Long-time tails in freely cooling granular gases

The long-time behavior of the current autocorrelation functions for the velocity, the shear stress, and the heat flux is investigated in freely cooling granular gases. It is found that the correlation functions for the velocity and the shear stress have the long-time tails obeying ${\tau }^{-d∕2}$, while the correlation function for heat flux decays as ${\tau }^{-(d+2)∕2}\exp (-{\zeta }^{*}\tau )$ with the dimensionless cooling rate ${\zeta }^{*}$, the spatial dimension $d$, and the scaled time $\tau$ in terms of the collision frequency. The result of our numerical simulation of the freely cooling granular gases is consistent with the theoretical prediction.

Figure 1

(a) $C_D\left(\tau \right)$ as the function of $\tau$ with $e=1.0$ and 0.9. (b) $C_D\left(\tau \right)$ as the function of $\tau$ with $e=0.8$. ${\tau }_H$ and ${\tau }_L$ are the time of the violation of Haff’s law and the time starting to obey $T\left(\tau \right)\sim {\tau }^{-1}$, respectively.

Figure 2

(a) The snapshot of the density field at $\tau =22$. (b) The snapshot of the density field at $\tau =70$. Both figures are obtained from simulation of $e=0.8$. The size of the system shown in this figure is 337σ.

Figure 3

$T\left(\tau \right)$ as the function of $\tau$ with $e=0.8$. See the caption of Fig. 1 for the definitions of ${\tau }_H$ and ${\tau }_L$.

Figure 4

${\tau }_H$ and ${\tau }_P$ as the function of $1-e^2$.

Figure 5

$C_{\lambda }\left(\tau \right)$ as the function of $\tau$ with $e=1.0$ and 0.95.