Physics:Characteristic state function

Short description: Particular relationship between the partition function of an ensemble

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

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

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

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