Nucleation model for diffusion-limited coalescence with finite reaction rates in one dimension

Abstract

We study the diffusion-limited coalescence model $A+A\ensuremath{\rightarrow}A$ with nucleation and a finite reaction rate in one dimension for the particle density decay by means of a Monte Carlo simulation and analytic modeling. In our model, one or more particles in a lattice site act as a nucleus for the particles that diffuse into the site without reacting. The master equation governing the time evolution of local particle number and the rate equation for the particle density are derived. We present an analytic approach for the early time regime (reaction-controlled limit), which is strongly dependent on the initial particle density. In this regime, the particle density decays faster than the classical (or second-order reaction) limit and lower than the exponential decay (or first-order reaction) limit. For the long time regime the diffusion-controlled limit is recovered. We show that the intermediate regime can be obtained as an interpolation between the initial decay and the diffusion limit. The numerical integration results from the analytic approach are compared with computer simulations.