Velocity and density profiles

A comparison of velocity profiles at instants having the same energy in the two models is presented in Fig. fig_shocks. The shocks in the velocity profile $v(x,\overline{t})$, correspond to high density clusters. The shocks are discontinuity of the velocity field, characterized by an almost continuous first derivative of the profile, which is averagely positive: this means that the particles in a shock are almost all non-colliding, as $(v_i-v_j) \times (x_i - x_j)>0$ almost always. In this frame there are also the so-called pre-shocks, i.e. parts of the velocity profile that are going to become a shock in the near future, but are still continuous.

In the second frame, it is shown the profile $v(i,\overline{t})\equiv v(x_i(\overline{t}))$ of the same Hard Rod model, where $i$ is the particle label. The difference between $v(x,\overline{t})$ and $v(i,\overline{t})$ in the Hard Rod system is in the sign of large gradients: $v(x,\overline{t})$ displays large negative gradients, while $v(i,\overline{t})$ has large positive gradients. This happens because of the density profile: most of the particles are sitting near the negative discontinuity of the field, so that when the profile is ``unrolled'' as a function of the particle index, this negative discontinuity becomes a very large negative gradient, compared to the positive gradient which competes to few particles.

Finally, the bottom frame displays the $v(i,\overline{\tau})$ profile of the Inelastic Lattice Gas which fairly compares with the analogous profile for the Hard Rod system.

It is important to note that the presence of shocks in the lattice model is due to the kinematic constraint, since they disappear, together with the Porod's tails, relaxing the constraint.

Figure: Portions of the instantaneous velocity profiles for the 1D MD (top $v(x,t)$, middle $v(i,t)$) and for the 1D lattice gas model (bottom, $v(i,\tau )$). In the middle frame we display the MD profile against the particle label in order to compare the shocks and preshocks structures with the lattice gas model (the dotted lines show how shocks and preshocks transform in the two representations for the MD). Data refers to $N=2\cdot 10^4$ particles, $r=0.99$ (both models).