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Functional Derivative Appendix Functional Derivative Appendix A: Calculus of Variation Consider a mapping from a vector space of functions to real number. Such mapping is called functional. One of the most representative examples of such a mapping is the action functional of analytical mechanics, A ,which is defined by Zt2 A½xtðÞ Lxt½ðÞ; vtðÞdt ðA:1Þ t1 where t is the independent variable, xtðÞis the trajectory of a particle, vtðÞ¼dx=dt is the velocity of the particle, and L is the Lagrangian which is the difference of kinetic energy and the potential: m L ¼ v2 À uxðÞ ðA:2Þ 2 Calculus of variation is the mathematical theory to find the extremal function xtðÞwhich maximizes or minimizes the functional such as Eq. (A.1). We are interested in the set of functions which satisfy the fixed boundary con- ditions such as xtðÞ¼1 x1 and xtðÞ¼2 x2. An arbitrary element of the function space may be defined from xtðÞby xtðÞ¼xtðÞþegðÞt ðA:3Þ where ε is a real number and gðÞt is any function satisfying gðÞ¼t1 gðÞ¼t2 0. Of course, it is obvious that xtðÞ¼1 x1 and xtðÞ¼2 x2. Varying ε and gðÞt generates any function of the vector space. Substitution of Eq. (A.3) to Eq. (A.1) gives Zt2 dx dg aðÞe A½¼xtðÞþegðÞt L xtðÞþegðÞt ; þ e dt ðA:4Þ dt dt t1 © Springer Science+Business Media Dordrecht 2016 601 K.S. Cho, Viscoelasticity of Polymers, Springer Series in Materials Science 241, DOI 10.1007/978-94-017-7564-9 602 Appendix: Functional Derivative From Eq. (A.4), we know that the extremal condition of Eq. (A.1) is equivalent to da ¼ 0 ðA:5Þ de e¼0 Hence, the introduction of Eq. (A.3) reduces the problems of extremal condition over a function space to those over real number. Evaluation of Eq. (A.5) can be done by the use of integration by parts: Zt2 da dg ¼ L1½xtðÞ; vtðÞgðÞþt L2½xtðÞ; vtðÞ dt de e¼0 dt t 1 ðA:6Þ Zt2 d ¼ L ½ÀxtðÞ; vtðÞ L ½xtðÞ; vtðÞ gðÞt dt 1 dt 2 t1 where @ @ L ¼ Lx½; v ; L ¼ Lx½; v ðA:7Þ 1 @x 2 @v Since the test function gðÞt is arbitrary, it is clear that Eq. (A.5) implies d @ @ Lx½; v ¼ Lx½; v ðA:8Þ dt @v @x This is the Euler–Lagrange equation. Substitution of Eq. (A.2) gives dv du m ¼À ðA:9Þ dt dx This is the equation of motion in Newtonian mechanics. When a binary mixture of fluid is considered, the free energy functional is given in terms of concentration profile /ðÞx as follows: Z c F½¼/ðÞx f ½þ/ðÞx kkr/ 2dV ðA:10Þ 2 X where f ðÞ/ is the free energy per unit volume for homogeneous mixture and γ is the material constant representing interfacial tension. Note that if /ðÞx is the concen- tration of one component, then that of the other component is 1 À /ðÞx . Thermodynamic theory addresses that the equilibrium concentration minimizes the free energy functional. It is the problem of calculus of variation to determine the Appendix: Functional Derivative 603 equilibrium concentration profile. As before, we define possible concentration fields by the equilibrium concentration and the test function: /ðÞ¼x /ðÞþx egðÞx ðA:11Þ The test function gðÞx is zero on the boundary @X of the domain of the mixture fluid. Substitution of Eq. (A.11) to Eq. (A.10) and differentiation with respect to ε give Z Z @ Âà F x x df x : @ /ðÞþegðÞ ¼ gðÞdV þ c r/ Árg dV ¼ 0 ðA 12Þ e e¼0 d/ X X Application of the divergence theorem to Eq. (A.12) gives Z df À c r2/ gðÞx dV ¼ 0 ðA:13Þ d/ X Since this equality must be hold for arbitrary domain, we have df ¼ cr2/ ðA:14Þ d/ This is the Euler–Lagrange equation for scalar field. It is more convenient to use the notation such that dxtðÞegðÞt for Eq. (A.3)or d/ðÞx egðÞx for Eq. (A.11). This notation is called variational notation. Consider a functional defined by Zt2 A ¼ Lt½; q1ðÞt ; ...; qnðÞt ; q_ 1ðÞt ; ...; q_ nðÞt dt ðA:15Þ t1 where q_ kðÞ¼t dqk=dt. The variation of the functional is defined as dA ¼ A½ÀqðÞþt dqðÞt ; q_ ðÞþt dq_ ðÞt A½qðÞt ; q_ ðÞt ðA:16Þ where qðÞ¼t ½q1ðÞt ; ...; qnðÞt and q_ ðÞ¼t ½q_ 1ðÞt ; ...; q_ nðÞt are short notation. It is implied that ddq dq_ ¼ k ðA:17Þ k dt 604 Appendix: Functional Derivative Then, Eq. (A.16) implies that Zt2 dA ¼ fgLt½À; qðÞþt dqðÞt ; q_ ðÞþt dq_ ðÞt Lt½; qðÞt ; q_ ðÞt dt t 1 ðA:18Þ Zt2 Xn @L @L ¼ dq þ dq_ dt @q k @q_ k k¼1 k k t1 Application of the integration by parts to the second integration gives Zt2 Xn @L t¼t2 Xn @L d @L dA ¼ dq þ À dq dt ðA:19Þ @q_ k @q dt @q_ k k¼1 k t¼t1 k¼1 k k t1 Here, we again use the boundary conditions for the variations such that dqkðÞ¼t1 dqkðÞ¼t2 0 for any k. Then, Eq. (A.19) becomes Zt2 Xn @L d @L dA ¼ À dq dt ðA:20Þ @q dt @q_ k k¼1 k k t1 Since we can take dqk independently, dA ¼ 0 implies that for any k, @L d @L À ¼ 0 ðA:21Þ @qk dt @q_ k For a multivariable function fxðÞ1; ...; xn , the differential of the function is given by Xn @f df ¼ dx ðA:22Þ @x k k¼1 k The stationary condition for the function is that for any k, @f ¼ 0 ðA:23Þ @xk Equation (A.20) is very similar to Eq. (A.22), and the Euler–Lagrange equation (A.21) is also similar to Eq. (A.23). Hence, it can be said that the Euler–Lagrange equation is the condition that the derivative of the functional A becomes zero. This analogy provides a clue to define functional derivative. We shall introduce a gen- eralized derivative called the Gâteaux derivative which is a generalization of directional derivative. Appendix: Functional Derivative 605 Appendix B: Gâteaux Derivative Derivative can be considered a linear mapping from the infinitesimal variation of domain to that of image. Consider a real-valued function whose domain is real number, y ¼ fxðÞ. From elementary calculus, we know that dy ¼ f 0ðÞx dx ðB:1Þ Note that dy is the infinitesimal variation of image, while dx is that of domain. If domain is position vector and image is scalar field, we know that when y ¼ f ðÞx , dy ¼rf Á dx ðB:2Þ The gradient rf is the derivative of f ðÞx . As for vector field, v ¼ vxðÞ, we know that dv ¼rðÞv T Á dx ðB:3Þ Then, the tensor ðÞrv T is the derivative of the vector field v. In Sect. 5.3,we learned the relation between directional derivative and gradient. Remember that d f ðÞx þ th ¼rf Á h ðB:4Þ dt t¼0 and d T vxðÞþ th ¼rðÞv Á h ðB:5Þ dt t¼0 Consider a vector space with a suitable norm which corresponds to the mag- nitude of vector. The vector space can be a set of numbers, vectors, tensors, or functions. For a mapping F from vector space X to vector space Y, the Gâteaux derivative of the mapping is defined as FðÞÀx þ e h FðÞx @ dGFðÞ¼x; h lim ¼ FðÞx þ e h ðB:6Þ e 0 @ ! e e e¼0 where x 2 X, h 2 X , and FðÞ2x Y. If x is a position vector and the image of F is a real number, then Eq. (B.6) implies the directional derivative in the direction to h. Higher-order Gâteaux derivative can be defined by @2 d2 FðÞ¼x; h ; h FðÞx þ e h þ e h ðB:7Þ G 1 2 @e @e 1 1 2 2 1 2 e1¼e2¼0 606 Appendix: Functional Derivative If F is a mapping from function to real number, the Gâteaux derivative is related to the functional derivative. As for the functional of Eq. (A.1), the Gâteaux derivative is given by Zt2 dA d A½¼xtðÞ; dxtðÞ dxtðÞdt ðB:8Þ G dxtðÞ t1 where we used the following notation: dA d2x duðÞn Àu0½ÀxtðÞ m with u0ðÞ¼n ðB:9Þ dxtðÞ d2t dn Analogy to Eqs. (B.4) and (B.5), it can be said that dA½xðÞ =dxtðÞ is the functional derivative. Hence, calculus of variation is to find the functional deriva- tive. Then, using this notation, we have Zt2 Zt2 d2A d2 A½¼xtðÞ; dxtðÞ dxtðÞdxtðÞ0 dt dt0 ðB:10Þ G dxtðÞdxtðÞ0 t1 t1 Note that a function can be considered as a special case of functional. Using the Dirac delta function, a function ftðÞcan be represented by Z1 ftðÞ¼ f ðÞs dsðÞÀ t ds F½ftðÞ ðB:11Þ À1 Application of the Gâteaux derivative gives dF½ftðÞ dftðÞ ¼ ¼ dðÞt À s ðB:12Þ df ðÞs df ðÞs As for the derivative of a function, we know that Z1 Z1 f ðÞs d0ðÞs À t ds ¼À f 0ðÞs dsðÞÀ t ds ðB:13Þ À1 À1 We have df 0ðÞt ¼Àd0ðÞt À s ðB:14Þ df ðÞs Appendix: Functional Derivative 607 With the help of Eqs. (B.12) and (B.13), it is obvious that d2A ¼Àu00ðÞxtðÞdðÞÀt À t0 md00ðÞt À t0 ðB:15Þ dxtðÞdxtðÞ0 Equation (B.6) implies that when x 2 X and h 2 X Z1 FðÞ¼x þ h FðÞþx dGFðÞx þ th; h dt ðB:16Þ 0 Then, the use of Eq.
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