Introduction

Ever since physics came into existence as a scientific discipline, it has been mainly concerned with identifying a minimal set of fundamental laws connecting a minimal number of different constituents, under the more or less implicit assumption that knowledge of these rules were a sufficient condition for explaining the world we live in. Such a program has had remarkable success, to the extent that processes occurring on scales that range from that of elementary particles up to those of stellar evolution can nowadays be satisfactorily described. Nevertheless, it has meanwhile become clear that the inverse approach is a source of unexpected richness and can hardly be included with the original microscopic equations. The existence of different phases of matter (gas, liquid, solid, and plasma, together with perhaps glasses, granular materials, and Bose–Einstein condensates) provides the most striking evidence that the adjustment of a parameter (e.g. temperature) can dramatically change the organization of the system.

Moreover, not only can the same set of microscopic laws give rise to different structures, but also the converse is true: different systems can, for example, converge toward the same crystalline configuration. One of the basic ingredients that makes the connection between different levels of description complex, though very intriguing, is the presence of nonlinearities. As long as each atom deviates slightly from its equilibrium position, the equations of motion can be linearized and the dynamics thereby decomposed into the sum of independent evolutions of so-called normal modes. In some cases, especially when disorder is present, such a decomposition cannot be easily performed (and very interesting physics can indeed arise, e.g. Anderson localization) but is, at least in principle, feasible.