  We investigate both analytically and by computer simulations the ensemble
averaged, time averaged, non-ergodic, and ageing properties of massive
particles diffusing in a medium with a time dependent diffusivity. We call this
stochastic diffusion process the (ageing) underdamped scaled Brownian motion
(UDSBM). We demonstrate how the mean squared displacement (MSD) and the time
averaged MSD of UDSBM are affected by the inertial term in the Langevin
equation, both at short, intermediate, and even long diffusion times. In
particular, we quantify the ballistic regime for the MSD and the time averaged
MSD as well as the spread of individual time averaged MSD trajectories. One of
the main effects we observe is that---both for the MSD and the time averaged
MSD---for superdiffusive UDSBM the ballistic regime is much shorter than for
ordinary Brownian motion. In contrast, for subdiffusive UDSBM the ballistic
region extends to much longer diffusion times. Therefore, particular care needs
to be taken when the overdamped limit indeed provides a correct description,
even in the long time limit. We also analyze to what extent ergodicity in the
Boltzmann-Khinchin sense in this non-stationary system is broken, both for
subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow
UDSBM is considered, with a mixed logarithmic and power law dependence of the
ensemble and time averaged MSDs of the particles. In the limit of strong
ageing, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly
in the short time ballistic limit. The approaches developed here open new ways
for considering other stochastic processes under physically important
conditions when a finite particle mass and ageing in the system cannot be
neglected.
