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# 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( − βQ) or P = exp( + βQ)

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.