Active matter on Riemannian manifolds.

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Brownian motion describes random particle movement.
• Active particles exhibit self-propulsion.
• Curved surfaces (2-manifolds) present unique diffusion challenges.

Purpose of the Study:

• To formulate the dynamics of overdamped Brownian active particles on any Riemannian 2-manifold.
• To derive analytical expressions for swimmer diffusion variance at short times.
• To generalize these dynamics to specific manifolds like spheroids and tori.

Main Methods:

• Formulation of active particle dynamics on Riemannian 2-manifolds.
• Derivation of analytical expressions for variance and probability density functions.
• Validation using Brownian dynamics simulations.

Main Results:

• An analytical expression for the variance of swimmers on any Riemannian 2-manifold at short times.
• Analytical expressions for long-time variances and steady angular marginal probability density functions on spheroids and tori.
• Successful validation of theoretical findings through simulations.

Conclusions:

• The developed model accurately describes active particle dynamics on curved surfaces.
• The analytical expressions provide valuable tools for understanding diffusion on 2-manifolds.
• This work offers a general framework applicable to various curved geometries.