Derivation of the classical theory of correlations in fluids by means of functional differentiation

The k-particle, infinite-volume distribution functions n̄k (r1, …, rk−1, γ) and modified Ursell correlation functions Ūk (r1, …, rk−1, γ) of a classical system of particles with the two-body potential q(r) + γνK(γr) are considered. The limiting values of the functions n̄k (r1, …, rk−1, γ), n̄k (S1/γ, …, Sk−1/γ, γ) and γ(1−k)ν Ūk (S1/γ, …, Sk−1/γ, γ) in the limit γ → 0 are calculated, under fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the rage of the potential γνK(γr). The direct correlation function c̄ (r, γ) is also considered and it is shown that for S ≠ 0, limγ→0 γ−νc̄ (Sγ, γ) = −βK (S), for all one-phase states, where β is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.