Cell Theories for Hard Particles

Abstract

The consequences of each of the three approximations made by Kirkwood in deriving the Lennard‐Jones—Devonshire cell theory are elucidated by a comparison with exact results in the one‐dimensional case of hard rods. This comparison gives an explicit calculation of the one‐dimensional communal entropy and shows that the single‐occupancy approximation is exact at close packing, although it does not lead to thermodynamic properties which are analytic functions of density. For two‐ and three‐dimensional systems, numerical results show that the cell‐theory entropy predictions are more accurate than in one dimension because the particles are more localized. The more‐than‐one‐particle‐per‐cell theories provide simple and rigorous lower bounds on the entropy, but the convergence to the thermodynamic limit by considering more and more particles is slow.