The search for fundamental particles and the mathematical formalism with which to describe their motions and interactions has in common with the search for the laws governing gravitational, electromagnetic, and other fields of force the aim of finding the most economical basis from which, in principle, theories of all other material processes may be derived. Some of these processes are simple—a single particle moving in a given field of force, for example—if the term refers to the nature of the system studied and not to the mathematical equipment that may sometimes be brought to bear. A complex process, on the other hand, is typically one in which many interacting particles are involved and for which it is hardly ever possible to proceed to a complete mathematical solution. A computer may be able to follow in detail the movement of thousands of atoms interacting in a specified way, but a wholly successful study along these lines does no more than display on a large scale and at an assimilable speed what nature achieves on its own. Much can be learned from these studies, but, if one is primarily concerned with discovering what will happen in given circumstances, it is frequently quicker and cheaper to do the experiment than to model it on a computer. In any case, computer modeling of quantum mechanical, as distinct from Newtonian, behaviour becomes extremely complicated as soon as more than a few particles are involved.

The art of analyzing complex systems is that of finding the means to extract from theory no more information than one needs. It is normally of no value to discover the speed of a given molecule in a gas at a given moment; it is, however, very valuable to know what fraction of the molecules possess a given speed. The correct answer to this question was found by Maxwell, whose argument was ingenious and plausible. More rigorously, Boltzmann showed that it is possible to proceed from the conservation laws governing molecular encounters to general statements, such as the distribution of velocities, which are largely independent of how the molecules interact. In thus laying the foundations of statistical mechanics, Boltzmann provided an object lesson in how to avoid recourse to the fundamental laws, replacing them with a new set of rules appropriate to highly complex systems. This point is discussed further in Entropy and disorder below.

The example of statistical mechanics is but one of many that together build up a hierarchical structure of simplified models whose function is to make practicable the analysis of systems at various levels of complexity. Ideally, the logical relationship between each successive pair of levels should be established so that the analyst may have confidence that the methods he applies to his special problem are buttressed by the enormous corpus of fact and theory that comprises physical knowledge at all levels. It is not in the nature of the subject for every connection to be proved with mathematical rigour, but, where this is lacking, experiment will frequently indicate what trust may be placed in the intuitive steps of the argument.

For instance, it is out of the question to solve completely the quantum mechanical problem of finding the stationary states in which an atomic nucleus containing perhaps 50 protons or neutrons can exist. Nevertheless, the energy of these states can be measured and models devised in which details of particle position are replaced by averages, such that when the simplified model is treated by the methods of quantum mechanics the measured energy levels emerge from the calculations. Success is attained when the rules for setting up the model are found to give the right result for every nucleus. Similar models had been devised earlier by the English physicist Douglas R. Hartree to describe the cloud of electrons around the nucleus. The increase in computing power made it feasible to add extra details to the model so that it agreed even better with the measured properties of atoms. It is worth noting that when the extranuclear electrons are under consideration it is frequently unnecessary to refer to details of the nucleus, which might just as well be a point charge; even if this is too simplistic, a small number of extra facts usually suffices. In the same way, when the atoms combine chemically and molecules in a gas or a condensed state interact, most of the details of electronic structure within the atom are irrelevant or can be included in the calculation by introducing a few extra parameters; these are often treated as empirical properties. Thus, the degree to which an atom is distorted by an electric field is often a significant factor in its behaviour, and the investigator dealing with the properties of assemblies of atoms may prefer to use the measured value rather than the atomic theorist’s calculation of what it should be. However, he knows that enough of these calculations have been successfully carried out for his use of measured values in any specific case to be a time-saver rather than a denial of the validity of his model.

These examples from atomic physics can be multiplied at all levels so that a connected hierarchy exists, ranging from fundamental particles and fields, through atoms and molecules, to gases, liquids, and solids that were studied in detail and reduced to quantitative order well before the rise of atomic theory. Beyond this level lie the realms of the Earth sciences, the planetary systems, the interior of stars, galaxies, and the Cosmos as a whole. And with the interior of stars and the hypothetical early universe, the entire range of models must be brought to bear if one is to understand how the chemical elements were built up or to determine what sort of motions are possible in the unimaginably dense, condensed state of neutron stars.

The following sections make no attempt to explore all aspects and interconnections of complex material systems, but they highlight a few ideas which pervade the field and which indicate the existence of principles that find little place in the fundamental laws yet are the outcome of their operation.