Multibody Multipole Methods

TL;DR

This paper introduces a fast, scalable tree-code algorithm for efficiently approximating multibody potentials involving more than two particles, enabling practical computations previously hindered by high complexity.

Contribution

It presents the first Barnes-Hut type algorithm for multibody interactions, combining deterministic and Monte Carlo approximation schemes with error guarantees.

Findings

Achieved significant speedups in computing three-body potentials.

Demonstrated the algorithm's accuracy with error bounds.

Applied method to the Axilrod-Teller potential with promising results.

Abstract

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.