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Encyclopedia > Functional derivative

In mathematics and theoretical physics, the functional derivative is a generalization of the directional 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 usual calculus derivative. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... For other uses, see Calculus (disambiguation). ... For other uses, see Derivative (disambiguation). ...


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 the Gâteaux derivative. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, the Gâteaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. ...


Given a manifold M representing (continuous/smooth/with certain boundary conditions/etc.) functions φ and a functional F defined as On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In mathematics, the term functional is applied to certain functions. ...

Fcolon M rightarrow mathbb{R} quad mbox{or} quad Fcolon M rightarrow mathbb{C} ,

the functional derivative of F, denoted δF / δφ(x), is a distribution δF[φ] such that for all test functions f, This article is about generalized functions in mathematical analysis. ... This page deals with mathematical distributions. ...

leftlangle delta F[phi], f rightrangle = left.frac{d}{depsilon}F[phi+epsilon f]right|_{epsilon=0}.

Sometimes physicists write the definition in terms of a limit and the Dirac delta function, δ: The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...

frac{delta F[phi(x)]}{delta phi(y)}=lim_{varepsilonto 0}frac{F[phi(x)+varepsilondelta(x-y)]-F[phi(x)]}{varepsilon}.

Contents

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 the Fréchet derivative, while one uses the Gâteaux derivative on more general locally convex spaces. Note that the well-known Hilbert spaces are special cases of Banach spaces. The more formal 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. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. ... In mathematics, the Gâteaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. ... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... For other uses, see Calculus (disambiguation). ... Analysis has its beginnings in the rigorous formulation of calculus. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...


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 functions 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. This page deals with mathematical distributions. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...


   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 r1, is affected by changing another quantity, say the density of electric charge at position r2. 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[rho(r')] := V(r) = frac{1}{4piepsilon_0} int frac{rho(r')}{|r-r'|} mathrm{d}^3r'.

That is, for each r, the potential V(r) is a functional of ρ(r'). We can apply either definition---here we apply the math definition:

begin{align} leftlangle delta F[rho(r')], varphi(r') rightrangle & {} = frac{d}{dvarepsilon} left. frac{1}{4piepsilon_0} int frac{rho(r') + varepsilon varphi(r')}{|r-r'|} mathrm{d}^3r' right|_{varepsilon=0} & {} = frac{1}{4piepsilon_0} int frac{varphi(r')}{|r-r'|} mathrm{d}^3r' & {} = leftlangle frac{1}{4piepsilon_0} frac{1}{|r-r'|}, varphi(r') rightrangle. end{align}

So,

frac{delta V(r)}{delta rho(r')} = frac{1}{4piepsilon_0}frac{1}{|r-r'|}.

Now, we can evaluate the functional derivative at r = r1 and r' = r2 to see how the potential at r1 is changed due to a small variation in the density at r2. 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 from physics. Another example in physics is the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ...


Formula for the integral of a function and its derivatives

Given a functional of the form

F[rho(mathbf{r})] = int f( mathbf{r}, rho(mathbf{r}), nablarho(mathbf{r}) ), d^3r,

with ρ vanishing at the boundaries of mathbf{r}, the functional derivative can be written

begin{align} leftlangle delta F[rho], phi rightrangle & {} = frac{d}{dvarepsilon} left. int f( mathbf{r}, rho + varepsilon phi, nablarho+varepsilonnablaphi ), d^3r right|_{varepsilon=0} & {} = int left( frac{partial f}{partialrho} phi + frac{partial f}{partialnablarho} cdot nablaphi right) d^3r & {} = int left[ frac{partial f}{partialrho} phi + nabla cdot left( frac{partial f}{partialnablarho} phi right) - left( nabla cdot frac{partial f}{partialnablarho} right) phi right] d^3r & {} = int left[ frac{partial f}{partialrho} phi - left( nabla cdot frac{partial f}{partialnablarho} right) phi right] d^3r & {} = leftlangle frac{partial f}{partialrho} - nabla cdot frac{partial f}{partialnablarho},, phi rightrangle, end{align}

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

delta F[rho] = frac{partial f}{partialrho} - nabla cdot frac{partial f}{partialnablarho}

or, writing the expression more explicitly,

frac{delta F[rho(mathbf{r})]}{deltarho(mathbf{r})} = frac{partial}{partialrho(mathbf{r})}f(mathbf{r}, rho(mathbf{r}), nablarho(mathbf{r})) - nabla cdot frac{partial}{partialnablarho(mathbf{r})}f(mathbf{r}, rho(mathbf{r}), nablarho(mathbf{r}))

The above example is specific to the particular case that the functional depends on the function rho(mathbf{r}) and its gradient nablarho(mathbf{r}) only. In the more general case that the functional depends on higher order derivatives, i.e. For other uses, see Gradient (disambiguation). ...

F[rho(mathbf{r})] = int f( mathbf{r}, rho(mathbf{r}), nablarho(mathbf{r}), nabla^2rho(mathbf{r}), dots, nabla^Nrho(mathbf{r})), d^3r,

where nabla^i is a tensor whose ni 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

begin{align} frac{delta F[rho]}{delta rho} = frac{partial f}{partialrho} - nabla cdot frac{partial f}{partial(nablarho)} + nabla^2 cdot frac{partial f}{partialleft(nabla^2rhoright)} - cdots cdots + (-1)^N nabla^N cdot frac{partial f}{partialleft(nabla^Nrhoright)} = sum_{i=0}^N (-1)^{i}nabla^i cdot frac{partial f}{partialleft(nabla^irhoright)}. 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: Year 1927 (MCMXXVII) was a common year starting on Saturday (link will display full calendar) of the Gregorian calendar. ... Fermi redirects here. ... In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. ... Density functional theory (DFT) is one of the most popular approaches to quantum mechanical many-body electronic structure calculations of molecular and condensed matter systems. ...

T_mathrm{TF}[rho] = C_mathrm{F} int rho^{5/3}(mathbf{r}) , d^3r.

T_mathrm{TF}[varrho] depends only on the charge density rho(mathbf{r}) and does not depend on its gradient, Laplacian, or other higher-order derivatives. Therefore,

frac{delta T_mathrm{TF}[rho]}{delta rho} = C_mathrm{F} frac{partial rho^{5/3}(mathbf{r})}{partial rho} = frac{5}{3} C_mathrm{F} rho^{2/3}(mathbf{r}).

Coulomb potential energy functional

For the classical part of the potential, Thomas and Fermi employed the Coulomb potential energy functional This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each...

J[rho] = frac{1}{2}intint frac{rho(mathbf{r}) rho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r d^3r' = int left(frac{1}{2}int frac{rho(mathbf{r}) rho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert} d^3r'right) d^3r = int j[mathbf{r},rho(mathbf{r})], d^3r.

Again, J[ρ] depends only on the charge density ρ and does not depend on its gradient, Laplacian, or other higher-order derivatives. Therefore,

frac{delta J[rho]}{delta rho} = frac{partial j}{partial rho} = frac{1}{2}int frac{partial}{partial rho}frac{rho(mathbf{r}) rho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r' = int frac{rho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r'

The second functional derivative of the Coulomb potential energy functional is

frac{delta^2 J[rho]}{delta rho^2} = frac{delta}{delta rho}int frac{rho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r' = frac{partial}{partial rho} frac{rho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert} = frac{1}{vert mathbf{r}-mathbf{r}' 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: 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ... Carl Friedrich Freiherr (Baron) von Weizsäcker (June 28, 1912, Kiel – 28 April 2007, Söcking near Starnberg) was a German physicist and philosopher. ...

T_mathrm{W}[rho] = frac{1}{8} int frac{nablarho(mathbf{r}) cdot nablarho(mathbf{r})}{ rho(mathbf{r}) }, d^3r = frac{1}{8} int frac{(nablarho(mathbf{r}))^2}{rho(mathbf{r})}, d^3r = int t[rho(mathbf{r}),nablarho(mathbf{r})], d^3r.

Now TW[ρ] depends on the charge density ρ and its gradient, therefore,

frac{delta T[rho]}{delta rho} = frac{partial t}{partial rho} - nablacdotfrac{partial t}{partial (nabla rho)} = -frac{1}{8} frac{(nablarho(mathbf{r}))^2}{rho(mathbf{r})^2} - nablacdotleft(frac{1}{4} frac{nablarho(mathbf{r})}{rho(mathbf{r})}right) = frac{1}{8} frac{(nablarho(mathbf{r}))^2}{rho^2(mathbf{r})} - frac{1}{4} frac{nabla^2rho(mathbf{r})}{rho(mathbf{r})}.

Writing a function as a functional

Finally, note that any function can be written in terms of a functional. For example,

rho(mathbf{r}) = int rho(mathbf{r}') delta(mathbf{r}-mathbf{r}'), d^3r'.

This functional is a function of ρ only, and thus, is in the same form as the above examples. Therefore,

frac{delta rho(mathbf{r})}{deltarho(mathbf{r}')}=frac{delta int rho(mathbf{r}') delta(mathbf{r}-mathbf{r}'), d^3r'}{delta rho(mathbf{r}')} = frac{partial left(rho(mathbf{r}') delta(mathbf{r}-mathbf{r}')right)}{partial rho} = delta(mathbf{r}-mathbf{r}').

Entropy

The entropy of a discrete random variable is a functional of the probability mass function. Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

H[p(x)] = − ∑ p(x)log2p(x)
x

Thus,

begin{align} leftlangle delta H, phi rightrangle & {} = sum_x delta H , varphi(x) & {} = frac{d}{depsilon} left. H[p(x) + epsilonphi(x)] right|_{epsilon=0} & {} = -frac{d}{dvarepsilon} left. sum_x [p(x) + varepsilonvarphi(x)] log_2 [p(x) + varepsilonvarphi(x)] right|_{varepsilon=0} & {} = displaystyle -sum_x [1+log_2 p(x)]varphi(x) & {} = leftlangle -[1+log_2 p(x)], varphi rightrangle. 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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