The equation of state of hard spheres and the approach to random closest packing

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Abstract
Data on the internal pressure of hard‐sphere and hard‐disk fluids have been available for some time from computer simulations, even at densities above the freezing density. These results for the metastable fluids suggest that the pressure diverges at the density of random closest packing. However, our examinations of these data indicates that the divergence is characterized by a fractional exponent. We show that incorporating this nonanalyticity not only enables us to construct a semiempirical equation of state which is accurate at densities well beyond that of the Carnahan–Starling equation of state, but it enables us to predict a finite entropy at random closest packing. We also show that this kind of thermodynamic singularity implies that the direct correlation function becomes infinitely long ranged with a critical exponent remarkably similar to the percolation exponent. Given the difficulties inherent in simulating hard spheres at such densities, however, we do suggest that these findings be regarded with some caution.