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

Davidchack, R.L.; Ouldridge, T.E.; Tretyakov, M.V.

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.

Citation

Davidchack, R., Ouldridge, T., & Tretyakov, M. (in press). Geometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions. Journal of Chemical Physics, 147(22), Article 224103. https://doi.org/10.1063/1.4999771

• Journal Article Type: Article
• Acceptance Date: Nov 26, 2017
• Online Publication Date: Dec 12, 2017
• Deposit Date: Dec 13, 2017
• Publicly Available Date: Dec 13, 2017
• Journal: Journal of Chemical Physics
• Print ISSN: 0021-9606
• Electronic ISSN: 1089-7690
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 147
• Issue: 22
• Article Number: 224103
• DOI: https://doi.org/10.1063/1.4999771
• Keywords: rigid body dynamics; quaternions; hydrodynamic interactions; Stokesian dynamics; canonical; ensemble; Langevin equations; stochastic differential equations; weak approximation; ergodic limits; stochastic; geometric integrators
• Public URL: https://nottingham-repository.worktribe.com/output/899501
• Publisher URL: http://aip.scitation.org/doi/full/10.1063/1.4999771
• Contract Date: Dec 13, 2017