The transition of the rigid-sphere gas

It is shown that, apart from the known exact solution, the Percus-Yevick integral equation for a rigid-sphere gas has infinitely many other solutions, which correspond to various types of ordered arrangement. These solutions are always mathematically possible but, below a certain density, are excluded physically because they would imply a two-particle distribution function with negative regions. For the same reason the disordered solution is excluded above a certain density.