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(- \beta Q)

P = \exp(+ \beta Q)

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 $G$ .

Statistical mechanics

Statistical ensembles	Microcanonical Canonical Grand canonical Isothermal–isobaric Isoenthalpic–isobaric Open

Statistical thermodynamics	Characteristic state functions

Partition functions	Translational Vibrational Rotational

Equations of state	Dieterici Van der Waals Ideal gas law Birch–Murnaghan

Entropy	Sackur–Tetrode equation Tsallis entropy Von Neumann entropy

Particle statistics	Maxwell–Boltzmann statistics Fermi–Dirac statistics Fermi–Dirac distribution Bose–Einstein statistics Bose–Einstein distribution

Statistical field theory	Conformal field theory Osterwalder–Schrader axioms

Quantum statistical mechanics	Gibbs canonical ensemble Density matrix Gibbs measure Partition function Weyl quantization Slater determinant

See also	Probability distribution Structureless particles