Equilibrium Statistical Mechanics of Inhomogeneous Fluids

The excess thermodynamic and molecular properties induced in a classical single-component fluid by static external force fields are examined from several points of view. By utilizing the techniques of cluster theory, a local ``pressure,'' p̄(r), is constructed whose spatial integral yields directly and precisely the logarithm of the grand partition function, including the result of interaction with container vessel walls as well as other external fields (present, for example, in gravitational or centrifugal equilibrium). In particular, it is remarked that the problem arising from cluster integral dependence upon vessel volume in the usual imperfect gas theory is solved. The density expansion for p̄ is transformed into an integral involving a modification (Xg) of the well-known ``direct-correlation function,'' or non-nodal cluster function. Subsequent construction of local free energies provides a condition under which these quantities may be represented by uniform fluid values (evaluated at the local density), plus corrections involving just density gradients (or Laplacians). When the external force field is generated by a fixed set of particles, the formalism leads to new integral equations for molecular distribution functions. Comparison with similar previously known integral equations, in the pair distribution case, yields an explicit, formally exact, expression in terms of Xg for the troublesome ``elementary diagram cluster sum,'' which has prevented exact solution to the pair distribution problem. Finally, a new fluctuation theorem is derived, which relates the density derivative of surface tension for a fluid next to a planar wall of its vessel, to molecular distribution at this interface.