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| In the [[calculus of variations]], a field of [[mathematical analysis]], the '''functional derivative''' (or '''variational derivative''')<ref name="GiaquintaHildebrandtP18">{{harv|Giaquinta|Hildebrandt|1996|p=18}}</ref> relates a change in a [[Functional (mathematics)|functional]] to a change in a [[Function (mathematics)|function]] that the functional depends on.
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| In the calculus of variations, functionals are usually expressed in terms of an [[integral]] of functions, their [[Argument of a function|arguments]], and their [[derivative]]s. In an integrand {{math|''L''}} of a functional, if a function {{math|''f''}} is varied by adding to it another function {{math|''δf''}} that is arbitrarily small, and the resulting {{math|''L''}} is expanded in powers of {{math|''δf''}}, the coefficient of {{math|''δf''}} in the first order term is called the functional derivative.
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| For example, consider the functional
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| :<math> J[f] = \int \limits_a^b L[ \, x, f(x), f \, '(x) \, ] \, dx \ , </math>
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| where {{math|''f'' ′(''x'') ≡ ''df/dx''}}. If {{math|''f''}} is varied by adding to it a function {{math|''δf''}}, and the resulting integrand {{math|''L''(''x, f +δf, f '+δf'' ′)}} is expanded in powers of {{math|''δf''}}, then the change in the value of {{math|''J''}} to first order in {{math|''δf''}} can be expressed as follows:<ref name="GiaquintaHildebrandtP18" /><ref Group = 'Note'>According to {{harvtxt|Giaquinta|Hildebrandt|1996|p=18}}, this notation is customary in [[Physics|physical]] literature.</ref>
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| :<math> \delta J = \int_a^b \frac{\delta J}{\delta f(x)} {\delta f(x)} dx \, . </math>
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| The coefficient of {{math|''δf(x)''}}, denoted as {{math|''δJ''/''δf(x)''}}, is called the '''functional derivative''' of {{math|''J''}} with respect to {{math|''f''}} at the point {{math|''x''}}.<ref name=ParrYangP246>{{harv|Parr|Yang|1989|p=246}}.</ref> For this example functional, the functional derivative is the left hand side of the [[Calculus_of_variations#Euler.E2.80.93Lagrange_equation|Euler-Lagrange equation]],<ref name=GelfandFominP28>{{harv|Gelfand|Fomin|2000|p=28}}</ref>
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| :<math> \frac{\delta J}{\delta f(x)} = \frac{\part L}{\part f} -\frac{d}{dx} \frac{\part L}{\part f'} \, . </math> | |
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| ==Definition==
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| The functional derivative is defined. Then the functional differential is defined in terms of the functional derivative.
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| ===Functional derivative===
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| Given a [[manifold]] ''M'' representing ([[continuous function (topology)|continuous]]/[[smooth function|smooth]]/with certain [[boundary condition]]s/etc.) functions ''ρ'' and a [[functional (mathematics)|functional]] ''F'' defined as
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| ::<math>F\colon M \rightarrow \mathbb{R} \quad \mbox{or} \quad F\colon M \rightarrow \mathbb{C} \, ,</math>
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| the '''functional derivative''' of {{math|''F''[}}''ρ''], denoted {{math|''δF/δ''}}''ρ'', is defined by<ref name=ParrYangP246A.2>{{harv|Parr|Yang|1989|loc= p. 246, Eq. A.2}}.</ref> | |
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| :<math>
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| \begin{align}
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| \int \frac{\delta F}{\delta\rho(x)} \ \phi(x) \ dx
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| &= \lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon \phi]-F[\rho]}{\varepsilon} \\
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| &= \left [ \frac{d}{d\epsilon}F[\rho+\epsilon \phi]\right ]_{\epsilon=0},
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| \end{align}
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| </math> | |
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| where <math>\epsilon \phi</math> is the variation of ''ρ'', and <math>\phi</math> is an arbitrary function.
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| ===Functional differential=== | |
| The differential (or variation or first variation) of the functional {{math|''F''}}[''ρ''] is,<ref name=ParrYangP246A.1> {{harv|Parr|Yang|1989|loc= p. 246, Eq. A.1}}.</ref> <Ref Group = 'Note' > Called ''differential'' in {{harv|Parr|Yang|1989|p=246}}, ''variation'' or ''first variation'' in {{harv|Courant|Hilbert|1953|p=186}}, and ''variation'' or ''differential'' in {{harv|Gelfand|Fomin|2000|loc= p. 11, § 3.2}}.</ref>
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| :<math> \delta F = \int \frac {\delta F} {\delta \rho(x)} \ \delta \rho(x) \ dx \ , </math>
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| where {{math|''δ''}}''ρ''{{math|(''x'') {{=}} ''εϕ''(''x'')}} is the variation of ''ρ''{{math|(''x'')}}. This is similar in form to the [[total differential]] of a function {{math|''F''}}(''ρ''<sub>1</sub>, ''ρ''<sub>2</sub>, ... , ''ρ''<sub>n</sub>),
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| :<math> dF = \sum_{i=1} ^n \frac {\partial F} {\partial \rho_i} \ d\rho_i \ ,</math>
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| where ''ρ<sub>1</sub>, ρ<sub>2</sub>, ... , ρ<sub>n</sub>'' are independent variables.
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| Comparing the last two equations, the functional derivative {{math|''δF''/''δ''}}''ρ''{{math|(''x'')}} has a role similar to that of the partial derivative {{math|''∂F/∂''}}''ρ<sub>i</sub>'' , where the variable of integration {{math|''x''}} is like a continuous version of the summation index {{math|''i''}}.<ref name=ParrYangP246 />
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| ===Formal description===
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| The definition of a functional derivative may be made more mathematically precise and formal by defining the [[topological vector space|space of functions]] more carefully. For example, when the space of functions is a [[Banach space]], the functional derivative becomes known as the [[Fréchet derivative]], while one uses the [[Gâteaux derivative]] on more general [[locally convex space]]s. Note that the well-known [[Hilbert space]]s are special cases of [[Banach space]]s. The more formal treatment allows many theorems from ordinary [[calculus]] and [[Mathematical analysis|analysis]] to be generalized to corresponding theorems in [[functional analysis]], as well as numerous new theorems to be stated.
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| ==Properties==
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| Like the derivative of a function, the functional derivative satisfies the following properties, where {{math|''F''}}[''ρ''] and {{math|''G''}}[''ρ''] are functionals:
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| * Linear:<ref name=ParrYangP247A.3>{{harv|Parr|Yang|1989|loc= p. 247, Eq. A.3}}.</ref>
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| : <math>\frac{\delta(\lambda F + \mu G)}{\delta \rho(x)} = \lambda \frac{\delta F}{\delta \rho(x)} + \mu \frac{\delta G}{\delta \rho(x)},\ \qquad \lambda,\mu</math> [[Constant (mathematics)|constant]],
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| * Product rule:<ref name=ParrYangP247A.4>{{harv|Parr|Yang|1989|loc= p. 247, Eq. A.4}}.</ref>
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| : <math>\frac{\delta(FG)}{\delta \rho(x)} = \frac{\delta F}{\delta \rho(x)} G + F \frac{\delta G}{\delta \rho(x)} \, , </math>
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| * Chain rules:
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| :If {{math|''f''}} is a differentiable function, then
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| :<math>\displaystyle\frac{\delta F[f(\rho)] }{\delta\rho(x)} = \frac{\delta F[f(\rho)]}{\delta f(\rho(x) )} \ \frac {df(\rho(x))} {d\rho(x)} \ , </math><ref name='ChainRule'> {{harv|Greiner|Reinhardt|1996|loc=p. 38, Eq. 7}}.</ref>
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| : <math>\frac{\delta f( F[\rho])}{\delta\rho(x)} = \frac {df(F[\rho])} {dF[\rho]} \ \frac{\delta F[\rho]}{\delta\rho(x)} \, . </math><ref name=ParrYangP251A.34>{{harv|Parr|Yang|1989|loc=p. 251, Eq. A.34}}.</ref>
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| ==Determining functional derivatives==
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| We give a formula to determine functional derivatives for a common class of functionals that 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 [[Joseph-Louis Lagrange|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).
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| ===Formula===
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| Given a functional
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| :<math>F[\rho] = \int f( \boldsymbol{r}, \rho(\boldsymbol{r}), \nabla\rho(\boldsymbol{r}) )\, d\boldsymbol{r},</math>
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| and a function {{math|''ϕ''}}('''{{math|''r''}}''') that vanishes on the boundary of the region of integration, from a previous section [[Functional_derivative#Definition|Definition]],
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| :<math>
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| \begin{align}
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| \int \frac{\delta F}{\delta\rho(\boldsymbol{r})} \, \phi(\boldsymbol{r}) \, d\boldsymbol{r}
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| & = \left [ \frac{d}{d\varepsilon} \int f( \boldsymbol{r}, \rho + \varepsilon \phi, \nabla\rho+\varepsilon\nabla\phi )\, d\boldsymbol{r} \right ]_{\varepsilon=0} \\
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| & = \int \left( \frac{\partial f}{\partial\rho} \, \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d\boldsymbol{r} \\
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| & = \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} \\
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| & = \int \left[ \frac{\partial f}{\partial\rho} \, \phi - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\boldsymbol{r} \\
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| & = \int \left( \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi(\boldsymbol{r}) \ d\boldsymbol{r} \, .
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| \end{align}
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| </math>
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| The second line is obtained using the [[total derivative]], where {{math|''∂f'' /''∂∇''}}''ρ'' is a [[Matrix_calculus#Scalar-by-vector|derivative of a scalar with respect to a vector]].<ref group="Note"> For a three-dimensional cartesian coordinate system,
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| :<math>
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| \begin{align}
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| \frac{\partial f}{\partial\nabla\rho} = \frac{\partial f}{\partial\rho_x} \mathbf{\hat{i}} + \frac{\partial f}{\partial\rho_y} \mathbf{\hat{j}} + \frac{\partial f}{\partial\rho_z} \mathbf{\hat{k}}\, , \qquad
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| & \text{where} \ \rho_x = \frac{\partial \rho}{\partial x}\, , \ \rho_y = \frac{\partial \rho}{\partial y}\, , \ \rho_z = \frac{\partial \rho}{\partial z}\, \\
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| & \text{and} \ \ \mathbf{\hat{i}}, \ \mathbf{\hat{j}}, \ \mathbf{\hat{k}} \ \ \text {are unit vectors along the x, y, z axes.}
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| \end{align}
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| </math> </ref> The third line was obtained by use of a [[Divergence#Properties|product rule for divergence]]. The fourth line was obtained using the [[divergence theorem]] and the condition that {{math|''ϕ''{{=}}0}} on the boundary of the region of integration. Since {{math|''ϕ''}} is also an arbitrary function, applying the [[fundamental lemma of calculus of variations]] to the last line, the functional derivative is
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| :<div style="{{divstylewhite}}; width:20em; margin:.3em"><center><math>
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| \frac{\delta F}{\delta\rho(\boldsymbol{r})} = \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho}
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| </math></center></div>
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| where ''ρ'' = ''ρ''('''{{math|''r''}}''') and {{math|''f'' {{=}} ''f'' ('''{{math|''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 [[Functional_derivative#Coulomb_potential_energy_functional|Coulomb potential energy functional]].)
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| 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,
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| :<math>
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| F[\rho(\boldsymbol{r})] = \int f( \boldsymbol{r}, \rho(\boldsymbol{r}), \nabla\rho(\boldsymbol{r}), \nabla^{(2)}\rho(\boldsymbol{r}), \dots, \nabla^{(N)}\rho(\boldsymbol{r}))\, d\boldsymbol{r},
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| </math>
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| where the vector {{math|'''''r''''' ∈ ℝ<sup>''n''</sup>}}, and {{math|∇<sup>(''i'')</sup>}} is a tensor whose {{math|''n<sup>i</sup>''}} components are partial derivative operators of order {{math|''i''}},
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| :<math> \left [ \nabla^{(i)} \right ]_{\alpha_1 \alpha_2 \cdots \alpha_i} = \frac {\partial^{\, i}} {\partial r_{\alpha_1} \partial r_{\alpha_2} \cdots \partial r_{\alpha_i} } \qquad \qquad \text{where} \quad \alpha_1, \alpha_2, \cdots, \alpha_i = 1, 2, \cdots , n \ . </math><ref group = 'Note'>For example, for the case of three dimensions ({{math|''n'' {{=}} 3}}) and second order derivatives ({{math|''i'' {{=}} 2}}), the tensor {{math|∇<sup>(2)</sup>}} has components,
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| :<math> \left [ \nabla^{(2)} \right ]_{\alpha \beta} = \frac {\partial^{\,2}} {\partial r_{\alpha} \, \partial r_{\beta}} \qquad \qquad \text{where} \quad \alpha, \beta = 1, 2, 3 \, . </math> </ref>
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| An analogous application of the definition of the functional derivative yields
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| :<math>
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| \begin{align}
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| \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)} \\
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| &{} = \frac{\partial f}{\partial\rho} + \sum_{i=1}^N (-1)^{i}\nabla^{(i)} \cdot \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} \ .
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| \end{align}
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| </math>
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| In the last two equations, the {{math|''n<sup>i</sup>''}} components of the tensor <math> \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} </math> are partial derivatives of {{math|''f''}} with respect to partial derivatives of ''ρ'',
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| :<math> \left [ \frac {\partial f} {\partial \left (\nabla^{(i)}\rho \right ) } \right ]_{\alpha_1 \alpha_2 \cdots \alpha_i} = \frac {\partial f} {\partial \rho_{\alpha_1 \alpha_2 \cdots \alpha_i} } \qquad \qquad \text{where} \quad \rho_{\alpha_1 \alpha_2 \cdots \alpha_i} \equiv \frac {\partial^{\, i}\rho} {\partial r_{\alpha_1} \, \partial r_{\alpha_2} \cdots \partial r_{\alpha_i} } \ , </math>
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| and the tensor scalar product is,
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| :<math> \nabla^{(i)} \cdot \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} = \sum_{\alpha_1, \alpha_2, \cdots, \alpha_i = 1}^n \ \frac {\partial^{\, i} } {\partial r_{\alpha_1} \, \partial r_{\alpha_2} \cdots \partial r_{\alpha_i} } \ \frac {\partial f} {\partial \rho_{\alpha_1 \alpha_2 \cdots \alpha_i} } \ . </math> <ref group = 'Note'>For example, for the case {{math|''n'' {{=}} 3}} and {{math|''i'' {{=}} 2}}, the tensor scalar product is,
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| :<math> \nabla^{(2)} \cdot \frac{\partial f}{\partial\left(\nabla^{(2)}\rho\right)} = \sum_{\alpha, \beta = 1}^3 \ \frac {\partial^{\, 2} } {\partial r_{\alpha} \, \partial r_{\beta} } \ \frac {\partial f} {\partial \rho_{\alpha \beta} } \qquad \text{where} \ \ \rho_{\alpha \beta} \equiv \frac {\partial^{\, 2}\rho} {\partial r_{\alpha} \, \partial r_{\beta} } \ . </math> </ref>
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| ===Examples===
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| ====Thomas–Fermi kinetic energy functional====
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| The [[Thomas–Fermi model]] of 1927 used a kinetic energy functional for a noninteracting uniform [[free electron model|electron gas]] in a first attempt of [[density-functional theory]] of electronic structure:
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| :<math>T_\mathrm{TF}[\rho] = C_\mathrm{F} \int \rho^{5/3}(\mathbf{r}) \, d\mathbf{r} \, .</math>
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| Since the integrand of {{math|''T''<sub>TF</sub>}}[''ρ''] does not involve derivatives of ''ρ''{{math|('''''r''''')}}, the functional derivative of {{math|''T''<sub>TF</sub>}}[''ρ''] is,<ref name=ParrYangP247A.6>{{harv|Parr|Yang|1989|loc=p. 247, Eq. A.6}}.</ref>
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| :<math>
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| \begin{align}
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| \frac{\delta T_{\mathrm{TF}}}{\delta \rho (\boldsymbol{r}) }
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| & = C_\mathrm{F} \frac{\partial \rho^{5/3}(\mathbf{r})}{\partial \rho(\mathbf{r})} \\
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| & = \frac{5}{3} C_\mathrm{F} \rho^{2/3}(\mathbf{r}) \, .
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| \end{align}
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| </math>
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| ====Coulomb potential energy functional====
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| For the '''electron-nucleus potential''', Thomas and Fermi employed the [[Coulomb's law|Coulomb]] potential energy functional
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| :<math>V[\rho] = \int \frac{\rho(\boldsymbol{r})}{|\boldsymbol{r}|} \ d\boldsymbol{r}.</math>
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| Applying the definition of functional derivative,
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| :<math>
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| \begin{align}
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| \int \frac{\delta V}{\delta \rho(\boldsymbol{r})} \ \phi(\boldsymbol{r}) \ d\boldsymbol{r}
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| & {} = \left [ \frac{d}{d\varepsilon} \int \frac{\rho(\boldsymbol{r}) + \varepsilon \phi(\boldsymbol{r})}{|\boldsymbol{r}|} \ d\boldsymbol{r} \right ]_{\varepsilon=0} \\
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| & {} = \int \frac {1} {|\boldsymbol{r}|} \, \phi(\boldsymbol{r}) \ d\boldsymbol{r} \, .
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| \end{align}
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| </math>
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| So,
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| :<math> \frac{\delta V}{\delta \rho(\boldsymbol{r})} = \frac{1}{|\boldsymbol{r}|} \ . </math>
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| For the classical part of the '''electron-electron interaction''', Thomas and Fermi employed the [[Coulomb's law|Coulomb]] potential energy functional
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| :<math>J[\rho] = \frac{1}{2}\iint \frac{\rho(\mathbf{r}) \rho(\mathbf{r}')}{\vert \mathbf{r}-\mathbf{r}' \vert}\, d\mathbf{r} d\mathbf{r}' \, .</math>
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| From the [[Functional_derivative#Functional_derivative|definition of the functional derivative]],
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| :<math>
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| \begin{align}
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| \int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r}
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| & {} = \left [ \frac {d \ }{d\epsilon} \, J[\rho + \epsilon\phi] \right ]_{\epsilon = 0} \\
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| & {} = \left [ \frac {d \ }{d\epsilon} \, \left ( \frac{1}{2}\iint \frac {[\rho(\boldsymbol{r}) + \epsilon \phi(\boldsymbol{r})] \, [\rho(\boldsymbol{r}') + \epsilon \phi(\boldsymbol{r}')] }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert}\, d\boldsymbol{r} d\boldsymbol{r}' \right ) \right ]_{\epsilon = 0} \\
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| & {} = \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}') \phi(\boldsymbol{r}) }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert}\, d\boldsymbol{r} d\boldsymbol{r}' +
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| \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}) \phi(\boldsymbol{r}') }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert}\, d\boldsymbol{r} d\boldsymbol{r}' \\
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| \end{align}
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| </math>
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| 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,
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| :<math> \int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r} = \int \left ( \int \frac {\rho(\boldsymbol{r}') }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert} d\boldsymbol{r}' \right ) \phi(\boldsymbol{r}) d\boldsymbol{r} </math>
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| and the functional derivative of the electron-electron coulomb potential energy functional {{math|''J''}}[''ρ''] is,<ref name=ParrYangP248A.11>{{harv|Parr|Yang|1989|loc=p. 248, Eq. A.11}}.</ref>
| |
| :<math> \frac{\delta J}{\delta\rho(\boldsymbol{r})} = \int \frac {\rho(\boldsymbol{r}') }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert} d\boldsymbol{r}' \, . </math>
| |
| | |
| The second functional derivative is
| |
| :<math>\frac{\delta^2 J[\rho]}{\delta \rho(\mathbf{r}')\delta\rho(\mathbf{r})} = \frac{\partial}{\partial \rho(\mathbf{r}')} \left ( \frac{\rho(\mathbf{r}')}{\vert \mathbf{r}-\mathbf{r}' \vert} \right ) = \frac{1}{\vert \mathbf{r}-\mathbf{r}' \vert}.
| |
| </math>
| |
| | |
| ====Weizsäcker kinetic energy functional====
| |
| In 1935 [[Carl Friedrich von Weizsacker|von Weizsäcker]] proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it suit better a molecular electron cloud:
| |
| :<math>T_\mathrm{W}[\rho] = \frac{1}{8} \int \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{ \rho(\mathbf{r}) } d\mathbf{r} = \int t_\mathrm{W} \ d\mathbf{r} \, ,</math>
| |
| where
| |
| :<math> t_\mathrm{W} \equiv \frac{1}{8} \frac{\nabla\rho \cdot \nabla\rho}{ \rho } \qquad \text{and} \ \ \rho = \rho(\boldsymbol{r}) \ . </math>
| |
| Using a previously derived [[Functional_derivative#Formula|formula]] for the functional derivative,
| |
| :<math>
| |
| \begin{align}
| |
| \frac{\delta T_\mathrm{W}}{\delta \rho(\boldsymbol{r})}
| |
| & = \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}
| |
| </math>
| |
| and the result is,<ref name=ParrYangP247A.9>{{harv|Parr|Yang|1989|loc= p. 247, Eq. A.9}}.</ref>
| |
| :<math> \frac{\delta T_\mathrm{W}}{\delta \rho(\boldsymbol{r})} = \ \ \, \frac{1}{8}\frac{\nabla\rho \cdot \nabla\rho}{\rho^2} - \frac{1}{4}\frac{\nabla^2\rho}{\rho} \ . </math>
| |
| | |
| ====Entropy====
| |
| The [[information entropy|entropy]] of a discrete [[random variable]] is a functional of the [[probability mass function]].
| |
| | |
| :<math>
| |
| \begin{align}
| |
| H[p(x)] = -\sum_x p(x) \log p(x)
| |
| \end{align}
| |
| | |
| </math>
| |
| Thus,
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \sum_x \frac{\delta H}{\delta p(x)} \, \phi(x)
| |
| & {} = \left[ \frac{d}{d\epsilon} H[p(x) + \epsilon\phi(x)] \right]_{\epsilon=0}\\
| |
| & {} = \left [- \, \frac{d}{d\varepsilon} \sum_x \, [p(x) + \varepsilon\phi(x)] \ \log [p(x) + \varepsilon\phi(x)] \right]_{\varepsilon=0} \\
| |
| & {} = \displaystyle -\sum_x \, [1+\log p(x)] \ \phi(x) \, .
| |
| \end{align}
| |
| </math>
| |
| | |
| Thus,
| |
| | |
| :<math>
| |
| \frac{\delta H}{\delta p(x)} = -1-\log p(x).
| |
| </math>
| |
| | |
| ==== Exponential ====
| |
| | |
| Let
| |
| :<math> F[\varphi(x)]= e^{\int \varphi(x) g(x)dx}.</math>
| |
| | |
| Using the delta function as a test function,
| |
| :<math>
| |
| \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}
| |
| </math>
| |
| | |
| Thus,
| |
| :<math> \frac{\delta F[\varphi(x)]}{\delta \varphi(y)} = g(y) F[\varphi(x)]. </math>
| |
| | |
| This is particularly useful in calculating the [[Correlation_function_(quantum_field_theory)|correlation functions]] from the [[Partition function (quantum field theory)|partition function]] in [[quantum field theory]].
| |
| | |
| ====Functional derivative of a function====
| |
| A function can be written in the form of an integral like a functional. For example,
| |
| :<math>\rho(\boldsymbol{r}) = F[\rho] = \int \rho(\boldsymbol{r}') \delta(\boldsymbol{r}-\boldsymbol{r}')\, d\boldsymbol{r}'.</math>
| |
| Since the integrand does not depend on derivatives of ''ρ'', the functional derivative of ''ρ''{{math|('''''r''''')}} is,
| |
| :<math>
| |
| \begin{align}
| |
| \frac {\delta \rho(\boldsymbol{r})} {\delta\rho(\boldsymbol{r}')} \equiv \frac {\delta F} {\delta\rho(\boldsymbol{r}')}
| |
| & = \frac{\partial \ \ }{\partial \rho(\boldsymbol{r}')} \, [\rho(\boldsymbol{r}') \delta(\boldsymbol{r}-\boldsymbol{r}')] \\
| |
| & = \delta(\boldsymbol{r}-\boldsymbol{r}').
| |
| \end{align}
| |
| </math>
| |
| | |
| ==Using the delta function as a test function==
| |
| In physics, it's common to use the [[Dirac delta function]] <math>\delta(x-y)</math> in place of a generic test function <math>\phi(x)</math>, for yielding the functional derivative at the point <math>y</math> (this is a point of the whole functional derivative as a [[partial derivative]] is a component of the gradient):
| |
| | |
| : <math>\frac{\delta F[\rho(x)]}{\delta \rho(y)}=\lim_{\varepsilon\to 0}\frac{F[\rho(x)+\varepsilon\delta(x-y)]-F[\rho(x)]}{\varepsilon}.
| |
| </math>
| |
| | |
| This works in cases when <math>F[\rho(x)+\varepsilon f(x)]</math> formally can be expanded as a series (or at least up to first order) in <math>\varepsilon</math>. The formula is however not mathematically rigorous, since <math>F[\rho(x)+\varepsilon\delta(x-y)]</math> is usually not even defined.
| |
| | |
| The definition given in a previous section is based on a relationship that holds for all test functions {{math|''ϕ''}}, so one might think that it should hold also when {{math|''ϕ''}} is chosen to be a specific function such as the [[Dirac delta function|delta function]]. However, the latter is not a valid test function.
| |
| | |
| In the definition, the functional derivative describes how the functional <math>F[\varphi(x)]</math> changes as a result of a small change in the entire function <math>\varphi(x)</math>. The particular form of the change in <math>\varphi(x)</math> is not specified, but it should stretch over the whole interval on which <math>x</math> is defined. Employing the particular form of the perturbation given by the delta function has the meaning that <math>\varphi(x)</math> is varied only in the point <math>y</math>. Except for this point, there is no variation in <math>\varphi(x)</math>.
| |
| | |
| ==Notes==
| |
| {{Reflist|group=Note}}
| |
| | |
| ==Footnotes==
| |
| {{reflist|29em}}
| |
| | |
| ==References==
| |
| *{{cite book | last1=Courant | first1=Richard | authorlink1=Richard Courant | last2=Hilbert | first2=David | authorlink2=David Hilbert | title = Methods of Mathematical Physics | volume = Vol. I | edition = First English | ref=harv | publisher = [[Interscience Publishers]], Inc | year = 1953 | location = New York, New York | chapter = Chapter IV. The Calculus of Variations | pages = 164–274 | isbn = 978-0471504474| mr = 0065391 | zbl = 0001.00501}}.
| |
| *{{Citation
| |
| | last = Frigyik
| |
| | first = Béla A.
| |
| | author-link =
| |
| | last2 = Srivastava
| |
| | first2 = Santosh
| |
| | author2-link =
| |
| | last3 = Gupta
| |
| | first3 = Maya R.
| |
| | author3-link =
| |
| | title = Introduction to Functional Derivatives
| |
| | place = Seattle, WA
| |
| | publisher = Department of Electrical Engineering at the University of Washington
| |
| | series = UWEE Tech Report
| |
| | volume = UWEETR-2008-0001
| |
| |date=January 2008
| |
| | pages = 7
| |
| | language =
| |
| | url = https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf
| |
| }}.
| |
| *{{Citation
| |
| | last = Gelfand
| |
| | first = I. M.
| |
| | author-link = Israel Gelfand
| |
| | last2 = Fomin
| |
| | first2 = S. V.
| |
| | author2-link = Sergei Fomin
| |
| | title = Calculus of variations
| |
| | place = Mineola, N.Y.
| |
| | publisher = [[Dover Publications]]
| |
| | series = translated and edited by Richard A. Silverman
| |
| | origyear = 1963
| |
| | year = 2000
| |
| | edition = Revised English
| |
| | url = http://store.doverpublications.com/0486414485.html
| |
| | doi =
| |
| | id =
| |
| | isbn = 978-0486414485
| |
| | mr = 0160139
| |
| | zbl = 0127.05402
| |
| }}.
| |
| *{{Citation
| |
| | last = Giaquinta
| |
| | first = Mariano
| |
| | author-link = Mariano Giaquinta
| |
| | last2 = Hildebrandt
| |
| | first2 = Stefan
| |
| | title = Calculus of Variations 1. The Lagrangian Formalism
| |
| | place = Berlin
| |
| | publisher = [[Springer-Verlag]]
| |
| | series = Grundlehren der Mathematischen Wissenschaften
| |
| | volume = 310
| |
| | year = 1996
| |
| | edition = 1st
| |
| | url =
| |
| | isbn = 3-540-50625-X
| |
| | mr = 1368401
| |
| | zbl = 0853.49001
| |
| }}.
| |
| *{{Citation
| |
| | last = Greiner
| |
| | first = Walter
| |
| | last2 = Reinhardt
| |
| | first2 = Joachim
| |
| | title = Field quantization
| |
| | place = Berlin–Heidelberg–New York
| |
| | publisher = Springer-Verlag
| |
| | series = With a foreword by D. A. Bromley
| |
| | volume =
| |
| | origyear =
| |
| | year = 1996
| |
| | edition =
| |
| | chapter = Section 2.3 – Functional derivatives
| |
| | chapterurl = http://www.physics.byu.edu/faculty/berrondo/wt752/functional%20derivative.pdf
| |
| | page =
| |
| | pages = 36–38
| |
| | language =
| |
| | url =
| |
| | doi =
| |
| | id =
| |
| | isbn = 3-540-59179-6
| |
| | mr = 1383589
| |
| | zbl = 0844.00006
| |
| }}.
| |
| *{{cite book |first1=R. G.|last1=Parr|first2=W.|last2=Yang| title = [http://books.google.com/?id=mGOpScSIwU4C&printsec=frontcover&dq=Density-Functional+Theory+of+Atoms+and+Molecules&cd=1#v=onepage&q Density-Functional Theory of Atoms and Molecules] | chapter = Appendix A, Functionals | pages = 246–254 | publisher = Oxford University Press | year = 1989 |location=New York| url = | ref=harv | isbn = 978-0195042795}}
| |
| | |
| {{Functional Analysis}}
| |
| | |
| [[Category:Differential calculus]]
| |
| [[Category:Topological vector spaces]]
| |
| [[Category:Differential operators]]
| |
| [[Category:Calculus of variations]]
| |
| [[Category:Variational analysis]]
| |
He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.
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He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.
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The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum
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