The experimental investigation of chaos in atomic physics began with the historic experiment on microwave ionization of Rydberg atoms reported by Bayfield and Koch in 1974. The central result is the existence of an ionization threshold as a function of the microwave field. At the time (1974) this result was totally unexpected since ionization thresholds, in analogy to the photo-electric effect, were thought to appear only as a function of the frequency. Nowadays, especially in the light of the material presented in Chapters 5 and 6, the existence of an ionization threshold in the microwave field is less surprising and may be attributed to the existence of a critical microwave field that marks the onset of global chaos in the classical analogue of the Bayfield-Koch experiments. But at the time the Bayfield-Koch experiment was conducted, a connection with chaos was not suspected. Leopold and Percival (1978, 1979) were the first to investigate the Bayfield-Koch experiments using purely classical mechanics. This is allowed on the basis of the correspondence principle, Leopold and Percival argued, since the quantum numbers involved in the ionization experiments are large. This line of thought turned out to be very fruitful. With the help of Monte Carlo simulations of the time evolution of classical trajectories in phase space, Leopold and Percival were able to reproduce the existence and location of the microwave thresholds established by the Bayfield-Koch experiment.

The emphasis in the previous part of the book was on mathematical tools that permit the classification of symbolic signals in a general and accurate way by means of a few numbers or functions. The study of complexity, however, cannot be restricted to the evaluation of indicators alone, since each of them presupposes a specific model which may or may not adhere to the system. For example, power spectra hint at a superposition of periodic components, fractal dimensions to a self-similar concentration of the measure, the thermodynamic approach to extensive Hamiltonian functions. It seems, hence, necessary to seek procedures for the identification of appropriate models before discussing definite complexity measures. Stated in such a general form, the project would be far too ambitious (tantamount to finding “meta-rules” that select physical theories). The symbolic encoding illustrated in the previous chapters provides a homogeneous environment which makes the general modelling question more amenable to an effective formalization.

Independent efforts in the study of neuron nets, compiler programs, mathematical logic, and natural languages have led to the construction of finite discrete models (automata) which produce symbolic sequences with different levels of complexity by performing elementary operations. The corresponding levels of computational power are well expressed by the Chomsky hierarchy, a list of four basic families of automata culminating in the celebrated Turing machine.

The core of the problem when dealing with a complex system is the difficulty in discerning elements of order in its structure. If the object of the investigation is a symbolic pattern, one usually examines finite samples of it. The extent to which these can be considered regular, however, depends both on the observer's demand and on their size. If strict periodicity is required, this might possibly be observed only in very small patches. A weaker notion of regularity permits the identification of larger “elementary” domains. This intrinsic indefiniteness, shared alike by concepts such as order and organization, seems to prevent us from attaining a definition of complexity altogether. This impasse can be overcome by noticing that the discovery of the inner rules of the system gives a clue as to how its description can be shortened. Intuitively, systems admitting a concise description are simple. More precisely, one tries to infer a model which constitutes a compressed representation of the system. The model can then be used to reproduce already observed patterns, as a verification, or even to “extend” the whole object beyond its original boundaries, thus making a prediction about its possible continuation in space or time.

As we shall see, a crucial distinction must be made at this point.

In this chapter, we present some of the most frequently quoted examples of “complex” behaviour observed in nature. Far from proposing a global explanation of such disparate systems within a unique theoretical framework, we select those common properties that do cast light on the ways in which complexity exhibits itself.

Natural macroscopic systems are usually characterized by intensive parameters (e.g., temperature T or pressure P) and extensive ones (volume V, number of particles N) which are taken into account by suitable thermodynamic functions, such as the energy E or the entropy S. When the only interaction of a system with its surroundings consists of a heat exchange with a thermal bath, an equilibrium state eventually results: the macroscopic variables become essentially time independent, since fluctuations undergo exponential relaxation. The equilibrium state corresponds to the minimum of the free energy F = E − TS and is determined by the interplay between the order induced by the interactions, described by E, and the disorder arising from the multiplicity of different macroscopic states with the same energy, accounted for by the entropy S.

The commonest case is, however, represented by systems that are open to interactions with the environment, which usually takes the form of a source of energy and a sink where this is dissipated.

The term “complex” is being used more and more frequently in science, often in a vague sense akin to “complication”, and referred to any problem to which standard, well-established methods of mathematical analysis cannot be immediately applied. The spontaneous, legitimate reaction of the careful investigator to this attitude can be summarized by the questions: “Why study complexity?”, “What is complexity?”.

In the first part of the book, we have illustrated several examples from various disciplines in which complexity purportedly arises, trying, on the one hand, to exclude phenomena which do not really call for new concepts or mathematical tools and, on the other, to find common features in the remaining cases which could be of guidance for a sound and sufficiently general formulation of the problem. While amply answering the former question, the observed variety of apparently complex behaviour renders the task of formalizing complexity, i.e., of answering the latter question, quite hard. This is the subject of the main body of the book.

Aware of the difficulty of developing a formalism which is powerful enough to yield meaningful answers in all cases of interest, we have presented a critical comparison among various approaches, with the help of selected examples, stressing their complementarity.

The scientific basis of the discussion about complexity is first exposed in general terms, with emphasis on the physical motivation for research on this topic. The genesis of the “classical” notion of complexity, born in the context of the early computer science, is then briefly reviewed with reference to the physical point of view. Finally, different methodological questions arising in the practical realization of effective complexity indicators are illustrated.

Statement of the problem

The success of modern science is the success of the experimental method. Measurements have reached an extreme accuracy and reproducibility, especially in some fields, thanks to the possibility of conducting experiments under well controlled conditions. Accordingly, the inferred physical laws have been designed so as to yield nonambiguous predictions. Whenever substantial disagreement is found between theory and experiment, this is attributed either to unforeseen external forces or to an incomplete knowledge of the state of the system. In the latter case, the procedure so far has followed a reductionist approach: the system has been observed with an increased resolution in the search for its “elementary” constituents. Matter has been split into molecules, atoms, nucleons, quarks, thus reducing reality to the assembly of a huge number of bricks, mediated by only three fundamental forces: nuclear, electro-weak and gravitational interactions.

The intuitive notion of complexity is well expressed by the usual dictionary definition: “a complex object is an arrangement of parts, so intricate as to be hard to understand or deal with” (Webster, 1986). A scientist, when confronted with a complex problem, feels a sensation of distress that is often not attributable to a definite cause: it is commonly associated with the inability to discriminate the fundamental constituents of the system or to describe their interrelations in a concise way. The behaviour is so involved that any specifically designed finite model eventually departs from the observation, either when time proceeds or when the spatial resolution is sharpened. This elusiveness is the main hindrance to the formulation of a “theory of complexity”, in spite of the generality of the phenomenon.

The problem of characterizing complexity in a quantitative way is a vast and rapidly developing subject. Although various interpretations of the term have been advanced in different disciplines, no comprehensive discussion has yet been attempted. The fields in which most efforts have been originally concentrated are automata and information theories and computer science. More recently, research in this topic has received considerable impulse in the physics community, especially in connection with the study of phase transitions and chaotic dynamics.