Thermodynamic bounds for diffusion in non-equilibrium systems with multiple timescales

We derive a Thermodynamic Uncertainty Relation bounding the mean squared displacement of a Gaussian process with memory, driven out of equilibrium by unbalanced thermal baths and/or by external forces. Our bound is tighter with respect to previous results and also holds at finite time. We apply our findings to experimental and numerical data for a many-body interacting granular fluid, characterized by regimes of anomalous diffusion. In some cases, our relation can distinguish between equilibrium and non-equilibrium behavior, a non-trivial inference task, particularly for Gaussian processes.