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 $k\mathrm{th}$ term is given in terms of the mean scalar product $\langle {\mathbf{r}}_1·{\mathbf{r}}_k\rangle$, 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 $\mathrm{\Omega }$. We derive an explicit analytical expression of $\mathrm{\Omega }$ 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 $1-\mathrm{\Omega }$. 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 outperform the predictions of the first-Sonine approximation to the MSD, achieving accuracy comparable to the second-Sonine approximation.

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• Brownian motion
• Classical statistical mechanics
• Diffusion
• Kinetic theory
• Granular gases
• Boltzmann theory
• Monte Carlo methods