Coarsening with nontrivial in-domain dynamics: Correlations and interface fluctuations

Using numerical simulations we investigate the space-time properties of a system in which spirals emerge within coarsening domains, thus giving rise to nontrivial internal dynamics. Initially proposed in the context of population dynamics, the studied six-species model exhibits growing domains composed of three species in a rock-paper-scissors relationship. Through the investigation of different quantities, such as space-time correlations and the derived characteristic length, autocorrelation, density of empty sites, and interface width, we demonstrate that the nontrivial dynamics inside the domains affects the coarsening process as well as the properties of the interfaces separating different domains. Domain growth, aging, and interface fluctuations are shown to be governed by exponents whose values differ from those expected in systems with curvature driven coarsening.

Figure 1

Interaction diagram for the (6,3) game. The arrows connect predators with their prey. On a two-dimensional lattice two teams of cyclically interacting species each form their own domains, see Fig. 2. The bold arrows indicate the two teams of three species that emerge from this interaction scheme.

Figure 2

First row: Snapshots of the (6,3) system with empty sites at various times on a $256\times 256$ lattice showing the generation of the two types of competing domains where in each domain the teams play a rock-paper-scissors game. Empty sites are indicated by black dots. Second row: Snapshots of the (6,3) system without empty sites.

Figure 3

(a) Snapshot of the (6,3) system with empty sites on a $256\times 256$ lattice at time $t=1000$. All individuals of one team are shown in red, whereas the individuals of the other team are colored in blue. Empty sites are indicated by black dots. (b) The same for the (6,3) system without empty sites. For this case the boundaries between domains are very diffuse. (c) Snapshot of the two-species (2,1) game with empty sites on a $256\times 256$ lattice at time $t=1000$.

Figure 4

Space- and time-dependent correlation function for the (6,3) game with empty sites in two space dimensions when (a) all six species are treated as separate and (b) species which make up a team are treated as one. (c) Time-dependent correlation lengths extracted from the space-time correlation. The values of $C_{6,0}$ and $C_{2,0}$ used for this are indicated by the horizontal line segments in (a) and (b) with the color and line type matching those in (c). The solid magenta lines in (c) indicate the slope 0.43. Also shown is the correlation length for the (2,1) system with the slope 0.5 at later times as expected for curvature driven coarsening. The data have been obtained for a system with $700\times 700$ sites and result from averaging over 7000 independent realizations of the noise.

Figure 5

Aging scaling of the two-times autocorrelation function $C_2(t,s)$ for different waiting times $s$: (a) a (6,3) system with empty sites and (b) a (2,1) system. The insets show the time dependence of the autocorrelation for $s=0$. The data for (6,3), obtained for systems with $700\times 700$ sites, result from averaging over 30 000 independent runs for $s>0$, whereas for $s=0$ the average is taken over 100 000 realizations of the noise. For (2,1) we used 2900 realizations for every value of $s$.

Figure 6

Time evolution of the density of empty sites in the (6,3) model with the reaction scheme (2). After some transient behavior the density of the empty sites created by reactions between members from one team (green line) approaches a constant value. On the other hand, the density of empty sites that result from interactions between the teams (red line) decays algebraically with an exponent $x_E=-0.25\left(1\right)$ as indicated by the dashed red line. The data, obtained for a system with $700\times 700$ sites, result from averaging over 7000 different runs. The inset: Comparison of the (6,3) density of empty sites resulting from interactions between teams with the density of empty sites obtained for the (2,1) model. The dashed blue line indicates a decay with an exponent $-0.5$.

Figure 7

First row: Snapshots at various times of an interface in the (6,3) system with empty sites on a $256\times 256$ lattice. At $t=0$ the system is separated in two halves where all the sites in one of the halves are occupied randomly by individuals from one team only. After formation of the spirals large interface fluctuations are observed. Second row: The same, but now for the (6,3) system without empty sites. Due to the absence of large structures in each half, the interface fluctuations are much less pronounced. Third row: The same but now for the (2,1) system with each species initially occupying one half of the system.

Figure 8

Interface width $W\left(t\right)$ for the different systems. (a) The (6,3) model with empty sites. The large-scale interface fluctuations induced by the spirals, see the first row of snapshots in Fig. 7, yield values for the growth exponent $\beta =0.43\left(1\right)$ and roughness exponent $\alpha =0.15\left(2\right)$ that differ from the standard Edwards-Wilkinson values of $\beta =1/4$ and $\alpha =1/2$. (b) The (6,3) model without empty sites for which the interface width displays the Edwards-Wilkinson scaling, see also the inset in the upper left corner. The inset in the lower right corner plots $L^{-1/2}W$ vs time for the (2,1) case for which we again find the Edwards-Wilkinson exponents. The system sizes are $500\times H$ where the different values of $H$ are given in the legend of panel (a). The data result from averaging over at least 8000 independent runs.