Quantum statistics: Fermi-Dirac and Bose-Einstein

Ordinary statistical mechanics, developed for classical gas particles, treats particles as distinguishable and allows any number of them to share the same energy state. Quantum particles violate both assumptions: identical quantum particles are fundamentally indistinguishable, and whether more than one of them can occupy the same single-particle quantum state depends entirely on which of two classes the particle belongs to.

Fermions — the class that includes electrons, protons, and neutrons — are subject to the Pauli exclusion principle: no two identical fermions can occupy the same quantum state. Building the statistical distribution that respects this restriction produces the Fermi-Dirac distribution, which gives the average occupation number of a state at a given energy and temperature, capped so that it never exceeds one particle per state. This single restriction has large consequences: it is the reason electrons in a solid or in an atom fill states from lowest energy upward instead of all crowding into the ground state, and the reason a Fermi gas at low temperature has most of its particles locked into filled low-energy states with only those near the topmost occupied level — the Fermi energy — free to respond to small perturbations.

Bosons — the class that includes photons — face no such restriction: any number of identical bosons can occupy the same state simultaneously. The resulting Bose-Einstein distribution allows occupation numbers to grow without the fermionic cap, and in fact makes states of the very lowest energy increasingly favoured as temperature drops, culminating at low enough temperature and high enough density in a macroscopic fraction of all particles collapsing into the single lowest-energy state — a Bose-Einstein condensate, a distinctly quantum phenomenon with no classical counterpart.

Both distributions converge on the classical Maxwell-Boltzmann distribution in the same limit: whenever the number of accessible quantum states vastly outnumbers the particles available to fill them, the probability of two particles ever competing for the same state becomes negligible, so whether or not multiple occupancy is technically permitted stops mattering. This is why ordinary gases at everyday densities and temperatures are well described by classical statistics even though every constituent particle is, in principle, a fermion or a boson — quantum statistics is the general theory, and the classical result is its dilute-gas limiting case.