Geometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions

TL;DR

This paper introduces a new Langevin model for rigid body motion with quaternion-based rotations and hydrodynamic interactions, along with a geometric integrator that preserves key properties of the continuous dynamics.

Contribution

It presents a novel Langevin-type equation framework with quaternion rotations and a second-order geometric integrator that maintains the system's geometric features.

Findings

The integrator accurately preserves quaternion lengths.

Numerical experiments demonstrate distinct behaviors due to inertia and coupling.

The model effectively samples from the canonical ensemble.

Abstract

We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces, and hydrodynamic coupling. In the absence of non-conservative forces the Langevin-type equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevin-type equations, we construct a weak 2nd order geometric integrator which preserves the main geometric features of the continuous dynamics. The integrator uses Verlet-type splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualitatively distinct behaviours.