ghf_monog_appx_C

ghf_monog_appx_C - APPENDIX C CALCULUS OF FUNCTIONALS In...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 minmax 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 ( x )] 2 ) (C.3) is a familiar functional that appears in the LandauGinzburg 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 f ( x ) K ( x,x ) f ( x ) (C.4) where the kernel K ( x,x ) is an arbitrary function of x and x . 393 394 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 LandauGinzburg 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 f ( x )....
View Full Document

This note was uploaded on 12/29/2011 for the course CHE 230a taught by Professor Ghf during the Fall '10 term at UCSB.

Page1 / 7

ghf_monog_appx_C - APPENDIX C CALCULUS OF FUNCTIONALS In...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online