Pattern Formation by Energy Minimization
Cyrill Muratov, New Jersey Institute of Technology
Pattern formation is a very broad term that refers to an extremely diverse set of observed phenomena, often of very different nature. Usually, by the word "pattern" one understands a spatially coherent structure that forms as a result of the physical processes acting on different length scales. Thus, a key attribute of all pattern forming systems is the appearance of spatial order.
Because of the very different physical mechanisms responsible for the formation of patterns, a general mathematical framework for their description is lacking. At the same time, many mathematical models of pattern forming systems come from variational principles dictated by the underlying physics. Perhaps the simplest and best-known example of such systems are crystalline solids. There the spatial order is expressed by the periodic lattice arrangement of atoms, which is mediated by the competition of repulsive and attractive interatomic forces and is achieved by the minimization of total potential energy. Other examples can be found in soft condensed matter systems, where patterns can be viewed as minimizers, either local or global, of certain energy functionals.
Despite their apparent simplicity, even the most basic models of pattern forming systems present mathematicians with formidable challenges. In this talk, I will overview some of the main issues in the analysis of spatial patterns, concentrating on systems with competing short-range attractive and long-range repulsive interactions. I will show how the same mathematical model can be applied to strikingly different physical systems, pointing out universality in the pattern forming behavior. Surprisingly, the model also provides a glimpse at pattern formation mechanisms in a different class of models that generally lack a variational structure, yet are crucial in understanding morphogenetic patterns in embryonic development.