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The characteristic state function in statistical mechanics refers to a particular relationship between the partition function of an ensemble.

In particular, if the partition function P satisfies

$$P = \exp(- \beta Q) or P = \exp(+ \beta Q)$$

in which Q is a thermodynamic quantity, then Q is known as the "characteristic state function" of the ensemble corresponding to "P". Beta refers to the thermodynamic beta.
Examples

The microcanonical ensemble satisfies $$\Omega(U,V,N) = e^{ \beta T S} \;\,$$ hence, its characteristic state function is TS.
The canonical ensemble satisfies $$Z(T,V,N) = e^{- \beta A} \,\;$$ hence, its characteristic state function is the Helmholtz free energy A.
The grand canonical ensemble satisfies \mathcal $$Z(T,V,\mu) = e^{-\beta \Phi} \,\; ,$$ so its characteristic state function is the Grand potential $$\Phi.$$
The isothermal-isobaric ensemble satisfies $$\Delta(N,T,P) = e^{-\beta G} \;\,$$ so its characteristic function is the Gibbs free energy G.

Physics Encyclopedia