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,

$p_n=\frac{1}{Z}\exp (-\frac{E_n}{k_{B}T})$,

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

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

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

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