Distribution functions of a one-dimensional phase equilibrium with an interface

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Abstract
We calculate the exact expression of the pair distribution function in the interface of the one-dimensional lattice gas, in which the separation of phases is achieved by imposing an external field which changes sign at the middle of the lattice. The exact distribution functions of arbitrary order can then be obtained from an earlier theorem of Percus. The pair distribution function is shown to decay exponentially as a function of interparticle distance, with a decay length equal to the correlation length of the spontaneous density fluctuations in either bulk liquid or gas phases. This exact result enables us to test approximate theories of the pair distribution function of nonuniform fluids. The exact expression of the direct correlation function is obtained from recent studies of Percus, and of Robert and Widom, and it also is used to test approximate theories of the direct correlation function of nonuniform fluids. Extremely severe disagreement is found between exact results and all commonly used approximations based on local thermodynamics à la van der Waals, while excellent, nearly perfect agreement is found for the dynamic capillary wave-like approximation of Kalos, Percus, and Rao.