The Reaction-Diffusion Front for $A+B \to\emptyset$ in One Dimension

TL;DR

This paper investigates the behavior of a reaction-diffusion front in a one-dimensional system with opposing particle currents, revealing a transition from mean field to noise-dominated Gaussian profiles depending on reaction rate and flux parameters.

Contribution

It provides a theoretical and numerical analysis of the reaction front, including a new prediction for the Gaussian profile in the high noise regime and insights into the front width scaling.

Findings

Mean field theory describes the front at low reaction rates.

High reaction rates induce a Gaussian profile due to noise.

Simulation results agree with theoretical predictions.

Abstract

We study theoretically and numerically the steady state diffusion controlled reaction $A+B\rightarrow\emptyset$, where currents $J$ of $A$ and $B$ particles are applied at opposite boundaries. For a reaction rate $\lambda$, and equal diffusion constants $D$, we find that when $\lambda J^{-1/2} D^{-1/2}\ll 1$ the reaction front is well described by mean field theory. However, for $\lambda J^{-1/2} D^{-1/2}\gg 1$, the front acquires a Gaussian profile - a result of noise induced wandering of the reaction front center. We make a theoretical prediction for this profile which is in good agreement with simulation. Finally, we investigate the intrinsic (non-wandering) front width and find results consistent with scaling and field theoretic predictions.