The mean field collision model

In order to shed some light on the relationship between the spatial clusterization and the anomalous velocity distributions observed above (Fig. fig_v_1d_MD, Fig. fig_v_1d_DSMC, Fig. fig_v_2d_DSMC, Fig. fig_v_tails) we present a simplified theoretical model. For sake of notation simplicity, we discuss only the $d=1$ case. Let us treat the collisions in a mean-field like fashion and modify the Langevin dynamics plus collision rules by the following set of coupled equations for the velocities ( $i \in [1,N]$):

$$\frac{dv_i}{dt}=-\frac{v_i}{\tau_b}+\frac{1}{N}\sum_{j=1}^N g(v_i-v_j) +\sqrt{\frac{2T_b}{\tau_b}}f_i(t)$$ (3.44)

where the second term in the r.h.s. determines the velocity change of the particle $i$ due to the collisions with the remaining particles and is chosen to mimic the inelastic behavior. This requirement poses some constraints about the form of the function $g(v-v')$:

• The momentum conservation dictates the antisymmetric property, $g(v-v')=-g(v'-v)$

• The inelasticity of the collision process requires $g(v-v')(v-v')\leq 0$

The Fokker-Planck equation of the process described by Eqs. (3.46) is

$$\begin{multline} \frac{\partial}{\partial t} P_N(v_1,...,v_N,t)-\frac{1}{\tau_b}... ... \frac{1}{N} \sum_{j=1}^N g(v_i-v_j) P_N(v_1,...,v_N,t) \right]=0 \end{multline}$$

From the above equation, using the fact that in the limit $N \rightarrow \infty$ the mean field approximation holds, assuming again Molecular Chaos, one can obtain a master evolution equation for the $1$-body velocity probability distribution. We omit this derivation, and give directly the master equation:

$$\frac{\partial P(v,t)}{\partial t}-\frac{1}{\tau_b}\frac{\partial... ...P(v,t)}{\partial v^2}+\frac{\partial}{\partial v}\int dv'P(v,t)P(v',t)g(v-v')=0$$ (3.45)

From Eq. (3.48) one observes that the quantity:

$$\int dv'P(v',t)g(v-v')=G(v)=-\frac{\partial U(v)}{\partial v}$$ (3.46)

which is a function of $v$ and a functional of $P(v)$, can be considered as an effective force acting on the particle generated by an effective potential $U$. Integrating once with respect to the velocity the stationary version of eq. (3.48) one can obtain the following equation:

$$\left(-\frac{v}{\tau_b}-\frac{T_b}{\tau_b}\frac{\partial}{\partial v}+ G(v) \right)P(v)=0$$ (3.47)

The solution of Eq. (3.50) is:

$$P(v) \propto \exp \left(-\frac{\tau_b}{T_b}\left(\frac{v^2}{2 \tau_b}+ U(v)\right) \right)$$ (3.48)

In order to make some progress we consider the qualitative shape of $g(v_i-v_j)$. In eq. (3.46) the effect of collisions between the particles $i$ and $j$ in the unit of time is given by:

$$\frac{d}{dt}(v_i-v_j)\vert _{coll}=\frac{2}{N}g(v_i-v_j)$$ (3.49)

The variation of momentum in an interval $dt$ can be rewritten as

$$\delta(v_i-v_j)\vert _{coll}=\delta q_{ij} \cdot (v_i-v_j)$$ (3.50)

where $\delta q_{ij}$ is the analogue of $q=1-r$ in the randomly driven granular gas model defined in the paragraph 3.1.1. The important difference is that here $\delta q_{ij}$ represents the effect of all the collisions during $dt$, and thus must be associated to an effective restitution coefficient. Eq. (3.53) may be rewritten as:

$$\frac{d}{dt}(v_i-v_j)\vert _{coll}=\chi_{ij} q \cdot (v_i-v_j)$$ (3.51)

where $\chi_{ij}$ is the number of collisions between the $i-th$ and the $j-th$ particles in the unit of time. Upon comparing Eqs. (3.52) and (3.54) one obtains an expression for $g(v_i-v_j)$:

$$g(v_i-v_j)=\frac{2 \chi_{ij} q}{N}(v_i-v_j)$$ (3.52)

Now it is easy to understand that $\chi_{ij}$ is a decreasing function of $\vert v_i-v_j\vert$: indeed, a great number of collisions occurs when the pair $i,j$ belongs to a cluster (where $\vert v_i-v_j\vert$ is small), whereas the two particles rarely collide when they are out of a cluster (and $\vert v_i-v_j\vert$ is high). We can, therefore, make a reasonable ansatz on $g(v_i-v_j)$, that is:

with $\beta'<1$. From Eq. (3.49) it appears that $G(v) \sim g(v-v')\vert _{v'=0}$ as the integration has to be performed with respect to the measure $P(v',t)dv'$ that is strongly peaked at $v'=0$. Finally, one can conclude from the same Eq. (3.49) that

where $\beta=\beta'+1<2$. It is clear now, looking at Eq. (3.51), that when $\tau_b < \tau_c$ (i.e., in the non-clusterized regime) the argument of the exponential is dominated by $v^2/\tau_b$ and therefore a Gaussian is expected for $P(v)$ with variance $T_b$. In the opposite regime, when $\tau_b > \tau_c$ the distribution is a Gaussian with variance $\frac{\tau_c} {\tau_b}T_b$ at low velocities, a simple exponential (if $\beta=1$) at high velocities, and a Gaussian with variance $T_b$ at extremely high velocities, but this very far tails practically cannot be observable. In Fig. fig_v_tails in case (c), when $\tau_b » \tau_c$ we observe a simple exponential tail, as we may expect from the argument presented above.