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

TL;DR

This paper critiques the traditional volume-based statistical mechanics for granular matter, proposing a connectivity-based approach that accurately accounts for all structural degrees of freedom and entropy, with explicit results and experimental validation.

Contribution

It introduces a novel connectivity function replacing the volume function, enabling a correct entropy calculation for granular systems.

Findings

The volume function underestimates entropy significantly.

The connectivity function accounts for all structural degrees of freedom.

Entropy is shown to be extensive in a 2D experimental system.

Abstract

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 ${\it connectivity}$ 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.