... $y(\vec{x})$¹

A functional is a function of a function; in this case, it takes a function and maps it to a scalar which is numerically equal to the total free energy of the system

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... values²

If $y$ represents a conserved quantity like $c$, then the variation $vt$ must not contribute to the total content of the system ( $\int vt dV = 0$), but we will satisfy this requirement automatically below.

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....³

This is one particular choice for the functional gradient, for which there are an infinite number of choices. This particular choice (the gradient in the $L2$-norm of functions) describes the physics of the problem.

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....⁴

For the general functional, $P[y] = \int Q[y(\vec{x}),\nabla y] dV$, the variational derivative of $P$ is

$$\frac{\partial Q}{\partial y} - \nabla \cdot \frac{\partial Q}{\partial \nabla y}$$

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... quantity.⁵

An extensive quantity is one which depend on the total size of a system, like the volume, enthalpy, or free energy. A molar extensive quantity is scaled by diving by the total number of moles in the system: $\overline{V}(c) = V(c)/(N_A + N_B) = \overline{V}(N_A/(N_A + N_B)) = \overline{V}(\overline{N}_A)$. A molal extensive quantity is scaled by dividing by the number of moles of a particular species.

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