Rapid Sampling of Stochastic Displacements in Brownian Dynamics Simulations with Stresslet Constraints

TL;DR

This paper introduces a computationally efficient method for Brownian Dynamics simulations of colloidal particles with stresslet constraints, enabling rapid and accurate modeling of hydrodynamic interactions with linear scaling.

Contribution

The authors extend the positively-split Ewald summation method to include stresslet constraints, allowing fast and scalable sampling of hydrodynamic displacements in constrained Brownian Dynamics.

Findings

Method achieves linear scaling with particle number.

Rapid sampling of Brownian displacements is possible.

Hydrodynamic interactions are accurately modeled under constraints.

Abstract

Brownian Dynamics simulations are an important tool for modeling the dynamics of soft matter. However, accurate and rapid computations of the hydrodynamic interactions between suspended, microscopic components in a soft material is a significant computational challenge. Here, we present a new method for Brownian Dynamics simulations of suspended colloidal scale particles subject to an important class of hydrodynamic constraints with practically linear scaling of the computational cost with the number of particles modeled. Specifically, we consider the "stresslet" constraint for which suspended particles resist local deformation. The presented method is an extension of the recently reported positively-split formulation for Ewald summation of the Rotne-Prager-Yamakawa (RPY) mobility tensor to higher order (dipole) terms in the hydrodynamic scattering series [Andrew M. Fiore et al. The Journal of Chemical Physics, 146(12):124116, 2017]. The hydrodynamic mobility tensor, which is proportional to the covariance of particle Brownian displacements, is constructed as an Ewald sum in a novel way that guarantees the real-space and wave-space sums are independently positive-definite for all possible particle configurations. This property is leveraged to rapidly sample the Brownian displacements from a superposition of statistically independent processes with the wave-space and real-space contributions as respective covariances. The cost of computing the Brownian displacements in this way is comparable to the cost of computing the deterministic displacements. Addition of a stresslet constraint to the over-damped particle equations of motion leads to a stochastic differential algebraic equation (SDAE) of index 1, which is integrated in time using a mid-point integration scheme that implicitly produces displacements consistent with the fluctuation-dissipation theorem for the constrained system.