Zur Quantentheorie des idealen Gases. Offprint from: Sitzungsberichte der Preussischen Akademie der Wissenschaften, Bd. 3, 1925.
Berlin: Königlich Akademie der Wissenschaften, 1925. First edition, very rare author’s presentation offprint (not to be confused with the more common trade separate – see below), from the library of the great German physicist Arnold Sommerfeld with his signature and annotations, of Einstein’s third paper on his quantum theory of the ideal gas of 1924–1925, Einstein’s “last major innovative contribution to physics” (Pais, Subtle is the Lord, p. 343). In 1924 Einstein received a copy of the Indian physicist S. N. Bose’s paper ‘Planck’s law and the hypothesis of light quanta.’ Einstein immediately recognized its importance, and had it published, shortly followed by a paper of his own applying Bose’s ideas to ideal gases rather than radiation (molecules rather than light quanta). These two papers laid the foundations of ‘Bose-Einstein statistics.’ Einstein published a second paper in which he showed that the new statistics led to the prediction of a new state of matter, the ‘Bose-Einstein condensate’ (the creation of which in the laboratory was the topic of the 2001 Nobel Prize in Physics). At the time, however, many of Einstein’s colleagues, in particular his close friend Paul Ehrenfest, were sceptical of the new statistics. Einstein therefore attempted in the present paper to justify his quantum theory of the ideal gas by more traditional methods, rather than the novel statistics he had used in his two previous papers. “It contains an attempt to extend and exhaust the characterization of the monatomic ideal gas without appealing to combinatorics. Its ambiguities illustrate Einstein’s confusion with his initial success in extending Bose’s results and in realizing the consequences of what later came to be called Bose–Einstein statistics … Its arguments are based on Einstein’s belief in the complete analogy between the thermodynamics of light quanta and of material particles and invoke considerations of adiabatic transformations as well as of dimensional analysis. These techniques were well known to Einstein from earlier work on Wien’s displacement law, Planck’s radiation theory and the specific heat of solids” (Pérez & Sauer). “In a letter to Ehrenfest he writes that, on his next visit in Leyden, ‘I shall then convince you completely of the gas-degeneracy-equation. I found another safe, though not entirely complete, approach to it, free of the incriminating statistics’. The arguments advanced in this third paper indeed do not make use of the new statistics. Instead, Einstein invokes arguments involving dimensional analysis and adiabatic compression” (Papers, p. lxx). The only other copy of this offprint listed on RBH is that in Einstein’s own collection (Christie’s, 2008). Provenance: Arnold Sommerfeld (1868-1951) (his signature and characteristic numbering in red pencil (‘44’) on front cover and three annotations in the text). The annotations consist of corrections to formulas (5) on p. 20, (11) and (12) on p. 21, (16a) on p. 23, and (19) on p. 24, to the equation on the last line of p. 22, and to two mathematical symbolson lines 2 and 3 of p. 25. “The son of a physician, Sommerfeld was educated at the University of Königsberg. After teaching briefly at the universities of Göttingen, Clausthal, and Aachen he was appointed professor of physics at the University of Münich in 1906. Sommerfeld should have retired in 1936 in favour of his pupil, Werner Heisenberg. Opposition from the Nazi party to Heisenberg’s appointment prolonged Sommerfeld’s tenure and it was not in fact until late 1939 that he finally retired, to be succeeded not by Heisenberg but by Wilhelm Müller, a Nazi aerodynamicist without a single publication in physics to his credit. Although Sommerfeld and Heisenberg were not Jewish, they were regarded by the Nazis as Jewish sympathizers. Sommerfeld, however, survived the war and returned to his Münich chair in 1945, continuing to work at physics until he died in a car accident in 1951” (Oxford Reference). “Arnold Sommerfeld was one of the most distinguished representatives of the transition period between classical and modern theoretical physics. The work of his youth was still firmly anchored in the conceptions of the nineteenth century; but when in the first decennium of the century the flood of new discoveries, experimental and theoretical, broke the dams of tradition, he became a leader of the new movement, and in combining the two ways of thinking he exerted a powerful influence on the younger generation. This combination of a classical mind, to whom clarity of conception and mathematical rigour are essential, with the adventurous spirit of a pioneer, are the roots of his scientific success, while his exceptional gift of communicating his ideas by spoken and written word made him a great teacher” (Max Born, p. 275). The motivation for the publication of this paper is given by Einstein in the opening paragraph (translation from Papers, Vol. 14, English Translation Supplement): ‘Stimulated by a derivation of Planck’s radiation formula originating from Bose, which consistently supports itself on the light-quantum hypothesis, I recently postulated a quantum theory for ideal gas. This theory seems legitimate when one starts out from the conviction that a light quantum (disregarding its polarization property) differs from a monatomic molecule essentially only in that the quantum’s mass at rest is vanishingly small. But because the presupposition of this analogy is certainly not accepted by all researchers, and furthermore, because the statistical method used by Mr. Bose and me is certainly not beyond doubt but rather just seems justified a posteriori by its success in the case of radiation, I looked for other considerations on the quantum theory of ideal gas that are as free of arbitrary hypotheses as possible. These considerations shall be communicated in the following. They provide an effective support for the theory postulated earlier, even though the results attained do not yield a full substitute for that theory. Here it is a matter of establishing considerations in the field of gas theory by a method and with results largely analogous to those in the field of radiation theory leading to Wien’s displacement law.’ “Einstein followed an approach in this paper that was not based only on statistical considerations and that was closer to thermodynamics. He tried to find general conditions that any theory of the ideal gas would have to satisfy, mainly by establishing and exploiting analogies with radiation, where the displacement law at least provided some hints as to what the radiation law should look like” (Pérez & Sauer). The problem Einstein wished to solve was to find the distribution function ρ = ρ(L, κT, V, m), where L is the kinetic energy, κ the Boltzmann constant, T the temperature, V the volume and m the mass of the molecules. The distribution law will be of the form dn = ρ(L, κT, V, m)Vdp1dp2dp3 / h3, where dn is the number of molecules whose Cartesian components of the momenta are in the range (p1, p2, p3) to (p1+ dp1, p2 + dp2, p3 + dp3) (h is Planck’s constant). Einstein did not assume that collisions between molecules are governed by the laws of mechanics. He asserted that if that were the case, one would arrive at the classical Maxwell’s distribution law. He neglected interactions among molecules, this being essentially the definition of an ideal gas. Einstein first used dimensional analysis to place a restriction on the possible forms of the distribution function. He had used similar arguments in his 1909 paper ‘Zum gegenwärtigen Stand des Strahlungsproblems’ (Physikalische Zeitschrift 10, 185–193) to deduce Wien’s displacement law, and in his 1911 paper on the quantum theory of solids ‘Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern’ (Annalen der Physik 35, 679–694). Since ρ is dimensionless (a pure number), it had to be a function only of dimensionless combinations of L, κT, V, m, and h. There are two independent such combinations, which means that ρ is reduced to a function of two variables rather than five: A = L/κT and B = m(V/N)2/3κT/h2. To reduce ρ to a function of a single variable, Einstein needed further restrictions. “Einstein proposed two of those: Einstein argued that these two properties should be valid disregarding collisions. But the neglect of intermolecular collisions made their assumption unprovable, even if they would be ‘very natural.’ In support of both, he announced they would lead not only to the same result, but also to a result according to which Maxwell’s distribution law is valid in the region where quantum effects can be neglected” (Pérez & Sauer). Einstein deduced from these assumptions that ρ = Ψ(A + χ(B)), (*) where Ψ and χ are universal functions of dimensionless variables. Einstein then looked at the case in which the constant h disappears from the expression for dn, i.e., at the classical limit. He found that ρ = Be–A, i.e., the Maxwell-Boltzmann law. In contrast, Einstein’s statistical theory had produced the expression ρ = B/(eA – 1). “Summarizing, Einstein pointed out that two aims have been achieved: ‘First, we found a general condition (equation (*)), which has to be satisfied by any theory of the ideal gas. Second, it follows from the above that the equation of state which I derived will not be changed by either adiabatic compression or by the existence of conservative force fields’” (Pérez & Sauer). Why was this paper little noticed by Einstein’s colleagues? “The practically immediate appearance of the revolutionary contributions of 1925 to quantum theory eclipsed any possible interest of Einstein’s paper. The arguments it contains only concern the ideal gas from a thermodynamic perspective. But what is more important, it includes hypotheses that were in open contradiction with the course quantum researches had taken. Many physicists had rejected already the laws of mechanics, and Einstein assumed their validity for describing the motions of the gas molecules. “The papers of the twenties that refer to Einstein’s theory usually mention all three instalments. This indicates that, in spite of the almost complete lack of comments on it, its existence was known. We are inclined to think that it simply was not of any interest to Einstein’s colleagues. Einstein justified the considerations of the non-statistical paper with the deep dissatisfaction over the statistical route by which he had arrived at the new distribution function. However, the problem was not whether his colleagues saw Bose’s statistics favourably, but that in the following months the physicists’ ideas around the quantum issues changed substantially. Bose’s statistics, in spite of implying a way of counting that was incompatible with classical statistics, led to an already accepted result. This was much more than could be said of other attempts of explaining, for example, the Zeeman effect or multielectronic spectra … “In retrospect, Einstein’s initial suspicion about Bose’s statistics will turn into one of the first symptoms of his later distancing himself from quantum mechanics. For this reason we find no justification for the neglect of Einstein’s paper by historians of physics. Perhaps we are dealing here with Einstein’s last attempt to contribute positively to the construction of the quantum theory, for which he had done so much. In addition, this paper closed the circle he initiated in 1905 with the hypothesis of energy quanta. First, the analogy was going one way, now, finally, it was also going the other way. The statistical dependence among light quanta which had limited the analogy with an ideal gas now was found also among molecules. Hence, for the first time the analogy was complete … “The last ‘positive contribution’ of Einstein to statistical physics includes a paper in which he offered arguments independent of the ‘incriminated statistics,’ because what nowadays is called Bose-Einstein’s statistics was not more, according to its creator, than a calculatory artifice absolutely devoid of any physical meaning. It was simply a consequence of using the wrong mechanics or of not considering some kind of interaction. As Einstein explained to Halpern, it ‘cannot be considered as giving a true theoretical basis to Planck’s law’” (Pérez & Sauer). This author’s presentation offprint is very rare and must be distinguished from other so-called ‘offprints’ of papers from the Berlin Sitzungsberichte, many of which are commonly available on the market. The celebrated bookseller Ernst Weil (1919-1981), in the introduction to his Einstein bibliography, wrote: “I have often been asked about the number of those offprints. It seems to be certain that there were few before 1914. They were given only to the author, and mostly ‘Überreicht vom Verfasser’ (Presented by the Author) is printed on the wrapper. Later on, I have no doubt, many more offprints were made, and also sold as such, especially by the Berlin Academy.” If the term ‘offprint’ means, as we believe it should, a separate printing of a journal article given (only) to the author for distribution to colleagues, then ‘offprints’ were not commercially available. Although there is certainly some truth in Weil’s remark, in our view it requires clarification and explanation. Until about 1916, most of Einstein’s papers were published in Annalen der Physik; from 1916 until he left Germany for the United States in 1933, most were published in the Berlin Sitzungsberichte. The Sitzungsberichte differed from other journals in which Einstein published in that it made separate printings of its papers commercially available. These separate printings have ‘Sonderabdruck’ printed on the front wrapper, the usual German term for offprint, but they are not offprints according to our definition. They were available to anyone; indeed a price list of these ‘trade offprints’ is printed on the rear wrapper. True author’s presentation offprints can be distinguished from these trade offprints by the presence of ‘Überreicht vom Verfasser’ on the front wrapper, as in the present offprint. In the period 1916 to 1919 or 1920, the Sitzungsberichte trade offprints are themselves rare: for example, RBH list only three ‘offprints’ of Einstein’s famous 1917 Sitzungsberichte paper ‘Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie’ (the auction records do not distinguish between trade and author’s presentation offprints). After 1919 or 1920, however, the trade offprints become much more common, although the author’s presentation offprints are still very rare. The reason for this change is that it was only in 1919 that Einstein became famous among the general public. Weil 145. Shields, “Writings of Albert Einstein” (in Albert Einstein: Philosopher-Scientist [1948], pp. 689-758), no. 195. Born, ‘Arnold Johannes Wilhelm Sommerfeld 1868-1951,’ Obituary Notices of Fellows of the Royal Society 8 (1952), pp. 275-296. The Collected Papers of Albert Einstein digital, Vol. 14: The Berlin Years: Writings & Correspondence, April 1923-May 1925. Pais, Subtle is the Lord, 1982. Pérez & Sauer, ‘Einstein’s quantum theory of the monatomic ideal gas: non-statistical arguments for a new statistics,’ Archive for History of Exact Sciences 64 (2010), pp. 561-612.
8vo (255 x 183 mm), pp. 18-25. Original orange printed wrappers.
Item #6417
Price: $5,000.00
