Editing Quantum statistical mechanics (section)

== Grand canonical ensemble ==

{{main|grand canonical ensemble}}

For open systems where the energy and numbers of particles may fluctuate, the system is described by the [[grand canonical ensemble]], described by the density matrix
<math display="block"> \rho = \frac{\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}}{\operatorname{Tr}\left(\mathrm{e}^{ \beta ( \sum_i \mu_iN_i - H)}\right)}. </math>
where the ''N''<sub>1</sub>, ''N''<sub>2</sub>, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Note that this is a density matrix including many more states (of varying N) compared to the canonical ensemble.

The grand partition function is
<math display="block">\mathcal Z(\beta, \mu_1, \mu_2, \cdots) = \operatorname{Tr}(\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}) </math>