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

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