Phase Transition of a Bethe Lattice Gas of Hard Molecules

Abstract

A pseudo-lattice, or homogeneous Husimi tree, is simpler statistically than a true lattice in two or three dimensions, since the pseudo-lattice contains only low-order cycles. The prototype is the ``Bethe lattice,'' or Cayley tree, containing no cycles at all. Reported here are the results of studies of the lattice statistics of the Bethe lattice of coordination number three, using the language of the hard molecule lattice gas; the size of an adsorbed molecule prevents simultaneous occupancy of the same site or any of the three nearest-neighbor sites. Using methods related to those used in enumerating graphs, a recursion relation is obtained which must be satisfied by the grand partition function. It is shown that the resulting equation of state is not the quasi-chemical equation expected because of the absence of cycles. There is no phase transition of lower than third order and in all likelihood none at all. The quasi-chemical equation of state is obtained only if ``surface'' effects are eliminated; even then the solution is valid only for activities z < 4, at which point the ``interior'' system undergoes a second-order transition to an ordered state with a finite discontinuity in compressibility.