Estimating angoricity and granular equations of state for monodisperse packings of frictional hard spheres in a wide range of densities

In this paper, we study the granular equation of state (EOS) for computer-generated three-dimensional mechanically stable packings of frictional monodisperse particles over a wide range of densities (packing fractions), $\phi =0.56$–0.72. As a statistical physics framework, we utilize the statistical ensemble for granular matter, specifically the "angoricity" ensemble, where the compressional component ${\Sigma }_p$ of the force-moment tensor serves as granular energy and angoricity $A_p$ is the corresponding granular "temperature." We demonstrate that the systems under study conform well to this statistical description, and the simple equation of state ${\Sigma }_p=2.8N{A}_p$ holds very well, where $N$ is the number of particles. We show that granular temperature exhibits a rapid drop around the random-close packing (RCP) limit $\phi \approx 0.64$–0.65, and, hence, one can say that granular packings "freeze" at the RCP limit. We repeat these calculation for shear angoricity $A_{\text{sh}}$ and shear component ${\Sigma }_{\text{sh}}$ of the force-moment tensor and obtain a similar EOS, ${\Sigma }_{\text{sh}}=0.85N{A}_{\text{sh}}$. Additionally, we measure the so-called keramicity, an inverse temperature variable corresponding to the determinant of the force-moment tensor, while pressure angoricity corresponds to its trace. We show that inverse keramicity ${\kappa }^{-1}$ and angoricity $A_p$ conform to an EOS $\frac{1}{A_p}\frac{{\Sigma }_p}{N}+0.11\kappa {\left(\frac{{\Sigma }_p}{N}\right)}^3=1.2$, whose form is predicted by mean-field theory. Finally, we demonstrate that the alternative statistical ensemble where Voronoi volumes serve as granular energy (and so-called compactivity serves as temperature) does not describe the systems under study well.