Fisher Waves in the Diffusion-Limited Coalescence Process A+A<-->A

TL;DR

This paper provides an exact, comprehensive analysis of Fisher waves in a diffusion-limited coalescence process, revealing that the wave front propagates with a biased random walk, maintaining its shape but appearing to broaden due to stochastic edge movement.

Contribution

It extends previous studies by deriving an exact description of the full hierarchy of correlation functions, showing the wave front's shape remains constant while its edge exhibits biased random walk behavior.

Findings

Wave propagates at constant speed from stable to unstable regions.

Wave front maintains its shape over time.

Edge of the wave performs a biased random walk.

Abstract

Fisher waves have been studied recently in the specific case of diffusion-limited reversible coalescence, A+A<-->A, on the line. An exact analysis of the particles concentration showed that waves propagate from a stable region to an unstable region at constant speed, just as in Fisher's "mean-field" theory; but also that the wave front fails to retain its initial shape and instead it broadens with time. Our present analysis encompasses the full hierarchy of multiple-point density correlation functions, and thus it provides a complete exact description of the same system. We find that as the wave propagates, the particles in the stable phase remain distributed exactly as in their initial (equilibrium) state. On the other hand, the leading particle---the one at the edge of the wave---advances as a biased random walk, rather than simply linearly with time. Thus the shape of the wave remains actually constant, but it is the "noisy" propagation of the wave's edge that causes its apparent broadening.