  The emerging diffusive dynamics in many complex systems shows a
characteristic crossover behaviour from anomalous to normal diffusion which is
otherwise fitted by two independent power-laws. A prominent example for a
subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer
membranes, while superdiffusive-diffusive crossovers occur in systems of
actively moving biological cells. We here consider the general dynamics of a
stochastic particle driven by so-called tempered fractional Gaussian noise,
that is noise with Gaussian amplitude and power-law correlations, which are cut
off at some mesoscopic time scale. Concretely we consider such noise with
built-in exponential or power-law tempering, driving an overdamped Langevin
equation (fractional Brownian motion) and fractional Langevin equation motion.
We derive explicit expressions for the mean squared displacement and
correlation functions, including different shapes of the crossover behaviour
depending on the concrete tempering, and discuss the physical meaning of the
tempering. In the case of power-law tempering we also find a crossover
behaviour from faster to slower superdiffusion and slower to faster
subdiffusion. As a direct application of our model we demonstrate that the
obtained dynamics quantitatively described the subdiffusion-diffusion and
subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that
a model of tempered fractional Brownian motion recently proposed by Sabzikar
and Meerschaert leads to physically very different behaviour with a seemingly
paradoxical ballistic long time scaling.
