Order-disorder transitions in lattice gases with annealed reactive constraints

We study equilibrium properties of catalytically-activated A + A -> circle divide reactions taking place on a lattice of adsorption sites. The particles undergo continuous exchanges with a reservoir maintained at a constant chemical potential mu and react when they appear at the neighbouring sites, provided that some reactive conditions are fulfilled. We model the latter in two different ways: in the Model I some fraction p of the bonds connecting neighbouring sites possesses special catalytic properties such that any two As appearing on the sites connected by such a bond instantaneously react and desorb. In the Model II some fraction p of the adsorption sites possesses such properties and neighbouring particles react if at least one of them resides on a catalytic site. For the case of annealed disorder in the distribution of the catalyst, which is tantamount to the situation when the reaction may take place at any point on the lattice but happens with a finite probability p, we provide an exact solution for both models for the interior of an infinitely large Cayley tree-the so-called Bethe lattice. We show that both models exhibit a rich critical behaviour: for the annealed Model I it is characterised by a transition into an ordered state and a re-entrant transition into a disordered phase, which both are continuous. For the annealed Model II, which represents a rather exotic model of statistical mechanics in which interactions of any particle with its environment have a peculiar Boolean form, the transition to an ordered state is always continuous, while the re-entrant transition into the disordered phase may be either continuous or discontinuous, depending on the value of p.