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

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 $\Lambda <{\Lambda }_{\ast }$ 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.

5 REVTEX pages, no figures