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We have discovered that two significant quantities within hard particlesystems: the probability of successfully inserting an additional particle atrandom and the scale distribution function, can be connected by a conciserelation. We anticipate that this relation holds universal applicability forconvex hard particles. Our investigations encompassed a range of particleshapes, including one-dimensional line segments, two-dimensional disks,equilateral and non-equilateral triangles, squares, rectangles, andthree-dimensional spheres. Remarkably, we have observed a close alignmentbetween the two sides of the relation in all cases we examined. Furthermore, weshow that this relation can be derived from the fundamental thermodynamicrelation that connects entropy, pressure, and chemical potential. Our studyunveils a geometrically rooted relation that underpins essential thermodynamicrelations, shedding light on the intricate interplay of geometry andthermodynamics in hard particle systems.

A geometry-originated universal relation for arbitrary convex hard particles