Abstract
Using functional variation methods an exact functional equation has been derived for the bridge function in an inhomogeneous fluid. Step by step the equation has been simplified using the virial theorem as a closure relation. As a result a closed but approximate equation has been given that expresses a fully consistent bridge function in terms of the correlation function and its derivatives. In the homogeneous limit new and exact global relations have been derived by requiring consistency with respect to the energy, the virial and the compressibility theorem.
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