A Generalized Approach to Improve Approximation of Inverse Langevin Function

Abstract

The inverse Langevin function has a crucial role in different research fields, such as polymer physics, para- or superpara-magnetism materials, molecular dynamics simulations, turbulence modeling, and solar energy conversion. The inverse Langevin function cannot be explicitly derived and thus, its inverse function is usually approximated using rational functions. Here, a generalized approach is proposed that can provide multiple approximation functions with a different degree of complexity/accuracy for the inverse Langevin function. While some special cases of our approach have already been proposed as approximation function, a generic approach to provide a family of solutions to a wide range of accuracy/complexity trade-off problems has not been available so far. By coupling a recurrent procedure with current estimation functions, a hybrid function with adjustable accuracy and complexity is developed. Four different estimation families based four estimation functions are presented here and their relative error is calculated with respect to the exact inverse Langevin function. The level of error for these simple and easy-to-use formulas can be reduced as low as 0.1%.