Failure of the Volume Function in Granular Statistical Mechanics and an Alternative Formulation

We first show that the currently accepted statistical mechanics for granular matter is flawed. The reason is that it is based on the volume function, which depends only on a minute fraction of all the structural degrees of freedom and is unaffected by most of the configurational microstates. Consequently, the commonly used partition function underestimates the entropy severely. We then propose a new formulation, replacing the volume function with a connectivity function that depends on all the structural degrees of freedom and accounts correctly for the entire entropy. We discuss the advantages of the new formalism and derive explicit results for two- and three-dimensional systems. We test the formalism by calculating the entropy of an experimental two-dimensional system, as a function of system size, and showing that it is an extensive variable.

Figure 1

A 2D granular pack, with $(\alpha -1)M=19$ boundary grains (shaded), of which $M=10$ contact the walls. The (solid and dashed) vectors $\overrightarrow{r}$ connect a grain’s nearest contacts, circulating clockwise. In 3D, the intercontact vectors circulate clockwise around the facet that faces a cell, as seen from inside the grain. The solid vectors in the figure form a nondirectional spanning tree: they are independent, representing the independent DOFs, they reach every contact and the dashed vectors are linear combinations of them. There are $\alpha M=29$ boundary vectors, and our choice of spanning tree includes all of them but one.

Figure 2

The volume function, Eq. (2), does not depend on the intercontact vectors $\overrightarrow{r}$ surrounding grain $A$ and therefore cannot distinguish between configurations $a$ and $b$.

Figure 3

Entropy vs number of particles of the 2D experimental granular systems of Ref. [23].