characteristic state function

{{short description|Particular relationship between the partition function of an ensemble}}

The characteristic state function or Massieu's potential{{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." in statistical mechanics refers to a particular relationship between the partition function of an ensemble.

In particular, if the partition function P satisfies

: $P = \exp(- \beta Q) \Leftrightarrow Q=-\frac{1}{\beta} \ln(P)$ or $P = \exp(+ \beta Q) \Leftrightarrow Q=\frac{1}{\beta} \ln(P)$

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

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

References

Category:Statistical mechanics