# Boltzmann Distribution

The **Maxwell-Boltzmann distribution function** is a function *f(E)* which gives the probability that a system in contact with a thermal bath at temperature *T* has energy *E*. This distribution is *classical* and is used to describe systems with *identical* but *distinguishable* particles.

\[f(E) \propto \Omega(E) \exp \left[ - E/k_B T \right] \]

where \(\Omega(E)\) is the degeneracy of the energy *E*; leading to

\[f(E) = \frac{1}{Z} \Omega(E) \exp \left[ -E/k_B T \right] \]

where

- \(k_B\) is the Boltzmann constant,
- \(T\) is the temperature, and
- the normalization constant \(Z\) is the partition function of the system.