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On the mean square displacement of intruders in freely cooling granular gases

We compute the mean square displacement (MSD) of intruders immersed in a freely cooling granular gas made up of smooth inelastic hard spheres. In general, intruders and particles of the granular gas are assumed to have different mechanical properties, implying that non-equipartition of energy must be accounted for in the computation of the diffusion coefficient $D$. In the hydrodynamic regime, the time decay of the granular temperature $T$ of the cooling granular gas is known to be dictated by Haff's law; the corresponding decay of the intruder's collision frequency entails a time decrease of the diffusion coefficient $D$. Explicit knowledge of this time dependence allows us to determine the MSD by integrating the corresponding diffusion equation. As in previous studies of self-diffusion (intruders mechanically equivalent to gas particles) and the Brownian limit (intruder's mass much larger than the grain's mass), we find a logarithmic time dependence of the MSD as a consequence of Haff's law. Beyond the logarithmic time growth, we find that the MSD depends on the mechanical system parameters in a highly complex way. To explain the observed behaviour, we analyze in detail the intruder's random walk, consisting of ballistic displacements interrupted by anisotropic deflections caused by the collisions with the hard spheres. We also show that the MSD can be thought of as arising from an equivalent random walk with isotropic, uncorrelated steps.