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 $Ω$. 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.

💡 Research Summary

The paper investigates the diffusion of a tracer (or intruder) particle immersed in a granular gas composed of inelastic hard spheres that evolves under the homogeneous cooling state (HCS). Traditional diffusion theory, dating back to Smoluchowski, assumes that after each collision the tracer’s direction is completely randomized, leading to the elementary mean‑square‑displacement (MSD) formula ⟨R²⟩ = 2N h_r², where N is the number of collisions and h_r the mean free path. However, in granular gases the “persistence of collisions” – a tendency for the post‑collision velocity to retain a component of the pre‑collision direction – invalidates this assumption, especially when the tracer mass differs significantly from that of the gas particles.

The authors formulate the tracer’s trajectory as a random walk: after N collisions the position is R_N = Σ_{i=1}^N r_i, with r_i the displacement between successive collisions. The MSD can be written as
⟨R²⟩ = Σ_i⟨r_i²⟩ + 2 Σ_{i<j}⟨r_i·r_j⟩.
The cross‑terms ⟨r_i·r_j⟩ embody the collision persistence. By evaluating the mean scalar product ⟨r_1·r_k⟩ for successive steps, the authors discover that the ratio
Ω ≡ ⟨r_1·r_k⟩ / ⟨r²⟩
is essentially independent of k and can be treated as a constant. Consequently, the series of cross‑terms forms a geometric series, and the MSD simplifies to the compact expression
⟨R²⟩ = N ⟨r²⟩ / (1 − Ω).
Here Ω coincides with the mean‑persistence ratio ⟨ω⟩ known from kinetic‑theory treatments of gases.

A central achievement of the work is the derivation of an explicit analytical formula for Ω in three dimensions, valid for arbitrary tracer‑to‑gas mass ratio, size ratio, and normal restitution coefficients (α for gas‑gas collisions, α₀ for tracer‑gas collisions). Introducing the temperature ratio β = (m₀ T)/(m T₀) and the geometric factor Υ that accounts for size differences, the authors obtain
Ω = (1 + β)^{−1}