Is the Landau Free Energy U-TS or βH? I'm having a hard time figuring out the physical meaning of the Landau Free Energy density: 
$$f(\phi,\nabla\phi,T) = \frac{1}{2}|\nabla\phi |^2 + \frac{a(T-T_c)}{2}|\phi |^2 + \frac{b}{4}|\phi |^4$$
If $\beta F =\beta \int f d^Dx$ is the Hamiltonian, so that $Z = \int D\phi e^{ F(\phi,\nabla\phi, T)}$, why is it a free energy in the sense that $F = U -TS$ and that $\frac{\partial F}{\partial T} = S$?
 A: In many situations in statistical mechanics, the configuration space that you sum over in your partition function is coarse-grained in a way that certain microscopic degrees of freedom are ignored. The weight of each coarse-grained configuration $c'$ in the partition function includes the sum of all weights of microscopic configurations $c$ that get mapped to $c'$ under coarse-graining. 
As an example, suppose that you start with a microscopic Hamiltonian $\beta \mathcal{H}[c]$. For example, $c$ could faithfully represent the configuration of each particle in a crystal (a complete set of many-body quantum numbers). We may be primarily interested in the behavior of spins, and only the average behavior of spins over relatively large domains. Our reduced degree of freedom could then be a simple spin field $s(x)$, whose Fourier transform is restricted to frequencies below some relatively low cutoff frequency $\Lambda$ (the Fourier transform has support in a ball of radius $\Lambda$ centered on the origin, so that $s(x)$ will be relatively smooth). 
To go from the microscopic picture to the coarse-grained one, we write
\begin{align}
Z = & \sum_c\exp(-\beta\mathcal H[c])\\=&\sum_{s(x)}\sum_{c\rightarrow s(x)}'\exp(-\beta\mathcal H[c]) \\ \equiv & \sum_{s(x)} \exp(-\beta F[s]),
\end{align}
where $\beta F[s]=-\ln(Z'[s])$, and $Z'[s]=\sum_{c\rightarrow s}e^{-\beta\mathcal H[c]}$.  The functional $F[s]$ can be thought of as the free energy of the system when the average local spin is restricted to a certain value $s(x)$. From locality and universality under renormalization group (i.e. a sort of functional 'central limit theorem' that is used routinely in physics), the functional $F[s]$ can typically be decomposed into an integral over local densities like the one you describe.
The 'entropy term' in $F$ comes from the multiplicity of microscopic configurations $c$ with roughly the same energy that get mapped to $s(x)$ under coarse-graining.
