Mesoscopic Physics/Mesoscopic Physics Glossary/Boltzmann Distribution

Given a quantum system with energy eigenvalues $E_{n}$, in equilibrium the probability of finding the system in state $n$ is given by the Boltzmann distribution,

$p_{n}={1 \over Z}\exp(-{E_{n} \over k_{B}T})$,

where $Z$ is the partition sum that makes the distribution normalized.

In terms of the system's density matrix, this is equivalent to saying

${\hat {\rho }}={1 \over Z}\exp(-{{\hat {H}} \over k_{B}T})$,

where ${\hat {H}}$ is the system's Hamiltonian.