Honeycomb conjecture for the Laplacian eigenvalues - tiling a plane in a dynamical process

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, that 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 original honeycomb problem. The functional which is minimized can be identified as the sum of Renyi entropy productions for individual components. Defined in this way, total entropy production decreases towards its stationary value. Both the minimization of the total entropy production, and the minimization of the sum of the Laplacian eigenvalues, lead to the same shapes of cells. (author)