The problem of scale separation

The attack carried against granular hydrodynamics by I. Goldhirsch [96,95,94,92,93] is less hopeless, as he takes in consideration all the recent literature on rapid granular flows while he does not discuss the dense and quasi-static situations: the limits of hydrodynamics have been intensively probed by means of simulations and experiments, and it seems that a range of good validity can be found (perhaps it is more difficult to predict it!). However he points out that even these successes are somehow lacking a rigorous foundation, or using is words that ``the notion of a hydrodynamic,or macroscopic description of granular materials is based on unsafe grounds and it requires further study''. He addresses two fundamental issues:

1. in granular materials a reference equilibrium state is missing;

2. in granular materials the spatial and temporal scales of the dynamics of the particles are not well separated from the relevant macroscopic scales;

The first problem is more evident than the second. If a molecular gas is left to itself it comes to an equilibrium state given by the stationary solution of the corresponding kinetic equation (rarefied gases follow the Boltzmann equation, dense gases follow the Enskog-Boltzmann equation or better the ring kinetic equations). If such an equilibrium state is well defined, perturbations around it can be used as solutions of non-equilibrium problems. Moreover, if external time scales are much larger than the microscopic time scale of relaxation to equilibrium, most of the degrees of freedom of the gas are rapidly averaged and only a few variables are needed for the description of the out of equilibrium dynamics, which obey to macroscopic equations such as Euler or Navier-Stokes equations. If a granular gas is left to itself, instead, the only equilibrium state is an asymptotic death of the motion of all the particles, but before it different kinds of correlations arise leading to strong inhomogeneities (clustering, vortices, shocks, collapse, and so on). In this sense the relaxation to equilibrium has a characteristic time which is infinite and many other characteristic times given by different instabilities, due to the non-conservative nature of the collisions. What reference state can be used in a perturbative method like the Chapman-Enskog expansion? In the first derivations of granular hydrodynamics the Maxwell-Boltzmann equilibrium was used, in the latest derivations a more rigorous Chapman-Enskog expansion has been followed using solutions of the Enskog-Boltzmann equation by means of a Sonine expansion (which again must be performed around a Maxwell distribution). Goldhirsch has observed however that the limit $(1-r) \to 0$ and $Kn \to 0$ (the Knudsen number, indicating the intensity of the gradients, see Eq. (2.38)) is smooth and non singular for the granular Boltzmann equation, since the relaxation to local equilibrium takes place in a few collisions per particle, while the effect of (low) inelasticity is relevant on the order of hundreds or thousands of collisions. This means that a perturbative (in $1-r^2$ and $Kn$) expansion may be applied to the Boltzmann equation around a well suited ``elastic'' equilibrium, but it is expected to breakdown as $(1-r)$ or $Kn$ are of order $\sim 1$.

In a stationary state the only hope is that the system fluctuates around a well defined ``most probable state'' (described by a well defined $n$-particles distribution function, hopefully $n=1$) and again an expansion around it can be performed. This program has not yet been realized: till now all hydrodynamic theories assume that the equilibrium reference state does not depend on the boundary conditions (e.g. the form of the external driving) and that both cooling and driven regimes can be described by the same set of equations.

The second issue raised by Goldhirsch stems from a more quantitative discussion. He stresses the fact that the lack of scales separation is not only a mere experimental problem: one can in principle think of experiments with an Avogadro number of grains and very large containers. Instead the lack of separation of scales is of fundamental nature in the framework of granular materials. This problem has been already recognized in molecular gases: indeed, when molecular gases are subject to large shear rates or large thermal gradients (i.e. when the velocity field or the temperature field changes significantly over the scale of a mean free path or the time defined by the mean free time) there is no scale separation between the microscopic and macroscopic scales and the gas can be considered mesoscopic. In this case the Burnett and super-Burnett corrections (and perhaps beyond) are of importance and the gas exhibits differences of the normal stress (e.g. $\mathcal{P}_{xx} \neq \mathcal{P}_{yy}$) and other properties characteristic of granular gases. Even if clusters are not expected in molecular gases, strongly sheared gases do exhibit ordering which violates the molecular-chaos assumption. In granular gases this kind of mesoscopicity is generic and not limited to strong forcing. Moreover, phenomena like clustering, collapse (and of course avalanches or oscillon excitations) pertain only to granular gases. In mesoscopic systems fluctuations are expected to be stronger and the ensemble averages need not to be representative of their typical values. Furthermore, like in turbulent systems or systems close to second-order phase transitions, in which scale separation is non-existent, one expects constitutive relations to be scale dependent, as it happens in granular gases.

The quantitative demonstration of the intrinsic mesoscopic nature of (cooling) granular gases follows from the relation [96]

$$T=C\frac{\gamma^2l_0^2}{1-r^2}$$ (2.157)

that relates the local granular temperature with the local shear rate $\gamma$ and the mean free path $l_0$. The above relation holds until $\gamma$ can be considered a slow varying (decaying) quantity in respect to the much more rapid decay of the temperature fluctuations (this can be observed by a linear stability analysis and also by the fact that shear modes decay slowly for small wave-numbers - a result of momentum conservation). From the Eq. (2.165) follows that the ratio between the change of macroscopic velocity over a distance of a mean free path $l_0\gamma$ and the thermal speed $\sqrt{T}$ is $\sqrt{1-r^2}/\sqrt{C}$, e.g. $\simeq 0.44$ for $r=0.9$, that is not small. Thus, except for very low values of $1-r^2$, the shear rate is always large and the Chapman-Enskog expansion should therefore carried out beyond the Navier-Stokes order. The above consideration is a simple consequence of the supersonic nature of granular gases [93]. It is clear that a collision between two particles moving in (almost) the same direction reduces the relative velocity, i.e. velocity fluctuations, but not the sum of their momenta, so that in a number of these collisions the magnitude of the velocity fluctuations may become very small with respect to the macroscopic velocities and their differences over the distance of a mean free path. Also the notion of mean free path may become useless: $l_0$ is defined as a Galilean invariant, i.e. as the product between the thermal speed $\sqrt{T}$ and the mean free time $\tau$; but in a shear experiment the average squared velocity of a particle is given by $\gamma^2y^2+T$ ($y$ is the direction of the increasing velocity field), so when $y \gg \sqrt{T}/\gamma$, the distance covered by the particle in the mean free time $\tau$ is $l(y)=yl_0\gamma/\sqrt{T}=y\sqrt{1-r^2}/\sqrt{C}$ and therefore can become much larger that the ``equilibrium'' mean free path $l_0$ and even of the length of the system in the streamwise direction.

Furtherly, the ratio between the mean free time $\tau=l_0/\sqrt{T}$ and the macroscopic characteristic time of the problem $1/\gamma$ , using expression (2.165), reads again $\sqrt{1-r^2}/\sqrt{C}$. This means that also the separation between microscopic and macroscopic time scales is guaranteed only for $r \to 1$. And this result is irrespective of the size of the system or the size of the grains. This lack of separation of time scales poses two serious problem: (a) the fast local equilibration that allows to use local equilibrium as zeroth order distribution function is not obvious; (b) the stability studies are usually performed linearizing hydrodynamic equations, but the characteristic times related to the (stable and unstable) eigenvalues must be of the order of the characteristic ``external'' time (e.g. $1/\gamma$) which, in this case, is of the order of the mean free time (as just derived), leading to the paradoxical conclusion that the hydrodynamic equations predict instabilities on time scales which they are not supposed to resolve.

Goldhirsch [93] has also shown that the lack of separation of space and time scales leads to scale dependence of fields and fluxes. In particular he has shown that the pressure tensor depends on the scale of the coarse graining used to take space-time averages. This is similar to what happens, for example, in turbulence, where the ``eddy viscosity£ is scale dependent. Pursuing this analogy, Goldhirsch has noted that an intermittent behavior can be observed in the time series of experimental and numerical measures of the components pressure tensor: single collisions, which are usually averaged over in molecular systems, appear as ``intermittent events'' in granular systems as they are separated by macroscopic times.