Natural phase transitions, shape memory alloys, and naturally optimal bio-materials form a novel area of application of the mathematical techniques under discussion. These problems, involving complicated materials, are in many respects similar to structural optimization. In both cases one deals with several materials or solid phases that are distributed in a domain in a specific way. The optimality requirement posed by a designer is parallel to a natural variational principle of minimization of the total energy of the system (Gibbs principle). The transformation to another phase is parallel to the use of another material in a design. In minimizing its energy the system exhibits phase separation and forms a sort of natural composite that possesses optimal microstructure.

These similarities suggest that similar approaches could be applied to describe natural mixtures with minimal energy. This concept was put forward and implemented in various systems in the works of Ball, Bhattacharia, Kohn, Ericksen, Greenfield, James, Luskin, Khachaturyan, Kinderlehrer, Roitburg, Rozakis, Truskinovsky, and others. The methods of quasiconvexity are successively implemented for an explanation of structures arriving at some natural phase transitions; we refer to the works of the above mentioned authors.

However, the mentioned natural phenomena are much knottier than the problems of structural optimization. Indeed, the best engineering system should reach the global minimum of the minimizing functional that represents the quality of the system. On the contrary, a steady state of a natural system corresponds to any local minimum of the energy. It should be mentioned that the energy of complicated natural systems typically is characterized by a large class of metastable local minima. The search for a distribution of local minima requires techniques different from those discussed here, and we do not touch on this subject in the book.

There are other differences, too. Contrary to an optimal engineering construction, a realizable steady state of a natural system corresponds to a stable dynamical process that has led to it. Finally, natural composites usually are a random mixture of the states that correspond to local minima. The search for a distribution of local minima requires different techniques than discussed here, and we do not touch this subject in the book.

An approach to this problem is discussed in this preprint

The exact soution of the corresponding problem in non-linear dynamics is found in this preprint.