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 , 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 on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an
integral of functions, their
arguments , and their
derivative s. 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
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
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
\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 n ... representing (
continuous /
smooth ) functions (with certain
boundary condition 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 \phi right ">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 \phi right ">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 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 + F rho \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
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[principle of least action in ](_blank)Lagrangian mechanics (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 ... Formula
Given a functional
F rho = \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 definitio ... \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 \right d\boldsymbol \\
& = \int \left \frac \, \phi - \left( \nabla \cdot \frac \right) \phi \right d\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
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ... fundamental lemma of calculus of variations 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
The Thomas–Fermi (TF) model,
named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equat ... electron gas in a first attempt of density-functional theory of electronic structure:
T_\mathrm rho = 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
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ... V rho = \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
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ... J rho = \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 ">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
The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple Preposition and postposition, preposition used by commoners that means ''of'' or ''from''.
Nobility directo ... T_\mathrm rho = \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
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ... \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
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ... 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, ...
Thus,
\begin
\sum_x \frac \, \phi(x)
& = \left \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
Fvarphi(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 voicele ...
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) Fvarphi(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 voicele ...
This is particularly useful in calculating the correlation functions
The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms.
D ... partition function in quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ... 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 = \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
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ... \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 . 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 Frho(x)
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 alphabet, Phoenician letter Resh, res . Its u ... \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
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External links
*
{{Analysis in topological vector spaces
Calculus of variations
Differential calculus
Differential operators
Topological vector spaces
Variational analysis
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