Surprising Aspects of Fluctuating "Pulled" Fronts

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.