Direct Correlation Functions and Their Derivatives with Respect to Particle Density

Abstract

An equation relating the many‐body distribution functions of a classical statistical mechanical system and their derivatives with respect to the particle density is stated and proved. It is found possible to define a heirarchy of symmetric functions of 2, 3, 4, etc. variables, of which the two‐variable function is the Ornstein— Zernike correlation function. These functions, termed the direct correlation functions, and their derivatives with respect to density satisfy a strikingly simple relation. The possibility of a closure of the equations is discussed and a superposition approximation suggested. The resulting equation for the two‐particle direct correlation function has the advantage that a knowledge of the function at one density predicts its values at another, thereby differing from most approximations, which require the solution of an integral equation in each case.