Exact equations and the theory of liquids

Publisher Website
Google Scholar
Abstract
It is shown that the exact systems of equations for the , l = 1, 2, …, distribution functions (BBGYK, Kirkwood-Salsburg equations, etc.) have the sense of constancy conditions for the chemical potentials of a group of l particles (l = 1, 2, …). The necessary condition for a solution of the BBGYK equations to exist has been obtained. It is shown that the solution of the exact equations by expansion powers of density n correctly reflects the shortrange behaviour of but yields erroneous results at longer ranges. A correlation parameter λ is introduced into the BBGYK equations and used to obtain an expansion correctly describing the long range behaviour of but involving a tangible error at shorter ranges. The Kirkwood-Fischer superposition hypothesis is formulated for the general case and is shown to be unsuitable for solving the conventional exact equation types in the case of condensed systems. The BBGYK equations are transformed into an exact system of two equations for the unary and binary () distribution functions, the right hand parts of which contain an infinite number of integrals of the binary correlation function. For liquids, the solution of this system can already be obtained by expansion in powers of λ, the first term of which (in the case of spatially homogeneous systems) is determined by the HNC equation. It is shown that this equation is derivable from the exact equation for via Kirkwood's superposition approximation = . The HNC equation is generalized to the case of spatially inhomogeneous systems for which , and a method is suggested for calculating corrections to the HNC equation.