  It is quite generally assumed that the overdamped Langevin equation provides
a quantitative description of the dynamics of a classical Brownian particle in
the long time limit. We establish and investigate a paradigm anomalous
diffusion process governed by an underdamped Langevin equation with an explicit
time dependence of the system temperature and thus the diffusion and damping
coefficients. We show that for this underdamped scaled Brownian motion (UDSBM)
the overdamped limit fails to describe the long time behaviour of the system
and may practically even not exist at all for a certain range of the parameter
values. Thus persistent inertial effects play a non-negligible role even at
significantly long times. From this study a general questions on the
applicability of the overdamped limit to describe the long time motion of an
anomalously diffusing particle arises, with profound consequences for the
relevance of overdamped anomalous diffusion models. We elucidate our results in
view of analytical and simulations results for the anomalous diffusion of
particles in free cooling granular gases.
