Determining whether two particle systems are similar is a common problem in particle simulations. When the comparison should be invariant under permutations, orthogonal transformations, and translations of the systems, special techniques are needed. We present an algorithm that can test particle systems of finite size for similarity and, if they are similar, can find the optimal alignment between them. Our approach is based on an invariant version of the root mean square deviation (RMSD) measure and is capable of finding the globally optimal solution in $O(n^3)$ operations where $n$ is the number of three-dimensional particles.

翻译：确定两个粒子系统是否相似是粒子模拟中常见的一个问题。 当比较应在变异、正方形变异和系统翻译中是无变的时,需要特殊技术。 我们提出一种算法,可以测试大小有限的粒子系统以相似性,如果它们相似,可以找到它们之间最佳的对齐。 我们的方法基于根平方偏差(RMSD)测量的变异性版本,并且能够以美元(n%3)的操作找到全球最佳解决方案,而美元就是三维粒子的数量。