Closed analytical form of many-body free volume and thermodynamics of monodisperse hard disks.

The hard-disk model is a fundamental reference system for excluded-volume physics, liquid structure, and two-dimensional 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 expressed analytically through the 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.