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arXiv:2602.04484

From Florence to Fermions: a historical reconstruction of the origins of Fermi's statistics one hundred years later

Roberto Casalbuoni, Daniele Dominici

physics.hist-phhep-phhep-thquant-phSubmitted Feb 4, 2026• Announced Feb 5, 2026
arXivHTMLPDF

Abstract

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.

Comments
19 pages, 8 figures

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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 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.
Abstract Planck’s original introduction of the quantum idea was based on a statistical mechanical treatment of black body radiation. S. Bose, a physicist working in India, found a new way of deriving Planck’s law. Einstein applied Bose’s method to a gas of massive particles. This led to the prediction of a new physical effect (Bose-Einstein condensation). With the introduction of the exclusion principle by Pauli, Fermi, and Dirac independently introduced a new kind of statistics. Considerations by Einstein and Schrödinger introduced a preliminary form of second quantization.
The field of cold atomic gases faced a revolution in 1995 when Bose-Einstein condensation was achieved. Since then, there has been an impressive progress, both experimental and theoretical. The quest for ultra-cold Fermi gases started shortly after the 1995 discovery, and quantum degeneracy in a gas of fermionic atoms was obtained in 1999. The Pauli exclusion principle plays a crucial role in many aspects of ultra-cold Fermi gases, including inhibited interactions with applications to precision measurements, and strong correlations. The path towards strong interactions and pairing of fermions opened up with the discovery in 2003 that molecules formed by fermions near a Feshbach resonance were surprisingly stable against inelastic decay, but featured strong elastic interactions.This remarkable combination was explained by the Pauli exclusion principle and the fact that only inelastic collisions require three fermions to come close to each other. The unexpected stability of strongly interacting fermions and fermion pairs triggered most of the research which was presented at this summer school. It is remarkable foresight (or good luck) that the first steps to organize this summer school were already taken before this discovery. It speaks for the dynamics of the field how dramatically it can change course when new insight is obtained. The contributions in this volume provide a detailed coverage of the experimental techniques for the creation and study of Fermi quantum gases, as well as the theoretical foundation for understanding the properties of these novel systems.
The concepts of statistical physics have been used to study fermionic systems rapidly after the birth of the “new" quamtum mechanics in 1925. This proved to be extremely fruitful and led to major advances in atomic and solid state physics. Fermi-Dirac statistics was introduced in 1926 by Enrico Fermi and Paul Dirac. It was used by Ralph Fowler in 1926 to describe the collapsed state of a star (a so-called white dwarf ) and by Arnold Sommerfeld in 1928 to describe conduction electrons in solids. This chapter first presents some general results and then discusses a choice of specific systems.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed