We now take a break from the theoretical developments of the previous chapters and consider applications of lattice gases to the study of complex flows through complex geometries. Our complex geometry is one of the most complicated nature has to offer—a porous rock. The flows we consider are either those of a simple fluid, such as water, or an immiscible two-fluid mixture, such as water and oil. The problems we shall illustrate are not only of intrinsic interest for physics but have applications in fields as diverse as hydrology, oil recovery, and biology, to name just a few.

Our objectives in this chapter are twofold. First, we wish to indicate the level of accuracy that one may expect from these kinds of flow simulations. Second, we wish to show what we can learn from such work. We begin with a brief introduction to the subject.

Geometric complexity

All rocks found on the earth's surface are porous, but those rocks that we call sandstones are usually more porous than others. Sandstones are formed from random assemblages of sand grains that are cemented together over geologic time. Fluids such as oil or water may then become trapped in the pore space between the cemented sand grains. There may be an economic interest in extracting these fluids, or, equally possible in modern times, we may wish to predict the rate at which some contaminant such as radioactive waste could migrate through such a medium.

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.