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

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

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 $\Omega (U,V,N)=e^{\beta TS}\;\,$ hence, its characteristic state function is $TS\;\,.$ This quantity roughly speaking, denotes the energy of the entropy at a particular temperature.
• The canonical ensemble satisfies $Z(T,V,N)=e^{-\beta A}\,\;$ hence, its characteristic state function is the Helmholtz free energy.
• The grand canonical ensemble satisfies $\Xi (T,V,\mu )=e^{\beta PV}\,\;$, so its characteristic state function is the total Pressure-volume work.
• The isothermal-isobaric ensemble satisfies $\Delta (N,T,P)=e^{-\beta G}\;\,$ so its characteristic function is the Gibbs free energy.