# Mesoscopic Physics/Mesoscopic Physics Glossary/Boltzmann Distribution

Given a quantum system with energy eigenvalues ${\displaystyle E_{n}}$, in equilibrium the probability of finding the system in state ${\displaystyle n}$ is given by the Boltzmann distribution,
${\displaystyle p_{n}={1 \over Z}\exp(-{E_{n} \over k_{B}T})}$,
where ${\displaystyle Z}$ is the partition sum that makes the distribution normalized.
${\displaystyle {\hat {\rho }}={1 \over Z}\exp(-{{\hat {H}} \over k_{B}T})}$,
where ${\displaystyle {\hat {H}}}$ is the system's Hamiltonian.