Bose-Einstein distribution

The Bose-Einstein distribution is a Probability distribution that describes the behavior of a system of $N$ non-interacting bosons in thermal equilibrium. Its Probability density function is

$\langle n_{i} \rangle$ is the average number of fermions in the $i$-th single-particle state of energy $\varepsilon_{i}$, $\mu$ is the system's chemical potential, $\beta=1/k_{B}T$ is the inverse temperature, with $k_{B}$ the Boltzmann constant and $T$ the temperature, and $z$ is the fugacity. The normalization constant is

Bosons are not subject to the Pauli exclusion principle, so $\langle n_{i} \rangle$ can be any positive real.

Bose-Einstein condensation#

Note that this function diverges if $e^{\beta(\varepsilon_{i}-\mu)}=1$, which is to say $\mu=0$ and $\varepsilon\to 0$. In this state, the ground state's occupation number blows up to infinity and all other states becomes unoccupied, which means that all bosons "collapsed" into the ground state. This is a phase transition called Bose-Einstein condensation and it only occurs in bosons due to the $-1$ term at the denominator. Neither fermions nor classical particles exhibit this behavior.