Inertia-chirality interplay in active Brownian motion: exact dynamics and phase maps

TL;DR

This paper provides an exact theoretical analysis of a two-dimensional chiral active Brownian particle with inertia, revealing how inertia and chirality influence its dynamics, phase behavior, and approach to equilibrium.

Contribution

It introduces a closed-form, time-resolved theory for inertial chiral active Brownian particles, including phase maps and fluctuation-dissipation relations, extending understanding beyond overdamped models.

Findings

Velocity autocorrelation factorizes into inertial and chiral envelopes.

Long-time diffusion matches overdamped behavior regardless of mass.

Chirality confines activity, affecting phase behavior and kurtosis.

Abstract

We present an exact, time-resolved theory for a two-dimensional chiral active Brownian particle (cABP) with translational inertia. Using a Laplace-transform moment hierarchy, we derive closed-form expressions for the mean velocity, velocity-orientation projections, velocity autocorrelation, mean-squared velocity, mean-squared displacement, and the fourth moment of velocity. These results agree quantitatively with simulations over all masses, activities, and chiralities. We show that the velocity autocorrelation factorizes into an inertial envelope and a chiral envelope. Despite rich transients in the velocity sector, the long-time positional diffusion equals the overdamped cABP value, independent of mass. From the steady mean-squared velocity, we define a kinetic temperature and a modified fluctuation-dissipation relation whose violation vanishes in two limits: large mass or large chirality, identifying chirality as an additional route to equilibrium-like behavior. The steady-state velocity excess kurtosis gives a phase map that exhibits a (Gaussian-like)-active(bimodal)-(Gaussian-like) re-entrance with mass; chirality confines activity and shrinks the active sector. A narrow positive-kurtosis window emerges at large mass and intermediate chirality, with analytic boundaries consistent with the heavy-mass asymptote.