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.

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.

Given a manifold "M" representing (continuous/smooth/with certain boundary conditions/etc.) functions φ and a functional "F" defined as ::$$Fcolon M ightarrow mathbb{R} quad mbox{or} quad Fcolon M ightarrow mathbb{C} ,

the functional derivative of "F", denoted $${delta F}/{deltaphi(x)}, is a distribution $$delta F [phi] such that for all test functions "f",

:$$leftlangle delta F [phi] , f ight angle = left.frac{d}{depsilon}F [phi+epsilon f] ight|_{epsilon=0}.

Sometimes physicists write the definition in terms of a limit and the Dirac delta function, δ:

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

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.

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.

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 $$r_1, is affected by changing another quantity, say the density of electric charge at position $$r_2. 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 [ ho(r')] := V(r) = frac{1}{4piepsilon_0} int frac{ ho(r')} mathrm{d}^3r' \& {} = leftlangle frac{1}{4piepsilon_0} frac{1}, varphi(r') ight angle.end{align}

So,

:$$frac{delta V(r)}{delta ho(r')} = frac{1}{4piepsilon_0}frac{1}.

Now, we can evaluate the functional derivative at $$r=r_1 and $$r'=r_2 to see how the potential at $$r_1 is changed due to a small variation in the density at $$r_2. 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.

Formula for the integral of a function and its derivatives

Given a functional of the form:$$F [ ho(mathbf{r})] = int f( mathbf{r}, ho(mathbf{r}), abla ho(mathbf{r}) ), d^3r,with $$ho vanishing at the boundaries of $$mathbf{r}, the functional derivative can be written

:$$egin{align}leftlangle delta F [ ho] , phi ight angle & {} = frac{d}{dvarepsilon} left. int f( mathbf{r}, ho + varepsilon phi, abla ho+varepsilon ablaphi ), d^3r ight|_{varepsilon=0} \& {} = int left( frac{partial f}{partial ho} phi + frac{partial f}{partial abla ho} cdot ablaphi ight) d^3r \& {} = int left [ frac{partial f}{partial ho} phi + abla cdot left( frac{partial f}{partial abla ho} phi ight) - left( abla cdot frac{partial f}{partial abla ho} ight) phi ight] d^3r \& {} = int left [ frac{partial f}{partial ho} phi - left( abla cdot frac{partial f}{partial abla ho} ight) phi ight] d^3r \& {} = leftlangle frac{partial f}{partial ho} - abla cdot frac{partial f}{partial abla ho},, phi ight angle,end{align}

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

:$$delta F [ ho] = frac{partial f}{partial ho} - abla cdot frac{partial f}{partial abla ho}

or, writing the expression more explicitly,

:$$frac{delta F [ ho(mathbf{r})] }{delta ho(mathbf{r})} = frac{partial}{partial ho(mathbf{r})}f(mathbf{r}, ho(mathbf{r}), abla ho(mathbf{r})) - abla cdot frac{partial}{partial abla ho(mathbf{r})}f(mathbf{r}, ho(mathbf{r}), abla ho(mathbf{r}))

The above example is specific to the particular case that the functional depends on the function $$ho(mathbf{r}) and its gradient $$abla ho(mathbf{r}) only. In the more general case that the functional depends on higher order derivatives, i.e.

:$$F [ ho(mathbf{r})] = int f( mathbf{r}, ho(mathbf{r}), abla ho(mathbf{r}), abla^2 ho(mathbf{r}), dots, abla^N ho(mathbf{r})), d^3r,

where $$abla^i is a tensor whose $$n^i 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

:$$egin{align}frac{delta F [ ho] }{delta ho} = frac{partial f}{partial ho} - abla cdot frac{partial f}{partial( abla ho)} + abla^2 cdot frac{partial f}{partialleft( abla^2 ho ight)} - cdots \cdots + (-1)^N abla^N cdot frac{partial f}{partialleft( abla^N ho ight)} = sum_{i=0}^N (-1)^{i} abla^i cdot frac{partial f}{partialleft( abla^i ho ight)}.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::$$T_mathrm{TF} [ ho] = C_mathrm{F} int ho^{5/3}(mathbf{r}) , d^3r.$$T_mathrm{TF} [varrho] depends "only" on the charge density $$ho(mathbf{r}) and does not depend on its gradient, Laplacian, or other higher-order derivatives. Therefore,:$$frac{delta T_mathrm{TF} [ ho] }{delta ho} = C_mathrm{F} frac{partial ho^{5/3}(mathbf{r})}{partial ho} = frac{5}{3} C_mathrm{F} ho^{2/3}(mathbf{r}).

Coulomb potential energy functional

For the classical part of the potential, Thomas and Fermi employed the Coulomb potential energy functional:$$J [ ho] = frac{1}{2}intint frac{ ho(mathbf{r}) ho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r d^3r' = int left(frac{1}{2}int frac{ ho(mathbf{r}) ho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert} d^3r' ight) d^3r = int j [mathbf{r}, ho(mathbf{r})] , d^3r.Again, $$J [ ho] depends "only" on the charge density $$ho and does not depend on its gradient, Laplacian, or other higher-order derivatives. Therefore, :$$frac{delta J [ ho] }{delta ho} = frac{partial j}{partial ho} = frac{1}{2}int frac{partial}{partial ho}frac{ ho(mathbf{r}) ho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r' = int frac{ ho(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 [ ho] }{delta ho^2} = frac{delta}{delta ho}int frac{ ho(mathbf{r}')}{vert mathbf{r}-mathbf{r}' vert}, d^3r' = frac{partial}{partial ho} frac{ ho(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::$$T_mathrm{W} [ ho] = frac{1}{8} int frac{ abla ho(mathbf{r}) cdot abla ho(mathbf{r})}{ ho(mathbf{r}) }, d^3r = frac{1}{8} int frac{( abla ho(mathbf{r}))^2}{ ho(mathbf{r})}, d^3r = int t [ ho(mathbf{r}), abla ho(mathbf{r})] , d^3r.Now $$T_mathrm{W} [ ho] depends on the charge density $$ho "and" its gradient, therefore,:$$frac{delta T [ ho] }{delta ho} = frac{partial t}{partial ho} - ablacdotfrac{partial t}{partial ( abla ho)} = -frac{1}{8} frac{( abla ho(mathbf{r}))^2}{ ho(mathbf{r})^2} - ablacdotleft(frac{1}{4} frac{ abla ho(mathbf{r})}{ ho(mathbf{r})} ight) = frac{1}{8} frac{( abla ho(mathbf{r}))^2}{ ho^2(mathbf{r})} - frac{1}{4} frac{ abla^2 ho(mathbf{r})}{ ho(mathbf{r})}.

Writing a function as a functional

Finally, note that any function can be written in terms of a functional. For example,:$$ho(mathbf{r}) = int ho(mathbf{r}') delta(mathbf{r}-mathbf{r}'), d^3r'.This functional is a function of $$ho only, and thus, is in the same form as the above examples. Therefore,:$$frac{delta ho(mathbf{r})}{delta ho(mathbf{r}')}=frac{delta int ho(mathbf{r}') delta(mathbf{r}-mathbf{r}'), d^3r'}{delta ho(mathbf{r}')} = frac{partial left( ho(mathbf{r}') delta(mathbf{r}-mathbf{r}') ight)}{partial ho} = delta(mathbf{r}-mathbf{r}').

Entropy

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

:$$H [p(x)] = -sum_x p(x) log_2 p(x)Thus,

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