Hybrid method for simulating front propagation in reaction-diffusion systems

We study the propagation of pulled fronts in the $A\leftrightarrow A+A$ microscopic reaction-diffusion process using Monte Carlo simulations. In the mean field approximation the process is described by the deterministic Fisher-Kolmogorov-Petrovsky-Piscounov equation. In particular, we concentrate on the corrections to the deterministic behavior due to the number of particles per correlated volume $\mathrm{\Omega }$. By means of a hybrid simulation scheme, we manage to reach large macroscopic values of $\mathrm{\Omega }$, which allows us to show the importance in the dynamics of microscopic pulled fronts of the interplay of microscopic fluctuations and their macroscopic relaxation.

Figure 1

Snapshot of the front profile for one realization of the hybrid method with $\mathrm{\Omega }={10}^{20}$ (upper panel). Below: schematic view of the front $N_i\left(t\right)$ close to $i*$. Open symbols are in the macroscopic region and full symbols in the microscopic region. For each lattice point $i$ we define the flux of outgoing $F_i^{+}\left(t\right)$ and incoming $F_i^{-}\left(t\right)$ particles. In the figure only those for $i*$ are shown. Lines are guides to the eye.

Figure 2

Asymptotic velocity correction as a function of the logarithm of the number of particles $\mathrm{\Omega }$. Open symbols are results for the full MC simulation while full symbols are for the hybrid scheme. Dashed line corresponds to the scaling given by Eq. (3) with ${\mathcal{K}}_v$ given by Eq. (9). Error bars are not shown when smaller than the symbol size. Inset: time evolution of the instantaneous velocity of the front (solid lines) with $\mathrm{\Omega }={10}^{20},{10}^{30}$, and ${10}^{40}$ from top to bottom. Dashed line is the prediction given by Eq. (2).

Figure 3

Diffusion coefficient of the front $D_f$ as a function of the logarithm of the number of particles $\mathrm{\Omega }$. Symbols are as in Fig. 2. Dashed line is a fit of the last points to the scaling form (5) with ${\mathcal{K}}_D=26.5$. Dot-dashed line is the scaling $D_f\sim {\Omega }^{-0.32}$ of [8]. Error bars are not shown when smaller than the symbol size. Inset: Velocity as a function of time for one realization of the algorithm with $\mathrm{\Omega }={10}^{20}$.

Figure 4

Velocity time correlation as a function of time for different values of the number of particles $\mathrm{\Omega }$. Inset shows the scaling given by Eq. (11).