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It is quite generally assumed that the overdamped Langevin equation providesa quantitative description of the dynamics of a classical Brownian particle inthe long time limit. We establish and investigate a paradigm anomalousdiffusion process governed by an underdamped Langevin equation with an explicittime dependence of the system temperature and thus the diffusion and dampingcoefficients. We show that for this underdamped scaled Brownian motion (UDSBM)the overdamped limit fails to describe the long time behaviour of the systemand may practically even not exist at all for a certain range of the parametervalues. Thus persistent inertial effects play a non-negligible role even atsignificantly long times. From this study a general questions on theapplicability of the overdamped limit to describe the long time motion of ananomalously diffusing particle arises, with profound consequences for therelevance of overdamped anomalous diffusion models. We elucidate our results inview of analytical and simulations results for the anomalous diffusion ofparticles in free cooling granular gases.

Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion