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

The continuity conditions of the radial distribution function g® and its close relative the cavity function y® are studied in the context of the Percus-Yevick (PY) integral equation for 3D square-well fluids. The cases corresponding to a well width, (w-1)d, equal to a fraction of the diameter of the hard core, d/m, with m=1,2,3 have been considered. In these cases, it is proved that the function y® and its first derivative are everywhere continuous but eventually the derivative of some order becomes discontinuous at the points (n+1)d/m, n=0,1,… . The order of continuity (the highest order derivative of y® being continuous at a given point) is found to be proportional to n in the first case (m=1) and to 2n in the other two cases (m=2,3), for large values of n. Moreover, derivatives of y® up to third order are continuous at r=d and r=w*d for w=3/2 and w=4/3 but only the first derivative is continuous for w=2. This can be understood as a non-linear resonance effect.