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 conditions 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

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 concentration 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 @ ÂÃ 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 generalized 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 derivative. 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

FðÞ¼x þ h FðÞþx dGFðÞx þ th; h dt ðB:16Þ 0

Then, the use of Eq. (B.16) gives the Taylor expansion of F such that 1 1 FðÞ¼x þ h FðÞþx d FðÞþx; h d2 Fðx; h Þþ ÁÁÁ þ dn FðÞþx; h R ðB:17Þ G 2! G n! G n þ 1 where Z1 1 R ðÞ1 À t ndn þ 1FðÞx þ th; h dt ðB:18Þ n þ 1 n! G 0

Consider the functional of Eq. (2.1) in Chap. 10. Assume that the functional of Eq. (2.1) in Chap. 10 has the following form:

Z t s¼t ei Á T ½ÁCtðÞx; s ek ¼ Pik½CtðÞx; s ds Sik½ðCtðÞx; s B:19Þ s¼À1 À1 where Pik½X is the ikth component of a tensor-valued function of a tensor X.We know that Z t dSik dGSik½¼XðÞs ; dXðÞs dXabðÞs ds dXabðÞs À1 Z t Z t d2S ðB:20Þ d2 T0 X s ; dX s ik dX s dX s ds ds0 G ik½¼ðÞ ðÞ 0 abðÞ cdðÞ dXabðÞs dXcdðÞs À1 À1 . . 608 Appendix: Functional Derivative

Then, the Taylor expansion of Sik½CtðÞx; s about I ¼ CtðÞx; t is given by

Z t S ÂÃ S C x; d ik ab x; ik½tðÞs Ct ðÞÀs dab ds dXabðÞs À1 Z t Z t 1 d2S ÂÃÂÃ ik Cab x; s d Ccd x; s d ds ds0 þ 0 t ðÞÀab t ðÞÀcd 2 dXabðÞs dXcdðÞs À1 À1 þ ÁÁÁ ðB:21Þ

This is the derivation of Eq. (2.4) in Chap. 10. Index

A Contact force, 112 Arrhenius equation, 443 Continuous retardation spectrum, 317 Constraction map, 376 B Contravariant base vectors, 30 Bagley correction, 504 Convolution theorem, 83 Bernstein basis polynomials, 368 Covariant base vectors, 30 Blatz model, 277 Cox-Merz relation, 505 Body force, 112 Creep compliance, 139, 290 Boltzmann superposition principle, 142, 288 Creep ringing, 311 Bond correlation function, 240 Creep test, 290 Branching parameter, 242 Critical molecular weight, 136 Brownian motion, 211 Cubic Hermite polynomial, 366 Bulk modulus, 125 Cumulant generating function, 180 Current configuration, 94 C Canonical ensemble, 191 D Caputo derivative, 329 Damping function, 519 Carreau-Yasuda model, 135 Davies and Anderssen limit, 425 Cauchy sequence, 11 Deborah number, 286 Cauchy stress, 117 Deformation gradient, 97 Cayley-Hamilton theorem, 66, 131 Deformation rate tensor, 103 Central limit theorem, 184 Deformed configuration, 94 Chandrasekhar equation, 227 Diffusion wave spectroscopy, 313 Chapman-Kolmogorov equation, 214 Dirac delta function, 138 Characteristic function, 180 Discrete fourier transform (DFT), 361 Characteristic ratio, 249 Discrete relaxation time spectrum, 316 Chebyshev nodes, 365 Discrete retardation time spectrum, 317 Clausius-Duhem inequality, 157 Displacement vector, 99 Clausius inequality, 152 Double reptation, 474 Coefficient of variation, 383 Dynamic moduli, 294 Cole-Cole model, 323 Cole-Cole plot, 323 E Conditional probability, 181 Eight-constant model, 514 Cone-and-plate fixture, 303 End-to-end vector, 239 Configuration, 94 Engineering strain, 99 Conformation tensor, 527 Entanglement concentration, 264 Constitutive equation, 113 Enthalpy, 153

Entropy production, 156 Hindered rotation chain model (HRC), 242 Equal a priori probability, 188 Hookean body, 136 Equipartition theorem, 204 Hydrodynamic interaction, 335 Ergodic hyperthesis, 205 Hydrodynamic screening length, 336 Euler-Lagrange equation, 602 Eulerian description, 95 I Eulerian objective vectors and tensors, 172 Ideal chain, 240 Extended irreversible thermodynamics, 161 Inaccessible statement of Caratheodory, 152 Extensional flow, 492 Incompressible fluid, 132 Incompressible solid, 127 F Inertial pressure scale, 133 Fading memory, 105, 138 Infinitesimal strain, 99 Fast Fourier transform (FFT), 314, 394 Information theory, 191 Finitely extensible nonlinear elastic model Internal energy, 119 (FENE ), 270, 379 Internal variables, 143, 161 Finite extensible nonlinear elasticity preaverage The intersection condition of coordinate lines, (FENE-P), 529 28 Fick's diffusion law, 215 Inverse integration with regularization (IIR), Final value theorem of Laplace transform, 291 386 Fixed-point iteration, 415 Inverse tensor, 62 Flory theorem, 335 Invertible tensor, 62 Flow direction, 134 Isotropic tensor, 69 Fluctuation-dissipation theorem, 213 Fokker-Planck equation, 187, 223 J Fourier Transform Rheology (FT-Rheology), Jaumann time derivative, 174 547 Jeffreys model, 140 Fractional derivative, 142 Joint probability density function, 181 Fractional models, 142 Fredholm integral equation of the 1st kind, 406 K Freely jointed chain model (FJC), 242 Kelvin-Voigt model, 138 Freely rotating chain model (FRC), 242 Kohlrausch-Williams-Watts (KWW) equation, Froude number, 133 142 Fuoss-Kirkwood relation, 398 Kramers-Kronig relations, 298 Functional, 601 Kronecker's delta, 8 Functional derivative, 606 L G Lagrangian description, 95 Gâteaux derivative, 605 Lagrangian objective tensors, 172 Gaussian distribution function, 184 Laminar flow, 492 Generalized Langevin equation, 222 Langevin equation, 187, 212 Generalized Maxwell model, 141 LAOStrain, 563 Generalized Voigt model, 141 LAOStress, 563 Gibbs free energy, 153 Large amplitude oscillatory shear (LAOS), 545 Gradient direction, 134 Least square method, 363 Gram-Schmidt orthogonalization, 15 Left Cauchy-Green tensor, 98 Grand canonical ensemble, 193 Legendre transform, 154 Levenberg-Marquardt method, 369 H Liouville equation, 209, 216 Hamiltonian, 178 Liouville operator, 217 Heat conductivity, 118 Lissajous-Bowditch loop (LB loop), 546 Heat flux, 118 Lodge equation, 511 Helmholtz free energy, 153 Long-range interaction, 240 Hilbert space, 11 Loss modulus, 294 Index 611

Loss tangent, 300 Plot of van Gurp and Palmen, 452 Lower-convected time derivative, 511 Poisson's ratio, 126 Lower-convective time derivative, 174 Polar decomposition, 97 Post-Widder formula, 411 M Power law fluid model, 136 Marginal distribution function, 181 Primitive chain, 347 Mark-Houwink equation, 260 Principle of causality, 138, 289 Material coordinate, 94 Principle of material frame-indifference, 129, Material description, 95 173 Material time derivative, 95 Proper transform, 30 Maxwell distribution, 226 Pseudo-inverse, 413 Maxwell model, 137 Pseudo inverse matrix, 387 Mean relaxation time, 297 Phan-Thien and Tanner (PTT) model, 533 Memory function, 145 Microcanonical ensemble, 188 R Microrheology, 315 Radius of gyration, 241 Microstate variables, 178 Rational thermodynamics, 160 Middle amplitude oscillatory shear (MAOS), Real chain, 240 548 Reference configuration, 94 Modulus, 125 Relative deformation gradient, 105 Molecular stress function model (MSF model), Relative finger deformation tensor, 106 553 Relative Piola deformation tensor, 106 Mooney-Rivlin model, 273 Relaxation modulus, 288 Mooney-Rivlin plot, 273 Relaxation time spectrum, 142 Retardation time, 139, 140 N Retardation time spectrum, 142 Navier equation, 129 Reynolds number, 133 Navier-Stokes equation, 132 Reynolds transport theorem, 112 Neo-Hookean model, 273 Riemann and Liouville derivative, 329 Newtonian fluid, 131 Right Cauchy-Green tensor, 98, 348 Nonlinear regularization, 414 Rigid body, 96 Norm, 11 Rivlin-Ericksen tensors, 107 Normal equations, 363 Root-mean-square error, 363 Nth cumulant, 180 Runge phenomenon, 366 Nth order Rivlin-Ericksen tensor, 508 S O Secondary critical molecular weight, 441 Objective time derivatives, 174 The second order fluid, 509 Ogden model, 277 Sharda model, 277 Orthogonal tensor, 61 Shear flow, 492 Overlap concentration, 262 Shear modulus, 126 Shear rate, 134 P Shear stress, 126 Pade approximation, 378 Shear-thinning fluids, 136 Parallel plate fixture, 303 Short-range interaction, 240 The parallelism of coordinate line, 29 Simple elongation, 126 Partition function, 189 Simple shear, 125 Phase angle, 300 Singular value, 387 Phase space, 178 Singular value decomposition, 387, 413 Piola-Kirchhoff stress of the 1st kind, 117 Small amplitude oscillatory shear (SAOS), 545 Piola-Kirchhoff stress of the 2nd kind, 117 Smoluchowski equation, 227, 525 Pipkin diagram, 287 Spatial description, 95 612 Index

Specific heat capacity, 120 Trouton relation, 506 Spin tensor, 103 True stress, 117 Standard solid model, 140 Tschoegl model, 277 State variables, 147 Statistical independence, 182 U Steady-state compliance, 293 Unit step function, 138 Stokes-Einstein equation, 213 Upper-convective Maxwell model, 168, 512 Stokes flow, 134 Upper-convective time derivative, 168 Storage modulus, 294 Strain-frequency superposition (SFS), 548, 563 V Strain-rate frequency superposition (SRFS), Valanis-Landel hypothesis, 274 548, 563 Valanis-Landel model, 277 Streamlines, 492 Velocity gradient, 102 Stress, 112 Viscometric flow, 493 Stress-controlled and strain-controlled Viscous pressure scale, 133 rheometers, 302 Voigt model, 138 Stress decomposition (SD), 547 Vorticity direction, 134 Stress-optical rule, 348 Stress power, 120 W Stress relaxation test, 288 Warner's approximation, 379 Stress vector, 112, 114 Weight-average molecular weight, 136 Strouhal number, 133 Weissenberg-Rabinowitsch equation, 504 Sum of state, 189 Wick's theorem, 186

T Y Tangent base vectors, 29 Young's modulus, 126 Thermodynamic limit, 201 Thermodynamic process, 147 Z Tikhonov regularization, 371, 386, 414 Zero-shear viscosity, 135, 293 Time-strain separable K-BKZ model, 517 Time-temperature superposition, 438