Diffusion-Limited Coalescence with Finite Reaction Rates in One Dimension

TL;DR

This paper develops an approximation scheme for diffusion-limited coalescence with finite reaction rates in one dimension, accurately describing early, intermediate, and long-time behaviors, and explores steady states with particle input.

Contribution

It introduces an interpolative approximation method based on IPDF for finite reaction rates, extending previous exact solutions for infinite rates, and analyzes steady states with back reactions and particle input.

Findings

Approximation is exact at early and long times.

System reaches a non-trivial steady state with finite particle concentration.

Theoretical predictions match computer simulations.

Abstract

We study the diffusion-limited process $A+A\to A$ in one dimension, with finite reaction rates. We develop an approximation scheme based on the method of Inter-Particle Distribution Functions (IPDF), which was formerly used for the exact solution of the same process with infinite reaction rate. The approximation becomes exact in the very early time regime (or the reaction-controlled limit) and in the long time (diffusion-controlled) asymptotic limit. For the intermediate time regime, we obtain a simple interpolative behavior between these two limits. We also study the coalescence process (with finite reaction rates) with the back reaction $A\to A+A$, and in the presence of particle input. In each of these cases the system reaches a non-trivial steady state with a finite concentration of particles. Theoretical predictions for the concentration time dependence and for the IPDF are compared to computer simulations. P. A. C. S. Numbers: 82.20.Mj 02.50.+s 05.40.+j 05.70.Ln