Active particles on spherical viscous interfaces

Microswimmers can navigate in complex environments with curved geometries. The intrinsic curvature and topology of the underlying space can affect their transport behavior in non-trivial ways. We formulate the dynamics and the Fokker-Planck equation of active Brownian particles (ABPs) on two-dimensional Riemannian surfaces using the Cartan moving frame method. Specifically, we focus on the case of a spherical geometry, where the configuration of an ABP is an element in the unit tangent bundle of the 2-sphere. This mathematical structure enables us to describe the ABP's configuration using a single quaternion, and to develop a robust and efficient numerical scheme for particle simulations. Moreover, the moments of the probability density function on a sphere correspond to spin-weighted functions, enabling a spectral method for the Fokker-Planck equation based on spin-weighted spherical harmonics (SWSHs). Within the same framework, we also explore the effects of hydrodynamic interactions of ABPs embedded in a viscous membrane surrounded by bulk fluids.