Three-dimensional space partition based on the first Laplacian eigenvalues in cells.

We determine a partition of three-dimensional space into cells by minimization of the sum of the first Laplacian eigenvalues over the cells. This partitioning scheme emerges as a stationary state of a reaction-diffusion process taking place in a system of n different species which mutually annihilate, and simultaneously are duplicated in an autocatalytic reaction, so that the number of particles is kept constant and equal for each species. The system is considered in the limit of strong reactivity, so that the species separate each other into cells with well-defined, sharp boundaries. For a given n and fixed sizes of a periodic simulation box, this partition minimizes the aforementioned sum of eigenvalues. Further minimization is done by changing n and the side ratio of the periodic box. The global minimum is obtained for the structure with A15 symmetry, similar to the Weaire-Phelan foam. Depending on n and the side ratio, there are also many local minima, in particular: hcp (hexagonal close packed), fcc (face centered cubic), the Kelvin structure, and Frank-Kasper sigma phase.

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.

Minimization of the Renyi entropy production in the space-partitioning process.

The spontaneous division of space in Fleming-Viot processes is studied in terms of non-extensive thermodynamics. We analyze a system of n different types of Brownian particles confined in a box. Particles of different types annihilate each other when they come into close contact. Each process of annihilation is accompanied by a simultaneous nucleation of a particle of the same type, so that the number of particles of each component remains constant. The system eventually reaches a stationary state, in which the available space is divided into n separate subregions, each occupied by particles of one type. Within each subregion, the particle density distribution minimizes the Renyi entropy production. We show that the sum of these entropy productions in the stationary state is also minimized, i.e., the resulting boundaries between different components adopt a configuration which minimizes the total entropy production. The evolution of the system leads to decreasing of the total entropy production monotonically in time, irrespective of the initial conditions. In some circumstances, the stationary state is not unique-the entropy production may have several local minima for different configurations. In the case of a rectangular box, the existence and stability of different stationary states are studied as a function of the aspect ratio of the rectangle.