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The decay of energy

Figure: Energy decay for $ r=0.9$ and $ r=0.2$ ($ 1024^2$ sites). The bold dashed line $ \sim 1/\tau $ is a guide to the eye for the asymptotic energy decay, while the light dashed line is the exponential fit corresponding to the Haff law $ \exp(-2 \gamma_0 \tau)$. The Haff regime is too short to be observed in the system with $ r=0.2$. The two indicated times $ \tau =24$ and $ \tau =860$ correspond to the plots of Fig. fig_tsigma.
\includegraphics[clip=true,width=7cm, height=12cm,keepaspectratio]{prl2-fig1-ene.eps}

One sees from Fig. fig_endecay that during the initial stage, the total energy per particle

$\displaystyle E(\tau)=\frac{\sum v_i(\tau)^2}{N}$ (5.58)

is dissipated at an exponential rate

$\displaystyle E(\tau)=E(0)\exp(-\tau/\tau_{Haff})$ (5.59)

with $ \tau_{Haff}^{-1}= (1-r^2)/d$, as indicated in formulas 5.6 and 5.5b. The Haff law can be deduced from Eq.(5.57) imposing that each 'spin' fluctuates independently of the others. For times larger than $ \tau_{cross}$, of the order of $ \tau_{Haff}$, the system displays a crossover to a different regime, where the cooperative effects become dominant and the average energy per particle decay as $ E(\tau) \sim
\tau^{-1}$. Such a behavior agrees with the mode-coupling theory [45] discussed above (see paragraph 5.1.4) and with the behavior suggested by inelastic hard spheres simulations (IHSS) reported in various papers [175,45]. As already discussed, and as shown below, the crossover from one regime to the other is due to the formation of a macroscopic velocity field, i.e. to the shear instability. This is analogous to the formation of domains in a standard quench processes of magnetic systems. After the formation stage these regions start to compete to homogenize: this causes a conversion of kinetic energy into heat by viscous heating. In the hydrodynamics description of Equation (5.8c), this is due to the third term, that is the term where the coupling between the stress tensor $ \mathcal{P}_{ij}$ and the velocity gradients $ \partial u_j/\partial r_i$ appear. The viscous heating acts against the collisional cooling and leads to a slower decay of the energy.

An interesting observation from Fig. fig_endecay is that the smaller the inelasticity, the higher is the cumulated dissipated energy measured in the second (correlated) regime. This, a phenomenon observed in inelastic hard spheres simulations [158] and is due to the fact that the Haff regime (which is more rapid in dissipating energy) lasts more for less dissipative systems.


next up previous contents
Next: The structure factors and Up: The Inelastic Lattice Gas: Previous: The Inelastic Lattice Gas   Contents
Andrea Puglisi 2001-11-14