- Functional derivative
In

mathematics and theoreticalphysics , the**functional derivative**is a generalization of thedirectional derivative . The difference is that the latter differentiates in the direction of a vector, while the former differentiates in the direction of a function. Both of these can be viewed as extensions of the usualcalculus derivative .Two possible, restricted definitions suitable for certain computations are given here. There are more general definitions of functional derivatives relying on ideas from

functional analysis , such as theGâteaux derivative .Given a

manifold "M" representing (continuous/smooth/with certainboundary condition s/etc.) functions φ and a functional "F" defined as ::$Fcolon\; M\; ightarrow\; mathbb\{R\}\; quad\; mbox\{or\}\; quad\; Fcolon\; M\; ightarrow\; mathbb\{C\}$,the

**functional derivative**of "F", denoted $\{delta\; F\}/\{deltaphi(x)\}$, is a distribution $delta\; F\; [phi]$ such that for alltest function s "f",:$leftlangle\; delta\; F\; [phi]\; ,\; f\; ight\; angle\; =\; left.frac\{d\}\{depsilon\}F\; [phi+epsilon\; f]\; ight|\_\{epsilon=0\}.$

Sometimes physicists write the definition in terms of a limit and the

Dirac delta function , δ:: $frac\{delta\; F\; [phi(x)]\; \}\{delta\; phi(y)\}=lim\_\{varepsilon\; o\; 0\}frac\{F\; [phi(x)+varepsilondelta(x-y)]\; -F\; [phi(x)]\; \}\{varepsilon\}.$

**Formal description**The definition of a functional derivative may be made much more mathematically precise and formal by defining the space of functions more carefully. For example, when the space of functions is a

Banach space , the functional derivative becomes known as theFréchet derivative , while one uses theGâteaux derivative on more generallocally convex space s. Note that the well-knownHilbert space s are special cases ofBanach space s. The more formal treatment allows many theorems from ordinarycalculus and analysis to be generalized to corresponding theorems infunctional analysis , as well as numerous new theorems to be stated.**Relationship between the mathematical and physical definitions**The mathematicians' definition and the physicists' definition of the functional derivative differ only in the physical interpretation. Since the mathematical definition is based on a relationship that holds for all

test function s "f", it should also hold when "f" is chosen to be a specific function. The only handwaving difficulty is that specific function was chosen to be a delta function---which is not a valid test function.In the mathematical definition, the functional derivative describes how the entire functional, $F\; [varphi(x)]$, changes as a result of a small change in the function $varphi(x)$. Observe that the particular form of the change in $varphi(x)$ is not specified. The physics definition, by contrast, employs a particular form of the perturbation --- namely, the delta function --- and the 'meaning' is that we are varying $varphi(x)$ only about some neighborhood of $y$. Outside of this neighborhood, there is no variation in $varphi(x)$.

Often, a physicist wants to know how one quantity, say the electric potential at position $r\_1$, is affected by changing another quantity, say the density of electric charge at position $r\_2$. The potential at a given position, is a functional of the density. That is, given a particular density function and a point in space, one can compute a number which represents the potential of that point in space due to the specified density function. Since we are interested in how this number varies across all points in space, we treat the potential as a function of $r$. To wit,

:$F\; [\; ho(r\text{'})]\; :=\; V(r)\; =\; frac\{1\}\{4piepsilon\_0\}\; int\; frac\{\; ho(r\text{'})\}\; mathrm\{d\}^3r\text{'}\; \backslash \&\; \{\}\; =\; leftlangle\; frac\{1\}\{4piepsilon\_0\}\; frac\{1\},\; varphi(r\text{'})\; ight\; angle.end\{align\}$

So,

:$frac\{delta\; V(r)\}\{delta\; ho(r\text{'})\}\; =\; frac\{1\}\{4piepsilon\_0\}frac\{1\}.$

Now, we can evaluate the functional derivative at $r=r\_1$ and $r\text{'}=r\_2$ to see how the potential at $r\_1$ is changed due to a small variation in the density at $r\_2$. In practice, the unevaluated form is probably more useful.

**Examples**We give a formula to derive a common class of functionals that can be written as the integral of a function and its derivatives (a generalization of the

Euler–Lagrange equation ), and apply this formula to three examples taken fromphysics . Another example in physics is the derivation of the Lagrange equation of the second kind from theprinciple of least action inLagrangian mechanics .**Formula for the integral of a function and its derivatives**Given a functional of the form:$F\; [\; ho(mathbf\{r\})]\; =\; int\; f(\; mathbf\{r\},\; ho(mathbf\{r\}),\; abla\; ho(mathbf\{r\})\; ),\; d^3r,$with $ho$ vanishing at the boundaries of $mathbf\{r\}$, the functional derivative can be written

:$egin\{align\}leftlangle\; delta\; F\; [\; ho]\; ,\; phi\; ight\; angle\; \{\}\; =\; frac\{d\}\{dvarepsilon\}\; left.\; int\; f(\; mathbf\{r\},\; ho\; +\; varepsilon\; phi,\; abla\; ho+varepsilon\; ablaphi\; ),\; d^3r\; ight|\_\{varepsilon=0\}\; \backslash \; \{\}\; =\; int\; left(\; frac\{partial\; f\}\{partial\; ho\}\; phi\; +\; frac\{partial\; f\}\{partial\; abla\; ho\}\; cdot\; ablaphi\; ight)\; d^3r\; \backslash \; \{\}\; =\; int\; left\; [\; frac\{partial\; f\}\{partial\; ho\}\; phi\; +\; abla\; cdot\; left(\; frac\{partial\; f\}\{partial\; abla\; ho\}\; phi\; ight)\; -\; left(\; abla\; cdot\; frac\{partial\; f\}\{partial\; abla\; ho\}\; ight)\; phi\; ight]\; d^3r\; \backslash \; \{\}\; =\; int\; left\; [\; frac\{partial\; f\}\{partial\; ho\}\; phi\; -\; left(\; abla\; cdot\; frac\{partial\; f\}\{partial\; abla\; ho\}\; ight)\; phi\; ight]\; d^3r\; \backslash \; \{\}\; =\; leftlangle\; frac\{partial\; f\}\{partial\; ho\}\; -\; abla\; cdot\; frac\{partial\; f\}\{partial\; abla\; ho\},,\; phi\; ight\; angle,end\{align\}$

where, in the third line, $phi=0$ is assumed at the integration boundaries. Thus,

:$delta\; F\; [\; ho]\; =\; frac\{partial\; f\}\{partial\; ho\}\; -\; abla\; cdot\; frac\{partial\; f\}\{partial\; abla\; ho\}$

or, writing the expression more explicitly,

:$frac\{delta\; F\; [\; ho(mathbf\{r\})]\; \}\{delta\; ho(mathbf\{r\})\}\; =\; frac\{partial\}\{partial\; ho(mathbf\{r\})\}f(mathbf\{r\},\; ho(mathbf\{r\}),\; abla\; ho(mathbf\{r\}))\; -\; abla\; cdot\; frac\{partial\}\{partial\; abla\; ho(mathbf\{r\})\}f(mathbf\{r\},\; ho(mathbf\{r\}),\; abla\; ho(mathbf\{r\}))$

The above example is specific to the particular case that the functional depends on the function $ho(mathbf\{r\})$ and its

gradient $abla\; ho(mathbf\{r\})$ only. In the more general case that the functional depends on higher order derivatives, i.e.:$F\; [\; ho(mathbf\{r\})]\; =\; int\; f(\; mathbf\{r\},\; ho(mathbf\{r\}),\; abla\; ho(mathbf\{r\}),\; abla^2\; ho(mathbf\{r\}),\; dots,\; abla^N\; ho(mathbf\{r\})),\; d^3r,$

where $abla^i$ is a tensor whose $n^i$ components $(mathbf\{r\}\; in\; mathbb\{R\}^n)$ are all partial derivative operators of order $i$, i.e. $partial^i/(partial\; r^\{i\_1\}\_1,\; partial\; r^\{i\_2\}\_2\; cdots\; partial\; r^\{i\_n\}\_n)$ with $i\_1+i\_2+cdots+i\_n\; =\; i$, an analogous application of the definition yields

:$egin\{align\}frac\{delta\; F\; [\; ho]\; \}\{delta\; ho\}\; =\; frac\{partial\; f\}\{partial\; ho\}\; -\; abla\; cdot\; frac\{partial\; f\}\{partial(\; abla\; ho)\}\; +\; abla^2\; cdot\; frac\{partial\; f\}\{partialleft(\; abla^2\; ho\; ight)\}\; -\; cdots\; \backslash cdots\; +\; (-1)^N\; abla^N\; cdot\; frac\{partial\; f\}\{partialleft(\; abla^N\; ho\; ight)\}\; =\; sum\_\{i=0\}^N\; (-1)^\{i\}\; abla^i\; cdot\; frac\{partial\; f\}\{partialleft(\; abla^i\; ho\; ight)\}.end\{align\}$

**Thomas-Fermi kinetic energy functional**In 1927 Thomas and Fermi used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of

density-functional theory of electronic structure::$T\_mathrm\{TF\}\; [\; ho]\; =\; C\_mathrm\{F\}\; int\; ho^\{5/3\}(mathbf\{r\})\; ,\; d^3r.$$T\_mathrm\{TF\}\; [varrho]$ depends "only" on the charge density $ho(mathbf\{r\})$ and does not depend on its gradient, Laplacian, or other higher-order derivatives. Therefore,:$frac\{delta\; T\_mathrm\{TF\}\; [\; ho]\; \}\{delta\; ho\}\; =\; C\_mathrm\{F\}\; frac\{partial\; ho^\{5/3\}(mathbf\{r\})\}\{partial\; ho\}\; =\; frac\{5\}\{3\}\; C\_mathrm\{F\}\; ho^\{2/3\}(mathbf\{r\}).$**Coulomb potential energy functional**For the classical part of the potential, Thomas and Fermi employed the Coulomb potential energy functional:$J\; [\; ho]\; =\; frac\{1\}\{2\}intint\; frac\{\; ho(mathbf\{r\})\; ho(mathbf\{r\}\text{'})\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\},\; d^3r\; d^3r\text{'}\; =\; int\; left(frac\{1\}\{2\}int\; frac\{\; ho(mathbf\{r\})\; ho(mathbf\{r\}\text{'})\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\}\; d^3r\text{'}\; ight)\; d^3r\; =\; int\; j\; [mathbf\{r\},\; ho(mathbf\{r\})]\; ,\; d^3r.$Again, $J\; [\; ho]$ depends "only" on the charge density $ho$ and does not depend on its gradient, Laplacian, or other higher-order derivatives. Therefore, :$frac\{delta\; J\; [\; ho]\; \}\{delta\; ho\}\; =\; frac\{partial\; j\}\{partial\; ho\}\; =\; frac\{1\}\{2\}int\; frac\{partial\}\{partial\; ho\}frac\{\; ho(mathbf\{r\})\; ho(mathbf\{r\}\text{'})\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\},\; d^3r\text{'}\; =\; int\; frac\{\; ho(mathbf\{r\}\text{'})\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\},\; d^3r\text{'}$

The second functional derivative of the Coulomb potential energy functional is:$frac\{delta^2\; J\; [\; ho]\; \}\{delta\; ho^2\}\; =\; frac\{delta\}\{delta\; ho\}int\; frac\{\; ho(mathbf\{r\}\text{'})\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\},\; d^3r\text{'}\; =\; frac\{partial\}\{partial\; ho\}\; frac\{\; ho(mathbf\{r\}\text{'})\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\}\; =\; frac\{1\}\{vert\; mathbf\{r\}-mathbf\{r\}\text{'}\; vert\}$

**Weizsäcker kinetic energy functional**In 1935 Weizsäcker proposed a gradient correction to the Thomas-Fermi kinetic energy functional to make it suit better a molecular electron cloud::$T\_mathrm\{W\}\; [\; ho]\; =\; frac\{1\}\{8\}\; int\; frac\{\; abla\; ho(mathbf\{r\})\; cdot\; abla\; ho(mathbf\{r\})\}\{\; ho(mathbf\{r\})\; \},\; d^3r\; =\; frac\{1\}\{8\}\; int\; frac\{(\; abla\; ho(mathbf\{r\}))^2\}\{\; ho(mathbf\{r\})\},\; d^3r\; =\; int\; t\; [\; ho(mathbf\{r\}),\; abla\; ho(mathbf\{r\})]\; ,\; d^3r.$Now $T\_mathrm\{W\}\; [\; ho]$ depends on the charge density $ho$ "and" its gradient, therefore,:$frac\{delta\; T\; [\; ho]\; \}\{delta\; ho\}\; =\; frac\{partial\; t\}\{partial\; ho\}\; -\; ablacdotfrac\{partial\; t\}\{partial\; (\; abla\; ho)\}\; =\; -frac\{1\}\{8\}\; frac\{(\; abla\; ho(mathbf\{r\}))^2\}\{\; ho(mathbf\{r\})^2\}\; -\; ablacdotleft(frac\{1\}\{4\}\; frac\{\; abla\; ho(mathbf\{r\})\}\{\; ho(mathbf\{r\})\}\; ight)\; =\; frac\{1\}\{8\}\; frac\{(\; abla\; ho(mathbf\{r\}))^2\}\{\; ho^2(mathbf\{r\})\}\; -\; frac\{1\}\{4\}\; frac\{\; abla^2\; ho(mathbf\{r\})\}\{\; ho(mathbf\{r\})\}.$

**Writing a function as a functional**Finally, note that any function can be written in terms of a functional. For example,:$ho(mathbf\{r\})\; =\; int\; ho(mathbf\{r\}\text{'})\; delta(mathbf\{r\}-mathbf\{r\}\text{'}),\; d^3r\text{'}.$This functional is a function of $ho$ only, and thus, is in the same form as the above examples. Therefore,:$frac\{delta\; ho(mathbf\{r\})\}\{delta\; ho(mathbf\{r\}\text{'})\}=frac\{delta\; int\; ho(mathbf\{r\}\text{'})\; delta(mathbf\{r\}-mathbf\{r\}\text{'}),\; d^3r\text{'}\}\{delta\; ho(mathbf\{r\}\text{'})\}\; =\; frac\{partial\; left(\; ho(mathbf\{r\}\text{'})\; delta(mathbf\{r\}-mathbf\{r\}\text{'})\; ight)\}\{partial\; ho\}\; =\; delta(mathbf\{r\}-mathbf\{r\}\text{'}).$

**Entropy**The entropy of a discrete

random variable is a functional of theprobability mass function .:$H\; [p(x)]\; =\; -sum\_x\; p(x)\; log\_2\; p(x)$Thus,

:$egin\{align\}leftlangle\; delta\; H,\; phi\; ight\; angle\; \{\}\; =\; sum\_x\; delta\; H\; ,\; varphi(x)\; \backslash \; \{\}\; =\; frac\{d\}\{depsilon\}\; left.\; H\; [p(x)\; +\; epsilonphi(x)]\; ight|\_\{epsilon=0\}\backslash \; \{\}\; =\; -frac\{d\}\{dvarepsilon\}\; left.\; sum\_x\; [p(x)\; +\; varepsilonvarphi(x)]\; log\_2\; [p(x)\; +\; varepsilonvarphi(x)]\; ight|\_\{varepsilon=0\}\; \backslash \; \{\}\; =\; displaystyle\; -sum\_x\; [1+log\_2\; p(x)]\; varphi(x)\backslash \; \{\}\; =\; leftlangle\; -\; [1+log\_2\; p(x)]\; ,\; varphi\; ight\; angle.end\{align\}$

Thus,

:$frac\{delta\; H\}\{delta\; p\}\; =\; -\; [1+log\_2\; p(x)]\; .$

**References*** R. G. Parr, W. Yang, “Density-Functional Theory of Atoms and Molecules”, Oxford university Press, Oxford 1989.

* B. A. Frigyik, S. Srivastava and M. R. Gupta, "Introduction to Functional Derivatives", UWEE Tech Report 2008-0001. http://www.ee.washington.edu/research/guptalab/publications/functionalDerivativesIntroduction.pdf

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