## Bose-Einstein statistics

Bose-

Einstein statistics are used to describe a

gas of

indistinguishable bosons.
Particles with integral spin must necessarily have

wave functions which are

symmetric under
particle exchange. That is, two configurations which differ
by only exchanging the positions and velocities of a pair of
bosons are given identical weights.

The average number of particles in state *s* is given by

<*n*_{s}> = 1/(exp((*E*_{s}-u)/
*k*_{B}T) - 1)

where *E*_{s} is the energy of a particle in *s*,
*u* is the chemical potential, *T* is the temperature,
and *k*_{B} is Boltzmann's constant. Compare this
to Fermi-Dirac statistics where the -1 is replaced by +1.
In the limit where exp((*E*_{s}-u)/
*k*_{B}T) >> 1, either due to large *T* or
large *u*, then the quantum nature of the gas is unimportant
and the system is described by classical
Maxwell-Boltzmann statistics.

On the other hand, for very cold temperatures (and in a
finite volume), the number
of particles allowed to be in excited energy states is bounded.
Therefore, if additional particles are put into the system, they
are pushed into the ground state. Under these circumstances
it is possible that a macroscopically large number
of particles can all be in a single quantum state (with
*E*_{0} = 0). This phenomenon is called
Bose-Einstein condensation.