Functional derivation

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The functional derivative is a generalized directional derivative of a functional. A functional is a mapping that assigns a number to a function. Because the underlying vector space in this case is a function space , “in the direction of a function” is derived. A related concept is the first variation .

The functional derivation is relevant in applied mathematics and theoretical physics . There it is used, among other things, in density functional theory and field theory .

definition

Be a function space over ; a (linear or non-linear) functional with or ; a function and . Furthermore, denote a space of test functions above . Be in the following . ${\displaystyle \Omega (D)}$ ${\displaystyle D\subset \mathbb {R} ^{n}}$ ${\displaystyle F\colon \Omega \rightarrow \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle y\in \Omega (D)}$ ${\displaystyle \varepsilon \in \mathbb {R} }$ ${\displaystyle {\mathcal {D}}(\Omega ):=\left\{h\in \Omega (D)\mid h(\partial D)=0\right\}}$ ${\displaystyle \Omega }$ ${\displaystyle h\in {\mathcal {D}}(\Omega )}$

As a functional derivative

${\displaystyle f(x)={\frac {\delta F[y]}{\delta y(x)}}}$

from one defines the function (or distribution ) which the equation ${\displaystyle F}$ ${\displaystyle f}$

${\displaystyle \forall h\in {\mathcal {D}}(\Omega ):\langle f,h\rangle _{\Omega (D)}=\int _{D}f(x)h(x)dx=\lim _{\varepsilon \to 0}{\frac {F[y+\varepsilon h]-F[y]}{\varepsilon }}={\frac {d}{d\varepsilon }}F[y+\varepsilon h]{\bigg \vert }_{\varepsilon =0}}$

Fulfills.

The functional derivation plays the role of a gradient , which is to be expressed by the notation . ${\displaystyle {\tfrac {\delta }{\delta y(x)}}}$

properties

1. The functional derivation is a linear map:

   ${\displaystyle {\frac {\delta }{\delta y(x)}}(\alpha F[y]+\beta G[y])=\alpha {\frac {\delta F[y]}{\delta y(x)}}+\beta {\frac {\delta G[y]}{\delta y(x)}}}$

2. The product rule applies to a functional product: ${\displaystyle H[y]=F[y]G[y]}$

   ${\displaystyle {\frac {\delta }{\delta y(x)}}(F[y]G[y])=F[y]{\frac {\delta G[y]}{\delta y(x)}}+{\frac {\delta F[y]}{\delta y(x)}}G[y]}$

3. The chain rule applies to local functionals : ${\displaystyle F[y]=\int _{\mathbb {R} }g(y(x))dx}$

   ${\displaystyle {\frac {\delta F[y]}{\delta y(x)}}=g^{\prime }(y(x))}$

4. If the functional has the form , then: ${\displaystyle F[y]=\int _{\mathbb {R} }g(y^{\prime }(x))dx}$

   ${\displaystyle {\frac {\delta F[y]}{\delta y(x)}}=-g^{\prime \prime }(y^{\prime }(x))y^{\prime \prime }(x)}$

5. If the functional is linear in form , then the functional derivation is particularly simple: ${\displaystyle F[y]=\int _{\mathbb {R} }y(x)g(x)dx}$

   ${\displaystyle {\frac {\delta F[y]}{\delta y(x)}}=g(x)}$

   Corollary: If is linear, then is . This is also a consequence of the Fréchet-Riesz representation theorem : Because there is a linear functional here, it can be represented as a scalar product . ${\displaystyle F}$ ${\displaystyle F[y]=\int _{\mathbb {R} }y(x){\frac {\delta F[y]}{\delta y(x)}}dx}$ ${\displaystyle F}$ ${\displaystyle \left\langle y,{\frac {\delta F[y]}{\delta y}}\right\rangle }$

Examples

1. The non-linear functional has the functional derivative , as can easily be shown with the help of the definition: ${\displaystyle F[y]=\int _{\mathbb {R} }y(x)^{2}g(x)dx}$ ${\displaystyle {\frac {\delta F[y]}{\delta y(x)}}=2y(x)g(x)}$

   ${\displaystyle {\begin{aligned}\int _{\mathbb {R} }{\frac {\delta F[y]}{\delta y(x)}}h(x)dx&=\lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}(F[y+\varepsilon h]-F[y])\\&=\lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}\left(\int _{\mathbb {R} }(y(x)+\varepsilon h(x))^{2}g(x)dx-\int _{\mathbb {R} }y(x)^{2}g(x)dx\right)\\&=\lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}\int _{\mathbb {R} }2y(x)\varepsilon h(x)g(x)+\varepsilon ^{2}h(x)^{2}g(x)dx\\&=\int _{\mathbb {R} }2y(x)g(x)h(x)dx\end{aligned}}}$

   Since this is for all test functions must apply it concludes . ${\displaystyle h}$ ${\displaystyle {\frac {\delta F[y]}{\delta y(x)}}=2y(x)g(x)}$

2. Another example comes from density functional theory. In the LDA approximation , there is the exchange energy

   ${\displaystyle E_{x}[\varrho ]:=c\int _{\mathbb {R} }\varrho (r)^{4/3}d^{3}r}$

   a functional of density . The associated exchange potential is ${\displaystyle \varrho }$

   ${\displaystyle V_{x}(r):={\frac {\delta E_{x}[\varrho ]}{\delta \varrho (r)}}=c{\frac {4}{3}}\varrho (r)^{1/3}}$.

3. Another multidimensional example from density functional theory is the electron-electron interaction as a functional of the density : ${\displaystyle F}$ ${\displaystyle \varrho }$

   ${\displaystyle F[\varrho ]={\frac {k}{2}}\iint _{\mathbb {R} ^{6}}{\frac {\varrho (\mathbf {r} )\varrho (\mathbf {r} ')}{\vert \mathbf {r} -\mathbf {r} '\vert }}\,d\mathbf {r} d\mathbf {r} '\,.}$

   We do the math:

   ${\displaystyle {\begin{aligned}\int _{\mathbb {R} ^{3}}{\frac {\delta F[\varrho ]}{\delta \varrho (\mathbf {r} )}}h(\mathbf {r} )d{\boldsymbol {r}}&=\lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}(F[\varrho +\varepsilon h]-F[\varrho ])\\&=\lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}{\frac {k}{2}}\left(\iint _{\mathbb {R} ^{6}}{\frac {[\varrho ({\boldsymbol {r}})+\epsilon h({\boldsymbol {r}})]\,[\varrho ({\boldsymbol {r}}')+\epsilon h({\boldsymbol {r}}')]}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'-\iint _{\mathbb {R} ^{6}}{\frac {\rho ({\boldsymbol {r}})\,\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'\right)\\&={\frac {k}{2}}\iint _{\mathbb {R} ^{6}}{\frac {\varrho ({\boldsymbol {r}}')h({\boldsymbol {r}})}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'+{\frac {k}{2}}\iint _{\mathbb {R} ^{6}}{\frac {\varrho ({\boldsymbol {r}})h({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'\\&={\frac {2k}{2}}\iint _{\mathbb {R} ^{6}}{\frac {\varrho ({\boldsymbol {r}}')h({\boldsymbol {r}})}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}'d{\boldsymbol {r}}\\&=\int _{\mathbb {R} ^{3}}\left(k\int _{\mathbb {R} ^{3}}{\frac {\varrho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}'\right)h({\boldsymbol {r}})d{\boldsymbol {r}}\end{aligned}}}$

   Since this is for all test functions must apply, we deduce the well-known result . ${\displaystyle h}$ ${\displaystyle {\frac {\delta F[y]}{\delta \varrho ({\boldsymbol {r}})}}=k\int _{\mathbb {R} ^{3}}{\frac {\varrho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}d{\boldsymbol {r}}'}$

4. The following example is useful in quantum field theory to calculate correlation functions from sums of state. That is functional

   ${\displaystyle F[y]=e^{\int _{\mathbb {R} }y(x)g(x)dx}.}$

   With the help of the limit it is easy to show: ${\displaystyle \lim _{\varepsilon \to 0}{\frac {e^{\varepsilon a}-1}{\varepsilon }}=\lim _{\varepsilon \to 0}{\frac {1+\varepsilon a+{\frac {\varepsilon ^{2}}{2}}a^{2}+...-1}{\varepsilon }}=a}$

   ${\displaystyle {\frac {\delta F[y]}{\delta y(x)}}=e^{\int _{\mathbb {R} }y(x)g(x)dx}g(x)=F[y]g(x).}$

5. It also allows distributions to, it can be a function using the Dirac delta function written as functional: . With that in mind is ${\displaystyle f}$ ${\displaystyle f(x)=F_{x}[f]:=\int _{\mathbb {R} }f(y)\delta (x-y)dy}$

   ${\displaystyle {\frac {\delta f(x)}{\delta f(y)}}=\delta (x-y)}$.

Relation to the first variation

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While the first variation is the Gâteaux differential of a functional at the point in the direction of a function , the functional derivative corresponds to the Fréchet derivative of the functional at the point . If Fréchet is differentiable, then Gâteaux is also differentiable and the Fréchet derivative corresponds to the Gâteaux derivative. The reverse is generally not true. For details see the connection between the Fréchet and Gâteaux derivation . ${\displaystyle F}$ ${\displaystyle y}$ ${\displaystyle h}$ ${\displaystyle F}$ ${\displaystyle y}$ ${\displaystyle F}$ ${\displaystyle F}$

See also

• variation
• Hamilton's principle
• Euler-Lagrange equation

Individual evidence

1. ↑ ^{a b c d} R. G. Parr, W. Yang Appendix A, Functionals. In: Density-Functional Theory of Atoms and Molecules . Oxford University Press, New York 1989, ISBN 978-0195042795 , pp. 246-254
2. ↑ ^{a b c d} Michael J. Gruber, University of Hanover, functional derivatives "in a nutshell" , accessed on April 7, 2016
3. ↑ Klaus Capelle, A bird's-eye view of density-functional theory , Version 5, November 2006, equation (83)