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

[math]\displaystyle{ P = \exp(- \beta Q) \Leftrightarrow Q=-\frac{1}{\beta} \ln(P) }[/math] or [math]\displaystyle{ P = \exp(+ \beta Q) \Leftrightarrow Q=\frac{1}{\beta} \ln(P) }[/math]

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

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

References

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