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 (HCS). Tracer and granular gas particles are in general mechanically different. Our approach uses a series representation of the MSD where the $k$-th term is given in terms of the mean scalar product $\langle \mathbf{r}_1\cdot\mathbf{r}_k \rangle$, with $\mathbf{r}_i$ denoting the displacements of the tracer between successive collisions. We find that this series approximates a geometric series with the ratio $\Omega$. We derive an explicit analytical expression of $\Omega$ 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 $1-\Omega$. 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 outperforms the predictions of the first-Sonine approximation to the MSD, achieving accuracy comparable to the second-Sonine approximation.