In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (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 derivatives. In an integrand 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.

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: where the variation in the derivative, was rewritten as the derivative of the variation , and integration by parts was used in these derivatives.

Definition

In this section, the functional differential (or variation or first variation) is defined. Then the functional derivative is defined in terms of the functional differential.

Functional differential

is a Banach space is a functional is the linear functional defined is the Fréchet derivative.

However, this notion of functional differential is so strong it may not exist, and in those cases a weaker notion, like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined Note that this notion of the functional differential can even be defined without a norm.

In a more general case the function space.

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Functional derivative

times another function (denoted then this function is called the functional derivative of at . continues to satisfy these conditions.

, and then this is similar in form to the total differential. One thinks of as the gradient of at the point , so the value measures how much the functional will change if the function is changed at the point .

Properties

Like the derivative of a function, the functional derivative satisfies the following properties, where and are functionals:

• Linearity: where are constants.

• Product rule:

Chain rules:

• If is a functional and another functional, then

• If is an ordinary differentiable function (local functional) , then this reduces to

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 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).

Formula

that vanishes on the boundary of the region of integration, from a previous section Definition

The second line is obtained using the total derivative, where is a derivative of a scalar with respect to a vector.

The third line was obtained by use of a product rule for divergence. The fourth line was obtained using the divergence theorem is also an arbitrary function, applying the fundamental lemma of calculus of variations

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.)

where the vector , and is a tensor whose components are partial derivative operators of order

In the last two equations, the are partial derivatives of with respect to partial derivatives of ρ

Examples

Thomas–Fermi kinetic energy functional

The Thomas–Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of density-functional theory Since the integrand of does not involve derivatives of , the functional derivative of is,

Coulomb potential energy functional

The electron–nucleus

The functional derivative of the classical part of the electron–electron interaction From the definition of the functional derivative The first and second terms on the right hand side of the last equation are equal, since and and the functional derivative of the electron-electron Coulomb potential energy functional [ρ] is,

von Weizsäcker kinetic energy functional

In 1935 von Weizsäcker Using a previously derived formula and the result is,

Entropy

The entropy of a discrete random variable is a functional of the probability mass function.

Exponential

This is particularly useful in calculating the correlation functions from the partition function in quantum field theory.

Functional derivative of a function

Since the integrand does not depend on derivatives of ρ, the functional derivative of ρ

Functional derivative of iterated function

Putting in

Using the delta function as a test function

In physics, it is common to use the Dirac delta function (this is a point of the whole functional derivative as a partial derivative is a component of the gradient):

is usually not even defined.

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).

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Notes

Footnotes

References

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External links