多体多极方法
Multibody Multipole Methods
摘要：
一个三体势函数可以解释三个粒子之间的相互作用，这些粒子不能被库仑势或伦纳-琼斯势等成对相互作用函数捕获。同样地，$n$阶的多体势可以解释未被低阶相互作用函数捕获的粒子的$n$元组之间的相互作用。到目前为止，由于其$O(N^n)$标度成本，计算大量粒子的多体势函数是不可能的。在本文中，我们描述了一个快速的树代码有效地近似多体潜力，可以分解为产品的两两距离的功能。第一次，我们展示了如何得到一个Barnes-Hut型算法处理两个以上的粒子之间的相互作用。我们的算法使用两种近似方案：1）基于确定性级数展开的方法; 2）基于中心极限定理的Monte Carlo近似。我们的方法保证了用户指定的绝对或相对误差在计算潜在的渐近概率保证的界限。我们提供了一个三体色散势，Axilrod-Teller势的加速结果。
英文摘要：
A three-body potential function can account for interactions among triples of particles which are uncaptured by pairwise interaction functions such as Coulombic or Lennard-Jones potentials. Likewise, a multibody potential of order $n$ can account for interactions among $n$-tuples of particles uncaptured by interaction functions of lower orders. To date, the computation of multibody potential functions for a large number of particles has not been possible due to its $O(N^n)$ scaling cost. In this paper we describe a fast tree-code for efficiently approximating multibody potentials that can be factorized as products of functions of pairwise distances. For the first time, we show how to derive a Barnes-Hut type algorithm for handling interactions among more than two particles. Our algorithm uses two approximation schemes: 1) a deterministic series expansion-based method; 2) a Monte Carlo-based approximation based on the central limit theorem. Our approach guarantees a user-specified bound on the absolute or relative error in the computed potential with an asymptotic probability guarantee. We provide speedup results on a three-body dispersion potential, the Axilrod-Teller potential.
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