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{{short description|Particular relationship between the partition function of an ensemble}}
{{Refimprove|date=July 2007}}
The '''characteristic state function''' or '''Massieu's potential'''<ref>{{Cite journal|date=2017-11-01|title=François Massieu and the thermodynamic potentials|journal=Comptes Rendus Physique|language=en|volume=18|issue=9–10|pages=526–530|doi=10.1016/j.crhy.2017.09.011|issn=1631-0705|doi-access=free|last1=Balian|first1=Roger|bibcode=2017CRPhy..18..526B}} "Massieu's potentials [...] are directly recovered as logarithms of partition functions."</ref> in [[statistical mechanics]] refers to a particular relationship between the [[partition function (statistical mechanics)|partition function]] of an [[Statistical ensemble (mathematical physics)|ensemble]].

In particular, if the partition function ''P'' satisfies

:<math>P = \exp(- \beta Q) \Leftrightarrow Q=-\frac{1}{\beta} \ln(P) </math> or <math>P = \exp(+ \beta Q) \Leftrightarrow Q=\frac{1}{\beta} \ln(P) </math>

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

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

== References ==
<references />

{{Statistical mechanics topics}}
[[Category:Statistical mechanics]]

{{thermodynamics-stub}}
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Characteristic state function - Revision history

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