Intrinsic Langevin dynamics of rigid inclusions on curved surfaces

TL;DR

This paper develops intrinsic Langevin equations for rigid inclusions on curved surfaces, accounting for curvature effects on dynamics, and provides a framework for simulating passive and active diffusion processes in biological and soft matter systems.

Contribution

It introduces a novel geometric formulation of Langevin dynamics on curved surfaces, including curvature-induced coupling and conditions for potential forces, extending stochastic modeling in complex geometries.

Findings

Derived intrinsic Langevin equations using Cartan's moving frames.

Identified curvature coupling between linear and angular momenta.

Provided overdamped equations for efficient Brownian dynamics simulations.

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.