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

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 orientation, mean velocity, velocity–orientation projections, velocity autocorrelation, mean-squared velocity, lag-time mean-squared displacement, second-order displacement moments, and the fourth moment of velocity. We further obtain the long-time diffusivity tensor, including its antisymmetric odd-diffusivity contribution. These results agree quantitatively with simulations over all masses, activities, and chiralities. We show that the velocity autocorrelation contains two distinct relaxation sectors: an inertial envelope and a chiral envelope with oscillatory modulation. Despite rich transients in the velocity sector, the long-time positional diffusion equals the overdamped cABP value, independent of mass. At the same time, the long-time transport is not solely determined by the scalar diffusion coefficient but also includes an odd-diffusivity term that captures the transverse diffusive response generated by chirality. 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–(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. The non-Gaussianity of the displacement sector is likewise characterized through the displacement excess kurtosis.