Numerical Methods with Coordinate Transforms for Efficient Brownian Dynamics Simulations

TL;DR

This paper introduces coordinate and time-rescaling transforms to improve the accuracy and efficiency of numerical simulations of Brownian dynamics with configuration-dependent diffusion, applicable to complex multibody systems.

Contribution

It develops a method combining coordinate and time-rescaling transforms to map variable-diffusion processes into constant-diffusion ones, enhancing simulation accuracy and efficiency.

Findings

Transforms improve convergence to invariant distribution.

Numerical simulations show increased efficiency.

Applicable to multibody, anisotropic diffusion in biophysics.

Abstract

Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.