FLUCTUATION THEORY FOR THE WAVE-VECTOR EXPANSION OF THE EXCESS GRAND POTENTIAL OF A LIQUID-VAPOR INTERFACE AND THE THEORY OF INTERFACIAL FLUCTUATIONS

We develop a theory for the wavevector expansion of the excess grand potential of a liquid-vapour interface emphasizing the need to minimize the grand potential density functional for a given collective coordinate l(y) denoting the position of a surface of fixed density rho(X) (magnetization m(X)). We rederive the Triezenberg-Zwanzig formula for the surface tension gamma which has a unique value independent of rho(X). Our analysis yields a new expression for a rigidity kappa(rho(X)) which is strongly dependent on the particular value of rho(X) used to define l(y). We show that the expressions derived for kappa(rho(X)) (and gamma) are precisely those that need to be adopted when using the recently developed Fisher-Jin method of deriving an effective interfacial Hamiltonian appropriate to an asymptotically free interface. From the set of effective Hamiltonians describing the fluctuations of surfaces of all possible fixed density/magnetization we derive the correct analytic mean-field expression for the position dependence of the spin-spin correlation function for a free interface modelled by a Landau-Ginzburg-Wilson Hamiltonian. We emphasize that allowing for the m(X)-dependence of the rigidity is essential in this study of interfacial correlations to achieve true thermodynamic consistency.