Grand canonical ensemble

At fixed temperature, volume, and chemical potential, the grand canonical ensemble permits both energy and particle number to fluctuate and weights a state by

e^{- β (E - μ N)}

.

Table of contents
- Grand canonical partition function Grand canonical ensemble
- Grand potential Grand canonical ensemble

Grand canonical partition function ( $Z$ )

The grand canonical partition function is

Z = \sum_{s} e^{- β (E_{s} - μ N_{s})} .

Grand potential ( $Φ$ )

The grand potential is

Φ = E - TS - μ ⟨ N ⟩ = - k_{B} T lo g Z .

For a homogeneous extensive system,

Φ = - p V

.

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