Statistical mechanical free energy

Definition (canonical free energy).
For a system with Hamiltonian $H$ in the canonical ensemble , the canonical partition function at inverse temperature β is $Z(\beta,V,N)$. The Helmholtz free energy in statistical mechanics is

It matches the thermodynamic Helmholtz free energy in the thermodynamic limit (up to conventions for additive constants).

Equivalently, one often uses the dimensionless free energy (Massieu potential)

which is especially convenient because its derivatives produce equilibrium observables from log Z .

Key formulas (generating properties).
Let $\langle \cdot \rangle$ denote the ensemble average in the canonical ensemble. Then

and the pressure can be extracted via pressure from the partition function .

Variational (Gibbs) principle.
Let $\rho$ range over all normalized probability densities on phase space (or over all probability measures on microstates). The functional

uses the Gibbs (Shannon) entropy $S_G[\rho]$. The canonical equilibrium state minimizes $\mathcal{F}[\rho]$, and its minimum value equals $F(T,V,N)$. This viewpoint is tightly connected to relative entropy and convex duality ideas such as the Legendre transform .

Other ensembles.
Analogous definitions hold in other ensembles: for the grand canonical ensemble with grand partition function $\Xi(\beta,\mu,V)$, the grand potential is

matching the thermodynamic grand potential . These relationships are organized conceptually by Legendre-transform constructions such as Legendre transform from entropy to free energy .