Hole Theories of the Liquid State

The hole theory of liquids may be considered as a special case of the cell theory including multiple cell occupations corresponding to the restriction 

In fact the hole theory of liquids was developed before the multiple cell occupation model (Cernuschi and Eyeing [1939], Ono [1947], Peek and Hill [1950], Rowlinson and Curtiss [1951], Mayer and Careri [1952], De Boer [1952]). 

The configurational partition function becomes (using the same assumptions as for (7.6.3)) 

In the limit of low temperatures and high densities, the number of holes must tend towards zero. On the other hand in the limit of low densities we obtain 

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Given the diversity of relevant applications, it is not surprising that the characterization of voids in disordered systems has an appreciable history, which can be traced back to primitive hole theories of the liquid state (Frenkel, 1955 Ono and Kondo, 1960). While the early theories offer an admittedly rudimentary lattice description of voids, recent computational advances permit an exact (and highly efficient) characterization of the continuum void geometry present in particle packings in two (Rintoul and Torquato, 1995) and three dimensions (Sastry et al., 1997a). 

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