Continuity Conditions for the Radial Distribution Function of Square-Well Fluids

The continuity properties of the radial distribution function g(r) and its close relative the cavity function y(r) ≡ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{\phi (r)/k_B T} g(r)$$ \end{document} are studied in the context of the Percus–Yevick (PY) integral equation for 3D square-well fluids. The cases corresponding to a well width (λ−1)σ equal to a fraction of the diameter of the hard core σ/m, with m=1, 2, 3, have been considered. In these cases, it is proved that the function y(r) and its first derivative are everywhere continuous, but eventually the derivative of some order becomes discontinuous at the points (n+1)σ/m, n=0, 1,.... The order of continuity [the highest order derivative of y(r) being continuous at a given point] κn is found to be κn∼n in the first case (m=1) and κn∼2n in the other two cases (m=2, 3), for n≫1. Moreover, derivatives of y(r) up to third order are continuous at r=σ and r=λσ for λ=3/2 and λ=4/3, but only the first derivative is continuous for λ=2. This can be understood as a nonlinear resonance effect.