Distributions of velocity

Figure: Distribution of rescaled horizontal velocities $v/\sqrt {T_w}$ for the 2D Inclined Plane Model with stochastic wall at different temperatures $T_w=50$, $T_w=100$, $T_w=250$. The other parameters are $N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$

Figure: Distribution of horizontal velocities, for the 2D Inclined Plane Model with stochastic wall, measured on stripes at different heights and rescaled by the average granular temperature at that height. The inset shows the normalized number density profile with the position of the chosen stripes. $N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$, $T_w=100$

In the Fig. fig:tflip_vglob and Fig. fig:tflip_vstripe we display the distributions of horizontal velocities for the 2D Inclined Plane Model with a bottom wall with stochastic vibrations. The first Fig. fig:tflip_vglob, shows the distributions taken from all over the system (i.e. counting particles independently of the height) for different $T_w$: the data collapse is obtained by rescaling the velocities by $\sqrt{T_w}$. The distributions are very different from a Gaussian.

Instead, in Fig. fig:tflip_vstripe we show the velocity distributions of particles contained in stripes at different heights from the wall, again rescaled by $\sqrt{T_g(y)}$ (their own variance) in order to obtain the data collapse. It again appears that the distributions are non-Gaussian and their broadening (that is the granular temperature $T_g(y)$) is height dependent. This dependence is shown in Fig. fig:tflip_t.

The case of periodically vibrated wall is illustrated in Fig. fig:vflip_v. One can see the distribution of horizontal velocities in two different regimes: for $g_e=-1$ a non-Gaussian distribution is obtained, while a distribution close to a Gaussian appears when $g_e=-100$. This trend toward a Gaussian, as the angle of inclination is raised up, reproduces exactly the experimental observation of Kudrolli and Henry [132] (where the angle of inclination of the plane was raised up from $\theta=0.1^o$ to $\theta=10^o$). This phenomena can be explained as an effect of the increase of the collision rate with the wall induced by the increase of the force of gravity: an higher collision rate ``randomizes'' the velocities in a more efficient way. If one accepts the analogy between the vibrating wall and the heating bath used to drive the granular gas in Chapter 3, the increase of collision rate with the wall is analogous to the decrease of the ratio $\tau_b/\tau_c$ (an increase of the ``heating rate''). In the Randomly Driven model the decrease of this ratio corresponds to the transition from the non-Gaussian regime to the Gaussian one.

In Fig. fig:vflip_v_tails we show a log-log plot of the tails of the collapsed distributions presented in Fig. fig:tflip_vstripe. An algebraic fit $P(v) \sim v^{-2.8}$ is proposed.

Figure: Distributions of horizontal velocities for the Inclined Plane Model with periodically vibrating wall for two different values of inclination, that is $g_e=-1$ and $g_e=-100$, while the other parameters are fixed: $N=500$, $N_w \approx 56$, $r=0.5$, $r_w=0.7$, $f_w=400\pi$, $A_w=0.1$

Figure: Zoom in log-log scale of the distribution of horizontal velocities, for the 2D Inclined Plane Model with stochastic wall, measured on stripes at different heights and rescaled by the average granular temperature at that height. The inset shows the normalized number density profile with the position of the chosen stripes. $N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$, $T_w=100$ (same as Fig. fig:tflip_vstripe). The fit indicates the power law behavior.