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

Challenges in thermodynamics: Irreversible processes, nonextensive entropies, and systems without equilibrium states

Recent works on evaporation and condensation demonstrate that even these simplest irreversible processes, studied for over 100 years, are not well understood. In the case of a liquid evaporating into its vapor, the liquid temperature is constant during evaporation and the evaporation flux is governed by the heat transfer from the hotter vapor into the colder liquid. Whether liquid evaporates into its own vapor or into the vacuum, the irreversible pathway in the process goes through a number of steps which quickly lead to the steady-state conditions with mechanical equilibrium in most parts of the system—the fact overlooked in all previous studies. Even less is known about general rules which govern systems far from equilibrium. Recently, it has been demonstrated that a work done in an irreversible process can be related to the free energy difference between equilibrium states joined by the process. Finally, a real challenge in thermodynamics is a description of living systems since they do not have equilibrium states, are nonextensive, (i.e., they cannot be divided into subsystems), and cannot be isolated. Thus, their proper description requires new paradigms in thermodynamics.

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.