FermiDirac

Fermi-Dirac statistics, named after Enrico Fermi and Paul Dirac, is a quantum statistical distribution that describes the probability that a given energy state is occupied by fermions at temperature T in thermal equilibrium. It applies to indistinguishable particles with half-integer spin that obey the Pauli exclusion principle, such as electrons, protons, neutrons, and other fermions. The mean occupation number of a single-particle state of energy ε is f(ε) = 1 / (exp[(ε − μ)/(k_B T)] + 1), where μ is the chemical potential and k_B is Boltzmann’s constant. The distribution ranges between 0 and 1.

In the zero-temperature limit, f(ε) becomes a step function: f(ε) = 1 for ε < μ and 0 for ε > μ. The energy ε at which f crosses 1/2 is related to the Fermi energy ε_F, a characteristic scale for fermionic systems. At high temperatures and low densities, Fermi-Dirac statistics converge to Boltzmann statistics.

Fermi-Dirac statistics are fundamental to the electronic structure of atoms, the behavior of conduction electrons in metals, and carrier statistics in semiconductors. They underpin calculations of electron density, specific heat, and electrical conductivity in degenerate systems, and are extended to relativistic and interacting many-body theories.

Historically, the distribution emerged from incorporating the Pauli exclusion principle into quantum statistics, with independent lines of development by Fermi and Dirac in the 1920s. See also Pauli exclusion principle, Fermi gas, Bose-Einstein statistics, and Boltzmann statistics.