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An Itinerant Oscillator model with cage inertia for mesorheological granular experiments

Recent experiments with a rotating probe immersed in weakly fluidized granular materials show a complex behavior on a wide range of timescales. Viscous-like relaxation at high frequency is accom- panied by an almost harmonic dynamical trapping at intermediate times, with possibly anomalous long time behavior in the form of super-diffusion. Inspired by the Itinerant Oscillator model for dif- fusion in molecular liquids, and other models with coupled thermostats acting at different timescales, here we discuss a new model able to account for fast viscous relaxation, dynamical trapping and super-diffusion at long times. The main difference with respect to liquids, is a non-negligible cage inertia for the surrounding (granular) fluid, which allows it to sustain a slow but persistent mo- tion for long times. The computed velocity power density spectra and mean-squared displacement qualitatively reproduce the experimental findings. We also discuss the linear response to external perturbations and the tail of the distribution of persistency time, which is associated with superdif- fusion, and whose cut-off time is determined by cage inertia.