Eur. Phys. J. Special Topics 224, 1377-1394 (2015)
https://doi.org/10.1140/epjst/e2015-02465-0

Regular Article

A geometric approach to self-propelled motion in isotropic & anisotropic environments

¹ Physikalisch-Technische Bundesanstalt, Abbestr. 2-12, 10587 Berlin, Germany
² Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, UMR 7351 CNRS, Parc Valrose, 06108 Nice Cedex 02, France

^a e-mail: grossmann@physik.hu-berlin.de

Received: 10 April 2015
Revised: 18 May 2015
Published online: 24 July 2015

Abstract

We propose a geometric perspective to describe the motion of self-propelled particles moving at constant speed in d dimensions. We exploit the fact that the vector that conveys the direction of motion of the particle performs a random walk on a (d − 1)-dimensional manifold. We show that the particle performs isotropic diffusion in d-dimensions if the manifold corresponds to a hypersphere. In contrast, we find that the self-propelled particle exhibits anisotropic diffusion if this manifold corresponds to a deformed hypersphere (e.g. an ellipsoid). This simple approach provides an unified framework to deal with isotropic as well as anisotropic diffusion of particles moving at constant speed in any dimension.