Free energy

A free energy is a thermodynamic potential whose decrease governs equilibrium under specified environmental constraints.

A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

At fixed temperature and volume, the Helmholtz free energy is $F=E-TS=-k_{B}T\log Z$ in the canonical ensemble.