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In the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, a field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, the functional derivative (or variational derivative) relates a change in a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
(a functional in this sense is a function that acts on functions) to a change in a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
on which the functional depends. In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their
arguments An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, and their derivatives. In an integral of a functional, if a function is varied by adding to it another function that is arbitrarily small, and the resulting integrand is expanded in powers of , the coefficient of in the first order term is called the functional derivative. For example, consider the functional J = \int_a^b L( \, x, f(x), f \, '(x) \, ) \, dx \ , where . If is varied by adding to it a function , and the resulting integrand is expanded in powers of , then the change in the value of to first order in can be expressed as follows:According to , this notation is customary in physical literature. \delta J = \int_a^b \left( \frac \delta f(x) + \frac \frac \delta f(x) \right) \, dx \, = \int_a^b \left( \frac - \frac \frac \right) \delta f(x) \, dx \, + \, \frac (b) \delta f(b) \, - \, \frac (a) \delta f(a) \, where the variation in the derivative, was rewritten as the derivative of the variation , and integration by parts was used.


Definition

In this section, the functional derivative is defined. Then the functional differential is defined in terms of the functional derivative.


Functional derivative

Given a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
representing ( continuous/
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
) functions (with certain
boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s etc.), and a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
defined as F\colon M \to \mathbb \quad \text \quad F\colon M \to \mathbb \, , the functional derivative of , denoted , is defined through. \begin \int \frac(x) \phi(x) \; dx &= \lim_\frac \\ &= \left \fracF[\rho+\varepsilon_\phiright_.html" ;"title="rho+\varepsilon_\phi.html" ;"title="\fracF[\rho+\varepsilon \phi">\fracF[\rho+\varepsilon \phiright ">rho+\varepsilon_\phi.html" ;"title="\fracF[\rho+\varepsilon \phi">\fracF[\rho+\varepsilon \phiright , \end where \phi is an arbitrary function. The quantity \varepsilon\phi is called the variation of . In other words, \phi \mapsto \left \fracF[\rho+\varepsilon_\phiright_.html" ;"title="rho+\varepsilon_\phi.html" ;"title="\fracF[\rho+\varepsilon \phi">\fracF[\rho+\varepsilon \phiright ">rho+\varepsilon_\phi.html" ;"title="\fracF[\rho+\varepsilon \phi">\fracF[\rho+\varepsilon \phiright is a linear functional, so one may apply the Riesz–Markov–Kakutani representation theorem to represent this functional as integration against some measure (mathematics), measure. Then is defined to be the Radon–Nikodym derivative of this measure. One thinks of the function as the gradient of at the point (that is, how much the functional will change if the function is changed at the point ) and \int \frac(x) \phi(x) \; dx as the directional derivative at point in the direction of . Then analogous to vector calculus, the inner product with the gradient gives the directional derivative.


Functional differential

The differential (or variation or first variation) of the functional F\left rho\right/math> is . Called ''differential'' in , ''variation'' or ''first variation'' in , and ''variation'' or ''differential'' in . \delta F rho; \phi= \int \frac (x) \ \phi(x) \ dx \ . Heuristically, \phi is the change in \rho, so we 'formally' have \phi = \delta\rho, and then this is similar in form to the
total differential In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is th ...
of a function F(\rho_1,\rho_2,\dots,\rho_n), dF = \sum_ ^n \frac \ d\rho_i \ , where \rho_1,\rho_2,\dots,\rho_n are independent variables. Comparing the last two equations, the functional derivative \delta F/\delta\rho(x) has a role similar to that of the partial derivative \partial F/\partial\rho_i, where the variable of integration x is like a continuous version of the summation index i..


Properties

Like the derivative of a function, the functional derivative satisfies the following properties, where and are functionals: Here the notation \frac(x) \equiv \frac is introduced. * Linearity:. \frac = \lambda \frac + \mu \frac, where are constants. * Product rule:. \frac = \frac G
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
+ F
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
\frac \, , * Chain rules: **If is a functional and another functional, then \frac = \int dx \frac_\cdot\frac \ . **If is an ordinary differentiable function (local functional) , then this reduces to \frac = \frac \ \frac \ .


Determining functional derivatives

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 The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
in
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
(18th century). The first three examples below are taken from density functional theory (20th century), the fourth from
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
(19th century).


Formula

Given a functional F
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
= \int f( \boldsymbol, \rho(\boldsymbol), \nabla\rho(\boldsymbol) )\, d\boldsymbol, and a function that vanishes on the boundary of the region of integration, from a previous section
Definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
, \begin \int \frac \, \phi(\boldsymbol) \, d\boldsymbol & = \left \frac \int f( \boldsymbol, \rho + \varepsilon \phi, \nabla\rho+\varepsilon\nabla\phi )\, d\boldsymbol \right \\ & = \int \left( \frac \, \phi + \frac \cdot \nabla\phi \right) d\boldsymbol \\ & = \int \left \frac \, \phi + \nabla \cdot \left( \frac \, \phi \right) - \left( \nabla \cdot \frac \right) \phi \rightd\boldsymbol \\ & = \int \left \frac \, \phi - \left( \nabla \cdot \frac \right) \phi \rightd\boldsymbol \\ & = \int \left( \frac - \nabla \cdot \frac \right) \phi(\boldsymbol) \ d\boldsymbol \, . \end The second line is obtained using the total derivative, where is a derivative of a scalar with respect to a vector.For a three-dimensional Cartesian coordinate system, \frac = \frac \mathbf + \frac \mathbf + \frac \mathbf\, , where \rho_x = \frac\, , \ \rho_y = \frac\, , \ \rho_z = \frac and \mathbf, \mathbf, \mathbf 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 on the boundary of the region of integration. Since is also an arbitrary function, applying the
fundamental lemma of calculus of variations In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal ze ...
to the last line, the functional derivative is \frac = \frac - \nabla \cdot \frac where and . This formula is for the case of the functional form given by 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 rho(\boldsymbol)= \int f( \boldsymbol, \rho(\boldsymbol), \nabla\rho(\boldsymbol), \nabla^\rho(\boldsymbol), \dots, \nabla^\rho(\boldsymbol))\, d\boldsymbol, where the vector , and is a tensor whose components are partial derivative operators of order , \left \nabla^ \right = \frac \qquad \qquad \text \quad \alpha_1, \alpha_2, \cdots, \alpha_i = 1, 2, \cdots , n \ . For example, for the case of three dimensions () and second order derivatives (), the tensor has components, \left \nabla^ \right = \frac \qquad \qquad \text \quad \alpha, \beta = 1, 2, 3 \, . An analogous application of the definition of the functional derivative yields \begin \frac & = \frac - \nabla \cdot \frac + \nabla^ \cdot \frac + \dots + (-1)^N \nabla^ \cdot \frac \\ & = \frac + \sum_^N (-1)^\nabla^ \cdot \frac \ . \end In the last two equations, the components of the tensor \frac are partial derivatives of with respect to partial derivatives of ''ρ'', \left \frac \right = \frac \qquad \qquad \text \quad \rho_ \equiv \frac \ , and the tensor scalar product is, \nabla^ \cdot \frac = \sum_^n \ \frac \ \frac \ . For example, for the case and , the tensor scalar product is, \nabla^ \cdot \frac = \sum_^3 \ \frac \ \frac \qquad \text \ \ \rho_ \equiv \frac \ .


Examples


Thomas–Fermi kinetic energy functional

The Thomas–Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform
electron gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. ...
in a first attempt of density-functional theory of electronic structure: T_\mathrm
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
= C_\mathrm \int \rho^(\mathbf) \, d\mathbf \, . Since the integrand of does not involve derivatives of , the functional derivative of is,. \begin \frac & = C_\mathrm \frac \\ & = \frac C_\mathrm \rho^(\mathbf) \, . \end


Coulomb potential energy functional

For the electron-nucleus potential, Thomas and Fermi employed the Coulomb potential energy functional V
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
= \int \frac \ d\boldsymbol. Applying the definition of functional derivative, \begin \int \frac \ \phi(\boldsymbol) \ d\boldsymbol & = \left \frac \int \frac \ d\boldsymbol \right \\ & = \int \frac \, \phi(\boldsymbol) \ d\boldsymbol \, . \end So, \frac = \frac \ . For the classical part of the electron-electron interaction, Thomas and Fermi employed the Coulomb potential energy functional J
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
= \frac\iint \frac\, d\mathbf d\mathbf' \, . From the definition of the functional derivative, \begin \int \frac \phi(\boldsymbol)d\boldsymbol & = \left \frac__\,_J[\rho_+_\epsilon\phi\right_.html" ;"title="rho_+_\epsilon\phi.html" ;"title="\frac \, J[\rho + \epsilon\phi">\frac \, J[\rho + \epsilon\phi\right ">rho_+_\epsilon\phi.html" ;"title="\frac \, J[\rho + \epsilon\phi">\frac \, J[\rho + \epsilon\phi\right \\ & = \left [ \frac \, \left ( \frac\iint \frac \, d\boldsymbol d\boldsymbol' \right ) \right ]_ \\ & = \frac\iint \frac \, d\boldsymbol d\boldsymbol' + \frac\iint \frac \, d\boldsymbol d\boldsymbol' \\ \end The first and second terms on the right hand side of the last equation are equal, since and in the second term can be interchanged without changing the value of the integral. Therefore, \int \frac \phi(\boldsymbol)d\boldsymbol = \int \left ( \int \frac d\boldsymbol' \right ) \phi(\boldsymbol) d\boldsymbol and the functional derivative of the electron-electron coulomb potential energy functional 'ρ''is,. \frac = \int \frac d\boldsymbol' \, . The second functional derivative is \frac = \frac \left ( \frac \right ) = \frac.


Weizsäcker kinetic energy functional

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_\mathrm
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
= \frac \int \frac d\mathbf = \int t_\mathrm \ d\mathbf \, , where t_\mathrm \equiv \frac \frac \qquad \text \ \ \rho = \rho(\boldsymbol) \ . Using a previously derived formula for the functional derivative, \begin \frac & = \frac - \nabla\cdot\frac \\ & = -\frac\frac - \left ( \frac \frac - \frac \frac \right ) \qquad \text \ \ \nabla^2 = \nabla \cdot \nabla \ , \end and the result is,. \frac = \ \ \, \frac\frac - \frac\frac \ .


Entropy

The entropy of a discrete random variable is a functional of the
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
. H
(x) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, m ...
= -\sum_x p(x) \log p(x) Thus, \begin \sum_x \frac \, \phi(x) & = \left \frac_H[p(x)_+_\epsilon\phi(x)\right.html" ;"title="(x)_+_\epsilon\phi(x).html" ;"title="\frac H[p(x) + \epsilon\phi(x)">\frac H[p(x) + \epsilon\phi(x)\right">(x)_+_\epsilon\phi(x).html" ;"title="\frac H[p(x) + \epsilon\phi(x)">\frac H[p(x) + \epsilon\phi(x)\right\\ & = \left [- \, \frac \sum_x \, [p(x) + \varepsilon\phi(x)] \ \log [p(x) + \varepsilon\phi(x)] \right]_ \\ & = -\sum_x \, [1+\log p(x)] \ \phi(x) \, . \end Thus, \frac = -1-\log p(x).


Exponential

Let F
varphi(x) Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicel ...
e^. Using the delta function as a test function, \begin \frac & = \lim_\frac\\ & = \lim_\frac\\ & = e^\lim_\frac\\ & = e^\lim_\frac\\ & = e^g(y). \end Thus, \frac = g(y) F
varphi(x) Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicel ...
This is particularly useful in calculating the correlation functions from the 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, \rho(\boldsymbol) = F
rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
= \int \rho(\boldsymbol') \delta(\boldsymbol-\boldsymbol')\, d\boldsymbol'. Since the integrand does not depend on derivatives of ''ρ'', the functional derivative of ''ρ'' is, \begin \frac \equiv \frac & = \frac \, rho(\boldsymbol') \delta(\boldsymbol-\boldsymbol')\\ & = \delta(\boldsymbol-\boldsymbol'). \end


Functional derivative of iterated function

The functional derivative of the iterated function f(f(x)) is given by: \frac = f'(f(x))\delta(x-y) + \delta(f(x)-y) and \frac = f'(f(f(x))(f'(f(x))\delta(x-y) + \delta(f(x)-y)) + \delta(f(f(x))-y) In general: \frac = f'( f^(x) ) \frac + \delta( f^(x) - y ) Putting in gives: \frac = - \frac


Using the delta function as a test function

In physics, it is common to use the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
\delta(x-y) in place of a generic test function \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): \frac=\lim_\frac. This works in cases when F rho(x)+\varepsilon f(x)/math> formally can be expanded as a series (or at least up to first order) in \varepsilon. The formula is however not mathematically rigorous, since 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 \phi(x), so one might think that it should hold also when \phi(x) is chosen to be a specific function such as the
delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. 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 rho(x)/math> changes as a result of a small change in the entire function \rho(x). The particular form of the change in \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 \rho(x) is varied only in the point y. Except for this point, there is no variation in \rho(x).


Notes


Footnotes


References

*. *. *. *. *. *


External links

* {{Analysis in topological vector spaces Calculus of variations Differential calculus Differential operators Topological vector spaces Variational analysis