FermiDiracstatistiek

Fermi-Dirac statistics describe the statistical distribution of indistinguishable fermions across quantum states in systems where particle exchange and number fluctuations obey the Pauli exclusion principle. The average occupation number of a single-particle state with energy ε is given by the Fermi-Dirac distribution f(ε) = 1 / [exp((ε − μ)/(kB T)) + 1], where μ is the chemical potential and kB is the Boltzmann constant. For fermions, each quantum state can be occupied by at most one particle per spin state, with two spins (up and down) per spatial orbital when considering degeneracy.

At zero temperature, f(ε) becomes a step function: f(ε) = 1 for ε < μ = EF and f(ε) = 0 for ε > μ, where EF is the Fermi energy. At finite temperature, the distribution smooths around μ over an energy scale of order kB T. In the high-temperature limit, the distribution approaches the classical Boltzmann form f(ε) ≈ exp(−(ε − μ)/(kB T)).

Fermi-Dirac statistics are central to many areas of physics. They underpin the behavior of electrons in metals and doped semiconductors, where the electronic density of states and the Fermi energy determine transport and optical properties. They also describe degenerate fermion gases in astrophysical contexts, such as white dwarfs and neutron stars, and in ultracold atomic systems with fermionic atoms. The theory naturally reduces to non-interacting limits, while real systems may require corrections for interactions, finite size, or reduced dimensionality.