Inertial and confined dynamics of a constant-speed active particle in three dimensions

We study a self-propelled particle moving at a constant speed in three spatial dimensions, where the orientation vector evolves via a rotational Langevin equation with Ornstein-Uhlenbeck-like statistics. This formulation ensures a unit propulsion direction while allowing for fully three-dimensional motion. The orientational noise is implemented orthogonally to both the propulsion axis and a fluctuating auxiliary unit vector, enabling reorientation without affecting speed. We analyze both underdamped and overdamped regimes, deriving analytical results for the particle's dynamics, including the time-dependent mean-squared displacement and alignment of velocity and propulsion direction. In the underdamped case, the dynamics exhibit a ballistic-to-diffusive crossover governed by the interplay between inertial and rotational timescales, independent of the initial velocity. The nonequilibrium nature of the system is characterized through the entropy production rate (EPR), where we derive explicit expressions and demonstrate that the finite misalignment between velocity and propulsion direction leads to a suppression of EPR, distinguishing our model from existing active Ornstein-Uhlenbeck and Brownian particle models. In the presence of harmonic confinement, the overdamped particle exhibits effective diffusion on the surface of a sphere, with a radius set by the interplay between propulsion, friction, and trap stiffness. Numerical simulations confirm our theoretical predictions, supporting the model's relevance for confined active systems in three dimensions.