Equations of motion and collisions

The randomly driven granular gas that we have introduced (Puglisi et al. [183,184]) consists of an assembly of $N$ identical hard objects (spheres, disks or rods) of mass $m$ and diameter $\sigma$. We put, for simplicity, $m=1$ and $k_B=1$ (the Boltzmann constant). Moreover, in dimension $d=1$ the diameter (the length of the rods) is irrelevant because the objects are undeformable, therefore the only physical quantities that determine the dynamics of the gas are the lengths of the free spaces between the grains and their relative velocities: a system of infinitesimal grains of diameter $\sigma=0$ is equivalent to a system with finite $\sigma$. Consequently, in dimension $d=1$, we use the coordinate $x$ to count only the free space (as the grains had $\sigma=0$): at every instant $t$ the true coordinates $X_i(t)$ of the centers of the grains with finite $\sigma$ can be obtained from their fictitious coordinates $x_i(t)$ by the invertible mapping

$$X_i(t)=x_i(t)+(2i+1)\frac{\sigma}{2}$$ (3.1)

The grains move in a box (a square in $d=2$, a segment in $d=1$) of volume $V=L^d$ ($L$ is the length of the sides of the box), with periodic boundary conditions, i.e. opposite borders of the box are identified. Therefore the topology of the ambient space is a $d$-dimensional torus (i.e. a circle in $d=1$).

The mean free path (calculated exactly in Eq. (2.29) for the case of an homogeneous gas of 3D hard spheres with a Maxwellian distribution of velocities) can be roughly estimated as

$$\lambda^*=\overline{v}^*\tau_c^*=\frac{\overline{v}^*}{\nu^*} \simeq \frac{\overline{v}^*}{n^*\overline{v}^*S^*}=\frac{1}{n^*S^*}$$ (3.2)

where $\overline{v}^*$ is the typical velocity of the particles $\sqrt{\langle v^2 \rangle}$, $\tau_c^*$ is the mean collision time, $\nu^*$ is the mean collision rate (inverse of $\tau_c^*$), $n^*=N/V$ is the mean number density and $S^*$ is the total scattering cross section. We stress the fact that $S^*$ has the dimensions of a surface in $d=3$ ( $S^* \sim \sigma^2$), of a line in $d=2$ ( $S^* \sim \sigma$) and no dimensions in $d=1$ (this is consistent with the fact that the diameter, in $d=1$ is irrelevant). The stars $^*$ indicate that all these quantities are global quantities: in this model we observe strong local fluctuations, that is the global quantities can be poorly representative of the effective dynamics.

The dynamics of the gas, as already discussed in the introduction to this chapter, is obtained as the byproduct of three physical phenomena: friction with the surroundings, random accelerations due to external driving, inelastic collisions among the grains. We model the first two ingredients in the shape of a Langevin equation with exact fulfillment of the Einstein relation (see for example[130]), for the evolution of the velocities of the grains in the free time between collisions. The inelastic collisions are, instead, considered at the kinetic level, i.e. they are instantaneous transformations of the velocities of two colliding particles (grains that are touching, with approaching relative velocity). The equations of motion for a particle $i$ that is not colliding with any other particle, are:

We call the parameters $\tau _b$ and $T_b$ characteristic time of the bath and temperature of the bath respectively. The function $\eta$$_i(t)$ is a stochastic process with average $<$$\eta$$_i(t)>=0$ and correlations $<\eta^\alpha_i(t)\eta^\beta_j(t')>=\delta(t-t')\delta_{ij}\delta_{\alpha \beta}$ ($\alpha$ and $\beta$ being component indexes) i.e. a standard white noise.

When two particles $i$ and $j$ satisfy the following conditions:

they are said to be colliding. The unit vector $\hat{\mathbf{n}}$ is defined by Eq. (3.4a). When two particles are colliding, their velocities are instantaneously changed into new velocities according to the following collision rule:

These are the rules for inelastic collisions, for $0 \leq r<1$ they reduce the longitudinal component of the relative velocity, while for $r=1$ this component is only inverted. The parameter $r$ is the normal restitution coefficient. After a collision, the two particles satisfy the following conditions:

We say that two such particles are anticolliding.