Thermodynamically consistent truncations of distribution function hierarchies. I. General formulation

In this paper we initiate a study of the closure problem in the statistical mechanical theory of fluids, with particular emphasis on the problem of identifying thermodynamically consistent truncations of distribution function hierarchies. Our approach is: (i) to choose an appropriately restricted set of variables for the reduced distribution functions ρ(n) and to generate corresponding hierarchies; (ii) to retain, for a kth order truncation, all those equations involving only the ρ(n) with n≤k; and (iii) to choose the truncation for which all these equations can be satisfied simultaneously. In this paper we focus on the problem of identifying self-consistent, third-order truncations for translationally invariant systems. We show that the analytical approach taken provides guidance in the construction of such closures and we comment on the factors involved in implementing numerically a specific choice of closure identified in our study.