Front Propagation Dynamics with Exponentially-Distributed Hopping

TL;DR

This paper investigates reaction-diffusion fronts with exponentially-distributed jumps, revealing how fluctuations and particle density influence front velocity, with analytical and simulation results showing unbounded velocity growth in certain conditions.

Contribution

It introduces a novel analysis of reaction-diffusion fronts with exponential jump distributions, including heuristic fluctuation treatment and velocity dependence on particle density.

Findings

Fluctuations significantly affect front propagation at low hopping rates.

Front velocity increases unboundedly with particle density in a reaction rate gradient.

Analytical expressions for velocity dependence are confirmed by simulations.

Abstract

We study reaction-diffusion systems where diffusion is by jumps whose sizes are distributed exponentially. We first study the Fisher-like problem of propagation of a front into an unstable state, as typified by the A+B $\to$ 2A reaction. We find that the effect of fluctuations is especially pronounced at small hopping rates. Fluctuations are treated heuristically via a density cutoff in the reaction rate. We then consider the case of propagating up a reaction rate gradient. The effect of fluctuations here is pronounced, with the front velocity increasing without limit with increasing bulk particle density. The rate of increase is faster than in the case of a reaction-gradient with nearest-neighbor hopping. We derive analytic expressions for the front velocity dependence on bulk particle density. Compute simulations are performed to confirm the analytical results.