Encounter-controlled coalescence and annihilation on a one-dimensional growing domain

TL;DR

This paper investigates how domain growth affects diffusion-limited coalescence and annihilation reactions in one dimension, revealing that fast growth halts reactions early and causes particles to freeze in a non-ordered state, with exact analytical results provided.

Contribution

The study extends the method of intervals to growing domains, providing exact solutions for reaction kinetics and spatial distributions in a non-static setting.

Findings

Fast domain growth halts reactions prematurely.

Survival probability approaches a finite value at long times.

Particles freeze in a non-fully ordered state.

Abstract

The kinetics of encounter-controlled processes in growing domains is markedly different from that in a static domain. Here, we consider the specific example of diffusion limited coalescence and annihilation reactions in one-dimensional space. In the static case, such reactions are among the few systems amenable to exact solution, which can be obtained by means of a well-known method of intervals. In the case of a uniformly growing domain, we show that a double transformation in time and space allows one to extend this method to compute the main quantities characterizing the spatial and temporal behavior. We show that a sufficiently fast domain growth brings about drastic changes in the behavior. In this case, the reactions stop prematurely, as a result of which the survival probability of the reacting particles tends to a finite value at long times and their spatial distribution freezes before reaching the fully self-ordered state. We obtain exact results for the survival probability and for key properties characterizing the degree of self-ordering induced by the chemical reactions, i.e., the interparticle distribution function and the pair correlation function. These results are confirmed by numerical simulations.