Characteristic State Function

The characteristic state function or Massieu's potential^[1] in statistical mechanics refers to a particular relationship between the partition function of an ensemble.

In particular, if the partition function P satisfies

${\displaystyle P=\exp (-\beta Q)\iff Q=-\frac{1}{\beta }\ln (P)}$ or ${\displaystyle P=\exp (+\beta Q)\iff Q=\frac{1}{\beta }\ln (P)}$

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.

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

State functions are those which tell about the equilibrium state of a system

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