Velocity fluctuations of stochastic reaction fronts propagating into an unstable state: strongly pushed fronts

TL;DR

This paper investigates velocity fluctuations of strongly pushed reaction fronts propagating into unstable states, revealing non-perturbative effects and anomalous fluctuations at intermediate times through theoretical analysis and Monte Carlo simulations.

Contribution

It introduces a non-perturbative correction to the diffusion constant of strongly pushed fronts and uncovers anomalous fluctuation behavior at intermediate times.

Findings

Effective diffusion constant scales as 1/N with corrections near transition points.

Front position fluctuations are diffusive only over very long timescales, with large non-diffusive fluctuations at intermediate times.

Monte Carlo simulations support the theoretical predictions.

Abstract

The empirical velocity of a reaction-diffusion front, propagating into an unstable state, fluctuates because of the shot noises of the reactions and diffusion. Under certain conditions these fluctuations can be described as a diffusion process in the reference frame moving with the average velocity of the front. Here we address pushed fronts, where the front velocity in the deterministic limit is affected by higher-order reactions and is therefore larger than the linear spread velocity. For a subclass of these fronts -- strongly pushed fronts -- the effective diffusion constant $D_f\sim 1/N$ of the front can be calculated, in the leading order, via a perturbation theory in $1/N \ll 1$, where $N\gg 1$ is the typical number of particles in the transition region. This perturbation theory, however, overestimates the contribution of a few fast particles in the leading edge of the front. We suggest a more consistent calculation by introducing a spatial integration cutoff at a distance beyond which the average number of particles is of order 1. This leads to a non-perturbative correction to $D_f$ which even becomes dominant close to the transition point between the strongly and weakly pushed fronts. At the transition point we obtain a logarithmic correction to the $1/N$ scaling of $D_f$. We also uncover another, and quite surprising, effect of the fast particles in the leading edge of the front. Because of these particles, the position fluctuations of the front can be described as a diffusion process only on very long time intervals with a duration $\Delta t \gg \tau_N$, where $\tau_N$ scales as $N$. At intermediate times the position fluctuations of the front are anomalously large and non-diffusive. Our extensive Monte-Carlo simulations of a particular reacting lattice gas model support these conclusions.