Josiah Willard Gibbs and the Foundations of Statistical Mechanics

Foundations of Physics

Volumn 28, Issue 12, 1998, Pages 1785-1815

  Mehra, Jagdish ^a  

^a UNIVERSITY OF HOUSTON   (United States)

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EID: 0032281301     PISSN: 00159018     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1018890903755     Document Type: Article

References (167)

1. E. B. Wilson, "A Letter From Lord Rayleigh to J. Willard Gibbs and his Reply," Proc. Natl. Acad. Sci. USA 31, 34-38 (1954), especially pp. 35, 38. See also Robert John Strutt, Fourth Baron Rayleigh, Life of John William Strutt, Third Baron Rayleigh, University of Wisconsin Press, 1968, pp. 172-3. Letter No. 118 in Gibbs' Scientific Correspondence.
2. E. B. Wilson, "A Letter From Lord Rayleigh to J. Willard Gibbs and his Reply," Proc. Natl. Acad. Sci. USA 31, 34-38 (1954), especially pp. 35, 38. See also Robert John Strutt, Fourth Baron Rayleigh, Life of John William Strutt, Third Baron Rayleigh, University of Wisconsin Press, 1968, pp. 172-3. Letter No. 118 in Gibbs' Scientific Correspondence.
3. E. B. Wilson, "A Letter From Lord Rayleigh to J. Willard Gibbs and his Reply," Proc. Natl. Acad. Sci. USA 31, 34-38 (1954), especially pp. 35, 38. See also Robert John Strutt, Fourth Baron Rayleigh, Life of John William Strutt, Third Baron Rayleigh, University of Wisconsin Press, 1968, pp. 172-3. Letter No. 118 in Gibbs' Scientific Correspondence.
4. E. B. Wilson wrote in "Reminiscences of Gibbs by a Student and Colleague'," Sci. Monthly 32, 210-227 (1931), on p. 221: "I do not believe that Gibbs kept much in the way of notes. I imagine that he wrote the closely reasoned and highly mathematical 'Statistical Mechanics' out of his head (rather than from notes accumulated during previous years) between the time in the autumn of 1900 when he had agreed to produce the book and the time in the summer of 1901 when he delivered the manuscript. The reason for this belief lies in the fewness and in the character of the papers he left when he died - there was practically no Nachlass. And yet he was known to be working on a program of publication." On the other hand, H. A. Bumstead, in his obituary remarked [Am. J. Sci. 16, 187-203 (1903), see in particular p. 200] "So far from publishing partial results, he seldom, if ever, spoke of what he was doing until it was practically in its final and complete form. This was his chief limitation as a teacher of advanced students; he did not take them into his confidence with regard to his current work, and even when he lectured upon a subject in advance of its publication (as was the case for a number of years before the appearance of the Statistical Mechanics) the work was really complete except for a few finishing touches."
5. E. B. Wilson wrote in "Reminiscences of Gibbs by a Student and Colleague'," Sci. Monthly 32, 210-227 (1931), on p. 221: "I do not believe that Gibbs kept much in the way of notes. I imagine that he wrote the closely reasoned and highly mathematical 'Statistical Mechanics' out of his head (rather than from notes accumulated during previous years) between the time in the autumn of 1900 when he had agreed to produce the book and the time in the summer of 1901 when he delivered the manuscript. The reason for this belief lies in the fewness and in the character of the papers he left when he died - there was practically no Nachlass. And yet he was known to be working on a program of publication." On the other hand, H. A. Bumstead, in his obituary remarked [Am. J. Sci. 16, 187-203 (1903), see in particular p. 200] "So far from publishing partial results, he seldom, if ever, spoke of what he was doing until it was practically in its final and complete form. This was his chief limitation as a teacher of advanced students; he did not take them into his confidence with regard to his current work, and even when he lectured upon a subject in advance of its publication (as was the case for a number of years before the appearance of the Statistical Mechanics) the work was really complete except for a few finishing touches."
6. Proc. Am. Assoc. Adv. Sci. 33, 57-58 (1884), Abstract.
7. Gibbs' handwritten notes on Statistical Mechanics are in the rare books and manuscripts archives of Yale University [The Beinecke Rare Book Library].
8. See, for instance, the remarks in J. W. Gibbs, "Rudolf Julius Emanuel Clausius, Obituary," Proc. Am. Acad. 26, 458-465 (1889).
9. J. W. Gibbs, Trans. Conn. Acad. Arts Sci. 2, 382-404 (1873).
10. We shall return later on to a discussion of these papers that laid the basis of statistical thermodynamics.
11. It is a remarkable fact that Gibbs, who would pursue some of the most abstract ideas in the history of physics, graduated with a paper on an extremely practical problem in engineering. His thesis was "On the Form of the Teeth of Wheels in Spur Gearing." For another practical work, on "An Improved Railway Car Brake," Gibbs was granted a U.S. Patent on April 17, 1866. He also invented a device to control the fluctuations in the power output of steam engines, called the Gibbs Governor.
12. Of the three older sisters Anna, Eliza, and Julia, Eliza had died in 1849 and the younger sister, Emily, died in 1864. Their mother had died in 1855.
13. Darboux lectured on the mathematical theory of heat, Liouville on rational mechanics, and Serret on celestial mechanics. From the few notes that Gibbs kept one does not see a direct influence of Darboux on his later work in thermodynamics or statistical mechanics, but they provided a larger outlook. (L. P. Wheeler, Josiah Willard Gibbs, Yale University Press, New Haven, Connecticut, 1952, pp. 40-41.)
14. There Julia married Addison Van Name and returned to New Haven, whereas Anna remained with Gibbs.
15. Gibbs' reading of European scientific and mathematical literature was extensive. He read Lagrange, Laplace, Poisson, Fresnel, and Cauchy; the memoirs of Gauss, Jacobi, Hamilton, Dirksen, and Clebsch on mechanics; the articles of Lord Kelvin, Carl Neumann, and Hankel on electricity; the works of Fresnel, Green, Ampère, Hamilton, Lorentz, and Kirchhoff in electrodynamics and the theory of light. In his notebooks one does not find the names of Maxwell and Clausius (Wheeler, l.c., pp. 43-44).
16. Wheeler, l.c., pp. 56-57. There had been a complete reorganization of the university, proposed by a committee on which Gibbs' brother-in-law, Van Name, and the mathematician-astronomer Hubert A. Newton, also served. Gibbs was appointed to the new position probably at the strong advice of Newton, in spite of a lack of publications.
17. Wheeler, l.c., p. 58.
18. The first offer came from Bowdoin College in Brunswick, Maine, in 1873 (Wheeler, l.c., p. 59).
19. Poisson's text was the "Traité de Mécanique;" Fresnel's work dealt with diffraction and polarization (Wheeler, l.c., pp. 61, 62).
20. It is surprising that in a paper on "The Fundamental Formulae of Dynamics," Am. J. Math. 2, 49-64 (1879), related to the subject he had not used the vector method. The lectures on the Elements of Sector Analysis were arranged for the use of students in physics, and privately printed in 1881 and 1884. It is remarkable that in this private edition no application to physics or mechanics was made.
21. At that time, he still did not receive a salary at Yale. But this offer brought him to the attention of the administration at his university, and everything was mobilized to keep him. He was even offered a salary, starting at $2000 a year! Gibbs stayed on in New Haven (Wheeler, l.c., pp. 91-93).
22. A fund raising among his friends was held between October 1875 and May 1876, to see through the publication of the first 140 pages of his Equilibrium of Heterogeneous Substances (B. Jaffe, Men of Science in America, Chapter 13, New York, 1944).
23. His principal mailing list finally contained 160 names in the United States and Canada, 119 names in Great Britain, 104 in Germany, 57 in France, 11 in Italy, 10 in Holland and Belgium, 10 in Austria, 8 in Russia, 8 in Switzerland, 6 in Norway and Sweden, 5 in Poland, 3 in Denmark, 2 in India, and one each in Spain, Brazil, China, and Japan - with a total of 507. Of his first thermodynamic papers, he sent out 73 reprints; complimentary copies of his Elementary Principles went to 21 individuals and 22 scientific journals and societies (Wheeler, l.c., p. 94).
24. When Irving Fisher, a student of Gibbs, went to Berlin to continue his studies there in 1893, the mathematicians there did not know of anybody from the Yale faculty, but when he mentioned Gibbs, they said: "Gibbs, Gibbs, jawohl, ausgezeichnet." (J. G. Crowther, Famous American Men of Science, London, 1937, p. 231).
25. J. J. Thomson remembered in his memoirs: "When a new university was founded in 1887 the newly elected President came over to Europe to find professors. He came to Cambridge and asked me if I could tell him of anyone who would make a good professor of Molecular Physics. I said, 'You need not come to England for that; the best man you could get is an American, Willard Gibbs.' 'Oh,' he said, 'you mean Wolcott Gibbs,' mentioning the prominent American chemist. 'No, I don't,' I said, 'I mean Willard Gibbs,' and told him something about Gibbs' work. He sat thinking for a minute or two and then said, 'I'd like you to give me another name. Willard Gibbs can't be a man of much personal magnetism or I should have heard of him" (Crowther, l.c., pp. 231-232).
26. J. J. Thomson in a letter dated 22 November 1901.
27. The papers have been published by H. A. Bumstead and R. Gibbs Van Name in two volumes as The Scientific Papers of J. Willard Gibbs, New York, 1906 [Dover reprint, New York, 1961].
28. J. W. Gibbs, Trans. Conn. Acad. Arts Sci. 2, 309-342, 382-404 (1873).
29. Diagrams had been used for the first time by James Watt, who used pressure-volume diagrams to determine the work done by a steam engine.
30. The reason is that both volume and entropy are proportional to the quantities of substances (extensive properties), and are thus particularly useful to deal with heterogeneous substances, with which Gibbs already dealt in his first paper.
31. In fact, Gibbs gave a fairly extensive discussion of his choice of scientific terms here, which he seldom did, referring to Tait, Rankine and, most important to Clausius (See Dover reprint, Vol. I, p. 2).
32. In his lecture on the "Dynamical Evidence of the Molecular Constitution of Bodies," he said: "The purely thermodynamical relations of the different states of matter do not belong to our subject, as they are independent of particular theories about molecules. I must not, however, omit to mention a most important American contribution to this part of thermodynamics by Prof. Willard Gibbs, of Yale College, U.S., who has given us a remarkably simple and thoroughly satisfactory method of representing the relations of the different states of matter by means of a model. By means of this model, problems which had long resisted the efforts of myself and others may be solved at once" (quoted after Crowther, l.c., p. 267).
33. J. W. Gibbs, "On the Equilibrium of Heterogeneous Substances," Trans. Conn. Acad. III. 108-248 (October 1875-May 1876), pp. 343-524 (May 1877-July 1878).
34. (29) Wilhelm Ostwald wrote, "...I cannot abstain from expressing my admiration at this place. If you had published this work during a longer period in single papers in an accessible journal then you would be regarded at present not only in the small circle of those who know your work but in the public, as the greatest scientist in thermodynamics since Clausius, even for surpassing the latter with respect to the security and versatility of your physical judgement in many cases." Ostwald had learned about Gibbs in the 1880's. He wrote his first letter to Gibbs on 26 April 1887, mentioning that together with J. H. van't Hoff he was going to publish a journal in physical chemistry, and requested Gibbs to contribute to it, as he was one of the "decisive researchers" in that field. In the same letter he also asked Gibbs whether he would agree to an extended German translation of his fundamental papers. Ostwald returned to the question of publication again in the fall of 1888 and proposed to include it in the series of Classical Scientific Papers (Klassiker der Exakten Wissenschaften) which he was editing, the first of which, Helmholtz' Conservation of Force, appeared in 1889. Though Gibbs feared that "the call for a German edition would hardly justify the labor and expense of the translator and publisher," he agreed to it, and after some negotiation with another publisher who had also been interested in Gibbs papers, Ostwald succeeded in bringing out an edition of the Thermodynamische Studien (Leipzig 1892), including all the three memoirs from the Transactions of the Connecticut Academy. The first part of Gibbs' 1876 paper also appeared in a French translation, prepared by H. Le Chatelier, under the title Équilibre des Systèmes Chimiques (Paris, 1899).
35. After the publication of this paper [29], Gibbs occasionally received letters from engineers working in mining and other applied fields asking him for advice in applying his rules. In his scientific correspondence there are, among others, requests from the U.S. geologist, G. F. Becker, and the Austrian mining chemist, F. Wald, from the iron plant at Kladno (Bohemia). [Letters 178, 179 of the Scientific Correspondence.] The most important applications of the new rules were made by the Dutch School of J. D. van der Waals, especially by B. Roozeboom in the late 1890s, by E. Cohen and A. Smits, and finally by F. A. H. Schreinemakers. Their work, as well as the papers and books by J. H. van't Hoff, W. Ostwald, and numerous others, made Gibbs' ideas available to the scientific community, because his own condensed and abstract style was not easy to read by people who worked in applied research. The first "popularisation" of Gibbs' results was contained in a booklet by W. Meyerhoffer from Vienna, entitled Die Phasenregel und ihre Anwendungen (Leipzig and Vienna, 1893), published one year after Ostwaldts translation, entitled Thermodynamisclie Studien, had come out. In a letter to Gibbs, dated 23 June 1893 (No. 150 of the Scientific Correspondence) W. Meyerhoffer announced his new booklet, saying that the "phase rule is one of the most fruitful generalizations in theoretical chemistry."
36. The first appeared in Am. J. Sci. (3) 18, 277-293, 371-387 (1879), treating the dependence of the vapor density of the partially dissociated substances on the temperature. The short note on osmotic pressure was published in Nature 55, 461-462 (1897). There is another note in Science 1, 160 (1883), "On an Alleged Exception to the Second Law of Thermodynamics," which disproved, of course, an assertion made in this regard. Gibbs also wrote two letters to Professor Oliver J. Lodge on "Electrochemical Thermodynamics," in which he drew consequences from his theory
37. (Rep. Br. Assoc. Adv. Sci. 1886, 388, 389; 1888, 343-346).
38. (Rep. Br. Assoc. Adv. Sci. 1886, 388, 389; 1888, 343-346).
39. Hamilton presented his new algebra in November 1843 before the Royal Irish Academy, and Grassmann published his Ausdehnungslehre in 1844.
40. J. W. Gibbs, "On Multiple Algebra," Proc. Am. Assoc. Adv. Sci. 35, 37-66 (1886). In this lecture, as in several notes to Nature, Gibbs expressed his concern with developing the more flexible and practical method of Grassmann, the unrecognized German mathematician and later on Sanskrit scholar.
41. See "On the Role of Quaternions in the Algebra of Vectors," Nature 43, 511-513 (1891); "Quaternions and the Ausdehnungslehre," Nature 44, 79-82 (1891); "Quaternions and the Algebra of Vectors," Nature 47, 463-464 (1893); "Quaternions and Vector Analysis," Nature 48, 364-367 (1893).
42. See "On the Role of Quaternions in the Algebra of Vectors," Nature 43, 511-513 (1891); "Quaternions and the Ausdehnungslehre," Nature 44, 79-82 (1891); "Quaternions and the Algebra of Vectors," Nature 47, 463-464 (1893); "Quaternions and Vector Analysis," Nature 48, 364-367 (1893).
43. See "On the Role of Quaternions in the Algebra of Vectors," Nature 43, 511-513 (1891); "Quaternions and the Ausdehnungslehre," Nature 44, 79-82 (1891); "Quaternions and the Algebra of Vectors," Nature 47, 463-464 (1893); "Quaternions and Vector Analysis," Nature 48, 364-367 (1893).
44. See "On the Role of Quaternions in the Algebra of Vectors," Nature 43, 511-513 (1891); "Quaternions and the Ausdehnungslehre," Nature 44, 79-82 (1891); "Quaternions and the Algebra of Vectors," Nature 47, 463-464 (1893); "Quaternions and Vector Analysis," Nature 48, 364-367 (1893).
45. 35a. In the preface to the third edition of Tait's Quaternions, there had occurred the following passage: "Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann." To this, Gibbs replied in a most effective and dignified manner as follows: "The merits or demerits of a pamphlet printed for private distribution a good many years ago do not constitute a subject of any great importance, but the assumptions implied in the sentence quoted are suggestive of certain reflections and inquiries which are of broader interest, and seem not untimely at a period when the methods and results of the various forms of multiple algebra are attracting so much attention. It seems to be... (see Vol. 2, p. 155) uncouth."
46. See the letters from the influential German textbook writer A. Föppl (dated 6 January 1902, No. 138 of the Correspondence) and by J. J. Thomson (No. 123 of the Correspondence; it is not entirely clear whether this letter refers to the 1884 or the 1901 book; it refers most probably to the 1884 treatise).
47. "Notes on the Electromagnetic Theory of Light "I. On Double Refraction and the Dispension of Colors in Perfectly Transparent Aledia," Am. J. Sci. 23, 262-275 (1882); "II. On the Double Refraction in Perfectly Transparent Media which Exhibit the Circular Polarization," Am. J. Sci. 23, 460-476 (1882); "III. On the General Equations of Mono- Chromatic Light in Media of Every Degree of Transparency," Am. J. Sci. 25, 107-118 (1883); "A Comparison of the Elastic and Electrical Theories of Light, with Respect to the Law of Double Refraction and the Dispersion of Colors," Am. J. Sci. 35, 467-475 (1888); "A Comparison of the Electrical Theory of Light with Sir William Thomson's Theory of a Quasi-Labile Ether," Am. J. Sci. 37, 129-144 (1889).
48. "Notes on the Electromagnetic Theory of Light "I. On Double Refraction and the Dispension of Colors in Perfectly Transparent Aledia," Am. J. Sci. 23, 262-275 (1882); "II. On the Double Refraction in Perfectly Transparent Media which Exhibit the Circular Polarization," Am. J. Sci. 23, 460-476 (1882); "III. On the General Equations of Mono- Chromatic Light in Media of Every Degree of Transparency," Am. J. Sci. 25, 107-118 (1883); "A Comparison of the Elastic and Electrical Theories of Light, with Respect to the Law of Double Refraction and the Dispersion of Colors," Am. J. Sci. 35, 467-475 (1888); "A Comparison of the Electrical Theory of Light with Sir William Thomson's Theory of a Quasi-Labile Ether," Am. J. Sci. 37, 129-144 (1889).
49. "Notes on the Electromagnetic Theory of Light "I. On Double Refraction and the Dispension of Colors in Perfectly Transparent Aledia," Am. J. Sci. 23, 262-275 (1882); "II. On the Double Refraction in Perfectly Transparent Media which Exhibit the Circular Polarization," Am. J. Sci. 23, 460-476 (1882); "III. On the General Equations of Mono-Chromatic Light in Media of Every Degree of Transparency," Am. J. Sci. 25, 107-118 (1883); "A Comparison of the Elastic and Electrical Theories of Light, with Respect to the Law of Double Refraction and the Dispersion of Colors," Am. J. Sci. 35, 467-475 (1888); "A Comparison of the Electrical Theory of Light with Sir William Thomson's Theory of a Quasi-Labile Ether," Am. J. Sci. 37, 129-144 (1889).
50. "Notes on the Electromagnetic Theory of Light "I. On Double Refraction and the Dispension of Colors in Perfectly Transparent Aledia," Am. J. Sci. 23, 262-275 (1882); "II. On the Double Refraction in Perfectly Transparent Media which Exhibit the Circular Polarization," Am. J. Sci. 23, 460-476 (1882); "III. On the General Equations of Mono- Chromatic Light in Media of Every Degree of Transparency," Am. J. Sci. 25, 107-118 (1883); "A Comparison of the Elastic and Electrical Theories of Light, with Respect to the Law of Double Refraction and the Dispersion of Colors," Am. J. Sci. 35, 467-475 (1888); "A Comparison of the Electrical Theory of Light with Sir William Thomson's Theory of a Quasi-Labile Ether," Am. J. Sci. 37, 129-144 (1889).
51. "Notes on the Electromagnetic Theory of Light "I. On Double Refraction and the Dispension of Colors in Perfectly Transparent Aledia," Am. J. Sci. 23, 262-275 (1882); "II. On the Double Refraction in Perfectly Transparent Media which Exhibit the Circular Polarization," Am. J. Sci. 23, 460-476 (1882); "III. On the General Equations of Mono- Chromatic Light in Media of Every Degree of Transparency," Am. J. Sci. 25, 107-118 (1883); "A Comparison of the Elastic and Electrical Theories of Light, with Respect to the Law of Double Refraction and the Dispersion of Colors," Am. J. Sci. 35, 467-475 (1888); "A Comparison of the Electrical Theory of Light with Sir William Thomson's Theory of a Quasi-Labile Ether," Am. J. Sci. 37, 129-144 (1889).
52. 37a. Am. J. Math. 2, 49-64 (1879).
53. 37b. Mem. Nat. Acad. Sci. 4, 79-104 (1889).
54. Elementary Principles in Statistical Mechanics, Yale University Press, 1902.
55. 38a. In his article, "J. Willard Gibbs, A Brief Biography and a Summary of his Contributions to Chemistry," Can. Chem. Metallurgy 8, 215-217 (1924), R. C. Cantelo wrote: "Anyone who has ever tried (and the writer uses 'tried' advisedly) to read Gibbs is filled with a feeling 'of admiration and despair.' At the same time there is the wish that Gibbs had thought a little more of posterity, and while taking nothing from the rigidity of his mathematics, yet had enlarged upon his ideas with the view of helping the earnest student over the more difficult points of the developments. A man after reading Gibbs flies to Planck for relief."
56. Letter No. 15 of the Scientific Correspondence. The letter is dated June 5, 1889. R. Clausius, a member of the Academy since 1873, had died on August 24, 1888.
57. Letter No. 16, Scientific Correspondence.
58. Letter No. 17, Scientific Correspondence.
59. See, for instance, M. J. Klein, "Gibbs on Clausius," Hist. Stud. Phys. Sci. 1, 127-149 (1969).
60. See Gibbs' obituary, "Rudolf Julius Emanuel Clausius," Proc. Am. Acad. 16, 458-465 (1889), reprinted in The Scientific Papers, Vol. II, Dover, especially p. 262.
61. See Gibbs' obituary, "Rudolf Julius Emanuel Clausius," Proc. Am. Acad. 16, 458-465 (1889), reprinted in The Scientific Papers, Vol. II, Dover, especially p. 262.
62. Ref. 43, p. 263.
63. R. Clausius, Pogg. Ann. 100, 353 (1857).
64. 45a. R. Clausius, "On the Average Paths which Are Made by the Single Molecules," Pogg. Ann. 105, 239 (1858), and "On the Heat Conduction of Gaseous Bodies," Pogg. Ann. 115, 1 (1862). See also "On the Application of the Theorem Concerning the Equivalence of the Changes to the Internal Work," Pogg. Ann. 116, 73 (1862).
65. 45a. R. Clausius, "On the Average Paths which Are Made by the Single Molecules," Pogg. Ann. 105, 239 (1858), and "On the Heat Conduction of Gaseous Bodies," Pogg. Ann. 115, 1 (1862). See also "On the Application of the Theorem Concerning the Equivalence of the Changes to the Internal Work," Pogg. Ann. 116, 73 (1862).
66. 45a. R. Clausius, "On the Average Paths which Are Made by the Single Molecules," Pogg. Ann. 105, 239 (1858), and "On the Heat Conduction of Gaseous Bodies," Pogg. Ann. 115, 1 (1862). See also "On the Application of the Theorem Concerning the Equivalence of the Changes to the Internal Work," Pogg. Ann. 116, 73 (1862).
67. Ref. 43, p. 265.
68. It is interesting to compare the treatment given by R. Clausius in his paper "On the Reduction of the Second Law of the Mechanical Theory of Heat to General Mechanical Principles," Pogg. Ann. 142, 433 (1870), with Boltzmann's derivation.
69. This paper had been read at the meeting at the British Association for the Advancement of Science at Aberdeen on 21 September 1859, and was published under the title, "On the Motions and Collisions of Perfectly Elastic Spheres," Phil. Mag. (4) 19, 19 (1860). Some people assume that the study of Saturn's rings, which had to do with the irregularity of mechanical motion, brought Maxwell to study the kinetic theory of heat.
70. Ref. 48, p. 20.
71. Ref. 48, p. 23.
72. See Maxwell's papers, "On the Dynamical Theory of Gases," Trans. Lond. Phil. Soc. 157, 49 (1866), and "On Boltzmann's Theorem as the Average Distribution of Energy in a System of Matter Points," Trans. Camb. Phil. Soc. 12, 547 (1879).
73. See Maxwell's papers, "On the Dynamical Theory of Gases," Trans. Lond. Phil. Soc. 157, 49 (1866), and "On Boltzmann's Theorem as the Average Distribution of Energy in a System of Matter Points," Trans. Camb. Phil. Soc. 12, 547 (1879).
74. L. Boltzmann, Sitz. Ber. Kais. Akad. Wiss. (Wien) 53, 195-220 (1866).
75. L. Bollzmann, Wien. Ber. 58, 517-560 (1868).
76. L. Boltzmann, Wien. Ber. 63, 397 418 (1871).
77. The entropy function (or the quantity H) is referred to as E in Wiener Berichte 66, 275, 370 (1872).
78. L. Boltzmann, Wien. Ber. 76, 373-435 (1877). Boltzmann made further comments on Maxwell's article, "On Boltzmann's Theorem as the Average Distribution of Energy in a System of Material Points," Phil. Mag. (5) 14, 299-312 (1882). Boltzmann stated (p. 300): "There is a difference in method between Maxwell and Boltzmann, in as much as Boltzmann measures the probability of a condition by the time during which the system possesses this condition on the average, whereas Maxwell considers innumerable similarly constituted systems with all possible initial conditions. The ratio of the number of systems which are in that condition to the total number of systems defines the probabilities in question."
79. L. Boltzmann, Wien. Ber. 76, 373-435 (1877). Boltzmann made further comments on Maxwell's article, "On Boltzmann's Theorem as the Average Distribution of Energy in a System of Material Points," Phil. Mag. (5) 14, 299-312 (1882). Boltzmann stated (p. 300): "There is a difference in method between Maxwell and Boltzmann, in as much as Boltzmann measures the probability of a condition by the time during which the system possesses this condition on the average, whereas Maxwell considers innumerable similarly constituted systems with all possible initial conditions. The ratio of the number of systems which are in that condition to the total number of systems defines the probabilities in question."
80. Scientific Papers of J. C. Maxwell, Dover edition, p. 715.
81. See the later papers of Boltzmann, e.g., "On Some Questions of the Kinetic Theory of Gases," Wien. Ber. 96, 891-918 (1887), and "On the Mechanical Analogy of the Heat Equilibrium of Two Bodies which are in Contact" (with G. H. Bryan), Wien. Ber. 103, 1125-2234 (1894), and Proc. Phys. Soc. (Lond.) 13, 485-493 (1895). The theory of gases was summarized by Boltzmann in the two volumes of Lectures on Gas Theory (Leipzig 1896, 1898; translated by S. G. Brush, Berkeley, 1964).
82. See the later papers of Boltzmann, e.g., "On Some Questions of the Kinetic Theory of Gases," Wien. Ber. 96, 891-918 (1887), and "On the Mechanical Analogy of the Heat Equilibrium of Two Bodies which are in Contact" (with G. H. Bryan), Wien. Ber. 103, 1125-2234 (1894), and Proc. Phys. Soc. (Lond.) 13, 485-493 (1895). The theory of gases was summarized by Boltzmann in the two volumes of Lectures on Gas Theory (Leipzig 1896, 1898; translated by S. G. Brush, Berkeley, 1964).
83. See the later papers of Boltzmann, e.g., "On Some Questions of the Kinetic Theory of Gases," Wien. Ber. 96, 891-918 (1887), and "On the Mechanical Analogy of the Heat Equilibrium of Two Bodies which are in Contact" (with G. H. Bryan), Wien. Ber. 103, 1125-2234 (1894), and Proc. Phys. Soc. (Lond.) 13, 485-493 (1895). The theory of gases was summarized by Boltzmann in the two volumes of Lectures on Gas Theory (Leipzig 1896, 1898; translated by S. G. Brush, Berkeley, 1964).
84. See the later papers of Boltzmann, e.g., "On Some Questions of the Kinetic Theory of Gases," Wien. Ber. 96, 891-918 (1887), and "On the Mechanical Analogy of the Heat Equilibrium of Two Bodies which are in Contact" (with G. H. Bryan), Wien. Ber. 103, 1125-2234 (1894), and Proc. Phys. Soc. (Lond.) 13, 485-493 (1895). The theory of gases was summarized by Boltzmann in the two volumes of Lectures on Gas Theory (Leipzig 1896, 1898; translated by S. G. Brush, Berkeley, 1964).
85. L. Boltzmann, "Reply to the Heat-Theoretical Considerations of E. Zermelo," Wiedemann's Ann. Phys. 57, 773-784 (1896), "On Zermelo's Paper 'On the Mechanical Explanation of Irreversible Processes,'" Wiedemann's Ann. Phys. 60, 392-398 (1897), and "On a Mechanical Law of H. Poincaré," Wien. Ber. 106, 12-20 (1897).
86. L. Boltzmann, "Reply to the Heat-Theoretical Considerations of E. Zermelo," Wiedemann's Ann. Phys. 57, 773-784 (1896), "On Zermelo's Paper 'On the Mechanical Explanation of Irreversible Processes,'" Wiedemann's Ann. Phys. 60, 392-398 (1897), and "On a Mechanical Law of H. Poincaré," Wien. Ber. 106, 12-20 (1897).
87. L. Boltzmann, "Reply to the Heat-Theoretical Considerations of E. Zermelo," Wiedemann's Ann. Phys. 57, 773-784 (1896), "On Zermelo's Paper 'On the Mechanical Explanation of Irreversible Processes,'" Wiedemann's Ann. Phys. 60, 392-398 (1897), and "On a Mechanical Law of H. Poincaré," Wien. Ber. 106, 12-20 (1897).
88. L. Boltzmarm, "On the So-Called H-Curve," Math. Ann. 50, 325-332 (1898).
89. See the obituary of Rudolf Julius Emanuel Clausius, in Gibbs' Scientific Papers, Vol. II, p. 266.
90. 61a. J. W. Gibbs, "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics," Proc. Am. Assoc. Adv. Sci. 33, 57-58 (1884).
91. Gibbs, Scientific Papers, Vol. II, p. 16.
92. Gibbs, Scientific Papers, Vol. II, pp. 149-154.
93. At the Beineke Rare Book Library at Yale University, there are several boxes (containing folders) of Gibbs' papers, manuscripts, and Scientific Correspondence. Not all of his manuscripts have been preserved, and what exists does not form a continuous set giving information about the problems on which Gibbs was working at a particular time. Some of the manuscripts are numbered and dated, with large gaps in between. Obviously Gibbs kept only few of his notes, throwing away the pages he did not need any more. Some of the papers which he preserved are rather informative, filling gaps in our knowledge about how Gibbs, who published so rarely, worked. It is perhaps only accidental that his working draft for the Elementary Principles has been preserved, because he died shortly after the book was completed. We shall quote from these sources occasionally, to correct and supplement the opinions hitherto held on the writing of the Elementary Principles. For the permission to use the materials I am very grateful to the Librarian of the Beineke Rare Book Library of Yale University.
94. He referred (see p. 11 of the Dover edition, Chapter I), to an extension-in-phase which is constant in time, and added the footnote: "If we regard a phase as represented by a point in space of 2n dimensions, the change which takes place in the course of time in our ensemble of systems will be represented by a current in such space. This current will be steady as long as the external coordinates are not varied. In any case the current will satisfy a law which in its various expressions is analogous to the hydrodynamic law which may be expressed by the phrases conservation of volumes or conservation of density about a moving point, or by the equation formula presented The analogue in statistical mechanics is,..."
95. See ref. 1.
96. See ref. 1.
97. Gibbs' handwritten notes, ref. 64.
98. We find this title on several sheets of his notes. L. P. Wheeler, in his book Josiah Willard Gibbs, p. 143, reported that Gibbs read a paper "at the New Haven meeting [of the National Academy of Sciences] in 1894, 'On a Certain Theorem in Theoretical Mechanics,' which formed part of the material for his Statistical Mechanics, published in 1902." We would think that this was essentially the same as Gibbs' note "On the Fundamental Formula of Statistical Mechanics with Application to Astronomy and Thermodynamics," given at the meeting of the American Association for the Advancement of Science in 1884. The sheets with the title "Molecular Mechanics and Thermodynamics" or "Molecular Thermodynamics" are not dated, but they were clearly written in preparation tor the final title. This assumption is substantiated by the content of the corresponding notes.
99. Some of his friends thought that this labor on the book broke him down physically, leading to his death (Wheeler, l.c., p. 196).
100. The final organization appears on a handwritten sheet dated June 1901.
101. Chapter numbers refer to the old organization.
102. The remarks in italics are the ones which Gibbs added later on in answer to his questions.
103. Gibbs had referred to a paper on the theory of solutions by Lord Rayleigh, Nature 55, 253 (1897). In the book, he did not elaborate on this point.
104. L. Boltzmann, "The Relations of Applied Mechanics," lecture at the St. Louis Conference in 1904, pp. 601-602.
105. From the Preface to the Elementary Principles, ref. 38, p. vii.
106. Elementary Principles, Dover edition, pp. 33-34.
107. At another point he remarked in his notes: "The increase of entropy is not really so much increase in time, as departure from the determining time."
108. He devoted only the very short Chapter XII to the whole subject, not even mentioning Boltzmann's collision equation. In contrast to the other chapters this one does not have a single equation. Nevertheless this chapter is regarded as Gibbs' profoundest in the book, and the questions he raised there have not yet been answered completely. (See my discussions with M. Kac and G. E. Uhlenbeck, "Conceptual Development of Statistical Mechanics," Harry Ransom Center, The University of Texas at Austin, 1973).
109. The comparison of the extension-in-phase with an incompressible fluid played an important role in Gibbs thinking, and he referred to it at various times in his handwritten notes. In the book he Summarized (p. 147 of the Dover edition): "The analogy between the motion of an ensemble of systems in an extension-in-phase and a steady current in an incompressible liquid, and the diagrammatic representation of the case of one degree of freedom, which appeals to our geometrical intuitions, may be sufficient to show the conservation of density in phase, which involves the conservation of the average value of the index of probability of phase, inconsistent with an approach to a limiting condition in which that average value is less. We might perhaps fairly infer from such considerations as have been adduced that an approach to a limiting condition of statistical equilibrium is the general rule, when the initial condition is not of that character."
110. The letter, dated April 1st, 1902, is No. 185 in the Scientific Correspondence.
111. See, e.g., H. A. Lorentz, Les Théories Statistiques en Thermodynamique (lectures in November 1912 at the Collège de France, edited by L. Dunoyer, Leipzig, 1916), and Lectures in Theoretical Physics, Vol. II (London, 1927). We might also mention that Lorentz in his writings almost completely took over the thermodynamical notations of Gibbs, including η for entropy etc.
112. See, e.g., H. A. Lorentz, Les Théories Statistiques en Thermodynamique (lectures in November 1912 at the Collège de France, edited by L. Dunoyer, Leipzig, 1916), and Lectures in Theoretical Physics, Vol. II (London, 1927). We might also mention that Lorentz in his writings almost completely took over the thermodynamical notations of Gibbs, including η for entropy etc.
113. Letter No. 168 of the Scientific Correspondence.
114. It was published in Leipzig in 1905.
115. M. Planck, Vorlesungen über Wärmestrahlung, (Leipzig, 1906). The first English translation by M. Masius appeared in Philadelphia in 1914, as Theory of Heat Radiation.
116. The idea of the phase-integral played an important role in the later formulation of the quantum conditions, especially in the hands of Sommerfeld.
117. In his handwritten notes, Gibbs came back at various placed to the statistical mechanics of the ether, without taking definite steps to discuss it. The question is addressed in the preface of Elementary Principles, on p. x: "Difficulties of this kind have deterred the author from attempting to explain the mysteries of nature, and have forced him to be contented with the more modest aim of deducing some of the more obvious propositions relating to the statistical branch of mechanics."
118. W. Nernst, "Commemorating the 100th Birthday of Willard Gibbs," Naturwissenschaften, 27, 393-394 (1939), in particular, p. 394.
119. It appeared in the second volume of the Commentary on the Scientific Writings of J. Willard Gibbs (New Haven, 1936), pp. 521-584, especially p. 521.
120. A. Einstein, "Kinetic Theory of Thermal Equilibrium and the Second Law of Thermodynamics," Ann. Phys. (Leipzig) 9, 417 (1902); "A Theory of the Foundations of Thermodynamics," Ann. Phys. (Leipzig) 11, 170 (1903); "On the General Molecular Theory of Heat," Ann. Phys. (Leipzig) 14, 354 (1904).
121. A. Einstein, "Kinetic Theory of Thermal Equilibrium and the Second Law of Thermodynamics," Ann. Phys. (Leipzig) 9, 417 (1902); "A Theory of the Foundations of Thermodynamics," Ann. Phys. (Leipzig) 11, 170 (1903); "On the General Molecular Theory of Heat," Ann. Phys. (Leipzig) 14, 354 (1904).
122. A. Einstein, "Kinetic Theory of Thermal Equilibrium and the Second Law of Thermodynamics," Ann. Phys. (Leipzig) 9, 417 (1902); "A Theory of the Foundations of Thermodynamics," Ann. Phys. (Leipzig) 11, 170 (1903); "On the General Molecular Theory of Heat," Ann. Phys. (Leipzig) 14, 354 (1904).
123. A. Einstein, ref. 90. l.c., p. 417.
124. In "Autobiographical Notes," in Albert Einstein: Philosopher-Scientist, P. A. Schilpp, ed. (Harper, New York, 1959), Vol. I, pp. 1-95, Einstein wrote (p. 47): "Not acquainted with the earlier investigations of Boltzmann and Gibbs, which had appeared earlier and actually exhausted the subject, I developed the statistical mechanics and the molecular-kinetic theory of thermodynamics which was based on the former."
125. It was introduced by L. Boltzmann, Wien. Ber. 63, 712 (1871).
126. See Einstein's lecture at the Gesellschaft Deutscher Naturforscher at Salzburg in 1909, "Development of our Concepts Concerning the Nature and Constitution of Radiation," Phys. Z. 10, 817-825 (1909).
127. Elementary Principles, p. 74. Boltzmann stated a similar opinion, on p. 318 of the Lectures on Gas Theory (English edition).
128. Elementary Principles, p. 74. Boltzmann stated a similar opinion, on p. 318 of the Lectures on Gas Theory (English edition).
129. Even in the review article of P. and T. Ehrenfest, "The Conceptual Foundations of the Statistical Approach in Mechanics," Encyl. Math. Wiss., Vol. IV, 2 II (Leipzig, 1911), translated by M. Moravcsik (Ithaca, N.Y., 1959); Einstein's papers are only quoted in connection with Gibbs' book.
130. P. Hertz, Ann. Phys. (4) 33, 225, 537 (1910). See also Hertz' article "Statistical Mechanics," Weber-Gans' Repertorium der Physik, Vol. I/II (Leipzig, 1916). Hertz criticized some of Einstein's derivations from the conceptual point of view (see, e.g., the remarks concerning Einstein's proof of the laws of thermal equilibrium). Einstein admitted in a note that his considerations had not been formulated sufficiently carefully, Ann. Phys. 34, 175-176 (1911).
131. For instance, the systems might be given by equations of motions.
132. R. T. Tolman, The Principles of Statistical Mechanics (Oxford, 1938).
133. A. Einstein, Ann. Phys. (Leipzig) 34, 176 (1911).
134. A. Einstein, Naturwissenschaften 4, 480-481 (1916).
135. Einstein, despite his remarks quoted in ref. 100, was perhaps so dissatisfied with Gibbs that he did not refer to him at any place positively. For instance, he lectured on statistical mechanics at the University of Berlin in 1917/18, without ever referring to Gibbs. (In the lecture notes taken of these lectures, there is no mention of Gibbs at all.) (Einstein Archive, Hebrew University of Jerusalem)
136. A. Münster, "Principles of Statistical Mechanics," Handbucd der Physik, S. Flügge, ed., Vol. III, Part 2 (Berlin, 1959), pp. 189-190 [in German].
137. This Festschrift, edited by Stefan Meyer (Leipzig, 1904), was dedicated to Ludwig Boltzmann on his 60th birthday and contained 117 contributions from friends and colleagues throughout the world, including those by W. Wien, J. H. van't Hoff, J. D. van der Waals, J. Stark, E. Mach, V. Bjerkness, J. E. Trevor (Ithaca, N.Y.), S. H. Burbury, J. Larmor, F. Hosenöhrl, H. A. Lorentz, H. Righi, A. Sommerfeld, S. Arrhenius, W. Nernst, and H. Nagaoka. A particularly important article was contributed by M. v. Smoluchowski, "On Irregularities in the Distribution of Gas Molecules and their. Influence on the Entropy of the Equation of State." This contribution, which was received on September 28, 1903, constituted the first theoretical study which finally led to the quantitative study of fluctuations, (it was applied later on in the theory of critical opalescence proposed by Smoluchowski in 1908 and improved by Einstein in 1910).
138. Grand canonical ensembles can describe systems with a varying number of particles. In this context Gibbs also introduced generic phases which allowed for the identity of particles and solved the mixing paradox.
139. M. Planck, l.c., pp. 121-122.
140. Note 96. Boltzmann, who had already prepared, together with J. Nabl, an article on the "Kinetic Theory of Matter" for the Encyclopaedia (Vol. V, part 1) in 1905, was persuaded to write on the foundations of statistical mechanics by F. Klein, who had threatened that otherwise E. Fermelo, Boltzmann's arch-critic, would get the job. Boltzmann, however, committed suicide on 6 September 1906. Toward the end of October 1906 Ehrenfest was invited by Klein to give a talk at the Mathematical Colloquium in Göttingen, and he talked on statistical mechanics. Hence he was invited to write the encyclopedia article alone, or in collaboration with his wife [M. J. Klein, Paul Ehrenfest, Vol. 1 (Amsterdam, 1970), p. 81]. The writing of the article proved to be a difficult work and it was not completed until 1911. This article, however, was to win Ehrenfest the succession to the chair of H. A. Lorentz in Leiden. (The letter is dated May 6, 1912. See M. J. Klein, l.c., p. 181.)
141. Note 96. Boltzmann, who had already prepared, together with J. Nabl, an article on the "Kinetic Theory of Matter" for the Encyclopaedia (Vol. V, part 1) in 1905, was persuaded to write on the foundations of statistical mechanics by F. Klein, who had threatened that otherwise E. Fermelo, Boltzmann's arch-critic, would get the job. Boltzmann, however, committed suicide on 6 September 1906. Toward the end of October 1906 Ehrenfest was invited by Klein to give a talk at the Mathematical Colloquium in Göttingen, and he talked on statistical mechanics. Hence he was invited to write the encyclopedia article alone, or in collaboration with his wife [M. J. Klein, Paul Ehrenfest, Vol. 1 (Amsterdam, 1970), p. 81]. The writing of the article proved to be a difficult work and it was not completed until 1911. This article, however, was to win Ehrenfest the succession to the chair of H. A. Lorentz in Leiden. (The letter is dated May 6, 1912. See M. J. Klein, l.c., p. 181.)
142. Note 96. Boltzmann, who had already prepared, together with J. Nabl, an article on the "Kinetic Theory of Matter" for the Encyclopaedia (Vol. V, part 1) in 1905, was persuaded to write on the foundations of statistical mechanics by F. Klein, who had threatened that otherwise E. Fermelo, Boltzmann's arch-critic, would get the job. Boltzmann, however, committed suicide on 6 September 1906. Toward the end of October 1906 Ehrenfest was invited by Klein to give a talk at the Mathematical Colloquium in Göttingen, and he talked on statistical mechanics. Hence he was invited to write the encyclopedia article alone, or in collaboration with his wife [M. J. Klein, Paul Ehrenfest, Vol. 1 (Amsterdam, 1970), p. 81]. The writing of the article proved to be a difficult work and it was not completed until 1911. This article, however, was to win Ehrenfest the succession to the chair of H. A. Lorentz in Leiden. (The letter is dated May 6, 1912. See M. J. Klein, l.c., p. 181.)
143. In their joint paper, "Remark on the Theory of Entropy Increase in the "Statistical Mechanics' of W. Gibbs," Wien. Ber. 115, 89 (1906), the Ehrenfests pointed out that Gibbs had not consistently maintained the distinction between the 'fine grained density' and its average over a small but finite volume element, the 'coarse grained density.' Thus his theory of irreversibility lacked foundation. (M. Klein, l.c., p. 133). Epstein took a different point of view in the article, "Application of Gibbs' Statistics," in the Commentary on the Scientific Writings of J. Willard Gibbs, p. 513. Experimentally, he said, we can measure only macro-elements of a phase space in systems with many degrees of freedom. Therefore one cannot tell a coarse from a fine density. And a coarse density always includes the tendency to equilibrium.
144. On p. 62 of the critical English translation, ref. 96, the statements of Gibbs and Boltzmann concerning the increase of entropy for irreversible processes in an isolated system are contrasted. Whereas Boltzmann let the quantity H control the motion of the system towards equilibrium, Gibbs does not have an equally well-defined quantity for which one can prove a quasi-periodic behavior.
145. M. J. Klein called Chapter XIV of the Elementary Principles, entitled "Discussion of Thermodynamic Analogies," the culmination of Gibbs' work, (Klein, l.c., p. 132). The Ehrenfests found Chapter XII, "On the Motion of Systems and Ensembles of Systems through Long Period of Time," rather preliminary.
146. D. Enskog, Phys. Z. 12, 56, 533 (1911); Inaugural Dissertation. Uppsala 1917; S. Chapman, Phil. Trans. R. Soc. Lond. A 216, 279 (1916); 217, 115 (1917); Proc. R. Soc. Lond. A 93, 1 (1916). See also D. Hilbert, Math. Ann. 72, 562 (1912).
147. D. Enskog, Phys. Z. 12, 56, 533 (1911); Inaugural Dissertation. Uppsala 1917; S. Chapman, Phil. Trans. R. Soc. Lond. A 216, 279 (1916); 217, 115 (1917); Proc. R. Soc. Lond. A 93, 1 (1916). See also D. Hilbert, Math. Ann. 72, 562 (1912).
148. D. Enskog, Phys. Z. 12, 56, 533 (1911); Inaugural Dissertation. Uppsala 1917; S. Chapman, Phil. Trans. R. Soc. Lond. A 216, 279 (1916); 217, 115 (1917); Proc. R. Soc. Lond. A 93, 1 (1916). See also D. Hilbert, Math. Ann. 72, 562 (1912).
149. D. Enskog, Phys. Z. 12, 56, 533 (1911); Inaugural Dissertation. Uppsala 1917; S. Chapman, Phil. Trans. R. Soc. Lond. A 216, 279 (1916); 217, 115 (1917); Proc. R. Soc. Lond. A 93, 1 (1916). See also D. Hilbert, Math. Ann. 72, 562 (1912).
150. D. Enskog, Phys. Z. 12, 56, 533 (1911); Inaugural Dissertation. Uppsala 1917; S. Chapman, Phil. Trans. R. Soc. Lond. A 216, 279 (1916); 217, 115 (1917); Proc. R. Soc. Lond. A 93, 1 (1916). See also D. Hilbert, Math. Ann. 72, 562 (1912).
151. D. Enskog, Phys. Z. 12, 56, 533 (1911); Inaugural Dissertation. Uppsala 1917; S. Chapman, Phil. Trans. R. Soc. Lond. A 216, 279 (1916); 217, 115 (1917); Proc. R. Soc. Lond. A 93, 1 (1916). See also D. Hilbert, Math. Ann. 72, 562 (1912).
152. In its original form, the H-theorem said that any system tends toward equilibrium if equilibrium does not initially exist. On the criticism of Loschmidt and Zermelo, Boltzmann became more careful and restated the H-theorem as a statement about the most probable behavior of a system.
153. The ergodic hypothesis, which Boltzmann introduced in 1887 [J. Math. 100, 2101 (1877)], and which Maxwell had called before the assumption of the continuity in path [Trans. Camb. Phil. Soc. 12, 547 (1879)], states in its strong form that a system passes in the course of time through any point of its phase space volume, consistent with the energy of the system.
154. The ergodic hypothesis, which Boltzmann introduced in 1887 [J. Math. 100, 2101 (1877)], and which Maxwell had called before the assumption of the continuity in path [Trans. Camb. Phil. Soc. 12, 547 (1879)], states in its strong form that a system passes in the course of time through any point of its phase space volume, consistent with the energy of the system.
155. A. Rosenthal, Ann. Phys. (Leipzig) 42, 796 (1913); M. Plancherel, Ann. Phys. (Leipzig) 42, 1061 (1913). 115. P. and T. Ehrenfest, Encyclopedia article, l.c., p. 22, and notes 98 and 99; E. Fermi, Phys. Z. 24, 261 (1923). The reason why one does not stay with the physical time ensemble lies in the fact that the average cannot be computed in normal case.
156. A. Rosenthal, Ann. Phys. (Leipzig) 42, 796 (1913); M. Plancherel, Ann. Phys. (Leipzig) 42, 1061 (1913). 115. P. and T. Ehrenfest, Encyclopedia article, l.c., p. 22, and notes 98 and 99; E. Fermi, Phys. Z. 24, 261 (1923). The reason why one does not stay with the physical time ensemble lies in the fact that the average cannot be computed in normal case.
157. A. Rosenthal, Ann. Phys. (Leipzig) 42, 796 (1913); M. Plancherel, Ann. Phys. (Leipzig) 42, 1061 (1913). 115. P. and T. Ehrenfest, Encyclopedia article, l.c., p. 22, and notes 98 and 99; E. Fermi, Phys. Z. 24, 261 (1923). The reason why one does not stay with the physical time ensemble lies in the fact that the average cannot be computed in normal case.
158. G. D. Birkhoff, Proc. Natl. Acad. Sci. 17, 650, 656 (1931); G. D. Birkhoff, and B. O. Koopman, Proc. Natl. Acad. Sci. 18, 279 (1932); J. v. Neumann, ibid. 18, 70, 263 (1932). Further research has shown that a wide class of systems satisfies these conditions [See, e.g., J. C. Oxtoby, and S. M. Ulam, Ann. Math. 42, 874 (1941)].
159. G. D. Birkhoff, Proc. Natl. Acad. Sci. 17, 650, 656 (1931); G. D. Birkhoff, and B. O. Koopman, Proc. Natl. Acad. Sci. 18, 279 (1932); J. v. Neumann, ibid. 18, 70, 263 (1932). Further research has shown that a wide class of systems satisfies these conditions [See, e.g., J. C. Oxtoby, and S. M. Ulam, Ann. Math. 42, 874 (1941)].
160. G. D. Birkhoff, Proc. Natl. Acad. Sci. 17, 650, 656 (1931); G. D. Birkhoff, and B. O. Koopman, Proc. Natl. Acad. Sci. 18, 279 (1932); J. v. Neumann, ibid. 18, 70, 263 (1932). Further research has shown that a wide class of systems satisfies these conditions [See, e.g., J. C. Oxtoby, and S. M. Ulam, Ann. Math. 42, 874 (1941)].
161. G. D. Birkhoff, Proc. Natl. Acad. Sci. 17, 650, 656 (1931); G. D. Birkhoff, and B. O. Koopman, Proc. Natl. Acad. Sci. 18, 279 (1932); J. v. Neumann, ibid. 18, 70, 263 (1932). Further research has shown that a wide class of systems satisfies these conditions [See, e.g., J. C. Oxtoby, and S. M. Ulam, Ann. Math. 42, 874 (1941)].
162. G. D. Birkhoff, Proc. Natl. Acad. Sci. 17, 650, 656 (1931); G. D. Birkhoff, and B. O. Koopman, Proc. Natl. Acad. Sci. 18, 279 (1932); J. v. Neumann, ibid. 18, 70, 263 (1932). Further research has shown that a wide class
163. G. D. Birkhoff, Proc. Natl. Acad. Sci. 17, 650, 656 (1931); G. D. Birkhoff, and B. O. Koopman, Proc. Natl. Acad. Sci. 18, 279 (1932); J. v. Neumann, ibid. 18, 70, 263 (1932). Further research has shown that a wide class of systems satisfies these conditions [See, e.g., J. C. Oxtoby, and S. M. Ulam, Ann. Math. 42, 874 (1941)].
164. See ref. 89. The eason why Gibbs' method is easier to generalize is that he does not speak about the constituents of his system and about the characteristic properties of the particles. Therefore, bosons and fermions can be included by simply making turther specifications. In the light of the last chapter of Elementary Principles we simply have to replace the phases by "generic" phases (See Epstein, l.c., ref. 89).
165. The problem which has to be solved in statistical mechanic is: By which ensemble, characterized by a macroscopic parameter, has a system to be represented?
166. A. Vibert Douglas, "Forty Minutes with Einstein," J. R. Astr. Soc. Canada 50, 99-102 (1956).
167. Ref. 120, p. 102.