的粒子体系，以单位球面（unit sphere）上的粒子局部最小排布与全局最小排布为示例开展计算。针对对数相互作用势（logarithmic potential），本文得到全新研究结果：当粒子数N≤65时，共发现45个新的局部极小值，以及2个新的全局极小值（对应N=19与N=46）；同时我们还获得了此前尚未被研究过的平方反比律相互作用势相关计算结果。本文提供了所有已发现极小值的完整汇总表格，并剔除了所有鞍点（saddle points）。该算法可在合理的运行时长内完成粒子数N>100的计算任务。