Inter-dependence of the volume and stress ensembles and equipartition in statistical mechanics of granular systems

TL;DR

This paper reveals the interdependence of volume and stress ensembles in granular matter, challenges previous assumptions, and introduces an equipartition principle and a method to quantify compactivity from macroscopic data.

Contribution

It demonstrates the interdependence of volume and stress ensembles, introduces an equipartition principle, and provides a way to measure compactivity from macroscopic observations.

Findings

Volume and stress ensembles are inter-dependent.

Structural properties depend on the angoricity tensor.

Stress-related quantities depend on the compactivity.

Abstract

We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are inter-dependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle - the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements.