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

TL;DR

This work develops a random-walk description for tracer diffusion in a freely cooling granular gas, recasting the tracer MSD as a collisional series that closely approximates a geometric series with ratio $\Omega$. By deriving a closed-form expression for $\Omega$ in three dimensions and validating it against DSMC simulations across a broad parameter space, the authors show that the MSD per collision is the mean free path squared divided by $1-\Omega$, yielding results that surpass the first-Sonine diffusion estimate and rival the second-Sonine approximation. The approach also links to Smoluchowski–Jeans heritage and polymer-physics concepts, providing intuition through persistence-length and Kuhn-length analogies. Overall, the method offers a robust, relatively simple framework that reproduces key kinetic-theory results and demonstrates cancellation of intermediate-approximation errors, with potential extensions to disks, moderate densities, mixtures, and non-rough hard-sphere models.

Abstract

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.

Figures (10)

• Figure 1: DSMC values of the first ratios $c_{k+1}/c_k$ of the reduced collisional series vs the (common) coefficient of restitution $\alpha=\alpha_0$ for $k=1, 2, 3, 4$ (circles, squares, up triangles, down triangles, respectively) and, for from top to bottom, $m_0/m=8, 6, 4, 2, 1, 1/2, 1/8$ with $\sigma_0/\sigma=1$. Here, and in the rest of the figures, the absence of error bars indicates that the standard error is smaller than the size of the symbols. Due to large statistical errors, especially at $\alpha=0.9$ and 1, we have not included simulation data for $c_5/c_4$ when $m_0/m=1/8$. The error bars shown for $m_0/m=1/8$ correspond to the uncertainty in $c_4/c_3$. The solid lines represent $\Omega$ defined by Eq. \ref{['Omegafinal']}.
• Figure 2: (Color online.) DSMC values of $c_{k+1}/c_k$ vs the (common) coefficient of restitution $\alpha=\alpha_0$ for $k=1, 2, 3, 4$ (circles, squares, up triangles, down triangles, respectively) and, for from top to bottom, $\{m_0/m, \sigma_0/\sigma\}=\{6, 1/2\}$, $\{6, 1\}$, $\{6, 2\}$, $\{2, 1/2\}$, $\{2, 1\}$, $\{2, 2\}$, $\{1, 1/2\}$, $\{1, 1\}$ and $\{1,2\}$. The solid lines represent $\Omega$ defined by Eq. \ref{['Omegafinal']}. The {blue, black, red} color is used for $\sigma_0/\sigma=\{1/2,1,2\}$.
• Figure 3: (Color online.) DSMC values of $c_{k+1}/c_k$ vs $\alpha_0$ with $\alpha=0.7$ for $k=1, 2, 3, 4$ (circles, squares, up triangles, down triangles, respectively) and, for from top to bottom, $\{m_0/m, \sigma_0/\sigma\}=\{6, 1\}$, $\{2, 1\}$ , $\{1, 1/2\}$, $\{1, 1\}$, $\{1, 2\}$ and $\{1/2,1\}$ The solid lines represent $\Omega$ defined by Eq. \ref{['Omegafinal']}. The {blue, black, red} color is used for $\sigma_0/\sigma=\{1/2,1,2\}$
• Figure 4: DSMC values of $\left\langle R^2\right\rangle/(\left\langle N \right\rangle \left\langle r^2\right\rangle)$ vs the (common) coefficient of restitution ($\alpha=\alpha_0$) for, from top to bottom, $\{m_0/m, \sigma_0/\sigma\}=\{2,1\}$ (squares), $\{1,1/2\}$ (stars), $\{1,1\}$ (down triangles), $\{1,2\}$ (up triangles), and $\{1/2,1\}$ (circles). The solid lines represent the function $1/(1-\Omega)$, where $\Omega$ is given by Eq. \ref{['Omegafinal']}.
• Figure 5: DSMC values of $\left\langle R^2\right\rangle/(\left\langle N \right\rangle \left\langle r^2\right\rangle)$ vs $\alpha_0$ with $\alpha=0.7$ and $\{m_0/m, \sigma_0/\sigma\}=\{2,1\}$ (squares), $\{1,1/2\}$ (stars), $\{1,1\}$ (down triangles), $\{1,2\}$ (up triangles), and $\{1/2,1\}$ (circles). The solid lines represent the function $1/(1-\Omega)$, with $\Omega$ given by Eq. \ref{['Omegafinal']}.