Distribution of velocities

As discussed above, the conjecture that the Inelastic Hard Rod model falls, asymptotically, in the universality class of the inviscid Burgers equation (that describes the sticky gas), independently of $r$, also predicts that the tails of the velocity distribution decay as $\exp(-(v/v_0(t))^3)$ [89]. On the other hand, an accurate numerical test of this prediction [22] seems problematic, whereas the bulk of the distribution does not deviate appreciably from a Gaussian. This can be seen in Fig. fig_vdist1d, right frame.

This Figure shows again (like in the discussion of the $d=2$ Inelastic Lattice Gas) that the asymptotic velocity distribution is a Gaussian, whereas it deviates sensibly from a Gaussian in the first Haff regime.

Figure: Rescaled velocity distributions for the 1D MD and in the 1D lattice gas, during the homogeneous (left) and the inhomogeneous phase (right). In the left frame is also shown the initial distribution (both models). The distributions refer to systems having the same energy. Data refer to $N=10^6$ (both models) particles with $r=0.99$ and $r=0.5$ (for the lattice model in the inhomogeneous regime).

To the best of our knowledge, exact analytic treatments of the Inelastic Hard Rod model are limited to the homogeneous regime, by means of asymptotic solutions of the Boltzmann equation. Caglioti et al. [25] have found analytic asymptotic solutions of the Boltzmann equation for a 1d inelastic hard rod gas in the limit $r \to 1^-$ and $\rho \to \infty$ (where $\rho$ is the density), for general starting distributions. They obtained a distribution, invariant if $v$ is rescaled with $v_0(t)$ which is the superposition of two delta functions. For MD numerical simulations, the observation of two distinct peaks in the rescaled velocity distribution was reported by Sela and Goldhirsch [197]. Our measurement agrees with this prediction, as evident in Fig. fig_vdist1d, left frame: it represents a precursor of that singular asymptotic distribution. However, the appearance of inhomogeneities and, eventually, the inelastic collapse prevent the approach to the expected limiting distribution.