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) = e(phi(r)/kBT)g(r) are studied in the context of the Percus-Yevick (PY) integral equation for 3D square-well fluids. The cases corresponding to a well width (lambda - 1) sigma equal to a fraction of the diameter of the hard core sigma/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) sigma/m, n = 0, 1,.... The order of continuity [the highest order derivative of y(r) being continuous at a given point] kappa(n) is found to be kappa(n) similar to n in the first case (m = 1) and kappa(n) similar to 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 = sigma and r = lambda sigma for lambda = 3/2 and lambda = 4/3, but only the first derivative is continuous for lambda= 2. This can be understood as a nonlinear resonance effect.