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