Optimal navigation strategies for microswimmers on curved manifolds

TL;DR

This paper develops an analytical framework for microswimmers navigating curved surfaces, showing their optimal paths are geodesics of a Randers metric, which outperforms direct navigation strategies and reveals complex geometric features.

Contribution

It introduces a novel formalism linking microswimmer navigation to Randers geodesics on curved manifolds, advancing understanding of optimal strategies in fluid environments.

Findings

Randers geodesics determine optimal swimmer paths

Randers policy outperforms direct navigation

Complex features like cusps and self-intersections in isochrones

Abstract

Finding the fastest path to a desired destination is a vitally important task for microorganisms moving in a fluid flow. We study this problem by building an analytical formalism for overdamped microswimmers on curved manifolds and arbitrary flows. We show that the solution corresponds to the geodesics of a Randers metric, which is an asymmetric Finsler metric that reflects the irreversible character of the problem. Using the example of a spherical surface, we demonstrate that the swimmer performance that follows this "Randers policy" always beats a more direct policy. A study of the shape of isochrones reveals features such as self-intersections, cusps, and abrupt nonlinear effects. Our work provides a link between microswimmer physics and geodesics in generalizations of general relativity.