Rapid Sampling of Stochastic Displacements in Brownian Dynamics Simulations

TL;DR

This paper introduces a novel, scalable method for efficiently sampling stochastic displacements in Brownian Dynamics simulations of colloidal particles, significantly improving computational speed and accuracy for large systems.

Contribution

The paper presents a new formulation for Ewald summation of the RPY tensor that enables linear scaling in sampling Brownian displacements for large particle systems.

Findings

Scales linearly for up to 4 million particles

Achieves over tenfold speedup compared to existing methods

Maintains positive-definiteness and symmetry in tensor decomposition

Abstract

We present a new method for sampling stochastic displacements in Brownian Dynamics (BD) simulations of colloidal scale particles. The method relies on a new formulation for Ewald summation of the Rotne-Prager-Yamakawa (RPY) tensor, which guarantees that the real-space and wave-space contributions to the tensor are independently symmetric and positive-definite for all possible particle configurations. Brownian displacements are drawn from a superposition of two independent samples: a wave-space (far-field or long-ranged) contribution, computed using techniques from fluctuating hydrodynamics and non-uniform Fast Fourier Transforms; and a real-space (near-field or short-ranged) correction, computed using a Krylov subspace method. The combined computational complexity of drawing these two independent samples scales linearly with the number of particles. The proposed method circumvents the super-linear scaling exhibited by all known iterative sampling methods applied directly to the RPY tensor that results from the power law growth of the condition number of tensor with the number of particles. Calculations for hard sphere dispersions and colloidal gels are illustrated and used to explore the role of microstructure on performance of the algorithm. In practice, the logarithmic part of the predicted scaling is not observed and the algorithm scales linearly for up to 4 million particles, obtaining speed ups of over an order of magnitude over existing iterative methods, and making the cost of computing Brownian displacements comparable the cost of computing deterministic displacements in BD simulations. A high-performance implementation employing non-uniform fast Fourier transforms implemented on graphics processing units and integrated with the software package HOOMD-blue is used for benchmarking.