Langevin equations and a geometric integration scheme for the overdamped limit of rotational Brownian motion of axisymmetric particles
arXiv:2403.04501 · doi:10.1103/wzdn-29p4
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.
accepted for publication
References in corpus (39)
- Active Particles in Complex and Crowded Environments
- Self-motile colloidal particles: from directed propulsion to random walk
- Physics of Microswimmers - Single Particle Motion and Collective Behavior
- Active Brownian Particles. From Individual to Collective Stochastic Dynamics
- Propulsion of a molecular machine by asymmetric distribution of reaction--products
- Microswimmers in Patterned Environments
- Sedimentation of active colloidal suspensions
- Emergent behavior in active colloids
- Active Brownian Motion Tunable by Light
- State-dependent diffusion: thermodynamic consistency and its path integral formulation
- Brownian motion of a self-propelled particle
- Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering
- Gravitaxis of asymmetric self-propelled colloidal particles
- Brownian Particles far from Equilibrium
- Influence of Noise on Force Measurements
- Phoretic Motion of Spheroidal Particles Due To Self-Generated Solute Gradients
- Intermediate scattering function of an anisotropic active Brownian particle
- Dynamics and efficiency of a self-propelled, diffusiophoretic swimmer
- Brownian motion and the hydrodynamic friction tensor for colloidal particles of complex shape
- The self-propelled Brownian spinning top: dynamics of a biaxial swimmer at low Reynolds numbers
- Coarse-Grained Modelling Out of Equilibrium
- Langevin dynamics in inhomogeneous media: Re-examining the Itô-Stratonovich Dilemma
- Gravity-induced dynamics of a squirmer microswimmer in wall proximity
- Emergence of molecular friction in liquids: bridging between the atomistic and hydrodynamic pictures
- Analytic Solution of an Active Brownian Particle in a Harmonic Well
- Brownian colloidal particles: Ito, Stratonovich or a different stochastic interpretation
- Ångström-scale chemically powered motors
- Enhanced Diffusion of a Needle in a Planar Course of Point Obstacles
- Effective Perrin theory for the anisotropic diffusion of a strongly hindered rod
- Active colloidal propulsion over a crystalline surface
- Entangled Dynamics of a Stiff Polymer
- Eliminating Inertia in a Stochastic Model of a Micro-Swimmer with Constant Speed
- Generalization of Stokes-Einstein relation to coordinate dependent damping and diffusivity: An apparent conflict
- Dynamically Crowded Solutions of Infinitely Thin Brownian Needles
- Tube Concept for Entangled Stiff Fibers Predicts Their Dynamics in Space and Time
- Resonant diffusion of a gravitactic circle swimmer
- Depinning transition of self-propelled particles
- Stochastic Dissipative Euler's equations for a free body
- Photocatalytic magnetic microgyroscopes with activity-tunable precessional dynamics