Calculation of y(r) inside the core for hard discs and spheres

Abstract

A simple method is used to calculate a set of correlation functions P _j(r) which give the probability that a point at distance r from a sphere centre has j, and only j, sphere centres within one diameter, σ. The function P ₁(r) gives the ‘cavity-cavity’ correlation function y(r) = g(r) exp (βu(r)) in the range 0 < r < σ, where P ₁(r) = y(r)/y(0), and it gives the pressure and chemical potential efficiently. The calculated values of y(r) conform to the simple expression In {y(r)/y(0)} = ΔV(r){a ₁ + a ₂ r} where ΔV(r) is the change in the volume excluded to the system when a pair of cavity sites are separated from r = 0 to r. a ₁ and a ₂ are determined by exact thermodynamic relations. Results are reported for the hard disc and sphere fluids.