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Bose-Einstein (or B-E) statistics are closely related to Maxwell-Boltzmann statistics (M-B) and Fermi-Dirac statistics (F-D). While F-D statistics holds for fermions, M-B statistics holds for classical particles, i.e. identical but distinguishable particles, and represents the classical or high-temperature limit of both F-D and B-E statistics. (M-B, B-E, and F-D statistics are all derived from the Boltzmann factor probability weight applied to the problem of classical particles and discrete energy quanta with boson/fermion behavior, respectively.)

Bosons, unlike fermions, are not subject to the Pauli exclusion principle: an unlimited number of particles may occupy the same state at the same time. This explain why, at low temperatures, bosons can behave very differently than fermions; all the particles will tend to congregate together at the same lowest-energy state, forming what is known as a Bose-Einstein condensate.

B-E statistics was introduced for photons in 1920 by Bose and generalized to atoms by Einstein in 1924.

The Bose-Einstein distribution function

The distribution function f_BE(E) is the expected number of particles in an energy state E for B-E statistics:

where:

E is the energyThis article is about the scientific concept. Energy use by humans is discussed in other articles''. Energy generally and qualitatively speaking, is the property (or the quantity of the property) of doing things or supplying power. The expressions energy

k_B is Boltzmann's constant

T is absolute temperatureTemperature is the physical property of a system which underlies the common notions of "hot" and "cold"; the material with the higher temperature is said to be hotter. General description The formal properties of temperature are studied in thermodynamics.

μ is the chemical potentialThe chemical potential of a thermodynamic system is the change in the energy of the sytem when an additional constituent particle is introduced, with the entropy and volume held fixed. If a system contains more than one species of particle, there is a sep

See also parastatisticsIn quantum mechanics, despite what many textbooks and articles erronously claim, the Bose-Einstein and Fermi-Dirac statistics (and Maxwell-Boltzmann statistics) are NOT the only alternatives. We can have parastatistics as well. In lower spacetime dimensio.