Clustering and velocity distributions in granular gases cooling by solid friction

We present large-scale molecular dynamics simulations to study the free evolution of granular gases. Initially, the density of particles is homogeneous and the velocity follows a Maxwell-Boltzmann (MB) distribution. The system cools down due to solid friction between the granular particles. The density remains homogeneous, and the velocity distribution remains MB at early times, while the kinetic energy of the system decays with time. However, fluctuations in the density and velocity fields grow, and the system evolves via formation of clusters in the density field and the local ordering of velocity field, consistent with the onset of plug flow. This is accompanied by a transition of the velocity distribution function from MB to non-MB behavior. We used equal-time correlation functions and structure factors of the density and velocity fields to study the morphology of clustering. From the correlation functions, we obtain the cluster size, $L$, as a function of time, $t$. We show that it exhibits power law growth with $L\left(t\right)\sim t^{1/3}$.

Figure 1

Evolution snapshots of the density field for an inelastic granular gas in $d=2$ at different times: (a) $t=50$; (b) $t=150$; (c) $t=300$; (d) $t=500$. These pictures are obtained for a system with particle number $N=250000$, number density $\sigma =0.30$ (packing fraction $\varphi \approx 0.236$), and friction coefficient $\mu =0.10$. The size of the system is ${912}^2$. For clarity, we have shown only a ${600}^2$ corner. The density field is obtained by directly drawing a black point at the center of particle. Void spaces represent, thus, regions free of particles.

Figure 2

Evolution snapshots of the coarse-grained velocity field at different times: (a) $t=50$; (b) $t=150$; (c) $t=300$; (d) $t=500$. For the sake of clarity, we have shown only a ${48}^2$ corner of the ${128}^2$ box.

Figure 3

The scaling form of the temperature decay as a function of scaled time $t/t_0$. The crossover time is defined in the text and is larger for smaller $\sigma$ and $\mu$. At early times, temperature decays as $\tilde{T}\left(t\right)\sim {\left(t/t_0\right)}^{-3/2}$. As time increases the decay slows down, leading eventually to a decay of the form $\tilde{T}\left(t\right)\sim {\left(t/t_0\right)}^{-1}$. Dashed line and solid line, respectively, represent algebraic decays with exponent 3/2 and 1.

Figure 4

Normalized velocity distribution function $f\left(v_i\right)$ plotted on a semi-log scale for (a) $t=50$, (b) $t=150$, (c) $t=300$, and (d) $t=500$. Data is shown for $\mu =0.10$ and $\sigma =0.30$ at different times. The open circles represent the data obtained from the numerical simulations and averaged over ten independent runs. The solid line represents the $P_{\text{MB}}$ at that time (with the corresponding time-dependent temperature). The statistical data deviates from $P_{\text{MB}}$ at later times, particularly in the tail region.

Figure 5

Time-dependence of the correlation length $L_{\psi }\left(t\right)$ corresponding to the clustering of the density field order-parameter, $\psi (\vec{r},t)$ field. Clearly, in the late stage, $L_{\psi }\left(t\right)$ follows a $t^{1/3}$ growth law. The solid line with exponent $\frac{1}{3}$ is shown, corresponds to diffusive growth of clustering.

Figure 6

Scaling plot of correlation functions and structure factors for the $\psi (\vec{r},t)$ field. We obtain spherically averaged $C_{\psi \psi }(r,t)$ and ${\tilde{S}}_{\psi }(k,t)$ as an average over ten independent runs on lattices of size ${128}^2$, after mapping the actual system of size ${912}^2$. (a) Plot of $C_{\psi \psi }(r,t)$ vs. $r/L_{\psi }$ at different times, denoted by the indicated symbols. The data collapse at different time corresponds to dynamical scaling. (b) Plot of ${\tilde{S}}_{\psi }(k,t)$ vs. $k{L}_{\psi }$ on a log-log scale at different times. The solid line labeled with $k^{-3}$ shows the Porod's law: ${\tilde{S}}_{\psi }(k,t)\sim k^{-3}$ in the long wave vector limit.

Figure 7

The correlation length scale $L_u\left(t\right)$ of the $\vec{u}(\vec{r},t)$ field as a function of time. Plot of $L_u\left(t\right)$ vs. $t$ on a log-log scale. The solid line with an exponent of $\frac{1}{3}$ is shown, corresponds to diffusive growth for ordering.

Figure 8

Scaling plot of spherically averaged correlation functions $C_{uu}(r,t)$ and structure factors ${\tilde{S}}_u(k,t)$ for $\vec{u}(\vec{r},t)$ field. (a) Plot of $C_{uu}(r,t)$ vs. $r/L_u$ at different time, denoted by the indicated symbols. The data collapse at different time confirms to dynamic self-similarity. (b) Plot of ${\tilde{S}}_u(k,t)$ vs. $k{L}_u$ on a log-log scale at different times. The solid line labeled with $k^{-4}$ represents the generalized Porod's law: $S_{uu}(k,t)\sim k^{-(d+n)}$ for $d=2$ and $n=2$. The rest of the details are the same as described in the caption of Fig. 6.