Fokker-Planck description of an active Brownian particle with rotational inertia

TL;DR

This paper introduces a perturbative method based on the Fokker-Planck equation to analytically compute the mean-squared displacement of active Brownian particles with rotational inertia, validated by simulations.

Contribution

It provides a novel analytical framework for calculating MSD of ABPs with inertia, extending previous models that neglected rotational inertia.

Findings

Derived explicit MSD expression as a function of inertia

Validated analytical results with numerical simulations

Enhanced understanding of inertia effects on particle dynamics

Abstract

We develop a perturbative framework to calculate the mean-squared displacement (MSD) of active Brownian particles (ABPs) with a finite moment of inertia. Starting from the corresponding Fokker-Planck equation, we employ a Fourier transform for the spatial coordinates and Hermite polynomials as eigenfunctions for the angular velocity, which enables a systematic perturbative expansion of the MSD order by order. By resumming the resulting series in Laplace space and performing the inverse transform, we obtain an explicit expression for the MSD as a function of the moment of inertia. The analytical results are further validated by comparison with numerical simulations.