## Fermi-Dirac statistics

Fermi-

Dirac statistics are used to describe a

gas of

indistinguishable fermions.
Particles with half-integral spin must necessarily have

wave functions which are

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

*times* -1.
This is a realization of the

Pauli exclusion principle.

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 Bose-Einstein 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.