Entropy production of active particles in underdamped regime

The present work investigates the effect of inertia on the entropy production rate $\Pi$ for all canonical models of active particles for different dimensions and the type of confinement. To calculate $\Pi$, the link between the entropy production and dissipation of heat rate is explored resulting in a simple and intuitive expression. By analyzing the Kramers equation, alternative formulations of $\Pi$ are obtained and the virial theorem for active particles is derived. Exact results are obtained for particles in an unconfined environment and in a harmonic trap. In both cases, $\Pi$ is independent of temperature. For the case of a harmonic trap, $\Pi$ attains a maximal value for $\tau = \omega^{-1}$ where $\tau$ is the persistence time and $\omega$ is the natural frequency of an oscillator. For active particles in 1D box, or other non-harmonic potentials, thermal fluctuations are found to reduce $\Pi$.