Diffusion-controlled annihilation $A + B \to 0$: The growth of an $A$ particle island from a localized $A$-source in the $B$ particle sea

TL;DR

This paper investigates the growth dynamics of an $A$ particle island in a sea of $B$ particles undergoing diffusion-controlled annihilation, revealing dimensional dependencies and critical source strengths for island growth and stationarity.

Contribution

It provides a comprehensive analysis of the growth behavior of $A$ islands across different dimensions, including critical conditions and scaling laws, which was not previously detailed.

Findings

In 1D, the $A$ island grows indefinitely regardless of source strength.

In 3D, the island only grows if the source strength exceeds a critical value.

In 2D, the island grows indefinitely but formation time varies exponentially with source strength.

Abstract

We present the growth dynamics of an island of particles $A$ injected from a localized $A$-source into the sea of particles $B$ and dying in the course of diffusion-controlled annihilation $A+B\to 0$. We show that in the 1d case the island unlimitedly grows at any source strength $\Lambda$, and the dynamics of its growth {\it does not depend} asymptotically on the diffusivity of $B$ particles. In the 3d case the island grows only at $\Lambda > \Lambda_{c}$, achieving asymptotically a stationary state ({\it static island}). In the marginal 2d case the island unlimitedly grows at any $\Lambda$ but at $\Lambda < \Lambda_{*}$ the time of its formation becomes exponentially large. For all the cases the numbers of surviving and dying $A$ particles are calculated, and the scaling of the reaction zone is derived.