Characteristic state function explained

The characteristic state function or Massieu's potential^[1] 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 =- (1)/(\beta) ln (P) or P = \exp( + \beta Q) \Leftrightarrow Q = (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 hence, its characteristic state function is TS .

• The canonical ensemble satisfies

Z(T,V,N) = e ^{-} hence, its characteristic state function is the Helmholtz free energy A .

• The grand canonical ensemble satisfies

l Z(T,V,\mu) = e ^{-\beta} , so its characteristic state function is the Grand potential \Phi .

• The isothermal-isobaric ensemble satisfies

\Delta(N,T,P) = e ^{-\beta} so its characteristic function is the Gibbs free energy G .

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

References

1. 2017-11-01. François Massieu and the thermodynamic potentials. Comptes Rendus Physique. en. 18. 9–10. 526–530. 10.1016/j.crhy.2017.09.011. 1631-0705. free. Balian. Roger. 2017CRPhy..18..526B. "Massieu's potentials [...] are directly recovered as logarithms of partition functions."