A random packing of hard particles represents a fundamental model for granular matter. Despite its importance, analytical modeling of random packings remains difficult due to the existence of strong correlations which preclude the development of a simple theory. Here, we take inspiration from liquid theories for the n-particle angular correlation function to develop a formalism of random packings of hard particles from the bottom up. A progressive expansion into a shell of particles converges in the large layer limit under a Kirkwood-like approximation of higher-order correlations. We apply the formalism to hard disks and predict the density of two-dimensional random close packing (RCP), φrcp = 0.85 ± 0.01, and random loose packing (RLP), φrlp = 0.67 ± 0.01. Our theory also predicts a phase diagram and angular correlation functions that are in good agreement with experimental and numerical data.
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Jin, Yuliang, James G. Puckett and Hernan A. Makse. "Statistical Theory of Correlations in Random Packings of Hard Particles." Physical Review E 89:052207 (May 2014).
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Original version is available from the publisher at: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.052207