Non-Gaussian behaviour of a self-propelled particle on a substrate

TL;DR

This paper analytically investigates the non-Gaussian dynamics of a self-propelled particle constrained to move along a line, influenced by internal forces, torque, and orientational diffusion, relevant for active particles like Janus particles and bacteria.

Contribution

It provides analytical solutions for displacement moments of an active particle with internal force and torque, revealing non-Gaussian behavior at finite times.

Findings

Non-Gaussian displacement distribution at finite times

Normalized kurtosis approaches zero as time increases

Analytical expressions for first four displacement moments

Abstract

The overdamped Brownian motion of a self-propelled particle which is driven by a projected internal force is studied by solving the Langevin equation analytically. The "active" particle under study is restricted to move along a linear channel. The direction of its internal force is orientationally diffusing on a unit circle in a plane perpendicular to the substrate. An additional time-dependent torque is acting on the internal force orientation. The model is relevant for active particles like catalytically driven Janus particles and bacteria moving on a substrate. Analytical results for the first four time-dependent displacement moments are presented and analysed for several special situations. For vanishing torque, there is a significant dynamical non-Gaussian behaviour at finite times t as signalled by a non-vanishing normalized kurtosis in the particle displacement which approaches zero for long time with a 1/t long-time tail.