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 $$ \Omega(U,V,N) = e^{ \beta T S} \;\, $$ 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