Exact results for an asymmetric annihilation process with open boundaries

TL;DR

This paper introduces an algebraic approach called the transfer matrix Ansatz to exactly solve a nonequilibrium reaction-diffusion model with open boundaries, revealing its steady state and correlation properties.

Contribution

It develops a novel transfer matrix technique for exact analysis of an asymmetric annihilation process with open boundaries, connecting system sizes and providing explicit steady state solutions.

Findings

Derived the steady state distribution explicitly

Established the transfer matrix Ansatz for system size embedding

Proposed a conjecture for the characteristic polynomial of Markov matrices

Abstract

We consider a nonequilibrium reaction-diffusion model on a finite one dimensional lattice with bulk and boundary dynamics inspired by Glauber dynamics of the Ising model. We show that the model has a rich algebraic structure that we use to calculate its properties. In particular, we show that the Markov dynamics for a system of a given size can be embedded in the dynamics of systems of higher sizes. This remark leads us to devise a technique we call the transfer matrix Ansatz that allows us to determine the steady state distribution and correlation functions. Furthermore, we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of Markov matrices. Lastly, we compare the transfer matrix Ansatz used here with the matrix product representation of the steady state of one-dimensional stochastic models.