Particles, trajectories and diffusion: random walks in cooling granular gases
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917227970691072 |
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| author | Yuste, Santos Bravo González, Rubén Gómez Garzó, Vicente |
| author_facet | Yuste, Santos Bravo González, Rubén Gómez Garzó, Vicente |
| contents | 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 $Ω$. 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 outperforms the predictions of the first-Sonine approximation to the MSD, achieving accuracy comparable to the second-Sonine approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Particles, trajectories and diffusion: random walks in cooling granular gases Yuste, Santos Bravo González, Rubén Gómez Garzó, Vicente Statistical Mechanics Soft Condensed Matter 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 $Ω$. 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 outperforms the predictions of the first-Sonine approximation to the MSD, achieving accuracy comparable to the second-Sonine approximation. |
| title | Particles, trajectories and diffusion: random walks in cooling granular gases |
| topic | Statistical Mechanics Soft Condensed Matter |
| url | https://arxiv.org/abs/2505.02777 |