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 $f_F$:

\begin{aligned} \boxed{ \expval{N} = f_F(\varepsilon) = \frac{1}{e^{\beta (\varepsilon - \mu)} + 1} } \end{aligned}

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

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