Complex Systems and the Evolution of Matter

Abstract

How can order arise from complex, irregular, and chaotic states of matter? In classical antiquity philosophers tried to take the complexity of natural phenomena back to first principles (Sect. 2.1). With Newton and Leibniz something new was added to the theory of kinetic models. The velocity vector field has become one of the basic concepts in dynamical systems theory (Sect. 2.2). But Poincaré discovered that those models may be non-computable in the long run (the many-body-problem). Even in a fully deterministic world, the assumption of a Laplacean demon which can calculate the universe in the long run was exposed as an illusionary fiction (Sect. 2.3). From a methodological point of view, nonlinearity is a necessary but not sufficient condition of chaos. It also allows the emergence of order (Sect. 2.4). The theory of complex systems opens new avenues of pattern formation in the nano world with applications for self-constructing materials in materials science (Sect. 2.5). From a methodological point of view, the question arises, how can we detect attractors of pattern formation in an immense variety of measured data? Time series analysis, fractals, and multifractals are challenges in the current theory of complex systems (Sect. 2.6).