Speed fluctuations of a stochastic Huxley-Zel'dovich front

TL;DR

This paper investigates the long-time speed fluctuations of a stochastic reaction-diffusion front modeled by a Huxley-Zel'dovich system, combining theoretical predictions with Monte Carlo simulations to understand its behavior and deviations.

Contribution

It provides a detailed analysis of the speed fluctuations and large deviations of a stochastic front, confirming theoretical scaling laws and revealing anomalous early-time behavior.

Findings

Speed fluctuation and systematic shift scale as 1/N with particle number N.

Front diffusion coefficient D_f can be perturbatively determined in 1/√N.

Monte Carlo simulations support theoretical asymptotic results and show long-lived anomalous behavior.

Abstract

The empirical speed of travelling reaction-diffusion fronts fluctuates due to the intrinsic shot noise of the reactions and diffusion. Here we study the long-time front speed fluctuations of a stochastic Huxley-Zel'dovich front. It involves a population of particles $A$ which perform a fast continuous-time random walk on a one-dimensional lattice and undergo reversible on-site reactions $2A \rightleftarrows 3A$. This front describes an invasion of $A$-particles into an initially empty region of space which, in a deterministic description, is marginally stable but nonlinearly unstable with a zero instability threshold. Typical fluctuations of this front can be described as front diffusion in a reference frame moving with the average front speed. According to the existing perturbation theory, the shot-noise-induced systematic shift of the average front speed, $\delta c$, and the front diffusion coefficient, $D_f$, are both expected to scale with $N$ as $1/N$, where $N \gg 1$ is the typical number of particles in the transition region. Furthermore, $D_f$ can be determined perturbatively in the small parameter $1/\sqrt{N}$. Our Monte Carlo simulations support these asymptotic results, but also reveal a long-lived anomalous behavior of the first few particles before they reach the expected diffusion asymptotic. We also study large deviations of the empirical speed of the front at long times. These are dominated by optimal histories of the system in the form of a propagating front which travels with a speed different from the average speed, or even travel in the wrong direction.