Fermi-Dirac distribution

The Fermi-Dirac distribution is a Probability distribution that describes the behavior of a system of $N$ non-interacting fermions in thermal equilibrium. Its Probability density function is

\langle n_{i} \rangle =\frac{1}{e^{\beta(\varepsilon_{i}-\mu)}+1}=\frac{1}{z^{-1}e^{\beta \varepsilon_{i}}+1}

$\langle n_{i} \rangle$ 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

\sum_{i}n_{i}=N

Since we are working with fermions, the Pauli exclusion principle must hold, which is to say that $\langle n_{i} \rangle< 1$ . In fact, at low temperatures, this causes the distribution to approach the Heaviside step function according to the limit

\lim_{ T \to 0 } \langle n_{i} \rangle=\Theta(\mu-\varepsilon_{i})

From David Tong's lecture notes on statistical physics. $E_{F}$ is the Fermi energy.