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

Evgeniy Khain, Baruch Meerson, Pavel Sasorov

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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 Df∼1/N of the front can be calculated, in the leading order, via a perturbation theory in 1/Nâ‰1, where N≫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 nonperturbative correction to Df 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 Df. 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 Δt≫τN, where τN scales as N. At intermediate times the position fluctuations of the front are anomalously large and nondiffusive. Our extensive Monte Carlo simulations of a particular reacting lattice gas model support these conclusions.

Original languageAmerican English
Article number022137
JournalPhysical Review E
Volume102
Issue number2
DOIs
StatePublished - Aug 2020

Bibliographical note

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© 2020 American Physical Society.

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