Velocity distribution of a driven inelastic one-component Maxwell gas

The nature of the velocity distribution of a driven granular gas, though well studied, is unknown as to whether it is universal or not, and, if universal, what it is. We determine the tails of the steady state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular gas where the rate of collision between particles is independent of the separation as well as the relative velocity. We show that the steady state velocity distribution is nonuniversal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution is shown to be identical to that of a noninteracting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modeled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.

Figure 1

The numerically calculated velocity distribution $P\left(v\right)$, obtained from the inverse Fourier transform of the characteristic function $Z\left(\lambda \right)$, for different noise distributions as described in Eq. (3) with (a) $\gamma =1/2$, (b) $\gamma =1$, (c) $\gamma =2$, and (d) $\gamma =3$ for $a=3$. $P\left(v\right)$ is computed for $r_w=1/2$ (dissipative driving) and $r_w=1$ (diffusive driving) and compared with the noise distribution.

Figure 2

The moment ratios [see Eq. (13)] for different noise distributions as described in Eq. (3) with (a) $\gamma =1/2$, (b) $\gamma =1$, (c) $\gamma =2$, and (d) $\gamma =3$ for $a=3$. In each panel the ratios are plotted for $r=1/4,1/2$, as well as $r_w=1/4,1/2$, corresponding to dissipative driving. These are compared with the moment ratios of the noninteracting system in which collisions are ignored, as well as the noise distribution (dashed green line).

Figure 3

The coefficient $b\left(n\right)$ obtained from Eq. (15) for (a) $\gamma =1/2$ and (b) $\gamma =2$ varies linearly with $n^{-1}$ for dissipative driving $r_w<1$. The choice of $r$ and $r_w$ is the same in both plots and labeled in (b). The corresponding $b\left(n\right)$ obtained for the noninteracting system are also shown. The variation of $b=b\left(\infty \right)$ with $r_w$ is shown for (c) $\gamma =1/2$, (d) $\gamma =1$, (e) $\gamma =2$, and (f) $\gamma =3$.

Figure 4

The moment ratios [see Eq. (13)] for different noise distributions as described in Eq. (3) with (a) $\gamma =1/2$, (b) $\gamma =1$, (c) $\gamma =2$, and (d) $\gamma =3$ for $a=3$. The data are for $r_w=1$ (diffusive driving), and the ratios are plotted for $r=1/4$ and $1/2$. These are compared with the noise distribution (dashed green line). In panels (b), (c), and (d), we also plot moment ratios for the exponential distribution with analytically obtained value of ${\lambda }^{*}$ [see Eqs. (16) and (17)].

Figure 5

The variation of the coefficient $b\left(n\right)$ with $n$ obtained from Eq. (15) for diffusive driving $r_w=1$ and for different values of $r$, for different noise distributions characterized by (a) $\gamma =1/2$, (b) $\gamma =1$, (c) $\gamma =2$, and (d) $\gamma =3$. The dashed line corresponds to the analytically obtained asymptotic value ${\lambda }^{*}$ [see Eqs. (16) and (17)].

Figure 6

Schematic diagram summarizing the results obtained in paper. The parameters $r_w\in [-1,1]$ is the coefficient of restitution of wall-particle collisions and $\gamma$ characterizes the noise distribution [see Eq. (3)]. When $r_w=-1$, the system does not reach a time-independent steady state. When $r_w=1,P\left(v\right)$ is universal when $\gamma \ge 1$ and has the same asymptotic behavior as the noise distribution when $\gamma <1$. When the driving is dissipative $\left(|r_w|<1\right)$, $P\left(v\right)$ has the same asymptotic behavior as the noise distribution for $\gamma \le 1$. When $\gamma >1$, the coefficient in the exponential gets modified.