{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ghf_monog_appx_C

ghf_monog_appx_C - APPENDIX C CALCULUS OF FUNCTIONALS In...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
APPENDIX C CALCULUS OF FUNCTIONALS In order to understand the field-based approach to modelling inhomogeneous fluids, it is necessary to have a basic familiarity with the calculus of functionals (Volterra, 1959). This broad subject includes topics such as functional differen- tiation, functional integration, and min–max problems. These are typically dis- cussed in texts on functional analysis, calculus of variations, optimization theory, and field theory. Here, we provide a brief tutorial. Physically oriented references where more details can be found include (Fetter and Walecka, 1980; Hansen and McDonald, 1986; Parr and Yang, 1989; Zee, 2003). C.1 Functionals In the simplest case, a functional is a mapping between a function f ( x ) defined over some interval x [ a, b ] and a number F that generally depends on the values of the function over all points of the interval. For example, a simple linear functional is just the integral F 1 [ f ] = Z b a dx f ( x ) (C.1) This formula associates a number F 1 with the integral of f ( x ) over a x b . We adopt the functional “square bracket” notation F 1 [ f ] to indicate that F 1 depends on f ( x ) at all points over the interval. An example of a nonlinear functional is F 2 [ f ] = Z b a dx [ f ( x )] 2 (C.2) Both F 1 [ f ] and F 2 [ f ] are referred to as local functionals because values of f ( x ) for different x contribute independently (additively) to the value of the functional. More generally, a functional can depend on the function and its derivatives over the interval. Such functionals are referred to as non-local . For example, F 3 [ f ] = Z b a dx ( - [ f ( x )] 2 + [ f ( x )] 4 + [ f 0 ( x )] 2 ) (C.3) is a familiar functional that appears in the Landau–Ginzburg theory of phase transitions (Chaikin and Lubensky, 1995; Goldenfeld, 1992). A second example of a non-local functional is the quadratic expression F 4 [ f ] = Z b a dx Z b a dx 0 f ( x ) K ( x, x 0 ) f ( x 0 ) (C.4) where the kernel K ( x, x 0 ) is an arbitrary function of x and x 0 . 393
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
394 CALCULUS OF FUNCTIONALS Functionals can also be defined for multi-variable functions f ( r ) that, e.g., could represent the chemical potential fields w ( r ) that are central to the subject of this monograph. For example, the extension of eqn (C.3) to functions defined in three dimensions is the Landau–Ginzburg “square gradient” functional F 5 [ f ] = Z d r ( - [ f ( r )] 2 + [ f ( r )] 4 + |∇ f | 2 ) (C.5) C.2 Functional differentiation The concept of differentiation of functionals is a straightforward extension of the notion of partial differentiation for multi-variable functions. Consider subjecting a function f ( x ) defined over x [ a, b ] to an arbitrary small perturbation δf ( x ). The perturbation δf ( x ) is itself a function defined over the same interval of x . For some functional F [ f ], we then consider its value F [ f + δf ] when f ( x ) f ( x )+ δf ( x ). This quantity can be Taylor-expanded in powers of the perturbation δf ( x ) to yield the general form F [ f + δf ] = F [ f ] + Z b a dx Γ 1 ( x ) δf ( x ) + 1 2!
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern