The asymptotic decay of energy

The initial energy decay, due to the dissipation in inelastic collisions in the homogeneous cooling regime, is very well described by the Haff law [103] $E(t)=E_{Haff}(t) \sim t^{-2} \sim \exp(-2\epsilon \tau)$. This law has largely been verified in numerical simulations. When correlations build up (in the velocity field, as well as in the density field), the energy decay changes: the shear instability (formation of vortices) discussed above slows it down because of the increased relevance of the shear heating term in the temperature balance. Shearing regions are heat sources that contrast the collisional cooling.

Analytical and numerical works on the energy decay in the correlated regime are very rare [96,95,45]. A mode-coupling analysis has been put forward by Brito and Ernst [45], for $d \geq 2$. They have studied the contribution to the energy decay due to unstable modes, obtaining that

$$E(T)=E_{Haff}(t)+E_{uu}(t)+E_{nT}(t)$$ (5.27)

where $E_{uu}(t)$ and $E_{nT}$ are the contributions to the kinetic energy due to macroscopic velocity flow and to the $n-T$ fluctuations respectively. Brito and Ernst have calculated their contributions to the energy decay:

and therefore the $n-T$ term is sub-leading in respect to the $u-u$ term. As a consequence of (5.30), the decay of the kinetic energy after the homogeneous phase is predicted to be of the diffusive form $\sim \tau^{-d/2}$. The crossover (internal) time $\tau_{cross}$ is determined by the equality $E_{Haff}(\tau)=E_{uu}(\tau)$. To obtain the physical time $t_{cross}$ is more difficult, as the relation $t(\tau)$ is not clear in the correlated phase. It must be stressed that in the homogeneous phase the main contribution to the energy decay is due to the short wavelengths, the standard-elastic range $kl_0>\gamma_0$ (see paragraph 5.1.2), while the diffusive behavior of the non-homogeneous phase is due to the long wavelength of the dissipative range $kl_0 < \gamma_0$. The predictions of this theory have been checked in $d=2$ [45,175]: it seems difficult to have good tests of the prediction $\tau^{-d/2}$, MD simulations are quite demanding and rarely give a sufficient statistics. There is agreement in a window of values of the restitution coefficient $r$, of the volume fraction $\phi$ and of the time $\tau$.