An Efficient Gauss-Newton-like Method for the Numerical Solution of the Ornstein-Zernike Integral Equation for a Class of Fluid Models

Abstract

A numerical algorithm for solving the Ornstein-Zernike (OZ) integral equation of statistical mechanics is described for the class of fluids composed of molecules with axially symmetric interactions. Since the OZ equation is a nonlinear second-kind Fredholm equation whose key feature for the class of problems of interest is the highly computationally intensive nature of the kernel, the general approach employed in this paper is thus potentially useful for similar problems with this characteristic. The algorithm achieves a high degree of computational efficiency by combining iterative linearization of the most complex portion of the kernel with a combination of Newton-Raphson and Picard iteration methods for the resulting approximate equation. This approach makes the algorithm analogous to the approach of the classical Gauss-Newton method for nonlinear regression, and we call our method the GN algorithm. An example calculation is given illustrating the use of the algorithm for the hard prolate ellipsoid fluid and its results are compared directly with those of the Picard iteration method. The GN algorithm is four to ten times as fast as the Picard method, and we present evidence that it is the most efficient general method currently available.