The diffusion equation arises in two different ways in above table and it is instructive to examine the principles from which they derive.
The homogeneous free energy density (that is, without contribution of the gradients)
can be expanded about its stable state: 
.
Because the integral of C(x) is conserved (or, at least up to its gradients integrated
over an arbitrary boundary), the 
inner product is expected to have the correct
behavior and, indeed, the diffusion equation is gradient flow in this inner product.
Alternatively, we may suppose a penalty exists for gradients alone: 
.
The concentration gradient is not a conserved quantity and naturally the 
inner product
gives the (isotropic, uniform mobility) diffusion equation as gradient flow.