## Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics.

New York; London: Scribners; E. Arnold, 1902.

First edition, **inscribed presentation copy** to the great French mathematician and mathematical physicist Henri Poincaré. “Of Gibbs [Einstein] wrote in 1918: ‘[His] book is … a masterpiece, even though it is hard to read and the main points are found between the lines’” (Pais, *Subtle is the Lord* (1983), p. 73). This book was “a major advance in statistical mechanics, the branch of science in which a purely mechanical view of natural phenomena is replaced by one combining mechanics with probability” (Norman). Gibbs’ book was “a triumph of the rigorous axiomatic method, which placed him beside Clausius, Maxwell, and Boltzmann as one of the principal founders of statistical mechanics” (Mehra, p. 1786). “Albert Einstein – who independently developed his own version of statistical mechanics from 1902 to 1904, having no knowledge of Gibbs’ work – remarked in 1910 ‘Had I been familiar with Gibbs’ book at that time, I would not have published those papers at all, but would have limited myself to the discussion of just a few points’” (Inaba, p. 102). “Gibbs' book on statistical mechanics became an instant classic and has remained so for almost a century” (Mehra). In this book, Gibbs formulated statistical mechanics in terms of ‘ensembles’ of systems, which were collections of large numbers of copies of the system of interest, all identical except for their physical properties (volume, temperature, etc.). “In most of his elegant *Principles in Statistical Mechanics* of 1902, [Gibbs] described the underlying mechanical system in a formal manner, by generalised coordinates subject to Hamilton’s equations … He introduced and systematically studied the three fundamental ensembles of statistical mechanics: the micro-canonical, the canonical, and the grand-canonical ensemble (in which the number of molecules may vary). He examined the relations between these three ensembles and their analogies with thermodynamic systems, including fluctuation formulas” (Buchwald & Fox, p. 784). “A year before his death, Einstein paid Gibbs the highest compliment. When asked who were the greatest men, the most powerful thinkers he had known, he replied ‘Lorentz’, and added ‘I never met Willard Gibbs; perhaps, had I done so, I might have placed him beside Lorentz’” (Pais, p. 73). According to Emilio Segré (*From Falling Bodies to Radio Waves* (1984), p. 250), “even Jules-Henri Poincaré found [*Elementary Principles*] difficult to digest” – the present copy is presumably the one Poincaré puzzled over. Although reasonably well represented in institutional collections, this is a very rare book on the market. ABPC/RBH lists only one other copy in the last 35 years (and that copy lacked the dust-jacket).

*Provenance*: Jules-Henri Poincaré (1854-1912), presentation inscription on front free endpaper: ‘M. J.-H. Poincaré with the respects of the author’. Poincaré was “one of the greatest mathematicians and mathematical physicists at the end of 19th century. He made a series of profound innovations in geometry, the theory of differential equations, electromagnetism, topology, and the philosophy of mathematics” (Britannica). Although Poincaré did not work directly on statistical mechanics, his work on the three-body problem in celestial mechanics had an important impact upon it. In 1890, he proved his ‘recurrence theorem’, according to which mechanical systems governed by Hamilton’s equations will, after a sufficiently long time, return to a state very close to the initial state. This theorem created serious difficulties for any mechanical explanation of the laws of thermodynamics, as it apparently contradicts the Second Law, which says that large dynamical systems evolve irreversibly towards states with higher entropy, so that if one starts with a low-entropy state, the system will never return to it.

“Josiah Willard Gibbs was born in 1839: his father was at that time a professor of sacred literature at Yale University. Gibbs graduated from Yale in 1858, after he had compiled a distinguished record as a student. His training in mathematics was good, mainly because of the presence of H. A. Newton on the faculty. Immediately after graduation he enrolled for advanced work in engineering and attained in 1863 the first doctorate in engineering given in the United States. After remaining at Yale as tutor until 1866, Gibbs journeyed to Europe for three years of study divided between Paris, Berlin, and Heidelberg. Not a great deal of information is preserved concerning his areas of concentration during these years, but it is clear that his main interests were theoretical science and mathematics rather than applied science. It is known that at this time he became acquainted with Möbius’ work in geometry, but probably not with the systems of Grassmann or Hamilton. Gibbs returned to New Haven in 1869 and two years later was made professor of mathematical physics at Yale, a position he held until his death [in 1903].

“His main scientific interests in his first year of teaching after his return seem to have been mechanics and optics. His interest in thermodynamics increased at this time, and his research in this area led to the publication of three papers, the last being his now classic ‘On the Equilibrium of Heterogeneous Substances,’ published in 1876 and 1878 in volume III of the *Transactions of the Connecticut Academy. *This work of over three hundred pages was of immense importance. When scientists finally realized its scope and significance, they praised it as one of the greatest contributions of the century” (Crowe, p. 151).

“During the academic year 1889–1890 Gibbs announced ‘A short course on the a priori Deduction of Thermodynamic Principles from the Theory of Probabilities,’ a subject on which he lectured repeatedly during the 1890s” (DSB).

“Lord Rayleigh, writing on 5 June 1892 about an optical problem to Josiah Willard Gibbs in New Haven, Connecticut, concluded his letter as follows:

‘Have you ever thought of bringing out a new edition of, or a treatise founded upon, your ‘Equilibrium of Heterogeneous Substances.’ The original version, though now attracting the attention it deserves, is too difficult and too condensed for most, I might say all, readers. The result is that, as has happened to myself, the idea is not grasped until the subject has come up in ones mind more or less independently. I am sure that there is no one who could write a book on Thermodynamics like yourself.’

“Gibbs replied on 27 June 1892.

‘I thank you very much for your kind interest in my ‘Equilib. Het. Subst.’ I myself had come to the conclusion that the fault was that it was too *long*. I do not think that I had any sense of the value of time, of my own or others, when I wrote it. Just now I an trying to get ready for publication something on thermodynamics from the *a priori* point of view, or rather on ‘Statistical Mechanics’ of which the principal interest would be in its application to thermodynamics – in the line therefore of the work of Maxwell and Boltzmann. I do not know that I shall have anything particularly new in substance, but shall be contented if I can so choose my standpoint (as it seems to me possible) as to get a simpler view of the subject.’

“These lines doubtless indicate that the *Statistical Mechanics* of Gibbs was not conceived and written within nine or twelve months prior to his death, as seems to be the general impression, to some extent created by Gibbs’ own students.

“An abstract entitled, ‘The Fundamental Formulas of Statistical Mechanics, with Applications to Astronomy and Thermodynamics,’ appeared in the *Proceedings of the American Association for the Advancement of Science* in 1884. Although it did not contain much information, it at least testified to the fact that Gibbs had been working on the problem of the statistical foundations of thermodynamics at the time. Moreover, detailed notes in Gibbs’ own hand exist, some of them dated since 1892, extending over many years and dealing with the organization and details of the subject which Gibbs had undertaken to treat. Gibbs, who became a professor of mathematical physics at Yale in the year 1871, the same year in which Maxwell became Cavendish professor at Cambridge, read the scientific papers of his European colleagues from the very beginning. Thus it seems very probable that, starting already in 1873, Gibbs thought about the statistical foundations simultaneously with his writing of his papers on thermodynamics. It was also partially during these years that the important papers of Maxwell and Boltzmann on statistical mechanics were published.

“There was much entirely new in what Gibbs was undertaking to write, not just a continuation of the ‘line of Maxwell and Boltzmann’ …

“The immediate occasion to publish his formulation of the statistical approach top thermodynamics was provided by the request of the university administration to contribute a book to commemorate the bicentennial of Yale College in 1901 …

“The first sketch of the book numbered nine chapters only: I. General Motions and Principles; II. Application to the Theory of Errors; III. Application to the Theory of Integration; IV. On Statistical Equilibrium and a Distribution in Phase Called Canonical Ensemble, Microcanonical Distribution; V. Maximum and Minimum Properties and Inequalities; VI. Effect of Time; VII. Various Processes; VIII. Thermodynamics; IX. Systems of Molecules.

“In the final book, Chapters I, II and III were the same with respect to contents, but the mammoth Chapter IV was split into seven chapters, including: IV. On the Distribution-in-Phase called Canonical; V. Average Values in a Canonical Ensemble of Systems; VI. Extension-in-Configuration and Extension-in-Velocity; VII. Further Discussion of Averages in a Canonical Ensemble of Systems; VIII. On a Certain Important Function of the Energy of a System; IX. The Function * * and the Canonical Distribution; X. On a Distribution in Phase Called Microcanonical in which all the Systems Have the Same Energy …

“The writing of the *Elementary Principles* seems to have proceeded very systematically and rather rapidly. Starting from the principal question of how the statistical equilibrium is to be described, and how the equilibrium can be reached in time, Gibbs developed section after section of his book, of which Boltzmann said: ‘The task of systematizing this science, of compiling it into a large book, and of giving it a characteristic name, was executed by one of the greatest American scholars, and in regard to abstract thinking, purely theoretic investigation, perhaps the greatest, Willard Gibbs, the recently deceased professor of Yale University. He called this science statistical mechanics.’

“Boltzmann and Maxwell had dealt with specific systems, those of more or less ideal gas molecules. Gibbs turned away from the restrictions imposed by specific systems. He imagined ‘a great number of systems of the same nature, but differing in the configuration and velocities which they have at a given instant, and differing not merely infinitesimally, but it may be so as to embrace every conceivable combination of configurations and velocities.’ Thus he developed the *canonical ensemble*, in which the equilibrium distribution depends in the most simple manner on the energy, namely ‘the index of probability is a linear function of the energy,’ the divisor of the energy being essentially the temperature.

“Having found the most natural, elegant or even ‘geometrical’ representation of the equilibrium situation, from which the results of Maxwell and Boltzmann, e.g., the famous energy-distribution of the latter, followed immediately, Gibbs went on to discuss the Second Law … Gibbs took great pains in defining and deriving properly the relations of equilibrium thermodynamics from his statistical mechanics.

“The sections on the approach to equilibrium, on the other hand, are largely of a qualitative character. The question of whether a system, which is in an arbitrary state, develops in a direction which one calls equilibrium. Gibbs treated the example of an incompressible fluid to which, in some initial far-from-equilibrium distribution, colored fluid is added. By stirring in any sense [i.e., direction] mixing will finally be achieved.

“To summarize, one might say that Gibbs succeeded very well in founding equilibrium thermodynamics on a statistical basis. But due to the great difficulties that existed at that time, and still do, he could not in the same way describe the gradual changes of the macroscopic properties in their approach to equilibrium …

“As usual, Gibbs sent copies of his book on the *Elementary Principles in Statistical Mechanics* to numerous colleagues and institutions. In his Scientific Correspondence, we find a few letters expressing warm thanks. Most of the distinguished recipients of Gibbs’ book wrote that they had not as yet had the opportunity of studying the book in detail, and in fact it is doubtful if many did so later on. However, two of them took a deeper interest; they were H. A. Lorentz and M. Planck …

“[Planck] said that Gibbs had given him great joy by sending the new work on statistical mechanics.

‘I need hardly mention that I shall study your book with the greatest interest, because as far as I can see from a perusal and from the preface, it deals with a question which seems to me to be of the greatest interest: the introduction of the methods of probability calculus into mechanics, *independent of the application to thermodynamics*. Because it is only along this path that we might hope once to gain a deeper insight into the laws of irreversible processes, which according to my conviction will move more and more into the forefront of theoretical interest.’

“Planck found the book so valuable that he persuaded his student and collaborator E. Zermelo to translate it. Planck also used Gibbs’ method to reformulate his quantum condition for the oscillator, shifting the emphasis from the quantum of energy to the quantum of action. In the Maxwell-Gibbs phase-space, which has two dimensions for a linear harmonic oscillator, the surfaces of constant energies are similar ellipses whose areas are measured in multiples of Planck’s quantum of action *h*. It was this from Gibbs’ Elementary Principles that Planck obtained a deeper insight into the phenomena of heat radiation” (Mehra).

“Later physicists regarded Gibbs’ theory as so general that its principles are still valid in quantum theory. It needed some adjustments, but its elementary features could be still mobilized, as a bridge that has been repaired can still serve to cross the river. In

this sense we can say that Gibbs provided an infrastructure that made it possible to lead to quantum statistics” (Inaba, p. 104).

Norman 900 (this copy). Crowe, *A History of Vector Analysis*, 1967. Inaba, ‘Reading Elementary Principles: Gibbs and the origin of statistical mechanics,’ *Annalen der Physik* 527 (2015), pp. A102-A104. Mehra, ‘Josiah Willard Gibbs and the Foundations of Statistical Mechanics,’ *Foundations of Physics* 28 (1998), pp. 1785-1815.

8vo (223 x 148 mm), pp. [i-vii], viii-xviii, [1-3], 4-207, [1]. Original blue cloth, gilt-lettered spine, printed dust-jacket (slight wear to jacket). A fine copy.

Item #4733

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Price:
$17,500.00
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