  We analyze the translational and rotational motion of an ellipsoidal Brownian
particle from the viewpoint of stochastic thermodynamics. The particle's
Brownian motion is driven by external forces and torques and takes place in an
heterogeneous thermal environment where friction coefficients and (local)
temperature depend on space and time. Our analysis of the particle's stochastic
thermodynamics is based on the entropy production associated with single
particle trajectories. It is motivated by the recent discovery that the
overdamped limit of vanishing inertia effects (as compared to viscous fricion)
produces a so-called "anomalous" contribution to the entropy production, which
has no counterpart in the overdamped approximation, when inertia effects are
simply discarded. Here, we show that rotational Brownian motion in the
overdamped limit generates an additional contribution to the "anomalous"
entropy. We calculate its specific form by performing a systematic singular
perturbation analysis for the generating function of the entropy production. As
a side result, we also obtain the (well-known) equations of motion in the
overdamped limit. We furthermore investigate the effects of particle shape and
give explicit expressions of the "anomalous entropy" for prolate and oblate
spheroids and for near-spherical Brownian particles.
