Asymptotic Scaling of the Diffusion Coefficient of Fluctuating "Pulled" Fronts

TL;DR

This paper provides a heuristic theoretical derivation showing that the diffusion coefficient of fluctuating pulled fronts decreases as the inverse cube of the logarithm of particle number, aligning with previous numerical results.

Contribution

It introduces a heuristic derivation for the asymptotic scaling of the diffusion coefficient of fluctuating pulled fronts, confirming the $1/ ext{ln}^3 N$ behavior.

Findings

$D_f$ approaches zero as $1/ ext{ln}^3 N$ for large $N$

Shape fluctuations at the front tip drive the diffusion behavior

The scaling law is model-independent

Abstract

We present a (heuristic) theoretical derivation for the scaling of the diffusion coefficient $D_f$ for fluctuating ``pulled'' fronts. In agreement with earlier numerical simulations, we find that as $N\to\infty$, $D_f$ approaches zero as $1/\ln^3N$, where $N$ is the average number of particles per correlation volume in the stable phase of the front. This behaviour of $D_f$ stems from the shape fluctuations at the very tip of the front, and is independent of the microscopic model.