Bose–Einstein statistics

Bose–Einstein statistics describes how indistinguishable bosons occupy available quantum states at thermal equilibrium. Unlike fermions, bosons can share a state without limit.

Bose–Einstein distribution: The formula giving the mean occupation of a bosonic energy state at a specified temperature and chemical potential. It gives the equilibrium occupation numbers that define Bose–Einstein statistics.
Boson: A particle with integer spin whose many-particle quantum state is symmetric under exchange. Bose–Einstein statistics applies to indistinguishable particles of this kind.
Fermi–Dirac statistics: The distribution governing indistinguishable fermions, which obey the Pauli exclusion principle. Its one-particle occupancy limit contrasts with unrestricted bosonic sharing.
Black-body radiation: Electromagnetic radiation emitted by an object in thermal equilibrium, with a spectrum determined by its temperature. Its spectrum is explained by bosonic photon occupations across electromagnetic modes.
Bose–Einstein condensation: A phase transition in which a macroscopic number of bosons occupy the lowest energy state. The distribution predicts its sudden ground-state population below a critical temperature.
Indistinguishable particles: Quantum particles that cannot be individually distinguished even in principle by their physical properties. Their state counting differs from classical counting of labeled particles.
Maxwell–Boltzmann statistics: The classical distribution for particles treated as distinguishable and dilute. It approximates bosonic statistics when quantum occupation effects are negligible.
Photon gas: A collection of photons treated as a gas in thermal equilibrium with matter or radiation boundaries. Photons obey Bose–Einstein statistics and have zero chemical potential in equilibrium.
Grand canonical ensemble: A statistical-mechanical ensemble in which a system exchanges energy and particles with a reservoir. It yields the standard occupation distribution when particle number can fluctuate.
Symmetric wavefunction: A many-particle wavefunction unchanged when any two identical particles are exchanged. Bosonic exchange symmetry permits multiple particles in the same state.

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