One-particle distribution of velocities

In the framework of a kinetic approach we are much interested in the behavior of the distribution of velocities $P(\mathbf{v})$, which is related to the one-particle probability density function $P_1(\mathbf{r},\mathbf{v},t)$ which appears in the kinetic equations of gases, i.e. the first equation of the BBGKY hierarchy (2.70), or the Boltzmann equation (2.74) if the Molecular Chaos can be reasonably assumed. In particular, in a stationary regime, after a transient time $t_0$ of the order of the maximum characteristic time of the problem ($\tau _b$ or $\tau_c$), and using an observation time $T \gg t_0$, we have:

$$P(\mathbf{v})=\frac{1}{V}\frac{1}{T-t_0}\int_V d\mathbf{r} \int_{t_0}^T dt P_1(\mathbf{r},\mathbf{v},t)$$ (3.18)

Figure: Rescaled distribution of velocities $v/v_0$ ( $v_0=\sqrt {T_g}$) in a Gaussian ( $\tau _b=0.01$, $r=0.99$, crosses) and a non-Gaussian regime ( $\tau _b=100$, $r=0.7$, pluses '+') for the MD simulations with $d=1$. In both cases $N=L=500$. The dot-dashed curve represents the Gaussian. The dashed line represents the fit discussed in paragraph 3.3

We have measured the distribution of velocities of the particles filling a histogram at many different times, after having verified the stationarity of the system. In Fig. fig_v_1d_MD we show the results of these measurements for Molecular Dynamics in $d=1$. Similar results are obtained in $d=1$ with DSMC algorithm simulations, presented in Fig. fig_v_1d_DSMC. We find again a strong difference between the collisionless regime, $\tau_c \gg \tau_b$ and the colliding regime $\tau_c \ll \tau_b$: in this case the collisionless regime is characterized by a Gaussian distribution of velocities, while in the colliding regime a non-Gaussian behavior appears. The non-Gaussian distributions are characterized by enhanced high-energy tails.

Figure: Rescaled distribution of velocities in a Gaussian ( $\tau _b=0.01$, $r=0.99$) and a non-Gaussian regime ( $\tau _b=100$, $r=0.5$) for the DSMC simulations with $d=1$. In both cases $N=L=2000$, $\tau _c=0.5$, $r_b=0.4$. The dashed curve represents the Gaussian. $v_0=\sqrt {T_g}$

We have repeated the measurements in $d=2$, using DSMC simulations. The results, shown in Fig. fig_v_2d_DSMC are in fair agreement with the ones in $d=1$, showing that this non-Gaussian behavior is robust not only against the Molecular Chaos, but also to the increase of dimensionality. Finally, to give a quantitative evaluation of the deviation from the Gaussian regime, we have fitted the tails of the observed distribution, in Fig. fig_v_tails. We have found evidence of $\sim \exp(-v^{3/2})$ tails, in agreement with the theoretical prediction of Ernst and van Noije [211] (which has been obtained with the assumption of spatial homogeneity and Molecular Chaos). We have also observed tails with a slower decay, e.g. $\sim \exp(-v)$ in a simulation with a very low restitution coefficient and a very high ratio $\rho=\tau_b/\tau_c$, i.e. when the system is approaching the ``cooling limit'' discussed in paragraph 3.1.2; we can consider it a stationary state because we measure time averages over very large time scales (larger than $\tau _b$), but the statistics of velocities appears to be in agreement to the one predicted by Ernst and van Noije for a cooling granular gas. Again we point out that the results of Ernst and van Noije were obtained with the assumptions of Molecular Chaos and absence of spatial gradients: in our clusterized regimes this is far from true, therefore we cannot rely on a direct comparison with the solutions of the homogeneous Enskog-Boltzmann equation. We can argue that the observed tails of the distributions of velocity are the sum of different effects: inelasticity as well as clusterization.

Figure: Rescaled distribution of velocities in a Gaussian ( $\tau _b=0.01$, $r=0.99$) and a non-Gaussian regime ( $\tau _b=100$, $r=0.5$) for the DSMC simulations with $d=2$. In both cases $N=L^2=10000$, $\tau _c=0.05$, $r_b=0.22$. The dashed curve represents the Gaussian. $v_0=\sqrt {T_g}$

Figure: Rescaled distributions of velocities (particular) for three different choices of parameters, in DSMC simulations with $d=2$: (a) $N=L^2=10000$, $\tau _b=0.01$, $r=0.99$, with Gaussian fit; (b) $N=L^2=3000$, $\tau _b=5$, $r=0.5$ with the fit $\sim \exp(-v^{3/2}/1.25)$; (c) $N=L^=10000$, $\tau _b=100$, $r=0.2$ with the fit $\sim \exp(-v/0.7)$. In the cases (a) and (c): $\tau _c=0.05$, $r_B=0.22$. In the case (b): $\tau _c=0.5$ and $r_B=0.63$. $v_0=\sqrt {T_g}$