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ghf_monog_appx_C

ghf_monog_appx_C - APPENDIX C CALCULUS OF FUNCTIONALS In...

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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

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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!
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