On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond

TL;DR

This paper provides exact and approximate solutions for the behavior of immobile reactants undergoing coalescence and annihilation reactions on Bethe and 2D square lattices, analyzing coverage and spatial distribution effects.

Contribution

It introduces an exact solution for Bethe lattices and an approximate method for 2D lattices, advancing understanding of immobile reactive systems on different lattice structures.

Findings

Exact coverage solutions for Bethe lattices, including 1D case

Approximate solutions for 2D square lattices

Analysis of initial condition dilution effects

Abstract

We study the behavior of the chemical reactions $A+A\to A+S$ and $A+A\to S+S$ (where the reactive species $A$ and the inert species $S$ are both assumed to be immobile) embedded on Bethe lattices of arbitrary coordination number $z$ and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact solutions for the coverage in the $A$ species in terms of the initial condition are obtained. In particular, our results hold for the important case of an infinite one-dimensional (1D) lattice ($z=2$). The method is based on an expansion in terms of conditional probabilities which exploits a Markovian property of these systems. Along the same lines, an approximate solution for the case of a 2D square lattice is developed. The effect of dilution in a random initial condition is discussed in detail, both for the lattice coverage and for the spatial distribution of reactants.