Combinatorics

Physics

Electron

Particle system

Mathematics

Kernel (algebra)

Fourier transform

Operating system

Statistical physics

Statistics

Ewald summation

Mathematical analysis

Composite material

Materials science

Monte Carlo method

Range (aeronautics)

Fourier series

Geometry

Computer science

Molecular dynamics

Quantum mechanics

Applied mathematics

Coulomb

Scaling

SIAM Journal on Scientific Computing

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 $N$-body systems. The RBE method is based on the Ewald splitting for the Coulomb kernel with a random “minibatch” 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--Hückel theory, the classical Ewald summation, and the particle-particle particle-mesh method, demonstrating that the proposed method has the attractiveness of being easy to implement with the linear scaling and is promising for many practical applications.

A Random Batch Ewald Method for Particle Systems with Coulomb Interactions