A random batch Ewald method for particle systems with Coulomb interactions

TL;DR

The paper introduces a random batch Ewald (RBE) method that efficiently simulates particle systems with Coulomb interactions, achieving linear complexity and reducing computational cost while maintaining accuracy.

Contribution

It presents a novel RBE method combining Ewald splitting with random mini-batches and importance sampling, enabling fast, unbiased Coulomb force calculations with controlled variance.

Findings

Achieves O(N) complexity per simulation step.

Demonstrates accuracy comparable to classical Ewald summation.

Shows significant efficiency improvements in charged system simulations.

Abstract

We develop a random batch Ewald (RBE) method for molecular dynamics simulations of particle systems with long-range Coulomb interactions, which achieves an $O(N)$ complexity in each step of simulating the $N$-body systems. The RBE method is based on the Ewald splitting for the Coulomb kernel with a random "mini-batch" type technique introduced to speed up the summation of the Fourier series for the long-range part of the splitting. Importance sampling is employed to reduce the induced force variance by taking advantage of the fast decay property of the Fourier coefficients. The stochastic approximation is unbiased with controlled variance. Analysis for bounded force fields gives some theoretic support of the method. Simulations of two typical problems of charged systems are presented to illustrate the accuracy and efficiency of the RBE method in comparison to the results from the Debye-H\"uckel theory and the classical Ewald summation, demonstrating that the proposed method has the attractiveness of being easy to implement with the linear scaling and is promising for many practical applications.