Foundations of Statistical Mechanics: A Deductive Treatment

by Oliver Penrose

This volume demonstrates the manner in which statistical mechanics can be built up deductively from a small number of well-defined physical assumptions. A solid basis for the deductive structure is provided by embodying these assumptions in a system of five postulates that describe an idealized model of real physical systems. These postulates play a theoretical role similar to that of the first and second laws in thermodynamics.The first chapter concerns the primary physical assumptions and their idealization in the form of postulates. The following three chapters examine the consequences of these postulates, culminating in the derivation of the fundamental formulas for calculating probabilities in terms of dynamical quantities. Two concluding chapters are devoted to an analysis of the notion of entropy, illustrating its links between statistical mechanics and thermodynamics and between statistical mechanics and communication theory. Because this book deals mostly with general principles, its only detailed considerations of physical applications are in terms of the system with the simplest possible the ideal classical gas, which is discussed both in its equilibrium and its nonequilibrium aspects.Intended for readers with a knowledge of physics at the advanced undergraduate and graduate levels, this volume considers topics of interest not only to physicists, but also to statisticians, communication theorists, chemists, and mathematicians.

Genres: Physics

272 pages, Paperback

First published February 11, 2005

Reviews

[William Bies · Rating 4 out of 5]

Einstein is famously (among physicists, that is) said once to have pointed to thermodynamics as the sole discipline in the edifice of theoretical physics that is apt to endure forever. Indeed, during his lifetime he himself contributed, as part of the two great revolutions of the twentieth century, to the downfall of Newtonian classical mechanics and theory of universal gravitation and their replacement by the quantum theory resp. general theory of relativity. Thermodynamics, on the other hand, has survived the transition from classical to quantum relatively unscathed. For, in the guise of its statistical-mechanical version, it has to do with the generic behavior of systems possessing a macroscopically large number of degrees of freedom (on the order of Avagadro’s constant, 6.02214086×10²³). As such, it is to a large degree independent of the nature of the physical laws governing the constituents – as one has a right to expect, from the viewpoint of structural realism in the philosophy of science. Instead, its results depend in large measure on the generic properties of random processes, such as form the subject of probability theory as it applies to physics.

The application of the methods of modern probability theory to statistical mechanics is by no means routine and – what is more – invokes concepts which themselves involve considerable subtley, such as expectation, fluctuations about the mean, correlation and clustering of correlations, the Markovian property, the ergodic hypothesis and other chaos-theoretic concepts such as mixing, the Anosov and K-properties, algorithmic complexity etc. With such a welter of ideas to master, the beginning student may well be perplexed as to where to begin, especially in view of the voluminous textbook literature. This reviewer’s preference, though, would be to seek textbook treatments exhibiting the greatest possible conceptual clarity and mathematical rigor. Now, the author of the present work, Oliver Penrose, is an old hand in the community of mathematical physicists who, beginning around the early 1960’s, first concerned themselves to flesh out the foundations of statistical thermodynamics in mathematically unexceptionable terms. Thus, Wolfgang Pauli’s celebrated lecture series in theoretical physics would be totally unacceptable. Other names associated with this movement would be Eliot Lieb, Walter Thirring and David Ruelle. Selecta from these scientists’ curricula vitae are indeed in print, but it would be somewhat cumbersome to try to learn the rudiments of the theory from them. Practically speaking then, we are left with just two good textbooks: the one by Oliver Penrose himself, entitled Foundations of Statistical Physics: A Deductive Treatment, and the one by David Ruelle, Statistical Mechanics: Rigorous Results (still in print now by the Imperial College Press under the World Scientific label).

Ruelle’s exposition, however, is significantly more demanding from the technical point of view (one would want not only to be familiar with operators in Banach spaces but also with the theory of von Neumann algebras). For a long while now, this reviewer has brooded over the thought of returning to Ruelle, perhaps to understand him for the first time (as T.S. Eliot suggests must be the case in his late poem, The Four Quartets!), but too many books, too little time! So a promised review of Ruelle will have to be postponed until well into the future. Nevertheless, the subject of this review is Penrose’s text, so let us turn to it without further ado.

In contrast to more popular treatments (say, by Richard Feynman), Penrose means business when he aims to articulate a conceptually clear foundation and to derive from it as much of the phenomena as he can. For the most part, his exposition is limited to classical statistical mechanics, as is only right. The mathematics is all clear and stated in rigorous terms, but the author mostly eschews the temptation to burden his text with overlong proofs. The present text is a place to acquire perspective, not professional competence in the very recondite mathematics behind, say, the KAM and Nekhoroshev theorems. If that is what one wants, try for starters Jürgen Moser, Stable and Random Motions in Dynamical Systems (now available as a reprint by the Princeton University Press).

Telegraphic review of contents: an initial chapter on the basics from the standpoint of the physics, not to be neglected. Then a second chapter on the basics from the standpoint of probability theory. Chapters three and four form the heart of the work. Here one engages the question as to what is meant by equilibrium and the approach to equilibrium. For the probabilities with which one reckons in equilibrium statistical mechanics ought to originate in the underlying dynamics, and Penrose tells us how. Chapters five and six take up the definition of entropy and its physical meaning, in Boltzmann’s theory and in general (not limited to dilute gases). Penrose omits any serious treatment of non-equilibrium statistical mechanics such as we know it from N.N. Bogoliubov and Ilya Prigogine’s classic papers and treatises. To this we rejoin: one has to start somewhere!

Penrose embellishes his text with a none-too-excessive number of homework problems. This reviewer worked all of them and as he recalls, they tend to fall more into the category of either routine verifications or practice with the material just skimmed than into that of potentially research-grade challenges. So attempt them all and be satisfied with gaining some facility; in any case, solutions are provided in an appendix.

If Einstein is right, quantum mechanics itself will dissolve someday into a more fundamental and complete theory – and this reviewer would be inclined to stand with him against the tide of trendier physicists who suppose, all-too quickly, that the quantum mechanics as we know it must be a final theory (pace Bell’s theorem and the Aspect experiment!). All the same, quantum statistical mechanics is by now too well established for it simply to be overturned. Rather, it will always persist as the first approximation to any true successor theory. In this respect, it differs from the classical statistical mechanics to which Penrose devotes the bulk of his energy [Fleiß]. For sometimes a strictly classical point of view is adequate, as in the kinetic theory of dilute gases, and other times it is altogether inadequate, as Fermi was the first to realize about the transport theory of metals and semiconductors. But since the classical will ever be closer to our unschooled intuition based on experiences of the world around us, it remains the logical place to start. Therefore, Penrose’s textbook is heartily to be recommended to any serious beginning graduate student in theoretical physics.

Let us close with a couple comments that still linger in this reviewer’s mind. First, is the curious Gibbs’ paradox (calling for division by N! when computing the entropy) somehow connected with the representation theory of the symmetric group of permutations on N particles? In other words, if we start from classical non-equilibrium statistical mechanics and project onto the time-invariant subspace, would the Gibbs paradox receive a natural solution rather than have to be resolved by force majeure in an appeal to indistinguishability of quantum particles? Unfortunately, this reviewer never had the leisure to dwell on a question such as this for a length of time sufficient to pick up, say, the Zwanzig-Mori theory well enough answer it.

Second, the student of Oliver Penrose’s crisp text herein reviewed will learn precious little either about real physical systems or about the history of the subject. An education in physics, after all, must be a drawn-out affair and, as to the former, everyone who keeps at it long enough will eventually assemble the needed expertise. As to the latter, we enjoy the good fortune of being able to advert to a collection of classic papers put together and commented upon by the excellent historian of science Stephen Brush, namely, The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary (also published by the Imperial College Press). Brush’s own historical discussions reprinted in this volume and elsewhere venture into more philosophical depth than Penrose is willing to broach in his rather dry exposition. Complement Oliver Penrose with Stephen Brush and one cannot go wrong!