Weakly nonlinear dynamics of chemically active particles near the threshold for spontaneous motion: general theory

We study the weakly nonlinear dynamics of an isotropic chemically active particle in the limit where the Péclet number approaches its critical value, beyond which the symmetric base state is unstable and the particle exhibits symmetry-breaking spontaneous motion. The theory consists of a nonlinear amplitude equation governing the slow-time dynamics of the particle's velocity, in which unsteadiness—associated with the particle interacting with its own concentration wake over large scales—is manifested in a temporally nonlocal term that nonlinearly depends on the history of the particle's motion. Our derivation involves matching a quasi-steady particle-scale expansion with an unsteady leading-order solution for the concentration field in a large-scale region, describing diffusion from a moving point source; it additionally relies on a novel adjoint formulation that enables us to straightforwardly treat fully 3D particle motion and include the effects of general weak perturbations, which can have a leading-order effect sufficiently near the threshold. Our weakly nonlinear theory allows inexpensive numerical simulation of one or many active particles over long times scales, as well as analysis of various physical effects (see following talk by Gunnar Peng).