Definitions

# 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

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

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 $Omega\left(U,V,N\right) = e^\left\{ beta T S\right\} ;,$ 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 $Z\left(T,V,N\right) = e^\left\{- beta A\right\} ,;$ hence, its characteristic state function is the Helmholtz free energy.
• The grand canonical ensemble satisfies $Xi\left(T,V,mu\right) = e^\left\{beta P V\right\} ,;$, so its characteristic state function is the total Pressure-volume work.
• The isothermal-isobaric ensemble satisfies $Delta\left(N,T,P\right) = e^\left\{-beta G\right\} ;,$ so its characteristic function is the Gibbs free energy.

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