FKPP fronts in quenched random media

TL;DR

This study numerically investigates FKPP front propagation in quenched random media, revealing universal linear velocity increase with disorder variance and quadratic scaling of front fluctuations.

Contribution

It provides the first quantitative analysis of how quenched randomness affects FKPP front velocity and fluctuations, establishing universal scaling laws.

Findings

Velocity increases linearly with disorder variance, with a universal coefficient.

Front position fluctuations are diffusive, with a quadratic dependence on disorder.

Results suggest a universal response of FKPP fronts to quenched heterogeneity.

Abstract

We study numerically the evolution of one-dimensional FKPP fronts initiated from steep initial conditions in the presence of a quenched random growth rate. Compared to both the homogeneous case (with velocity $v_0$) and deterministic disorder, quenched randomness increases the average propagation speed. We show that the velocity shift relative to the homogeneous case scales linearly with the disorder variance $\sigma^2$, with a universal prefactor -- independent of the specific distribution of the disorder -- such that $v = v_0 + a \sigma^2$, with $a \approx 0.02432 \pm 0.00002$. Moreover, the front position exhibits diffusive fluctuations across disorder realizations. The corresponding effective diffusion coefficient scales quadratically with $\sigma$, $D = \frac{b^2 \sigma^2}{2}$, with $b \approx 0.223 \pm 0.002$. These results suggest a universal statistical response of FKPP fronts to quenched heterogeneity.