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

In particular, if the partition function P satisfies

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

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

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

References

1. Balian, Roger (2017-11-01). "François Massieu and the thermodynamic potentials". Comptes Rendus Physique. 18 (9–10): 526–530. Bibcode:2017CRPhy..18..526B. doi:10.1016/j.crhy.2017.09.011. ISSN 1631-0705. "Massieu's potentials are directly recovered as logarithms of partition functions."

v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine

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