Intrinsic Langevin dynamics of rigid inclusions on curved surfaces

TL;DR

This work develops intrinsic Langevin dynamics for rigid inclusions constrained to curved surfaces by employing Cartan's moving frames to construct a coordinate-invariant Hamiltonian formalism. It derives phase-space stochastic equations, the corresponding geometric Fokker-Planck equation, and a careful Smoluchowski limit via adiabatic elimination of momenta, including curvature-induced coupling between translation and rotation and non-conservative (active) forces. The framework is specialized to the sphere, yielding explicit local-coordinate SDEs, Fokker-Planck, and Smoluchowski equations, and it extends naturally to multiple inclusions and active Brownian motion on curved manifolds. The results enable accurate, intrinsic Brownian dynamics simulations on curved surfaces and offer a pathway to study collective and active phenomena on membranes and interfaces with geometric fidelity.

Abstract

The stochastic dynamics of a rigid inclusion constrained to move on a curved surface has many applications in biological and soft matter physics, ranging from the diffusion of passive or active membrane proteins to the motion of phoretic particles on liquid-liquid interfaces. Here we construct intrinsic Langevin equations for an oriented rigid inclusion on a curved surface using Cartan's method of moving frames. We first derive the Hamiltonian equations of motion for the translational and rotational momenta in the body frame. Surprisingly, surface curvature couples the linear and angular momenta of the inclusion. We then add to the Hamiltonian equations linear friction, white noise and arbitrary configuration-dependent forces and torques to obtain intrinsic Langevin equations of motion in phase space. We provide the integrability conditions, made non-trivial by surface curvature, for the forces and torques to admit a potential, thus distinguishing between passive and active stochastic motion. We derive the corresponding Fokker-Planck equation in geometric form and obtain fluctuation-dissipation relations that ensure Gibbsian equilibrium. We extract the overdamped equations of motion by adiabatically eliminating the momenta from the Fokker-Planck equation, showing how a peculiar cancellation leads to the naively expected Smoluchowski limit. The overdamped equations can be used for accurate and efficient intrinsic Brownian dynamics simulations of passive, driven and active diffusion processes on curved surfaces. Our work generalises to the collective dynamics of many inclusions on curved surfaces.

Figures (2)

• Figure 1: Illustration of moving frames and coordinate systems on the surface $S$.
• Figure 2: Representation of an inclusion on a curved surface $S$.