Exact hard-disk free volumes

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Abstract
Properties of exact hard‐disk free volumes are determined by a combination of analytical and numerical techniques. Both the high‐density fluid phase and the lower‐density fluid phase are treated. These one‐particle free volumes are used to verify known thermodynamic information for hard disks and to calculate the shear modulus for the hard‐disk solid phase. The free volumes are also compared to approximate free‐volume estimates made from the known hard‐disk entropy. The statistical distributions of free volume and free surface (perimeter of the free volume) are studied. The percolation transition, at which the free‐volume changes from extensive to intensive, is found to occur at about one‐third of the freezing density.

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