  The passive and active motion of micron-sized tracer particles in crowded
liquids and inside living biological cells is ubiquitously characterised by
"viscoelastic" anomalous diffusion, in which the increments of the motion
feature long-ranged negative and positive correlations. While viscoelastic
anomalous diffusion is typically modelled by a Gaussian process with correlated
increments, so-called fractional Gaussian noise, an increasing number of
systems are reported, in which viscoelastic anomalous diffusion is paired with
non-Gaussian displacement distributions. Following recent advances in Brownian
yet non-Gaussian diffusion we here introduce and discuss several possible
versions of random-diffusivity models with long-ranged correlations. While all
these models show a crossover from non-Gaussian to Gaussian distributions
beyond some correlation time, their mean squared displacements exhibit
strikingly different behaviours: depending on the model crossovers from
anomalous to normal diffusion are observed, as well as unexpected dependencies
of the effective diffusion coefficient on the correlation exponent. Our
observations of the strong non-universality of random-diffusivity viscoelastic
anomalous diffusion are important for the analysis of experiments and a better
understanding of the physical origins of "viscoelastic yet non-Gaussian"
diffusion.
