Optimal navigation strategies for active particles

¹ Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf D-40225 Düsseldorf, Germany
² Institut für Festkörperphysik, Technische Universität Darmstadt - D-64289 Darmstadt, Germany

^(b) liebchen@fkp.tu-darmstadt.de
^(c) hlowen@hhu.de

Received: 14 June 2019
Accepted: 22 July 2019

Abstract

The quest for the optimal navigation strategy in a complex environment is at the heart of microswimmer applications like cargo carriage or drug targeting to cancer cells. Here, we formulate a variational Fermat's principle for microswimmers determining the optimal path towards a given target regarding travelling time, energy dissipation or fuel consumption. For piecewise constant forces (or flow fields), the principle leads to Snell's law, showing that the optimal path is piecewise linear, as for light rays, but with a generalized refraction law. For complex environments, like general 1D, shear or vortex fields, we obtain exact analytical expressions for the optimal path, showing, for example, that microswimmers sometimes have to temporarily navigate away from their target to reach it fastest. Our results apply to idealized microswimmers which can instantaneously steer, are fast enough so that translational noise is unimportant and might be useful, e.g., to benchmark algorithmic schemes for optimal navigation.