Categories: Physics, Quantum mechanics, Statistics.

# Fermi-Dirac distribution

**Fermi-Dirac statistics** describe how identical **fermions**,
which obey the Pauli exclusion principle,
distribute themselves across the available states in a system at equilibrium.

Consider one single-particle state $s$, which can contain $0$ or $1$ fermions. Because the occupation number $N$ is variable, we turn to the grand canonical ensemble, whose grand partition function $\mathcal{Z}$ is as follows, where $\varepsilon$ is the energy of $s$ and $\mu$ is the chemical potential:

$\begin{aligned} \mathcal{Z} = \sum_{N = 0}^1 \Big( e^{-\beta (\varepsilon - \mu)} \Big)^N = 1 + e^{-\beta (\varepsilon - \mu)} \end{aligned}$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 **Fermi-Dirac distribution** or **Fermi function** $f_F$:

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.