Fluctuation theory for the wavevector expansion of the excess grand potential of a liquid-vapour interface and the theory of interfacial fluctuations

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Abstract
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 mX). 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 K( rho X) (and y) 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 mx-dependence of the rigidity is essential in this study of interfacial correlations to achieve true thermodynamic consistency.