Equilibrium properties of two-species reactive lattice gases on random catalytic chains

TL;DR

This paper investigates the thermodynamic properties of a two-species reactive lattice gas with catalytic reactions on a one-dimensional lattice, providing exact solutions for pressure in both annealed and quenched disorder scenarios using combinatorial and matrix methods.

Contribution

It introduces exact analytical methods for calculating the pressure of reactive lattice gases with catalytic reactions under quenched and annealed disorder, using combinatorial and matrix approaches.

Findings

Exact expressions for pressure in annealed disorder cases.

Representation of pressure as a trace of random matrix products.

Analytical solutions for both uncorrelated and correlated disorder models.

Abstract

We focus here on the thermodynamic properties of adsorbates formed by two-species $A+B \to \oslash$ reactions on a one-dimensional infinite lattice with heterogeneous "catalytic" properties. In our model hard-core $A$ and $B$ particles undergo continuous exchanges with their reservoirs and react when dissimilar species appear at neighboring lattice sites in presence of a "catalyst." The latter is modeled by supposing either that randomly chosen bonds in the lattice promote reactions (Model I) or that reactions are activated by randomly chosen lattice sites (Model II). In the case of annealed disorder in spatial distribution of a catalyst we calculate the pressure of the adsorbate by solving three-site (Model I) or four-site (Model II) recursions obeyed by the corresponding averaged grand-canonical partition functions. In the case of quenched disorder, we use two complementary approaches to find $\textit{exact}$ expressions for the pressure. The first approach is based on direct combinatorial arguments. In the second approach, we frame the model in terms of random matrices; the pressure is then represented as an averaged logarithm of the trace of a product of random $3 \times 3$ matrices -- either uncorrelated (Model I) or sequentially correlated (Model II).