Geometry-originated universal relation for arbitrary convex hard particles

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.