Order-disorder transitions in lattice gases with annealed reactive constraints

TL;DR

This paper provides exact solutions for lattice gas models with annealed reactive constraints, revealing complex phase transitions including continuous and re-entrant behaviors on a Bethe lattice.

Contribution

It introduces and solves two novel lattice gas models with annealed catalytic constraints, analyzing their critical behavior and phase transitions.

Findings

Model I exhibits a continuous transition to an ordered state and a re-entrant transition back to disorder.

Model II shows a continuous transition with re-entrant behavior that can be either continuous or discontinuous.

Both models demonstrate rich critical phenomena on the Bethe lattice.

Abstract

We study equilibrium properties of catalytically-activated $A + A \to \oslash$ 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 {\em bonds} connecting neighbouring sites possesses special catalytic properties such that any two $A$s appearing on the sites connected by such a bond instantaneously react and desorb. In the Model II some fraction $p$ of the adsorption {\em sites} possesses such properties and neighbouring particles react if at least one of them resides on a catalytic site. For the case of \textit{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 $p$.