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

TL;DR

This historical study traces how Enrico Fermi integrated concerns about the absolute entropy constant with Sommerfeld quantization for identical particles to generalize Pauli’s Exclusion Principle to non‑interacting gases, yielding a quantum statistics framework that presaged Fermi–Dirac statistics. It documents the intellectual pathway from Fermi’s formative Florentine years through his Göttingen and Leiden experiences, the Florence appointment, and the pivotal 1925–1926 work in which occupancy numbers $N_s$ and degeneracy factors $Q_s$ lead to a distribution with $\beta=\frac{h\nu}{kT}$ and $N_s=Q_s \frac{\alpha e^{-\beta s}}{1+\alpha e^{-\beta s}}$. The paper clarifies how, although rooted in the old quantum theory, Fermi’s approach anticipated later wave‑mechanics formalisms and influenced Dirac, who recognized the priority of his results and popularized the terminology of fermions. The work underscores the broad significance of Fermi–Dirac statistics for dense matter, astrophysics, and modern electronics, including semiconductors, and situates it within a rich historical context from the Sackur–Tetrode framework to contemporary quantum many‑body theory.

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.

Figures (8)

• Figure 1: The three friends Enrico Fermi, Nello Carrara and Franco Rasetti on a trip to the Apuan Alps.
• Figure 2: Illustration of Sommerfeld's quantization rules for three distinguishable particles (on the left) and three identical particles (on the right).
• Figure 3: Two-dimensional representation of the volume of gas divided into $N$ cells with one molecule per cell (left) and into $N/2$ cells with two molecules per cell (right).
• Figure 4: Two-dimensional representation of the volume of gas divided into $N/2$ cells with two different molecules per cell.
• Figure 5: Fermi's letter to Ettore Bortolotti, from guerrarobotti.