The break of equipartition

In 1995 Du, Li and Kadanoff [77] have published the results of the simulation of a minimal model of granular gas in one dimension. In this model $N$ hard rods (i.e. hard particles in one dimension) move on a segment of length $L$ interacting by instantaneous binary inelastic collisions with a restitution coefficient $r < 1$. To avoid the cooling down of the system (due to inelasticity) a thermal wall is placed at the one of the boundaries, i.e. when the leftmost particle bounces against the left extreme ($x=0$) of the segment, it is reflected with a new velocity taken out from a Gaussian distribution with variance $T$. This particle carries the energy to the rest of the system. The main finding of the authors was that even at very small dissipation $1-r«1$ the profiles predicted by general hydrodynamic equations (they used constitutive relations of Haff [103] or Jenkins and Richman [120]) were not able to reproduce the essential features of the simulation. In particular the stationary state predicted by hydrodynamics is a flow of heat from the left wall to the right (it goes to zero at the right wall), with no macroscopic velocity flow ($u(x,t)=0$), a temperature profile $T(x,t)$ which decreases from $x=0$ to $x=L$, and a density profile inversely proportional to the temperature (as the pressure $p=nT$ is constant throughout the system). In Fig. fig_kadanoff some snapshots of the system (i.e. the position of the grains at different instants) are shown. The system is in an ``extraordinary'' state with almost all the particle moving slowly and very near the right wall, while almost all the kinetic energy is concentrated in the leftmost particle. The system cannot be considered in a stationary state, even if its kinetic energy is statistically stationary (i.e. fluctuates around a well defined average which is time translational invariant). Moreover, reducing the dissipativity $1-r$ at fixed $N$ the cluster near the wall becomes smaller and smaller. If the heat bath is replaced by a sort of saw-tooth vibrating wall which reflects the leftmost particle always with the same velocity $v_0$, the evolution of the baricentrum changes in a stationary oscillation very near to the rightmost wall, so that this clustering instability does not disappear. The authors also point out the fact that the Boltzmann Equation can give a qualitative prediction of this clustering phenomena in the limit $N \to \infty$, $1-r \to 0$ with $N(1-r) \sim 1$. We have reproduced the results of Du et al. and have discovered the this model has no proper thermodynamic limit, i.e. when $N,L \to \infty$ with $N/L \sim 1$ the mean kinetic energy and the mean dissipated power reduce to zero. This is consistent with the scenario suggested by the authors: the equipartition of energy (i.e. local equilibration of the different degrees of freedom) is broken and the description of the system in terms of macroscopic (slowly varying) quantities no more holds. In this scenario, usual thermodynamic quantities such as mean kinetic energy or mean dissipated power, are not extensive quantities.

Figure 2.3: Some snapshots of the 1D system from the work of Du et al. [77]

L. P. Kadanoff has also addressed, in a recent review article [122], a set of experimental situations in which hydrodynamics seems useless. We have already discussed a well known experiment by Jaeger, Knight, Liu and Nagel [117] where a container full of sand is shaken from the bottom, when the shaking is very rapid. The observations indicate that there is a boundary layer of a thickness of few grains near the bottom that is subject to a very rapid dynamics with sudden changes of motion of the particles. At the top of the container, instead, the particles move ballistically encountering very few collisions in their trajectory. Both the top and the bottom of the container cannot be described by hydrodynamics, as the assumption of slow variation of fields or that of scale separation between times (the mean free time must be orders of magnitude lower than the characteristic macroscopic times, as the vibration period) are not satisfied. On the other hand, the slow dynamics regime has been studied, when the vibration is reduced to a rare tapping, so that the system reaches mechanical equilibrium (stop of motion) between successive tappings [126]. The equilibrium is reached at different densities, and - as the tapping is carried on - the ``equilibrium'' density slowly changes and its evolution depends on many previous instants and not on the very last tap, i.e. is history dependent. This non-locality in time cannot be described by a set of partial differential equations, therefore the hydrodynamic description here again fails.

The simulation of cooling granular gases have also interested L. P. Kadanoff in his excursus of the limits of hydrodynamics. The clustering instabilities and the inelastic collapse are clear signatures of the failure of macroscopic description. Moreover, in the inelastic collapse Kadanoff and Zhou [228] have pointed out (see Fig. fig_zhou) that there is a correlation between velocity directions of the particles involved in the collapse: in particular collapse is favored by parallel velocities (because they cannot escape in perpendicular directions). This situation implies a dramatic breakdown of Molecular Chaos assumption and gives evidence of the fact that Inelastic Collapse cannot be described even by a Boltzmann equation.

Figure 2.4: A snapshot from a MD simulation of cooling inelastic hard spheres [228]. The particles in black are those that have participated in the last collisions, just before a collapse