Impact of torque on active Brownian particle: exact moments in two and three dimensions

In this work, we investigate the effects of torque, accounting for translational diffusion, on active Brownian particles (ABPs) in two (2d) and three (3d) dimensions. The torque is equivalent to chirality in 2d, well known as chiral active Brownian particles (cABPs), whereas in 3d, this is introduced as an external torque on the ABPs. Despite the inherent complexity in solving the Fokker–Planck equation, we demonstrate a Laplace transform method to precisely calculate the temporal evolution of various dynamic moments. Our analysis yields explicit expressions for multiple moments, such as the second and fourth moments of displacement, revealing the impact of persistence and chirality/torque. These moments exhibit oscillatory behavior, and excess kurtosis indicates deviations from the Gaussian distribution during intermediate time intervals.