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 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 $v^{\ast }$ for $N\to \infty$ is extremely slow -- going only as ${\ln }^{-2}N$. 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 ${\ln }^{-2}N$ asymptotics to be valid.

26 pages, 11 figures, to appear in Phys. rev. E

References in corpus (4)

• Internal Fluctuations Effects on Fisher Waves
• The universality class of fluctuating pulled fronts
• The Weakly Pushed Nature of "Pulled" Fronts with a Cutoff
• Fronts with a Growth Cutoff but Speed Higher than $v^{\ast }$

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• Effects of Fluctuations on Propagating Fronts
• A Langevin equation for high energy evolution with pomeron loops
• Asymptotic Scaling of the Diffusion Coefficient of Fluctuating "Pulled" Fronts
• Front Propagation and Diffusion in the A <--> A + A Hard-core Reaction on a Chain
• Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics
• Stochastic drift in discrete waves of non-locally interacting-particles