The Conceptual Foundations of the Statistical Approach in Mechanics

by Paul Ehrenfest, Tatiana Ehrenfest

In this concise classic, Paul Ehrenfest ― one of the twentieth century's greatest physicists ― reformulated the foundations of the statistical approach in mechanics. Originally published in 1912, this classic has lost little of its scientific and didactic value, and is suitable for advanced undergraduate and graduate students of physics and historians of science.
Part One describes the older formulation of statistico-mechanical investigations (kineto-statistics of the molecule). Part Two takes up the modern formulation of kineto-statistics of the gas model, and Part Three explores W. B. Gibbs's major work, Elementary Principles in Statistical Mechanics and its coverage of such topics as the problem of axiomatization in kineto-statistics, the introduction of canonical and microcanonical distributions, and the analogy to the observable behavior of thermodynamic systems. The book concludes with the authors' original notes, a series of useful appendixes, and a helpful bibliography.

Genres: Physics

128 pages, Paperback

First published March 1, 1990

Reviews

[William Bies · Rating 5 out of 5]

Poor Ludwig Boltzmann, along with James Clerk Maxwell originator of the modern statistical approach to the kinetic theory of near-ideal gases, could not, during the late phase of his career in the first decade of the twentieth century, tolerate the vehemence of his opponents, who, fortified by a methodologically sound Machian instrumentalism, wanted to deny the very existence of the atoms upon which his kinetic theory was founded, as empirically unconfirmed and therefore superfluous. Despite the fact that Boltzmann does, in his late papers, supply cogent replies to the incisive criticisms of Loschmidt (1876, 1877) and Zermelo (1896, 1900, 1906), he was in the end driven to commit suicide in September 1906, subject – apparently – to a bipolar disorder. If only history had played itself out a little differently! For the present work, entitled The Conceptual Foundations of the Statistical Approach in Mechanics, written in 1911 and appearing originally as an installment in the German encyclopedia of the mathematical sciences in 1912, soon attained the stature of a classic and could have allayed Boltzmann’s fears, had it appeared but a few years earlier, during his lifetime. Its coauthors, the Dutch physicist Paul Ehrenfest and his wife, the Russian mathematician Tatyana Afanasyeva, summarize the status quaestionis and then go on to lay out an original exposition of the conceptual problems associated with the statistical mechanics in its classical guise.

Succinct overview of contents: the encyclopedia article itself occupies just seventy pages. A three-page introduction sets the context for Boltzmann’s contentious demonstration of a molecular basis of irreversible behavior. Part I frames the problem from an intentionally old-school point of view, roughly speaking, what Boltzmann’s opponents would have known of his pioneering theory by reading its landmark publications from 1868 onwards. Part II enters into an extended discussion that aims to give substance to the point of view arrived at by the mature Boltzmann, when arguing in response to his critics. Briefly put, the claim will be that the second law of thermodynamics has a statistical character only: entropy increases monotonically in time merely in conditional expectation, not necessarily in fact – what his contemporaries found it very difficult to reconcile themselves with, as it would tend to overthrow the deterministic world-view then in vogue. Lastly, Part III addresses the American physicist J. Williard Gibbs’ putative axiomatization of statistical mechanics in his innovative monograph Elementary Principles in Statistical Mechanics from 1902.

Critical reaction: perusing this text makes for rather hard going, but for that all the more rewarding. Remember, it concerns probability theory as much as the physics itself. The reader may be in for a surprise: he will find little in the way of advanced mathematics, and much in the way of philosophically-oriented analysis of the problem. Thus, this text cuts against the grain of everything for which the overwhelming majority of present-day theoretical physicists stand.

The English translation from the German original and end-notes by Mark Kac and George E. Uhlenbeck come across as fine and obviate the need to think about technically difficult material in a foreign language. There are also a retrospective preface to the translation and several appendices, from 1959, added by the widow Afanasyeva. Here, she sharpens the text’s engagement with critical responses to its first publication.

Therefore, the present Dover reprint of the Ehrenfests’ celebrated work is to be prized as an historical document in its own right and – what is more – as an indispensable complement to Oliver Penrose’s Foundations of Statistical Mechanics: A Deductive Treatment, which we have just reviewed here.

Excursus: the contemporary controversy, however, is not quite so simple. With the advantage of hindsight, we know that the existence of atoms was seemingly confirmed experimentally by Jean Baptiste Perrin in 1911, basing himself on Einstein’s 1905 microscopic theory of the diffusion coefficient under Brownian motion. But it went deeper, namely, the period in question witnessed the heyday of the energeticism of Wilhelm Ostwald (late 1890 or early 1891 through 1907) and Georg Helm (1898), along with the flourishing of the electromagnetic world-view of Wilhelm Wien (1900), in light of which the authors’ perspective in the present work may appear naïve. For it is true that they elaborate an approach to classical statistical mechanics of great lucidity, but it remains clear nonetheless that the Newtonian world-picture of pointlike atoms executing a solution to the canonical ordinary differential equations of Hamilton forms an unavoidable underpinning to their work. Yet, at the time, the proponents of the electromagnetic world-view envisioned that these equations would have to be replaced with some modification of the partial differential equations of Maxwellian electrodynamics. Concurrently, the energeticists were advocating an even more radical revision of theoretical physics, in which the very concept of a solution to a differential equation – central to the classical vision ever since Newton – was to be suspended. Either possibility calls fundamentally into question the applicability of the Ehrenfests’ analysis to the real world. Thus, even if correct as an idealized mathematical model of the kinetic theory, it may fail the test of being an accurate description of the world of our experience and thus might not be good physics after all, one could maintain. In the aftermath of the discovery of the quantum mechanics, we today face a similar state of affairs [Sachlage]. There do exist approaches to a quantum statistical mechanics that build upon the foundations of Maxwell, Boltzmann, Gibbs and the Ehrenfests. The authors cannot fairly be faulted for failing fully to appreciate, in 1911, the significance of the quantum revolution then underway. But it remains a cause of regret that they omit to frame their theory profoundly enough for it to be capable of reckoning with the challenge posed in their day by Wien and the energeticists. The lesson? Perhaps the persistent recalcitrance of our current understanding of the quantum mechanics towards yielding a satisfactory solution to the measurement problem indicates a deeper property of nature: that the energeticists may well have been right to suspect that the processes of nature are only approximatively or incompletely to be described in terms of a differential equation such as Schrödinger’s, or indeed any differential equation whatsoever, be it ordinary, partial, evolutionary, Banach-space valued etc. If we are to make headway, possibly the place to start would be to seek a more trenchant grasp of the so-called theory of weak measurement.