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. This dependence extends well beyond the two aforementioned cases, as it holds in all spatial dimensions for arbitrary values of the mechanical parameters of the system (masses and diameters of intruders and grains, as well as their coefficients of normal restitution). Our result for self-diffusion in a three-dimensional granular gas agrees qualitatively, but not quantitatively, with that recently obtained by Blumenfeld [arXiv: 2111.06260] in the framework of a random walk model. Beyond the logarithmic time growth, we find that the MSD depends on the mechanical system parameters in a highly complex way. We carry out a comprehensive analysis from which interesting features emerge, such a non-monotonic dependence of the MSD on the coefficients of normal restitution and on the intruder-grain mass ratio. 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. Finally, we derive some results for the MSD of an intruder inmersed in a driven granular gas and compare them with those obtained for the freely cooling case. In general, we find significant quantitative differences in the dependence of the scaled diffusion coefficient on the coefficient of normal restitution for the grain-grain collisions.