Free-energy density ($f$)

A free-energy density is free energy per unit volume, with units of pressure or energy density. For a homogeneous state, $F=Vf$ when surface contributions are neglected. A function $f(\varphi ,T)$ of an order parameter compares the thermodynamic cost of candidate uniform states at a fixed temperature.

For example, $f=f_0+a{\varphi }^2/2+b{\varphi }^4/4$ with $b>0$ has a minimum at $\varphi =0$ when $a>0$, and minima at $\varphi =\pm \sqrt{-a/b}$ when $a<0$. This is a simple Landau theory description of a change in preferred state. Stability requires the appropriate constrained minimum: a conserved total order parameter or coexistence of phases changes the minimization problem. A spatially varying field additionally requires gradient and boundary terms, so minimizing only a local free-energy density cannot determine an interface profile.