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

Felix Höfling, Arthur V. Straube

TL;DR

The paper develops a rigorous geometric framework for overdamped rotational Brownian motion of axisymmetric particles, clarifying the equivalence of Itō and Stratonovich forms through a construction on the unit sphere via infinitesimal rotations. It introduces a normalization-preserving geometric integration scheme that advances orientation by finite random rotations, achieving weak order 1 convergence and, in an advanced variant, weak exactness for large time steps. The authors extend the formalism to axisymmetric particles with anisotropic friction, derive the corresponding Fokker–Planck operator, and prove detailed balance and Boltzmann equilibrium under potential torques, illustrating with a dipole in a homogeneous field. Compared with projection-based methods, the geometric scheme offers larger stable time steps and improved accuracy, enabling efficient simulations of self-propelled Janus and related colloids, with open-source applicability in HAL’s MD package.

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 Itō'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.

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

TL;DR

The paper develops a rigorous geometric framework for overdamped rotational Brownian motion of axisymmetric particles, clarifying the equivalence of Itō and Stratonovich forms through a construction on the unit sphere via infinitesimal rotations. It introduces a normalization-preserving geometric integration scheme that advances orientation by finite random rotations, achieving weak order 1 convergence and, in an advanced variant, weak exactness for large time steps. The authors extend the formalism to axisymmetric particles with anisotropic friction, derive the corresponding Fokker–Planck operator, and prove detailed balance and Boltzmann equilibrium under potential torques, illustrating with a dipole in a homogeneous field. Compared with projection-based methods, the geometric scheme offers larger stable time steps and improved accuracy, enabling efficient simulations of self-propelled Janus and related colloids, with open-source applicability in HAL’s MD package.

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 Itō'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.
Paper Structure (27 sections, 101 equations, 2 figures, 1 table)

This paper contains 27 sections, 101 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Heuristic derivation of the overdamped limit
  3. Langevin equation of rotational Brownian motion with inertia
  4. Heuristic derivation of the overdamped limit
  5. Two-dimensional case.
  6. Itō form of the overdamped equation
  7. Geometric construction of the overdamped Langevin equation
  8. Geometric construction and derivation of the Itō form
  9. Discussion of the drift term
  10. Orientational autocorrelation function
  11. Equivalence of the Itō and Stratonovich forms
  12. Two-dimensional case
  13. Geometric numerical integration scheme
  14. Algorithm
  15. Estimation of the numerical error
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Numerically obtained correlation functions $\widehat{C}_\ell(t)$ for $\ell=1$ (circles) and $\ell=2$ (diamonds) using the geometric integrator (algorithm in \ref{['tab:algorithm']}, closed symbols) and the projection scheme [Eqs. \ref{['eq:adhoc-scheme']}, open symbols], which combines Euler--Maruyama discretization of the Langevin equation \ref{['eq:kinematic-Str']} with a posteriori normalization of the orientation vector. Colors distinguish different integration time steps $\Delta t$, numerical data are available only for correlation times $t$ that are a multiple of $\Delta t$, and colored lines serve as guides to the eye. The solid and dashed black lines show the exact solutions $C_\ell(t)$ for $\ell = 1$ and $\ell = 2$, respectively [\ref{['eq:corr-exact']}].
  • Figure 2: Probability distribution $p(\vartheta, \Delta t)$ of the angle $\vartheta$ between the initial and final orientations after an integration time step of length $\Delta t=0.3 D_\text{R}^{-1}$ for the geometric scheme with Gaussian rotational vectors [\ref{['eq:theta-dist-gauss']}, green line] and for Briels' projection scheme, combining an Euler--Maruyama step with an orthogonal projection [\ref{['eq:theta-dist-adhoc']}, red line]. The gray line shows the exact propagator of rotational Brownian motion [\ref{['eq:theta-dist']}].