Diffusion-annihilation dynamics in one spatial dimension

TL;DR

This paper analyzes a one-dimensional reaction-diffusion model with asymmetric hopping and annihilation, providing exact solutions for density evolution and correlations using free fermion representation and field theory.

Contribution

It introduces an exact analytical framework for the long-term and time-dependent behavior of a driven reaction-diffusion system in one dimension.

Findings

Exact long-time density profiles for step initial conditions

Time-dependent density for uncorrelated initial states

Calculation of higher order correlation functions

Abstract

We discuss a reaction-diffusion model in one dimension subjected to an external driving force. Each lattice site may be occupied by at most one particle. The particles hop with asymmetric rates (the sum of which is one) to the right or left nearest neighbour site if it is vacant, and annihilate with rate one if it is occupied. We compute the long time behaviour of the space dependent average density in states where the initial density profiles are step functions. We also compute the exact time dependence of the particle density for uncorrelated random initial conditions. The representation of the uncorrelated random initial state and also of the step function profile in terms of free fermions allows the calculation of time-dependent higher order correlation functions. We outline the procedure using a field theoretic approach.