Velocity structure factors in 2D

In a recent paper Trizac and coworkers [215] have studied the formation of structures in driven granular gases, using some statistical tools already developed for the case of cooling granular gases [212]. They have studied a different version of the randomly driven granular gas, as it does not take into account the systematic viscosity (that is: the grains follow Eq. (3.3) without the viscous term $v_i(t)/\tau_B$). The model without this term has been studied by many authors ([226,67,205,211,31,178]. The absence of the viscous term arises many problems: typically it happens that total momentum has large fluctuations and, to avoid it, the simulations have artificial ``re-scaling'' mechanisms that keep the momentum constant. We have also evidence that in $d=1$ the total kinetic energy logarithmically increases with time.

The study of large structures on this version of randomly driven granular gases has shown that there are no instabilities (all modes remain stable if linearly perturbed), at odds with the cooling case, where the existence of shear and cluster instabilities is well established (see Chapter V for a detailed discussion). The velocity correlators, for large distances, in $d=2$ show a power law decay and in $d=3$ a logarithmic decay:

We have measured the sphericized structure factors and velocity correlators using our DSMC simulations for the randomly driven granular gas model, in $d=2$, with and without viscosity, in order to make a comparison with the theoretical prediction of van Noije et al. [215]. The results are shown in Fig. fig_structures. The choice of using the Direct Simulation Monte Carlo may affect the results of these numerical experiments, as certain structures expected to be originated at short scales are completely ruled out by the Molecular Chaos assumption. However we have already seen that large scale density structures appear and can be measured by means of Grassberger and Procaccia fractal dimension (see paragraph 3.2.3).

Figure: Velocity structure factors and correlators in randomly driven granular gases: DSMC simulations of the model introduced in this chapter without viscosity (circles) and with viscosity (squares $\tau _b=10$ and diamonds $\tau _b=1$). In all the simulations: $N=L^2=5000$, $T_b=1$, $r=0.5$, $\tau _c \approx 0.1$, and the coarse-graining boxes used to compute the structure factors are $100 \times 100$. The data for the case without viscosity have been fitted in the intermediate range with power laws

The main observation from Fig. fig_structures is about the effect of viscosity. When the viscous term is present, the structure factors are almost flat and the correlators have rapid decays. When the viscous term is suppressed (and the regularization of total momentum is implemented) we measure in an intermediate range for $k$ or $\vert\mathbf{r}\vert$, for the sphericized factors and correlators:

with $\beta \approx 0.59$ for a restitution coefficient $r=0.5$ and absence of viscous term. This observation can be relevant to give a criterion for the choice of the correct model in particular experimental situations: the measure of velocity correlations can be of use to determine if the granular gas can be modeled neglecting viscosity or if viscosity (for example from the contact with the vibrating plate) plays any role.