Fundamental problems in statistical physics of jammed packings

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Abstract

For packed i.e., “jammed”, hard and rough objects kinetic energy is a minor and ignorable quantity, as is elastic strain. Hence in the static case, the stress equations need supplementing by “missing equations” depending solely on configurations. A different pathway of analysis is the calculation of the probability distribution of interparticle forces. This paper presents the mini-review of recently obtained results in this field and poses a number of fundamental problems which are yet to be solved.

Section snippets

The problem

The crucial granular system concerns packed, hard and rough objects: packed means no kinetic energy, hard means no elastic deformation and rough means all motion is confined to sliding and rolling after friction threshold is overcome. A granular material can have different packing fractions according to its history of preparation, and the application of external forces causes forces to exist between the grains. It has been suggested that in appropriate circumstances the central concept of

Probability distributions

For infinitely hard bodies there is no enthalpy and external forces have no effect on distribution functions. This means that if P is the probability distribution of configurations and of intergranular forces, it must separate into two partsP=Pc(configurations)Pf(forces).We now consider Pc which is the function of geometrical characteristics of the system. Let us assume that the set of contact points Cαβ is the total geometrical specification for a static packing. We define the centroid of

Slow dynamics

The basic movement is when two grains are subjected to sufficient force to overcome friction when sliding and rolling result. The picture in Fig. 9 is a standard problem in Ninteenth century dynamics textbooks and is very complex. Without friction it is resolved by the Gibbs–Appell equations (Desloge [22], Pars [23]; Whittaker [24]); with friction it involves putting Gibbs–Appell in Rayleighan form which we have not seen in the textbooks. A progression of these ideas is given in Fig. 8.

The

Acknowledgments

The authors have benefited from discussions with Prof. R. C. Ball, Dr. R. Blumenfeld, Dr. I. Hopkinson, Prof. D. Levine and Dr. H. Makse. D.V.G. acknowledges the Oppenheimer Fellowship from Sir Ernest Oppenheimer Fund and a Junior Research Fellowship from Wolfson College (Cambridge).

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