Categories: Physics, Quantum mechanics, Statistics.

# Bose-Einstein distribution

**Bose-Einstein statistics** describe how bosons,
which do not obey the Pauli exclusion principle,
distribute themselves across the available states
in a system at equilibrium.

Consider a single-particle state $s$, which can contain any number of bosons. Since the occupation number $N$ is variable, we use the grand canonical ensemble, whose grand partition function $\mathcal{Z}$ is as shown below, where $\varepsilon$ is the energy per particle, and $\mu$ is the chemical potential. We evaluate the sum in $\mathcal{Z}$ as a geometric series:

$\begin{aligned} \mathcal{Z} = \sum_{N = 0}^\infty \Big( e^{-\beta (\varepsilon - \mu)} \Big)^{N} = \frac{1}{1 - e^{-\beta (\varepsilon - \mu)}} \end{aligned}$The corresponding thermodynamic potential is the Landau potential $\Omega$, given by:

$\begin{aligned} \Omega = - k T \ln{\mathcal{Z}} = k T \ln\!\big( 1 - e^{-\beta (\varepsilon - \mu)} \big) \end{aligned}$The average number of particles $\expval{N}$ in $s$ is then found by taking a derivative of $\Omega$:

$\begin{aligned} \expval{N} = - \pdv{\Omega}{\mu} = k T \pdv{\ln{\mathcal{Z}}}{\mu} = \frac{e^{-\beta (\varepsilon - \mu)}}{1 - e^{-\beta (\varepsilon - \mu)}} \end{aligned}$By multiplying both the numerator and the denominator by $e^{\beta(\varepsilon \!-\! \mu)}$,
we arrive at the standard form of the **Bose-Einstein distribution** $f_B$:

This gives the expected occupation number $\expval{N}$ of state $s$ with energy $\varepsilon$, given a temperature $T$ and chemical potential $\mu$.

## References

- H. Gould, J. Tobochnik,
*Statistical and thermal physics*, 2nd edition, Princeton.