Characteristic State Function

The characteristic state function or Massieu's potential "Massieu's potentials ..are directly recovered as logarithms of partition functions." in statistical mechanics refers to a particular relationship between the partition function of an ensemble.

In particular, if the partition function ''P'' satisfies

:$P = \exp(- \beta Q) \Leftrightarrow Q=-\frac \ln(P)$ or $P = \exp(+ \beta Q) \Leftrightarrow Q=\frac \ln(P)$

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.

Examples

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

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

References

Statistical mechanics

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