A discussion on the approximations: dilute density and absence of tangential forces

Under the assumption of molecular chaos, i.e. $P_2({\bf x},{\bf x}+\sigma{\bf\hat{n}},{\bf v}_1,{\bf v}_2,t)=P({\bf x},{\bf v}_1,t) P({\bf x}+\sigma{\bf\hat{n}},{\bf v}_2,t)$ where $P_2$ and $P$ are the probability density functions for two particles and one particle respectively, it is possible to write down the Boltzmann Equation (2.98), which can be solved by means of Monte Carlo methods. Here we used a simplified (but still efficient) version of the Direct Simulation Monte Carlo scheme proposed by Bird [30]. A detailed discussion of this method can be found in Appendix A. Here we briefly anticipate a difference with respect to the original version of the algorithm: the clock which determines the collision rate is replaced by an a-priori fixed collision rate via a constant collision probability $p_c$ given to every disk at every time-step $\Delta t$ of the simulation, in such a way that the single-particle collision rate is $\chi \sim p_c/\Delta t$. The colliding particle then seeks its collision partner among the other particles in a neighborhood of radius $\sigma_B$, choosing it randomly with a probability proportional to their relative velocities. Moreover in this approximation the diameter $\sigma$ is no more explicitly relevant but it is directly related to the choices of $p_c$ and $\sigma_B$ in a non trivial way: in fact the Bird algorithm allows the particles to pass through each others, so that a precise diameter cannot be defined and estimated as a function of $p_c$ and $\sigma_B$. The assumption of Molecular Chaos is reasonable at low densities and low inelasticities, when colliding particles can be considered uncorrelated. We have discussed what happens at medium and higher densities in elastic hard spheres gases (see section 2.2.7).

Furtherly the chosen collision rule excludes the presence of tangential forces [76,181,196], and hence the rotational degrees of freedom do not contribute to the description of the dynamics.

The agreement between our simulations and the inspiring experiments, justifies the simplifying assumptions considered for our model, i.e. assuming molecular chaos and neglecting tangential forces. Nevertheless, as a partial check, in section 4.3.2 we try a modified version of the 2D Inclined Channel Model where the tangential forces may affect the post-collisional velocities of the particles. As reported below, the introduction of such forces does not change the behavior of the measured quantities.