Tiling a Plane in a Dynamical Process and its Applications to Arrays of Quantum Dots, Drums, and Heat Transfer

We present a reaction-diffusion system consisting of $N$ components. The evolution of the system leads to the partition of the plane into cells, each occupied by only one component. For large $N$, the stationary state becomes a periodic array of hexagonal cells. We present a functional of the densities of the components, which decreases monotonically during the evolution and attains its minimal value in the stationary state. This value is equal to the sum of the first Laplacian eigenvalues for all cells. Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem.

Figure 1

The stationary states of $N$ components in the square with absorbing sides, for $N=4$, 6, 16, and 320.

Figure 2

The lowest $\tilde{\sigma }$ [see Eq. (7)] among stationary states found for $N$ components in absorbing square frame, as a function of $N$. The solid line shows the algebraic function: ${\tilde{\sigma }}_{\mathrm{hex}}+\alpha /\sqrt{N}$, with $\alpha =4.8$. This power-law behavior corresponds to the fact that the deviation from ${\tilde{\sigma }}_{\mathrm{hex}}$ is mainly due to the components that stay in contact with the frame. Their relative number is of the order of $1/\sqrt{N}$.

Figure 3

Snapshots of the evolution of the system of $N=4$ components in the periodic rectangle with the aspect ratio $L_y/L_x=\sqrt{3}/2$.

Figure 4

Time dependence of the partition cost functional, $\tilde{\sigma }$, during the evolution of the system shown in Fig. 3. The inset shows the rapid, algebraic decay of $\tilde{\sigma }$ for small time. The values of ${\tilde{\sigma }}_{\mathrm{rect}}$ and ${\tilde{\sigma }}_{\mathrm{hex}}$ are given in Table .

Figure 5

Snapshots of the evolution of the system of $N=8$ components in the periodic rectangle with the aspect ratio $L_y/L_x=\sqrt{3}/2$.

Figure 6

Time dependence of $\tilde{\sigma }$ during the evolution of the system shown in Fig. 5. The insets show the same graph in the region of small time, and near the transition to the stationary state.

Figure 7

The partition cost functional $\tilde{\sigma }$ as a function of $\ln (L_y/L_x)$ for stationary states of $N$ components in the periodic box of size $L_y\times L_x$. Solid, dotted, and dashed lines are for $N=4$, 5, and 6, respectively.