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 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:

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

where:

E is the energy

k_B is Boltzmann's constant

T is absolute temperature

μ is the chemical potential