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The European Physical Journal H

Indistinguishable elements in the origins of quantum statistics. The case of Fermi–Dirac statistics

Enric Pérez Canals, Joana Ibáñez

Published Jan 5, 2022• 1 citation
Publisher

Abstract

Abstract In this paper, we deal with the historical origins of Fermi–Dirac statistics, focusing on the contribution by Enrico Fermi of 1926. We argue that this statistics, as opposed to that of Bose–Einstein, has been somewhat overlooked in the usual accounts of the old quantum theory. Our main objective is to offer a critical analysis of Fermi’s seminal paper and its immediate impact. Secondly, we are also interested in assessing the status of the particle concept in the years 1926–1927, especially regarding the germ of quantum indistinguishability. We will see, for example, that the first applications of the Fermi–Dirac statistics to the study of metals or stellar matter had a technical nature, and that their main instigators barely touched upon interpretative matters. Finally, we will discuss the reflections and remarks made in these respects in two famous events in physics of 1927, the Como conference and the fifth Solvay congress.

DOI
10.1140/epjh/s13129-021-00033-6

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•Jan 1, 2012•0 citations
Abstract In this paper, we deal with the historical origins of Fermi–Dirac statistics, focusing on the contribution by Enrico Fermi of 1926. We argue that this statistics, as opposed to that of Bose–Einstein, has been somewhat overlooked in the usual accounts of the old quantum theory. Our main objective is to offer a critical analysis of Fermi’s seminal paper and its immediate impact. Secondly, we are also interested in assessing the status of the particle concept in the years 1926–1927, especially regarding the germ of quantum indistinguishability. We will see, for example, that the first applications of the Fermi–Dirac statistics to the study of metals or stellar matter had a technical nature, and that their main instigators barely touched upon interpretative matters. Finally, we will discuss the reflections and remarks made in these respects in two famous events in physics of 1927, the Como conference and the fifth Solvay congress.
The principal mathematical idea behind the statistical properties of black-body radiation (photons) was introduced already by L. Boltzmann (1877/2015) and used by M. Planck (1900; 1906) to derive the frequency distribution of radiation (Planck’s law) when its discrete (quantum) structure was additionally added to the reasoning. The fundamental physical idea – the principle of indistinguishability of the quanta (photons) – had been somewhat hidden behind the formalism and evolved slowly. Here the role of P. Debye (1910), H. Kamerlingh Onnes and P. Ehrenfest (1914) is briefly elaborated and the crucial role of W. Natanson (1911a; 1911b; 1913) is emphasized. The reintroduction of this Natanson’s statistics by S. N. Bose (1924/2009) for light quanta (called photons since the late 1920s), and its subsequent generalization to material particles by A. Einstein (1924; 1925) is regarded as the most direct and transparent, but involves the concept of grand canonical ensemble of J. W. Gibbs (1902/1981), which in a way obscures the indistinguishability of the particles involved. It was ingeniously reintroduced by P. A. M. Dirac (1926) via postulating (imposing) the transposition symmetry onto the many-particle wave function. The above statements are discussed in this paper, including the recent idea of the author (Spałek 2020) of transformation (transmutation) – under specific conditions – of the indistinguishable particles into the corresponding to them distinguishable quantum particles. The last remark may serve as a form of the author’s post scriptum to the indistinguishability principle.
We discuss another of Einstein’s statistical contributions to quantum theory: his work on quantum statistics. It was prompted by a paper that S. Bose sent to him in 1924 when the physics community had finally accepted the photon idea upon the discovery and analysis of Compton scattering. We present in detail Bose’s derivation of Planck’s distribution, using a particle approach. Einstein extended Bose’s analysis of radiation to systems of matter particles. Here he made the discovery of the astounding phenomenon of Bose–Einstein condensation. The ultimate understanding of Planck’s spectral distribution came about in modern quantum mechanics with its notion of indistinguishable particles, and the connection between spin and statistics. We also provide a modern discussion of Bose–Einstein condensation.
Aim of this paper is to retrace the path that led the young Enrico Fermi to write his paper on the statistics of an ideal monatomic gas. This discovery originated in his interest, which he had shown since his formative years, in the absolute entropy constant and in the problems he highlighted in Sommerfeld's quantization in the case of identical particle systems. The fundamental step taken by Fermi in writing his work on statistics was to apply the Exclusion Principle, formulated for electrons in an atom and which could therefore have been a pure effect due to dynamics, to a system of non-interacting particles.
Abstract Indistinguishability of like particles, and the fermion and boson exchange symmetries discussed.Pauli exclusion principle and features of multi-electron atoms, including selection rules are discussed. Degeneracy pressure and the formation of compact stellar objects is analysed. Quantum exchange force between electrons and its contribution to ferromagnetism is outlined. Fermi-Dirac and Bose-Einstein statistics, includng the chemical potential are derived. The conditions for Bose-Einstein condensation are deduced; condensates and their stability are considered.
Quantum statistical mechanics is the arithmetic of indistinguishable particles: bosons that pile into a single state and fermions that refuse to share---statistics whose consequences range from the Bose-Einstein condensate, a macroscopic quantum state, to the Fermi gas's pressure that holds white dwarfs against collapse. This article presents a narrative review of the primary literature of that statistics, from Bose's Planck's law and the light quantum hypothesis and Einstein's quantum theory of the monatomic ideal gas with its 1925 second paper predicting the condensation, through Fermi's quantization of the monatomic perfect gas and Dirac's theory of quantum mechanics, which independently founded the Fermi-Dirac statistics, London's identification of the lambda-phenomenon of liquid helium with the Bose-Einstein degeneracy, the experimental vindication of Anderson, Ensher, Matthews, Wieman, and Cornell's observation of Bose-Einstein condensation in a dilute atomic vapor and Davis and colleagues' sodium condensate, DeMarco and Jin's onset of Fermi degeneracy in a trapped atomic gas, and the syntheses of Leggett's fundamental concepts, Pitaevskii and Stringari's Bose-Einstein condensation, and Bloch, Dalibard, and Zwerger's many-body physics with ultracold gases. The synthesis is organized around three themes: the statistics' invention, in which indistinguishability's consequences were computed before any condensate existed; the condensate's realization, in which the dilute atomic vapors delivered Einstein's sixty-year-old prediction; and the many-body era, in which the ultracold gases became quantum simulators of matter's collective behavior. It is concluded that quantum statistics is the theory whose consequences were verified in reverse---predicted abstractly, found in liquid helium, created in atomic vapors---and that its bosons and fermions remain the universe's two grammars.
It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and ‘indistinguishability’ of particles has come to be required. However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject. Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace. In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the twentieth century the mutual influence between both disciplines was profuse. We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.