Crossover from Rate-Equation to Diffusion-Controlled Kinetics in Two-Particle Coagulation

TL;DR

This paper introduces an analytical approximation scheme for one-dimensional coagulation reactions that effectively captures the transition from mean-field to fluctuation-dominated kinetics over time.

Contribution

The authors develop a diffusion-equation-based approximation that accurately describes the crossover from rate-equation to fluctuation-controlled behavior in 1D coagulation reactions.

Findings

Accurately models the crossover from mean-field to fluctuation-dominated regimes.

Quantitatively accurate for small initial densities and fast reactions.

Provides correct descriptions for initial and large-time behaviors, with some limitations in intermediate times.

Abstract

We develop an analytical diffusion-equation-type approximation scheme for the one-dimensional coagulation reaction A+A->A with partial reaction probability on particle encounters which are otherwise hard-core. The new approximation describes the crossover from the mean-field rate-equation behavior at short times to the universal, fluctuation-dominated behavior at large times. The approximation becomes quantitatively accurate when the system is initially close to the continuum behavior, i.e., for small initial density and fast reaction. For large initial density and slow reaction, lattice effects are nonnegligible for an extended initial time interval. In such cases our approximation provides the correct description of the initial mean-field as well as the asymptotic large-time, fluctuation-dominated behavior. However, the intermediate-time crossover between the two regimes is described only semiquantitatively.