Jamming I: A volume function for jammed matter

TL;DR

This paper introduces a volume function based on Voronoi volumes to describe jammed matter geometrically, providing an analytical formula and a statistical theory to connect microscopic contact networks with macroscopic properties.

Contribution

It presents the first analytical and statistical framework linking microscopic contact networks to macroscopic properties in jammed systems.

Findings

Voronoi volume is key to describing jammed matter.

Derived an analytical formula for Voronoi volume in any dimension.

Discovered a mesoscopic volume function inversely related to coordination number.

Abstract

We introduce a "Hamiltonian"-like function, called the volume function, indispensable to describe the ensemble of jammed matter such as granular materials and emulsions from a geometrical point of view. The volume function represents the available volume of each particle in the jammed systems. At the microscopic level, we show that the volume function is the Voronoi volume associated to each particle and in turn we provide an analytical formula for the Voronoi volume in terms of the contact network, valid for any dimension. We then develop a statistical theory for the probability distribution of the volumes in 3d to calculate an average volume function coarse-grained at a mesoscopic level. The salient result is the discovery of a mesoscopic volume function inversely proportional to the coordination number. Our analysis is the first step toward the calculation of macroscopic observables and equations of state using the statistical mechanics of jammed matter, when supplemented by the condition of mechanical equilibrium of jamming that properly defines jammed matter at the ensemble level.