Statistical mechanics of a nonuniform fluid with long-range attractions

A recursive method is described for generating the Helmholtz free energy and paircorrelation function of a nonuniform fluid, in powers of the Kac inverse-range parameter $\gamma$. The results agree with earlier calculations using graphical and functional-integral methods. The ordering of the free energy is functionally consistent with a similar ordering of the density profile. The lowest-order or "van der Waals" theory is equivalent to a mean-field average of the attractive intermolecular energies and use of local thermodynamics in a reference hard-sphere fluid. The next level of approximation is the "random-phase approximation," which is shown to produce logarithmic anomalies in the free energy and density profile similar to those expected on the basis of capillary-wave theory. The occurrence of these anomalies results from the existence of a longwavelength divergence in the transverse Fourier transform of the lowest-order paircorrelation function.