Exact correlations in the one-dimensional coagulation-diffusion process by the empty-interval method

TL;DR

This paper extends the empty-interval method to calculate exact space-time-dependent correlation functions in the one-dimensional coagulation-diffusion process, revealing detailed fluctuation effects and dynamic scaling for various initial states.

Contribution

The authors generalize the empty-interval method to include two-interval probabilities, enabling exact calculation of correlations in the coagulation-diffusion process for arbitrary initial conditions.

Findings

Exact correlation functions derived for the process.

Analysis of dynamic scaling behavior.

Solution of boundary conditions for arbitrary initial states.

Abstract

The long-time dynamics of reaction-diffusion processes in low dimensions is dominated by fluctuation effects. The one-dimensional coagulation-diffusion process describes the kinetics of particles which freely hop between the sites of a chain and where upon encounter of two particles, one of them disappears with probability one. The empty-interval method has, since a long time, been a convenient tool for the exact calculation of time-dependent particle densities in this model. We generalise the empty-interval method by considering the probability distributions of two simultaneous empty intervals at a given distance. While the equations of motion of these probabilities reduce for the coagulation-diffusion process to a simple diffusion equation in the continuum limit, consistency with the single-interval distribution introduces several non-trivial boundary conditions which are solved for the first time for arbitrary initial configurations. In this way, exact space-time-dependent correlation functions can be directly obtained and their dynamic scaling behaviour is analysed for large classes of initial conditions.