Surprising Aspects of Fluctuating "Pulled" Fronts

TL;DR

This paper investigates how stochastic effects influence the speed of pulled fronts in a simple lattice model, revealing a halt-and-go dynamics at the front tip and providing a probabilistic framework that matches simulations better than previous asymptotic predictions.

Contribution

It introduces a new probabilistic approach to model stochastic front dynamics, capturing finite particle effects and the halt-and-go behavior at the front tip.

Findings

The stochastic front speed deviates significantly from mean-field predictions for moderate N.

The halt-and-go dynamics at the front tip are crucial for understanding front propagation.

The probabilistic model matches well with numerical simulations for smaller N.

Abstract

Recently it has been shown that when an equation that allows so-called pulled fronts in the mean-field limit is modelled with a stochastic model with a finite number $N$ of particles per correlation volume, the convergence to the speed $v^*$ for $N \to \infty$ is extremely slow -- going only as $\ln^{-2}N$. However, this convergence is seen only for very high values of $N$, while there can be significant deviations from it when $N$ is not too large. Pulled fronts are fronts that propagate into an unstable state, and the asymptotic front speed is equal to the linear spreading speed $v^*$ of infinitesimal perturbations around the unstable state. In this paper, we consider front propagation in a simple stochastic lattice model. The microscopic picture of the front dynamics shows that for the description of the far tip of the front, one has to abandon the idea of a uniformly translating front solution. The lattice and finite particle effects lead to a ``halt-and-go'' type dynamics at the far tip of the front, while the average front behind it ``crosses over'' to a uniformly translating solution. In this formulation, the effect of stochasticity on the asymptotic front speed is coded in the probability distribution of the times required for the advancement of the ``foremost occupied lattice site''. These probability distributions are obtained by matching the solution of the far tip with the uniformly translating solution behind in a mean-field type approximation, and the results for the probability distributions compare well to the results of stochastic numerical simulations. This approach allows one to deal with much smaller values of $N$ than it is required to have the $\ln^{-2}N$ asymptotics to be valid.