Condensed matter physics

Bose–Einstein statistics

The Bose–Einstein distribution gives the mean occupation of bosonic states at thermal equilibrium. Bosons are not subject to Pauli exclusion and can share a state; under suitable density and temperature, macroscopic ground-state occupation can occur.

At equilibrium, state occupation depends on energy and chemical potential. For bosons, the number in a state is not limited by Pauli exclusion; the denominator approaches zero as energy approaches the chemical potential from above.

Temperature limits and condensation

At high temperature and low density, n̄ᵢ ≪ 1 and the distribution approaches Maxwell–Boltzmann. For a uniform three-dimensional ideal Bose gas below its critical temperature, particles exceeding the excited-state capacity can occupy the ground state: Bose–Einstein condensation.

λₜ is the thermal de Broglie wavelength. nλₜ³ of order unity signals collective quantum behavior; the value 2.612 applies to a uniform ideal 3D Bose gas at the condensation threshold. Interactions and trap geometry modify details.

A deeper view uses the fugacity $z=e^{\mu/(k_BT)}$ and the grand partition function. Each mode has $\mathcal Z_i=(1-ze^{-\beta\epsilon_i})^{-1}$; differentiating with respect to chemical potential gives its mean occupation. As $z$ approaches its largest allowed value, excited modes cannot accommodate more particles at the same temperature and density. In a finite system the transition is rounded: condensation means macroscopic ground-state occupation, not an absolute singularity.

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