Particles, trajectories, and diffusion: Random walks in cooling granular gases

We study the mean-square displacement (MSD) of a tracer particle diffusing in a granular gas of inelastic hard spheres under homogeneous cooling state. Tracer and granular gas particles are in general mechanically different. Our approach uses a series representation of the MSD where the kth term is given in terms of the mean scalar product 〈r_{1}·r_{k}〉, with r_{i} denoting the displacements of the tracer between successive collisions. We find that this series approximates a geometric series with the ratio Ω. We derive an explicit analytical expression of Ω for granular gases in three dimensions and validate it through a comparison with the numerical results obtained from the direct simulation Monte Carlo (DSMC) method. Our comparison covers a wide range of masses, sizes, and inelasticities. From the geometric series, we find that the MSD per collision is simply given by the mean-square free path of the particle divided by 1-Ω. The analytical expression for the MSD derived here is compared with DSMC data and with the first- and second-Sonine approximations to the MSD obtained from the Chapman-Enskog solution of the Boltzmann equation. Surprisingly, despite their simplicity, our results outperform the predictions of the first-Sonine approximation to the MSD, achieving accuracy comparable to the second-Sonine approximation.