Active motion on curved surfaces

TL;DR

This paper develops a theoretical framework for active particle motion on curved surfaces using a generalized Telegrapher's equation, providing explicit solutions and analyzing motion on a sphere with oscillatory behavior.

Contribution

It introduces a generalized Telegrapher's equation for active particles on curved surfaces and derives explicit solutions, extending previous flat-surface models.

Findings

Explicit solutions for active motion on curved surfaces.

Oscillations in mean squared geodesic-displacement on a sphere.

Weak curvature limit expressions for probability density.

Abstract

A theoretical analysis of active motion on curved surfaces is presented in terms of a generalization of the Telegrapher's equation. Such generalized equation is explicitly derived as the polar approximation of the hierarchy of equations obtained from the corresponding Fokker-Planck equation of active particles diffusing on curved surfaces. The general solution to the generalized telegrapher's equation is given for a pulse with vanishing current as initial data. Expressions for the probability density and the mean squared geodesic-displacement are given in the limit of weak curvature. As an explicit example of the formulated theory, the case of active motion on the sphere is presented, where oscillations observed in the mean squared geodesic-displacement are explained.