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---
title: "Fermi-Dirac distribution"
date: 2021-07-11
categories:
- Physics
- Statistics
- Quantum mechanics
layout: "concept"
---
**Fermi-Dirac statistics** describe how identical **fermions**,
which obey the [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/),
will 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](/know/concept/grand-canonical-ensemble/),
whose grand partition function $\mathcal{Z}$ is as follows,
where we sum over all microstates of $s$:
$$\begin{aligned}
\mathcal{Z}
= \sum_{N = 0}^1 \exp(- \beta N (\varepsilon - \mu))
= 1 + \exp(- \beta (\varepsilon - \mu))
\end{aligned}$$
Where $\mu$ is the chemical potential,
and $\varepsilon$ is the energy contribution per particle in $s$,
i.e. the total energy of all particles $E = \varepsilon N$.
The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/)
is the Landau potential $\Omega$, given by:
$$\begin{aligned}
\Omega
= - k T \ln{\mathcal{Z}}
= - k T \ln\!\Big( 1 + \exp(- \beta (\varepsilon - \mu)) \Big)
\end{aligned}$$
The average number of particles $\Expval{N}$
in state $s$ is then found to be as follows:
$$\begin{aligned}
\Expval{N}
= - \pdv{\Omega}{\mu}
= k T \pdv{\ln{\mathcal{Z}}}{\mu}
= \frac{\exp(- \beta (\varepsilon - \mu))}{1 + \exp(- \beta (\varepsilon - \mu))}
\end{aligned}$$
By multiplying both the numerator and the denominator by $\exp(\beta (\varepsilon \!-\! \mu))$,
we arrive at the standard form of
the **Fermi-Dirac distribution** or **Fermi function** $f_F$:
$$\begin{aligned}
\boxed{
\Expval{N}
= f_F(\varepsilon)
= \frac{1}{\exp(\beta (\varepsilon - \mu)) + 1}
}
\end{aligned}$$
This tells the expected occupation number $\Expval{N}$ of state $s$,
given a temperature $T$ and chemical potential $\mu$.
The corresponding variance $\sigma^2$ of $N$ is found to be:
$$\begin{aligned}
\boxed{
\sigma^2
= k T \pdv{\Expval{N}}{\mu}
= \Expval{N} \big(1 - \Expval{N}\big)
}
\end{aligned}$$
## References
1. H. Gould, J. Tobochnik,
*Statistical and thermal physics*, 2nd edition,
Princeton.
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