Journal of Statistical Physics, Vol. 46, Nos. 1/2, 1987. p. 67-85.
The partition function is studied for an array of axisymmetric, hard bodies (capped cylinders, etc.) with each fixed at the base on a regular one-dimensional lattice. It is shown that if a phase transition occurs in a system of n molecules, then it also occurs in a system of two molecules for the same value of the spacing
parameter. With certain additional technical assumptions the converse is also true. Results are reported specifically for a system of thin, hard rods. Necessary and sufficient conditions are shown for a first-order transition to occur in the thermodynamic limit; there is only one transition and that happens when the
spacing parameter is equal to the length of the rod. As expected, there is no phase transition when the rod is contracted to a point.