the Fermi-Dirac distribution
/ FAIR-mee dee-RAK /
Imagine a staircase of energy levels and a crowd of electrons that refuse to double up — the Pauli exclusion principle allows at most one particle per quantum state. At absolute zero the electrons fill the staircase from the bottom, one per step, up to a sharp waterline. Warm them slightly and only the electrons near that waterline can jump. The Fermi-Dirac distribution is the precise rule for the average number of fermions occupying a state of energy epsilon at temperature T.
The mean occupation of a single-particle state of energy epsilon is n(epsilon) = 1 / (exp((epsilon - mu)/kT) + 1), where k is Boltzmann's constant and mu is the chemical potential. Because of the +1 in the denominator, n always lies between 0 and 1 — you literally cannot fit more than one fermion in a state. At T = 0 the distribution becomes a perfect step: n = 1 for epsilon < mu and n = 0 for epsilon > mu, and this zero-temperature value of mu is called the Fermi energy E_F. At finite T the step softens into an S-shaped curve, blurred over an energy width of order kT around mu, so only electrons within about kT of the Fermi surface participate in thermal and transport processes. The formula follows from the grand canonical ensemble applied mode by mode, summing 0 and 1 particle only.
The Fermi-Dirac distribution governs conduction electrons in metals, electrons in white dwarfs and neutron stars, protons and neutrons in nuclei, and carriers in semiconductors. An essential caveat about its relation to classical physics: it reduces to the Maxwell-Boltzmann distribution only in the dilute, high-temperature limit where exp((epsilon - mu)/kT) is large so the +1 is negligible and mean occupancies are far below 1. Whenever occupancies approach 1 — a 'degenerate' gas — the quantum statistics are essential and classical counting fails badly.
In copper at room temperature the Fermi energy is about 7 eV while kT is only about 0.025 eV, so kT/E_F is roughly 0.004. The Fermi-Dirac step is razor-sharp: only a tiny fraction of electrons, those within about kT of E_F, are thermally active, which is why a metal's electronic heat capacity is far smaller than the classical equipartition estimate.
Only electrons within about kT of the Fermi energy respond to heating.
The single sign difference from Bose-Einstein — a +1 instead of -1 in the denominator — is the whole content of the exclusion principle at the level of statistics. Do not confuse the chemical potential mu with the Fermi energy: they coincide only at T = 0, and mu drifts with temperature.