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Free 'volume' of the one dimensional hard rod and the two dimensional hard disk systems is obtained analytically and compared to that of the three dimensional hard sphere system obtained numerically. The logarithm of the free volume plotted against the number of holes makes an S-shape in cases of hard sphere and hard disk systems, in contrast with the hard rod system in which case it is a monotonical convex function. Similarity between these dependences of hard sphere and hard disk systems is shown to be closely connected not only with the melting transition, but also with the existence of three phases in the case of hard core systems with attractive potentials.

Note on the Hole Theory of Liquids: Free Volume of the Hard Particle Systems and Its Relation to Melting