Speed fluctuations of a stochastic Huxley-Zel'dovich front

Abstract

The empirical speed of travelling reaction-diffusion fronts fluctuates due to the intrinsic shot noise of the reactions and diffusion. Here we study the long-time front speed fluctuations of a stochastic Huxley-Zel'dovich front. It involves a population of particles $A$ which perform a fast continuous-time random walk on a one-dimensional lattice and undergo reversible on-site reactions $2A \rightleftarrows 3A$. This front describes an invasion of $A$-particles into an initially empty region of space which, in a deterministic description, is marginally stable but nonlinearly unstable with a zero instability threshold. Typical fluctuations of this front can be described as front diffusion in a reference frame moving with the average front speed. According to the existing perturbation theory, the shot-noise-induced systematic shift of the average front speed, $δc$, and the front diffusion coefficient, $D_f$, are both expected to scale with $N$ as $1/N$, where $N \gg 1$ is the typical number of particles in the transition region. Furthermore, $D_f$ can be determined perturbatively in the small parameter $1/\sqrt{N}$. Our Monte Carlo simulations support these asymptotic results, but also reveal a long-lived anomalous behavior of the first few particles before they reach the expected diffusion asymptotic. We also study large deviations of the empirical speed of the front at long times. These are dominated by optimal histories of the system in the form of a propagating front which travels with a speed different from the average speed, or even travel in the wrong direction.

Figures (7)

• Figure 1: The optimal density $u(x)$ for the standing noisy front, $c=0$, as described by the exact solution (\ref{['qzeroc']}).
• Figure 2: The TFS found numerically for $c=-0.5$: $Q(\xi)$ (top left), $P(\xi)$ (top right), and $u(\xi) = Q(\xi) [1+P(\xi)]$ (bottom).
• Figure 3: The rate function $f(c)$, which describes large deviations of the empirical front speed $c$, see Eq. (\ref{['ldf']}), vs. $c/c_0$, where $c_0=1/\sqrt{2}$. Shown are numerical results for $-c_{\text{cr}}\leq c\leq c_{\text{cr}}$ (the blue line) and the Gaussian asymptotic (\ref{['reducedaction']}) (the red dash-dotted line). The three fat points show the exact results $f(c=c_0)=0$, $f(c=0)=1/6$, and $f(c=-c_0)=c_0$.
• Figure 4: The density profiles of a stochastic HZ front at times $t=100$ (red) and $t=600$ (blue), observed in a single MC simulation. The gray lines show the deterministic TFS (\ref{['mffront']}) at the respective times. The rescaled parameters are $K=20$ and $N \simeq 173$.
• Figure 5: Finite-$N$ correction to the average front speed $c_0$. Plotted is the difference $1-c_*(N)/c_0$ vs. $N$. Blue points: the results of six different sets of simulations: $K=1$ and $N=10$, $K=1$ and $N\simeq14$, $K=1$ and $N=20$, $K=2$ and $N=20$, $K=2$ and $N=40$, and $K=5$ and $N\simeq 87$. Straight line: the fit $c_*(N)=c_0(1-0.8/N)$.