Characteristic state function

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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

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

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

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