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

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.