A statistical mechanics framework for static granular matter

TL;DR

This paper develops a statistical mechanics framework for static granular matter, introducing a new ensemble based on stress conservation, and links it with a mean-field jamming transition theory, advancing understanding of granular packings.

Contribution

It introduces a novel statistical ensemble for static granular materials based on stress conservation, distinct from Edwards ensemble, and connects it with a mean-field jamming transition theory.

Findings

Point J exhibits diverging pressure fluctuations.

The new ensemble applies to deformable grain packings.

Linking stress-based and contact number theories enhances understanding.

Abstract

The physical properties of granular materials have been extensively studied in recent years. So far, however, there exists no theoretical framework which can explain the observations in a unified manner beyond the phenomenological jamming diagram [1]. This work focuses on the case of static granular matter, where we have constructed a statistical ensemble [2] which mirrors equilibrium statistical mechanics. This ensemble, which is based on the conservation properties of the stress tensor, is distinct from the original Edwards ensemble and applies to packings of deformable grains. We combine it with a field theoretical analysis of the packings, where the field is the Airy stress function derived from the force and torque balance conditions. In this framework, Point J characterized by a diverging stiffness of the pressure fluctuations. Separately, we present a phenomenological mean-field theory of the jamming transition, which incorporates the mean contact number as a variable. We link both approaches in the context of the marginal rigidity picture proposed by [3, 4].