Diffusion-limited Reactions of hard-core Particles in one-dimension

TL;DR

This paper explores three methods to analyze diffusion-limited reactions of hard-core particles in one dimension, providing asymptotic behaviors, exact correlation functions, and a new path integral formulation for multi-species systems.

Contribution

It extends existing approaches, derives exact correlation functions, and introduces a novel path integral formalism for multi-species diffusion-limited reactions.

Findings

Asymptotic behavior of two-point correlation functions calculated.

Exact two-point correlation functions obtained for single and two-time cases.

A new path integral formulation for multi-species reactions proposed.

Abstract

We investigate three different methods to tackle the problem of diffusion-limited reactions (annihilation) of hard-core classical particles in one dimension. We first extend an approach devised by Lushnikov and calculate for a single species the asymptotic long-time and/or large distance behavior of the two-point correlation function. Based on a work by Grynberg et al., which was developed to treat stochastic adsorption-desorption models, we provide in a second step the exact two-point correlation function (both for one and two-time) of Lushnikov's model. We then propose a new formulation of the problem in terms of path integrals for pseudo-fermions. This formalism can be used to advantage in the multi-species case, specially when applying perturbative renormalization group techniques.