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.