# Ginzburg-Landau theory: why Helmholtz energy is used?

I don't understand why the Helmholtz energy is used in the Ginzburg-Landau theory of superconductivity: $$\mathcal{F} = \mathcal{F}_n + \alpha\left|\Psi\right|^2 + \frac{\beta}{2}\left|\Psi\right|^4 + \frac{1}{2m^*}\left|\left(-i\overrightarrow{\nabla}-e^*\overrightarrow{A}\right)\Psi\right|^2 + \frac{\left|\overrightarrow{\nabla}\times\overrightarrow{A}\right|^2}{2}$$

I would say that this is the energy of the system, not the Helmholtz free energy, I may be losing something.

Ginzburg-Landau theory is really a theory of phase transition, grounded in the Landau theory of phase transitions, with the superconducting wave function, $$\Psi$$ serving as the order parameters. Thus, this is neither Helmholtz energy, nor the Hamiltonian, but the Landau function. In fact, the first three terms are a usual Landau function for describing a second-order phase transition, whereas the remaining terms are added to describe the variation of the order parameters in space (and sometimes in time) - this implies that this variation is adiabatic, i.e., the potentials are changing slowly on the scales of the cooper pair size.
• Read "On the theory of superconductivity" at "Collected Papers of L. D. Landau", where Landau and Ginzburg say "Thus near $T_C$ we may write for the free energy..." and then they write equation (6) on the paper, where you have the three first terms of the energy in question. May 20, 2021 at 6:16