Statistical Thermodynamics of Nonuniform Fluids

We have developed a general formalism for obtaining the low-order distribution functions nq(r1, …, rq) and the thermodynamic parameters of nonuniform equilibrium systems where the nonuniformity is caused by a potential U(r). Our method consists of transforming from an initial (uniform) density n0 to the final desired density n(r) via a functional Taylor expansion. When n0 is chosen to be the density in the neighborhood of the r's we obtain nq as an expansion in the gradients of the density. The expansion parameter is essentially the ratio of the microscopic correlation length to the scale of the inhomogeneities. Our analysis is most conveniently done in the the grand ensemble formalism where the corresponding thermodynamic potential serves as the generating functional with U(r) as the variable for the nq. The transition from U(r) to n(r) as the relevant variable is accomplished via the direct correlation function which enters very naturally, relating the change in U at r2 due to a change in n at r1. It is thus essentially the matrix inverse of the two-particle Ursell function. The recent results of Stillinger and Buff on the thermodynamic potentials for non-uniform systems follow as a special case of our analysis without any recourse to the virial expansion. Thus, they hold also in the liquid region. In a succeeding paper we apply our analysis to obtain the asymptotic behavior of the radial distribution function in a uniform system.