The models

The Inelastic Hard Rods model

We introduce a system of hard rods (the length of the rods $\sigma$ is not relevant, as already discussed in paragraph 3.1.1), confined to a ring and colliding inelastically. Two rods collide if and only if they are at contact and their relative velocity is of opposite sign with respect to the relative position, i.e. $(v_i-v_j)(r_i-r_j)<0$.

After a binary collision the scalar velocities of the particles change according to:

where $r$ is the restitution coefficient, which takes the value $1$ for perfectly elastic systems and 0 for completely inelastic particles (in this case, $d=1$, equivalent to the sticky gas model). Few years ago McNamara and Young [158], and Sela and Goldhirsch [197], simulated such a model and observed a universal algebraic decay of the kinetic energy, $E(t)\propto t^{-2}$ (Haff's Law [103]), together with an anomalous behavior for the global velocity distribution, even in the early homogeneous regime. Furthermore, strong inhomogeneities appeared as the precursor of a numerically catastrophic event, the inelastic collapse: particles perform an infinite number of collisions in finite time interval [157].

As already discussed, the renewal of interest on one-dimensional granular flows, has been generated by the recent work of Ben-Naim et al [22]. They have circumvented the inconvenience of the inelastic collapse, by means of a sort of ``regularization'': they assumed that when two particles collide with an absolute relative velocity lesser than an arbitrary cut-off $\vert v_i-v_j\vert < \delta$, the collision occurs elastically. This is not too different from the proposed visco-elastic model (see paragraph 2.1.5), where the restitution coefficient depends on the relative velocity of the colliding objects. Moreover, the authors have verified that the main results (e.g. the asymptotic energy decay) do not depend on the choice of the cut-off $\delta$. Furtherly, we have checked that these (and other) results can be reproduced using the visco-elastic regularization, and they are quite independent from the details of the functional form $r(\vert v_i-v_j\vert)$. The choice of regularization, i.e. of the value of $\delta$, of course is relevant for the length of the simulation: the system in fact behaves mostly as an elastic gas when a large part of the molecules have reached a velocity of the order of $\delta$. This quasi-elastic final stage is $\delta$-dependent.

However this regularizations allows the system to enter into a dynamical regime which was not investigated in the past (many authors, to prevent the collapse, chose smaller systems, but this prevented also the long wavelength instabilities). As discussed before, during such a regime, the kinetic energy decays as $E(t)\propto t^{-2/3}$, for any $r < 1$. A direct inspection of the hydrodynamic profiles, shows that such a regime is highly inhomogeneous, with density clusters and shocks in the velocity field. Accordingly, they suggest that inelastic systems behave asymptotically as a sticky ($r=0$) gas [57] which is known to be described by the Burgers equation in the inviscid limit [199]. Such a behavior reflects the fact that the asymptotics is dominated by the dynamics of clusters of particles, which move through the system and coalesce, similarly to sticky objects. As already pointed out, one of the main predictions of the Burgers conjecture is the $\sim \exp(-\vert v/v_0(t)\vert^3)$ scaling distribution, which could not be verified in the work of Ben-Naim and co-workers.

The 1d Inelastic Lattice Gas model

Alternatively, one may attack the Boltzmann equation, by assuming a simpler form for the scattering cross section in the collision integral, i.e. taking the relative velocity of the colliding pair to be proportional to the average thermal velocity: $\vert v-v'\vert \sim \sqrt{E}$. This is the so called pseudo-Maxwell model [33], discussed in detail in paragraph 5.1.5.

The energy factor $\sqrt{E}$ can be eliminated via a time reparametrization, and one obtains a simpler equation:

$$\frac{\partial P(v,\tau)}{\partial\tau} +P(v,\tau)= \frac{1}{1-\gamma^*}\int duP(u,\tau)P\left(\frac{v-\gamma^* u}{1-\gamma^*},\tau\right)$$ (5.38)

where $\gamma^*=(1-r)/2$ and the $\tau$ counts the number of collisions per particle. Eq. (5.40) is the master equation of the inelastic version of Ulam's scalar model: at each step an arbitrary pair is selected and the scalar velocities are transformed according to the rule of Eq. (5.39).

This model accounts for mean field behavior, i.e. disregards spatial correlation. Therefore it cannot give reliable results in the correlated phase that one expects after the end of the Homogeneous Cooling. Moreover, it does not reproduce the behavior of the Hard Rod model even in the homogeneous phase, as it is demonstrated in the next section (this in general means that the so-called ``homogeneous'' phase is not really homogeneous, even if it is characterized by a Haff-like decay of the energy which, in $d>1$ is found only in a non-correlated system). To reinstate spatial correlation we considered the $d=1$ version of the Inelastic Lattice Gas model introduced in the previous section. $N$ sites disposed on a line with periodic boundary conditions have associated a ``velocity'' $v_i$. At every step of the evolution a pair of neighboring sites is chosen randomly and undergoes an inelastic collision according to Eq.(5.39) if $(v_i-v_j)\times (i-j)<0$. The latter condition, which avoids collisions between particles ``moving'' far from each other, is to be be referred below as the kinematic constraint and plays a key role in the formation of structures during the inhomogeneous phase. A unit of time $\tau$ correspond to $N$ collisions (i.e. to $1$ collision per particle on average).