  We have discovered that two significant quantities within hard particle
systems: the probability of successfully inserting an additional particle at
random and the scale distribution function, can be connected by a concise
relation. We anticipate that this relation holds universal applicability for
convex hard particles. Our investigations encompassed a range of particle
shapes, including one-dimensional line segments, two-dimensional disks,
equilateral and non-equilateral triangles, squares, rectangles, and
three-dimensional spheres. Remarkably, we have observed a close alignment
between the two sides of the relation in all cases we examined. Furthermore, we
show that this relation can be derived from the fundamental thermodynamic
relation that connects entropy, pressure, and chemical potential. Our study
unveils a geometrically rooted relation that underpins essential thermodynamic
relations, shedding light on the intricate interplay of geometry and
thermodynamics in hard particle systems.
