Quantum statistical mechanics

Summary

Quantum statistical mechanics is statistical mechanics for systems obeying quantum mechanics. Probability distributions over microstates are replaced by density operators on a Hilbert space, and averages by traces. The thermal equilibrium state at inverse temperature $\beta$ is the Gibbs state $e^{-\beta H} / \tr e^{-\beta H}$, and for infinite systems it is replaced by a KMS state. It explains phenomena with no classical analogue, such as Bose–Einstein condensation and the specific heat of solids at low temperature.