Grand potential ( $Φ$ )

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

For a homogeneous extensive system,

Φ = - p V

Fermi-window convolution of the grand potential ( $Φ (T, μ) = \int_{0}^{\infty} Φ_{0} (ϵ) [- \partial_{ϵ} n_{F}] d ϵ$ )

For noninteracting fermions with density of states

g

Φ_{0} (x) = - \int_{0}^{x} (x - y) g (y) d y

\int_{0}^{\infty} g (ϵ) F (ϵ) d ϵ

, where

F = k_{B} T ln (1 - n_{F})

F^{'} = n_{F}

. Integrating by parts twice gives the title expression, since

Φ_{0} (0) = Φ_{0}^{'} (0) = 0

and upper boundary terms vanish for ordinary polynomially growing densities of states. The Fermi window therefore thermally averages the zero-temperature grand potential as a function of trial Fermi energy.

Differentiating the grand partition function at fixed temperature and volume gives

⟨ N ⟩ = - \partial_{μ} Φ

and

Var N = - k_{B} T \partial_{μ}^{2} Φ = k_{B} T \partial_{μ} ⟨ N ⟩

. It follows because derivatives of the logarithm of the state sum generate cumulants of particle number. Nonnegative variance makes the grand potential concave in chemical potential in a stable grand-canonical description.

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The Grand potential is a thermodynamic potential used primarily in the context of statistical mechanics and quantum mechanics. It is particularly useful for systems where the number of particles can vary, such as in grand canonical ensembles, where both energy and particle number can fluctuate.