Nonmonotonic pressure as a function of the density in a fluid without attractive forces - Henderson, D. (Condensed Matter Physics, 2013)
A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is extended in a straightforward manner by adding additional terms that are of the same form as Rennert's formula. The resulting expression is moderately accurate but its accuracy does not necessarily improve as additional terms are included. This expression has the interesting consequence that the pressure can have a maximum, as the density increases, which is consistent with the freezing of the hard spheres. This occurs solely as a consequence of repulsive interactions. Only the Born-Green-Yvon and Kirkwood theories show such behavior for hard spheres and they require the numerical solution of an integral equation. The procedure outlined here is ad hoc but is, perhaps, useful just as the popular Carnahan-Starling equation for the hard sphere pressure is also ad hoc but useful.