## Offprints of Einstein’s three papers on Bose-Einstein statistics: (1) Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Bd. 22, 1924. (2) Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung, Ibid., Bd. 1, 1925. (3) Zur Quantentheorie des idealen Gases, Ibid., Bd. 3, 1925.

Berlin: Königlichen Akademie der Wissenschaften, 1924; 1925; 1925.

First editions, the rare author’s presentation offprint issues with ‘Ūberreicht vom Verfasser’ printed on front wrappers, of Einstein’s three papers on Bose-Einstein statistics, Einstein’s “last major innovative contribution to physics” (Pais, p. 343), containing, among other things, Einstein’s derivation of the Bose-Einstein condensate. Einstein wrote these papers after translating and publishing S. N. Bose’s ‘Planck’s law and the hypothesis of light quanta,’ which Bose had sent him from India in 1924. The Bose-Einstein statistics laid the foundation of quantum statistics; they mark the transition between the old quantum theory of Planck, Bohr and Einstein and the new quantum mechanics developed by Dirac, Schrödinger, Heisenberg and others. “As long as Einstein lived, he never ceased to struggle with quantum physics. As far as his constructive contributions to this subject are concerned, they came to an end with a triple of papers, the first published in September 1924, the last two in early 1925. In the true Einsteinian style, their conclusions are once again reached by statistical methods, as was the case for all his important earlier contributions to the quantum theory. The best-known result is his derivation of the Bose-Einstein condensation phenomenon … After the papers by Bose and the first one by Einstein came out, Ehrenfest and others objected (so we read in Einstein’s second paper) that ‘the quanta and molecules, respectively, are not treated as statistically independent, a fact that is not particularly emphasized in our papers.’ Einstein replied, ‘This [objection] is entirely correct.’ He went on to stress that the differences between the Boltzmann and the Bose-Einstein counting ‘express indirectly a certain hypothesis on a mutual influence of the molecules which for the time being is of a quite mysterious nature.’ With this remark, Einstein came to the very threshold of the quantum mechanics of identical particle systems (Pais, pp. 428, 430). It was later understood that Bose–Einstein statistics apply only to those particles that do not obey the Pauli exclusion principle restrictions; such particles have integer values of spin and are now named bosons. Particles with half-integer spin, which do obey the exclusion principle, are called fermions and obey Fermi-Dirac statistics (1926), developed first by Enrico Fermi in the context of the old quantum theory, and then by Paul Dirac in quantum mechanics. The Bose-Einstein condensate is now considered as the fifth state of matter in which separate atoms or subatomic particles, cooled to near absolute zero coalesce into a single quantum mechanical entity – that is, one that can be described by a wave function – on a near-microscopic scale. For producing the Bose-Einstein condensate in the laboratory, Eric Cornell, Wolfgang Ketterle and Carl Wiemann shared the 2001 Nobel Prize in Physics. ABPC/RBH list four copies: in Einstein’s own collection of his offprints (Christie’s, June 17, 2008, lot 100); in the collection of Einstein’s son Hans Albert (Christie’s New York, June 14, 2006, lot 264 – first two papers only); in the Richard Green collection of Einstein offprints (Christie’s, June 17, 2008, lot 101 – first paper only); and Swann, February 10, 1994, lot 19, $1840.

“Sometime in June 1924, … Einstein received a letter from a young, little-known Indian scholar, Satyendra Nath Bose. Enclosed with the letter was a manuscript in which Bose gave the first statistical derivation of Planck’s radiation law on the basis of light quanta without invoking results that were derived on wave-theoretical assumptions.

“At the time, Bose was a thirty-year-old reader at the newly established University of Dakha who had previously published a couple of papers in the *Philosophical Magazine* on the classical equation of state and on the quantum theory of Rydberg’s law. He had also been involved in the English translation of [Lorentz, Einstein & Minkowski, *Das Relativitätsprinzip*, 1922], published by the University of Calcutta, and had translated, in particular, Einstein’s 1916 review paper on the general theory of relativity.

“Bose recalled that he first conceived of the idea for his paper while reflecting on how best to teach the new quantum and relativity physics in a logical and straightforward manner, and without getting caught up in theoretical inconsistencies. One such inconsistency, repeatedly lamented by Einstein and others, was that derivations of Planck’s radiation formula typically rested on two kinds of assumptions that were mutually incompatible …

“In his 1916/17 papers, Einstein had made an advance on this problem by assuming that the resonators are Bohr atoms that interact with the radiation field according to Bohr’s frequency condition, and by introducing coefficients for induced absorption and spontaneous emission as well as induced emission (negative absorption). This allowed him to derive an expression for [the radiation density] as a function of the coefficients and the temperature. But in order to recover Planck’s law, he had to invoke the validity of Wien’s law, and conceded that ‘it will only be possible to compute the constants […] when we have an electrodynamics that has been modified in the sense of the quantum hypothesis’.

“In his manuscript, Bose took seriously the assumption that light quanta are particles with energy and momentum, and [assumed] that their phase space was quantized into cells of size *h*^{3}, where *h* is Planck’s constant, in the same way as had been done for material oscillators. On the basis of this assumption, he succeeded in deriving Planck’s formula by a straightforward application of Boltzmann’s principle by maximizing an expression for the entropy, i.e., by maximizing an expression for the probability of distributing the light quanta over the discrete cells of the phase space volume.

“Einstein immediately realized the importance of Bose’s procedure as being the first derivation of Planck’s radiation law using statistical mechanics for a system of light particles. He quickly set out to translate the paper into German and submitted it only a few days later to the *Zeitschrift für Physik*, not without adding a comment: ‘In my opinion Bose’s derivation of the Planck formula signifies an important advance. The method used also yields the quantum theory of the ideal gas, as I shall work out in detail elsewhere.’

“Einstein’s decision to apply the same statistical method to an ideal gas of molecules was probably the motivation for his quick action regarding Bose’s paper. On 10 July 1924, he presented to the Prussian Academy a paper in which he applied Bose’s procedure to the case of a monatomic ideal gas and derived its equation of state [paper (1)]. He verified that in this framework the relationship between pressure and kinetic energy is the same as in the classical theory, and he looked at the classical limit of the expressions obtained. He observed that in this new approach Nernst’s theorem would automatically be satisfied, since for vanishing temperature the entropy, too, would vanish. He concluded by pointing out that the method did not allow him to solve Gibbs’s paradox, the solution of which he posed as an open problem.

“Neither Bose nor Einstein may initially have realized how profoundly innovative their respective derivations would turn out to be. It was perhaps pointed out to Einstein after publication of the two papers that the use of the statistical procedure in the derivation implied a nontrivial fundamental assumption. In the case of a material gas, that assumption was particularly puzzling. As became clear in correspondence with Otto Halpern, a distribution of light quanta or gas molecules over cells of phase space leaves open the question which configurations are to be considered equally probable. In the case of independent particles, configurations in which two particles exchange places over different cells are considered to be different. In Bose’s and Einstein’s counting, however, it only mattered how many particles would sit in each cell. In the latter case, the particles were in effect indistinguishable, a feature that had been noted before in the case of distributing vibration modes (*hν*-quanta) over resonators but was an entirely new and non-classical assumption for material particles. The difference between the two ways of counting effectively resulted in a higher probability of configurations in which many particles share the same cell. It is as if the Bose-Einstein counting indicated a kind of attractive interaction between particles of the same energy and momentum. In his response to Halpern, Einstein states that the decision between the two alternatives can only be made by experience.

“Einstein’s correspondence with Ehrenfest provides some clues about his developing thoughts on the quantum theory of the ideal gas and his confidence in the theory. In July 1924, he had praised Bose’s work: ‘The Indian Bose gave a beautiful derivation of Planck’s law and of its constant, on the basis of the loose light quanta. The derivation is elegant but its nature remains dark. I applied his theory to the ideal gas. A rigid theory of ‘degeneracy.’ No zero point energy, and at the top no energy defect. God knows if this is so. The theory does not account for the deviations from the law of corresponding states’ …

“It is likely that Einstein was made aware of the ‘counting’ implications of the ideal gas paper by Ehrenfest, an expert on statistical physics and author of studies that had explicitly investigated the statistical assumptions in derivations of Planck’s formula. They probably discussed these issues when Einstein visited Leyden in October 1924. In a letter to Elsa at the time, Einstein wrote: ‘It is also not true that I made a mistake in my paper, which is built on Bose’s papers. Ehrenfest only disputes a nuance, Laue doesn’t dispute anything.’

“By late November 1924, he reported to Ehrenfest that he had found another intriguing consequence of the theory: ‘I examined with Grommer in more detail the degeneracy function. From a certain temperature on, the molecules ‘condense’ without attractive forces, i.e., they pile up at zero velocity. The theory is pretty, but is there any truth to it?’

“He added that he would try to investigate whether a connection to the behavior of thermo-forces at low temperatures could be found. Four days later, he again wrote to Ehrenfest: ‘The matter of the quantum gas is becoming very interesting. It seems to me more and more that there is much that is true and deep at the bottom of it. I look forward to our arguing about it.’

“At the end of the year, he eventually wrote a second paper on the topic and presented it to the Prussian Academy on 8 January. The paper is written formally as a continuation of the first paper, but Einstein now goes back to an analysis of the statistical method that he had employed: ‘Herr Ehrenfest and other colleagues have found fault with Bose’s theory of radiation and my analogous theory of the ideal gas in that these theories do not treat the quanta (or the molecules) as statistically independent entities, without our having especially addressed this circumstance in our papers. This is entirely correct.’ In the relevant section, Einstein now proceeds to compare explicitly the statistical assumptions underlying the different viewpoints.

“But this second paper on the quantum theory of the ideal gas contains yet another insight that would prove to be of great significance. In its first section, numbered §6, he observes that the equation of state that he had obtained does not allow for a straightforward limit in which, for a given number of molecules and given temperature, the volume may be made arbitrarily small. He asserts that ‘in this case, with increasing total density, a growing number of molecules go over into the 1st quantum state (a state without kinetic energy), while the rest of the molecules behave according to the parameter value *λ* = 1.” Therefore, something occurs that is similar to the isothermal compression of vapor beyond the saturation volume. A separation takes place whereby one part ‘condenses’ while the rest remains a ‘saturated ideal gas.’

“The predicted phenomenon of ‘condensation’ – according to which, below a certain critical temperature and depending on the mass and density of the molecules, a finite fraction of the molecules condense into the cell corresponding to zero energy – is perhaps the most significant result of Einstein’s new theory. This fraction of condensed gas increases with decreasing temperature, and Einsteinproposed that it might be observable in hydrogen or helium. This extraordinary phenomenon had to wait seven decades for its experimental discovery …

“It is perhaps not surprising that, like Ehrenfest, Adolf Smekal, Erwin Schrödinger, and others also questioned Einstein’s new statistical treatment of the ideal gas. But Einstein held on to his new idea. He told Schrödinger in crystal clear terms how different the new statistics is, and should be: ‘In the Bose statistics that I have used, the quanta, i.e., the molecules, are not treated as mutually independent. I missed stating clearly that here a special statistics is being applied. For the time being, there will be no other verification for it than its success’. He went on to explain this new statistics in some detail.

“But as these examples show, Einstein felt he needed to find further arguments in support of his approach, and to justify its results. He therefore presented a third paper, ‘On the Quantum Theory of the Ideal Gas’, to the Prussian Academy on 29 January 1925, only three weeks after his second one. This installment is quite different from its predecessors. 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.

“Einstein’s quantum theory of the monatomic ideal gas was a revolutionary step for several reasons. First, while the Planck distribution for radiation frequencies was already well established, no such distribution existed for an ideal gas. Second, he effectively introduced a novel concept into physics, that of indistinguishable material particles. The concept of light quanta was still considered doubtful and elusive, and thus treating them as lacking identity did not offend common understanding. There were earlier allusions to Bose’s method for treating radiation. Dealing with resonators comes close to the idea of cells with occupation numbers. In earlier papers, Ehrenfest and De Broglie had pointed out the necessity of quanta-grouping in order to obtain Planck’s, rather than Wien’s, distribution. But nothing like that existed for molecules, which were well-established massive particles. Maxwell-Boltzmann statistics, also well established, had dealt successfully with a conserved number of identifiable particles. Indeed Einstein had extensively used the Maxwell-Boltzmann statistics for gases (e.g., in his 1916/17 papers on the quantum theory of radiation). Applying the new statistics here had no precursor and had not been anticipated. Third, Einstein realized that the new equation of state implied a new, macroscopic, quantum phenomenon, and a new state of matter, that would come to be called ‘Bose-Einstein condensate’” (*Papers*, pp. lxvi-lxxi).

Weil *142, *144, 145. Shields, “Writings of Albert Einstein” (in *Albert Einstein: Philosopher-Scientist *[1948], pp. 689-758), nos. 185, 194, 195; nos. 185 and 194 also included in Shields’ “Chronological list principal works” on p. 757. *The Collected Papers of Albert Einstein**, Vol. 14: The Berlin Years: Writings & Correspondence, April 1923-May 1925*. Pais, *Subtle is the Lord,* 1982.

Together three offprints, 8vo (255 x 183 mm), pp. 261-267; 3-14; 18-25. Original printed wrappers, very slight soiling, tiny stains in lower margins. Extensive calculations and diagram to rear wrapper of the 2nd paper. Very good to fine. Former owner’s signature on front wrappers.

Item #5065

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Price:
$22,500.00
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