The convolution model

Now, we can go a step further relating the clustering properties of the system to the velocity distribution. In order to do that we consider the following quantities: the distribution of boxes (of volume $V/M$), $f_M(m)$, containing a given number $m$ of particles and the velocity variance $T_M(m)$, in a box occupied by $m$ particles. We also observe that restricting the statistics of the velocities to boxes of volume $V/M$ containing only a fixed number of particles, i.e. studying a sort of fixed density velocity distribution, the velocity distribution appears closer to a Maxwellian, even if the global distribution is strongly non-Gaussian. We show this measure in Fig. fig_vdist_fixdens

We consider first the non-clusterized case ( $\tau \ll \tau_c$ and $r \simeq 1$). Within this regime we find from the simulations that:

By assuming in each box a Gaussian velocity distribution with a constant variance $T_M^{hom}(m)$ it turns out that the global velocity distribution $P^{hom} (v)$ is Gaussian. Let us recall that the Poisson distribution is the one associated with a process of putting independently $\langle m \rangle M$ particles into $M$ boxes.

Let us turn to the clusterized case. If $\tau=100$ and $r=0.7$, in MD simulations with $d=1$, considering only the occupied boxes ($m > 0$), we measure the following relations

with $\beta\simeq 0.5$ and $c_{cl} \simeq 0.14$.

Let us compute from these scalings the global velocity distribution. Taking into account that the spatial probability distribution of the particles is $f_M^{clust}(m)$ and assuming that their local velocity distribution is Gaussian, but with a variance $T_M^{clust}(m) \simeq m^{-\beta}$ which depends on the occupancy, we obtain, for the global velocity distribution $P_{clust}(v)$, and in the continuum limit:

$$P_{inel}(v) \simeq \sum_{m=1}^{\infty} e^{(- \frac{v^2 m^{\beta}}{2})} e^{-c_{cl} m}.$$ (3.30)

We stress how the the distributions measured in the simulations are in very good agreement (see the dashed line in Fig. fig_v_1d_MD) with the numerical computation of eq. (3.31), which, in summary, has been obtained under only the following hypothesis:

(i)

non-Poissonian distribution for the box occupancy $f_M(m) \propto e^{-c_{cl} m}/m$;

(ii)

Gaussian distribution of velocities in each box with a density-dependent variance $T_M(m) \propto m^{-\beta}$.

The hypothesis about the scaling relation between the velocity variance (i.e. $T_M(m)$) and the local density, apart from being justified numerically, can be understood in the following way. The stationarity and the scale-invariance of the cluster distribution, implies a certain distribution of lifetimes for the clusters. In particular each cluster has a lifetime which is inversely proportional to its size. The scale-invariant cluster-size distribution thus implies a scale-invariant distribution for the lifetimes. The cluster lifetime is strictly related to the variance of the velocity distribution inside the cluster itself. In order to ensure the stability of a cluster in a stationary state we have to require that the velocities of the particles belonging to it are not too different, or equivalently that the variance of the distribution is smaller the higher the density. So, given a scale-invariant distribution of clusters one would expect a scale-invariant distribution of variances, that is $T_M(m) \sim m^{-\beta}$.