I gather that for the landau free energy, $L = \int d^{d}r (a\phi^2 + b\phi^4$) +... it makes sense to also include gradient terms like $(\nabla \phi)^2$, as the gradient represents the costs of domain interfaces. I have also read that gradient terms take into account local variation, and that near the critical temperature, "fluctuations" vary on long scales (on the order of the correlation length?), so powers of the gradient are small and their inclusion not needed.

How can we characterize the scale of fluctuations, and then of higher powers of gradient terms, to see what actually should be included? Also, what is the connection between "fluctuations" and derivatives of $\phi$?



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