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# Characteristic state function

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.

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