Slow reorientation dynamics of symmetry-breaking self-propelling motion

An isotropic diffusiophoretic active particle that emits a chemical and interacts with it to drive flow is an example of an object that can exhibit symmetry-breaking spontaneous motion in an arbitrary direction, essentially due to being repelled by its own chemical wake. We analyse how this self-propulsion is affected by weak perturbations, such as a small applied force or background chemical gradient (which could be due to the presence of other particles). The arbitrariness of the self-propulsion direction allows it to be slowly reoriented by the weak perturbation over a long timescale. By expanding to linear order in the perturbation amplitude and imposing a solubility condition, we derive the resulting reduced dynamics for the direction of the velocity vector, including a numerically calculated coefficient that depends on the parameters of the problem. We argue that the same form of result would be obtained generically for many different cases of weakly perturbed symmetry-breaking motion.