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)

or

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 $\Omega (U,V,N)=e^{\beta TS}\;\,$ 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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

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