Langevin equations and a geometric integration scheme for the overdamped limit of rotational Brownian motion of axisymmetric particles

TL;DR

This paper develops a geometric approach to model and simulate the rotational Brownian motion of axisymmetric particles, providing a scheme that preserves constraints and improves simulation efficiency.

Contribution

It introduces a geometric derivation of Langevin equations for rotational Brownian motion and proposes an efficient, constraint-preserving integration scheme with proven accuracy and stability.

Findings

The scheme converges weakly at order 1 with simple implementation.

Gaussian random rotations enable larger time steps than traditional methods.

The method satisfies detailed balance and converges to the correct equilibrium distribution.

Abstract

The translational motion of anisotropic or self-propelled colloidal particles is closely linked with the particle's orientation and its rotational Brownian motion. In the overdamped limit, the stochastic evolution of the orientation vector follows a diffusion process on the unit sphere and is characterized by an orientation-dependent (``multiplicative'') noise. As a consequence, the corresponding Langevin equation attains different forms depending on whether It\=o's or Stratonovich's stochastic calculus is used. We clarify that both forms are equivalent and derive them in a top-down appraoch from a geometric construction of Brownian motion on the unit sphere, based on infinitesimal random rotations. Our approach suggests further a geometric integration scheme for rotational Brownian motion, which preserves the normalization constraint of the orientation vector exactly. We show that a simple implementation of the scheme, based on Gaussian random rotations, converges weakly at order 1 of the integration time step, and we outline an advanced variant of the scheme that is weakly exact for an arbitrarily large time step. Due to a favorable prefactor of the discretization error, already the Gaussian scheme allows for integration time steps that are one order of magnitude larger compared to a commonly used algorithm for rotational Brownian dynamics simulations based on projection on the constraining manifold. For torques originating from constant external fields, we prove by virtue of the Fokker-Planck equation that the constructed diffusion process satisfies detailed balance and converges to the correct equilibrium distribution. The analysis is restricted to time-homogeneous rotational Brownian motion (i.e., a single rotational diffusion constant), which is relevant for axisymmetric particles and also chemically anisotropic spheres, such as self-propelled Janus particles.