Bose-Einstein statistics

In statistical thermodynamics, Bose-Einstein statistics determines the
statistical distribution of identical indistinguishable bosons over the
energy states in thermal equilibrium.

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 particules, and represents the classical or
high-temperature limit of both F-D and B-E statistics.

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 particules they will tend to congregate together at
the same lowest-energy state, forming what is 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(E) is the probability that a particle is in
energy state E, for B-E statistics the following hold:

     [f_{BE}(E) = \frac{1}{A \exp({E}/{k_B T}) - 1}]

where:

     E is the energy
     kB is Boltzmann's constant
     T is absolute temperature
     A is a normalization constant

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