Characterizing exact dynamics of a trapped active Brownian particle under torque in two and three dimensions

TL;DR

This paper develops an exact analytical framework to characterize the transient and steady-state dynamics of chiral active Brownian particles under harmonic confinement in two and three dimensions, revealing how chirality, propulsion, and confinement influence non-Gaussian fluctuations.

Contribution

It introduces a Laplace-transform based solution of the Fokker-Planck equation for exact moments, providing new insights into the non-Gaussian behavior of confined chiral active particles.

Findings

Excess kurtosis in 2D shows damped oscillations with re-entrant crossovers.

In 2D, increasing trap stiffness suppresses oscillations and positive kurtosis.

In 3D, excess kurtosis remains negative, indicating a robust non-Gaussian state.

Abstract

The interplay of chirality, self-propulsion, and spatial confinement generates striking non-equilibrium fluctuations whose higher-order statistics carry information about the dynamics and shape of the position distribution. Here, we present an exact analytical framework, based on a Laplace-transform solution of the Fokker-Planck equation, for the transient dynamics of a chiral active Brownian particle in a harmonic trap, in both two and three dimensions. We obtain closed-form expressions for all time-dependent moments up to fourth order, enabling a complete characterization of the excess kurtosis throughout the transient and steady-state regimes. In two dimensions, the excess kurtosis exhibits a damped oscillatory response with multiple re-entrant crossovers, evolving from negative values that reflect active off-centered ring-like position distributions to positive values characteristic of heavy-tailed fluctuations. This damped oscillatory excess kurtosis appears both for free and harmonic confinement, although increasing the trapping stiffness progressively suppresses it, and the positive excess kurtosis eventually vanishes at sufficiently high stiffness. In contrast, in three dimensions, the excess kurtosis remains negative, indicating a robustly active non-Gaussian state characterized by half-ring-like to band-like position distributions in the two-dimensional plane spanned by the torque axis and its normal radial direction. Our results demonstrate how chirality, propulsion, and confinement, together with dimensionality, shape transient dynamics, while providing experimentally accessible signatures of confined chiral active dynamics.