Diffusion-limited annihilation in inhomogeneous environments

TL;DR

This paper provides an exact analysis of diffusion-limited pair annihilation and fusion processes in inhomogeneous environments, revealing their equivalence and deriving explicit correlation functions for one-dimensional systems.

Contribution

It introduces a duality approach and fermionic representation to exactly compute density correlations in inhomogeneous diffusion-limited reactions.

Findings

Exact correlation functions for density decay in 1D systems

Equivalence of annihilation and fusion processes in inhomogeneous settings

Explicit formulas for universal and non-universal decay behaviors

Abstract

We study diffusion-limited (on-site) pair annihilation $A+A\to 0$ and (on-site) fusion $A+A\to A$ which we show to be equivalent for arbitrary space-dependent diffusion and reaction rates. For one-dimensional lattices with nearest neighbour hopping we find that in the limit of infinite reaction rate the time-dependent $n$-point density correlations for many-particle initial states are determined by the correlation functions of a dual diffusion-limited annihilation process with at most $2n$ particles initially. By reformulating general properties of annihilating random walks in one dimension in terms of fermionic anticommutation relations we derive an exact representation for these correlation functions in terms of conditional probabilities for a single particle performing a random walk with dual hopping rates. This allows for the exact and explicit calculation of a wide range of universal and non-universal types of behaviour for the decay of the density and density correlations.