Asymptotic front behavior in an $A+B\rightarrow 2A$ reaction under subdiffusion

TL;DR

This paper investigates the behavior of reaction fronts in a subdiffusive $A+B ightarrow 2A$ system, revealing two distinct scaling regimes and the limitations of continuous models at large times.

Contribution

It introduces a crossover analysis of front propagation under subdiffusion, identifying regimes where continuous equations fail to describe atomically sharp fronts.

Findings

Front solutions decelerate and narrow over time in continuous models.

At large times, fronts become atomically sharp and continuous models break down.

Front velocity decays faster than predicted by continuous equations at large times.

Abstract

We discuss the front propagation in the $A+B\rightarrow 2A$ reaction under subdiffusion which is described by continuous time random walks with a heavy-tailed power law waiting time probability density function. Using a crossover argument, we discuss the two scaling regimes of the front propagation: an intermediate asymptotic regime given by the front solution of the corresponding continuous equation, and the final asymptotics, which is fluctuation-dominated and therefore lays out of reach of the continuous scheme. We moreover show that the continuous reaction subdiffusion equation indeed possesses a front solution that decelerates and becomes narrow in the course of time. This continuous description breaks down for larger times when the front gets atomically sharp. We show that the velocity of such fronts decays in time faster than in the continuous regime.