Possible Discussion Topics

1. In the kinetic regime, the approach has always been inherently statistical. The element of dissipation in granular collisions leads to effects that are not present in molecular gases. For instance, it appears that for different species, granular temperatures are different... Also, correlations appear to be induced during collisions that have no analogue in molecular systems, but which affect the the distributions of velocities.
2. In the dense regime, friction inherently leads to ignorance concerning contact forces. Thus, complete positional information on particles does not determine force balance. Rather, there is a degree of ignorance about the tangential forces (and therefore in a collection of particles, the normal forces). This ignorance is comparable to the same level of ignorance we face when we try to deal with large numbers of molecules. It is simply impossible to know information at that level of detail, so we must take an ensemble point of view.
3. Given 2, and also typically contact reduncancy, what are the macroscopic parameters that one must specify in order to determine distribution functions? Here, a useful example is Couette shear flow: experimentally or computationally, one can establish well definied distributions for any parameter that one wants, but it is not clear what macroscopic parameters really determine these distributions.
4. What microscopic statistical behavior must one invoke in order to predict the mean behavior for force transmission? What are the appropriate distributions?

Edwards ensembles. 

It is now clear that there are at least two cases:

• Strong taps where the packing makes a random walk both in force space AND in packing space. Are all stable configurations equiprobable?
• Gentle taps, where the packing has a fixed geometry, and only contact forces rearrange. Are all configurations equiprobable?

This could be tested in a simple geometry: a wedge with a single ball. On the other hand, recent experiments by Kabla and Debregeas show that under gentle taps the system ages even without noticeable changes of density. Does that imply some non uniformity in the space of configurations?

• In what circumstances an `as poured' system can be thought of as a typical member of a tapped ensemble?

1. Force chains/force network models

Assuming e.g. an Edwards measure on forces, can one derive the macroscopic stresses in a granular system, and its response function? This has been tried by Edwards and Grinev, but it is fair to say that very few people really understand what they have done. But it is clear that they asked the right question! Using simplifying assumptions (linear Boltzmann, or 6-fold lattices) one finds conflicting results on the large scale structure of the response function (elliptic/hyperbolic). My impression is that

a) the answer depends on the degree of anisotropy of the packing. Isotropic packings should, nearly by symmetry, by ellipical, whereas a sufficient degree of anisotropy might help keeping the response hyperbolic (see, e.g. the simulations of Head, Tkachenko, Witten; or Zucker Claudin Clement).

b) there is a subtlety in the way the zero force limit is taken before or after the large height limit, that might affect the experimental results we have.

    2.    Cooperativity in slow dynamics.

Many groups try to understand if there is a growing length accompnying the slowing down of dynamics, both in granular material and in glassy materials. Measuring the so called 4-point susceptibility, or other similar correlation functions that try to extract a cooperative length from the images, seem to me extremely exciting. Coming back to the Kabla/Debregeas experiment (or the memory experiment of Josserand et al.), can one understand what is going on in terms of purely local subsystems with a complicated phase space (as in the trap model used by Head, or by Kabla/Debregeas), or should one look for a growing dynamical length scale?

1. In MD simulations of a gas, one can obtain surprising accuracy with a relatively small system by treating the walls as a thermal bath. I.e., each time a particle collides with the wall, a random increment (positive or negative) is added to the energy of the particle, the distribution of these increments being determined by the wall temperature. 

Q: Is there an analogous device for accelerating simulation of forces in dense granular systems in the slow-flow regime? This question might provide some clues about how to define the granular temperature in the slow-flow regime.

2. The double-Y model gives rise to a Boltzmann-type equation to describe the statistics of force chains in a static granular medium. Explicit solutions of this equation have been obtained for discrete idealizations of the model: i.e., force chains act only in a finite set of directions and can assume only a discrete set of magnitudes. These discrete solutions exhibit paradoxical behavior. 

Q: Can the paradoxes be eliminated by solving the full model, in which forces have a continuous distribution, both in direction and magnitude?

3. The term "thermoplasticity" refers to a theory, which was developed by Ziegler, Lubliner, Maugin, Collins, Houlsby, and others, in which the complete response of materials undergoing (rate-independent) plastic deformation, including frictional materials, may be described by specifying two functions: a thermodynamic potential (such as the Helmholz free energy) and the dissipation function. Their approach is elegant and natural, and it avoids the inconsistencies that sometime sneak into conventional formulations of plasticity.

Q: Might use of this theory facilitate the passage from the statistics of micromechanical behavior to continuum constitutive laws? For example, in their formalism, the yield function is obtained from the dissipation function by a Legender transformation; it may be easier to construct the dissipation function than the yield function since the former is closer to the basic micromechanical physics than the latter.

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