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

${\displaystyle P=\exp(-\beta Q)\Leftrightarrow Q=-{\frac {1}{\beta }}\ln(P)}${\displaystyle P=\exp(-\beta Q)\Leftrightarrow Q=-{\frac {1}{\beta }}\ln(P)} or ${\displaystyle P=\exp(+\beta Q)\Leftrightarrow Q={\frac {1}{\beta }}\ln(P)}${\displaystyle P=\exp(+\beta Q)\Leftrightarrow 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 ${\displaystyle \Omega (U,V,N)=e^{\beta TS}\;,円}${\displaystyle \Omega (U,V,N)=e^{\beta TS}\;,円} hence, its characteristic state function is ${\displaystyle TS}${\displaystyle TS}.
• The canonical ensemble satisfies ${\displaystyle Z(T,V,N)=e^{-\beta A},円\;}${\displaystyle Z(T,V,N)=e^{-\beta A},円\;} hence, its characteristic state function is the Helmholtz free energy ${\displaystyle A}${\displaystyle A}.
• The grand canonical ensemble satisfies ${\displaystyle {\mathcal {Z}}(T,V,\mu )=e^{-\beta \Phi },円\;}${\displaystyle {\mathcal {Z}}(T,V,\mu )=e^{-\beta \Phi },円\;}, so its characteristic state function is the Grand potential ${\displaystyle \Phi }${\displaystyle \Phi }.
• The isothermal-isobaric ensemble satisfies ${\displaystyle \Delta (N,T,P)=e^{-\beta G}\;,円}${\displaystyle \Delta (N,T,P)=e^{-\beta G}\;,円} so its characteristic function is the Gibbs free energy ${\displaystyle G}${\displaystyle G}.

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

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