Edwards entropy and compactivity in a model of granular matter

TL;DR

This paper develops a complete statistical mechanics framework for a model of granular matter, demonstrating the validity of equal probability assumptions and analyzing entropy and compactivity in jammed packings.

Contribution

It provides a fully characterized model of granular packings, including enumeration of jammed states and analysis of entropy and compactivity.

Findings

Jammed packings are independent of defect distribution.

All packings are isostatic.

Entropy maximization leads to equal compactivities at equilibrium.

Abstract

Formulating a statistical mechanics for granular matter remains a significant challenge, in part, due to the difficulty associated with a complete characterization of the systems under study. We present a fully characterized model of a granular material consisting of $N$ two-dimensional, frictionless, hard discs, confined between hard walls, including a complete enumeration of all possible jammed structures. We show the properties of the jammed packings are independent of the distribution of defects within the system and that all the packings are isostatic. This suggests the assumption of equal probability for states of equal volume, which provides one possible way of constructing the equivalent of a microcanonical ensemble, is likely to be vaild for our model. An application of the second law of thermodynamics involving two subsystems in contact shows that the expected spontaneous equilibration of defects between the two is accompanied by an increase in entropy and that the equilibirum, obtained by entropy maximization, is characterized by the equality of compactivities. Finally, we explore the properties of the equivalent to the canonical ensemble for this system.