Inertial dynamics of an active Brownian particle

TL;DR

This paper develops a comprehensive theoretical framework for active Brownian particles that are asymmetric and inertial, deriving equations and analyzing trajectories to understand their motion beyond idealized models.

Contribution

It consolidates previous findings into a general description of inertial, asymmetric active particles, deriving Langevin and Fokker-Planck equations, and provides a method to reconstruct hydrodynamic resistance.

Findings

Inertial effects lead to a universal transition to helical trajectories.

Trajectory analysis allows reconstruction of particle hydrodynamics.

Inertial particles exhibit a transition phase before settling into circular helices.

Abstract

Active Brownian motion commonly assumes spherical overdamped particles. However, self-propelled particles are often neither symmetric nor overdamped yet underlie random fluctuations from their surroundings. Active Brownian motion has already been generalized to include asymmetric particles. Separately, recent findings have shown the importance of inertial effects for particles of macroscopic size or in low-friction environments. We aim to consolidate the previous findings into the general description of a self-propelled asymmetric particle with inertia. We derive the Langevin equation of such a particle as well as the corresponding Fokker-Planck equation. Furthermore, a formula is presented that allows reconstructing the hydrodynamic resistance matrix of the particle by measuring its trajectory. Numerical solutions of the Langevin equation show that, independently of the particle's shape, the noise-free trajectory at zero temperature starts with an inertial transition phase and converges to a circular helix. We discuss this universal convergence with respect to the helical motion that many microorganisms exhibit.