# Characteristic state function

The characteristic state function in statistical mechanics refers to a particular relationship between the partition function of an ensemble.

In particular, if the partition function P satisfies

${\displaystyle P=\exp(-\beta Q)}$ or ${\displaystyle P=\exp(+\beta Q)}$

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

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