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.

The setting of a theory of complexity is greatly facilitated if it is carried out within a discrete framework. Most physical and mathematical problems, however, find their natural formulation in the real or complex field. Since the transformation of continuous quantities into a symbolic form is much more straightforward than the converse, it is convenient to adopt a common representation for complex systems based on integer arithmetics. This choice, in fact, does not restrict the generality of the approach, as this chapter will show. Moreover, discrete patterns actually occur in relevant physical systems and in mathematical models: consider, for example, magnets, alloys, crystals, DNA chains, and cellular automata. We recall, however, that a proposal for a theory of computational complexity over the real and complex fields has been recently advanced (Blum, 1990).

The symbolic representation of continuous systems also helps to elucidate the relationship between chaotic phenomena and random processes, although it is by no means restricted to nonlinear dynamics. Indeed, von Neumann's discrete automaton (von Neumann, 1966) was introduced to model natural organisms, which are mixed, “analogue–digital” systems: the genes are discrete information units, whereas the enzymes they control function analogically. Fluid configurations of the kind reproduced in Fig. 2.2 also lend themselves to discretization: owing to the constancy of the wavelength (complexity being associated with the orientation of the subdomains), a one-dimensional cut through the pattern yields a binary signal (high-low).

Most of the physical processes illustrated in the previous chapter are conveniently described by a set of partial differential equations (PDEs) for a vector field Ψ(x, t) which represents the state of the system in phase space X. The coordinates of Ψ are the values of observables measured at position x and time t: the corresponding field theory involves an infinity of degrees of freedom and is, in general, nonlinear.

A fundamental distinction must be made between conservative and dissipative systems: in the former, volumes in phase space are left invariant by the flow; in the latter, they contract to lower dimensional sets, thus suggesting that fewer variables may be sufficient to describe the asymptotic dynamics. Although this is often the case, it is by no means true that a dissipative model can be reduced to a conservative one acting in a lower-dimensional space, since the asymptotic trajectories may wander in the whole phase space without filling it (see, e.g., the definition of a fractal measure in Chapter 5).

A system is conceptually simple if its evolution can be reduced to the superposition of independent oscillations. This is the integrable case, in which a suitable nonlinear coordinate change permits expression of the equations of motion as a system of oscillators each having its own frequency.

When the metal complex [Ru(phen)2(dppz)]2+ is bound to DNA it can luminesce. If the metal complex [Rh(phi)2(phen)]3+ is nearby on the strand, the luminescence is quenched by electron transfer. By varying concentrations and by varying the DNA it is possible to probe the distribution of complexes in this one-dimensional (1D) system, and to gather information about the electron transfer length and interparticle forces. Our model assumes random deposition with allowance for interactions among the complexes. Long strands of calf thymus (CT) DNA and short strands of a synthetic 28-mer were used in the experiments and, for fixed [Ru(phen)2(dppz)]2+ concentration, quenching was measured as a function of [Rh(phi)2(phen)]3+ concentration. In previous work, to be cited later, we reported an electron transfer length based on the CT-DNA data. However, the short-strand (28-mer) experiments show a remarkable difference from the previously analyzed data. In particular, the electron-transfer quenching upon irradiation is enhanced by a factor of approximately four. This requires the consideration of new physical effects on the short strands. Our proposal is to introduce complexcomplex repulsion as an additional feature. This allows a reasonable fit within the context of the random-deposition model, although it does not take into account changes in the structure of the 28-mer introduced by the metal complexes during the loading process.

Editor's note

Kinetic Ising models in 1D provide a gallery of exactly solvable systems with nontrivial dynamics. The emphasis has traditionally been on their exact solvability, although much attention has also been devoted to models with conservation laws that have to be treated by numerical and approximation methods.

Chapter 4 reviews these models with emphasis on steady states and the approach to steady-state behavior. Chapter 5 puts the simplest 1D kinetic Ising models into a wider framework of the evaluation of dynamical critical behavior, analytically, in 1D, and numerically, for general dimension. Finally, Ch. 6 describes low-temperature nonequilibrium properties such as domain growth and freezing.

For a general description of dynamical critical behavior, not limited to 1D, as well as an excellent review and classification of various types of dynamics, the reader is directed to the classical work [1]. Certain probabilistic cellular automata are equivalent to kinetic Ising models.