Maxwell-Boltzmann statistic

The Maxwell-Boltzmann statistic or distribution (not to be confused with the other Maxwell-Boltzmann distribution) is a real, continuous Probability distribution that describes the behavior of a system of $N$ classical particles in thermal equilibrium. Its Probability density function is

$\langle n_{i} \rangle$ is the average number of particles in the $i$-th single-particle state of energy $\varepsilon_{i}$, $\mu$ is the system 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

This distribution is closely related to its quantum equivalents, the Fermi-Dirac distribution and the Bose-Einstein distribution. These converge to the Maxwell-Boltzmann statistic at high temperature.