arXiv2026
The hard disk model is a fundamental reference system for excluded volume physics, liquid structure, and 2D ordering. Yet, despite its apparent simplicity, an analytical formulation of its thermodynamics has remained difficult because the free volume available to disks centers has a highly nontrivial many-body geometry. Here we show that this geometry admits a closed and exact representation: for any given hard disk configuration, the free volume can be analytically expressed through intersection areas of up to five exclusion disks. This provides a direct geometrical route from particle coordinates to the configurational partition function and entropy. We prove that the N disk partition function factorizes into a product of conditional free volumes and identify its two limiting asymptotic forms: the low density fluid regime controlled by extensive cavities and the high density fluid regime controlled by intensive private cells. Using the geometric measures computed from representative equilibrium particle configurations, the resulting theory reproduces the established hard disk equation of state over almost the entire density range up to close packing. At intermediate densities, the analysis reveals a mixed fluid regime in which localized caged defects provide an additional configurational entropy within the density interval preceding liquid hexatic coexistence. Finally, the five disk intersection area is shown to be a sensitive local scalar measure of hexagonal ordering, complementing the orientational order parameter and emphasizing the central role of many-body correlations.