Characteristic state function

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

In particular, if the partition function P satisfies

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

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.

Examples

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

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