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
<ns> = 1/(exp((Es-u)/
kBT) + 1)
where Es is the energy of a particle in s,
u is the chemical potential, T is the temperature,
and kB is Boltzmann's constant. Compare this
to Bose-Einstein statistics where the +1 is replaced by -1.
In the limit where exp((Es-u)/
kBT) >> 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.