Differences between MD and DSMC

In the next paragraph we give the results of the simulations of the randomly driven granular gas model. We have used event driven Molecular Dynamics simulations in $d=1$: in these simulations the position of the particles are precisely known, the collisions happen if and only if the conditions (3.4) are verified, in other words the model is exactly reproduced by the simulator. In some of the runs performed, at low values of $r$ and high values of the ratio $\rho=\tau_b/\tau_c$ (approaching the cooling limit) we have observed a dramatic increase of the collision rate resulting in very poor performances of the algorithm: we have discarded the results of these simulations considering them the evidence of a sort of inelastic collapse (this fact is reflected in the figures, where some points are missing). To simulate a larger number of particles in more than one dimension, we have performed a number of Direct Simulation Monte Carlo: this is an algorithm introduced (by Bird, see for example [30]) to reproduce the solution of the Boltzmann equation and assumes the Molecular Chaos, i.e. disregards correlations of colliding particles. In Appendix A we give a detailed description of Molecular Dynamics of hard spheres in $d=1$ and of Direct Simulation Monte Carlo in $d=1$ and $d=2$. Here we just warn the reader to be careful and notice the captions of the figures. Where ``DSMC'' appears, it must be clear that an important assumption (Molecular Chaos) has been made. However the compatibility of the results with MD in $d=1$ and also many agreements with experimental measurements has convinced us that the Bird algorithm is very well suited to study qualitatively dilute granular gases. Finally, we stress the fact that even if a granular gas is well described by the Boltzmann equation (this is the message/anticipation of this brief introduction), nothing is said about the hydrodynamic description, where further assumption are needed (e.g. low values of the gradients, scale separation, etc).