Fermi–Dirac statisticsprinciple

Imagine a party where each guest demands their own unique chair and cannot stand neighbors on the same seat. Fermions (electrons, protons) behave exactly like that: at most one particle can occupy a quantum state. Fermi–Dirac statistics describes how these 'individualists' distribute themselves over energies: at low temperature they fill the lowest levels first, just as guests would occupy all chairs in the front row before moving to the back rows.

In 1926, Enrico Fermi and Paul Dirac, building on the Pauli exclusion principle, independently derived the distribution for particles with half-integer spin. Fermions cannot occupy the same quantum state more than once — like passengers on a train where each seat is strictly individual. The distribution function shows the probability that a level with energy E is occupied, and at zero temperature it becomes a step function: all states below the Fermi energy are filled, above — empty. As temperature rises, the step 'blurs'.

Fermi–Dirac statistics determines the equilibrium distribution of identical fermions (particles with half-integer spin) over energy levels in an ideal gas at thermodynamic equilibrium. The distribution function: f(E) = 1 / (e^{(E-μ)/(kT)} + 1), where f(E) is the probability of occupation of a state with energy E, μ is the chemical potential, k is Boltzmann's constant, T is absolute temperature. The Pauli principle is expressed in that the maximum occupation of each state is one, and as T → 0 the function becomes a step function (Fermi step).