Characteristic state function

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

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⁡(−βQ)⇔Q=−1βln⁡(P) or P=exp⁡(+βQ)⇔Q=1βln⁡(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 Ω(U,V,N)=eβTS hence, its characteristic state function is TS.
* The canonical ensemble satisfies Z(T,V,N)=e−βA hence, its characteristic state function is the Helmholtz free energy A.
* The grand canonical ensemble satisfies 𝒵(T,V,μ)=e−βΦ, so its characteristic state function is the Grand potential Φ.
* The isothermal-isobaric ensemble satisfies Δ(N,T,P)=e−βG 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. ↑ Balian, Roger (2017-11-01). "François Massieu and the thermodynamic potentials" (in en). Comptes Rendus Physique 18 (9–10): 526–530. doi:10.1016/j.crhy.2017.09.011. ISSN 1631-0705. Bibcode: 2017CRPhy..18..526B. "Massieu's potentials [...] are directly recovered as logarithms of partition functions."

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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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