User:Bob K31416/FD

.

A functional derivative relates a change in a functional to a change in a function that the functional depends on. For example, consider a functional F [ρ] that depends on the function ρ(x). Consider a small change in ρ(x) that adds to it a function δρ(x). The corresponding change in F [ρ] is,

$\delta F[\rho ]=F[\rho +\delta \rho ]-F[\rho ]$.

The functional derivative δF/δρ relates δF to δρ.

.

Given a manifold M representing (continuous/smooth/with certain boundary conditions/etc.) functions ρ and a functional F defined as

$F\colon M\rightarrow \mathbb {R} \quad {\mbox{or}}\quad F\colon M\rightarrow \mathbb {C} \,,$

the functional derivative of F[ρ], denoted δF/δρ, is defined by

${\begin{aligned}\int {\frac {\delta F}{\delta \rho (x)}}\ \phi (x)\ dx&=\lim _{\varepsilon \to 0}{\frac {F[\rho +\varepsilon \phi ]-F[\rho ]}{\varepsilon }}\\&=\left[{\frac {d}{d\epsilon }}F[\rho +\epsilon \phi ]\right]_{\epsilon =0},\end{aligned}}$

where εφ is the variation of ρ, and φ is an arbitrary function. These equations come from the expansion of the integrand of a functional in a power series. For example, consider the functional

$F[\rho ]=\int \limits _{a}^{b}L[\,x,\rho (x),\rho \,'(x)\,]\,dx\ ,$

where ρ′(x) ≡ dρ/dx. If ρ is varied by adding to it εφ, then the resulting integrand expanded in a Taylor series in powers of εφ is,

${\begin{aligned}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')&=L(x,\rho ,\rho ')+\left[{\frac {d}{d(\rho +\epsilon \phi )}}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')\right]_{\epsilon \phi =0}\epsilon \phi +{\frac {1}{2}}\left[{\frac {d^{2}}{d(\rho +\epsilon \phi )^{2}}}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')\right]_{\epsilon \phi =0}(\epsilon \phi )^{2}+\cdots \\&=L(x,\rho ,\rho ')+\left[{\frac {d}{d\epsilon }}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')\right]_{\epsilon =0}\epsilon +{\frac {1}{2}}\left[{\frac {d^{2}}{d\epsilon ^{2}}}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')\right]_{\epsilon =0}\epsilon ^{2}+\cdots \end{aligned}}$

Integrating both sides,

$F[\rho +\epsilon \phi ]=F[\rho ]+\int \left\{\left[{\frac {d}{d\epsilon }}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')\right]_{\epsilon =0}\epsilon +{\frac {1}{2}}\left[{\frac {d^{2}}{d\epsilon ^{2}}}L(x,\rho +\epsilon \phi ,\rho '+\epsilon \phi ')\right]_{\epsilon =0}\epsilon ^{2}+\cdots \right\}\ dx$

${\frac {F[\rho +\epsilon \phi ]-F[\rho ]}{\epsilon }}=\left[{\frac {d}{d\epsilon }}F[\rho +\epsilon \phi ]\right]_{\epsilon =0}+{\frac {1}{2}}\left[{\frac {d^{2}}{d\epsilon ^{2}}}F[\rho +\epsilon \phi ]\right]_{\epsilon =0}\epsilon +\cdots$

$\lim _{\epsilon \to 0}{\frac {F[\rho +\epsilon \phi ]-F[\rho ]}{\epsilon }}=\left[{\frac {d}{d\epsilon }}F[\rho +\epsilon \phi ]\right]_{\epsilon =0}$

${\begin{aligned}\delta F&=\int _{a}^{b}{\frac {\delta F}{\delta \rho (x)}}\ \delta \rho (x)\ dx\\&=\int _{a}^{b}{\frac {\delta F}{\delta \rho (x)}}\ \epsilon \phi (x)\ dx\end{aligned}}$

---

then the change in the value of J to first order in δf can be expressed as

$\delta J=\int _{a}^{b}{\frac {\delta J}{\delta f(x)}}{\delta f(x)}dx\,.$

The coefficient of δf(x), denoted as δJ/δf(x), is called the functional derivative of J with respect to f at the point x.

---

The differential (or variation or first variation) of the functional F[ρ] is,

$\delta F=\int {\frac {\delta F}{\delta \rho (x)}}\ \delta \rho (x)\ dx\ ,$

where δρ(x) = εϕ(x) is the variation of ρ(x). This is similar in form to the total differential of a function F(ρ₁, ρ₂, ... , ρ_n),

$dF=\sum _{i=1}^{n}{\frac {\partial F}{\partial \rho _{i}}}\ d\rho _{i}\ ,$

where ρ₁, ρ₂, ... , ρ_n are independent variables. Comparing the last two equations, the functional derivative δF/δρ(x) has a role similar to that of the partial derivative ∂F/∂ρ_i , where the variable of integration x is like a continuous version of the summation index i.

Definition

There are several approaches to defining the functional derivative: coefficient in a first order term; derivative of a function; ratio to an area.

References

• Parr, R. G.; Yang, W. (1989). "Appendix A, Functionals". Density-Functional Theory of Atoms and Molecules. New York: Oxford University Press. pp. 246–54. ISBN 978-0195042795. {{cite book}}: External link in |title= (help)

• Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A., translator (ed.). Calculus of variations (Unabridged reprint ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0486414485. {{cite book}}: |editor1-first= has generic name (help)CS1 maint: multiple names: editors list (link)

Footnotes

---

---

Parr, R. G.; Yang, W. (1989). Density-Functional Theory of Atoms and Molecules. New York: Oxford University Press. ISBN 978-0195042795. —

_________________

• p. 246

The differential of a functional is the part of the difference F [ f + δf ] − F [ f ] that depends on δf  linearly. Each δf (x) may contribute to this difference, so we write, for very small δf,

$\delta F=\int {\frac {\delta F}{\delta f(x)}}\ \delta f(x)\ dx\,,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\text{(A.1)}}$

where the quantity δF / δf (x) is the functional derivative of F with respect to f at the point x. Equation (A.1) is the fule for operating on δf (x) to give a number δF (x), and is the extension to continuous variables of the formula for the total differential of a function F (f₁, f₂, ... ):  dF = Σ_i (∂F / ∂f_i ) df_i .

---

Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A. (ed.). Calculus of variations (Unabridged repr. ed.). Mineola, N.Y.: Dover Publications. p. 27. ISBN 978-0486414485. —

_________________

• pp. 11–12

  3.2. We now introduce the concept of the variation (or differential) of a functional. Let J[ y] be a functional defined on some normed linear space, and let

$\Delta J[h]=J[y+h]-J[y]$

be its increment, corresponding to the increment h = h(x) of the "independent variable" y = y(x). If y is fixed, ΔJ[h] is a functional of h, in general a nonlinear functional. Suppose that

$\Delta J[h]=\phi [h]+\epsilon \|h\|\,,$

where φ[h] is a linear functional and ε → 0 as ||h|| → 0. Then the functional J[ y] is said to be differentiable, and the principal linear part of the increment ΔJ[h], i.e., the linear functional φ[h] which differs from ΔJ[h] by an infinitesimal of order higher than 1 relative to ||h||, is called the variation (or differential) of J[ y] and is denoted by δJ[h].

_________________

• p. 27

In Sec. 3.2 we introduced the concept of the differential of a functional. We now introduce the concept of the variational ( or functional) derivative, which plays the same role for functionals as the concept of the partial derivative plays for functions of n variables.

_________________

• pp.28–29

In the general case, the variational derivative is defined as follows: Let J[ y] be a functional depending on the function y(x), and suppose we give y(x) an increment h(x) which is different from zero only in the neighborhood of a point x₀. Dividing the corresponding increment J[ y + h] − J[ y] of the functional by the area Δσ lying between the curve y = h(x) and the x-axis, we obtain the ratio

${\frac {J[y+h]-J[y]}{\Delta \sigma }}\qquad \qquad \qquad \qquad \qquad \qquad \qquad {\text{(34)}}$

Next, we let the area Δσ go to zero in such a way that both max |h(x)| and the length of the interval in which h(x) is nonvanishing go to zero. Then, if the ratio (34) converges to a limit as Δσ → 0, this limit is called the variational derivative of the functional J[ y] at the point x₀ [for the curve y = y(x)], and is denoted by

$\left[{\frac {\delta J}{\delta y}}\right]_{x\,=\,x_{0}}\ .$

---

Mathews, Jon; Walker, RL (1965). Mathematical Methods of Physics. New York, New York.{{cite book}}: CS1 maint: location missing publisher (link) —

_________________

• p. 306

${\frac {\partial F}{\partial y}}-{\frac {d}{dx}}{\frac {\partial F}{\partial y'}}=0\qquad \qquad \qquad \qquad \qquad \qquad \qquad {\text{(12-8)}}$

The left-hand side of this equation is often written δF / δy, and called the variational, or functional, derivative of F with respect to y.

.

03:02, 21 May 2006‎ 66.41.8.178 :

Given a functional of the form

$F[\rho ]=\int f(\mathbf {r} ,\rho ,\nabla \rho ,\nabla ^{2}\rho ,\cdots )d^{3}r,$

the functional derivative can be written as

${\frac {\delta F[\rho ]}{\delta \rho }}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial (\nabla \rho )}}+\nabla ^{2}{\frac {\partial f}{\partial (\nabla ^{2}\rho )}}-\cdots .$

19:55, 20 August 2006‎ Md2perpe: Removed

05:59, 29 September 2006‎ DrF :

In the more general case that the functional depends on higher order derivatives, i.e.,

$F[\rho ]=\int f(\mathbf {r} ,\rho (\mathbf {r} ),\nabla \rho (\mathbf {r} ),\nabla ^{2}\rho (\mathbf {r} ),\cdots ,\nabla ^{N}\rho (\mathbf {r} ))\,d^{3}r,$

an analogous application of the definition yields

${\frac {\delta F[\rho ]}{\delta \rho (\mathbf {r} )}}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}+\nabla ^{2}\cdot {\frac {\partial f}{\partial \left(\nabla ^{2}\rho \right)}}-\cdots \nabla ^{N}\cdot {\frac {\partial f}{\partial \left(\nabla ^{N}\rho \right)}}=\sum _{i=0}^{N}(-1)^{i}\nabla ^{i}\cdot {\frac {\partial f}{\partial \left(\nabla ^{i}\rho \right)}}$

00:53, 7 November 2006‎ GuidoGer:

In the more general case that the functional depends on higher order derivatives, i.e.,

$F[\rho ]=\int f(\mathbf {r} ,\rho (\mathbf {r} ),\nabla \rho (\mathbf {r} ),\nabla ^{2}\rho (\mathbf {r} ),\cdots ,\nabla ^{N}\rho (\mathbf {r} ))\,d^{3}r,$

where $\nabla ^{i}$ is a vector whose $N*N$ components are all partial derivative operators of order $i$, i.e. $\partial ^{i}/(\partial r_{1}^{i_{1}}\partial r_{2}^{i_{2}}...\partial r_{N}^{i_{N}})$ with $i_{1}+i_{2}+...+i_{N}=i$, an analogous application of the definition yields

${\frac {\delta F[\rho ]}{\delta \rho (\mathbf {r} )}}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial (\nabla \rho )}}+\nabla ^{2}\cdot {\frac {\partial f}{\partial \left(\nabla ^{2}\rho \right)}}-\cdots +(-1)^{N}\nabla ^{N}\cdot {\frac {\partial f}{\partial \left(\nabla ^{N}\rho \right)}}=\sum _{i=0}^{N}(-1)^{i}\nabla ^{i}\cdot {\frac {\partial f}{\partial \left(\nabla ^{i}\rho \right)}}.$

---

â î ê ĵ

ê₁, ê₂, ê₃

${\hat {\mathbf {i} }}$

    where    

In terms of unit vectors $\mathbf {\hat {i}} ,\,\mathbf {\hat {j}} ,\,\mathbf {\hat {k}}$ along the axes of a cartesian coordinate system and partial derivatives ρ_x = ∂ρ/∂x, ρ_y = ∂ρ/∂y, ρ_z = ∂ρ/∂z,

---

F[ρ] — {{math|''F''[}}''ρ'']

F[ρ] — {{math|''F''[}}ρ]

F[ρ] — {{math|''F''}}[ρ]

F[ρ] — {{math|''F''[ρ]}}

---

$\rho =\rho (\mathbf {r} )$

$T_{W}[\rho ]={\frac {1}{8}}\int {\frac {\nabla \rho \cdot \nabla \rho }{\rho }}\ d\,^{3}r$

${\begin{aligned}T_{W}[\rho +\delta \rho ]&={\frac {1}{8}}\int {\frac {\nabla (\rho +\delta \rho )\cdot \nabla (\rho +\delta \rho )}{\rho +\delta \rho }}\,d\,^{3}r\\&={\frac {1}{8}}\int {\frac {\nabla \rho \cdot \nabla \rho +2\nabla \rho \cdot \nabla \delta \rho +\nabla \delta \rho \cdot \nabla \delta \rho }{\rho +\delta \rho }}\ d\,^{3}r\\\end{aligned}}$

Expanding the integrand in a Taylor series in powers of δρ and ∇δρ,

$T_{W}[\rho +\delta \rho ]={\frac {1}{8}}\int \left({\frac {\nabla \rho \cdot \nabla \rho }{\rho }}\ -\ {\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\,\delta \rho \ +\ 2\,{\frac {\nabla \rho \cdot \nabla \delta \rho }{\rho }}\ +\ R_{2}\right)d\,^{3}r$

where R₂ is the remainder after the first order terms. Using an identity for ${\frac {\nabla \delta \rho }{\rho }}\,,$ the integral of the third term in the integrand is,

$I_{3}\equiv \int 2\,{\frac {\nabla \rho \cdot \nabla \delta \rho }{\rho }}\ d\,^{3}r=\int 2\,\nabla \rho \cdot \left[\nabla \left({\frac {\delta \rho }{\rho }}\right)+{\frac {\delta \rho \ \nabla \rho }{\rho ^{2}}}\right]\ d\,^{3}r$

Using Green's first identity and the condition δρ = 0 on the boundary of the region of integration,

$I_{3}=\int \left(-\,2\,{\frac {\nabla ^{2}\rho }{\rho }}\,\delta \rho \ +\ 2\,{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\,\delta \rho \right)d\,^{3}r$

Substituting the integrand of I₃ for the third term in the integrand of T_W[ρ + δρ],

${\begin{aligned}T_{W}[\rho +\delta \rho ]&={\frac {1}{8}}\int \left({\frac {\nabla \rho \cdot \nabla \rho }{\rho }}\ -\ {\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\,\delta \rho \ -\,2\,{\frac {\nabla ^{2}\rho }{\rho }}\,\delta \rho \ +\ 2\,{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\,\delta \rho \ +\ R_{2}\right)d\,^{3}r\\&=T_{W}[\rho ]\ +\ {\frac {1}{8}}\int \left(-\,2\,{\frac {\nabla ^{2}\rho }{\rho }}\,\delta \rho \ +\ {\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\,\delta \rho \right)d\,^{3}r\ +\ {\frac {1}{8}}\int R_{2}\ d\,^{3}r\end{aligned}}$

Then the difference T_W[ρ+δρ] − T_W[ρ] to first order in δρ is,

$\delta T_{W}=\int \left(-\,{\frac {1}{4}}\,{\frac {\nabla ^{2}\rho }{\rho }}\ +\ {\frac {1}{8}}\,{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\right)\delta \rho \ d\,^{3}r$

The functional derivative is the coefficient of δρ in the integrand,

${\frac {\delta T_{W}}{\delta \rho }}=-\,{\frac {1}{4}}\,{\frac {\nabla ^{2}\rho }{\rho }}\ +\ {\frac {1}{8}}\,{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\ .$