The scaling of the local temperature with the cluster mass

Figure: Granular temperature vs. cluster mass 1d, MD, with $r=0.99$, $\tau _b=0.01$, $\tau _c=0.1$, and $r=0.7$ and $\tau _b=100$, $\tau _c=0.1$ with $N=L=300$ and $M=100$. The Gaussian case is constant, while the non-Gaussian case is fitted by $\sim m^{-0.5}$

Figure: Granular temperature vs. cluster mass 1d, DSMC, with $r=0.99$, $\tau _b=0.01$, $\tau _c=0.5$, and $r=0.5$ and $\tau _b=100$, $\tau _c=0.5$ with $N=L=500$ and $M=12000$. The Gaussian case is constant, while the non-Gaussian case is fitted by $\sim m^{-0.8}$

Figure: Granular temperature vs. cluster mass 2d, DSMC, with $r=0.99$, $\tau _b=0.01$, $\tau _c=0.05$, and $r=0.5$ and $\tau _b=100$, $\tau _c=0.05$ with $N=L^2=10000$ and $M=3200$. The Gaussian case is constant, while the non-Gaussian case is fitted by $\sim m^{-0.8}$

Figure: Velocity distribution in limited density 2d, DSMC: pluses (+) represent velocities only from boxes with $m=1$, while crosses ($\times$) represent velocities with $m=5$, with $r=0.5$, $N=L^2=10000$, $\tau _b=100$, $\tau _c=0.05$. $v_0=\sqrt {T_g}$

In paragraph 2.5 we have discussed the arguments of I. Goldhirsch against the possibility of a very general hydrodynamical description. One of the argument started by a heuristic estimate of the local temperature as a function of density and local shear rate (valid on time scales shorter than the time of decay of the shear rate), given in Eq. (2.165). In this formula it appeared that $T \propto \l _0^2 \propto n^{-2}$, so that the local scalar pressure $p=nT \propto n^{-1}$: this is the reason of the clustering instability, i.e. in small regions an increase of the number of particles corresponds to a decrease of the local pressure, so that more particles can enter the region. We have studied, parametrically, the relation between local temperature and local density, plotting the mean square velocity in a box $T_M$ as a function of the number of grains $m$ in that box, after having divided the total volume in $M$ boxes. As already discussed, the distribution of the cluster masses (i.e., the number of particles in a box) in the clusterized regime presents a power law decay with an exponential finite size cut-off. The plots of $T_M$ vs. $m$ are given in Fig. fig_tempscaling_1d_MD, Fig. fig_tempscaling_1d_DSMC and Fig. fig_tempscaling_2d_DSMC: these are $d=1$ Molecular Dynamics, $d=1$ DSMC simulations, and $d=2$ DSMC simulations respectively.

In all the cases we have studied a collisionless situation ( $\tau_b \ll \tau_c$) and a colliding situation ( $\tau_b \gg \tau_c$), obtaining different results. In the collisionless situation the local temperature does not depend upon the cluster mass $m$. In the colliding situation the local temperature appears to be a power of the cluster mass, i.e. $T_M(m) \sim m^{-\beta}$ with $0 < \beta < 1$. This relation ensures that the scalar pressure $p=nT \propto n^{1-\beta}=n^{\beta^*}$ with $\beta^*>0$: the clustering ``catastrophe'' (particles falling in an inverted pressure region) is absent. Moreover, we can give an estimate of the scale dependence of the temperature, using the previous result on the fractal correlation dimension $d_2$ (defined in Eq. 3.18) and assuming that the scaling relation for the temperature is valid at different scales replacing the density to the number of particles, i.e. $T(m) \sim n^{-\beta}$. The results on the fractal correlation dimension suggest that the local density has the following scale dependence $n(R) \sim R^{-(d-d_2)}$ so that the local temperature follows the law $T(R) \sim R^{\beta(d-d_2)}$ and the local pressure $p(R) \sim R^{-(1-\beta)(d-d_2)}$. The density and the pressure decrease with $R$, while the temperature increases. The scale dependence of the macroscopic fields is evidently at odds with the possibility of separating a mesoscopic scale from the microscopic one and therefore the hydrodynamical description cannot even be tempted.