Conclusion: The Impact of Statistical Physics Applications of Statistical Physics The different examples which we have treated in the present book have shown the role played by statistical mechanics in other branches of physics: as a theory of matter in its various forms it serves as a bridge between macroscopic physics, which is more descriptive and experimental, and microscopic physics, which aims at finding the principles on which our understanding of Nature is based. This bridge can be crossed fruitfully in both directions, to explain or predict macroscopic properties from our knowledge of microphysics, and inversely to consolidate the microscopic principles by exploring their more or less distant macroscopic consequences which may be checked experimentally. The use of statistical methods for deductive purposes becomes unavoidable as soon as the systems or the effects studied are no longer elementary. Statistical physics is a tool of common use in the other natural sciences. Astrophysics resorts to it for describing the various forms of matter existing in the Universe where densities are either too high or too low, and tempera­ tures too high, to enable us to carry out the appropriate laboratory studies. Chemical kinetics as well as chemical equilibria depend in an essential way on it. We have also seen that this makes it possible to calculate the mechan­ ical properties of fluids or of solids; these properties are postulated in the mechanics of continuous media. Even biophysics has recourse to it when it treats general properties or poorly organized systems. If we turn to the domain of practical applications it is clear that the technology of materials, a science aiming to develop substances which have definite mechanical (metallurgy, plastics, lubricants), chemical (corrosion), thermal (conduction or insulation), electrical (resistance, superconductivity), or optical (display by liquid crystals) properties, has a great interest in statis­ tical physics. For instance, ferromagnetism, a phenomenon which is of great importance as it has enabled the development of electrical engineering (alter­ nators, transformers, motors), is the result of an order-disorder phase transi­ tion; we have sketched its theory in Chap.9. Here, observations and practical applications preceded understanding. However, we have shown in Chap.11 how the theory of another kind of substances, the semiconductors, has led to the construction of devices such as diodes or transistors and everybody knows their innumerable applications (micro-electronics, computers, energetics). 572 Conclusion: The Impact of Statistical Physics Through thermodynamics, which has become its extension, but also di­ rectly, statistical physics contributes to the development of various energy techniques, whether the extraction, transformation, transport, storage, sav­ ing, or efficient use of thermal, electrical, radiative, chemical, mechanical, or nuclear energy. In passing we have mentioned some applications to the quest for new energy sources, such as fusion or solar energy. Methods and Concepts of Statistical Physics Statistical physics has also played a major role in the realm of ideas. Like rela­ tivity and quantum mechanics, and maybe even more deeply, it has changed our world picture through its philosophical impact and its epistemological implications. As we have seen several times it has accustomed us, even out­ side the framework of quantum theory, to incorporate in science a partially subjective, or at least a relative, aspect inherent in the probability concept. We see now scientific progress as a devolution of more and more probable, more and more faithful, images of reality, but the belief in an absolute sci­ entific Truth, which was so widespread amongst scientists and philosophers in the nineteenth century, can hardly be found nowadays: more or less great likelihood, yes, but no final certainty. Also noteworthy are the contributions of statistical physics to the devel­ opment of methods and concepts. It was one of the first disciplines where the statistical method, that is the art of making predictions when one has little knowledge, was worked out. A macroscopic substance is a typical example of a system which is too complicated to be described in all detail. The proper­ ties in which we are interested concern only a few degrees of freedom, or are average effects; the statistical treatment allows us to discard those aspects that are too complicated, not well known, or of no interest. This fruitful ap­ proach is met with in all applications of statistics. We have also illustrated another remarkable possibility opened up by statistics, to wit, the possibil­ ity to m^ake nearly certain predictions even though the system studied has a random nature, provided it is large. Another method, examples of which can be found in Chaps.7 or 15, dealing with transport, and 13, dealing with Kirchhoff's laws, consists in making up a detailed balance of exchanges which can take place between one part of the system and another. Conservation laws, such as those for energy, momentum, or particle numbers, play an important role in establishing this balance and this enables us to reduce the problem. These ideas can be transposed to economic sciences. We also saw in Chap.14 how fruitful is the systematic exploitation of symmetry and invariance properties. Statistical physics is a field where the use of models is a nearly perma­ nent feature. The ideal method, which would consist of starting from a known Hamiltonian to use a Boltzmann-Gibbs distribution for deriving the proper- Methods and Concepts of Statistical Physics 573 ties of a system, often turns out to be intractable. Therefore, as we did on many occasions in this book, one does not start from first principles, but one uses a model, that is, a simple, idealized structure, which is meant to depict consistently a more complicated reality. It even happens that one introduces contradictory models to describe different aspects of the same system. One knows that the social sciences, such as economics and sociology, make great use of models. Statistical physics can help us understand the significance of their use. In fact, in fields where theory is not so well developed, one is prepared to justify a model by its operational fertility and by more or less intuitive similarity arguments. In statistical physics, we can go further, as we saw at the end of Chap.l, by constructing a series of more and more refined models which are based partly upon more and more precise experiments, and partly upon known "first principles" from which approximations enable us to construct models. These nested models are successive steps in the capture of Nature. Several concepts introduced in statistical physics have already passed into everyday usage. It suffices to mention energy, its transformations and the consequences of its conservation: the limitation of our energy resources, on the one hand, and the inevitability of "thermal pollution" if one uses energy, on the other hand, have become commonplace. We have studied the relations between the concepts of entropy, informa­ tion, and disorder. Energy degradation, and irreversibility, are manifestations of the general tendency towards the loss of order. We also saw that when we increase our knowledge about a system, we put order into it. These kinds of considerations by and large go beyond the framework of physics. Especially, information theory, the birth of which was partly due to statistical physics, enables us to improve the efiiciency of transmission methods by using a suit­ able treatment of messages which avoids their being tangled up by noise. Prom this point of view, a sizeable part of biology appears as the study of mechanisms through which living organisms create order by using the infor­ mation contained in their genetic code. Is even intellectual activity nothing but a struggle against chaos? A set of remarkable effects, which may also be found outside statistical physics, is connected with the establishing of order in a system. We have been able to appreciate the diversity of the forms that order can take, from mag­ netism to the superfluidity of helium. The huge variety of crystal structures provide us with a spectacular example of this wealth of types of ordering. We saw also (§§ 11.1.2 and 12.2.3) that different forms of order may compete and that depending on the situation, one or other of them dominates. The a pri­ ori most surprising phenomenon is the sudden appearance of order: we have encountered several examples of phase transitions where a qualitative change is suddenly produced when the temperature is gradually lowered. Statisti­ cal physics enables us to understand the origin of such discontinuities which occur only thanks to the large size of the systems studied. 574 Conclusion: The Impact of Statistical Physics Finally, statistical physics clearly illustrates the concept of hierarchical structures, which is fashionable in human sciences. Depending on the scale in which we are interested, the laws of physics differ: a substance is on the microscopic scale a set of electrons and nuclei interacting through Coulomb forces; on the macroscopic scale, it is a gas or a solid with an order which we could not expect from the microscopic laws. However, the physicist believes in the existence of universal fundamental laws, or at least in the possibility of gradually unifying the various pieces of information available at a given time. Statistical physics helps us to resolve this contradiction between the unity of Nature and the change of her laws depending on the scale, by enabling us to derive a large number of macroscopic laws from the underlying microscopic laws. On the level of principles, the latter, which are simpler, appear to be more fundamental. On the practical level, the fact that this derivation is pos­ sible in relatively simple cases, all the same, does not transform macroscopic physics or chemistry into disciplines with the logical and consistent structure of mathematics: the constituent elements of reality are simple, but combining them leads to enormously rich complex phenomena.