Fluctuating pulled fronts: The origin and the effects of a finite particle cutoff

Abstract

Recently, it has been shown that when an equation that allows the so-called pulled fronts in the mean-field limit is modeled with a stochastic model with a finite number N of particles per correlation volume, the convergence to the speed [formula presented] for [formula presented] is extremely slow—going only as [formula presented] Pulled fronts are fronts that propagate into an unstable state, and the asymptotic front speed is equal to the linear spreading speed [formula presented] of small linear perturbations about the unstable state. In this paper, we study the front propagation in a simple stochastic lattice model. A detailed analysis of 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 “stop-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 bin.” We derive expressions of these probability distributions by matching the solution of the far tip with the uniformly translating solution behind. This matching includes various correlation effects in a mean-field type approximation. Our results for the probability distributions compare well to the results of stochastic numerical simulations. This approach also allows us to deal with much smaller values of N than it is required to have the [formula presented] asymptotics to be valid. Furthermore, we show that if one insists on using a uniformly translating solution for the entire front ignoring its breakdown at the far tip, then one can obtain a simple expression for the corrections to the front speed for finite values of N, in which various subdominant contributions have a clear physical interpretation. © 2002 The American Physical Society.