Functional derivative


In the calculus of variations, a field of mathematical analysis, the functional derivative relates a change in a functional 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 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
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.

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 representing functions , and a functional defined as
the functional derivative of ρ], denoted, is defined by
where is an arbitrary function. The quantity is called the variation of. In other words,
is a linear functional, so one may apply the Riesz–Markov–Kakutani representation theorem to represent this functional as integration against some measure.
Then is defined to be the Radon-Nikodym derivative of this measure.
We think of the function as the gradient of at the point and
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 of the functional is Called differential in, variation or first variation in, and variation or differential in.
Heuristically, is the change in, so we 'formally' have, and then
this is similar in form to the total differential of a function,
where are independent variables.
Comparing the last two equations, the functional derivative has a role similar to that of the partial derivative, where the variable of integration is like a continuous version of the summation index.

Rigorous description

The definition of a functional derivative may be made more mathematically precise and rigorous 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 the Fréchet derivative, while one uses the Gateaux derivative on more general locally convex spaces. Note that Hilbert spaces are special cases of Banach spaces. The more rigorous treatment allows many theorems from ordinary calculus and analysis to be generalized to corresponding theorems in functional analysis, as well as numerous new theorems to be stated.

Properties

Like the derivative of a function, the functional derivative satisfies the following properties, where and are functionals:
where are constants.
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. The first three examples below are taken from density functional theory, the fourth from statistical mechanics.

Formula

Given a functional
and a function 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 ρ 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 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 to the last line, the functional derivative is
where ρ = ρ. 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.
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,
where the vector, and is a tensor whose components are partial derivative operators of order,
An analogous application of the definition of the functional derivative yields
In the last two equations, the components of the tensor are partial derivatives of with respect to partial derivatives of ρ,
and the tensor scalar product is,

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 of electronic structure:
Since the integrand of does not involve derivatives of ρ, the functional derivative of is,

Coulomb potential energy functional

For the electron-nucleus potential, Thomas and Fermi employed the Coulomb potential energy functional
Applying the definition of functional derivative,
So,
For the classical part of the electron-electron interaction, Thomas and Fermi employed the Coulomb potential energy functional
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 in the second term can be interchanged without changing the value of the integral. Therefore,
and the functional derivative of the electron-electron coulomb potential energy functional is,
The second functional derivative is

Weizsäcker kinetic energy functional

In 1935 von Weizsäcker proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it suit better a molecular electron cloud:
where
Using a previously derived formula for the functional derivative,
and the result is,

Entropy

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

Exponential

Let
Using the delta function as a test function,
Thus,
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,
Since the integrand does not depend on derivatives of ρ, the functional derivative of ρ is,

Functional derivative of iterated function

The functional derivative of the iterated function is given by:
and
In general:
Putting in N=0 gives:

Using the delta function as a test function

In physics, it is common to use the Dirac delta function in place of a generic test function, for yielding the functional derivative at the point :
This works in cases when formally can be expanded as a series in. The formula is however not mathematically rigorous, since is usually not even defined.
The definition given in a previous section is based on a relationship that holds for all test functions, so one might think that it should hold also when is chosen to be a specific function such as the delta function. However, the latter is not a valid test function.
In the definition, the functional derivative describes how the functional changes as a result of a small change in the entire function. The particular form of the change in is not specified, but it should stretch over the whole interval on which is defined. Employing the particular form of the perturbation given by the delta function has the meaning that is varied only in the point. Except for this point, there is no variation in.

Footnotes