Essay: Sulla quantizzazione del gas perfetto monoatomico

Background and Aim
Enrico Fermi formulated a quantum-statistical description of an ideal monoatomic gas at a moment when quantum ideas were rapidly reshaping physics. The Pauli exclusion principle, newly proposed in 1925, demanded that no two identical fermions occupy the same quantum state. Fermi adopted that principle as the organizing constraint for a gas of identical particles and sought a consistent counting and thermodynamic treatment that honored quantum indistinguishability and the discrete nature of states in phase space.
The resulting essay introduces a statistical mechanics for particles that are mutually exclusive in single-particle states, shifting the foundation of gas theory away from classical Maxwell-Boltzmann counting. The focus is on deriving equilibrium occupation numbers, thermodynamic quantities, and limiting behaviors that connect quantum and classical regimes.

Method and quantization rule
Fermi imposed a phase-space quantization in which each quantum state occupies a finite cell of volume h^3, establishing a one-to-one relation between phase-space cells and single-particle quantum states. The Pauli exclusion principle was implemented by allowing at most one particle per quantum cell (per internal state), so occupation numbers could only be 0 or 1. Particle indistinguishability enters through a combinatorial counting that differs fundamentally from classical permutations.
The equilibrium distribution follows from maximizing the number of microscopic arrangements consistent with fixed total particle number and energy. Lagrange multipliers enforce those macroscopic constraints, producing an occupation law expressed in terms of energy, a temperature-like multiplier, and a chemical potential that enforces particle number conservation.

Derivation of the distribution and limiting cases
The maximization yields the characteristic occupancy formula now called Fermi-Dirac statistics, in which the mean occupation of a single-particle state of energy E is 1/(exp[(E − μ)/kT] + 1). This compact expression encapsulates the essential influence of the exclusion principle: no state can be occupied by more than one particle, and occupation smoothly interpolates from unity for low-energy states at low temperature to an exponentially small value for high-energy states at high temperature.
Classical behavior emerges as the high-temperature, low-density limit in which exp[(E − μ)/kT] ≫ 1 and the formula reduces to the Maxwell-Boltzmann factor. In the opposite, low-temperature limit, the distribution becomes a step function, with all states below a characteristic Fermi energy filled and those above essentially empty, producing the concept of a degenerate Fermi gas.

Physical implications and early applications
The statistics predict thermodynamic features that deviate sharply from classical expectations when degeneracy is significant. Pressure and energy acquire contributions from the zero-temperature occupancy of states up to the Fermi energy, giving rise to a temperature-independent baseline that becomes dominant at low temperatures. Such behavior immediately suggested relevance to the conduction electrons in metals, where a dense sea of fermions exhibits quantum degeneracy even at room temperature, and later to compact astrophysical objects where degeneracy pressure supports matter against gravitational collapse.
Fermi's formulation offered a concrete, quantitative tool for computing heat capacities, chemical potentials, and equations of state for systems of identical fermions, and it clarified when classical approximations break down.

Historical significance and legacy
The essay established a new branch of quantum statistics that, together with a contemporaneous independent derivation by Paul Dirac, created the theoretical framework known as Fermi-Dirac statistics. Its introduction enabled subsequent developments such as Sommerfeld's application to the electron theory of metals, detailed studies of electronic heat capacity and conductivity, and the microscopic understanding of white-dwarf structure and neutron-star matter decades later.
Beyond specific applications, the work cemented the principle that quantum indistinguishability and the Pauli exclusion principle fundamentally alter macroscopic thermodynamic behavior, making Fermi's quantization of the ideal monoatomic gas a cornerstone of modern statistical physics.