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Unformatted text preview: APPENDIX C CALCULUS OF FUNCTIONALS In order to understand the fieldbased 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 nonlocal . 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 nonlocal 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 multivariable 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 multivariable 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.
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