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