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## Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic,and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call thisstochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate howthe mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term inthe Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the bal-listic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD tra-jectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffu-sive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusiveUDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be takenunder what conditions the overdamped limit indeed provides a correct description, even in the long time limit. Wealso analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary 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 aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in theshort time ballistic limit. The approaches developed here open ways for considering other stochastic processesunder physically important conditions when a finite particle mass and aging in the system cannot be neglected