Functional derivative

In this section, the functional differential (or variation or first variation) Called first variation in {{harv|Giaquinta|Hildebrandt|1996|p=3}}, variation or first variation in {{harv|Courant|Hilbert|1953|p=186}}, variation or differential in {{harv|Gelfand|Fomin|2000|loc= p. 11, § 3.2}} and differential in {{harv|Parr|Yang|1989|p=246}}. is defined. Then the functional derivative is defined in terms of the functional differential.

Suppose $B$ is a Banach space and $F$ is a functional defined on $B$.

The differential of $F$ at a point $\backslash rho\backslash in\; B$ is the linear functional $\backslash delta\; F[\backslash rho,\backslash cdot]$ on $B$ defined{{harvp|Gelfand|Fomin|2000|p=11}}. by the condition that, for all $\backslash phi\backslash in\; B$,

F[\rho+\phi] - F[\rho]

=

\delta F [\rho; \phi] + \varepsilon \left\|\phi\right\|

where $\backslash varepsilon$ is a real number that depends on $\backslash |\backslash phi\backslash |$ in such a way that $\backslash varepsilon\backslash to\; 0$ as $\backslash |\backslash phi\backslash |\backslash to\; 0$. This means that $\backslash delta\; F[\backslash rho,\backslash cdot]$ is the Fréchet derivative of $F$ at $\backslash rho$.

However, this notion of functional differential is so strong it may not exist,{{harvp|Giaquinta|Hildebrandt|1996|p=10}}. and in those cases a weaker notion, like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined{{harvp|Giaquinta|Hildebrandt|1996|p=10}}. as the directional derivative

\begin{align}

\delta F[\rho,\phi]

&= \lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon \phi]-F[\rho]}{\varepsilon} \\[1ex]

&= \left [ \frac{d}{d\varepsilon}F[\rho+\varepsilon \phi]\right ]_{\varepsilon=0}.

\end{align}

Note that this notion of the functional differential can even be defined without a norm.

In many applications, the domain of the functional $F$ is a space of differentiable functions $\backslash rho$ defined on some space $\backslash Omega$ and $F$ is of the form

F[\rho]

=

\int_\Omega L(x,\rho(x),D\rho(x))\,dx

for some function $L(x,\backslash rho(x),D\backslash rho(x))$ that may depend on $x$, the value $\backslash rho(x)$ and the derivative $D\backslash rho(x)$.

If this is the case and, moreover, $\backslash delta\; F[\backslash rho,\backslash phi]$ can be written as the integral of $\backslash phi$ times another function (denoted {{math|δF/δρ}})

$$\backslash delta\; F\; [\backslash rho,\; \backslash phi]\; =\; \backslash int\_\backslash Omega\; \backslash frac\; \{\backslash delta\; F\}\; \{\backslash delta\; \backslash rho\}(x)\; \backslash \; \backslash phi(x)\; \backslash \; dx$$

then this function {{math|δF/δρ}} is called the functional derivative of {{math|F}} at {{math|ρ}}.{{harvp|Parr|Yang|1989|loc= p. 246, Eq. A.2}}.{{harvp|Greiner|Reinhardt|1996|p=36,37}}. If $F$ is restricted to only certain functions $\backslash rho$ (for example, if there are some boundary conditions imposed) then $\backslash phi$ is restricted to functions such that $\backslash rho+\backslash varepsilon\backslash phi$ continues to satisfy these conditions.

Heuristically, $\backslash phi$ is the change in $\backslash rho$, so we 'formally' have $\backslash phi\; =\; \backslash delta\backslash rho$, and then this is similar in form to the total differential of a function $F(\backslash rho\_1,\backslash rho\_2,\backslash dots,\backslash rho\_n)$,

$$dF\; =\; \backslash sum\_\{i=1\}\; ^n\; \backslash frac\; \{\backslash partial\; F\}\; \{\backslash partial\; \backslash rho\_i\}\; \backslash \; d\backslash rho\_i\; ,$$

where $\backslash rho\_1,\backslash rho\_2,\backslash dots,\backslash rho\_n$ are independent variables.

Comparing the last two equations, the functional derivative $\backslash delta\; F/\backslash delta\backslash rho(x)$ has a role similar to that of the partial derivative $\backslash partial\; F/\backslash partial\backslash rho\_i$, where the variable of integration $x$ is like a continuous version of the summation index $i$.{{harvp|Parr|Yang|1989|p=246}}. One thinks of {{math|δF/δρ}} as the gradient of {{math|F}} at the point {{math|ρ}}, so the value {{math|δF/δρ(x)}} measures how much the functional {{math|F}} will change if the function {{math|ρ}} is changed at the point {{math|x}}. Hence the formula

$$\backslash int\; \backslash frac\{\backslash delta\; F\}\{\backslash delta\backslash rho\}(x)\; \backslash phi(x)\; \backslash ;\; dx$$

is regarded as the directional derivative at point $\backslash rho$ in the direction of $\backslash phi$. This is analogous to vector calculus, where the inner product of a vector $v$ with the gradient gives the directional derivative in the direction of $v$.

Like the derivative of a function, the functional derivative satisfies the following properties, where {{math|F[ρ]}} and {{math|G[ρ]}} are functionals:

Here the notation

$$\backslash frac\{\backslash delta\{F\}\}\{\backslash delta\backslash rho\}(x)\; \backslash equiv\; \backslash frac\{\backslash delta\{F\}\}\{\backslash delta\backslash rho(x)\}$$

is introduced.

• Linearity:{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.3}}. $$\backslash frac\{\backslash delta(\backslash lambda\; F\; +\; \backslash mu\; G)[\backslash rho\; ]\}\{\backslash delta\; \backslash rho(x)\}\; =\; \backslash lambda\; \backslash frac\{\backslash delta\; F[\backslash rho]\}\{\backslash delta\; \backslash rho(x)\}\; +\; \backslash mu\; \backslash frac\{\backslash delta\; G[\backslash rho]\}\{\backslash delta\; \backslash rho(x)\},$$ where {{math|λ, μ}} are constants.
• Product rule:{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.4}}. $$\backslash frac\{\backslash delta(FG)[\backslash rho]\}\{\backslash delta\; \backslash rho(x)\}\; =\; \backslash frac\{\backslash delta\; F[\backslash rho]\}\{\backslash delta\; \backslash rho(x)\}\; G[\backslash rho]\; +\; F[\backslash rho]\; \backslash frac\{\backslash delta\; G[\backslash rho]\}\{\backslash delta\; \backslash rho(x)\}\; \backslash ,\; ,$$
• Chain rules:
• If {{math|F}} is a functional and {{math|G}} another functional, then{{harvp|Greiner|Reinhardt|1996|loc=p. 38, Eq. 6}}. $$\backslash frac\{\backslash delta\; F[G[\backslash rho]]\; \}\{\backslash delta\backslash rho(y)\}\; =\; \backslash int\; dx\; \backslash frac\{\backslash delta\; F[G]\}\{\backslash delta\; G(x)\}\_\{G\; =\; G[\backslash rho]\}\backslash cdot\backslash frac\; \{\backslash delta\; G[\backslash rho](x)\}\; \{\backslash delta\backslash rho(y)\}\; \backslash \; .$$
• If {{math|G}} is an ordinary differentiable function (local functional) {{math|g}}, then this reduces to{{harvp|Greiner|Reinhardt|1996|loc=p. 38, Eq. 7}}. $$\backslash frac\{\backslash delta\; F[g(\backslash rho)]\; \}\{\backslash delta\backslash rho(y)\}\; =\; \backslash frac\{\backslash delta\; F[g(\backslash rho)]\}\{\backslash delta\; g[\backslash rho(y)\; ]\}\; \backslash \; \backslash frac\; \{dg(\backslash rho)\}\; \{d\backslash rho(y)\}\; \backslash \; .$$

A formula to determine functional derivatives for a common class of functionals can be written as the integral of a function and its derivatives. This is a generalization of the Euler–Lagrange equation: indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics (18th century). The first three examples below are taken from density functional theory (20th century), the fourth from statistical mechanics (19th century).

Given a functional

$$F[\backslash rho]\; =\; \backslash int\; f(\; \backslash boldsymbol\{r\},\; \backslash rho(\backslash boldsymbol\{r\}),\; \backslash nabla\backslash rho(\backslash boldsymbol\{r\})\; )\backslash ,\; d\backslash boldsymbol\{r\},$$

and a function $\backslash phi(\backslash boldsymbol\{r\})$ that vanishes on the boundary of the region of integration, from a previous section Definition,

$$\backslash begin\{align\}$$

\int \frac{\delta F}{\delta\rho(\boldsymbol{r})} \, \phi(\boldsymbol{r}) \, d\boldsymbol{r}

& = \left [ \frac{d}{d\varepsilon} \int f( \boldsymbol{r}, \rho + \varepsilon \phi, \nabla\rho+\varepsilon\nabla\phi )\, d\boldsymbol{r} \right ]_{\varepsilon=0} \\

& = \int \left( \frac{\partial f}{\partial\rho} \, \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d\boldsymbol{r} \\

& = \int \left[ \frac{\partial f}{\partial\rho} \, \phi + \nabla \cdot \left( \frac{\partial f}{\partial\nabla\rho} \, \phi \right) - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\boldsymbol{r} \\

& = \int \left[ \frac{\partial f}{\partial\rho} \, \phi - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\boldsymbol{r} \\

& = \int \left( \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi(\boldsymbol{r}) \ d\boldsymbol{r} \, .

\end{align}

The second line is obtained using the total derivative, where {{math|∂f /∂∇ρ}} is a derivative of a scalar with respect to a vector.For a three-dimensional Cartesian coordinate system,

$$\backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash nabla\backslash rho\}\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash rho\_x\}\; \backslash mathbf\{\backslash hat\{i\}\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash rho\_y\}\; \backslash mathbf\{\backslash hat\{j\}\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash rho\_z\}\; \backslash mathbf\{\backslash hat\{k\}\}\backslash ,\; ,$$

where $\backslash rho\_x\; =\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; x\}\backslash ,\; ,\; \backslash \; \backslash rho\_y\; =\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; y\}\backslash ,\; ,\; \backslash \; \backslash rho\_z\; =\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; z\}$ and $\backslash mathbf\{\backslash hat\{i\}\}$, $\backslash mathbf\{\backslash hat\{j\}\}$, $\backslash mathbf\{\backslash hat\{k\}\}$ are unit vectors along the x, y, z axes.

The third line was obtained by use of a product rule for divergence. The fourth line was obtained using the divergence theorem and the condition that $\backslash phi=0$ on the boundary of the region of integration. Since $\backslash phi$ is also an arbitrary function, applying the fundamental lemma of calculus of variations to the last line, the functional derivative is

$$\backslash frac\{\backslash delta\; F\}\{\backslash delta\backslash rho(\backslash boldsymbol\{r\})\}\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash rho\}\; -\; \backslash nabla\; \backslash cdot\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash nabla\backslash rho\}$$

where {{math|1=ρ = ρ(r)}} and {{math|1=f = f (r, ρ, ∇ρ)}}. This formula is for the case of the functional form given by {{math|F[ρ]}} at the beginning of this section. For other functional forms, the definition of the functional derivative can be used as the starting point for its determination. (See the example Coulomb potential energy functional.)

The above equation for the functional derivative can be generalized to the case that includes higher dimensions and higher order derivatives. The functional would be,

$$F[\backslash rho(\backslash boldsymbol\{r\})]\; =\; \backslash int\; f(\; \backslash boldsymbol\{r\},\; \backslash rho(\backslash boldsymbol\{r\}),\; \backslash nabla\backslash rho(\backslash boldsymbol\{r\}),\; \backslash nabla^\{(2)\}\backslash rho(\backslash boldsymbol\{r\}),\; \backslash dots,\; \backslash nabla^\{(N)\}\backslash rho(\backslash boldsymbol\{r\}))\backslash ,\; d\backslash boldsymbol\{r\},$$

where the vector {{math|r ∈ Rⁿ}}, and {{math|∇⁽ⁱ⁾}} is a tensor whose {{math|nⁱ}} components are partial derivative operators of order {{math|i}},

$$\backslash left\; [\; \backslash nabla^\{(i)\}\; \backslash right\; ]\_\{\backslash alpha\_1\; \backslash alpha\_2\; \backslash cdots\; \backslash alpha\_i\}\; =\; \backslash frac\; \{\backslash partial^\{\backslash ,\; i\}\}\; \{\backslash partial\; r\_\{\backslash alpha\_1\}\; \backslash partial\; r\_\{\backslash alpha\_2\}\; \backslash cdots\; \backslash partial\; r\_\{\backslash alpha\_i\}\; \}\; \backslash qquad\; \backslash qquad\; \backslash text\{where\}\; \backslash quad\; \backslash alpha\_1,\; \backslash alpha\_2,\; \backslash dots,\; \backslash alpha\_i\; =\; 1,\; 2,\; \backslash dots\; ,\; n\; \backslash \; .$$For example, for the case of three dimensions ({{math|1=n = 3}}) and second order derivatives ({{math|1=i = 2}}), the tensor {{math|∇⁽²⁾}} has components,

$$\backslash left\; [\; \backslash nabla^\{(2)\}\; \backslash right\; ]\_\{\backslash alpha\; \backslash beta\}\; =\; \backslash frac\; \{\backslash partial^\{\backslash ,2\}\}\; \{\backslash partial\; r\_\{\backslash alpha\}\; \backslash ,\; \backslash partial\; r\_\{\backslash beta\}\}$$where $\backslash alpha$ and $\backslash beta$ can be $1,2,3$.

An analogous application of the definition of the functional derivative yields

$$\backslash begin\{align\}$$

\frac{\delta F[\rho]}{\delta \rho} &{} = \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial(\nabla\rho)} + \nabla^{(2)} \cdot \frac{\partial f}{\partial\left(\nabla^{(2)}\rho\right)} + \dots + (-1)^N \nabla^{(N)} \cdot \frac{\partial f}{\partial\left(\nabla^{(N)}\rho\right)} \\

&{} = \frac{\partial f}{\partial\rho} + \sum_{i=1}^N (-1)^{i}\nabla^{(i)} \cdot \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} \ .

\end{align}

In the last two equations, the {{math|nⁱ}} components of the tensor $\backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash left(\backslash nabla^\{(i)\}\backslash rho\backslash right)\}$ are partial derivatives of {{math|f}} with respect to partial derivatives of ρ,

$$\backslash left\; [\; \backslash frac\; \{\backslash partial\; f\}\; \{\backslash partial\; \backslash left\; (\backslash nabla^\{(i)\}\backslash rho\; \backslash right\; )\; \}\; \backslash right\; ]\_\{\backslash alpha\_1\; \backslash alpha\_2\; \backslash cdots\; \backslash alpha\_i\}\; =\; \backslash frac\; \{\backslash partial\; f\}\; \{\backslash partial\; \backslash rho\_\{\backslash alpha\_1\; \backslash alpha\_2\; \backslash cdots\; \backslash alpha\_i\}\; \}$$

where $\backslash rho\_\{\backslash alpha\_1\; \backslash alpha\_2\; \backslash cdots\; \backslash alpha\_i\}\; \backslash equiv\; \backslash frac\; \{\backslash partial^\{\backslash ,i\}\backslash rho\}\; \{\backslash partial\; r\_\{\backslash alpha\_1\}\; \backslash ,\; \backslash partial\; r\_\{\backslash alpha\_2\}\; \backslash cdots\; \backslash partial\; r\_\{\backslash alpha\_i\}\; \}$, and the tensor scalar product is,

$$\backslash nabla^\{(i)\}\; \backslash cdot\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash left(\backslash nabla^\{(i)\}\backslash rho\backslash right)\}\; =\; \backslash sum\_\{\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash cdots,\; \backslash alpha\_i\; =\; 1\}^n\; \backslash \; \backslash frac\; \{\backslash partial^\{\backslash ,\; i\}\; \}\; \{\backslash partial\; r\_\{\backslash alpha\_1\}\; \backslash ,\; \backslash partial\; r\_\{\backslash alpha\_2\}\; \backslash cdots\; \backslash partial\; r\_\{\backslash alpha\_i\}\; \}\; \backslash \; \backslash frac\; \{\backslash partial\; f\}\; \{\backslash partial\; \backslash rho\_\{\backslash alpha\_1\; \backslash alpha\_2\; \backslash cdots\; \backslash alpha\_i\}\; \}\; \backslash \; .$$ For example, for the case {{math|1=n = 3}} and {{math|1=i = 2}}, the tensor scalar product is,

$$\backslash nabla^\{(2)\}\; \backslash cdot\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\backslash left(\backslash nabla^\{(2)\}\backslash rho\backslash right)\}\; =\; \backslash sum\_\{\backslash alpha,\; \backslash beta\; =\; 1\}^3\; \backslash \; \backslash frac\; \{\backslash partial^\{\backslash ,\; 2\}\; \}\; \{\backslash partial\; r\_\{\backslash alpha\}\; \backslash ,\; \backslash partial\; r\_\{\backslash beta\}\; \}\; \backslash ,\; \backslash frac\; \{\backslash partial\; f\}\; \{\backslash partial\; \backslash rho\_\{\backslash alpha\; \backslash beta\}\; \}\; ,$$where $\backslash rho\_\{\backslash alpha\; \backslash beta\}\; \backslash equiv\; \backslash frac\; \{\backslash partial^\{\backslash ,\; 2\}\backslash rho\}\; \{\backslash partial\; r\_\{\backslash alpha\}\; \backslash ,\; \backslash partial\; r\_\{\backslash beta\}\; \}$.

The Thomas–Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of density-functional theory of electronic structure:

$$T\_\backslash mathrm\{TF\}[\backslash rho]\; =\; C\_\backslash mathrm\{F\}\; \backslash int\; \backslash rho^\{5/3\}(\backslash mathbf\{r\})\; \backslash ,\; d\backslash mathbf\{r\}\; \backslash ,\; .$$

Since the integrand of {{math|T_TF[ρ]}} does not involve derivatives of {{math|ρ(r)}}, the functional derivative of {{math|T_TF[ρ]}} is,{{harvp|Parr|Yang|1989|loc=p. 247, Eq. A.6}}.

$$\backslash frac\{\backslash delta\; T\_\{\backslash mathrm\{TF\}\}\}\{\backslash delta\; \backslash rho\; (\backslash boldsymbol\{r\})\; \}$$

= C_\mathrm{F} \frac{\partial \rho^{5/3}(\mathbf{r})}{\partial \rho(\mathbf{r})}

= \frac{5}{3} C_\mathrm{F} \rho^{2/3}(\mathbf{r}) \, .

The electron-nucleus potential energy is

$$V[\backslash rho]\; =\; \backslash int\; \backslash frac\{\backslash rho(\backslash boldsymbol\{r\})\}$$

Applying the definition of functional derivative,

$$\backslash begin\{align\}$$

\int \frac{\delta V}{\delta \rho(\boldsymbol{r})} \ \phi(\boldsymbol{r}) \ d\boldsymbol{r}

& {} = \left [ \frac{d}{d\varepsilon} \int \frac{\rho(\boldsymbol{r}) + \varepsilon \phi(\boldsymbol{r})}

& {} = \int \frac {\phi(\boldsymbol{r})}

\end{align}

So,

$$\backslash frac\{\backslash delta\; V\}\{\backslash delta\; \backslash rho(\backslash boldsymbol\{r\})\}\; =\; \backslash frac\{1\}$$

The functional derivative of the classical part of the electron-electron interaction (often called Hartree energy) is

$$J[\backslash rho]\; =\; \backslash frac\{1\}\{2\}\backslash iint\; \backslash frac\{\backslash rho(\backslash mathbf\{r\})\; \backslash rho(\backslash mathbf\{r\}\text{'})\}$$

From the definition of the functional derivative,

$$\backslash begin\{align\}$$

\int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r}

& {} = \left [ \frac {d \ }{d\varepsilon} \, J[\rho + \varepsilon\phi] \right ]_{\varepsilon = 0} \\

& {} = \left [ \frac {d \ }{d\varepsilon} \, \left ( \frac{1}{2}\iint \frac {[\rho(\boldsymbol{r}) + \varepsilon \phi(\boldsymbol{r})] \, [\rho(\boldsymbol{r}') + \varepsilon \phi(\boldsymbol{r}')] }

& {} = \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}') \phi(\boldsymbol{r}) }

\end{align}

The first and second terms on the right hand side of the last equation are equal, since {{math|r}} and {{math|r′}} in the second term can be interchanged without changing the value of the integral. Therefore,

$$\backslash int\; \backslash frac\{\backslash delta\; J\}\{\backslash delta\backslash rho(\backslash boldsymbol\{r\})\}\; \backslash phi(\backslash boldsymbol\{r\})d\backslash boldsymbol\{r\}\; =\; \backslash int\; \backslash left\; (\; \backslash int\; \backslash frac\; \{\backslash rho(\backslash boldsymbol\{r\}\text{'})\; \}$$

and the functional derivative of the electron-electron Coulomb potential energy functional {{math|J}}[ρ] is,{{harvp|Parr|Yang|1989|loc=p. 248, Eq. A.11}}.

$$\backslash frac\{\backslash delta\; J\}\{\backslash delta\backslash rho(\backslash boldsymbol\{r\})\}\; =\; \backslash int\; \backslash frac\; \{\backslash rho(\backslash boldsymbol\{r\}\text{'})\; \}$$

The second functional derivative is

$$\backslash frac\{\backslash delta^2\; J[\backslash rho]\}\{\backslash delta\; \backslash rho(\backslash mathbf\{r\}\text{'})\backslash delta\backslash rho(\backslash mathbf\{r\})\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash rho(\backslash mathbf\{r\}\text{'})\}\; \backslash left\; (\; \backslash frac\{\backslash rho(\backslash mathbf\{r\}\text{'})\}$$

In 1935 von Weizsäcker proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it better suit a molecular electron cloud:

$$T\_\backslash mathrm\{W\}[\backslash rho]\; =\; \backslash frac\{1\}\{8\}\; \backslash int\; \backslash frac\{\backslash nabla\backslash rho(\backslash mathbf\{r\})\; \backslash cdot\; \backslash nabla\backslash rho(\backslash mathbf\{r\})\}\{\; \backslash rho(\backslash mathbf\{r\})\; \}\; d\backslash mathbf\{r\}\; =\; \backslash int\; t\_\backslash mathrm\{W\}(\backslash mathbf\{r\})\; \backslash \; d\backslash mathbf\{r\}\; \backslash ,\; ,$$

where

$$t\_\backslash mathrm\{W\}\; \backslash equiv\; \backslash frac\{1\}\{8\}\; \backslash frac\{\backslash nabla\backslash rho\; \backslash cdot\; \backslash nabla\backslash rho\}\{\; \backslash rho\; \}\; \backslash qquad\; \backslash text\{and\}\; \backslash \; \backslash \; \backslash rho\; =\; \backslash rho(\backslash boldsymbol\{r\})\; \backslash \; .$$

Using a previously derived formula for the functional derivative,

$$\backslash begin\{align\}$$

\frac{\delta T_\mathrm{W}}{\delta \rho}

& = \frac{\partial t_\mathrm{W}}{\partial \rho} - \nabla\cdot\frac{\partial t_\mathrm{W}}{\partial \nabla \rho} \\

& = -\frac{1}{8}\frac{\nabla\rho \cdot \nabla\rho}{\rho^2} - \left ( \frac {1}{4} \frac {\nabla^2\rho} {\rho} - \frac {1}{4} \frac {\nabla\rho \cdot \nabla\rho} {\rho^2} \right ) \qquad \text{where} \ \ \nabla^2 = \nabla \cdot \nabla \ ,

\end{align}

and the result is,{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.9}}.

$$\backslash frac\{\backslash delta\; T\_\backslash mathrm\{W\}\}\{\backslash delta\; \backslash rho\}\; =\; \backslash \; \backslash \; \backslash ,\; \backslash frac\{1\}\{8\}\backslash frac\{\backslash nabla\backslash rho\; \backslash cdot\; \backslash nabla\backslash rho\}\{\backslash rho^2\}\; -\; \backslash frac\{1\}\{4\}\backslash frac\{\backslash nabla^2\backslash rho\}\{\backslash rho\}\; \backslash \; .$$

The entropy of a discrete random variable is a functional of the probability mass function.

$$H[p(x)]\; =\; -\backslash sum\_x\; p(x)\; \backslash log\; p(x)$$

Thus,

$$\backslash begin\{align\}$$

\sum_x \frac{\delta H}{\delta p(x)} \, \phi(x)

& {} = \left[ \frac{d}{d\varepsilon} H[p(x) + \varepsilon\phi(x)] \right]_{\varepsilon=0}\\

& {} = \left [- \, \frac{d}{d\varepsilon} \sum_x \, [p(x) + \varepsilon\phi(x)] \ \log [p(x) + \varepsilon\phi(x)] \right]_{\varepsilon=0} \\

& {} = -\sum_x \, [1+\log p(x)] \ \phi(x) \, .

\end{align}

Thus,

$$\backslash frac\{\backslash delta\; H\}\{\backslash delta\; p(x)\}\; =\; -1-\backslash log\; p(x).$$

Let

$$F[\backslash varphi(x)]=\; e^\{\backslash int\; \backslash varphi(x)\; g(x)dx\}.$$

Using the delta function as a test function,

$$\backslash begin\{align\}$$

\frac{\delta F[\varphi(x)]}{\delta \varphi(y)}

& {} = \lim_{\varepsilon\to 0}\frac{F[\varphi(x)+\varepsilon\delta(x-y)]-F[\varphi(x)]}{\varepsilon}\\

& {} = \lim_{\varepsilon\to 0}\frac{e^{\int (\varphi(x)+\varepsilon\delta(x-y)) g(x)dx}-e^{\int \varphi(x) g(x)dx}}{\varepsilon}\\

& {} = e^{\int \varphi(x) g(x)dx}\lim_{\varepsilon\to 0}\frac{e^{\varepsilon \int \delta(x-y) g(x)dx}-1}{\varepsilon}\\

& {} = e^{\int \varphi(x) g(x)dx}\lim_{\varepsilon\to 0}\frac{e^{\varepsilon g(y)}-1}{\varepsilon}\\

& {} = e^{\int \varphi(x) g(x)dx}g(y).

\end{align}

Thus,

$$\backslash frac\{\backslash delta\; F[\backslash varphi(x)]\}\{\backslash delta\; \backslash varphi(y)\}\; =\; g(y)\; F[\backslash varphi(x)].$$

This is particularly useful in calculating the correlation functions from the partition function in quantum field theory.

A function can be written in the form of an integral like a functional. For example,

$$\backslash rho(\backslash boldsymbol\{r\})\; =\; F[\backslash rho]\; =\; \backslash int\; \backslash rho(\backslash boldsymbol\{r\}\text{'})\; \backslash delta(\backslash boldsymbol\{r\}-\backslash boldsymbol\{r\}\text{'})\backslash ,\; d\backslash boldsymbol\{r\}\text{'}.$$

Since the integrand does not depend on derivatives of ρ, the functional derivative of ρ{{math|(r)}} is,

$$\backslash frac\; \{\backslash delta\; \backslash rho(\backslash boldsymbol\{r\})\}\; \{\backslash delta\backslash rho(\backslash boldsymbol\{r\}\text{'})\}\; \backslash equiv\; \backslash frac\; \{\backslash delta\; F\}\; \{\backslash delta\backslash rho(\backslash boldsymbol\{r\}\text{'})\}$$

= \frac{\partial \ \ }{\partial \rho(\boldsymbol{r}')} \, [\rho(\boldsymbol{r}') \delta(\boldsymbol{r}-\boldsymbol{r}')]

= \delta(\boldsymbol{r}-\boldsymbol{r}').

The functional derivative of the iterated function $f(f(x))$ is given by:

$$\backslash frac\{\backslash delta\; f(f(x))\}\{\backslash delta\; f(y)\; \}\; =\; f\text{'}(f(x))\backslash delta(x-y)\; +\; \backslash delta(f(x)-y)$$

and

$$\backslash frac\{\backslash delta\; f(f(f(x)))\}\{\backslash delta\; f(y)\; \}\; =\; f\text{'}(f(f(x))(f\text{'}(f(x))\backslash delta(x-y)\; +\; \backslash delta(f(x)-y))\; +\; \backslash delta(f(f(x))-y)$$

In general:

$$\backslash frac\{\backslash delta\; f^N(x)\}\{\backslash delta\; f(y)\}\; =\; f\text{'}(\; f^\{N-1\}(x)\; )\; \backslash frac\{\; \backslash delta\; f^\{N-1\}(x)\}\{\backslash delta\; f(y)\}\; +\; \backslash delta(\; f^\{N-1\}(x)\; -\; y\; )$$

Putting in {{math|1=N = 0}} gives:

$$\backslash frac\{\backslash delta\; f^\{-1\}(x)\}\{\backslash delta\; f(y)\; \}\; =\; -\; \backslash frac\{\; \backslash delta(f^\{-1\}(x)-y\; )\; \}\{\; f\text{'}(f^\{-1\}(x))\; \}$$

In physics, it is common to use the Dirac delta function $\backslash delta(x-y)$ in place of a generic test function $\backslash phi(x)$, for yielding the functional derivative at the point $y$ (this is a point of the whole functional derivative as a partial derivative is a component of the gradient):{{harvp|Greiner|Reinhardt|1996|p=37}}

$$\backslash frac\{\backslash delta\; F[\backslash rho(x)]\}\{\backslash delta\; \backslash rho(y)\}=\backslash lim\_\{\backslash varepsilon\backslash to\; 0\}\backslash frac\{F[\backslash rho(x)+\backslash varepsilon\backslash delta(x-y)]-F[\backslash rho(x)]\}\{\backslash varepsilon\}.$$

This works in cases when $F[\backslash rho(x)+\backslash varepsilon\; f(x)]$ formally can be expanded as a series (or at least up to first order) in $\backslash varepsilon$. The formula is however not mathematically rigorous, since $F[\backslash rho(x)+\backslash varepsilon\backslash delta(x-y)]$ is usually not even defined.

The definition given in a previous section is based on a relationship that holds for all test functions $\backslash phi(x)$, so one might think that it should hold also when $\backslash phi(x)$ is chosen to be a specific function such as the delta function. However, the latter is not a valid test function (it is not even a proper function).

In the definition, the functional derivative describes how the functional $F[\backslash rho(x)]$ changes as a result of a small change in the entire function $\backslash rho(x)$. The particular form of the change in $\backslash rho(x)$ is not specified, but it should stretch over the whole interval on which $x$ is defined. Employing the particular form of the perturbation given by the delta function has the meaning that $\backslash rho(x)$ is varied only in the point $y$. Except for this point, there is no variation in $\backslash rho(x)$.

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{{Functional analysis}}

{{Analysis in topological vector spaces}}

Category:Calculus of variations

Category:Differential calculus

Category:Differential operators

Category:Topological vector spaces

Category:Variational analysis