Power-law decay of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state

The hydrodynamic part of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state has been calculated by using mode-coupling theory for a finite system with periodic boundary conditions. The existence of the shearing instability, leading to a divergent behavior of the velocity flow fluctuations, is taken into account. A time region in which the velocity autocorrelation function exhibits a power-law decay, when time is measured by the number of collisions per particle, has been been identified. Also the explicit form of the exponential asymptotic long time decay has been obtained. The theoretical prediction for the power-law decay is compared with molecular dynamics simulation results, and a good agreement is found, after taking into account finite size corrections. The effects of approaching the shearing instability are also explored.

Figure 1

Dimensionless normalized VACF $f_{\omega }\left(s\right)$ for a system of inelastic hard disks in the HCS. Time $s$ is measured in units of ${[\tilde{T}\left(0\right)/m{\sigma }^2]}^{1/2}$, where $\tilde{T}\left(0\right)$ is the initial granular temperature of the system. The coefficient of normal restitution is $\alpha =0.99$ and the density is $n{\sigma }^2=0.385$. Different numbers of particles (and sizes of the system) have been used in the simulations, as indicated in the inset.

Figure 2

Values of the dimensionless normalized VACF $f_{\omega }$ as a function of the inverse of the number of particles $N$ of the system. The coefficient of normal restitution is $\alpha =0.99$ and the density $n{\sigma }^2=0.385$. The symbols are simulation results at three different times, namely $s=2.6$ (black), $s=5.0$ (red), and $s=7.2$ (green), from top to bottom. Time $s$ is measured in units of ${[\tilde{T}\left(0\right)/m{\sigma }^2]}^{1/2}$, where $\tilde{T}\left(0\right)$ is the initial granular temperature of the system. The straight lines are linear fits of the simulation data. The values of the slopes of the three lines are very close.

Figure 3

The same as in Fig. 1, but now the finite size correction given in Eq. (70) has been added to each curve.

Figure 4

VACF for an inelastic system of hard disks with a coefficient of normal restitution $\alpha =0.99$ and two different values of the density, as indicated in the inset. The results have been obtained in a system with 1000 particles, and finite size effects have been corrected as discussed in the main text.

Figure 5

Tail amplitude ${\alpha }_D$ as a function of the number density of the system. The symbols are simulation results: the (black) circles are for a system of elastic hard disks and the (red) squares for a system of inelastic hard disks with a coefficient of normal restitution $\alpha =0.99$. The solid line is the theoretical prediction using the mode-coupling theory developed in the main text and the transport coefficients obtained from the revised Enskog theory. On the scale of the figure, the predictions for the elastic and the inelastic systems are undistinguishable.

Figure 6

Values of the dimensionless normalized VACF $f_{\omega }$ as a function of the inverse of the number of particles $N$ of the system. The coefficient of normal restitution is $\alpha =0.98$ and the density $n{\sigma }^2=0.3$. The symbols are simulation results at three different times, namely $s=2.4$ (black), $s=5$ (red), and 10 (green), from top to bottom. Time $s$ is measured in units of ${[\tilde{T}\left(0\right)/m{\sigma }^2]}^{1/2}$, where $\tilde{T}\left(0\right)$ is the initial granular temperature of the system. The straight lines are linear fits of the simulation data in the region where this behavior is observed. The values of the slopes of the three lines are very close.