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Instabilities and Self-Organization in MaterialsVolume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology$

Nasr Ghoniem and Daniel Walgraef

Print publication date: 2008

Print ISBN-13: 9780199298686

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199298686.001.0001

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(p.473) APPENDIX B FUNCTIONAL DERIVATIVES

(p.473) APPENDIX B FUNCTIONAL DERIVATIVES

Source:
Instabilities and Self-Organization in Materials
Publisher:
Oxford University Press

In several applications, it has been necessary to work with the derivative of a functional. This occurs, in particular, in systems where the dynamics has a gradient structure, and is related to the functional derivative of a potential. This potential may be the free energy in equilibrium systems of a Lyapounov functional in nonequilibrium systems. How does one compute the derivative of a functional is a basic topic of the calculus of variations. Although this is often considered as rather mysterious, it is no more than a generalization of computing the derivative of multi-variable functions, as shown in this Appendix. The basics of the method rely on the fact that a functional F[y(x)] may be viewed as a function of a large collection of variables, namely the values of its argument y(x) at each point x of its existence interval [a, b]. One can thus consider y(x) as a very long vector, built on the values of y(x) at each point of [a, b]. If these points are labeled with an index i, so may be y(x) (of course, some caution has to be taken with multivalued functions). This may be formalized as follows. Consider a function of a set of variables, y = {y 1,…, y n}, and a differentiable function F of these variables, F = F ({y 1,…, y n}). The variation of F, due to a change from the point y 0 = { y 1 0 , , y n 0 } to the point y 0 + d y = { y 1 0 + d y 1 , y n 0 + d y n } is then

(B.1)
d F = F y 1 | y 0 d y 1 + F y 2 | y 0 d y 2 + = n = 1 N F y n | y 0 d y n .
In the limit N → ∞, the function F becomes a function of the function y(x), in the sense that it depends on all the values of y(x) in the interval [a, b]. On rewriting equation (B.1) as
(B.2)
d F = n = 1 N ε ( 1 ε F y n | y 0 ) d y n
and on applying the definition of an integral
(B.3)
a b d x y ( x ) = lim ε 0 n = 1 N ε y ( x n )
(p.474) where x n = a + , one has:
(B.4)
d F = a b d x δ F δ y ( x ) | y 0 ( x ) δ y ( x )
where dy n = δy(x). Equation (B.4) is thus the definition of the derivative of a functional F[y]. However, the method of calculus is still complicated, since, for each derivation, one would have to define a mesh {x n} and the function F of the set {y n}. Fortunately, one usually deals with functionals F[y] of functions y of a continuous variable x, and the definition (B.4) can be used quite straightforwardly. A simple example is given by
(B.5)
F [ y ] = a b d x y 2 ( x )
for which
(B.6)
F [ y + δ y ] = a b d x [ y ( x ) + δ y ( x ) ] 2 = a b d x [ y 2 ( x ) + 2 y ( x ) δ y ( x ) + ( δ y ( x ) ) 2 ] = F [ y ] + a b d x [ 2 y ( x ) δ y ( x ) + ( δ y ( x ) ) 2 ]
and the infinitesimal change in F, due to δy, is, for δy → 0:
(B.7)
d F = F [ y + δ y ] F [ y ] = a b d x 2 y ( x ) δ y ( x )
and
(B.8)
δ F δ y ( x ) = 2 y ( x ) .
More generally, when the functional is a simple integral, as it is the case for Landau free energies and Lyapounov functionals encountered in this book, the functional derivative may be computed as follows. Let us consider, for example,
(B.9)
F [ y ] = a b d x L ( x , y ( x ) )
(p.475) which gives
(B.10)
F [ y + δ y ] = a b d x L ( x , y + δ y ) = a b d x [ L ( x , y ) + L ( x , y ) y δ y ] | δ y 1 = F [ y ] + a b d x L ( x , y ) y δ y
and
(B.11)
δ F δ y ( x ) = L ( x , y ) y .
In this case, the functional derivative may thus be taken as a simple partial derivative. The situation is a bit more complicated if L depends on derivatives of y. When
(B.12)
F [ y ] = a b d x L ( x , y , y )
where y = dy/dx, one has
(B.13)
F [ y + δ y ] = a b d x L ( x , y + δ y , y + δ y ) = F [ y ] + a b d x ( L ( x , y , y ) y δ y + L ( x , y , y ) y δ y )
which does not yet correspond to the definition (B.4). The problem is to go from δy to δy in last term of (B.13). This is solved by integrating this term by parts (noting that δ y = d δ y d x )
(B.14)
a b d x L ( x , y , y ) y δ y = [ L ( x , y , y ) y δ y ( x ) ] a b a b d d x ( L ( x , y , y ) y ) δ y d x
and
(B.15)
d F = a b d x ( L ( x , y , y ) y d d x ( L ( x , y , y ) y ) ) δ y d x + boundary terms .
If x is not at a boundary of the interval, one has
(B.16)
δ F δ y = L ( x , y , y ) y d d x ( L ( x , y , y ) y ) .
(p.476) A simple example of this property is given by
(B.17)
F = + d x ( d y d x ) 2
for which
(B.18)
δ F δ y = 2 d 2 y d x 2 .
Generalizations to higher derivative dependence and higher space dimensions are obvious. For example,
(B.19)
F [ y ] = a b d x L ( x , y , y , y ) δ D δ y = L ( x , y , y , y ) y d d x ( L ( x , y , y , y ) y ) + d 2 d x 2 ( L ( x , y , y ) y )
and
(B.20)
F [ y ] = d x ( y ( b f x ) ) 2 δ F δ y = 2 2 y ( x )
where
= x 1 x + y 1 y + z 1 z
and
2 = 2 x 2 + 2 y 2 + 2 z 2 .