Integral equations and inequalities in the theory of fluids

Using the method of functional Taylor expansion developed previously, an extensive set of equations is obtained for the distribution functions and Ursell functions in a classical fluid. These include in a systematic way many previously derived relations, e.g., Mayer-Montroll and Kirkwood-Salsburg equations. By terminating the Taylor expansion after a finite number of terms and retaining the remainder, we also obtain inequalities for the distribution functions and thermodynamic parameters of the fluid. For the case of positive interparticle potentials, we recover the inequalities first found by Lieb. For nonpositive potentials, new inequalities (some also obtained by Penrose) are derived. These inequalities are applied to the case of a hard-sphere fluid in three dimensions where they are compared with the results of machine computations and approximate theories. Different inequalities, not obtainable from the above considerations, and some properties of the fugacity expansions, are also derived.